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1 International Mathematics Olympiad
8
1.1
1st IMO, Romania, 1959 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.2
2nd IMO, Romania, 1960 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.3
3rd IMO, Hungary, 1961 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.4
4th IMO, Czechoslovakia, 1962 . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.5
5th IMO, Poland, 1963 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.6
6th IMO, USSR, 1964 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.7
7th IMO, West Germany, 1965 . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.8
8th IMO, Bulgaria, 1966 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.9
9th IMO, Yugoslavia, 1967 . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.10 10th IMO, USSR, 1968 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
1.11 11th IMO, Romania, 1969 . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.12 12th IMO, Hungary, 1970 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.13 13th IMO, Czechoslovakia, 1971 . . . . . . . . . . . . . . . . . . . . . . . .
20
1
2
CONTENTS 1.14 14th IMO, USSR, 1972 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.15 15th IMO, USSR, 1973 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
1.16 16th IMO, West Germany, 1974 . . . . . . . . . . . . . . . . . . . . . . . .
22
1.17 17th IMO, Bulgaria, 1975 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
1.18 18th IMO, Austria, 1976 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
1.19 19th IMO, Yugoslavia, 1977 . . . . . . . . . . . . . . . . . . . . . . . . . .
25
1.20 20th IMO, Romania, 1978 . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
1.21 21st IMO, United Kingdom, 1979 . . . . . . . . . . . . . . . . . . . . . . .
27
1.22 22nd IMO, Washington, USA, 1981 . . . . . . . . . . . . . . . . . . . . . .
28
1.23 23rd IMO, Budapest, Hungary, 1982 . . . . . . . . . . . . . . . . . . . . . .
29
1.24 24th IMO, Paris, France, 1983 . . . . . . . . . . . . . . . . . . . . . . . . .
30
1.25 25th IMO, Prague, Czechoslovakia, 1984 . . . . . . . . . . . . . . . . . . .
31
1.26 26th IMO, Helsinki, Finland, 1985 . . . . . . . . . . . . . . . . . . . . . . .
32
1.27 27th IMO, Warsaw, Poland, 1986 . . . . . . . . . . . . . . . . . . . . . . .
33
1.28 28th IMO, Havana, Cuba , 1987 . . . . . . . . . . . . . . . . . . . . . . . .
34
1.29 29th IMO, Camberra, Australia, 1988 . . . . . . . . . . . . . . . . . . . . .
35
1.30 30th IMO, Braunschweig, West Germany, 1989 . . . . . . . . . . . . . . . .
36
1.31 31st IMO, Beijing, People’s Republic of China, 1990 . . . . . . . . . . . . .
38
1.32 32nd IMO, Sigtuna, Sweden, 1991 . . . . . . . . . . . . . . . . . . . . . . .
39
1.33 33rd IMO, Moscow, Russia, 1992 . . . . . . . . . . . . . . . . . . . . . . . .
40
CONTENTS
3
1.34 34th IMO, Istambul, Turkey, 1993 . . . . . . . . . . . . . . . . . . . . . . .
41
1.35 35th IMO, Hong Kong, 1994 . . . . . . . . . . . . . . . . . . . . . . . . . .
42
1.36 36th IMO, Toronto, Canada, 1995 . . . . . . . . . . . . . . . . . . . . . . .
43
1.37 37th IMO, Mumbai, India, 1996 . . . . . . . . . . . . . . . . . . . . . . . .
44
1.38 38th IMO, Mar del Plata, Argentina, 1997 . . . . . . . . . . . . . . . . . .
45
1.39 39th IMO, Taipei, Taiwan, 1998 . . . . . . . . . . . . . . . . . . . . . . . .
46
1.40 40th IMO, Bucharest, Romania, 1999 . . . . . . . . . . . . . . . . . . . . .
47
1.41 41st IMO, Taejon, South Korea, 2000 . . . . . . . . . . . . . . . . . . . . .
48
1.42 42nd IMO, Washington DC, USA, 2001 . . . . . . . . . . . . . . . . . . . .
49
1.43 43rd IMO, Glascow, United Kingdom, 2002 . . . . . . . . . . . . . . . . . .
50
1.44 44th IMO, Tokyo, Japan, 2003 . . . . . . . . . . . . . . . . . . . . . . . . .
51
2 Iberoamerican Mathematics Olympiad
52
2.1
1st Iberoamerican Olympiad, Villa de Leyva, Colombia, 1985 . . . . . . . .
52
2.2
2nd Iberoamerican Olympiad, Salto y Paysand´ u, Uruguay, 1987 . . . . . . .
53
2.3
3rd Iberoamerican Olympiad, Lima, Per´ u, 1988 . . . . . . . . . . . . . . . .
54
2.4
4th Iberoamerican Olympiad, La Habana, Cuba, 1989 . . . . . . . . . . . .
55
2.5
5th Iberoamerican Olympiad, Valladolid, Spain, 1990 . . . . . . . . . . . .
56
2.6
6th Iberoamerican Olympiad, C´ordoba, Argentina, 1991 . . . . . . . . . . .
57
2.7
7th Iberoamerican Olympiad, Caracas, Venezuela, 1992 . . . . . . . . . . .
58
2.8
8th Iberoamerican Olympiad, Ciudad de M´exico, M´exico, 1993 . . . . . . .
59
4
CONTENTS 9th Iberoamerican Olympiad, Fortaleza, Cear´a, Brazil, 1994 . . . . . . . . .
60
2.10 10th Iberoamerican Olympiad, Region V, Chile, 1995 . . . . . . . . . . . .
61
2.11 11th Iberoamerican Olympiad, Limon, Costa Rica, 1996 . . . . . . . . . . .
62
2.12 12th Iberoamerican Olympiad, Guadalajara, M´exico, 1997 . . . . . . . . . .
64
2.13 13th Iberoamerican Olympiad, Puerto Plata, Rep´ ublica Domincana, 1998 .
65
2.14 14th Iberoamerican Olympiad, La Habana, Cuba, 1999 . . . . . . . . . . .
66
2.15 15th Iberoamerican Olympiad, M´erida, Venezuela, 2000 . . . . . . . . . . .
67
2.16 16th Iberoamerican Olympiad, Minas, Uruguay, 2001 . . . . . . . . . . . .
68
2.17 17th Iberoamerican Olympiad, San Salvador, El Salvador, 2002 . . . . . . .
69
2.18 18th Iberoamerican Olympiad, Buenos Aires, Argentina, 2003 . . . . . . . .
70
2.9
3 William Lowell Putnam Competition
73
3.1
46th Anual William Lowell Putnam Competition, 1985 . . . . . . . . . . .
73
3.2
47th Anual William Lowell Putnam Competition, 1986 . . . . . . . . . . .
75
3.3
48th Anual William Lowell Putnam Competition, 1987 . . . . . . . . . . .
77
3.4
49th Anual William Lowell Putnam Competition, 1988 . . . . . . . . . . .
79
3.5
50th Anual William Lowell Putnam Competition, 1989 . . . . . . . . . . .
81
3.6
51th Anual William Lowell Putnam Competition, 1990 . . . . . . . . . . .
83
3.7
52th Anual William Lowell Putnam Competition, 1991 . . . . . . . . . . .
85
3.8
53th Anual William Lowell Putnam Competition, 1992 . . . . . . . . . . .
87
3.9
54th Anual William Lowell Putnam Competition, 1993 . . . . . . . . . . .
89
CONTENTS
5
3.10 55th Anual William Lowell Putnam Competition, 1994 . . . . . . . . . . .
91
3.11 56th Anual William Lowell Putnam Competition, 1995 . . . . . . . . . . .
92
3.12 57th Anual William Lowell Putnam Competition, 1996 . . . . . . . . . . .
94
3.13 58th Anual William Lowell Putnam Competition, 1997 . . . . . . . . . . .
96
3.14 59th Anual William Lowell Putnam Competition, 1998 . . . . . . . . . . .
98
3.15 60th Anual William Lowell Putnam Competition, 1999 . . . . . . . . . . .
99
3.16 61st Anual William Lowell Putnam Competition, 2000 . . . . . . . . . . . .
102
3.17 62nd Anual William Lowell Putnam Competition, 2001 . . . . . . . . . . .
103
3.18 63rd Anual William Lowell Putnam Competition, 2002 . . . . . . . . . . .
104
3.19 64th Anual William Lowell Putnam Competition, 2003 . . . . . . . . . . .
106
4 Asiatic Pacific Mathematical Olympiads
109
4.1
1st Asiatic Pacific Mathematical Olympiad, 1989 . . . . . . . . . . . . . . .
109
4.2
2nd Asiatic Pacific Mathematical Olympiad, 1990 . . . . . . . . . . . . . .
110
4.3
3rd Asiatic Pacific Mathematical Olympiad, 1991 . . . . . . . . . . . . . .
111
4.4
4th Asiatic Pacific Mathematical Olympiad, 1992 . . . . . . . . . . . . . . .
111
4.5
5th Asiatic Pacific Mathematical Olympiad, 1993 . . . . . . . . . . . . . . .
112
4.6
6th Asiatic Pacific Mathematical Olympiad, 1994 . . . . . . . . . . . . . . .
113
4.7
7th Asiatic Pacific Mathematical Olympiad, 1995 . . . . . . . . . . . . . . .
114
4.8
8th Asiatic Pacific Mathematical Olympiad, 1996 . . . . . . . . . . . . . . .
115
4.9
9th Asiatic Pacific Mathematical Olympiad, 1997 . . . . . . . . . . . . . . .
116
6
CONTENTS 4.10 10th Asiatic Pacific Mathematical Olympiad, 1998 . . . . . . . . . . . . . .
117
4.11 11th Asiatic Pacific Mathematical Olympiad, 1999 . . . . . . . . . . . . . .
118
4.12 12th Asiatic Pacific Mathematical Olympiad, 2000 . . . . . . . . . . . . . .
118
4.13 13th Asiatic Pacific Mathematical Olympiad, 2001 . . . . . . . . . . . . . .
119
4.14 14th Asiatic Pacific Mathematical Olympiad, 2002 . . . . . . . . . . . . . .
120
4.15 15th Asiatic Pacific Mathematical Olympiad, 2003 . . . . . . . . . . . . . .
121
4.16 15th Asiatic Pacific Mathematical Olympiad, 2003 . . . . . . . . . . . . . .
122
5 USA Mathematical Olympiad
123
5.1
18th USA Mathematical Olympiad, 1989 . . . . . . . . . . . . . . . . . . .
123
5.2
19th USA Mathematical Olympiad, 1990 . . . . . . . . . . . . . . . . . . .
124
5.3
20th USA Mathematical Olympiad, 1991 . . . . . . . . . . . . . . . . . . .
125
5.4
21st USA Mathematical Olympiad, 1992 . . . . . . . . . . . . . . . . . . .
125
5.5
22nd USA Mathematical Olympiad, 1993 . . . . . . . . . . . . . . . . . . .
126
5.6
23rd USA Mathematical Olympiad, 1994 . . . . . . . . . . . . . . . . . . .
127
5.7
24th USA Mathematical Olympiad, 1995 . . . . . . . . . . . . . . . . . . .
128
5.8
25th USA Mathematical Olympiad, 1996 . . . . . . . . . . . . . . . . . . .
129
5.9
26th USA Mathematical Olympiad, 1997 . . . . . . . . . . . . . . . . . . .
129
5.10 27th USA Mathematical Olympiad, 1998 . . . . . . . . . . . . . . . . . . .
130
5.11 28th USA Mathematical Olympiad, 1999 . . . . . . . . . . . . . . . . . . .
131
5.12 29th USA Mathematical Olympiad, 2000 . . . . . . . . . . . . . . . . . . .
132
CONTENTS
7
5.13 30th USA Mathematical Olympiad, 2001 . . . . . . . . . . . . . . . . . . .
133
5.14 31st USA Mathematical Olympiad, 2002 . . . . . . . . . . . . . . . . . . .
134
5.15 32nd USA Mathematical Olympiad, 2003 . . . . . . . . . . . . . . . . . . .
135
5.16 33rd USA Mathematical Olympiad, 2004 . . . . . . . . . . . . . . . . . . .
136
6 Canadian Mathematical Olympiad
138
6.1
30th Canadian Mathematical Olympiad, 1998 . . . . . . . . . . . . . . . . .
138
6.2
31st Canadian Mathematical Olympiad, 1999 . . . . . . . . . . . . . . . . .
139
6.3
32nd Canadian Mathematical Olympiad, 2000 . . . . . . . . . . . . . . . .
139
6.4
33rd Canadian Mathematical Olympiad, 2001 . . . . . . . . . . . . . . . .
140
6.5
34th Canadian Mathematical Olympiad, 2002 . . . . . . . . . . . . . . . . .
142
6.6
35th Canadian Mathematical Olympiad, 2003 . . . . . . . . . . . . . . . . .
143
6.7
36th Canadian Mathematical Olympiad, 2004 . . . . . . . . . . . . . . . . .
143
Chapter 1 International Mathematics Olympiad 1.1
1st IMO, Romania, 1959
1. Prove that the fraction
21n + 4 14n + 3 is irreductible for every natural number n
2. For what real values of x is q
q √ √ x + 2x − 1 + x − 2x − 1 = A
√ given (a) A = 2, (b) A = 1, (c) A = 2, where only non-negative real numbers are admitted for square roots? 3. Let a, b, c be real numbers. Consider the quadratic equation in cos x: a cos 2 x + b cos x + c = 0. Using the numbers a, b and c, form a quadratic ecuation in cos 2x, whose roots are the same as those of the original ecuation. Compare the ecuations in cos x and cos 2x for a = 4, b = 2 and c = −1 4. Construct a right triangle with hypotenuse c such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle. 5. An arbitrary point M is selected in the interior of the segment AB. The squares AM CD and M BEF are constructed on the same side of AB, sith the segments AM and M B as their respective bases. The circles circumscribed abut these squares, 8
1.2. 2N D IMO, ROMANIA, 1960
9
with centers P and Q intersect at M and also at another point N . Let N 0 denote the intersection of the straight lines AF and BC. (a) Prove that the points N and N 0 coinside. (b) Prove that the straight lines M N pass throught a fixed point S independent of the choice of M . (c) Find the locus of the midpoints of the the segment P Q as M varies between A and B. 6. Two planes, P and Q, intersect along the line p. The point A is given in the plane P , and the point C in the plane Q; neither of these points lies on the straight line p. Construct an isosceles trapezoid ABCD (with AB parallel to CD) in which a circle can be inscribed, and with vertices B and D lying in the planes P and Q respectively.
1.2
2nd IMO, Romania, 1960
1. Determine all three-digit numbers N having the property that N is divisible by 11, N and 11 is equal to the sum of the squares of the digits of N . 2. For what values of the variable x does the following inequality hold?
4x2 2 < 2x + 9 √ 1 − 1 + 2x
3. In a given right triangle 4ABC, the hypotenuse BC, of lenght a, is dividen into n equal parts (n an odd integer). Let α be the acute angle subtending, from A, that segment which contains the middle point of the hypotenuse. Let h be the lenght of the altitude to the hypotenuse of the triangle. Prove: tan α =
4nh − 1) a
(n2
4. Construct a triangle 4ABC, given ha , hb (the altitudes fron A and B) and ma , the median from vertex A. 5. Consider the cube ABCDA0 B 0 C 0 D 0 (whith face ABCD directly above face A0 B 0 C 0 D 0 ). (a) Find the locus of the midpoints of segment XY , where X is any point of AC and Y is any point of B 0 D 0 .
10
CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD (b) Find the locus of points Z which lie on the segment XY of part (a) with ZY = 2XZ. 6. Considere a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. Let V1 be the volume of the cone and V2 the volumen of the cilinder. (a) Prove that V1 6= V2 .
(b) Find the smallest number k for which V1 = kV2 , for this case, construct the angle subtended by a diameter of the base of the cone at the vertex of the cone.
7. An isosceles trapezoid with bases a and c, and altitude h is given. (a) On the axis of symmetry of this trapezoid, find all points P such that both legs of the trapezoid subtended right angles at P . (b) Calculate the distance of P from either base. (c) Determine under what conditions such points P actually exist. (Discuss varius case that might arise)
1.3
3rd IMO, Hungary, 1961
1. Solve the system of equations: x+y+z = a x + y 2 + z 2 = b2 xy = z 2 2
where a and b are constants. Give the conditions that a and b must satisfy so that x, y, z (the solutions of the system) are distinct positive numbers. √ 2. Let a, b, c the sides of a triangle, and T its area. Prove: a2 + b2 + c2 ≥ 4 3T . In what case does the equality hold? 3. Solve the equation cosn x − sinn x = 1, where n is a natural number. 4. Consider the triangle 4P1 P2 P3 and a point P within the triangle. Lines P P1 , P P2 , P P3 intersect the opposite side in points Q1 , Q2 , Q3 respectively. Prove that, of the numbers PP1QP1 , PP2QP2 , PP3QP3 at least one is less than or equal to 2 and at least one is grater than or equal to 2.
1.4. 4T H IMO, CZECHOSLOVAKIA, 1962
11
5. Construct triangle 4ABC if AC = b, AB = c and ^AM B = ω, where M is the midpoint of the segment BC and ω < 90◦ . Prove that a solution exists and only if b tan ω2 ≥ c < b. In what case does the equality hold? 6. Considere a plane ε and three non-collinear points A, B, C on the same side of ε; suppose the plane determined by these three points is not parallel to ε. In plane a take three arbitrary points A0 , B 0 , C 0 . Let L, M, N be the midpoints of segments AA0 , BB 0 , CC 0 ; let G the centroid of triangle 4LM N (We will not considere positions of A0 , B 0 , C 0 such that the points L, M , N do not form a triangle) What is the locus of point G as A0 , B 0 , C 0 range independently over the plane ε?
1.4
4th IMO, Czechoslovakia, 1962
1. Find the smallest natural number n which has the following properties: (a) Its decimal representation has 6 as the last digit. (b) If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as large as the original number n 2. Determine all real number x which satisfy the inequality: √ √ 1 3−x− x+1> 2 3. Consider the cube ABCDA0 B 0 C 0 D 0 (ABCD and A0 B 0 C 0 D 0 are the upper and lower bases, respectively, and edges AA0 , BB 0 , CC 0 , DD 0 are parallel) The point X moves at constant speed along the perimeterof the square ABCD in the direction ABCDA, and the point Y moves at the same rate along the perimeter of the square B 0 C 0 CB in the direction B 0 C 0 CBB 0 . Points X and Y begin their motion at the same instant from the starting position A and B 0 , respectively. Determine and draw the locus of the midpoints of the segment XY . 4. Solve the ecuation cos2 x + cos2 2x + cos2 3x = 1 5. On the circle K there are given three distinct points A, B, C. Construct (using only straightedge and compasses) a fourth point D on K such that a circle can be inscribed in the cuadrilateral thus obtained.
12
CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 6. Considere an isosceles triangle. let r be the radius of its circumscribed circle and ρ the radius of its inscribed circle. Prove that the distance d between the centers of these two circles is q d = r (r − 2ρ) 7. The tetrahedon SABC has the following propoerty: there exists five spheres, each tangent to the edges SA, SB, SC, BC, CA, AB or their extentions. (a) Prove that the tetrahedron SABC is regular. (b) Prove conversely that for every regular tetrahedron five such spheres exist.
1.5
5th IMO, Poland, 1963
1. Find all real roots of the equation eter.
√ √ 2 x − p + 2 x2 − 1 = x, where p is a real param-
2. Point A and segment BC are given. Determine the locus of points in space which are vertices of right angles with one side passing throught A, and the other side intersecting the segment BC. 3. In an n−gon all of whose interior angles are equal, the lenght of consecutive sides satisfy the relation a1 ≥ a2 ≥ · ≥ an . Prove that a1 = a2 = · = an . 4. Find all solution x1 , x2 , x3 , x4 , x5 of the system (1) (2) (3) (4) (5)
x 5 + x2 x 1 + x3 x 2 + x4 x 3 + x5 x 4 + x1
= = = = =
yx1 yx2 yx3 yx4 yx5
where y is a parameter 5. Prove that cos
2π 3π 1 π − cos + cos = 7 7 7 2
6. Five students, A, B, C, D, E, took part in a contest. One prediction was that contestants would finish in the order ABCDE. This prediction was very poor. In fact no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction has the contestants finishing
1.6. 6T H IMO, USSR, 1964
13
in the order DAECB. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.
1.6
6th IMO, USSR, 1964
1. (a) Find all positive integers n for which 2n − 1 is divisible by 7.
(b) Prove that there is not positive integer n such that 2n + 1 is dibisible by 7.
2. Let a, b, c be the sides of a triangle. Prove that a2 (b + c − a) + b2 (c + a − b) + c2 (a + b − c) ≤ 3abc 3. A circle is inscribed in triangle 4ABC with sides a, b, c. Tangents to the circle parallel to the sides of the triangle are constructed. Each of these tangents cuts off a triangle from 4ABC. In each of these triangle, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of a, b, c) 4. Seventeen people correspond by mail with one another, each one with all the rest. In their letters only three different topics are discussed. Each pair of correspondent deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic. 5. Suppose five points in a plane are situated so that no two of the straight lines joining the other four points. Determine the maximum number of intersections that these perpendiculars can have. 6. In tetrahedron ABCD, vertex D is connected with D0 the centroid of 4ABC. Lines parallel to DD0 are drawn through A, B and C. These lines intersect the planes BCD, CAD and ABD in points A1 , B1 and C1 , respectively. Prove that the volume of ABCD is one third the volume of A1 B1 C1 D0 . Is the result true if point D0 is selected anywhere within 4ABC?
1.7
7th IMO, West Germany, 1965
1. Determine all value x in the interval 0 ≤ x ≤ 2π which satisfy the inequality √ √ √ 2 cos x ≤ 1 + sin 2x − 1 − sin 2x ≤ 2
14
CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 2. Consider the system of equations a11 x1 + a12 x2 + a13 x3 = 0 a21 x1 + a22 x2 + a23 x3 = 0 a31 x1 + a32 x2 + a33 x3 = 0 with unknowns x1 , x2 , x3 . The coefficient satisfy the conditions: (a) a11 , a22 , a33 are positive numbers; (b) the remaining coefficients are negative numbers; (c) in each equation, the sum of the coefficient is positive . Prove that the given system has only the solution x1 = x2 = x3 = 0. 3. Given the tetrahedron ABCD whose edges AB and CD have lenght a and b respectively. The distance between the skew lines AB and CD is d, and the angle between them is ω. Tetrahedron ABCD is divided into two solid by plane ε, parallel to lines AB and CD. The ratio of the distances of ε from AB and CD is equal to k. Compute the ratio of the volumes of the two solids obtained. 4. Find all sets of four real numbers x1 , x2 , x3 , x4 such that the sum of any one and the product of the other three is equal to 2. 5. Consider 4OAB with acute angle ^AOB. Through a point M 6= O perpendiculars are dawn to OA and OB, the feet of which are P and Q respectively. The point of intersection of the altitudes of 4OP Q is H. What is the locus of H if M is permitted to range over (a) the side AB? (b) the interior of 4OAB? 6. In a plane a set of n points (n ≥ 3) is given. Each pair of points is connected by a segment. Let d be the length of the longest of these segment. We define a diameter of the set to be any connecting segment of length d. Prove that the number of diameters of the given set is at most n.
1.8
8th IMO, Bulgaria, 1966
1. In a mathematical contest, three problems, A, B, C were posed. Among the participants there were 25 students who solved at least one problem each. Of all the
1.9. 9T H IMO, YUGOSLAVIA, 1967
15
contestants who did not solve problem A, the number who solved B was twice the number who solved C. The number of students who solved only problem A was one more than the number of students who solved A and at least one other problem. How many students solved only problem B? 2. Let a, b, c be the lengths of the sides of a triangle and α, β, γ, respectively, the angles opposite these sides. Prove tat if a + b = tan γ2 (a tan α + b tan β), the triangle is isosceles. 3. Prove: The sum of the distances of the vertices of a regular tetrahedron from the centre of its circumscribed sphere is less than the sum of the distances of these vertices from any other poin in space. 4. Prove that for every natural number n, and for every real number x 6= non-negative integer and k any integer),
kπ 2t
(t any
1 1 1 + +···+ = cot x − cot 2n x sin 2x sin 4x sin 2n x 5. Solve the system of equations |a1 − a2 |x2 + |a1 − a3 |x3 + |a1 − a4 |x4 |a2 − a1 |x2 + |a2 − a3 |x3 + |a2 − a4 |x4 |a3 − a1 |x1 + |a3 − a2 |x2 + |a3 − a4 |x4 |a4 − a1 |x1 + |a4 − a2 |x2 + |a4 − a3 |x3
= = = =
1 1 1 1
where a1 , a2 , a3 , a4 are four different real numbers. 6. In the interior of sides BC, CA, AB of triangle 4ABC, any points K, L, M , respectively, are selected. Prove that the area of at least one of the triangle 4AM L, 4BKM , 4CLK is less than or equal to one quarter of the area of 4ABC
1.9
9th IMO, Yugoslavia, 1967
1. Let ABCD be a parallelogram with side lengths AB = a, AD = 1, and with ^BAD = α. If 4ABD is acute, prove that the four circles of√radius 1 with centers A, B, C, D cover the parallelogram if and only if a ≤ cos α + 3 sin α. 2. Prove that if one and only one edge of a tetrahedron is greater than 1, then its volume is smaller than or equal to 81
16
CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 3. Let k, m, n be natural numbers such that m + k + 1 is a prime greater than n + 1. Let cs = s (s + 1). Prove that the product (cm+1 − ck ) (cm+2 − ck ) · · · (cm+n − ck ) is divisible by the product c1 c2 · · · cn . 4. Let 4A0 B0 C0 and 4A1 B1 C1 be any two acute-angled triangles. Consider all triangles 4ABC that are similar to 4A1 B1 C1 and circumscribed about triangle 4A0 B0 C0 (where A0 lies on BC, B0 on CA and C0 on AB) Of all such triangles, determine the one with maximum area, and construct it. 5. Consider the sequence {cn }, where c1 = a 1 + a 2 + · · · + a 8 c2 = a21 + a22 + · · · + a28 .. . cn = an1 + an2 + · · · + an8 .. . in which a1 , a2 , . . . , a8 are real numbers not all equal to zero. Suppose that an infinite number of terms of the sequence {cn } are equal to zero. Find all natural numbers for which cn = 0. 6. In a sport contest, there were m medals awarded on n successive days (n > 1). On the first day, one medal and 71 of the remaining medals were awarded. On the second day, two medals and 71 of the now remaining medals were awarded; and so on. On the n-th and last day, the remaining n medals were awarded. How many days did the contest last. and how many medals were awarded altogether?
1.10
10th IMO, USSR, 1968
1. Prove that there is one and only one triangle whose side lengths are consecutive integers, and one of whose angles is twice as large as another. 2. Find all natural numbers x such that the product of their digits (in decimal notation) is equal to x2 − 10x − 22.
1.10. 10T H IMO, USSR, 1968
17
3. Consider the system of equations: ax21 ax22
+ +
bx1 bx2
+ c = x2 + c = x3 .. .
ax2n−1 + bxn−1 + c = xn ax2n + bxn + c = x1 with unknowns x1 , x2 , . . . , xn , where a, b, c are real and a 6= 0. Let 4 = (b − 1)2 − 4ac. Prove that for this system (a) If 4 < 0, ther is no solution, (b) If 4 = 0, ther is exactly one solution, (c) If 4 > 0, ther is more than one solution. 4. Prove than in every tetrahedon there is a vertex such that the three edges meeting there have lengths which are the sides of a triangle. 5. Let f be a real-valued function defined for all real numbers x such that, for some positive constant a, the equation f (x + a) =
1 q + f (x) − [f (x)]2 2
holds for all x (a) Prove that the function f is periodic (i.e. there exists a positive number b such that f (x + b) = f (x) for all x) (b) For a = 1, give an example of a non-constant function with the requiered properties. 6. For every natural number n, evaluate the sum ∞ X
k=0
$
%
$
%
n+1 n+2 n + 2k n + 2k = +··· + + · · · + 2k+1 2 4 2k+1
(the symbol bxc denotes the greatest integer not exceding x).
18
CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD
1.11
11th IMO, Romania, 1969
1. Prove that there are infinitely many numbers a with the following property: the number z = n4 + a is not prime for any natural number n. 2. Let a1 , a2 , . . . , an be real variable, and f (x) = cos (a1 + x) +
1 1 1 cos (a2 + x) + cos (a3 + x) + · · · + n−1 cos (an + x) 2 4 2
Given that f (x1 ) = f (x2 ) = 0, prove that x2 − x1 = mπ for some integer m. 3. For each value of k = 1, 2, 3, 4, 5, find necessary and sufficient conditions on the number a > 0 so that there exist a tetrahedron with k edges of length a, and the remaining 6 − k edges of lenght 1. 4. A semicircular arc γ is drawn on AB as diameter. C is a point on γ other than A and B, and D is the foot of the perpendicular from C to AB. We consider three circles γ1 , γ2 , γ3 , all tangent to the line AB. Of these, γ1 is inscrived in 4ABC, while γ2 and γ3 are both tangent to CD and to γ, one on each side of CD. Prove that γ1 , γ2 and γ3 have a second tangent in common. 5. Given n > 4 points in the plane such that no three are collinear. Prove that there n−3 are at least 2 convex quadrilaterals whose vertices are four of the given points.
6. Prove that for all real numbers x1 , x2 , y1 , y2 , z1 , z2 with x1 > 0, x2 > 0, x1 y1 − z12 > 0, x2 y2 − z22 > 0, the inequality 1 8 1 + 2 ≤ 2 x1 y1 − z1 x2 y2 − z22 (x1 + x2 ) (y1 + y2 ) − (z1 + z2 ) is satisfied. Give necessary and sufficient conditions for equality.
1.12
12th IMO, Hungary, 1970
1. Let M be a point on the sede AB of 4ABC. Let r1 , r2 and r be the radii of the inscribed circles of the triangles 4AM C, 4BM C and 4ABC. Let q1 , q2 and q be the radii of the excribed circles of the same triangles that lie in the angle 4ACB. Prove that r r1 r2 · = q1 q2 q
1.12. 12T H IMO, HUNGARY, 1970
19
2. Let a, b and n be integers greater than 1, and let a and b be the two bases of two number systems. An−1 and An are numbers in the system with base a and Bn−1 and Bn are numbers in the system with base b; these are related as follows: An = xn xn−1 · · · x0 Bn = xn xn−1 · · · x0
An−1 = xn−1 xn−2 · · · x0 Bn−1 = xn−1 xn−2 · · · x0
such that xn 6= 0 and xn−1 6= 01 . Prove that Bn−1 An−1 < ⇐ ⇒a > b An Bn 3. The real numbers a0 , a1 , . . . , an , . . . satisfy the condition 1 = a0 ≤ a1 ≤ a2 ≤ · · · ≤ an ≤ ·. The numbers b1 , b2 , . . . , bn , . . . are defined by bn =
n X
k=1
ak−1 1− ak
1 √ ak
(a) Prove that 0 ≤ bn < 2 for all n. (a) Given c with 0 ≤ c < 2, prove that there exist numbers a0 , a1 , . . . such that bn > c for large enough n. 4. Find the set of all positive integers n with the property that the set {n, n + 1, n + 2, n + 3, n + 4, n + 5} can be partitioned into sets such that the product of the numbers in one set equals the product of the numbers in the other set 5. In the tetrahedron ABCD, the angle ^BDC is a right angle. Suppose that the foot H of the perpendicular from D to the plane ABC is the intersection of the altitudes of 4ABC. Prove that
(AB + BC + CA)2 ≤ 6 AD 2 + BD 2 + CD 2
For what tetrahedra does equality hold? 6. In the plane are 100 points, no three of them are collinear. Consider all posible triangles having these points as vertices. Prove that no more than 70% of these triangles are acute-angled. 1
The xi ’s are the digits in the respective bases, and of course, all of them are lower than the lowest base
20
CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD
1.13
13th IMO, Czechoslovakia, 1971
1. Prove that the following assertion is true for n = 3 and n = 5, and that it is false for every other natural number n > 2. If a1 , a2 , . . . , an are arbitrary real numbers, then: (a1 − a2 ) (a1 − a3 ) · · · (a1 − an ) + (a2 − a1 ) (a2 − a3 ) · · · (a2 − an ) + · · · + (an − a1 ) (an − a2 ) · · · (an − an−1 ) ≤ 0 2. Consider a convex polyhedron P1 with nine vertices A1 A2 , ..., A9 ; let Pi be the polyhedron obtained from P1 by a translation that moves vertex A1 to Ai (i = 2, 3, ..., 9). Provethat at least two of the polyhedra P1 , P2 , ..., P9 have an interiorpoint in common. 3. Prove that the set of integers of the form 2k − 3(k = 2, 3, ...) contains an infinite subset in which every two members are relatively prime. 4. All the faces of tetrahedron ABCD are acute-angled triangles. We consider all closed polygonal paths of the form XY ZT X defined as follows: X is a point on edge AB distinct from A and B; similarly, Y, Z, T are interior points of edges BCCD, DA, respectively. Prove: (a) If ^DAB + ^BCD 6= ^CDA + ^ABC, then among thepolygonal paths, there is none of minimal length. (b) If ^DAB + ^BCD = ^CDA + ^ABC, then there areinfinitely many shortest polygonal paths, their common length being 2AC sin(α/2), where α = ^BAC + ^CAD + ^DAB. 5. Prove that for every natural number m, there exists a finite set S of points in a plane with the following property: For every point A in S, there are exactly m points in S which are at unit distance from A. 6. Let A = (aij )(i, j = 1, 2, ..., n) be a square matrix whose elements are non-negative integers. Suppose that whenever an element aij = 0, the sum of the elements in the ith row and the jth column is ≥ n. Prove that the sum of all the elements of the matrix is ≥ n2 /2.
1.14
14th IMO, USSR, 1972
1. Prove that from a set of ten distinct two-digit numbers (in the decimalsystem), it is possible to select two disjoint subsets whose members havethe same sum.
1.15. 15T H IMO, USSR, 1973
21
2. Prove that if n ≥ 4, every quadrilateral that can be inscribed in acircle can be dissected into n quadrilaterals each of which is inscribablein a circle. 3. Let m and n be arbitrary non-negative integers. Prove that (2m)!(2n)! m!n!(m + n)! is an integer. (0! = 1) 4. Find all solutions (x1 , x2 , x3 , x4 , x5 ) of the system of inequalities
(x21 − x3 x5 )(x22 − x3 x5 ) (x22 − x4 x1 )(x23 − x4 x1 ) (x23 − x5 x2 )(x24 − x5 x2 ) (x24 − x1 x3 )(x25 − x1 x3 ) (x25 − x2 x4 )(x21 − x2 x4 )
≤ ≤ ≤ ≤ ≤
0 0 0 0 0
where x1 , x2 , x3 , x4 , x5 are positive real numbers. 5. Let f and g be real-valued functions defined for all real values of xand y, and satisfying the equation f (x + y) + f (x − y) = 2f (x)g(y) for all x, y. Prove that if f (x) is not identically zero, and if |f (x)| ≤ 1 for all x, then |g(y)| ≤ 1 for all y. 6. Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.
1.15
15th IMO, USSR, 1973
−−→ −−→ −−→ 1. Point O lies on line g; OP1 , OP2 , . . . , OPn are unit vectors such that points P1 , P2 , ..., Pn all lie in a plane containing g and on one side of g. Prove that if n is odd, −−→ −−→ −−→ OP1 + OP2 + · · · + OPn
−−→ −−→ Here OM denotes the length of vector OM .
≥1
22
CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 2. Determine whether or not there exists a finite set M of points in spacenot lying in the same plane such that, for any two points A and B of M,one can select two other points C and D of M so that lines AB and CD are parallel and not coincident. 3. Let a and b be real numbers for which the equation x4 + ax3 + bx2 + ax + 1 = 0 has at least one real solution. For all such pairs (a, b), find the minimum value of a2 + b 2 . 4. A soldier needs to check on the presence of mines in a region having theshape of an equilateral triangle. The radius of action of his detector isequal to half the altitude of the triangle. The soldier leaves from one vertex of the triangle. What path should he follow in order to travel the least possible distance and still accomplish his mission? 5. G is a set of non-constant functions of the real variable x of the form f (x) = ax + b, a and b are real numbers, and G has the following properties: (a) If f and g are in G, then g ◦ f is in G; here (g ◦ f )(x) = g[f (x)].
(b) If f is in G, then its inverse f −1 is in G; here the inverse of f (x) = ax + b is f −1 (x) = (x − b)/a. (c) For every f in G, there exists a real number xf such that f (xf ) = xf .
Prove that there exists a real number k such that f (k) = k for all f in G. 6. Let a1 , a2 , ..., an be n positive numbers, and let q be a givenreal number such that 0 < q < 1. Find n numbers b1 , b2 , ..., bn forwhich (a) ak < bk for k = 1, 2, · · · , n,
(b) q <
bk+1 bk
<
1 q
for k = 1, 2, ..., n − 1,
(c) b1 + b2 + · · · + bn <
1.16
1+q (a1 1−q
+ a2 + · · · + an ).
16th IMO, West Germany, 1974
1. Three players A, B and C play the following game: On each of three cardsan integer is written. These three numbers p, q, r satisfy 0 < p < q < r. Thethree cards are shuffled and one is dealt to each player. Each then receivesthe number of counters indicated by the card he holds. Then the cards areshuffled again; the counters remain with the players.
1.17. 17T H IMO, BULGARIA, 1975
23
This process (shuffling, dealing, giving out counters) takes place for at least two rounds. After the last round, A has 20 counters in all, B has 10 and C has 9. At the last round B received r counters. Who received q counters on the first round? 2. In the triangle 4ABC prove that there is a point D on side AB suchthat CD is the geometric mean of AD and DB if and only if sin A sin B ≤ sin2 3. Prove that the number
Pn
k=0
2n+1 2k+1
C . 2
23k is not divisible by 5 for any integer n ≥ 0.
4. Consider decompositions of an 8 × 8 chessboard into p non-overlapping rectangles subject to the following conditions: (i) Each rectangle has as many white squares as black squares. (ii) If ai is the number of white squares in the i-th rectangle, then a1 < a2 < · · · < ap . Find the maximum value of p for which such a decomposition is possible. For this value of p, determine all possible sequences a1 , a2 , · · · , ap . 5. Determine all possible values of S=
a b c d + + + a+b+d a+b+c b+c+d a+c+d
where a, b, c, d are arbitrary positive numbers. 6. Let P be a non-constant polynomial with integer coefficients. If n(P ) isthe number of distinct integers k such that (P (k))2 = 1, prove that n(P ) − deg(P ) ≤ 2, where deg(P ) denotes the degree of the polynomial P.
1.17
17th IMO, Bulgaria, 1975
1. Let xi , yi (i = 1, 2, ..., n) be real numbers such that x1 ≥ x2 ≥ · · · ≥ xn and y1 ≥ y2 ≥ · · · ≥ yn Prove that, if z1 , z2 , · · · , zn is any permutation of y1 , y2 , · · · , yn , then n X i=1
(xi − yi )2 ≤
n X i=1
(xi − zi )2
24
CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 2. Let a1 , a2 , a3, · · · be an infinite increasing sequence of positive integers. Prove that for every p ≥ 1 there are infinitely many am which can be written in the form am = xap + yaq with x, y positive integers and q > p. 3. On the sides of an arbitrary triangle ABC, triangles ABR, BCP, CAQ areconstructed externally with ^CBP = ^CAQ = 45◦ , ^BCP = ^ACQ = 30◦ , ^ABR = ^BAR = 15◦ . Prove that ^QRP = 90◦ and QR = RP. 4. When 44444444 is written in decimal notation, the sum of its digits is A. Let B be the sum of the digits of A. Find the sum of the digits of B. (A and B are written in decimal notation.) 5. Determine, with proof, whether or not one can find 1975 points on the circumference of a circle with unit radius such that the distance between any two of them is a rational number. 6. Find all polynomials P, in two variables, with the following properties: (i) for a positive integer n and all real t, x, y P (tx, ty) = tn P (x, y) (that is, P is homogeneous of degree n), (ii) for all real a, b, c, P (b + c, a) + P (c + a, b) + P (a + b, c) = 0 (iii) P (1, 0) = 1.
1.18
18th IMO, Austria, 1976
1. In a plane convex quadrilateral of area 32, the sum of the lengths of two opposite sides and one diagonal is 16. Determine all possible lengths ofthe other diagonal. 2. Let P1 (x) = x2 − 2 and Pj (x) = P1 (Pj−1 (x)) for j = 2, 3, · · ·.Show that, for any positive integer n, the roots of the equation Pn (x) = x are real and distinct.
1.19. 19T H IMO, YUGOSLAVIA, 1977
25
3. A rectangular box can be filled completely with unit cubes. If one places as many cubes as possible, each with volume 2, in the box, so that their edges are parallel to the edges of the box, one can fill exactly 40% ofthe box. Determine the possible dimensions of all such boxes. 4. Determine, with proof, the largest number which is the product of positiveintegers whose sum is 1976. 5. Consider the system of p equations in q = 2p unknowns x1 , x2 , · · · , xq : a11 x1 + a12 x2 + · · · + a1q xq = 0 a21 x1 + a22 x2 + · · · + a2q xq = 0 ··· ap1 x1 + ap2 x2 + · · · + apq xq = 0 with every coefficient aij member of the set {−1, 0, 1}. Prove that the system has a solution (x1 , x2 , · · · , xq ) such that
(a) all xj (j = 1, 2, ..., q) are integers,
(b) there is at least one value of j for which xj 6= 0,
(c) |xj | ≤ q(j = 1, 2, ..., q).
6. A sequence {un } is defined by u0 = 2, u1 = 5/2, un+1 = un (u2n−1 − 2) − u1 for n = 1, 2, · · · Prove that for positive integers n, [un ] = 2[2
n −(−1)n ]/3
where [x] denotes the greatest integer ≤ x.
1.19
19th IMO, Yugoslavia, 1977
1. Equilateral triangles 4ABK, 4BCL, 4CDM, 4DAN are constructed inside the square ABCD. Prove that the midpoints of the four segments KL, LM, M N, N K and the midpoints of the eight segments AK, BK, BL, CL, CM, DM, DN, AN are the twelve vertices of a regular dodecagon.
26
CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 2. In a finite sequence of real numbers the sum of any seven successive terms is negative, and the sum of any eleven successive terms is positive.Determine the maximum number of terms in the sequence. 3. Let n be a given integer > 2, and let Vn be the set of integers 1+kn, where k = 1, 2, .... A number m ∈ Vn is called indecomposable in Vn if there do not exist numbers p, q ∈ Vn such that pq = m. Prove that there exists a number r ∈ Vn that can be expressed as the product of elements indecomposable in Vn in more than one way. (Products which differ only in the order of their factors will be considered the same.) 4. Four real constants a, b, A, B are given, and f (θ) = 1 − a cos θ − b sin θ − A cos 2θ − B sin 2θ Prove that if f (θ) ≥ 0 for all real θ, then a2 + b2 ≤ 2 and A2 + B 2 ≤ 1. 5. Let a and b be positive integers. When a2 + b2 is divided by a + b,the quotient is q and the remainder is r. Find all pairs (a, b) suchthat q 2 + r = 1977. 6. Let f (n) be a function defined on the set of all positive integers and having all its values in the same set. Prove that if f (n + 1) > f (f (n)) for each positive integer n, then f (n) = n for each n.
1.20
20th IMO, Romania, 1978
1. m and n are natural numbers with 1 ≤ m < n. In their decimal representations, the last three digits of 1978m are equal, respectively, to the last three digits of 1978n . Find m and n such that m + n has its least value. 2. P is a given point inside a given sphere. Three mutually perpendicular rays from P intersect the sphere at points U, V, and W ; Q denotes the vertex diagonally opposite to P in the parallelepiped determined by P U, P V, and P W. Find the locus of Q for all such triads of rays from P . 3. The set of all positive integers is the union of two disjoint subsets {f (1), f (2), ..., f (n), ...}, {g(1), g(2), ..., g(n), ...}, where f (1) < f (2) < · · · < f (n) < · · ·
1.21. 21ST IMO, UNITED KINGDOM, 1979
27
g(1) < g(2) < · · · < g(n) < · · · and g(n) = f (f (n)) + 1 for all n ≥ 1 Determine f (240). 4. In triangle 4ABC, AB = AC. A circle is tangent internally to thecircumcircle of triangle ABC and also to sides AB, AC at P, Q, respectively. Prove that the midpoint of segment P Q is the center of the incircle of triangle 4ABC. 5. Let {ak }(k = 1, 2, 3, ..., n, ...) be a sequence of distinct positive integers. Prove that for all natural numbers n, n n X X 1 ak ≥ 2 k=1 k k=1 k 6. An international society has its members from six different countries. The list of members contains 1978 names, numbered 1, 2, ..., 1978. Prove that there is at least one member whose number is the sum of thenumbers of two members from his own country, or twice as large as the numberof one member from his own country.
1.21
21st IMO, United Kingdom, 1979
1. Let p and q be natural numbers such that p 1 1 1 1 1 = 1− + − +···− + q 2 3 4 1318 1319 Prove that p is divisible by 1979. 2. A prism with pentagons A1 A2 A3 A4 A5 and B1 B2 B3 B4 B5 as top and bottom faces is given. Each side of the two pentagons and each of the line-segments Ai Bj for all i, j = 1, ..., 5, is colored either red or green. Every triangle whose vertices are vertices of the prism and whose sides have all been colored has two sides of a different color. Show that all 10 sides of the top and bottom faces are the same color. 3. Two circles in a plane intersect. Let A be one of the points of intersection. Starting simultaneously from A two points move with constant speeds, each point travelling along its own circle in the same sense. The two points return to A simultaneously after one revolution. Prove that there is a fixed point P in the plane such that, at any time, the distances from P to the moving points are equal.
28
CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 4. Given a plane π, a point P in this plane and a point Q not in π, find all points R in π such that the ratio (QP + P A)/QR is a maximum. 5. Find all real numbers a for which there exist non-negative real numbers x1 , x2 , x3 , x4 , x5 satisfying the relations 5 X
k=1
kxk = a,
5 X
k 3 xk = a 2 ,
k=1
5 X
k 5 xk = a 3
k=1
6. Let A and E be opposite vertices of a regular octagon. A frog starts jumping at vertex A. From any vertex of the octagon except E, it may jump to either of the two adjacent vertices. When it reaches vertex E, the frog stops and stays there.. Let a n be the number of distinct paths of exactly n jumps ending at E. Prove that a2n−1 = 0, 1 a2n = √ (xn−1 − y n−1 ), n = 1, 2, 3, · · · , 2 √ √ where x = 2 + 2 and y = 2 − 2
Note. A path of n jumps is a sequence of vertices (P0 , ..., Pn ) such that (i) P0 = A, Pn = E; (ii) for every i, 0 ≤ i ≤ n − 1, Pi is distinct from E;
(iii) for every i, 0 ≤ i ≤ n − 1, Pi and Pi+1 are adjacent.
1.22
22nd IMO, Washington, USA, 1981
1. P is a point inside a given triangle ABC.D, E, F are the feet of the perpendiculars from P to the lines BC, CA, AB respectively. Find all P for which BC CA AB + + PD PE PF is least. 2. Let 1 ≤ r ≤ n and consider all subsets of r elements of theset {1, 2, ..., n}. Each of these subsets has a smallest member. Let F (n, r) denote the arithmetic mean of these smallest numbers; prove that F (n, r) =
n+1 r+1
1.23. 23RD IMO, BUDAPEST, HUNGARY, 1982
29
3. Determine the maximum value of m3 + n3 ,where m and n are integers satisfying m, n ∈ {1, 2, ..., 1981} and (n2 − mn − m2 )2 = 1 4. (a) For which values of n > 2 is there a set of n consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining n − 1 numbers? (b) For which values of n > 2 is there exactly one set having the stated property? 5. Three congruent circles have a common point O and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle and the point O are collinear. 6. The function f (x, y) satisfies (1) f (0, y) = y + 1, (2)f (x + 1, 0) = f (x, 1), (3) f (x + 1, y + 1) = f (x, f (x + 1, y)), for all non-negative integers x, y. Determine f (4, 1981).
1.23
23rd IMO, Budapest, Hungary, 1982
1. The function f (n) is defined for all positive integers n and takes on non-negative integer values. Also, for all m, n f (m + n) − f (m) − f (n) = 0 or 1 f (2) = 0, f (3) > 0, and f (9999) = 3333 Determine f (1982). 2. A non-isosceles triangle A1 A2 A3 is given with sides a1 , a2 , a3 (ai is the side opposite Ai ). For all i = 1, 2, 3, Mi is the midpoint of side ai , and Ti . is the pointwhere the incircle touches side ai . Denote by Si the reflection of Ti in the interior bisector of angle Ai . Prove that the lines M1 , S1 , M2 S2 , and M3 S3 are concurrent. 3. Consider the infinite sequences {xn } of positive real numbers with the following properties: x0 = 1, and for all i ≥ 0, xi+1 ≤ xi
30
CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD (a) Prove that for every such sequence, there is an n ≥ 1 such that x20 x21 x2 + + · · · + n−1 ≥ 3.999 x1 x2 xn (b) Find such a sequence for which x2 x20 x21 + + · · · + n−1 < 4 x1 x2 xn 4. Prove that if n is a positive integer such that the equation x3 − 3xy 2 + y 3 = n has a solution in integers (x, y), then it has at least three suchsolutions. Show that the equation has no solutions in integers when n = 2891 5. The diagonals AC and CE of the regular hexagon ABCDEF are divided by the inner points M and N , respectively, so that CN AM = =r AC CE Determine r if B, M, and N are collinear. 6. Let S be a square with sides of length 100, and let L be a path within S which does not meet itself and which is composed of line segments A0 A1 , A1 A2 , · · · , An−1 An with A0 6= An . Suppose that for every point P of the boundary of S there is a point of L at a distance from P not greater than 1/2. Prove that there are two points X and Y in L such that the distance between X and Y is not greater than 1, and the length of that part of L which lies between X and Y is not smaller than 198.
1.24
24th IMO, Paris, France, 1983
1. Find all functions f defined on the set of positive real numbers which take positive real values and satisfy the conditions: (i) f (xf (y)) = yf (x) for all positive x, y; (ii) f (x) → 0 as x → ∞ 2. Let A be one of the two distinct points of intersection of two unequal coplanar circles C1 and C2 with centers O1 and O2 , respectively. One of the common tangents to the circles touches C1 at P1 and C2 at P2 , while the other touches C1 at Q1 and C2 at Q2 . Let M1 be the midpoint of P1 Q1 ,and M2 be the midpoint of P2 Q2 . Prove that ^O1 AO2 = ^M1 AM2 .
1.25. 25T H IMO, PRAGUE, CZECHOSLOVAKIA, 1984
31
3. Let a, b and c be positive integers, no two of which have a common divisor greater than 1. Show that 2abc − ab − bc − ca is the largest integer which cannot be expressed in the form xbc + yca + zab,where x, y and z are non-negative integers. 4. Let ABC be an equilateral triangle and E the set of all points contained in the three segments AB, BC and CA (including A, B and C). Determine whether, for every partition of E into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle. Justify your answer. 5. Is it possible to choose 1983 distinct positive integers, all less than or equal to 10 5 , no three of which are consecutive terms of an arithmetic progression? Justify your answer. 6. Let a, b and c be the lengths of the sides of a triangle. Prove that a2 b(a − b) + b2 c(b − c) + c2 a(c − a) ≥ 0 Determine when equality occurs.
1.25
25th IMO, Prague, Czechoslovakia, 1984
1. Prove that
7 27 where x, y and z arenon-negative real numbers for which x + y + z = 1. 0 ≤ yz + zx + xy − 2xyz ≤
2. Find one pair of positive integers a and b such that: (i) ab(a + b) is not divisible by 7; (ii) (a + b)7 − a7 − b7 is divisible by 77 . Justify your answer. 3. In the plane two different points O and A are given. For each point X of the plane, other than O, denote by a(X) the measure of the angle between OA and OX in radians, counterclockwise from OA(0 ≤ a(X) < 2π). Let C(X) be the circle with center O and radius of length OX + a(X)/OX. Each point of the plane is colored by one of a finite number ofcolors. Prove that there exists a point Y for which a(Y ) > 0 such that its color appears on the circumference of the circle C(Y ).
32
CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 4. Let ABCD be a convex quadrilateral such that the line CD is a tangent to the circle on AB as diameter. Prove that the line AB is a tangent to the circle on CD as diameter if and only if the lines BC and AD are parallel. 5. Let d be the sum of the lengths of all the diagonals of a plane convex polygon with n vertices (n > 3), and let p be its perimeter. Prove that
n 2d < n−3< p 2
n+1 −2 2
where [x] denotes the greatest integer not exceeding x. 6. Let a, b, c and d be odd integers such that 0 < a < b < c < d and ad = bc. Prove that if a + d = 2k and b + c = 2m for some integers k and m, then a = 1.
1.26
26th IMO, Helsinki, Finland, 1985
1. A circle has center on the side AB of the cyclic quadrilateral ABCD. The other three sides are tangent to the circle. Prove that AD + BC = AB 2. Let n and k be given relatively prime natural numbers, k < n. Each number in the set M = {1, 2, ..., n − 1} is colored either blue or white. It is given that (i) for each i ∈ M, both i and n − i have the same color;
(ii) for each i ∈ M, i 6= k both i and |i − k| have the same color. Prove that all numbers in M must have the same color. 3. For any polynomial P (x) = a0 + a1 x + · · · + ak xk with integer coefficients, the number of coefficients which are odd is denoted by w(P ). For i = 0, 1, ..., let Qi (x) = (1 + x)i . Prove that if i1 i2 , ..., in are integers such that 0 ≤ i1 < i2 < · · · < in then w(Qi1 + Qi2 + · · · + Qin ) ≥ w(Qi1 ) 4. Given a set M of 1985 distinct positive integers, none of which has a prime divisor greater than 26. Prove that M contains at least one subset of four distinct elements whose product is the fourth power of an integer. 5. A circle with center O passes through the vertices A and C of triangle ABC and intersects the segments AB and BC again at distinct points K and N, respectively. The circumscribed circles of the triangles 4ABC and 4EBN intersect at exactly two distinct points B and M. Provethat angle ^OM B is a right angle.
1.27. 27T H IMO, WARSAW, POLAND, 1986
33
6. For every real number x1 , construct the sequence x1 , x2 , ... by setting
xn+1 = xn xn +
1 for each n ≥ 1 n
Prove that there exists exactly one value of x1 for which 0 < xn < xn+1 < 1 for every n.
1.27
27th IMO, Warsaw, Poland, 1986
1. Let d be any positive integer not equal to 2, 5, or 13. Show that one canind distinct a, b in the set {2, 5, 13, d} such that ab − 1 is not perfect square. 2. A triangle 4A1 A2 A3 and a point P0 are given in the plane. We define As = As−3 for all s ≥ 4. We construct a set of points P1 , P2 , P3 , . . . , such that Pk+1 is the image of Pk under a rotation withenter Ak+1 through angle 120◦ clockwise (for k = 0, 1, 2,ldots). Prove that if P1986 = P0 , then the triangle 4A1 A2 A3 equilateral. 3. To each vertex of a regular pentagon an integer is assigned in such a way that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers x, y, z respectively and y < 0 then the following operation is allowed: the numbers x, y, z are replaced by x + y, −y, z + y respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to and end after a finite number of steps. 4. Let A, B be adjacent vertices of a regular n-gon (n ≥ 5) in thelane having center at O. A triangle XY Z, which is congruent to andnitially conincides with OAB, moves in the plane in such a way that Y and Z each trace out the whole boundary of the polygon, X remaining inside the polygon. Find the locus of X. 5. Find all functions f , defined on the non-negative real numbers and taking nonnegative real values, such that: (i) f (xf (y))f (y) = f (x + y) for all x, y ≥ 0,
(ii) f (2) = 0,
(iii) f (x) 6= 0 for 0 ≤ x < 2.
34
CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 6. One is given a finite set of points in the plane, each point having integeroordinates. Is it always possible to color some of the points in the set rednd the remaining points white in such a way that for any straight line Larallel to either one of the coordinate axes the difference (in absolutealue) between the numbers of white point and red points on L is not greaterhan 1?
1.28
28th IMO, Havana, Cuba , 1987
1. Let pn (k) be the number of permutations of the set {1, . . . , n}, n ≥ 1, which have exactly k fixed points. Prove that n X
k=0
k · pn (k) = n!
(Remark: A permutation f of a set S is a one-to-one mapping of S onto itself. An element i in S is called a fixed point of the permutation f if f (i) = i.) 2. In an acute-angled triangle 4ABC the interior bisector of the angle ^A intersects BC at L and intersects the circumcircle of 4ABC again at N . From point L perpendiculars are drawn to AB and AC, the feet of theseerpendiculars being K and M respectively. Prove that the quadrilateral AKN M and the triangle ABC have equal areas. 3. Let x1 , x2 , . . . , xn be real numbers satisfying x21 + x22 + · · · + x2n = 1. Prove that for every integer k ≥ 2 there are integers a1 , a2 , . . . , an , not all 0, such that |ai | ≤ k − 1 For all i and √ (k − 1) n |a1 x1 + a1 x2 + · · · + an xn | ≤ kn − 4. Prove that there is no function f from the set of non-negative integers into itself such that f (f (n)) = n + 1987 for every n. 5. Let n be an integer greater than or equal to 3. Prove that there is a set of n points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area. 6. Let n be an integer greater thanq or equal to 2. Prove that if k 2 + k + n is prime for all integers k such that 0 ≤ k ≤ n3 , then k 2 + k + n is prime for all integers k such that 0 ≤ k ≤ n − 2.
1.29. 29T H IMO, CAMBERRA, AUSTRALIA, 1988
1.29
35
29th IMO, Camberra, Australia, 1988
1. Consider two coplanar circles of radii R and r (R > r) with the same center. Let P be a fixed point on the smaller circle and B a variable point on the larger circle. The line BP meets the larger circle again at C. The perpendicular l to BP at P meets the smaller circle again at A. (If l is tangent to the circle at P then A = P .) (i) Find the set of values of BC 2 + CA2 + AB 2 . (ii) Find the locus of the midpoint of BC. 2. Let n be a positive integer and let A1 , A2 , . . . , A2n+1 be subsets of a set B. Suppose that (a) Each Ai has exactly 2n elements, (b) Each Ai ∩ Aj (1 ≤ i < j ≤ 2n + 1) contains exactly one element, and (c) Every element of B belongs to at least two of the Ai .
For which values of n can one assign to every element of B one of the numbers 0 and 1 in such a way that Ai has 0 assigned to exactly n of its elements? 3. A function f is defined on the positive integers by f (1) f (2n) f (4n + 1) f (4n + 3)
= = = =
1, f (3) = 3, f (n), 2f (2n + 1) − f (n), 3f (2n + 1) − 2f (n),
for all positive integers n. Determine the number of positive integers n, less than or equal to 1988, for which f (n) = n. 4. Show that set of real numbers x which satisfy the inequality 70 X
k 5 ≥ 4 k=1 x − k is a union of disjoint intervals, the sum of whose lengths is 1988. 5. 4ABC is a triangle right-angled at A, and D is the foot of the altituderom A. The straight line joining the incenters of the triangles 4ABD, 4ACD intersects the sides AB, AC at the points K, L respectively. S and T denote the areas of the triangles 4ABC and 4AKL respectively.how that S ≥ 2T .
36
CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 6. Let a and b be positive integers such that ab + 1 divides a2 + b2 . Show that a2 + b 2 ab + 1 is the square of an integer.
1.30
30th IMO, Braunschweig, West Germany, 1989
1. Prove that the set {1, 2, . . . , 1989} can be expressed as the disjoint union of subsets Ai (i = 1, 2, . . . , 117) such that: (i) Each Ai contains 17 elements; (ii) The sum of all the elements in each Ai is the same. 2. In an acute-angled triangle ABC the internal bisector of angle A meets their cumcircle of the triangle again at A1 . Points B1 and C1 are defined similarly. Let A0 be the point of intersection of the line AA1 with the external bisectors of angles B and C. Points B0 and C0 are defined similarly. Prove that: (i) The area of the triangle A0 B0 C0 is twice the area of the hexagon AC1 BA1 CB1 . (ii) The area of the triangle A0 B0 C0 is at least four times the area of the triangle ABC. 3. Let n and k be positive integers and let S be a set of n points in the plane such that (i) No three points of S are collinear, and (ii) For any point P of S there are at least k points of S equidistant from P . Prove that: k<
1 √ + 2n 2
4. Let ABCD be a convex quadrilateral such that the sides AB, AD, BC satisfy AB = AD + BC. There exists a point P inside the quadrilateral at a distance h from the line CD such that AP = h + AD and BP = h + BC. Show that: 1 1 1 √ ≥√ +√ h AD BC 5. Prove that for each positive integer n there exist n consecutive positive integers none of which is an integral power of a prime number.
1.30. 30T H IMO, BRAUNSCHWEIG, WEST GERMANY, 1989
37
6. A permutation (x1 , x2 , . . . , xm ) of the set {1, 2, . . . , 2n}, where n is a positive integer, is said to have property P if |xi − xi+1 | = n for at least one i in {1, 2, . . . , 2n − 1}. Show that, for each n, there are more permutations with property P than without.
38
CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD
1.31
31st IMO, Beijing, People’s Republic of China, 1990
1. Chords AB and CD of a circle intersect at a point E inside the circle. Let M be an interior point of the segment EB. The tangent line at E to the circle through D, E, and M intersects the lines BC and AC at F and G, respectively. If AM = t, AB find
EG EF
in terms of t. 2. Let n ≥ 3 and consider a set E of 2n − 1 distinct points on a circle. Suppose that exactly k of these points are to be colored black. Such a coloring is “good” if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly n points from E. Find the smallest value of k so that every such coloring of k points of E is good. 3. Determine all integers n > 1 such that 2n + 1 n2 is an integer. 4. Let Q+ be the set of positive rational numbers. Construct a function f : Q+ → Q+ such that f (x) f (xf (y)) = y + for all x, y in Q . 5. Given an initial integer n0 > 1, two players, A and B, choose integers n1 , n2 , n3 , . . . alternately according to the following rules: Knowing n2k , A chooses any integer n2k+1 such that n2k ≤ n2k+1 ≤ n22k . Knowing n2k+1 , B chooses any integer n2k+2 such that n2k+1 n2k+2
1.32. 32N D IMO, SIGTUNA, SWEDEN, 1991
39
is a prime raised to a positive integer power. Player A wins the game by choosing the number 1990; player B wins by choosing the number 1. For which n0 does: (a) A have a winning strategy?
(b) B have a winning strategy?
(c) Neither player have a winning strategy?
6. Prove that there exists a convex 1990-gon with the following two properties: (a) All angles are equal. (b) The lengths of the 1990 sides are the numbers 12 , 22 , 32 , . . . , 19902 in some order.
1.32
32nd IMO, Sigtuna, Sweden, 1991
1. Given a triangle ABC, let I be the center of its inscribed circle. The internal bisectors of the angles A, B, C meet the opposite sides in A0 , B 0 , C 0 respectively. Prove that AI · BI · CI 8 1 < ≤ 0 0 0 4 AA · BB · CC 27
2. Let n > 6 be an integer and a1 , a2 , . . . , ak be all the natural numbers less than n and relatively prime to n. If a2 − a1 = a3 − a2 = · · · = ak − ak−1 > 0 prove that n must be either a prime number or a power of 2. 3. Let S = {1, 2, 3, . . . , 280}. Find the smallest integer n such that each n-element subset of S contains five numbers which are pairwise relatively prime. 4. Suppose G is a connected graph with k edges. Prove that it is possible to label the edges 1, 2, . . . , k in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1. [A graph consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices u, v belongs to at most one edge. The graph G is connected if for each pair of distinct vertices x, y there is some sequence of vertices x = v0 , v1 , v2 , . . . , vm = y such that each pair vi , vi+1 (0 ≤ i < m) is joined by an edge of G.]
40
CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 5. Let 4ABC be a triangle and P an interior point of 4ABC. Show that at least one of the angles ^P AB, ^P BC, ^P CA is less than or equal to 30◦ . 6. An infinite sequence x0 , x1 , x2 , . . . of real numbers is said to be bounded if there is a constant C such that |xi | ≤ C for every i ≥ 0.
Given any real number a > 1, construct a bounded infinite sequence x0 , x1 , x2 , . . . such that |xi − xj ||i − j|a ≥ 1 for every pair of distinct nonnegative integers i, j.
1.33
33rd IMO, Moscow, Russia, 1992
1. Find all integers a, b, c with 1 < a < b < c such that (a − 1)(b − 1)(c − 1)
is a divisor of abc − 1
2. Let R denote the set of all real numbers. Find all functions f : R → R such that
f x2 + f (y) = y + (f (x))2
forall x, y ∈ R
3. Consider nine points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of n such that whenever exactly n edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color. 4. In the plane let C be a circle, L a line tangent to the circle C, and M a point on L. Find the locus of all points P with the following property: there exists two points Q, R on L such that M is the midpoint of QR and C is the inscribed circle of triangle 4P QR. 5. Let S be a finite set of points in three-dimensional space. Let Sx , Sy , Sz be the sets consisting of the orthogonal projections of the points of S onto the yz-plane, zx-plane, xy -plane, respectively. Prove that |S|2 ≤ |Sx | · |Sy | · |Sz | where |A| denotes the number of elements in the finite set |A|. (Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane.)
1.34. 34T H IMO, ISTAMBUL, TURKEY, 1993
41
6. For each positive integer n, S(n) is defined to be the greatest integer such that, for every positive integer k ≤ S(n), n2 can be written as the sum of k positive squares. (a) Prove that S(n) ≤ n2 − 14 for each n ≥ 4.
(b) Find an integer n such that S(n) = n2 − 14.
(c) Prove that there are infintely many integers n such that S(n) = n2 − 14.
1.34
34th IMO, Istambul, Turkey, 1993
1. Let f (x) = xn + 5xn−1 + 3, where n > 1 is an integer. Prove that f (x) cannot be expressed as the product of two nonconstant polynomials with integer coefficients. 2. Let D be a point inside acute triangle ABC such that ∠ADB = ^ACB + π/2 and AC · BD = AD · BC. (a) Calculate the ratio (AB · CD)/(AC · BD).
(b) Prove that the tangents at C to the circumcircles of 4ACD and 4BCD are perpendicular. 3. On an infinite chessboard, a game is played as follows. At the start, n2 pieces are arranged on the chessboard in an n by n block of adjoining squares, one piece in each square. A move in the game is a jump in a horizontal or vertical direction over an adjacent occupied square to an unoccupied square immediately beyond. The piece which has been jumped over is removed. Find those values of n for which the game can end with only one piece remaining on the board. 4. For three points P, Q, R in the plane, we define m(P QR) as the minimum length of the three altitudes of 4P QR. (If the points are collinear, we set m(P QR) = 0.) Prove that for points A, B, C, X in the plane, m(ABC) ≤ m(ABX) + m(AXC) + m(XBC) 5. Does there exist a function f : N → N such that f (1) = 2, f (f (n)) = f (n) + n for all n ∈ N, and f (n) < f (n + 1) for all n ∈ N?
42
CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 6. There are n lamps L0 , . . . , Ln−1 in a circle (n > 1), where we denote Ln+k = Lk . (A lamp at all times is either on or off.) Perform steps s0 , s1 , . . . as follows: at step si , if Li−1 is lit, switch Li from on to off or vice versa, otherwise do nothing. Initially all lamps are on. Show that: (a) There is a positive integer M (n) such that after M (n) steps all the lamps are on again; (b) If n = 2k , we can take M (n) = n2 − 1;
(c) If n = 2k + 1, we can take M (n) = n2 − n + 1.
1.35
35th IMO, Hong Kong, 1994
1. Let m and n be positive integers. Let a1 , a2 , . . . , am be distinct elements of {1, 2, . . . , n} such that whenever ai +aj ≤ n for some i, j, 1 ≤ i ≤ j ≤ m, there exists k, 1 ≤ k ≤ m, with ai + aj = ak . Prove that a1 + a 2 + · · · + a m n+1 ≥ m 2 2. 4ABC is an isosceles triangle with AB = AC. Suppose that (a) M is the midpoint of BC and O is the point on the line AM such that OB is perpendicular to AB; (b) Q is an arbitrary point on the segment BC different from B and C; (c) E lies on the line AB and F lies on the line AC such that E , Q, F are distinct and collinear. Prove that OQ is perpendicular to EF if and only if QE = QF . 3. For any positive integer k, let f (k) be the number of elements in the set {k + 1, k + 2, . . . , 2k} whose base 2 representation has precisely three 1s. (a) Prove that, for each positive integer m, there exists at least onepositive integer k such that f (k) = m. (b) Determine all positive integers m for which there exists exactly one k with f (k) = m.
1.36. 36T H IMO, TORONTO, CANADA, 1995
43
4. Determine all ordered pairs (m, n) of positive integers such that n3 + 1 mn − 1 is an integer. 5. Let S be the set of real numbers strictly greater than −1. Find all functions f : S → S satisfying the two conditions: (a) f (x + f (y) + xf (y)) = y + f (x) + yf (x) for all x and y in S; (b)
f (x) x
is strictly increasing on each of the intervals −1 < x < 0 and 0 < x.
6. Show that there exists a set A of positive integers with the following property: For any infinite set S of primes there exist two positive integers m ∈ A and n ∈ / A each of which is a product of k distinct elements of S for some k ≥ 2.
1.36
36th IMO, Toronto, Canada, 1995
1. Let A, B, C, D be four distinct points on a line, in that order. The circles with diameters AC and BD intersect at X and Y . The line XY meets BC at Z. Let P be a point on the line XY other than Z. The line CP intersects the circle with diameter AC at C and M , and the line BP intersects the circle with diameter BD at B and N . Prove that the lines AM, DN, XY are concurrent. 2. Let a, b, c be positive real numbers such that abc = 1. Prove that 1 1 1 3 + 3 + 3 ≥ + c) b (c + a) c (a + b) 2
a3 (b
3. Determine all integers n > 3 for which there exist n points A1 , . . . , An in the plane, no three collinear, and real numbers r1 , . . . , rn such that for 1 ≤ i < j < k ≤ n, the area of 4Ai Aj Ak is ri + rj + rk . 4. Find the maximum value of x0 for which there exists a sequence x0 , x1 . . . , x1995 of positive reals with x0 = x1995 , such that for i = 1, . . . , 1995, xi−1 +
2 xi−1
= 2xi +
1 xi
44
CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD 5. Let ABCDEF be a convex hexagon with AB = BC = CD and DE = EF = F A, such that ^BCD = ^EF A = π/3. Suppose G and H are points in theinterior of the hexagon such that ^AGB = ^DHE = 2π/3. Provethat AG + GB + GH + DH + HE ≥ CF . 6. Let p be an odd prime number. How many p-element subsets A of {1, 2, . . . 2p} are there, the sum of whose elements is divisible by p?
1.37
37th IMO, Mumbai, India, 1996
1. We are given a positive integer r and a rectangular board ABCD with dimensions |AB| = 20, |BC| = 12. The rectangle is divided into a grid of 20 × 12 unit squares. The following moves are permitted on the board: one can move from√one square to another only if the distance between the centers of the two squares is r. The task is to find a sequence of moves leading from the square with A as a vertex to the square with B as a vertex. (a) Show that the task cannot be done if r is divisible by 2 or 3. (b) Prove that the task is possible when r = 73. (c) Can the task be done when r = 97? 2. Let P be a point inside triangle 4ABC such that ^AP B − ^ACB = ^AP C − ^ABC Let D, E be the incenters of triangles AP B, AP C, respectively. Show that AP, BD, CE meet at a point. 3. Let S denote the set of nonnegative integers. Find all functions f from S to itself such that f (m + f (n)) = f (f (m)) + f (n) ∀m, n ∈ S 4. The positive integers a and b are such that the numbers 15a + 16b and 16a − 15b are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? 5. Let ABCDEF be a convex hexagon such that AB is parallel to DE, BC is parallel to EF , and CD is parallel to F A. Let RA , RC , RE denote the circumradii of triangles F AB, BCD, DEF , respectively, and let P denote the perimeter of the hexagon.
1.38. 38T H IMO, MAR DEL PLATA, ARGENTINA, 1997 Prove that RA + R C + R E ≥
45
P 2
6. Let p, q, n be three positive integers with p + q < n. Let (x0 , x1 , . . . , xn ) be an (n + 1)tuple of integers satisfying the following conditions: (a) x0 = xn = 0. (b) For each i with 1 ≤ i ≤ n, either xi − xi−1 = p or xi − xi−1 = −q. Show that there exist indices i < j with (i, j) 6= (0, n), such that xi = xj .
1.38
38th IMO, Mar del Plata, Argentina, 1997
1. In the plane the points with integer coordinates are the vertices of unit squares. The squares are colored alternately black and white (as on a chessboard). For any pair of positive integers m and n, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths m and n,ie along edges of the squares. Let S1 be the total area of the black part of the triangle and S2 be the total area of thehite part. Let f (m, n) = |S1 − S2 | (a) Calculate f (m, n) for all positive integers m and n which are eitheroth even or both odd. (b) Prove that f (m, n) ≤
1 2
max{m, n} for all m and n.
(c) Show that there is no constant C such that f (m, n) < C for all m and n. endenumerate
2. The angle at A is the smallest angle of triangle 4ABC. The points B and C divide the circumcircle of the triangle into two arcs. Let U be an interior point of the arc between B and C which does not contain A. The perpendicular bisectors of AB and AC meet the line AU at V and W , respectively. The lines BV and CW meet at T . Show that AU = T B + T C 3. Let x1 , x2 , . . . , xn be real numbers satisfying the conditions |x1 + x2 + · · · + xn | = 1
46
CHAPTER 1. INTERNATIONAL MATHEMATICS OLYMPIAD and
n+1 i = 1, 2, . . . , n 2 Show that there exists a permutation y1 , y2 , . . . , yn of x1 , x2 , . . . , xn such that |xi | ≤
|y1 + 2y2 + · · · + nyn | ≤
n+1 2
4. An n × n matrix whose entries come from the set S = {1, 2, . . . , 2n − 1} is called a silver matrix if, for each i = 1, 2, . . . , n, the ith row and the ith column together contain all elements of S. Show that (a) there is no silver matrix for n = 1997; (b) silver matrices exist for infinitely many values of n. 5. Find all pairs (a, b) of integers a, b ≥ 1 that satisfy the equation 2
ab = b a 6. For each positive integer n , let f (n) denote the number of ways of representing n as a sum of powers of 2 with nonnegative integer exponents.epresentations which differ only in the ordering of their summands are considered to be the same. For instance, f (4) = 4, because the number 4 can be represented in the following four ways: 4; 2 + 2; 2 + 1 + 1; 1 + 1 + 1 + 1. Prove that, for any integer n ≥ 3, 2n
1.39
2 /4
< f (2n ) < 2n
2 /2
.
39th IMO, Taipei, Taiwan, 1998
1. In the convex quadrilateral ABCD, the diagonals AC and BD are perpendicular and the opposite sides AB and DC are not parallel. Suppose that the point P , where the perpendicular bisectors of AB and DC meet, is inside ABCD. Prove that ABCD is a cyclic quadrilateral if and only if the triangles ABP and CDP have equal areas. 2. In a competition, there are a contestants and b judges, where b ≥ 3 is an odd integer. Each judge rates each contestant as either pass or fail. Suppose k is a number such that, for any two judges, their ratings coincide for at most k contestants. Prove that k/a ≥ (b − 1)/(2b).
1.40. 40T H IMO, BUCHAREST, ROMANIA, 1999
47
3. For any positive integer n, let d(n) denote the number of positive divisors of n (including 1 and n itself). Determine all positive integers k such that d(n2 )/d(n) = k for some n. 4. Determine all pairs (a, b) of positive integers such that ab2 + b + 7 divides a2 b + a + b. 5. Let I be the incenter of triangle ABC. Let the incircle of ABC touch the sides BC, CA, and AB at K, L, and M , respectively. The line through B parallel to M K meets the lines LM and LK at R and S, respectively. Prove that angle RIS is acute. 6. Consider all functions f from the set N of all positive integers into itself satisfying f (t2 f (s)) = s(f (t))2 for all s and t in N . Determine the least possible value of f (1998).
1.40
40th IMO, Bucharest, Romania, 1999
1. Determine all finite sets S of at least three points in the plane which satisfy the following condition: for any two distinct points A and B in S, the perpendicular bisector of the line segment AB is an axis of symmetry for S. 2. Let n be a fixed integer, with n ≥ 2. (a) Determine the least constant C such that the inequality X
1≤i b > c > d > 0. Suppose that ac + bd = (b + d + a − c) (b + d − a + c) Prove ab + cd is not a prime.
1.43
43rd IMO, Glascow, United Kingdom, 2002
1. S is the set of all (h, k) with h, k non-negative integers such that h + k < n. Each element of S is colored red or blue, so that if (h, k) is red and h0 ≤ h, k 0 ≤ k, then (h0 , k 0 ) is also red. A type 1 subset of S has n blue elements with different first member and a type 2 subset of S has n blue elements with different second member. Show that there are the same number of type 1 and type 2 subsets. 2. BC is a diameter of a circle center O. A is any point on the circle with ∠AOC > 60o . EF is the chord which is the perpendicular bisector of AO. D is the midpoint of the minor arc AB. The line through O parallel to AD meets AC at J. Show that J is the incenter of triangle CEF . 3. Find all pairs of integers m > 2, n > 2 such that there are infinitely many positive integers k for which k n + k 2 − 1 divides k m + k − 1. 4. The positive divisors of the integer n > 1 are d1 < d2 < . . . < dk , so that d1 = 1, dk = n. Let d = d1 d2 + d2 d3 + · · · + dk−1 dk . (a) Prove that D < n2 . (b) Determine all n for which D is a divisor of n2 .
1.44. 44T H IMO, TOKYO, JAPAN, 2003
51
5. Find all functions f from the set R of real numbers to itself such that (f (x) + f (z))(f (y) + f (t)) = f (xy − zt) + f (xt + yz) for all x, y, z, t in R. 6. n > 2 circles of radius 1 are drawn in the plane so that no line meets more than two P of the circles. Their centers are O1 , O2 , · · · , On . Show that 1≤i pq > b
c d
Show that: (a) q ≥ b + d (b) Si q = b + d, then p = a + c. 3. Show that between all triangles such that the distance from their vertices to a given point P are 3, 5 and 7, the one with the greatest perimeter has P as incenter. 4. Let ABC a triangle with sides length a, b, c. Each side of ABC is divided in n equal segments. Let S the sum of the squares of the distance of each vertex to each one of the points of division of the opositive side. Show that a2 +bS2 +c2 is a rational number. 5. Consider all the expression of the form x + yt + zt2 with x, y, z rational numbers and t3 = 2. Show that if x + yt + zt2 6= 0 then there exist ratinal numbers u, v, w such that x + yt + zt2 u + vt + wt2 = 1 6. Consider the sets of n natural numbers diferent from zero in which there is no three elements in arithmetic progresion. Show that in one of those sets the sum is maximal.
2.4. 4T H IBEROAMERICAN OLYMPIAD, LA HABANA, CUBA, 1989
2.4
55
4th Iberoamerican Olympiad, La Habana, Cuba, 1989
1. Find all triples if real numbers that satisfy the following equation system: x + y − z = −1 x2 − y 2 + z 2 = 1 −x3 + y 3 + z 3 = −1 2. Let x, y, z three real numbers such that 0 < x < y < z < π2 . Show that π + 2 sin x cos y + 2 sin y cos z > sin 2x + sin 2y + sin 2z 2 3. Let a, b, c be the length of the sides of a triangle. Show that: a − b a + b
1 b − c c − a < + + b + c c + a 16
4. The inscrite circumference in the triangle ABC, is tangente to the sides AC and BC in the points M y N respectively. The angle bisectors of the angles A and B intersect to M N in the points P and Q respectively. Let O be the incenter of the triangle ABC. Show that M P · OA = BC · OQ. 5. Let f be a function defined on the set of al positive integer numbers {1, 2, 3, . . .} by: f (1) = 1 f (2n + 1) = f (2n) + 1 f (2n) = 3f (n). Find the set of values that f tales. 6. Show that there are infinite many pairs of natural numbers (x, y) solutions of the equation 2x2 − 3x − 3y 2 − y + 1 = 0.
56
CHAPTER 2. IBEROAMERICAN MATHEMATICS OLYMPIAD
2.5
5th Iberoamerican Olympiad, Valladolid, Spain, 1990
1. Let f be a function defined on the set of the numbers greater than or equal to zero, and, i) If n = 2j − 1 for j = 0, 1, 2, . . . then f (n) = 0.
ii) If n 6= 2j − 1 for j = 0, 1, 2, . . . then f (n + 1) = f (n) − 1. (a) Show that for every nonnegative integer n there exists a nonnegative integer k such that f (n) + n = 2k − 1.
(b) Calculate f (21990 ).
2. Let ABC a triangle, and let I be the center of the circumference inscrite and D, E, F its tangent points with BC, CA and AB respectively. Let P be the other point of intersection of the line AD with the circumference inscrite. If M is the mid point of EF , show that the for points P , I, M , D are either on the same circumference or they are colinear. 3. Let f (x) = (x + b)2 − c be a polynomial with b and c integers. (a) If p is a prime number such that p divides c and p2 do not divide c, show that for any integer number n, p2 do not divide to f (n). (b) Let q be a prime number distict from 2 such that q do not divide to c. If q divide to f (n) for some integer n, show that for every positive integer r there exists an positive integer number n0 such that q r divide to f (n0 ). 4. Let C1 be a circumference, AB one of its diameters, t its tangent in B, and M a point on C1 distinct of A and B. It is constructed a circumference C2 tangent to C1 on M , and to the line t. (a) Find the point of tangency P to t and C2 , and find the locus of the centers of the circumferences C2 when M varies. (b) Show that there exists a circumference ortogonal to all the circumferences C2 . Note: Two circumferences are ortogonal one to the other if they intersect and the respective tangents to the point of intersection are ortogonal.
´ 2.6. 6T H IBEROAMERICAN OLYMPIAD, CORDOBA, ARGENTINA, 1991
57
5. Let A and B opositive vertices on a chessboard of n × n squares (n ≥ 1), on each square it is drawn the diagonal parallel to AB, forming then 2n2 equal triangles. A token moves along a path from A to B formed by segments of the board, and each time that a segment is passed by, a seed is put on each of the triangles that admits that segment as side. Theway is followed in a way that the token do not pass by any segment more than once, and after that there are exactly two seeds on each of the 2n2 triangles on the board. For what values of n is this situation possible? 6. Let f (x) be a polynomial of degree 3 with rational coeficients. Show that if the graphic of f is tangente to the x-axis, then f (x) has all its roots rational.
2.6
6th Iberoamerican Olympiad, C´ ordoba, Argentina, 1991
1. To each vertex of a cube it is assigned the value +1 or −1, and to each face it is assigned the product of its vertices. What values may take the sum of the 14 numbers obtained on this way? 2. Two perpendicular lines divide a square in for parts, three of them has area equal to 1. Show that the area of the full square is four. 3. Let F be an increasing function defined for all real number x such that 0 ≤ x ≤ 1, and F (0) = 0 F Find F
x 3
=
18 1991
F (x) 2
.
4. Find a number N of five diferent, non zero digits, such that it is equal to the sum of all the numbers of three diferent digits that can be formed with the digits of N (in any order). 5. Let P (X, Y ) = 2X2 −6 XY + 5Y 2 . An integer number A is a value of P if there exist integer numbers B and C such that A = P (B, C). i) Determine how many elements of {1, 2, . . . , 100} are values of P .
ii) Show that the product of two values of P is a value of P .
58
CHAPTER 2. IBEROAMERICAN MATHEMATICS OLYMPIAD 6. Given three non colinear points M, N, P , if M and N are the middle points of the sides of a triangle and P is the intersection point of the highs of such triangle. Show that such triangle is constructible with compas and ruler.
2.7
7th Iberoamerican Olympiad, Caracas, Venezuela, 1992
1. For each positive integer n, let an be the last digit of the number 1+2+···+n . Calculate a1 + a2 + · · · + a1992 2. Given n real numbers a1 , . . . , an such that 0 < a1 < a2 < · · · < an and given the function a1 a2 an f (x) = + +···+ x + a1 x + a2 x + an find the sum of the lengths of the disjoint intervals formed for al the values x such that f (x) > 1. 3. In a equilateral triangle of length 2, it is inscribed a circumference Γ. (a) Show that for all point P of Γ the sum of the squares of the distance of the vertices A, B and C is 5. (b) Show that for all point P of Γ it is possible to construct a triangle such√that its sides has the length of the segments AP , BP and CP , and its area is 43 . 4. Let (an ) and (bn ) be two sequences of integer numbers such that: (a) a0 = 0, b0 = 8 (b) an+2 = 2an+1 − an + 2, bn+2 = 2bn+1 − bn (c) a2n + b2n is a perfect square for all n.
Find at least two values of the pair (a1992 , b1992 ). 5. Let Γ be a circumference and let h, m positive numbers such that there exists a trapecious ABCD inscrite in Γ, of high h and such that the sum of the bases AB+CD es m. Construct the trapecious ABCD.
´ ´ 2.8. 8T H IBEROAMERICAN OLYMPIAD, CIUDAD DE MEXICO, MEXICO, 1993 59 6. Given a triangle T of vertices A, B and C it is constructed an hexagon H of vertices A1 , A2 , B1 , B2 , C1 and C2 as in the picture. Show that the area of the exagon H is greates than or equal to three times the area of the triangle T . A2
A1 a
a c
b B1
B
A a
c
b
C2
C c
b B2
2.8
C1
8th Iberoamerican Olympiad, Ciudad de M´ exico, M´ exico, 1993
1. A number is called capicua if when it is written in decimal notation, it can be read equal from left to right as from right to left. Examples: 8, 23432, 6446. Let x1 < x2 < · · · < xi < xi+1 , · · · the sequence of al capicua numbers. For each i it is defined yi = xi+1 − xi . How many distinct primes contains the set {y1 , y2 , . . .}? 2. Show that for every convex poligon with area less than or equal to one there exists a paralelogram with area equal to two that contains it. 3. Let N∗ = {1, 2, . . .}. Find al the functions f : N∗ → N∗ such that (a) If x < y then f (x) < f (y) (b) f (yf (x)) = x2 f (xy) for all the x, y ∈ N∗ . 4. Let ABC an equilateral triangle and Γ its inscribed circle. If D and E are points in the AD AE sides AB and AC respectively, such that DE is tangent to Γ, show that DB + EC = 1. 5. Let P and Q be two distinct points in the plane. Let us denote by m(P Q) the segment bisector of P Q. Let S be a finite subset of the plane, with more than one element, that satisfies the following porpoerties: (a) If P and Q are in S, then m(P Q) intersects S.
60
CHAPTER 2. IBEROAMERICAN MATHEMATICS OLYMPIAD (b) If P1 Q1 , P2 Q2 , P3 Q3 are three diferent segments such that its endpoints are points of S, then, there is non point in S such that it intersects the three lines m(P1 Q1 ), m(P2 Q2 ), and m(P3 Q3 ) Find the number of points that S may contain. 6. Two non negative integers a and b are tuanis if the decimal expresion of a+b contains only 0 and 1 as digits. Let A and B be two infinite sets of non negative integers such that B is the set of all the tuanis numbers to elements of the set A and A the set of all the tuanis numbers to elements of the set B. Show that in at least one of the sets A and B there is an infinite number of pairs (x, y) such that x − y = 1.
2.9
9th Iberoamerican Olympiad, Fortaleza, Cear´ a, Brazil, 1994
1. A number n is said to be nice if it exists an integer r > 0 such that the expresion of n in base r has all its digits equal. For example, 62 and 15 are nice because 62 is 222 in base 5, and 15 is 33 in base 4. Show that 1993 is not nice, but 1994 is. 2. Let ABCD a cuadrilateral inscribed in a circumference. Suppose that there is a semicircumference with its center on AB, that is tangent to the other three sides of the cuadrilateral. i Show that AB = AD + BC. ii Calculate, in term of x = AB and y = CD, the maximal area that can be reached for such cuadrilateral. 3. In each sqare of an n × n grid there is a lamp. If the lamp is touched it changes its state every lamp in the same row and every lamp in the same column (the one that are on are turned off and viseversa). At the begining, all the lamps are off. Show that always is posible, with an appropiated sequence of touches, that all the the lamps on the board end on and find, in function of n the minimal number of touches that are necesary to turn on evey lamp. 4. Let A, B and C be given points on a circumference K such that the triangle 4ABC is acute. Let P be a point in the interior of K. X, Y and Z be the other intersection of AP, BP and CP with the circumference. Determine the position of P to obtain 4XY Z equilateral
2.10. 10T H IBEROAMERICAN OLYMPIAD, REGION V, CHILE, 1995
61
5. Let n and r two positive integers. It is wanted to make r subsets A1 , A2 , . . . , Ar from the set {0, 1, · · · , n−1} such that all those subsets contain exactly k elements and such that, for all integer x with 0 ≤ x ≤ n − 1 there exist x1 ∈ A1 , x2 ∈ A2 . . . , xr ∈ Ar (an element of each set) with x = x1 + x2 + · · · + x r Find the minimum value of k in termis of n and r. 6. Show that evey natural number n ≤ 21 000 000 can be obtained begining with 1 doing less than 1 100 000 sums; more precisely, there is a finite sequence of natural numbers x0 , x1 , . . . , xk with k ≤ 1 100 000, x0 = 1, xk = n such that for all i = 1, 2, . . . , k there exist r, s with 0 ≤ r ≤ s < i such that xi = x r + x s .
10th Iberoamerican Olympiad, Region V, Chile, 1995
2.10
1. Find all the possible values of the sum of the digits of all the perfect squares1 . 2. Let n be a positive integer greater than 1. Determine all the collections or real numbers x1 , x2 , . . . , xn ≥ 1 and xn+1 ≤ 0 such that the next two conditions hold: 1
3
n− 21
i x12 + x22 + · · · + xn
ii
x1 +x2 +···+xn n
1 2 = nxn+1
= xn+1
3. Let r and s two orthogonal lines that does not lay on the same plane. Let AB be their common perpendicular, where A ∈ r and B ∈ s2 . Consider the sphere of diameter AB. The points M ∈ r and N ∈ s varies with the condition that M N is tangent to the sphere on the point T . Find the locus of T . 1 Sorry for the coment, but I translated this problem as good as I could. This is the problem worst redacted that I have ever seen, and it says the same in Spanish and Portuges. After I looked on the solution, I would rewrite it as follows: Let f : N → N such that f (n) is the sum of all the digits of the number n2 . Find the image of f (where, by image it is understood the set of all x such that exists an n with f (n) = x) 2 The plane that contains B and r is perpendicular to s
62
CHAPTER 2. IBEROAMERICAN MATHEMATICS OLYMPIAD 4. In a m × n grid are there are token. Every token dominates every square on its same row (↔), its same column (l), and diagonal (& -)(Note that the token does not dominate the diagonal (% .), determine the lowest number of tokens that must be on the board to dominate all the squares on the board. 5. The circumference inscribed on the triangle 4ABC is tangent go BC, CA and AB on the points D, E and F respectively. Suppose that such circumference cout again AD on its midpoint X, this means, AX = XD. The lines XB and XC intersect the circumference in Y and Z respectively. Show that EY = F Z 6. A function f : N → N is circular if for every p ∈ N there exists n ∈ N, n ≤ p such that f n (p) = p (f composed with itself n times) The function f has repulsion degree k > 0 if for every p ∈ N f i (p) 6= p for every i = 1, 2, . . . , b kp c3 . Determine the maximun repulsion degree can have a circular function.
2.11
11th Iberoamerican Olympiad, Limon, Costa Rica, 1996
1. Let n be a natural number. A cube of edge n may be divided in 1996 cubes whose edges length are also natural numbers. Find the minimum possible value for n. 2. Let 4ABC be a triangle, D the midpoint of BC, and M be the midpoint of AD. The line BM intersects the side AC on the point N . Show that AB is tangent to the circumference circunscrite to the triangle 4N BC if and only if the following equality is true (BC)2 BM = MN (BN )2 3. We have a grid of k 2 − k + 1 rows and k 2 − k + 1 columns, where k = p + 1 and p is prime. For each prime p, give a method to put the numbers 0 and 1, one number for each square in the grid, such that on each row there are exactly k 0’s,on each column there are exactly k 0s, and there is non rectangle with sides parallel to the sides of the grid with 0s on each four vertices. 4. Given a natural number n ≥ 2, consider all the fractions of the form b are natural numbers, relative primes and such that an . Show that for each n, the sum of all this fractions are 21 . 5. Three tokens A, B, C are, each one in a vertex of an equilateral triangle of side n. Its divided on equilateral triangles of side 1, such as it is shown in the figure for the case n = 3 C
B
A
Initially, all the lines of the figure are painted blue. The tokens are moving along the lines painting them of red, following the next two rules: (a) First A moves, after that B moves, and then C, by turns. On each turn, the token moves over exactly one line of one of the little triangles, form one side to the other. (b) Non token moves over a line that is already painted red, but it can rest on one endpoint of a side that is already red, even if there is another token there wating its turn. Show that for every positive ingeter n it is possible to paint red all the sides of the little triangles. 6. There are n diferent points A1 , . . . , An in the plain and each point Ai it is assigned a real number λi distinct from zero in such way that (Ai Aj )2 = λi + λj for all the i, j with i 6= j Show that (a) n ≤ 4 (b) If n = 4, then
1 λ1
+
1 λ2
+
1 λ3
+
1 λ4
= 0.
64
CHAPTER 2. IBEROAMERICAN MATHEMATICS OLYMPIAD
2.12
12th Iberoamerican Olympiad, Guadalajara, M´ exico, 1997
1. Let r ≥ 1 be areal number that holds with the property that for each pair of positive integer numbers m and n, with n a multiple of m, it is true that bnrc is multiple of bmrc. Show that r has to be an integer number4 . itemIn a triangle ABC, it is drawn a circumference with center in the incenter I and that intersect twice each of the sides of the triangle: the segment BC on D and P (where D is nearer two B); the segment CA on E and Q (where E is nearer to C); and the segment AB on F and R ( where F is nearer to A). Let S be the point of intersection of the diagonals of the cuadrilateral EQF R. Let T be the point of intersection of the diagonals of the cuadrilateral F RDP . Let U be the point of intersection of the diagonals of the quadrilateral DP EQ. Show that the circumferences circumscrites to the triangle 4F RT , 4DP U and 4EQS have a unique point in common. 2. Let n ≥ 2 be an integer number and Dn the set of all the points (x, y) in the plane such that its coordinates are integer numbers with −n ≤ x ≤ n and −n ≤ y ≤ n.
There are three posible colors in which the points of Dn are painted with (each point has a unique color). Show that with any distribution of the colors, there are always two points of Dn with the same color such that the line that contains them does not go thrue any other point of Dn . Find a way to paint the points of Dn with 4 colors such that if a line contains exactly two points of Dn , then, this points have diferent colors.
3. Let n be a positive integer. Consider the sum x1 y1 + x2 y2 + · · · + xn yn , where tha values of the variables x1 , x2 , . . . , xn , y1 , y2 , . . . , yn are either 0 or 1. Let I(n) be the number of 2n-tuples (x1 , x2 , . . . , xn , y1 , y2 , . . . , yn ) such that the sum of the number is odd, and let P (n) be the number of 2n-tuples (x1 , x2 , . . . , xn , y1 , y2 , . . . , yn ) such that the sum is an even number. Shot that P (n) 2n + 1 = n I(n) 2 −1 4. In an acute triangle 4ABC, let AE and BF be highs of it, and H its orthocenter. The symetric line of AE with respect to the angle bisector of ^A and the symetric 4
If x is a real number, bxc is the greatest integer lower than or equal to x
´ 2.13. 13T H IBEROAMERICAN OLYMPIAD, PUERTO PLATA, REPUBLICA DOMINCANA, 199865 line of BF with respect to the angle bisector of ^B intersect each other on the point O. The lines AE and AO intersect again the circumscrite circumference to 4ABC on the points M and N respectively. Let P be the intersection of BC with HN ; R the intersection of BC with OM ; and S the intersection of HR with OP . Show that AHSO is a paralelogram. 5. Let P = {P1 , P2 , ..., P1997 } be a set of 1997 points in the interior of a circle of radius 1, where P1 is the center of the circle. For each k = 1. . . . , 1997, let xk be the distance of Pk to the point of P closer to Pk , but diferent from it. Show that (x1 )2 + (x2 )2 + ... + (x1997 )2 ≤ 9.
2.13
13th Iberoamerican Olympiad, Puerto Plata, Rep´ ublica Domincana, 1998
1. There are 98 points given over a circumference. Mary and Joseph play alternatively in the next way: each one draw a segment joining two points that have not been joined before. The game ends when the 98 points have been used as end points of a segments at least once. The winner is the person that draw the last segment. If Joseph starts the game, who can assure that is going to win the game. 2. The circumference inscribed on the triangle ABC is tangent to the sides BC, CA and AB on the points D, E and F , respectively. AD intersect the circumference on the point Q. Show that the line EQ intersect the segment AF on its midpoint if and only if AC = BC. 3. Find the minimum natural number n with the following property: between any collection of n distinct natural numbers in the set {1, 2, . . . , 999} it is possible to choose four diferent a, b, c, d such that a + 2b + 3c = d. 4. There are representants from n diferent contries sit around a circular talbe (n ≥ 2), in such way that if two representants are from the same country, then, their neighbors to the right are not from the same country. Find, for every n, the maximal number of people that can be sit around the table. 5. Find the maximal posible value of n such that there exist points P1 , P2 , P3 , . . . , Pn in the plane and real numbers r1 , r2 , . . . , rn such that the distance between any two diferent points Pi and Pj is ri + rj .
66
CHAPTER 2. IBEROAMERICAN MATHEMATICS OLYMPIAD 6. Let λ the positive root of the equation t2 − 1998t − 1 = 0. It is defined the sequence x0 , x1 , x2 , . . . , xn , . . . by x0 = 1, xn+1 = bλxn c for n = 1, 2 . . . Find the remainder of the division of x1998 by 1998. Note: bxc is the greatest integer less than or equal to x.
2.14
14th Iberoamerican Olympiad, La Habana, Cuba, 1999
1. Find all the positive integers less than 1000 such that the cube of the sum of its digits is equal to the square of such integer. 2. Given two circumferences M and N , we say that M bisects N if they intersect in two points and the common chord is a diameter of N . Consider two fixed circumferences C1 and C2 not concentric. (a) Show that there exists infinite many circumferences B such that B bisects both C1 and C2 (b) Find the locus of the centers of such circumferences B. 3. Let P1 , P2 , . . . , Pn be n disctinct points over a line in the plane (n ≥ 2). Consider all the circumferences with diameters Pi Pj (1 ≤ i, j ≤ n) and they are painted with k given colors. Lets call this configuration a (n, k)-cloud. For each positive integer k, find all the positive integers n such that every posible (n, k)-cloud has two mutually exterior tangent circumferences of the same color. 4. Let B be an integer greater than 10 such that everyone of its digits belongs to the set {1, 3, 7, 9}. Show that B has a factor greater than or equal to 11. 5. An acute triangle 4ABC is inscribed in a circumference of center O. The highs of the triangle are AD, BE and CF . The line EF cut the circumference on P and Q. (a) Show that OA is perpendicular to P Q. (b) If M is the midpoint of BC, show that AP 2 = 2AD · OM .
´ 2.15. 15T H IBEROAMERICAN OLYMPIAD, MERIDA, VENEZUELA, 2000
67
6. Let A and B points in the plane and C a point in the segment bisector of AB. It is constructed a sequence of points C1 , C2 , . . . , Cn , . . . in the following way: C1 = C and for n ≥ 1, if Cn does not belongs to AB, then Cn+1 is the circumcenter of the triangle 4ABCn . Find all the points C such that the sequence C1 , C2 , . . . is defined for all n and turns eventually periodic5
2.15
15th Iberoamerican Olympiad, M´ erida, Venezuela, 2000
1. It is constructed a regular polygon of n sides (n ≥ 3) and its vertex are numbered from 1 to n. One draws all the diagonals of the polygon. Show that if n is odd, it is posible to assign to each side and to each diagonal an integer number between 1 and n, such that the next two conditions are simultaneusly satisfied: (a) The number assigned to each side or diagonal is diferent to the number assigned to any of the vertices that is endpoint of it. (b) For each vertex, all the sides and diagonals that have it as an endpoint, have diferent number assigned. 2. Let S1 and S2 be two circumferences, with centers O1 and O2 respectively, and secants on M and N . The line t is the common tangent to S1 and S2 closer to M . The points A and B are the intersection points of t with S1 and S2 , C is the point such that BC is a diameter of S2 , and D the intersection point of the line O1 O2 with the perpendicular line to AM thru B. Show that M , D and C are colinear 3. Find all the solutions of the equation (x + 1)y − xz = 1 For x, y, z integers greater than 1. 4. From an infinite arithmetic progression 1, a1 , a2 , . . . of real numbers some terms are deleted, obtaining an infinite geometric progression 1, b1 , b2 , . . . whose ratio is q. Find all the possible values of q. 5 A sequence C1 , C2 , . . . is called eventually periodic if there exist positive integers k and p such that Cn+p = cn for all n ≥ k
68
CHAPTER 2. IBEROAMERICAN MATHEMATICS OLYMPIAD 5. There are a buch of 2000 stones. Two players play alternatively, following the next rules: (a) On each turn, the player can take 1, 2, 3, 4 or 5 stones or the bunch. (b) On each turn, the player has forbiden to take the exact same amount of stones that the other player took just before of him in the last play. The player that on its turn can not do a valid play. Determine which player has winning strategy and give such strategy. 6. A convex hexagon is called pretty if it has for diagonals of length 1, such that their endpoints are all the vertex of the hexagon. (a) Given any real number k with 0 < k < 1 find a pretty hexagon with area equal to k (b) Show that the area of any pretty hexagon is less than 1.
2.16
16th Iberoamerican Olympiad, Minas, Uruguay, 2001
1. We say that a natural number n is bogus if it satisfy simultaneously the next conditions: • Every digit of n is greater than 1.
• Every time that four digits of n are multiplied, it is obtained a divisor of n
Show that every natural number k there exists a bogus number with more than k digits. 2. The inscrite circumference of the triangle 4ABC has center at O and it is tangent to the sides BC, AC and AB at the points X, Y and Z, respectively. The lines BO and CO intersect the line Y Z at the points P and Q, respectively. Show that if the segments XP and XQ has the same lenght, then the triangle 4ABC is isosceles. 3. Let S be a set of n elements and S1 , S2 , . . . , Sk are subsets of S (k ≥ 2), such that every one of them has at least r elements.
2.17. 17T H IBEROAMERICAN OLYMPIAD, SAN SALVADOR, EL SALVADOR, 200269 Show that there exists i and j, with 1 ≤ i < j ≤ k such that |Si
\
Sj | ≤ r −
nk 4(k − 1)
4. Find the maximum number or increasing arithmetic progresions that can have a finite sequence of real numbers a1 < a2 < · · · < an of n real numbers.
Note: Three terms on ai , aj , ak of a sequence of real numbers are in increasing arithmetic progresion if i < j < k, ai < aj < ak and aj − ai = ak − aj
5. In a board of 2000 × 2001 squares with integer coordinates (x, y), 0 ≤ x ≤ 1999 and 0 ≤ y ≤ 2000. An ship in the table moves in the next way: before each movement, the ship is in position (x, y) and has a velocity of (h, v) where h and v are integers. The ship choosea new velocity (h0 , v 0 ) such that h0 − h is either -1, 0 or 1, and v 0 − v is either -1, 0 or 1. The new position of the ship will be (x0 , y 0 ) where x0 is the remainder of the division of x + h0 by 2000 and y 0 is the remainder of the division of y + v 0 by 2001. There are two ships in the board: The Martian ship and the Terrestrian ship. The Terrestrian ship wants to catch the Martian ship. Initially each ship is in a different square and has velocity (0, 0). The first ship to move is the Terrestrian, and they continue moving alternatively. There exists a strategy such that the Terrestrian ship always catch to the Martian ship, whatever are the initial positions? Note: The Terrestrial ship always see the Martian ship and it catch it if after a movement it reaches the same position of the Martian ship. 6. Show that it is impossible to cover a unit sqare with five equal squares with side s ≤ 21 .
2.17
17th Iberoamerican Olympiad, San Salvador, El Salvador, 2002
1. The integer numbers from 1 to 2002, are written in a chalkboard on increasing order 1, 2,. . . , 2001, 2002. After that, somebody erase the numbers in the (3k + 1th places (1,4,7,. . . ). After that, the same person erase the numbers in the (3k + 1)th positions of the new list (in this case, 2, 5, 9,. . . ). This process is repited all the times that it is necesary to erase all the numbers in the list. What is the last number to be erased?
70
CHAPTER 2. IBEROAMERICAN MATHEMATICS OLYMPIAD 2. Given any set of 9 points in the plane such that there is no 3 of them colinear, show that for each point P of the set, the number of triangles with its vertices on the other 8 points and that contain P on its interior is even. 3. Let P be a point in the interior of the equilateral triangle 4ABC such that ^AP C = 120◦ . Let M be the intersection of CP with AB, and N the intersection of AP and BC. Find the locus of the circumcenter of the triangle 4M BN when P varies. 4. In a triangle 4ABC with all its sides of diferent length, D is on the side AC, such that BD is the angle bisector of ^ABC. Let E and F , respectively, be the feet of the perpendicular drawn from A and C to the line BD and let M be the point on BC such that DM is perpendicular to BC. Show that ^EM D = ^DM F . 5. The sequence of real numbers a1 , a2 , . . . is defined as follows: a1 = 56 and an+1 = an −
1 for n ≥ 1 an
Show that there is an integer k, 1 ≤ k ≤ 2002 such that ak < 0. 6. A policeman try to catch a rober in a board of 2001 × 2001 squares. They play alternatively, and each one of them on its turn should move a space on each one of the next directions: ↓(down); → (right); - (up-left diagonal).
If the policeman is on the square in the right-down corner, he can go directly to the square in the left-up corner (the rober can not do this). Initially the pliceman is in the central square, and the rober is in the square over the diagonal from the right-up to the left-down corners, that is contiguos (right-up) to the square where the policeman is. Show that: (a) The rober may move at least 10000 times before the policeman catch it. (b) The policeman has an strategy such that he will eventually catch the rober Note: The policeman catch the rober if he reaches the square where the rober is. If the rober reaches the policeman square, there is not catch.
2.18
18th Iberoamerican Olympiad, Buenos Aires, Argentina, 2003
1. (a) There are two sequences of numbers, with 2003 consecutive integers each, and a
2.18. 18T H IBEROAMERICAN OLYMPIAD, BUENOS AIRES, ARGENTINA, 2003 71 table of 2 rows and 2003 columns ······ ······ Say if it is always positive to distribut the numbers on the first sequence in the first row and the second sequence in the second row, such that the sequence obtained of the 2003 columnwise sums form a new sequence of 2003 consecutive integers. (b) Same question, if 2003 is replaced with 2004? In both cases, if the answer is affirmative, explain how are the numbers distributed, and if the answer is negative, explain why. 2. Let C and D be two points on the semicircumference with diameter AB such that B and C are in distinct semiplanes with respect to the line AD. Denote by M , N and P the midpoints of AC, BD and CD respectively. Let OA and OB the circumcenters of the triangles 4ACP and 4BDP . Show that the lines OA OB and M N are parallel. 3. Pablo is copying form the chalkboard the next problem: Consider all the sequences of 2004 real numbers (x1 , x2 , . . . , x2003 ) such that x0 = 1 0 ≤ x1 ≤ 2x0 0 ≤ x2 ≤ 2x1 .. . 0 ≤ x2003 ≤ 2x2002 From all this sequences, determine that one where the next expression reaches its maximum value S = ··· When Pablo was going to copy the expression, it was erased from the board. The only thing that he could remember was that S was of the form S = ±x1 ± x2 ± · · · ± x2002 + x2003 where the last term had coeficient +1 and the others has coeficients either +1 or -1. Show that, even when Pablo does not have the complete statement, he can determine the solution of the problem.
72
CHAPTER 2. IBEROAMERICAN MATHEMATICS OLYMPIAD 4. Let M = {1, 2, . . . , 49} the set of the first 49 positive integers. Determine the maximum integer k such that the set M has a subset of k elements such that there is no 6 consecutive integers in such subset. For that value of k, calculate the amount of subset of M with k elements and the given property. 5. In a sqare ABCD, let P and Q be points on the sides BC and CD respectively, diferent from its endpoints, such that BP = CQ. Consider points X and Y such that X 6= Y , in the segments AP and AQ respectively. Show that, for every X and X choosen, there exists a triangle whose sides have lengths BX, XY and DY . 6. The sequences (an )n≥0 , (bn )n≥0 are defined as follows: a0 = 1, b0 = 4 and for all n ≥ 0 an+1 = a2001 + bn , bn+1 = b2001 + an n n Show that 2003 is not divisor of any of the terms in these two sequences.
Chapter 3 William Lowell Putnam Competition 3.1
46th Anual William Lowell Putnam Competition, 1985
1. Determine, with proof, the number of ordered triples (A1 , A2 , A3 ) of sets which have the property that (i) A1 ∪ A2 ∪ A3 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, and
(ii) A1 ∩ A2 ∩ A3 = ∅.
Express your answer in the form 2a 3b 5c 7d , where a, b, c, d are nonnegative integers. 2. Let T be an acute triangle. Inscribe a rectangle R in T with one side along a side of T . Then inscribe a rectangle S in the triangle formed by the side of R opposite the side on the boundary of T , and the other two sides of T , with one side along the side of R. For any polygon X, let A(X) denote the area of X. Find the maximum value, or show that no maximum exists, of A(R)+A(S) , where T ranges over all triangles and A(T ) R, S over all rectangles as above. 3. Let d be a real number. For each integer m ≥ 0, define a sequence {am (j)}, j = 0, 1, 2, . . . by the condition am (0) = d/2m , am (j + 1) = (am (j))2 + 2am (j), Evaluate limn→∞ an (n). 73
j ≥ 0.
74
CHAPTER 3. WILLIAM LOWELL PUTNAM COMPETITION 4. Define a sequence {ai } by a1 = 3 and ai+1 = 3ai for i ≥ 1. Which integers between 00 and 99 inclusive occur as the last two digits in the decimal expansion of infinitely many ai ? 5. Let Im = Im 6= 0?
R 2π 0
cos(x) cos(2x) · · · cos(mx) dx. For which integers m, 1 ≤ m ≤ 10 is
6. If p(x) = a0 + a1 x + · · · + am xm is a polynomial with real coefficients ai , then set Γ(p(x)) = a20 + a21 + · · · + a2m . Let F (x) = 3x2 + 7x + 2. Find, with proof, a polynomial g(x) with real coefficients such that (i) g(0) = 1, and (ii) Γ(f (x)n ) = Γ(g(x)n ) for every integer n ≥ 1. 7. Let k be the smallest positive integer for which there exist distinct integers m1 , m2 , m3 , m4 , m5 such that the polynomial p(x) = (x − m1 )(x − m2 )(x − m3 )(x − m4 )(x − m5 ) has exactly k nonzero coefficients. Find, with proof, a set of integers m1 , m2 , m3 , m4 , m5 for which this minimum k is achieved. 8. Define polynomials fn (x) for n ≥ 0 by f0 (x) = 1, fn (0) = 0 for n ≥ 1, and d fn+1 (x) = (n + 1)fn (x + 1) dx for n ≥ 0. Find, with proof, the explicit factorization of f100 (1) into powers of distinct primes. 9. Let
a1,1 a1,2 a1,3 a2,1 a2,2 a2,3 a3,1 a3,2 a3,3 .. .. .. . . .
... ... ... .. .
be a doubly infinite array of positive integers, and suppose each positive integer appears exactly eight times in the array. Prove that am,n > mn for some pair of positive integers (m, n).
3.2. 47T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1986
75
10. Let C be the unit circle x2 +y 2 = 1. A point p is chosen randomly on the circumference C and another point q is chosen randomly from the interior of C (these points are chosen independently and uniformly over their domains). Let R be the rectangle with sides parallel to the x and y-axes with diagonal pq. What is the probability that no point of R lies outside of C? √ R R ∞ −x2 −1 11. Evaluate 0∞ t−1/2 e−1985(t+t ) dt. You may assume that −∞ e dx = π.
12. Let G be a finite set of real n × n matrices {Mi }, 1 ≤ i ≤ r, which form a group P under matrix multiplication. Suppose that ri=1 tr(Mi ) = 0, where tr(A) denotes the Pr trace of the matrix A. Prove that i=1 Mi is the n × n zero matrix.
3.2
47th Anual William Lowell Putnam Competition, 1986
1. Find, with explanation, the maximum value of f (x) = x3 − 3x on the set of all real numbers x satisfying x4 + 36 ≤ 13x2 . 2. What is the units (i.e., rightmost) digit of $
%
1020000 ? 10100 + 3
P
2 3. Evaluate ∞ n=0 Arccot(n + n + 1), where Arccot t for t ≥ 0 denotes the number θ in the interval 0 < θ ≤ π/2 with cot θ = t.
4. A transversal of an n × n matrix A consists of n entries of A, no two in the same row or column. Let f (n) be the number of n × n matrices A satisfying the following two conditions: (a) Each entry αi,j of A is in the set {−1, 0, 1}.
(b) The sum of the n entries of a transversal is the same for all transversals of A. An example of such a matrix A is
−1 0 −1 A = 0 1 0 . 0 1 0
76
CHAPTER 3. WILLIAM LOWELL PUTNAM COMPETITION Determine with proof a formula for f (n) of the form f (n) = a1 bn1 + a2 bn2 + a3 bn3 + a4 , where the ai ’s and bi ’s are rational numbers. 5. Suppose f1 (x), f2 (x), . . . , fn (x) are functions of n real variables x = (x1 , . . . , xn ) with continuous second-order partial derivatives everywhere on Rn . Suppose further that there are constants cij such that ∂fj ∂fi − = cij ∂xj ∂xi for all i and j, 1 ≤ i ≤ n, 1 ≤ j ≤ n. Prove that there is a function g(x) on Rn such that fi + ∂g/∂xi is linear for all i, 1 ≤ i ≤ n. (A linear function is one of the form a0 + a1 x1 + a2 x2 + · · · + an xn .) 6. Let a1 , a2 , . . . , an be real numbers, and let b1 , b2 , . . . , bn be distinct positive integers. Suppose that there is a polynomial f (x) satisfying the identity n
(1 − x) f (x) = 1 +
n X
a i xb i .
i=1
Find a simple expression (not involving any sums) for f (1) in terms of b1 , b2 , . . . , bn and n (but independent of a1 , a2 , . . . , an ). 7. Inscribe a rectangle of base b and height h in a circle of radius one, and inscribe an isosceles triangle in the region of the circle cut off by one base of the rectangle (with that side as the base of the triangle). For what value of h do the rectangle and triangle have the same area? 8. Prove that there are only a finite number of possibilities for the ordered triple T = (x − y, y − z, z − x), where x, y, z are complex numbers satisfying the simultaneous equations x(x − 1) + 2yz = y(y − 1) + 2zx + z(z − 1) + 2xy, and list all such triples T .
9. Let Γ consist of all polynomials in x with integer coefficienst. For f and g in Γ and m a positive integer, let f ≡ g (mod m) mean that every coefficient of f − g is an integral multiple of m. Let n and p be positive integers with p prime. Given that f, g, h, r and s are in Γ with rf + sg ≡ 1 (mod p) and f g ≡ h (mod p), prove that there exist F and G in Γ with F ≡ f (mod p), G ≡ g (mod p), and F G ≡ h (mod pn ).
3.3. 48T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1987
77
√ 10. For a positive real number r, let G(r) be the minimum value of |r − m2 + 2n2 | for all integers m and n. Prove or disprove the assertion that limr→∞ G(r) exists and equals 0. 11. Let f (x, y, z) = x2 + y 2 + z 2 + xyz. Let p(x, y, z), q(x, y, z), r(x, y, z) be polynomials with real coefficients satisfying f (p(x, y, z), q(x, y, z), r(x, y, z)) = f (x, y, z). Prove or disprove the assertion that the sequence p, q, r consists of some permutation of ±x, ±y, ±z, where the number of minus signs is 0 or 2. 12. Suppose A, B, C, D are n × n matrices with entries in a field F , satisfying the conditions that AB T andCD T are symmetric and AD T − BC T = I. Here I is the n × n identity matrix, and if M is an n × n matrix, M T is its transpose. Prove that AT D + C T B = I.
3.3
48th Anual William Lowell Putnam Competition, 1987
1. Curves A, B, C and D are defined in the plane as follows: (
)
x A = (x, y) : x − y = 2 , x + y2 ) ( y =3 , B = (x, y) : 2xy + 2 x + y2 C = D = Prove that A ∩ B = C ∩ D.
n
n
2
2
o
(x, y) : x3 − 3xy 2 + 3y = 1 , o
(x, y) : 3x2 y − 3x − y 3 = 0 .
2. The sequence of digits 123456789101112131415161718192021 . . . is obtained by writing the positive integers in order. If the 10n -th digit in this sequence occurs in the part of the sequence in which the m-digit numbers are placed, define f (n) to be m. For example, f (2) = 2 because the 100th digit enters the sequence in the placement of the two-digit integer 55. Find, with proof, f (1987).
78
CHAPTER 3. WILLIAM LOWELL PUTNAM COMPETITION 3. For all real x, the real-valued function y = f (x) satisfies y 00 − 2y 0 + y = 2ex . (a) If f (x) > 0 for all real x, must f 0 (x) > 0 for all real x? Explain. (b) If f 0 (x) > 0 for all real x, must f (x) > 0 for all real x? Explain. 4. Let P be a polynomial, with real coefficients, in three variables and F be a function of two variables such that P (ux, uy, uz) = u2 F (y − x, z − x) for all real x, y, z, u, and such that P (1, 0, 0) = 4, P (0, 1, 0) = 5, and P (0, 0, 1) = 6. Also let A, B, C be complex numbers with P (A, B, C) = 0 and |B − A| = 10. Find |C − A|. 5. Let ~ G(x, y) =
!
−y x , ,0 . x2 + 4y 2 x2 + 4y 2
Prove or disprove that there is a vector-valued function F~ (x, y, z) = (M (x, y, z), N (x, y, z), P (x, y, z)) with the following properties: (i) M, N, P have continuous partial derivatives for all (x, y, z) 6= (0, 0, 0);
(ii) Curl F~ = ~0 for all (x, y, z) 6= (0, 0, 0);
~ (iii) F~ (x, y, 0) = G(x, y).
6. For each positive integer n, let a(n) be the number of zeroes in the base 3 representation of n. For which positive real numbers x does the series ∞ X
xa(n) 3 n=1 n converge? 7. Evaluate Z
4 2
q
q
ln(9 − x) dx
ln(9 − x) +
q
ln(x + 3)
.
3.4. 49T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1988
79
8. Let r, s and t be integers with 0 ≤ r, 0 ≤ s and r + s ≤ t. Prove that s 0 t r
+
s 1 t r+1
+···+
s s t r+s
=
t+1 (t + 1 − s)
t−s r
9. Let F be a field in which 1 + 1 6= 0. Show that the set of solutions to the equation x2 + y 2 = 1 with x and y in F is given by (x, y) = (1, 0) and (x, y) =
r 2 − 1 2r , r2 + 1 r2 + 1
!
where r runs through the elements of F such that r 2 6= −1. 10. Let (x1 , y1 ) = (0.8, 0.6) and let xn+1 = xn cos yn − yn sin yn and yn+1 = xn sin yn + yn cos yn for n = 1, 2, 3, . . .. For each of limn→∞ xn and limn→∞ yn , prove that the limit exists and find it or prove that the limit does not exist. 11. Let On be the n-dimensional vector (0, 0, · · · , 0). Let M be a 2n × n matrix of complex numbers such that whenever (z1 , z2 , . . . , z2n )M = On , with complex zi , not all zero, then at least one of the zi is not real. Prove that for arbitrary real numbers r1 , r2 , . . . , r2n , there are complex numbers w1 , w2 , . . . , wn such that
w1 . re M .. = wn
r1 .. . . rn
(Note: if C is a matrix of complex numbers, re(C) is the matrix whose entries are the real parts of the entries of C.) 12. Let F be the field of p2 elements, where p is an odd prime. Suppose S is a set of (p2 − 1)/2 distinct nonzero elements of F with the property that for each a 6= 0 in F , exactly one of a and −a is in S. Let N be the number of elements in the intersection S ∩ {2a : a ∈ S}. Prove that N is even.
3.4
49th Anual William Lowell Putnam Competition, 1988
1. Let R be the region consisting of the points (x, y) of the cartesian plane satisfying both |x| − |y| ≤ 1 and |y| ≤ 1. Sketch the region R and find its area.
80
CHAPTER 3. WILLIAM LOWELL PUTNAM COMPETITION 2. A not uncommon calculus mistake is to believe that the product rule for derivatives 2 says that (f g)0 = f 0 g 0 . If f (x) = ex , determine, with proof, whether there exists an open interval (a, b) and a nonzero function g defined on (a, b) such that this wrong product rule is true for x in (a, b). 3. Determine, with proof, the set of real numbers x for which ∞ X 1
n=1
n
csc
1 −1 n
x
converges. 4. (a) If every point of the plane is painted one of three colors, do there necessarily exist two points of the same color exactly one inch apart? (b) What if three is replaced by nine? 5. Prove that there exists a unique function f from the set R+ of positive real numbers to R+ such that f (f (x)) = 6x − f (x) and f (x) > 0 for all x > 0. 6. If a linear transformation A on an n-dimensional vector space has n + 1 eigenvectors such that any n of them are linearly independent, does it follow that A is a scalar multiple of the identity? Prove your answer. 7. A composite (positive integer) is a product ab with a and b not necessarily distinct integers in {2, 3, 4, . . .}. Show that every composite is expressible as xy + xz + yz + 1, with x, y, z positive integers. 8. Prove or disprove: If x and y are real numbers with y ≥ 0 and y(y + 1) ≤ (x + 1)2 , then y(y − 1) ≤ x2 . 9. For every√n in the set N = {1, 2, . . .} of positive integers, let rn be the minimum value of |c − d 3| for all nonnegative integers c and d with c + d = n. Find, with proof, the smallest positive real number g with rn ≤ g for all n ∈ N. P
10. Prove that if ∞ n=1 an is a convergent series of positive real numbers, then so is P∞ n/(n+1) (a ) . n=1 n
3.5. 50T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1989
81
11. For positive integers n, let Mn be the 2n + 1 by 2n + 1 skew-symmetric matrix for which each entry in the first n subdiagonals below the main diagonal is 1 and each of the remaining entries below the main diagonal is -1. Find, with proof, the rank of Mn . (According to one definition, the rank of a matrix is the largest k such that there is a k × k submatrix with nonzero determinant.) One may note that
0 −1 1 M1 = 1 0 −1 −1 1 0 M2 =
0 −1 −1 1 1 1 0 −1 −1 1 1 1 0 −1 −1 −1 1 1 0 −1 −1 −1 1 1 0
12. Prove that there exist an infinite number of ordered pairs (a, b) of integers such that for every positive integer t, the number at + b is a triangular number if and only if t is a triangular number. (The triangular numbers are the tn = n(n + 1)/2 with n in {0, 1, 2, . . .})
3.5
50th Anual William Lowell Putnam Competition, 1989
1. How many primes among the positive integers, written as usual in base 10, are alternating 1’s and 0’s, beginning and ending with 1? 2. Evaluate
Z
a 0
Z
b 0
emax{b
2 x2 ,a2 y 2 }
dy dx where a and b are positive.
3. Prove that if 11z 10 + 10iz 9 + 10iz − 11 = 0, then |z| = 1. (Here z is a complex number and i2 = −1.) 4. If α is an irrational number, 0 < α < 1, is there a finite game with an honest coin such that the probability of one player winning the game is α? (An honest coin is one for which the probability of heads and the probability of tails are both 12 . A game is finite if with probability 1 it must end in a finite number of moves.)
82
CHAPTER 3. WILLIAM LOWELL PUTNAM COMPETITION 5. Let m be a positive integer and let G be a regular (2m + 1)-gon inscribed in the unit circle. Show that there is a positive constant A, independent of m, with the following property. For any points p inside G there are two distinct vertices v1 and v2 of G such that 1 A | |p − v1 | − |p − v2 | | < − 3. m m Here |s − t| denotes the distance between the points s and t. 6. Let α = 1 + a1 x + a2 x2 + · · · be a formal power series with coefficients in the field of two elements. Let
an =
1
if every block of zeros in the binary expansion of n has an even number of zeros in the block
0
otherwise.
(For example, a36 = 1 because 36 = 1001002 and a20 = 0 because 20 = 101002 .) Prove that α3 + xα + 1 = 0. 7. A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point √ hit a b+c , is nearer to the center than to any edge. Express your answer in the form d where a, b, c, d are integers. 8. Let S be a non-empty set with an associative operation that is left and right cancellative (xy = xz implies y = z, and yx = zx implies y = z).ssume that for every a in S the set {an : n = 1, 2, 3, . . .} is inite. Must S be a group? 9. Let f be a function on [0, ∞), differentiable and satisfying f 0 (x) = −3f (x) + 6f (2x) √
for x > 0. Assume that |f (x)| ≤ e− x for x ≥ 0 (so that f (x) tends rapidly to 0 as x increases). For n a non-negative integer, define µn =
Z
∞ 0
xn f (x) dx
(sometimes called the nth moment of f ). a) Express µn in terms of µ0 . n
b) Prove that the sequence {µn 3n! } always converges, and that the limit is 0 only if µ0 = 0.
3.6. 51T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1990
83
10. Can a countably infinite set have an uncountable collection of non-empty subsets such that the intersection of any two of them is finite? 11. Label the vertices of a trapezoid T (quadrilateral with two parallel sides) inscribed in the unit circle as A, B, C, D so that AB is parallel to CD and A, B, C, D are in counterclockwise order. Let s1 , s2 , and d denote the lengths of the line segments AB, CD, and OE, where E is the point of intersection of the diagonals of T , and O 2 is the center of the circle. Determine the least upper bound of s1 −s over all such T d for which d 6= 0, and describe allases, if any, in which it is attained. 12. Let (x1 , x2 , . . . xn ) be a point chosen at random from the n-dimensional region defined by 0 < x1 < x2 < · · · < xn < 1. Let f be a continuous function on [0, 1] with f (1) = 0. Set x0 = 0 and xn+1 = 1. Show that the expected value of the Riemann sum n X i=0
(xi+1 − xi )f (xi+1 )
R
is 01 f (t)P (t) dt, where P is a polynomial of degree n, independent of f , with 0 ≤ P (t) ≤ 1 for 0 ≤ t ≤ 1.
3.6
51th Anual William Lowell Putnam Competition, 1990
1. Let T0 = 2, T1 = 3, T2 = 6, and for n ≥ 3,
Tn = (n + 4)Tn−1 − 4nTn−2 + (4n − 8)Tn−3 .
The first few terms are 2, 3, 6, 14, 40, 152, 784, 5168, 40576. Find, with proof, a formula for Tn of the form Tn = An + Bn , where {An } and {Bn } are well-known sequences. √ √ √ 2. Is 2 the limit of a sequence of numbers of the form 3 n − 3 m (n, m = 0, 1, 2, . . .)? 3. Prove that any convex pentagon whose vertices (no three of which are collinear) have integer coordinates must have area greater than or equal to 25 .
84
CHAPTER 3. WILLIAM LOWELL PUTNAM COMPETITION 4. Consider a paper punch that can be centered at any point of the plane and that, when operated, removes from the plane precisely those points whose distance from the center is irrational. How many punches are needed to remove every point? 5. If A and B are square matrices of the same size such that ABAB = 0, does it follow that BABA = 0? 6. If X is a finite set, let X denote the number of elements in X. Call an ordered pair (S, T ) of subsets of {1, 2, . . . , n} admissible if s > |T | for each s ∈ S, and t > |S| for each t ∈ T . How many admissible ordered pairs of subsets of {1, 2, . . . , 10} are there? Prove your answer. 7. Find all real-valued continuously differentiable functions f on the real line such that for all x, Z x 2 (f (x)) = [(f (t))2 + (f 0 (t))2 ] dt + 1990. 0
8. Prove that for |x| < 1, |z| > 1, 1+
∞ X
(1 + xj )Pj = 0,
j=1
where Pj is
(1 − z)(1 − zx)(1 − zx2 ) · · · (1 − zxj−1 ) . (z − x)(z − x2 )(z − x3 ) · · · (z − xj )
9. Let S be a set of 2 × 2 integer matrices whose entries aij (1) are all squares of integers and, (2) satisfy aij ≤ 200. Show that if S has more than 50387 (= 154 − 152 − 15 + 2) elements, then it has two elements that commute. 10. Let G be a finite group of order n generated by a and b. Prove or disprove: there is a sequence g1 , g2 , g3 , . . . , g2n such that (1) every element of G occurs exactly twice, and (2) gi+1 equals gi a or gi b for i = 1, 2, . . . , 2n. (Interpret g2n+1 as g1 .) 11. Is there an infinite sequence a0 , a1 , a2 , . . . of nonzero real numbers such that for n = 1, 2, 3, . . . the polynomial pn (x) = a0 + a1 x + a2 x2 + · · · + an xn has exactly n distinct real roots?
3.7. 52T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1991
85
12. Let S be a nonempty closed bounded convex set in the plane. Let K be a line and t a positive number. Let L1 and L2 be support lines for S parallel to K1 , and let L be the line parallel to K and midway between L1 and L2 . Let BS (K, t) be the band of points whose distance from L is at most (t/2)w, where w is the distance between L1 and L2 . What is the smallest t such that S∩
\ K
BS (K, t) 6= ∅
for all S? (K runs over all lines in the plane.)
3.7
52th Anual William Lowell Putnam Competition, 1991
1. A 2 × 3 rectangle has vertices as (0, 0), (2, 0), (0, 3), and (2, 3). It rotates 90 ◦ clockwise about the point (2, 0). It then otates 90◦ clockwise about the point (5, 0), then 90◦ clockwise about the point (7, 0), and finally, 90◦ clockwise about the point (10, 0). (The side originally on the x-axis is now back on the x-axis.) Find the area of the region above the x-axis and below the curve traced out by the point whose initial position is (1,1). 2. Let A and B be different n×n matrices with real entries. If A3 = B 3 and A2 B = B 2 A, can A2 + B 2 be nvertible? 3. Find all real polynomials p(x) of degree n ≥ 2 for which there exist real numbers r1 < r2 < · · · < rn such that (a) p(ri ) = 0, (b) p0
ri +ri+1 2
i = 1, 2, . . . , n, and
=0
i = 1, 2, . . . , n − 1,
where p0 (x) denotes the derivative of p(x). 4. Does there exist an infinite sequence of closed discs D1 , D2 , D3 , . . . in the plane, with centers c1 , c2 , c3 , . . ., respectively, such that (a) the ci have no limit point in the finite plane, (b) the sum of the areas of the Di is finite, and (c) every line in the plane intersects at least one of the Di ?
86
CHAPTER 3. WILLIAM LOWELL PUTNAM COMPETITION 5. Find the maximum value of Z
y 0
q
x4 + (y − y 2 )2 dx
for 0 ≤ y ≤ 1. 6. Let A(n) denote the number of sums of positive integers a1 + a 2 + · · · + a r which add up to n with a1 > a 2 + a 3 , a2 > a 3 + a 4 , . . . ar−2 > ar−1 + ar , ar−1 > ar Let B(n) denote the number of b1 + b2 + · · · + bs which add up to n, with (a) b1 ≥ b2 ≥ . . . ≥ bs ,
(b) each bi is in the sequence 1, 2, 4, . . . , gj , . . . defined by g1 = 1, g2 = 2, and gj = gj−1 + gj−2 + 1, and (c) if b1 = gk then every element in {1, 2, 4, . . . , gk } appears at least once as a bi . Prove that A(n) = B(n) for each n ≥ 1.
(For example, A(7) = 5 because the relevant sums are 7, 6 + 1, 5 + 2, 4 + 3, 4 + 2 + 1, and B(7) = 5 because the relevant sums are 4 + 2 + 1, 2 + 2 + 2 + 1, 2 + 2 + 1 + 1 + 1, 2 + 1 + 1 + 1 + 1 + 1, 1 + 1 + 1 + 1 + 1 + 1 + 1.)
7. For each integer n ≥ 0, let S(n) = n − m2 , where m is the greatest integer with m2 ≤ n. Define a sequence (ak )∞ k=0 by a0 = A and ak+1 = ak + S(ak ) for k ≥ 0. For what positive integers A is this sequence eventually constant? 8. Suppose f and g are non-constant, differentiable, real-valued functions defined on (−∞, ∞). Furthermore, suppose that for each pair of real numbers x and y, f (x + y) = f (x)f (y) − g(x)g(y), g(x + y) = f (x)g(y) + g(x)f (y). If f 0 (0) = 0, prove that (f (x))2 + (g(x))2 = 1 for all x. 9. Does there exist a real number L such that, if m and n are integers greater than L, then an m × n rectangle may be expressed as a nion of 4 × 6 and 5 × 7 rectangles, any two of which intersect at most along their boundaries?
3.8. 53T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1992
87
10. Suppose p is an odd prime. Prove that p X
j=0
p j
!
!
p+j ≡ 2p + 1 (mod p2 ). j
11. Let p be an odd prime and let Zp denote (the field of) integers modulo p. How many elements are in the set {x2 : x ∈ Zp } ∩ {y 2 + 1 : y ∈ Zp }? 12. Let a and b be positive numbers. Find the largest number c, in terms of a and b, such that sinh u(1 − x) sinh ux +b ax b1−x ≤ a sinh u sinh u for all u with 0 < |u| ≤ c and for all x, 0 < x < 1. (Note: sinh u = (eu − e−u )/2.)
3.8
53th Anual William Lowell Putnam Competition, 1992
1. Prove that f (n) = 1 − n is the only integer-valued function defined on the integers that satisfies the following conditions. (i) f (f (n)) = n, for all integers n; (ii) f (f (n + 2) + 2) = n for all integers n; (iii) f (0) = 1. 2. Define C(α) to be the coefficient of x1992 in the power series about x = 0 of (1 + x)α . Evaluate ! Z 1 1992 X 1 C(−y − 1) dy. 0 k=1 y + k 3. For a given positive integer m, find all triples (n, x, y) of positive integers, with n relatively prime to m, which satisfy (x2 + y 2 )m = (xy)n .
88
CHAPTER 3. WILLIAM LOWELL PUTNAM COMPETITION 4. Let f be an infinitely differentiable real-valued function defined on the real numbers. If 1 n2 , n = 1, 2, 3, . . . , f = 2 n n +1 compute the values of the derivatives f (k) (0), k = 1, 2, 3, . . .. 5. For each positive integer n, let an = 0 (or 1) if the number of 1’s in the binary representation of n is even (or odd), respectively. Show that there do not exist positive integers k and m such that ak+j = ak+m+j = ak+m+2j , for 0 ≤ j ≤ m − 1. 6. Four points are chosen at random on the surface of a sphere. What is the probability that the center of the sphere lies inside the tetrahedron whose vertices are at the four points? (It is understood that each point is independently chosen relative to a uniform distribution on the sphere.) 7. Let S be a set of n distinct real numbers. Let AS be the set of numbers that occur as averages of two distinct elements of S. For a given n ≥ 2, what is the smallest possible number of elements in AS ? 8. For nonnegative integers n and k, define Q(n, k) to be the coefficient of xk in the expansion of (1 + x + x2 + x3 )n . Prove that Q(n, k) =
k X
j=0
where a ≥ 0,
a
b a b
n j
!
!
n , k − 2j
is the standard binomial coefficient. (Reminder: For integers a and b with =
a! b!(a−b)!
for 0 ≤ b ≤ a, with
a b
= 0 otherwise.)
9. For any pair (x, y) of real numbers, a sequence (an (x, y))n≥0 is defined as follows: a0 (x, y) = x, (an (x, y))2 + y 2 , an+1 (x, y) = 2
for n ≥ 0.
Find the area of the region {(x, y)|(an (x, y))n≥0 converges}.
3.9. 54T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1993
89
10. Let p(x) be a nonzero polynomial of degree less than 1992 having no nonconstant factor in common with x3 − x. Let d1992 dx1992
p(x) x3 − x
!
f (x) g(x)
=
for polynomials f (x) and g(x). Find the smallest possible degree of f (x). 11. Let Dn denote the value of the (n − 1) × (n − 1) determinant
Is the set
n
Dn n!
o
n≥2
3 1 1 1 .. .
1 4 1 1 .. .
1 1 5 1 .. .
1 1 1 6 .. .
··· ··· ··· ··· .. .
1 1 1 1 .. .
1 1 1 1 ··· n+1
.
bounded?
12. Let M be a set of real n × n matrices such that (i) I ∈ M, where I is the n × n identity matrix;
(ii) if A ∈ M and B ∈ M, then either AB ∈ M or −AB ∈ M, but not both;
(iii) if A ∈ M and B ∈ M, then either AB = BA or AB = −BA;
(iv) if A ∈ M and A 6= I, there is at least one B ∈ M such that AB = −BA.
Prove that M contains at most n2 matrices.
3.9
54th Anual William Lowell Putnam Competition, 1993
1. The horizontal line y = c intersects the curve y = 2x − 3x3 in the first quadrant as in the figure. Find c so that the areas of the two shaded regions are equal. [Figure not included. The first region is bounded by the y-axis, the line y = c and the curve; the other lies under the curve and above the line y = c between their two points of intersection.] 2. Let (xn )n≥0 be a sequence of nonzero real numbers such that x2n − xn−1 xn+1 = 1 for n = 1, 2, 3, . . .. Prove there exists a real number a such that xn+1 = axn − xn−1 for all n ≥ 1.
90
CHAPTER 3. WILLIAM LOWELL PUTNAM COMPETITION 3. Let Pn be the set of subsets of {1, 2, . . . , n}. Let c(n, m) be the number of functions f : Pn → {1, 2, . . . , m} such that f (A ∩ B) = min{f (A), f (B)}. Prove that c(n, m) =
m X
j n.
j=1
4. Let x1 , x2 , . . . , x19 be positive integers each of which is less than or equal to 93. Let y1 , y2 , . . . , y93 be positive integers each of which is less than or equal to 19. Prove that there exists a (nonempty) sum of some xi ’s equal to a sum of some yj ’s. 5. Show that Z
−10 −100
x2 − x x3 − 3x + 1
!2
dx +
Z
1 11 1 101
x2 − x x3 − 3x + 1
!2
dx +
Z
11 10 101 100
x2 − x x3 − 3x + 1
!2
dx
is a rational number. 6. The infinite sequence of 2’s and 3’s 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, . . . as the property that, if one forms a second sequence that records the number of 3’s between successive 2’s, the result is identical to the iven sequence. Show that there exists a real number r such that, for ny n, the nth term of the sequence is 2 if and only if n = 1 + brmc for some nonnegative integer m. (Note: bxrf loor denotes the largest integer less than or equal to x.) 7. Find the smallest positive integer n such that for every integer m with 0 < m < 1993, there exists an integer k for which m k m+1 < < . 1993 n 1994 8. Consider the following game played with a deck of 2n cards numbered from 1 to 2n. The deck is randomly shuffled and n cards are dealt to each of two players. Beginning with A, the players take turns discarding one of their remaining cards and announcing its number. The game ends as soon as the sum of the numbers on the discarded cards is divisible by 2n + 1. The last person to discard wins the game. Assuming optimal strategy by both A and B, what is the probability that A wins? 9. Two real numbers x and y are chosen at random in the interval (0,1) with respect to the uniform distribution. What is the probability that he closest integer to x/y is even? Express the answer in the form r + sπ, where r and s are rational numbers.
3.10. 55T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1994
91
10. The function K(x, y) is positive and continuous for 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and the functions f (x) and g(x) are positive and continuous for 0 ≤ x ≤ 1. Suppose that for all x, 0 ≤ x ≤ 1, Z 1
0
and
Z
1 0
f (y)K(x, y) dy = g(x)
g(y)K(x, y) dy = f (x).
Show that f (x) = g(x) for 9 ≤ x ≤ 1. 11. Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers. 12. Let S be a set of three, not necessarily distinct, positive integers. Show that one can transform S into a set containing 0 by a finite number of applications of the following rule: Select two of the three integers, say x and y, where x < y and replace them with 2x and y − x.
3.10
55th Anual William Lowell Putnam Competition, 1994
1. Let (an ) be a sequence of positive reals such that, for all n, an ≤ a2n + a2n+1 . Prove P that ∞ n=1 an diverges.
2. Find the positive value of m such that the area in the first quadrant enclosed by the 2 ellipse x9 + y 2 = 1, the x-axis, and the line y = 2x/3 is equal to the area in the first 2 quadrant enclosed by the ellipse x9 + y 2 = 1, the y-axis, and the line y = mx.
3. Prove that the points of an isosceles triangle of side length √ 1 annot be colored in four colors such that no two points at distance at least 2 − 2 from each other receive the same color. 4. Let A and B be 2 × 2 matrices with integer entries such that each of A, A + B, A + 2B, A + 3B, A + 4B has an inverse with integer entries. Prove that the same must be true of A + 5B. 5. Let (rn ) be a sequence of positive reals with limit 0. Let S be the set of all numbers expressible in the form ri1 + . . . + ri1994 for positive integers i1 < i2 < . . . < i1994 . Prove that every interval (a, b) contains a subinterval (c, d) whose intersection with S is empty.
92
CHAPTER 3. WILLIAM LOWELL PUTNAM COMPETITION 6. Let f1 , . . . , f10 be bijections of the integers such that for every integer n, there exists a sequence i1 , . . . , ik for some k such that fi1 ◦ . . . ◦ fik (0) = n. Prove that if A is any nonempty finite set, there exist at most 512 sequences (e1 , . . . , e10 ) of zeroes and e10 ones such that f1e1 ◦ . . . ◦ f10 maps A to A. (Here f 1 = f and f 0 means the identity function.) 7. Find all positive integers n such that |n − m2 | ≤ 250 for exactly 15 nonnegative integers m. 8. Find all c such that the graph of the function x4 + 9x3 + cx2 + ax + b meets some line in four distinct points. 9. Let f (x) be a positive-valued function over the reals such that f 0 (x) > f (x) for all x. For what k must there exist N such that f (x) > ekx for x > N ? !
3 2 10. Let A be the matrix and for positive integers n, define dn as the greatest 4 2 common divisor of the entries of An − I, where I = ((10)(01)). Prove that dn → ∞ as n → ∞. 11. Fix n a positive integer. For α real, define fα (i) as the greatest integer less than or equal to αi, and write f k for the k-th iterate of f (i.e. f 1 = f and f k+1 = f ◦ f k ). Prove there exists α such that fαk (n2 ) = fαk (n2 ) = n2 − k for k = 1, . . . , n. 12. Suppose a, b, c, d are integers with 0 ≤ a ≤ bleq99, 0 ≤ c ≤ d ≤ 99. For any integer i, let ni = 101i + 1002i . Show that if na + nb is congruent to nc + nd mod 10100, then a = c and b = d.
3.11
56th Anual William Lowell Putnam Competition, 1995
1. Let S be a set of real numbers which is closed under multiplication (that is, if a and b are in S, then so is ab). Let T and U be disjoint subsets of S whose union is S. Given that the product of any three (not necessarily distinct) elements of T is in T and that the product of any three elements of U is in U , show that at least one of the two subsets T, U is closed under multiplication. 2. For what pairs (a, b) of positive real numbers does the improper integral Z
∞ b
q√
x+a−
√
x−
q√
x−
√
x − b dx
3.11. 56T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1995
93
converge? 3. The number d1 d2 . . . d9 has nine (not necessarily distinct) decimal digits. The number e1 e2 . . . e9 is such that each of the nine 9-digit numbers formed by replacing just one of the digits di is d1 d2 . . . d9 by the corresponding digit ei (1 ≤ i ≤ 9) is divisible by 7. The number f1 f2 . . . f9 is related to e1 e2 . . . e9 is the same way: that is, each of the nine numbers formed by replacing one of the ei by the corresponding fi is divisible by 7. Show that, for each i, di −fi is divisible by 7. [For example, if d1 d2 . . . d9 = 199501996, then e6 may be 2 or 9, since 199502996 and 199509996 are multiples of 7.] 4. Suppose we have a necklace of n beads. Each bead is labeled with an integer and the sum of all these labels is n − 1. Prove that we can cut the necklace to form a string whose consecutive labels x1 , x2 , . . . , xn satisfy k X i=1
xi ≤ k − 1
for k = 1, 2, . . . , n.
5. Let x1 , x2 , . . . , xn be differentiable (real-valued) functions of a single variable f which satisfy dx1 = a11 x1 + a12 x2 + · · · + a1n xn dt dx2 = a21 x1 + a22 x2 + · · · + a2n xn dt .. .. . . dxn = an1 x1 + an2 x2 + · · · + ann xn dt for some constants aij > 0. Suppose that for all i, xi (t) → 0 as t → ∞. Are the functions x1 , x2 , . . . , xn necessarily linearly dependent? 6. Suppose that each of n people writes down the numbers 1,2,3 in random order in one column of a 3 × n matrix, with all orders equally likely and with the orders for different columns independent of each other. Let the row sums a, b, c of the resulting matrix be rearranged (if necessary) so that a ≤ b ≤ c. Show that for some n ≥ 1995, it is at least four times as likely that both b = a + 1 and c = a + 2 as that a = b = c. 7. For a partition π of {1, 2, 3, 4, 5, 6, 7, 8, 9}, let π(x) be the number of elements in the part containing x. Prove that for any two partitions π and π 0 , there are two distinct numbers x and y in {1, 2, 3, 4, 5, 6, 7, 8, 9} such that π(x) = π(y) and π 0 (x) = π 0 (y). [A partition of a set S is a collection of disjoint subsets (parts) whose union is S.]
94
CHAPTER 3. WILLIAM LOWELL PUTNAM COMPETITION 8. An ellipse, semi-axes have lengths a and b, rolls without slipping on the curve whose x y = c sin a . How are a, b, c related, given that the ellipse completes one revolution when it traverses one period of the curve? 9. To each positive integer with n2 decimal digits, we associate the determinant of the matrix obtained by writing the digits in order across the rows. For example, for ! 8 6 n = 2, to the integer 8617 we associate det = 50. Find, as a function of n, 1 7 the sum of all the determinants associated with n2 -digit integers. (Leading digits are assumed to be nonzero; for example, for n = 2, there are 9000 determinants.)
10. Evaluate
v u u 8 t 2207 −
Express your answer in the form
√ a+b c , d
1 . 1 2207 − 2207−... where a, b, c, d are integers.
11. A game starts with four heaps of beans, containing 3,4,5 and 6 beans. The two players move alternately. A move consists of taking either a) one bean from a heap, provided at least two beans are left behind in that heap, or b) a complete heap of two or three beans. The player who takes the last heap wins. To win the game, do you want to move first or second? Give a winning strategy. 12. For a positive real number α, define S(α) = {bnαc : n = 1, 2, 3, . . .}. Prove that {1, 2, 3, . . .} cannot be expressed as the disjoint union of three sets S(α), S(β) and S(γ). [As usual, bxc is the greatest integer ≤ x.]
3.12
57th Anual William Lowell Putnam Competition, 1996
1. Find the least number A such that for any two squares of combined area 1, a rectangle of area A exists such that the two squares can be packed in the rectangle (without interior overlap). You may assume that the sides of the squares are parallel to the sides of the rectangle.
3.12. 57T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1996
95
2. Let C1 and C2 be circles whose centers are 10 units apart, and whose radii are 1 and 3. Find, with proof, the locus of all points M for which there exists points X on C1 and Y on C2 such that M is the midpoint of the line segment XY . 3. Suppose that each of 20 students has made a choice of anywhere from 0 to 6 courses from a total of 6 courses offered. Prove or disprove: there are 5 students and 2 courses such that all 5 have chosen both courses or all 5 have chosen neither course. 4. Let S be the set of ordered triples (a, b, c) of distinct elements of a finite set A. Suppose that (a) (a, b, c) ∈ S if and only if (b, c, a) ∈ S;
(b) (a, b, c) ∈ S if and only if (c, b, a) ∈ / S;
(c) (a, b, c) and (c, d, a) are both in S if and only if (b, c, d) and (d, a, b) are both in S.
Prove that there exists a one-to-one function g from A to R such that g(a) < g(b) < g(c) implies (a, b, c) ∈ S. Note: R is the set of real numbers. 5. If p is a prime number greater than 3 and k = b2p/3c, prove that the sum !
!
p p p + +···+ 1 2 k
!
of binomial coefficients is divisible by p2 . 6. Let c > 0 be a constant. Give a complete description, with proof, of the set of all continuous functions f : R → R such that f (x) = f (x2 + c) for all x ∈ R. Note that R denotes the set of real numbers. 7. Define a selfish set to be a set which has its own cardinality (number of elements) as an element. Find, with proof, the number of subsets of {1, 2, . . . , n} which are minimal selfish sets, that is, selfish sets none of whose proper subsets is selfish. 8. Show that for every positive integer n,
2n − 1 e
2n−1 2
2n + 1 < 1 · 3 · 5 · · · (2n − 1) < e
2n+1 2
.
9. Given that {x1 , x2 , . . . , xn } = {1, 2, . . . , n}, find, with proof, the largest possible value, as a function of n (with n ≥ 2), of x1 x2 + x2 x3 + · · · + xn−1 xn + xn x1 .
96
CHAPTER 3. WILLIAM LOWELL PUTNAM COMPETITION
10. For any square matrix A, we can define sin A by the usual power series: ∞ X
(−1)n sin A = A2n+1 . n=0 (2n + 1)! Prove or disprove: there exists a 2 × 2 matrix A with real entries such that sin A =
1 1996 0 1
!
.
11. Given a finite string S of symbols X and O, we write ∆(S) for the number of X’s in S minus the number of O’s. For example, ∆(XOOXOOX) = −1. We call a string S balanced if every substring T of (consecutive symbols of) S has −2 ≤ ∆(T ) ≤ 2. Thus, XOOXOOX is not balanced, since it contains the substring OOXOO. Find, with proof, the number of balanced strings of length n. 12. Let (a1 , b1 ), (a2 , b2 ), . . . , (an , bn ) be the vertices of a convex polygon which contains the origin in its interior. Prove that there exist positive real numbers x and y such that (a1 , b1 )xa1 y b1 + (a2 , b2 )xa2 y b2 + · · · + (an , bn )xan y bn = (0, 0)
58th Anual William Lowell Putnam Competition, 1997
3.13
1. A rectangle, HOM F , has sides HO = 11 and OM = 5. A triangle ABC has H as the intersection of the altitudes, O the center of the circumscribed circle, M the midpoint of BC, and F the foot of the altitude from A. What is the length of BC? 2. Players 1, 2, 3, . . . , n are seated around a table, and each has a single penny. Player 1 passes a penny to player 2, who then passes two pennies to player 3. Player 3 then passes one penny to Player 4, who passes two pennies to Player 5, and so on, players alternately passing one penny or two to the next player who still has some pennies. A player who runs out of pennies drops out of the game and leaves the table. Find an infinite set of numbers n for which some player ends up with all n pennies. 3. Evaluate Z
∞ 0
!
x5 x7 x3 + − +··· x− 2 2·4 2·4·6
!
x2 x4 x6 1 + 2 + 2 2 + 2 2 2 + · · · dx. 2 2 ·4 2 ·4 ·6
3.13. 58T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1997
97
4. Let G be a group with identity e and φ : G → G a function such that φ(g1 )φ(g2 )φ(g3 ) = φ(h1 )φ(h2 )φ(h3 ) whenever g1 g2 g3 = e = h1 h2 h3 . Prove that there exists an element a ∈ G such that ψ(x) = aφ(x) is a homomorphism (i.e. ψ(xy) = ψ(x)ψ(y) for all x, y ∈ G). 5. Let Nn denote the number of ordered n-tuples of positive integers (a1 , a2 , . . . , an ) such that 1/a1 + 1/a2 + . . . + 1/an = 1. Determine whether N10 is even or odd. 6. For a positive integer n and any real number c, define xk recursively by x0 = 0, x1 = 1, and for k ≥ 0, cxk+1 − (n − k)xk . xk+2 = k+1 Fix n and then take c to be the largest value for which xn+1 = 0. Find xk in terms of n and k, 1 ≤ k ≤ n. 7. Let {x} denote the distance between the real number x and the nearest integer. For each positive integer n, evaluate Fn =
6n−1 X
min({
m=1
m m }, { }). 6n 3n
(Here min(a, b) denotes the minimum of a and b.) 8. Let f be a twice-differentiable real-valued function satisfying f (x) + f 00 (x) = −xg(x)f 0 (x), where g(x) ≥ 0 for all real x. Prove that |f (x)| is bounded. P
9. For each positive integer n, write the sum nm=1 1/m in the form pn /qn , where pn and qn are relatively prime positive integers. Determine all n such that 5 does not divide qn . 10. Let am,n denote the coefficient of xn in the expansion of (1 + x + x2 )m . Prove that for all [integers] k ≥ 0, 0≤
b 2k c 3
X i=0
(−1)i ak−i,i ≤ 1.
11. Prove that for n ≥ 2, n − 1 terms
n terms z }| {
2
2
···2
≡
z }| {
22
···2
(mod n).
98
CHAPTER 3. WILLIAM LOWELL PUTNAM COMPETITION
12. The dissection of the 3–4–5 triangle shown below (into four congruent right triangles similar to the original) has diameter 5/2. Find the least diameter of a dissection of this triangle into four parts. (The diameter of a dissection is the least upper bound of the distances between pairs of points belonging to the same part.)
3.14
59th Anual William Lowell Putnam Competition, 1998
1. A right circular cone has base of radius 1 and height 3. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side-length of the cube? 2. Let s be any arc of the unit circle lying entirely in the first quadrant. Let A be the area of the region lying below s and above the x-axis and let B be the area of the region lying to the right of the y-axis and to the left of s. Prove that A + B depends only on the arc length, and not on the position, of s. 3. Let f be a real function on the real line with continuous third derivative. Prove that there exists a point a such that f (a) · f 0 (a) · f 00 (a) · f 000 (a) ≥ 0. 4. Let A1 = 0 and A2 = 1. For n > 2, the number An is defined by concatenating the decimal expansions of An−1 and An−2 from left to right. For example A3 = A2 A1 = 10, A4 = A3 A2 = 101, A5 = A4 A3 = 10110, and so forth. Determine all n such that 11 divides An . 5. Let F be a finite collection of open discs in R2 whose union contains a set E ⊆ R2 . Show that there is a pairwise disjoint subcollection D1 , . . . , Dn in F such that E ⊆ ∪nj=1 3Dj . Here, if D is the disc of radius r and center P , then 3D is the disc of radius 3r and center P . 6. Let A, B, C denote distinct points with integer coordinates in R2 . Prove that if (|AB| + |BC|)2 < 8 · [ABC] + 1 then A, B, C are three vertices of a square. Here |XY | is the length of segment XY and [ABC] is the area of triangle ABC.
3.15. 60T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1999
99
7. Find the minimum value of (x + 1/x)6 − (x6 + 1/x6 ) − 2 (x + 1/x)3 + (x3 + 1/x3 ) for x > 0. 8. Given a point (a, b) with 0 < b < a, determine the minimum perimeter of a triangle with one vertex at (a, b), one on the x-axis, and one on the line y = x. You may assume that a triangle of minimum perimeter exists. 9. let H be the unit hemisphere {(x, y, z) : x2 + y 2 + z 2 = 1, z ≥ 0}, C the unit circle {(x, y, 0) : x2 + y 2 = 1}, and P the regular pentagon inscribed in C. Determine the surface area of that portion of H lying over the planar region inside P , and write your answer in the form A sin α + B cos β, where A, B, α, β are real numbers. 10. Find necessary and sufficient conditions on positive integers m and n so that mn−1 X
(−1)bi/mc+bi/nc = 0.
i=0
11. Let N be the positive integer with 1998 decimal digits, all of them 1; that is, N = 1111 · · · 11. Find the thousandth digit after the decimal point of
√
N.
12. Prove that, for any integers a, b, c, there exists a positive integer n such that is not an integer.
3.15
√
n3 + an2 + bn + c
60th Anual William Lowell Putnam Competition, 1999
1. Find polynomials f (x),g(x), and h(x), if they exist, such that for all x,
−1 if x < −1 3x + 2 if −1 ≤ x ≤ 0 |f (x)| − |g(x)| + h(x) = −2x + 2 if x > 0.
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CHAPTER 3. WILLIAM LOWELL PUTNAM COMPETITION
2. Let p(x) be a polynomial that is nonnegative for all real x. Prove that for some k, there are polynomials f1 (x), . . . , fk (x) such that p(x) =
k X
(fj (x))2 .
j=1
3. Consider the power series expansion ∞ X 1 = a n xn . 1 − 2x − x2 n=0
Prove that, for each integer n ≥ 0, there is an integer m such that a2n + a2n+1 = am . 4. Sum the series
∞ X ∞ X
m2 n . m m n m=1 n=1 3 (n3 + m3 )
5. Prove that there is a constant C such that, if p(x) is a polynomial of degree 1999, then Z 1 |p(0)| ≤ C |p(x)| dx. −1
6. The sequence (an )n≥1 is defined by a1 = 1, a2 = 2, a3 = 24, and, for n ≥ 4, an =
6a2n−1 an−3 − 8an−1 a2n−2 . an−2 an−3
Show that, for all n, an is an integer multiple of n. 7. Right triangle ABC has right angle at C and ∠BAC = θ; the point D is chosen on AB so that |AC| = |AD| = 1; the point E is chosen on BC so that ∠CDE = θ. The perpendicular to BC at E meets AB at F . Evaluate limθ→0 |EF |. 8. Let P (x) be a polynomial of degree n such that P (x) = Q(x)P 00 (x), where Q(x) is a quadratic polynomial and P 00 (x) is the second derivative of P (x). Show that if P (x) has at least two distinct roots then it must have n distinct roots. 9. Let A = {(x, y) : 0 ≤ x, y < 1}. For (x, y) ∈ A, let S(x, y) =
X
1 ≤m ≤2 2 n
xm y n ,
3.15. 60T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 1999
101
where the sum ranges over all pairs (m, n) of positive integers satisfying the indicated inequalities. Evaluate lim
(x,y)→(1,1),(x,y)∈A
(1 − xy 2 )(1 − x2 y)S(x, y).
10. Let f be a real function with a continuous third derivative such that f (x), f 0 (x), f 00 (x), f 000 (x) are positive for all x. Suppose that f 000 (x) ≤ f (x) for all x. Show that f 0 (x) < 2f (x) for all x. 11. For an integer n ≥ 3, let θ = 2π/n. Evaluate the determinant of the n × n matrix I +A, where I is the n×n identity matrix and A = (ajk ) has entries ajk = cos(jθ+kθ) for all j, k. 12. Let S be a finite set of integers, each greater than 1. Suppose that for each integer n there is some s ∈ S such that gcd(s, n) = 1 or gcd(s, n) = s. Show that there exist s, t ∈ S such that gcd(s, t) is prime.
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CHAPTER 3. WILLIAM LOWELL PUTNAM COMPETITION
3.16
61st Anual William Lowell Putnam Competition, 2000
1. Let A be a positive real number. What are the possible values of P x0 , x1 , . . . are positive numbers for which ∞ j=0 xj = A?
P∞
j=0
x2j , given that
2. Prove that there exist infinitely many integers n such that n, n + 1, n + 2 are each the sum of the squares of two integers. [Example: 0 = 02 + 02 , 1 = 02 + 12 , 2 = 12 + 12 .] 3. The octagon P1 P2 P3 P4 P5 P6 P7 P8 is inscribed in a circle, with the vertices around the circumference in the given order. Given that the polygon P1 P3 P5 P7 is a square of area 5, and the polygon P2 P4 P6 P8 is aectangle of area 4, find the maximum possible area of the octagon. 4. Show that the improper integral lim
Z
B→∞ 0
B
sin(x) sin(x2 ) dx
converges. 5. Three distinct points with integer coordinates lie in the plane on a circle of radius r > 0. Show that two of these points are separated by a distance of at least r 1/3 . 6. Let f (x) be a polynomial with integer coefficients. Define a sequence a0 , a1 , . . . of integers such that a0 = 0 and an+1 = f (an ) for all n ≥ 0. Prove that if there exists a positive integer m for which am = 0 then either a1 = 0 or a2 = 0. 7. Let aj , bj , cj be integers for 1 ≤ j ≤ N . Assume for each j, at least one of aj , bj , cj is odd. Show that there exist integers r, s, t such that raj + sbj + tcj is odd for at least 4N/7 values of j, 1 ≤ j ≤ N . 8. Prove that the expression gcd(m, n) n n m
!
is an integer for all pairs of integers n ≥ m ≥ 1. P
9. Let f (t) = N j=1 aj sin(2πjt), where each aj is real and aN is not equal to 0. Let Nk k denote the number of zeroes (including multiplicities) of ddtkf . Prove that N0 ≤ N1 ≤ N2 ≤ · · · and lim Nk = 2N. k→∞
[Editorial clarification: only zeroes in [0, 1) should be counted.]
3.17. 62N D ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 2001
103
10. Let f (x) be a continuous function such that f (2x2 − 1) = 2xf (x) for all x. Show that f (x) = 0 for −1 ≤ x ≤ 1. 11. Let S0 be a finite set of positive integers. We define finite sets S1 , S2 , . . . of positive integers as follows: the integer a is in Sn+1 if and only if exactly one of a − 1 or a is in Sn . Show that there exist infinitely many integers N for which SN = S0 ∪ {N + a : a ∈ S0 }. n+1
12. Let B be a set of more than 2 n distinct points with coordinates of the form (±1, ±1, . . . , ±1) in n-dimensional space with n ≥ 3. Show that there are three distinct points in B which are the vertices of an equilateral triangle.
3.17
62nd Anual William Lowell Putnam Competition, 2001
1. Consider a set S and a binary operation ∗, i.e., for each a, b ∈ S, a ∗ b ∈ S. Assume (a ∗ b) ∗ a = b for all a, b ∈ S. Prove that a ∗ (b ∗ a) = b for all a, b ∈ S. 2. You have coins C1 , C2 , . . . , Cn . For each k, Ck is biased so that, when tossed, it has probability 1/(2k+1) of falling heads. If the n coins are tossed, what is the probability that the number of heads is odd? Express the answer as a rational function of n. 3. For each integer m, consider the polynomial Pm (x) = x4 − (2m + 4)x2 + (m − 2)2 . For what values of m is Pm (x) the product of two non-constant polynomials with integer coefficients? 4. Triangle ABC has an area 1. Points E, F, G lie, respectively, on sides BC, CA, AB such that AE bisects BF at point R, BF bisects CG at point S, and CG bisects AE at point T . Find the area of the triangle RST . 5. Prove that there are unique positive integers a, n such that an+1 − (a + 1)n = 2001. 6. Can an arc of a parabola inside a circle of radius 1 have a length greater than 4? 7. Let n be an even positive integer. Write the numbers 1, 2, . . . , n2 in the squares of an n × n grid so that the k-th row, from left to right, is (k − 1)n + 1, (k − 1)n + 2, . . . , (k − 1)n + n.
104
CHAPTER 3. WILLIAM LOWELL PUTNAM COMPETITION Color the squares of the grid so that half of the squares in each row and in each column are red and the other half are black (a checkerboard coloring is one possibility). Prove that for each coloring, the sum of the numbers on the red squares is equal to the sum of the numbers on the black squares.
8. Find all pairs of real numbers (x, y) satisfying the system of equations 1 x 1 x
+ −
1 2y 1 2y
= (x2 + 3y 2 )(3x2 + y 2 ) = 2(y 4 − x4 ).
9. For any positive integer n, let hni denote the closest integer to
√
n. Evaluate
∞ X
2hni + 2−hni . 2n n=1 10. Let S denote the set of rational numbers different from {−1, 0, 1}. Define f : S → S by f (x) = x − 1/x. Prove or disprove that ∞ \
n=1
f (n) (S) = ∅,
where f (n) denotes f composed with itself n times. 11. Let a and b be real numbers in the interval (0, 1/2), and let g be a continuous realvalued function such that g(g(x)) = ag(x) + bx for all real x. Prove that g(x) = cx for some constant c. 12. Assume that (an )n≥1 is an increasing sequence of positive real numbers such that lim an /n = 0. Must there exist infinitely many positive integers n such that an−i + an+i < 2an for i = 1, 2, . . . , n − 1?
3.18
63rd Anual William Lowell Putnam Competition, 2002
1. Let k be a fixed positive integer. The n-th derivative of where Pn (x) is a polynomial. Find Pn (1).
1 xk −1
has the form
Pn (x) (xk −1)n+1
2. Given any five points on a sphere, show that some four of them must lie on a closed hemisphere.
3.18. 63RD ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 2002
105
3. Let n ≥ 2 be an integer and Tn be the number of non-empty subsets S of {1, 2, 3, . . . , n} with the property that the average of the elements of S is an integer. Prove that Tn −n is always even. 4. In Determinant Tic-Tac-Toe, Player 1 enters a 1 in an empty 3 × 3 matrix. Player 0 counters with a 0 in a vacant position, and play continues in turn until the 3 × 3 matrix is completed with five 1’s and four 0’s. Player 0 wins if the determinant is 0 and player 1 wins otherwise. Assuming both players pursue optimal strategies, who will win and how? 5. Define a sequence by a0 = 1, together with the rules a2n+1 = an and a2n+2 = an +an+1 for each integer n ≥ 0. Prove that every positive rational number appears in the set
an−1 :n≥1 = an
1 1 2 1 3 , , , , ,... . 1 2 1 3 2
6. Fix an integer b ≥ 2. Let f (1) = 1, f (2) = 2, and for each n ≥ 3, define f (n) = nf (d), where d is the number of base-b digits of n. For which values of b does ∞ X
1 n=1 f (n) converge? 7. Shanille O’Keal shoots free throws on a basketball court. She hits the first and misses the second, and thereafter the probability that she hits the next shot is equal to the proportion of shots she has hit so far. What is the probability she hits exactly 50 of her first 100 shots? 8. Consider a polyhedron with at least five faces such that exactly three edges emerge from each of its vertices. Two players play the following game: Each player, in turn, signs his or her name on a previously unsigned face. The winner is the player who first succeeds in signing three faces that share a common vertex. Show that the player who signs first will always win by playing as well as possible. 9. Show that, for all integers n > 1,
1 1 1 < − 1− 2ne e n
n
<
1 . ne
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CHAPTER 3. WILLIAM LOWELL PUTNAM COMPETITION
10. An integer n, unknown to you, has been randomly chosen in the interval [1, 2002] with uniform probability. Your objective is to select n in an odd number of guesses. After each incorrect guess, you are informed whether n is higher or lower, and you must guess an integer on your next turn among the numbers that are still feasibly correct. Show that you have a strategy so that the chance of winning is greater than 2/3. 11. A palindrome in base b is a positive integer whose base-b digits read the same backwards and forwards; for example, 2002 is a 4-digit palindrome in base 10. Note that 200 is not a palindrome in base 10, but it is the 3-digit palindrome 242 in base 9, and 404 in base 7. Prove that there is an integer which is a 3-digit palindrome in base b for at least 2002 different values of b. 12. Let p be a prime number. Prove that the determinant of the matrix
x y z p p y zp x p2 p2 p2 x y z is congruent modulo p to a product of polynomials of the form ax+by+cz, where a, b, c are integers. (We say two integer polynomials are congruent modulo p if corresponding coefficients are congruent modulo p.)
3.19
64th Anual William Lowell Putnam Competition, 2003
1. Let n be a fixed positive integer. How many ways are there to write n as a sum of positive integers, n = a1 + a2 + · · · + a k with k an arbitrary positive integer and a1 ≤ a2 ≤ · · · ≤ ak ≤ a1 + 1? For example, with n = 4, there are four ways: 4, 2+2, 1+1+2, 1+1+1+1. 2. Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn be nonnegative real numbers. Show that 1
1
1
(a1 a2 · · · an ) n + (b1 b2 · · · bn ) n ≤ ((a1 + b1 ) (a2 + b2 ) · · · (an + bn )) n
3.19. 64T H ANUAL WILLIAM LOWELL PUTNAM COMPETITION, 2003
107
3. Find the minimum value of | sin x + cos x + tan x + cot x + sec x + csc x| for real numbers x. 4. Suppose that a, b, c, A, B, C are real numbers, a 6= 0 and A 6= 0, such that |ax2 + bx + c| ≤ |Ax2 + Bx + B| for all real numbers x. Show that |b2 − 4ac| ≤ |B 2 − 4AC| 5. A Dyck n-path is a lattice path of n upsteps (1,1) and n downsteps (1,-2) that starts at the origin O and never dips below the x-axis. A return is a maximal sequence of contigouous downsteps that terminates on the x-axis. For example, the Dyck 5-path illustrated has two returns, of length 3 and 1 respectively. u u u
O
u u
u u
u
u u
u
Show that there is a one-to-one correspondence between the Dyck n-paths with no returns of even length and the Dyck (n − 1)-paths. 6. For a set S of nonnegative integers, let rS (n) denote the number of ordered pairs (s1 , s2 ) such that s1 ∈ S, s2 ∈ S, s1 6= s2 and s1 + s2 = n. Is it possible to partition the nonnegative integers into two sets A and B in such way that rA (n) = rB (n) for all n? 7. Do They exist polynomials a(x), b(x), c(y), d(y) such that 1 + xy + x2 y 2 = a(x)c(y) + b(x)d(y) holds identically?
108
CHAPTER 3. WILLIAM LOWELL PUTNAM COMPETITION
8. Let n be a positive integer. Starting with the sequence 1 1 1 , ,..., 2 3 n
1, form a new sequence of n − 1 entries
3 5 2n − 1 , ,..., 4 12 2n(n − 1) by taking the average of two consecutive entries in the first sequence. Repeat the averaging of neigbors on sthe second sequence to obtain a third sequence of n − 2 entries and continue until the final sequence produced consists of a single number xn . Show that xn < n2 . 9. Show that for each positive integer n, n! =
n Y
i=1
lcm 1, 2, . . . ,
n i
(Here lcm denotes the least common multiple and bxc denote the greatest integer ≤ x.) 10. Let f (z) = az 4 +bz 3 +cz 2 +dz +e = a(z −r1 )(z −r2 )(z −r3 )(z −r4 ) where a, b, c, d, e are integers, a 6= 0. Show that if r1 + r2 is a rational number, and if r1 + r2 6= r3 + r4 , then r1 r2 is a rational number. 11. Let A, B and C be qeuidistant points on the circumference of a circle of unit radius centered at O, and let P be any point in the circle’s interior. Let a, b, c be the distances from P to A, B, C respectively. Show that there is a triangle with sides lengths a, b, c, and that the area of this triangle depends only on the distance from P to O. 12. Let f (x) be a continuous real-valued function defined on the interval [0, 1]. Show that Z
1 0
Z
1 0
|f (x) + f (y)|dx dy ≤
Z
1 0
|f (x)|dx
Chapter 4 Asiatic Pacific Mathematical Olympiads 4.1
1st Asiatic Pacific Mathematical Olympiad, 1989
1. Let x1 , x2 , . . . , xn be positive real numbers, and let S = x1 + x2 + · · · + x n .
Prove that
(1 + x1 )(1 + x2 ) · · · (1 + xn ) ≤ 1 + S +
Sn S2 S3 + +···+ . 2! 3! n!
2. Prove that the equation 6(6a2 + 3b2 + c2 ) = 5n2 has no solutions in integers except a = b = c = n = 0. 3. Let A1 , A2 , A3 be three points in the plane, and for convenience, let A4 = A1 , A5 = A2 . For n = 1, 2, and 3, suppose that Bn is the midpoint of An An+1 , and suppose that Cn is the midpoint of An Bn . Suppose that An Cn+1 and Bn An+2 meet at Dn , and that An Bn+1 and Cn An+2 meet at En . Calculate the ratio of the area of triangle 4D1 D2 D3 to the area of triangle 4E1 E2 E3 . 4. Let S be a set consisting of m pairs (a, b) of positive integers with the property that 1 ≤ a < b ≤ n. Show that there are at least 2
(m − n4 ) 4m · 3n 109
110
CHAPTER 4. ASIATIC PACIFIC MATHEMATICAL OLYMPIADS triples (a, b, c) such that (a, b), (a, c), and (b, c) belong to S.
5. Determine all functions f from the reals to the reals for which (1) f (x) is strictly increasing, (2) f (x) + g(x) = 2x for all real x, where g(x) is the composition inverse function to f (x). (Note: f and g are said to be composition inverses if f (g(x)) = x and g(f (x)) = x for all real x.)
4.2
2nd Asiatic Pacific Mathematical Olympiad, 1990
1. Given triagnle ABC, let D, E, F be the midpoints of BC, AC, AB respectively and let G be the centroid of the triangle. For each value of ∠BAC, how many non-similar triangles are there in which AEGF is a cyclic quadrilateral? 2. Let a1 , a2 , . . . , an be positive real numbers, and let Sk be the sum of the products of a1 , a2 , . . . , an taken k at a time. Show that Sk Sn−k
n ≥ k
!2
a1 a2 · · · an
for k = 1, 2, . . . , n − 1. 3. Consider all the triangles ABC which have a fixed base AB and whose altitude from C is a constant h. For which of these triangles is the product of its altitudes a maximum? 4. A set of 1990 persons is divided into non-intersecting subsets in such a way that: (a) No one in a subset knows all the others in the subset, (b) Among any three persons in a subset, there are always at least two who do not know each other, and (c) For any two persons in a subset who do not know each other, there is exactly one person in the same subset knowing both of them. a Prove that within each subset, every person has the same number of acquaintances. b Determine the maximum possible number of subsets.
4.3. 3RD ASIATIC PACIFIC MATHEMATICAL OLYMPIAD, 1991
111
Note: It is understood that if a person A knows person B, then person B will know person A; an acquaintance is someone who is known. Every person is assumed to know one’s self. 5. Show that for every integer n ≥ 6, there exists a convex hexagon which can be dissected into exactly n congruent triangles.
4.3
3rd Asiatic Pacific Mathematical Olympiad, 1991
1. Let G be the centroid of triangle 4ABC and M be the midpoint of BC. Let X be on AB and Y on AC such that the points X, Y , and G are collinear and XY and BC are parallel. Suppose that XC and GB intersect at Q and Y B and GC intersect at P . Show that triangle 4M P Q is similar to triangle 4ABC. 2. Suppose there are 997 points given in a plane. If every two points are joined by a line segment with its midpoint coloured in red, show that there are at least 1991 red points in the plane. Can you find a special case with exactly 1991 red points? 3. Let a1 , a2 , . . . , an , b1 , b2 , . . . , bn be positive real numbers such that a1 +a2 +· · ·+an = b1 + b2 + · · · + bn . Show that a21 a22 a2n a1 + a 2 + · · · + a n + +···+ ≥ a1 + b 1 a2 + b 2 an + b n 2
4. During a break, n children at school sit in a circle around their teacher to play a game. The teacher walks clockwise close to the children and hands out candies to some of them according to the following rule. He selects one child and gives him a candy, then he skips the next child and gives a candy to the next one, then he skips 2 and gives a candy to the next one, then he skips 3, and so on. Determine the values of n for which eventually, perhaps after many rounds, all children will have at least one candy each. 5. Given are two tangent circles and a point P on their common tangent perpendicular to the lines joining their centres. Construct with ruler and compass all the circles that are tangent to these two circles and pass through the point P .
4.4
4th Asiatic Pacific Mathematical Olympiad, 1992
1. A triangle with sides a, b, and c is given. Denote by s the semiperimeter, that is
112
CHAPTER 4. ASIATIC PACIFIC MATHEMATICAL OLYMPIADS s = a+b+c . Construct a triangle with sides s − a, s − b, and s − c. This process is 2 repeated until a triangle can no longer be constructed with the side lengths given. For which original triangles can this process be repeated indefinitely?
2. In a circle C with centre O and radius r, let C1 , C2 be two circles with centres O1 , O2 and radii r1 , r2 respectively, so that each circle Ci is internally tangent to C at Ai and so that C1 , C2 are externally tangent to each other at A.Prove that the three lines OA, O1 A2 , and O2 A1 are concurrent. 3. Let n be an integer such that n > 3. Suppose that we choose three numbers from the set {1, 2, . . . , n}. Using each of these three numbers only once and using addition, multiplication, and parenthesis, let us form all possible combinations. (a) Show that if we choose all three numbers greater than n/2, then thealues of these combinations are all distinct. √ (b) Let p be a prime number such that p ≤ n. Show that the number of ways of choosing three numbers so that the smallest one is p and the values of the combinations are not all distinct is precisely the number of positive divisors of p − 1. 4. Determine all pairs (h, s) of positive integers with the following property: If one draws h horizontal lines and another s lines which satisfy: i they are not horizontal, ii no two of them are parallel, iii no three of the h + s lines are concurrent, then the number of regions formed by these h + s lines is 1992. 5. Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.
4.5
5th Asiatic Pacific Mathematical Olympiad, 1993
1. Let ABCD be a quadrilateral such that all sides have equal length and angle ^ABC is 60 deg. Let l be a line passing through D and not intersecting the quadrilateral (except at D). Let E and F be the points of intersection of l with AB and BC respectively. Let M be the point of intersection of CE and AF . Prove that CA2 = CM · CE.
4.6. 6T H ASIATIC PACIFIC MATHEMATICAL OLYMPIAD, 1994
113
2. Find the total number of different integer values the function f (x) = [x] + [2x] + [
5x ] + [3x] + [4x] 3
takes for real numbers x with 0 ≤ x ≤ 100. 3. Let
(
f (x) = an xn + an−1 xn−1 + · · · + a0 and g(x) = cn+1 xn+1 + cn xn + · · · + c0
be non-zero polynomials with real coefficients such that g(x) = (x + r)f (x) for some real number r. If a = max(|an |, . . . , |a0 |) and c = max(|cn+1 |, . . . , |c0 |), prove that a ≤ n + 1. c 4. Determine all positive integers n for which the equation xn + (2 + x)n + (2 − x)n = 0 has an integer as a solution. 5. Let P1 , P2 , . . . , P1993 = P0 be distinct points in the xy-plane with the following properties: i both coordinates of Pi are integers, for i = 1, 2, . . . , 1993; ii there is no point other than Pi and Pi+1 on the line segment joining Pi with Pi+1 whose coordinates are both integers, for i = 0, 1, . . . , 1992. Prove that for some i, 0 ≤ i ≤ 1992, there exists a point Q with coordinates (qx , qy ) on the line segment joining Pi with Pi+1 such that both 2qx and 2qy are odd integers.
4.6
6th Asiatic Pacific Mathematical Olympiad, 1994
1. Let f : R → R be a function such that: i For all x, y ∈ R, f (x) + f (y) + 1 ≥ f (x + y) ≥ f (x) + f (y) ii For all x ∈ [0, 1), f (0) ≥ f (x),
iii −f (−1) = f (1) = 1.
114
CHAPTER 4. ASIATIC PACIFIC MATHEMATICAL OLYMPIADS Find all such functions f .
2. Given a nondegenerate triangle 4ABC, with circumcentre O, orthocentre H, and circumradius R, prove that |OH| < 3R. 3. Let n be an integer of the form a2√+ b2 , where a and b are relatively prime integers and such that if p is a prime, p ≤ n, then p divides ab. Determine all such n. 4. Is there an infinite set of points in the plane such that no three points are collinear, and the distance between any two points is rational? 5. You are given three lists A, B, and C. List A contains the numbers of the form 10k in base 10, with k any integer greater than or equal to 1. Lists B and C contain the same numbers translated into base 2 and 5 respectively: A 10 100 1000 .. .
B 1010 1100100 1111101000 .. .
C 20 400 13000 .. .
Prove that for every integer n > 1, there is exactly one number in exactly one of the lists B or C that has exactly n digits.
4.7
7th Asiatic Pacific Mathematical Olympiad, 1995
1. Determine all sequences of real numbers a1 , a2 , . . . , a1995 which satisfy: q
2 an − (n − 1) ≥ an+1 − (n − 1), for n = 1, 2, . . . 1994, and
√ 2 a1995 − 1994 ≥ a1 + 1.
2. Let a1 , a2 , . . . , an be a sequence of integers with values between 2 and 1995 such that: i Any two of the ai ’s are realtively prime, ii Each ai is either a prime or a product of primes. Determine the smallest possible values of n to make sure that the sequence will contain a prime number.
4.8. 8T H ASIATIC PACIFIC MATHEMATICAL OLYMPIAD, 1996
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3. Let P QRS be a cyclic quadrilateral such that the segments P Q and RS are not parallel. Consider the set of circles through P and Q, and the set of circles through R and S. Determine the set A of points of tangency of circles in these two sets. 4. Let C be a circle with radius R and centre O, and S a fixed point in the interior of C. Let AA0 and BB 0 be perpendicular chords through S. Consider the rectangles SAM B, SBN 0 A0 , SA0 M 0 B 0 , and SB 0 N A. Find the set of all points M , N 0 , M 0 , and N when A moves around the whole circle. 5. Find the minimum positive integer k such that there exists a function f from the set Z of all integers to {1, 2, . . . k} with the property that f (x) 6= f (y) whenever |x − y| ∈ {5, 7, 12}.
4.8
8th Asiatic Pacific Mathematical Olympiad, 1996
1. Let ABCD be a quadrilateral AB = BC = CD = DA. Let M N and P Q be two segments perpendicular to the diagonal BD and such that the distance between them is d > BD/2, with M ∈ AD, N ∈ DC, P ∈ AB, and Q ∈ BC. Show that the perimeter of hexagon AM N CQP does not depend on the position of M N and P Q so long as the distance between them remains constant. 2. Let m and n be positive integers such that n ≤ m. Prove that 2n n! ≤
(m + n)! ≤ (m2 + m)n (m − n)!
3. Let P1 , P2 , P3 , P4 be four points on a circle, and let I1 be the incentre of the triangle P2 P3 P4 ; I2 be the incentre of the triangle P1 P3 P4 ; I3 be the incentre of the triangle P1 P2 P4 ; I4 be the incentre of the triangle P1 P2 P3 . Prove that I1 , I2 , I3 , I4 are the vertices of a rectangle. 4. The National Marriage Council wishes to invite n couples to form 17 discussion groups under the following conditions: (a) All members of a group must be of the same sex; i.e. they are either all male or all female. (b) The difference in the size of any two groups is 0 or 1. (c) All groups have at least 1 member. (d) Each person must belong to one and only one group.
116
CHAPTER 4. ASIATIC PACIFIC MATHEMATICAL OLYMPIADS Find all values of n, n ≤ 1996, for which this is possible. Justify your answer.
5. Let a, b, c be the lengths of the sides of a triangle. Prove that √ √ √ √ √ √ a+b−c+ b+c−a+ c+a−b≤ a+ b+ c , and determine when equality occurs.
4.9
9th Asiatic Pacific Mathematical Olympiad, 1997
1. Given S =1+
1 1+
1 3
+
1 1+
1 3
+
1 6
+···+
1+
1 3
1 + +···+ 1 6
1 1993006
where the denominators contain partial sums of the sequence of reciprocals of triangular numbers (i.e. k = n(n + 1)/2 for n = 1, 2, . . . , 1996). Prove that S > 1001. 2. Find an integer n, where 100 ≤ n ≤ 1997, such that 2n + 2 n is also an integer. 3. Let 4ABC be a triangle inscribed in a circle and let la =
mb mc ma , lb = , lc = , Ma Mb Mc
where ma , mb , mc are the lengths of the angle bisectors (internal to the triangle) and Ma , Mb , Mc are the lengths of the angle bisectors extended until they meet the circle. Prove that la lb lc + + ≥ 3, 2 2 sin A sin B sin2 C and that equality holds iff ABC is an equilateral triangle. 4. Triangle 4A1 A2 A3 has a right angle at A3 . A sequence of points is now defined by the following iterative process, where n is a positive integer. From An (n ≥ 3), a perpendicular line is drawn to meet An−2 An−1 at An+1 . (a) Prove that if this process is continued indefinitely, then one and only one point P is interior to every triangle An−2 An−1 An , n ≥ 3.
4.10. 10T H ASIATIC PACIFIC MATHEMATICAL OLYMPIAD, 1998
117
(b) Let A1 and A3 be fixed points. By considering all possible locations of A2 on the plane, find the locus of P . 5. Suppose that n people A1 , A2 , . . ., An , (n ≥ 3) are seated in a circle and that Ai has ai objects such that a1 + a2 + · · · + an = nN, where N is a positive integer. In order that each person has the same number of objects, each person Ai is to give or to receive a certain number of objects to or from its two neighbours Ai−1 and Ai+1 . (Here An+1 means A1 and An means A0 .) How should this redistribution be performed so that the total number of objects transferred is minimum?
4.10
10th Asiatic Pacific Mathematical Olympiad, 1998
1. Let F be the set of all n−tuples (A1 , . . . , An ) such that each Ai is a subset of {1, 2, . . . , 1998}. Let |A| denote the number of elements of the set A. Find: X
(A1 ,...,An )∈F
|A1 ∪ · · · ∪ An |
2. Show that for any positive integers a and b, (36a + b)(a + 36b) can not be a power of 2. 3. Let a, b, c be positive real numbers. Pruve that:
a 1+ b
b 1+ c
!
c a+b+c 1+ ≥2 1+ √ 3 a abc
!
4. Let 4ABC be a triangle and D the foot of the altitude from A. Let E and F lie on a line through D such that AE is perpendicular to BC, AF is perpendicular to CF , and E and F are different from D. Let M and N be the midpoint of the segments BC and EF , respectively. Prove that AN is perpendicular to N M . 5. √ Find the largest integer n such that n is divisible by all positive integers less than 3 n.
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CHAPTER 4. ASIATIC PACIFIC MATHEMATICAL OLYMPIADS
4.11
11th Asiatic Pacific Mathematical Olympiad, 1999
1. Find the smallest positive integer n with the following property: there does not exist an arithmetic progression of 1999 real numbers containing exactly n integers. 2. Let a1 , a2 , . . . be a sequence of real numbers satisfying ai+j ≤ ai + aj for all i, j = 1, 2, . . .. Prove that a2 a3 an a1 + + +···+ ≥ an 2 3 n for each positive integer n. 3. Let Γ1 and Γ2 be two circles intersecting at P and Q. The common tangent, closer to P , of Γ1 and Γ2 touches Γ1 at A and Γ2 at B. The tangent of Γ1 at P meets Γ2 at C, which is different from P , and the extension of AP meets BC at R. Prove that the circumcircle of triangle P QR is tangent to BP and BR. 4. Determine all pairs (a, b) of integers with the property that the numbers a2 + 4b and b2 + 4a are both perfect squares. 5. Let S be a set of 2n + 1 points in the plane such that no three are collinear and no four concyclic. A circle will be called good if it has 3 points of S on its circumference, n − 1 points in its interior and n − 1 points in its exterior. Prove that the number of good circles has the same parity as n.
4.12
12th Asiatic Pacific Mathematical Olympiad, 2000
1. Compute the sum S=
101 X
x3i 2 i=0 1 − 3xi + 3xi
for xi =
i 101
2. Given the following triangular arrangement of circles:
4.13. 13T H ASIATIC PACIFIC MATHEMATICAL OLYMPIAD, 2001
119
Each of the numbers 1, . . . , 9 is to be written into one of these circles, so that each circle contain exactly one of these numbers and: i the sum of the four numbers on each side of the triangle are equal; ii the sum of the squares of the four numbers on each side of the triangle are equal. Find all ways in which this can be done. 3. Let 4ABC be a triangle. Let M and N be the points in which the median and the angle bisector, respectively, at A meet the side BC. Let Q and P be the point in which the perpendicular at N to N A meets M A and BA, respectively, and O the point in which the perpendicular at P to BA meets AN produced. Prove that QO is perpendicular to BC. 4. Let n, k be given positive integers with n > k. Prove that n! 1 nn nn < · < n + 1 k k (n − k)n−k k! (n − k)! k k (n − k)n−k 5. Given a permutation (a0 , a1 , . . . , an ) of the sequence 0, 1, . . . , n. A transposition of ai with aj is called legal if ai = 0 for i > 0, and ai−1 + 1 = aj . The permutation (a0 , . . . , an ) is called regular if after a number of transpositions it becomes (1, 2, . . . , n, 0). For which numbers n is the permutation (1, n, n − 1, . . . , 3, 2, 0) regular?
4.13
13th Asiatic Pacific Mathematical Olympiad, 2001
1. For a positive integer n let S(n) be the sum of digits in the decimal representation of n. Any positive integer obtained by removing several (at least one) digits from the
120
CHAPTER 4. ASIATIC PACIFIC MATHEMATICAL OLYMPIADS right-hand end of the decimal representation of n is called a stump of n. Let T (n) be the sum of all stumps of n. Prove that n = S(n) + 9T (n).
2. Find the largest positive integer N so that the number of integers in the set {1, 2, . . . , N } which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both). 3. Let two equal regular n-gons S and T be located in the plane such that their intersection is a 2n-gon (n ≥ 3). The sides of the polygon S are coloured in red and the sides of T in blue. Prove that the sum of the lengths of the blue sides of the polygon S ∩ T is equal to the sum of the lengths of its red sides. 4. A point in the plane with a cartesian coordinate system is called a mixed point if one of its coordinates is rational and the other one is irrational. Find all polynomials with real coefficients such that their graphs do not contain any mixed point. 5. Find the greatest integer n, such that there are n+4 points A, B, C, D, X1 , . . . , Xn in the plane with AB 6= CD that satisfy the following condition: for each i = 1, 2, . . . , n triangles ABXi and CDXi are equal.
4.14
14th Asiatic Pacific Mathematical Olympiad, 2002
1. Let a1 , a2 , a3 , . . . , an be a sequence of non-negative integers, where n is a positive integer. Let a1 + a 2 + · · · + a n An = . n Prove that a1 !a2 ! . . . an ! ≥ (bAn c!)n , where bAn c is the greatest integer less than or equal to An , and a! = 1 × 2 × · · · × a for a ≥ 1 (and 0! = 1). When does equality hold? 2. Find all positive integers a and b such that a2 + b b2 − a
and
b2 + a a2 − b
are both integers. 3. Let 4ABC be an equilateral triangle. Let P be a point on the side AC and Q be a point on the side AB so that both triangles 4ABP and 4ACQ are acute. Let R be
4.15. 15T H ASIATIC PACIFIC MATHEMATICAL OLYMPIAD, 2003
121
the orthocentre of triangle ABP and S be the orthocentre of triangle ACQ. Let T be the point common to the segments BP and CQ. Find all possible values of ^CBP and ^BCQ such that triangle 4T RS is equilateral. 4. Let x, y, z be positive numbers such that 1 1 1 + + = 1. x y z Show that √
x + yz +
√ √ √ √ √ √ y + zx + z + xy ≥ xyz + x + y + z.
5. Let R denote the set of all real numbers. Find all functions f from R to R satisfying: (i) there are only finitely many s in R such that f (s) = 0, and (ii) f (x4 + y) = x3 f (x) + f (f (y)) for all x, y in R.
4.15
15th Asiatic Pacific Mathematical Olympiad, 2003
1. Let a, b, c, d, e, f be real numbers such that the polynomial p(x) = x8 − 4x7 + 7x6 + ax5 + bx4 + cx3 + dx2 + ex + f factorises into eight linear factors x − xi , with xi > 0 for i = 1, 2, . . . , 8. Determine all possible values of f . 2. Suppose ABCD is a square piece of cardboard with side length a. On a plane are two parallel lines `1 and `2 , which are also a units apart. The square ABCD is placed on the plane so that sides AB and AD intersect `1 at E and F respectively. Also, sides CB and CD intersect `2 at G and H respectively. Let the perimeters of 4AEF and 4CGH be m1 and m2 respectively. Prove that no matter how the square was placed, m1 + m2 remains constant. 3. Let k ≥ 14 be an integer, and let pk be the largest prime number which is strictly less than k. You may assume that pk ≥ 3k/4. Let n be a composite integer. Prove: (a) if n = 2pk , then n does not divide (n − k)! ; (b) if n > 2pk , then n divides (n − k)! 4. Let a, b, c be the sides of a triangle, with a + b + c = 1, and let n ≥ 2 be an integer. Show that √ n √ √ √ 2 n n n a n + b n + bn + c n + cn + a n < 1 + . 2
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CHAPTER 4. ASIATIC PACIFIC MATHEMATICAL OLYMPIADS
5. Given two positive integers m and n, find the smallest positive integer k such that among any k people, either there are 2m of them who form m pairs of mutually acquainted people or there are 2n of them forming n pairs of mutually unacquainted people.
4.16
15th Asiatic Pacific Mathematical Olympiad, 2003
1. Determine all finite nonempty sets S of positive integers satisfying i+j (i, j)
is an element of S for all i, j in S
where (i, j) is the greatest common divisor of i and j. 2. Let O be the circumcentre and H the orthocentre of an acute triangle 4ABC. Prove that the area of one of the triangles 4AOH, 4BOH and 4COH is equal to the sum of the areas of the other two. 3. Let a set S of 2004 points in the plane be given, no three of which are collinear. Let L denote the set of all lines (extended indefinitely in both directions) determined by pairs of points from the set. Show that it is possible to colour the points of S with at most two colours, such that for any points p, q of S, the number of lines in L which separate p from q is odd if and only if p and q have the same colour. Note: A line ` separates two points p and q if p and q lie on opposite sides of ` with neither point on `. 4. For a real number x, let bxc stand for the largest integer that is less than or equal to x. Prove that $ % (n − 1)! n(n + 1) is even for every positive integer n. 5. Prove that (a2 + 2)(b2 + 2)(c2 + 2) ≥ 9(ab + bc + ca) for all real numbers a, b, c > 0.
Chapter 5 USA Mathematical Olympiad 5.1
18th USA Mathematical Olympiad, 1989
1. For each positive integer n, let 1 1 1 + +···+ , 2 3 n = S1 + S2 + S3 + · · · + S n , Tn T1 T2 T3 + + +···+ . = 2 3 4 n+1
Sn = 1 + Tn Un
Find, with proof, integers 0 < a, b, c, d < 1000000 such that T1988 = aS1989 − b and U1988 = cS1989 − d. 2. The 20 members of a local tennis club have scheduled exactly 14 two-person games among themselves, with each member playing in at least one game. Prove that within this schedule there must be a set of 6 games with 12 distinct players. 3. Let P (z) = z n + c1 z n−1 + c2 z n−2 + · · · + cn be a polynomial in the complex variable z, with real coefficients ck . Suppose that |P (i)| < 1. Prove that there exist real numbers a and b such that P (a + bi) = 0 and (a2 + b2 + 1)2 < 4b2 + 1. 4. Let ABC be an acute-angled triangle whose side lengths satisfy the inequalities AB < AC < BC. If point I is the center of the inscribed circle of triangle ABC and point O is the center of the circumscribed circle, prove that line IO intersects segments AB and BC. 123
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CHAPTER 5. USA MATHEMATICAL OLYMPIAD
5. Let u and v be real numbers such that (u + u2 + u3 + · · · + u8 ) + 10u9 = (v + v 2 + v 3 + · · · + v 10 ) + 10v 11 = 8. Determine, with proof, which of the two numbers, u or v, is larger.
5.2
19th USA Mathematical Olympiad, 1990
1. A certain state issues license plates consisting of six digits (from 0 through 9). The state requires that any two plates differ in at least two places. (Thus the plates 027592 and 020592 cannot both be used.) Determine, with proof, the maximum number of distinct license plates that the state can use. 2. A sequence of functions {fn (x)} is defined recursively as follows: √ f1 (x) = x2 + 48, and fn+1 (x) =
q
x2 + 6fn (x) for n ≥ 1.
√ (Recall that is understood to represent the positive square root.) For each positive integer n, find all real solutions of the equation fn (x) = 2x . 3. Suppose that necklace A has 14 beads and necklace B has 19. Prove that for any odd integer n ≥ 1, there is a way to number each of the 33 beads with an integer from the sequence {n, n + 1, n + 2, . . . , n + 32} so that each integer is used once, and adjacent beads correspond to relatively prime integers. (Here a “necklace” is viewed as a circle in which each bead is adjacent to two other beads.) 4. Find, with proof, the number of positive integers whose base-n representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by ±1 from some digit further to the left. (Your answer should be an explicit function of n in simplest form.) 5. An acute-angled triangle ABC is given in the plane. The circle with diameter AB intersects altitude CC 0 and its extension at points M and N , and the circle with diameter AC intersects altitude BB 0 and its extensions at P and Q . Prove that the points M, N, P, Q lie on a common circle.
5.3. 20T H USA MATHEMATICAL OLYMPIAD, 1991
5.3
125
20th USA Mathematical Olympiad, 1991
1. In triangle ABC, angle A is twice angle B, angle C is obtuse, and the three side lengths a, b, c are integers. Determine, with proof, the minimum possible perimeter. 2. For any nonempty set S of numbers, let σ(S) and π(S) denote the sum and product, respectively, of the elements of S . Prove that X
σ(S) 1 1 1 (n + 1), = (n2 + 2n) − 1 + + + · · · + π(S) 2 3 n
where “Σ” denotes a sum involving all nonempty subsets S of {1, 2, 3, . . . , n}. 3. Show that, for any fixed integer n ≥ 1, the sequence 2
22
2, 22 , 22 , 22 , . . . (mod n) is eventually constant. [The tower of exponents is defined by a1 = 2, ai+1 = 2ai . Also ai (mod n) means the remainder which results from dividing ai by n.] 4. Let a = (mm+1 + nn+1 )/(mm + nn ), where m and n are positive integers. Prove that am + an ≥ mm + nn .
[You may wish to analyze the ratio (aN − N N )/(a − N ), for real a ≥ 0 and integer N ≥ 1.]
5. Let D be an arbitrary point on side AB of a given triangle ABC, and let E be the interior point where CD intersects the external common tangent to the incircles of triangles CD and BCD. As D assumes all positions between A, and B , prove that the point E traces the arc of a circle.
5.4
21st USA Mathematical Olympiad, 1992
1. Find, as a function of n, the sum of the digits of
n
9 × 99 × 9999 × · · · × 102 − 1 , where each factor has twice as many digits as the previous one.
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CHAPTER 5. USA MATHEMATICAL OLYMPIAD
2. Prove
1 1 cos 1◦ 1 . + + · · · + = cos 0◦ cos 1◦ cos 1◦ cos 2◦ cos 88◦ cos 89◦ sin2 1◦
3. For a nonempty set S of integers, let σ(S) be the sum of the elements of S. Suppose that A = {a1 , a2 , . . . , a11 } is a set of positive integers with a1 < a2 < · · · < a11 and that, for each positive integer n ≤ 1500, there is a subset S of A for which σ(S) = n. What is the smallest possible value of a10 ? 4. Chords AA0 , BB 0 , CC 0 of a sphere meet at an interior point P but are not contained in a plane. The sphere through A, B, C, P is tangent to the sphere through A0 , B 0 , C 0 , P . Prove that AA0 = BB 0 = CC 0 . 5. Let P (z) be a polynomial with complex coefficients which is of degree 1992 and has distinct zeros. Prove that there exist complex numbers a1 , a2 , . . . , a1992 such that P (z) divides the polynomial
5.5
2
· · · (z − a1 ) − a2
2
· · · − a1991
2
− a1992 .
22nd USA Mathematical Olympiad, 1993
1. For each integer n ≥ 2, determine, with proof, which of the two positive real numbers a and b satisfying an = a + 1, b2n = b + 3a is larger. 2. Let ABCD be a convex quadrilateral such that diagonals AC and BD intersect at right angles, and let E be their intersection. Prove that the reflections of E across AB, BC, CD, DA are concyclic. 3. Consider functions f : [0, 1] → R which satisfy (i) f (x) ≥ 0 for all x in [0, 1],
(ii) f (1) = 1,
(iii) f (x) + f (y) ≤ f (x + y) whenever x, y, and x + y are all in [0, 1]. Find, with proof, the smallest constant c such that f (x) ≤ cx for every function f satisfying (i)-(iii) and every x in [0, 1].
5.6. 23RD USA MATHEMATICAL OLYMPIAD, 1994
127
4. Let a, b be odd positive integers. Define the sequence (fn ) by putting f1 = a, f2 = b, and by letting fn for n ≥ 3 be the greatest odd divisor of fn−1 + fn−2 . Show that fn is constant for n sufficiently large and determine the eventual value as a function of a and b. 5. Let a0 , a1 , a2 , . . . be a sequence of positive real numbers satisfying ai−1 ai+1 ≤ a2i for i = 1, 2, 3, . . . . (Such a sequence is said to be log concave.) Show that for each n > 1, a0 + · · · + an a1 + · · · + an−1 a0 + · · · + an−1 a1 + · · · + an · ≥ · . n+1 n−1 n n
5.6
23rd USA Mathematical Olympiad, 1994
1. Let k1 < k2 < k3 < · · · be positive integers, no two consecutive, and let sm = k1 + k2 + · · · + km for m = 1, 2, 3, . . . . Prove that, for each positive integer n, the interval [sn , sn+1 ) contains at least one perfect square. 2. The sides of a 99-gon are initially colored so that consecutive sides are red, blue, red, blue, . . . , red, blue, yellow. We make a sequence of modifications in the coloring, changing the color of one side at a time to one of the three given colors (red, blue, yellow), under the constraint that no two adjacent sides may be the same color. By making a sequence of such modifications, is it possible to arrive at the coloring in which consecutive sides are red, blue, red, blue, red, blue, . . . , red, yellow, blue? 3. A convex hexagon ABCDEF is inscribed in a circle such that AB = CD = EF and diagonals AD, BE, and CF are concurrent. Let P be the intersection of AD and CE. Prove that CP/P E = (AC/CE)2 . √ P 4. Let a1 , a2 , a3 , . . . be a sequence of positive real numbers satisfying nj=1 aj ≥ n for all n ≥ 1. Prove that, for all n ≥ 1, n X
j=1
a2j
1 1 1 > 1+ +···+ . 4 2 n
5. Let |U |, σ(U ) and π(U ) denote the number of elements, the sum, and the product, respectively, of a finite set U of positive integers. (If U is the empty set, |U| = 0, σ(U ) = 0, π(U ) = 1.) Let S be a finite set of positive integers. As usual, let nk n! denote k! (n−k)! . Prove that X
U ⊆S
(−1)
|U |
!
m − σ(U ) = π(S) |S|
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CHAPTER 5. USA MATHEMATICAL OLYMPIAD for all integers m ≥ σ(S).
5.7
24th USA Mathematical Olympiad, 1995
1. Let p be an odd prime. The sequence (an )n≥0 is defined as follows: a0 = 0, a1 = 1, . . . , ap−2 = p − 2 and, for all n ≥ p − 1, an is the least positive integer that does not form an arithmetic sequence of length p with any of the preceding terms. Prove that, for all n, an is the number obtained by writing n in base p − 1 and reading the result in base p. 2. A calculator is broken so that the only keys that still work are the sin, cos, tan, sin −1 , cos−1 , and tan−1 buttons. The display initially shows 0. Given any positive rational number q, show that pressing some finite sequence of buttons will yield q. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians. 3. Given a nonisosceles, nonright triangle ABC, let O denote the center of its circumscribed circle, and let A1 , B1 , and C1 be the midpoints of sides BC, CA, and AB, respectively. Point A2 is located on the ray OA1 so that ∆OAA1 is similar to ∆OA2 A. Points B2 and C2 on rays OB1 and OC1 , respectively, are defined similarly. Prove that lines AA2 , BB2 , and CC2 are concurrent, i.e. these three lines intersect at a point. 4. Suppose q0 , q1 , q2 , . . . is an infinite sequence of integers satisfying the following two conditions: (i) m − n divides qm − qn for m > n ≥ 0, (ii) there is a polynomial P such that |qn | < P (n) for all n. Prove that there is a polynomial Q such that qn = Q(n) for all n. 5. Suppose that in a certain society, each pair of persons can be classified as either amicable or hostile. We shall say that each member of an amicable pair is a friend of the other, and each member of a hostile pair is a foe of the other. Suppose that the society has n persons and q amicable pairs, and that for every set of three persons, at least one pair is hostile. Prove that there is at least one member of the society whose foes include q(1 − 4q/n2 ) or fewer amicable pairs.
5.8. 25T H USA MATHEMATICAL OLYMPIAD, 1996
5.8
129
25th USA Mathematical Olympiad, 1996
1. Prove that the average of the numbers n sin n◦ (n = 2, 4, 6, . . . , 180) is cot 1◦ . 2. For any nonempty set S of real numbers, let σ(S) denote the sum of the elements of S. Given a set A of n positive integers, consider the collection of all distinct sums σ(S)s S ranges over the nonempty subsets of A. Prove that thisollection of sums can be partitioned into n classes so that in each class, the ratio of the largest sum to the smallest sum does not exceed 2. 3. Let ABC be a triangle. Prove that there is a line ` (in the plane of triangle ABC) such that the intersection of the interior of triangle ABC and the interior of its reflection A0 B 0 C 0 in ` has area more than 2/3 the area of triangle ABC. 4. An n-term sequence (x1 , x2 , . . . , xn ) in which each term is either 0 or 1 is called a binary sequence of length n. Let an be the number of binary sequences of length n containing no three consecutive terms equal to 0, 1, 0 in that order. Let bn be the number of binary sequences of length n that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that bn+1 = 2an for all positive integers n. 5. Triangle ABC has the following property: there is an interior point P such that ∠P AB = 10◦ , ∠P BA = 20◦ , ∠P CA = 30◦ , and ∠P AC = 40◦ . Prove that triangle ABC is isosceles. 6. Determine (with proof) whether there is a subset X of the integers with the following property: for any integer n there is exactly one solution of a + 2b = n with a, b ∈ X.
5.9
26th USA Mathematical Olympiad, 1997
1. Let p1 , p2 , p3 , . . . be the prime numbers listed in increasing order, and let x0 be a real number between 0 and 1. For positive integer k, define (
xk =
0 ) pk xk−1
if xk−1 = 0, if xk−1 6= 0,
where {x} denotes the fractional part of x. (The fractional part of x is given by x − bxc where bxc is the greatest integer less than or equal to x.) Find, with proof, all x0 satisfying 0 < x0 < 1 for which the sequence x0 , x1 , x2 , . . . eventually becomes 0.
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CHAPTER 5. USA MATHEMATICAL OLYMPIAD
2. Let ABC be a triangle, and draw isosceles triangles BCD, CAE, ABF externally to ABC, with BC, CA, AB as their respective bases. Prove that the lines through ←→ ←→ ←→ A, B, C perpendicular to the lines EF , F D, DE, respectively, are concurrent. 3. Prove that for any integer n, there exists a unique polynomial Q with coefficients in {0, 1, . . . , 9} such that Q(−2) = Q(−5) = n. 4. To clip a convex n-gon means to choose a pair of consecutive sides AB, BC and to replace them by the three segments AM, M N , and N C, where M is the midpoint of AB and N is the midpoint of BC. In other words, one cuts off the triangle M BN to obtain a convex (n + 1)-gon. A regular hexagon P6 of area 1 is clipped to obtain a heptagon P7 . Then P7 is clipped (in one of the seven possible ways) to obtain an octagon P8 , and so on. Prove that no matter how the clippings are done, the area of Pn is greater than 1/3, for all n ≥ 6. 5. Prove that, for all positive real numbers a, b, c, (a3 + b3 + abc)−1 + (b3 + c3 + abc)−1 + (c3 + a3 + abc)−1 ≤ (abc)−1 . 6. Suppose the sequence of nonnegative integers a1 , a2 , . . . , a1997 satisfies ai + aj ≤ ai+j ≤ ai + aj + 1 for all i, j ≥ 1 with i + j ≤ 1997. Show that there exists a real number x such that an = bnxc (the greatest integer ≤ nx) for all 1 ≤ n ≤ 1997.
5.10
27th USA Mathematical Olympiad, 1998
1. Suppose that the set {1, 2, · · · , 1998} has been partitioned into disjoint pairs {ai , bi } (1 ≤ i ≤ 999) so that for all i, |ai − bi | equals 1 or 6. Prove that the sum |a1 − b1 | + |a2 − b2 | + · · · + |a999 − b999 | ends in the digit 9. 2. Let C1 and C2 be concentric circles, with C2 in the interior of C1 . From a point A on C1 one draws the tangent AB to C2 (B ∈ C2 ). Let C be the second point of intersection of AB and C1 , and let D be the midpoint of AB. A line passing through A intersects C2 at E and F in such a way that the perpendicular bisectors of DE and CF intersect at a point M on AB. Find, with proof, the ratio AM/M C.
5.11. 28T H USA MATHEMATICAL OLYMPIAD, 1999
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3. Let a0 , a1 , · · · , an be numbers from the interval (0, π/2) such that π π π tan(a0 − ) + tan(a1 − ) + · · · + tan(an − ) ≥ n − 1. 4 4 4 Prove that tan a0 tan a1 · · · tan an ≥ nn+1 . 4. A computer screen shows a 98 × 98 chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minimum number of mouse clicks needed to make the chessboard all one color. 5. Prove that for each n ≥ 2, there is a set S of n integers such that (a − b)2 divides ab for every distinct a, b ∈ S. 6. Let n ≥ 5 be an integer. Find the largest integer k (as a function of n) such that there exists a convex n-gon A1 A2 . . . An for which exactly k of the quadrilaterals Ai Ai+1 Ai+2 Ai+3 have an inscribed circle. (Here An+j = Aj .)
5.11
28th USA Mathematical Olympiad, 1999
1. Some checkers placed on an n × n checkerboard satisfy the following conditions: (a) every square that does not contain a checker shares a side with one that does; (b) given any pair of squares that contain checkers, there is a sequence of squares containing checkers, starting and ending with the given squares, such that every two consecutive squares of the sequence share a side. Prove that at least (n2 − 2)/3 checkers have been placed on the board. 2. Let ABCD be a cyclic quadrilateral. Prove that |AB − CD| + |AD − BC| ≥ 2|AC − BD|. 3. Let p > 2 be a prime and let a, b, c, d be integers not divisible by p, such that {ra/p} + {rb/p} + {rc/p} + {rd/p} = 2 for any integer r not divisible by p. Prove that at least two of the numbers a + b, a + c, a + d, b + c, b + d, c + d are divisible by p. (Note: {x} = x − bxc denotes the fractional part of x.)
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CHAPTER 5. USA MATHEMATICAL OLYMPIAD
4. Let a1 , a2 , . . . , an (n > 3) be real numbers such that a1 + a 2 + · · · + a n ≥ n
and
a21 + a22 + · · · + a2n ≥ n2 .
Prove that max(a1 , a2 , . . . , an ) ≥ 2. 5. The Y2K Game is played on a 1×2000 grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning strategy. 6. Let ABCD be an isosceles trapezoid with AB k CD. The inscribed circle ω of triangle BCD meets CD at E. Let F be a point on the (internal) angle bisector of ∠DAC such that EF ⊥ CD. Let the circumscribed circle of triangle ACF meet line CD at C and G. Prove that the triangle AF G is isosceles.
5.12
29th USA Mathematical Olympiad, 2000
1. Call a real-valued function f very convex if
f (x) + f (y) x+y ≥ ()f + |x − y| 2 2 holds for all real numbers x and y. Prove that no very convex function exists. 2. Let S be the set of all triangles ABC for which !
1 3 1 1 6 5 − + + = , AP BQ CR min{AP, BQ, CR} r where r is the inradius and P, Q, R are the points of tangency of the incircle with sides AB, BC, CA, respectively. Prove that all triangles in S are isosceles and similar to one another. 3. A game of solitaire is played with R red cards, W white cards, and B blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of R, W, and B, the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
5.13. 30T H USA MATHEMATICAL OLYMPIAD, 2001
133
4. Find the smallest positive integer n such that if n squares of a 1000 × 1000 chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board. 5. Let A1 A2 A3 be a triangle and let ω1 be a circle in its plane passing through A1 and A2 . Suppose there exist circles ω2 , ω3 , . . . , ω7 such that for k = 2, 3, . . . , 7, ωk is externally tangent to ωk−1 and passes through Ak and Ak+1 , where An+3 = An for all n ≥ 1. Prove that ω7 = ω1 . 6. Let a1 , b1 , a2 , b2 , . . . , an , bn be nonnegative real numbers. Prove that n X
i,j=1
5.13
min{ai aj , bi bj } ≤
n X
i,j=1
min{ai bj , aj bi }.
30th USA Mathematical Olympiad, 2001
1. Each of eight boxes contains six balls. Each ball has been colored with one of n colors, such that no two balls in the same box are the same color, and no two colors occur together in more than one box. Determine, with justification, the smallest integer n for which this is possible. 2. Let ABC be a triangle and let ω be its incircle. Denote by D1 and E1 the points where ω is tangent to sides BC and AC, respectively. Denote by D2 and E2 the points on sides BC and AC, respectively, such that CD2 = BD1 and CE2 = AE1 , and denote by P the point of intersection of segments AD2 and BE2 . Circle ω intersects segment AD2 at two points, the closer of which to the vertex A is denoted by Q. Prove that AQ = D2 P . 3. Let a, b, and c be nonnegative real numbers such that a2 + b2 + c2 + abc = 4. Prove that 0 ≤ ab + bc + ca − abc ≤ 2. 4. Let P be a point in the plane of triangle ABC such that the segments P A, P B, and P C are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to P A. Prove that ∠BAC is acute. 5. Let S be a set of integers (not necessarily positive) such that
134
CHAPTER 5. USA MATHEMATICAL OLYMPIAD (a) there exist a, b ∈ S with gcd(a, b) = gcd(a − 2, b − 2) = 1;
(b) if x and y are elements of S (possibly equal), then x2 − y also belongs to S. Prove that S is the set of all integers. 6. Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number.
5.14
31st USA Mathematical Olympiad, 2002
1. Let S be a set with 2002 elements, and let N be an integer with 0 ≤ N ≤ 22002 . Prove that it is possible to color every subset of S either black or white so that the following conditions hold: (a) the union of any two white subsets is white; (b) the union of any two black subsets is black; (c) there are exactly N white subsets. 2. Let ABC be a triangle such that
A cot 2
2
B + 2 cot 2
2
C + 3 cot 2
2
6s = 7r
2
,
where s and r denote its semiperimeter and its inradius, respectively. Prove that triangle ABC is similar to a triangle T whose side lengths are all positive integers with no common divisors and determine these integers. 3. Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree n with real coefficients is the average of two monic polynomials of degree n with n real roots. 4. Let R be the set of real numbers. Determine all functions f : R → R such that f (x2 − y 2 ) = xf (x) − yf (y) for all pairs of real numbers x and y.
5.15. 32N D USA MATHEMATICAL OLYMPIAD, 2003
135
5. Let a, b be integers greater than 2. Prove that there exists a positive integer k and a finite sequence n1 , n2 , . . . , nk of positive integers such that n1 = a, nk = b, and ni ni+1 is divisible by ni + ni+1 for each i (1 ≤ i < k). 6. I have an n×n sheet of stamps, from which I’ve been asked to tear out blocks of three adjacent stamps in a single row or column. (I can only tear along the perforations separating adjacent stamps, and each block must come out of the sheet in one piece.) Let b(n) be the smallest number of blocks I can tear out and make it impossible to tear out any more blocks. Prove that there are real constants c and d such that 1 2 1 n − cn ≤ b(n) ≤ n2 + dn 7 5 for all n > 0.
5.15
32nd USA Mathematical Olympiad, 2003
1. Prove that for every positive integer n there exists an n-digit number divisible by 5n all of whose digits are odd. 2. A convex polygon P in the plane is dissected into smaller convex polygons by drawing all of its diagonals. The lengths of all sides and all diagonals of the polygon P are rational numbers. Prove that the lengths of all sides of all polygons in the dissection are also rational numbers. 3. Let n 6= 0. For every sequence of integers A = a 0 , a1 , a2 , . . . , an satisfying 0 ≤ ai ≤ i, for i = 0, . . . , n, define another sequence t(A) = t(a0 ), t(a1 ), t(a2 ), . . . , t(an ) by setting t(ai ) to be the number of terms in the sequence A that precede the term ai and are different from ai . Show that, starting from any sequence A as above, fewer than n applications of the transformation t lead to a sequence B such that t(B) = B. 4. Let ABC be a triangle. A circle passing through A and B intersects segments AC and BC at D and E, respectively. Lines AB and DE intersect at F while lines BD and CF intersect at M . Prove that M F = M C if and only if M B · M D = M C 2 .
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CHAPTER 5. USA MATHEMATICAL OLYMPIAD
5. Let a, b, c be positive real numbers. Prove that (2a + b + c)2 (2b + c + a)2 (2c + a + b)2 + + ≤ 8. 2a2 + (b + c)2 2b2 + (c + a)2 2c2 + (a + b)2 6. At the vertices of a regular hexagon are written six nonnegative integers whose sum is 2003. Bert is allowed to make moves of the following form: he may pick a vertex and replace the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Prove that Bert can make a sequence of moves, after which the number 0 appears at all six vertices.
5.16
33rd USA Mathematical Olympiad, 2004
1. Let ABCD be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60◦ . Prove that 1 |AB 3 − AD 3 | ≤ |BC 3 − CD 3 | ≤ 3|AB 3 − AD 3 |. 3 When does equality hold? 2. Suppose a1 , . . . , an are integers whose greatest common divisor is 1. Let S be a set of integers with the following properties. (a) For i = 1, . . . , n, ai ∈ S.
(b) For i, j = 1, . . . , n (not necessarily distinct), ai − aj ∈ S. (c) For any integers x, y ∈ S, if x + y ∈ S, then x − y ∈ S.
Prove that S must be equal to the set of all integers. 3. For what real values of k > 0 is it possible to dissect a 1×k rectangle into two similar, but noncongruent, polygons? 4. Alice and Bob play a game on a 6 by 6 grid. On his or her turn, a player chooses a rational number not yet appearing in the grid and writes it in an empty square of the grid. Alice goes first and then the players alternate. When all squares have numbers written in them, in each row, the square with the greatest number in that row is colored black. Alice wins if she can then draw a line from the top of the grid to the bottom of the grid that stays in black squares, and Bob wins if she can’t. (If two squares share a vertex, Alice can draw a line from one to the other that stays in those two squares.) Find, with proof, a winning strategy for one of the players.
5.16. 33RD USA MATHEMATICAL OLYMPIAD, 2004
137
5. Let a, b and c be positive real numbers. Prove that (a5 − a2 + 3)(b5 − b2 + 3)(c5 − c2 + 3) ≥ (a + b + c)3 . 6. A circle ω is inscribed in a quadrilateral ABCD. Let I be the center of ω. Suppose that (AI + DI)2 + (BI + CI)2 = (AB + CD)2 . Prove that ABCD is an isosceles trapezoid.
Chapter 6 Canadian Mathematical Olympiad 6.1
30th Canadian Mathematical Olympiad, 1998
1. Determine the number of real solutions a to the equation
1 1 1 a + a + a =a. 2 3 5
Here, if x is a real number, then [ x ] denotes the greatest integer that s less than or equal to x. 2. Find all real numbers x such that
1 x= x− x
1/2
1 + 1− x
1/2
.
3. Let n be a natural number such that n ≥ 2. Show that
1 1 1 1 1 1 1 + +···+ 1+ +···+ > . n+1 3 2n − 1 n 2 4 2n 4. Let 4ABC be a triangle with ∠BAC = 40◦ and ∠ABC = 60◦ . Let D and E be the points lying on the sides AC and AB, respectively,uch that ∠CBD = 40◦ and ∠BCE = 70◦ . Let F be the point of intersection of the lines BD and CE. Show that the line AF is perpendicular to the line BC. 138
6.2. 31ST CANADIAN MATHEMATICAL OLYMPIAD, 1999
139
5. Let m be a positive integer. Define the sequence a0 , a1 , a2 , . . . by a0 = 0, a1 = m, and an+1 = m2 an − an−1 for n = 1, 2, 3, . . . . Prove that an ordered pair (a, b) of non-negative integers, with a ≤ b, gives a solution to the equation a2 + b 2 = m2 ab + 1 if and only if (a, b) is of the form (an , an+1 ) for some n ≥ 0.
6.2
31st Canadian Mathematical Olympiad, 1999
1. Find all real solutions to the equation 4x2 − 40[x] + 51 = 0.
Here, if x is a real number, then [x] denotes the greatest integer that is less than or equal to x.
2. Let ABC be an equilateral triangle of altitude 1. A circle with radius 1 and center on the same side of AB as C rolls along the segment AB. Prove that the arc of the circle that is inside the triangle always has the same length. 3. Determine all positive integers n with the property that n = (d(n))2 . Here d(n) denotes the number of positive divisors of n. 4. Suppose a1 , a2 , . . . , a8 are eight distinct integers from {1, 2, . . . , 16, 17}. Show that there is an integer k > 0 such that the equation ai − aj = k has at least three different solutions. Also, find a specific set of 7 distinct integers from {1, 2, . . . , 16, 17} such that the equation ai − aj = k does not have three distinct solutions for any k > 0. 5. Let x, y, and z be non-negative real numbers satisfying x + y + z = 1. Show that x2 y + y 2 z + z 2 x ≤
4 , 27
and find when equality occurs.
6.3
32nd Canadian Mathematical Olympiad, 2000
1. At 12:00 noon, Anne, Beth and Carmen begin running laps around a circular track of length three hundred meters, all starting from the same point on the track. Each
140
CHAPTER 6. CANADIAN MATHEMATICAL OLYMPIAD jogger maintains a constant speed in one of the two possible directions for an indefinite period of time. Show that if Anne’s speed is different from the other two speeds, then at some later time Anne will be at least one hundred meters from each of the other runners. (Here, distance is measured along the shorter of the two arcs separating two runners.)
2. A permutation of the integers 1901, 1902, . . . , 2000 is a sequence a1 , a2 , . . . , a100 in which each of those integers appears exactly once. Given such a permutation, we form the sequence of partial sums s1 = a1 , s2 = a1 + a2 , s3 = a1 + a2 + a3 , . . . , s100 = a1 + a2 + · · · + a100 . How many of these permutations will have no terms of the sequence s1 , . . . , s100 divisible by three? 3. Let A = (a1 , a2 , . . . , a2000 ) be a sequence of integers each lying in the interval [−1000, 1000]. Suppose that the entries in A sum to 1. Show that some nonempty subsequence of A sums to zero. 4. Let ABCD be a convex quadrilateral with
and
∠CBD = 2∠ADB, ∠ABD = 2∠CDB AB = CB.
Prove that AD = CD. 5. Suppose that the real numbers a1 , a2 , . . . , a100 satisfy
and
a1 ≥ a2 ≥ · · · ≥ a100 ≥ 0, a1 + a2 ≤ 100 a3 + a4 + · · · + a100 ≤ 100.
Determine the maximum possible value of a21 + a22 + · · · + a2100 , and find all possible sequences a1 , a2 , . . . , a100 which achieve this maximum.
6.4
33rd Canadian Mathematical Olympiad, 2001
1. Randy: “Hi Rachel, that’s an interesting quadratic equation you have written down. What are its roots?”
6.4. 33RD CANADIAN MATHEMATICAL OLYMPIAD, 2001
141
Rachel: “The roots are two positive integers. One of the roots is my age, and the other root is the age of my younger brother, Jimmy.” Randy: “That is very neat! Let me see if I can figure out how old you and Jimmy are. That shouldn’t be too difficult since all of your coefficients are integers. By the way, I notice that the sum of the three coefficients is a prime number.” Rachel: “Interesting. Now figure out how old I am.” Randy: “Instead, I will guess your age and substitute it for x in your quadratic equation . . . darn, that gives me −55, and not 0.” Rachel: “Oh, leave me alone!” (a) Prove that Jimmy is two years old. (b) Determine Rachel’s age. 2. There is a board numbered −10 to 10 as shown. Each square is coloured either red or white, and the sum of the numbers on the red squares is n. Maureen starts with a token on the square labeled 0. She then tosses a fair coin ten times. Every time she flips heads, she moves the token one square to the right. Every time she flips tails, she moves the token one square to the left. At the end of the ten flips, the probability that the token finishes on a red square is a rational number of the form ab . Given that a + b = 2001, determine the largest possible value for n. −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 3. Let ABC be a triangle with AC > AB. Let P be the intersection point of the perpendicular bisector of BC and the internal angle bisector of ∠A. Construct points X on AB (extended) and Y on AC such that P X is perpendicular to AB and P Y is perpendicular to AC. Let Z be the intersection point of XY and BC. Determine the value of BZ . ZC 4. Let n be a positive integer. Nancy is given a rectangular table in which each entry is a positive integer. She is permitted to make either of the following two moves: (a) select a row and multiply each entry in this row by n. (b) select a column and subtract n from each entry in this column. Find all possible values of n for which the following statement is true: Given any rectangular table, it is possible for Nancy to perform a finite sequence of moves to create a table in which each entry is 0. 5. Let P0 , P1 , P2 be three points on the circumference of a circle with radius 1, where P1 P2 = t < 2. For each i ≥ 3, define Pi to be the centre of the circumcircle of 4Pi−1 Pi−2 Pi−3 .
142
CHAPTER 6. CANADIAN MATHEMATICAL OLYMPIAD (a) Prove that the points P1 , P5 , P9 , P13 , . . . are collinear. (b) Let x be the distance from P1 to P1001 , andqlet y be the distance from P1001 to P2001 . Determine all values of t for which 500 x/y is an integer.
6.5
34th Canadian Mathematical Olympiad, 2002
1. Let S be a subset of {1, 2, . . . , 9}, such that the sums formed by adding each unordered pair of distinct numbers from S are all different. For example, the subset {1, 2, 3, 5} has this property, but {1, 2, 3, 4, 5} does not, since the pairs {1, 4} and {2, 3} have the same sum, namely 5. What is the maximum number of elements that S can contain? 2. Call a positive integer n practical if every positive integer less than or equal to n can be written as the sum of distinct divisors of n. For example, the divisors of 6 are 1, 2, 3, and 6. Since 1=1, 2=2, 3=3, 4=1+3, 5=2+ 3,
6=6,
we see that 6 is practical. Prove that the product of two practical numbers is also practical. 3. Prove that for all positive real numbers a, b, and c, a3 b3 c3 + + ≥ a + b + c, bc ca ab and determine when equality occurs. 4. Let Γ be √ a circle with radius r. Let A and B be distinct points on Γ such that AB < 3r. Let the circle with centre B and radius AB meet Γ again at C. Let P be the point inside Γ such that triangle ABP is equilateral. Finally, let the line CP meet Γ again at Q. Prove that P Q = r. 5. Let N = {0, 1, 2, . . .}. Determine all functions f : N → N such that xf (y) + yf (x) = (x + y)f (x2 + y 2 ) for all x and y in N .
6.6. 35T H CANADIAN MATHEMATICAL OLYMPIAD, 2003
6.6
143
35th Canadian Mathematical Olympiad, 2003
1. Consider a standard twelve-hour clock whose hour and minute hands move continuously. Let m be an integer, with 1 ≤ m ≤ 720. At precisely m minutes after 12:00, the angle made by the hour hand and minute hand is exactly 1◦ . Determine all possible values of m. 2. Find the last three digits of the number 20032002
2001
.
3. Find all real positive solutions (if any) to x3 + y 3 + z 3 = x + y + z, and x2 + y 2 + z 2 = xyz. 4. Prove that when three circles share the same chord AB, every line through A different from AB determines the same ratio XY : Y Z, where X is an arbitrary point different from B on the first circle while Y and Z are the points where AX intersects the other two circles (labelled so that Y is between X and Z). 5. Let S be a set of n points in the plane such that any two points of S are at least 1 unit apart. Prove there is a subset T of S with at least n7 points such that any two √ points of T are at least 3 units apart.
6.7
36th Canadian Mathematical Olympiad, 2004
1. Find all ordered triples (x, y, z) of real numbers which satisfy the following system of equations: xy = z − x − y xz = y − x − z yz = x − y − z 2. How many ways can 8 mutually non-attacking rooks be placed on the 9×9 chessboard (shown here) so that all 8 rooks are on squares of the same colour? [Two rooks are said to be attacking each other if they are placed in the same row or column of the board.]
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CHAPTER 6. CANADIAN MATHEMATICAL OLYMPIAD
3. Let A, B, C, D be four points on a circle (occurring in clockwise order), with AB < AD and BC > CD. Let the bisector of angle ^BAD meet the circle at X and the bisector of angle ^BCD meet the circle at Y .onsider the hexagon formed by these six points on the circle.f four of the six sides of the hexagon have equal length, prove that BD must be a diameter of the circle. 4. Let p be an odd prime. Prove that p−1 X
k=1
k 2p−1 ≡
p(p + 1) 2
(mod p2 ).
[Note that a ≡ b (mod m) means that a − b is divisible by m.] 5. Let T be the set of all positive integer divisors of 2004100 . What is the largest possible number of elements that a subset S of T can have if no element of S is an integer multiple of any other element of S?
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