360 Problems for Mathematical Contests

TITD ANDREESCD

DORIN ANDRICA

360 Problems for Mathematical Contests

TITU ANDREESCU

DORIN ANDRICA

360 Problems for Mathematical Contests

GIL Publishing House

© GIL Publishing House

ISBN 973-9417-12-4

360 Problems for Mathematical Contests Authors: Titu Andreescu, Dorin Andrica Copyright © 2003 by Gil. All rights reserved. National Library of Romania CIP Description

ANDREESCU, TITU 360 Problems for Mathematical Contests/ Titu Andreescu,

Dorin Andrica. - Zalau: Gil, 2003

Contents FOREWORD 3 FROM THE AUTHORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Chapter 1 . ALGEBRA 7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Solutions . . . . . 17 Chapter 2. NUMBER THEORY . . . . . . 47 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Solutions . . . . .. .. .. 57 Chapter 3. GEOMETRY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Chapter 4. TRIGONOMETRY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Problems . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Solutions . .. 147 Chapter 5. MATHEMATICAL ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Chapter 6. COMPREHENSIVE PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

p. ;cm.

B iblio gr. ISBN 973-9417-12-4

. . . . . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . .

1. Andrica, Dorin 51(075.35)(076)

. . . . .

GIL Publishing House

P.o. Box 44, Post Office 3, 4700, Zalau, Romania, tel. (+40) 260/616314 fax. (+40) 260/616414 e-mail: [email protected]

www.gil.ro

IMPRIMERIA

ARTA IV GRAFICA I ��LIBRIS �

Calea $erbanVodft I33,S.4,Cod 70517,BUCURE$TI Tel.:3362911 Fax:337 0735

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . .

. . . . . . . . . .

. .

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . .

. . . . . . . . . . . . . . .

. . .

. . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

.

FOREWORD I take great pleasure in recommending to all readers - Romanians or from abroad - the book of professors Titu Andreescu and Dorin Andrica. This book is the fruit of a prodigious activity of the two authors, well-known creators of mathematics questions for Olympiads and other mathematical contests. They have published innumerable original problems in various mathematical journals. The book is organized in six chapters: algebra, number theory, geometry, trigonometry, analysis and comprehensive problems. In addition, other fields of math­ ematics found their place in this book, for example, combinatorial problems can be found in the last chapter, and problems involving complex numbers are included in the trigonometry section. Moreover, in all chapters of this book the serious reader can find numerous challenging inequality problems. All featured problems are interesting, with an increased level of difficulty; some of them are real gems that will give great satisfaction to any math lover attempting to solve or even extend them. Through their outstanding work as jury members of the National Mathematical Olympiad, the Balkan Mathematics Contest (BMO) , and the International Math­ ematical Olympiad (IMO) , the authors also supported the excellent results of the Romanian contestants in these competitions. A great effort was given in preparing lectures for summer and winter training camps and also for creating original problems to be used in selection tests to search for truly gifted mathematics students. To support the claim that the Romanian students selected to represent the country were really the ones to deserve such honor, we note that only two mathematicians of Romanian origin, both former IMO gold-medalists, were invited recently to give conferences at the International Mathematical Congress: Dan Voiculescu (Zurich, 1994) and Daniel Tataru (Beijing, 2002). The Romanian mathematical community unanimously recog­ nized this outstanding activity of professors Titu Andreescu and Dorin Andrica. As a consequence, Titu Andreescu, at that time professor at Loga Academy in Timi§oara and having students on the team participating in the IMO, was appointed to serve as deputy leader of the national team. Nowadays, Titu's potential, as with other Ro­ manians in different fields, has been fully realized in the United States, leading the USA team in the IMO, coordinating the training and selection of team contestants and serving as member of several national and regional mathematical contest juries.

One more time, I strongly express my belief that the 360 mathematics problems featured in this book will reveal the beauty of mathematics to all students and it will be a guide to their teachers and professors. Professor loan Tomescu Department of Mathematics and Computer Science University of Bucharest Associate member of the Romanian Academy

FROM THE AUTHORS This book is intended to help students preparing for all rounds of Mathematical Olympiads or any other significant mathematics contest. Teachers will also find this work useful in training young talented students. Our experience as contestants was a great asset in preparing this book. To this we added our vast personal experience from the other side of the " barricade" , as creators of problems and members of numerous contest committees. All the featured problems are supposed to be original. They are the fruit of our collaboration for the last 30 years with several elementary mathematics journals from all over the world. Many of these problems were used in contests throughout these years, from the first round to the international level. It is possible that some problems are already known, but this is not critical. The important thing is that an educated - to a certain extent - reader will find in this book problems that bring something new and will teach new ways of dealing with key mathematics concepts, a variety of methods, tactics, and strategies. The problems are divided in chapters, although this division is not firm, for some of the problems require background in several fields of mathematics. Besides the traditional fields: algebra, geometry, trigonometry and analysis, we devoted an entire chapter to number theory, because many contest problems require knowledge in this field. The comprehensive problems in the last chapter are also intended to help under­ graduate students participating in mathematics contests hone their problem solving skills. Students and teachers can find here ideas and questions that can be interesting topics for mathematics circles. Due to the difficulty level of the problems contained in this book, we deemed it appropriate to give a very clear and complete presentation of all solutions. In many cases, alternative solutions are provided. As a piece of advice to all readers, we suggest that they try to find their own solutions to the problems before reading the given ones. Many problems can be solved in multiple ways and pertain to interesting extensions.

4

This edition is significantly different from the 2002 Romanian edition. It features more recent problems, enhanced solutions, along with references for all published problems. We wish to extend our gratitude to everyone who influenced in one way or another the final version of this book. We will gladly receive any observation from the readers. The authors

Chapter 1 ALGEBRA

6

PROBLEMS

C2 , ,cn }.

be a set of n characters { Cl ' We call word a string of at most m characters, m ::; n, that does not start nor end with Cl . How many words can be formed with the characters of the set C? 1. Let

C

. . •

2. The numbers 1, 2, , 5n are divided into two disjoint sets. Prove that these sets contain at least n pairs such that the number is also an element of the set which contains the pair. . . .

(x, y), x > y,

x-y

al, a2 , . . . ,an

3. Let be distinct numbers from the interval permutation of {I, 2, . . . , n } . Define the function f : -t as follows:

[a, b] [a, b] x �i' i f(x) { xa i ifotherwIse q(

=

=

)

[a, b] and let

0'

be a

= I, n

Prove that there is a positive integer h such that

fafa... aJ. � htimes

x, where flh]

flh](X)

x, y, z are nonzero real numbers with x y z 0, then xx2 yy2 yy2 Z2 Z2 xx2 xyz3 y3 xyZ3 zx Z Z d Prove that Let a, b, c, d be complex numbers with a b a3 b3 c3 d3 3(abc bcd cda dab). Let a, b, c be nonzero real numbers such that a b c 0 and a3 b3 c3 a5 b5 c5• Prove that 4.

Prove that if

+

+

-- + +

+ +

--

+

+ +

--

5.

=

+

+

+

+

=

+

+ C + = o. +

=

+ +

+

7. Let a, b, c, d be integers. Prove that a + b + c + d divides

2(a4 b4 c4 +

+

+ at) -

(

a2 b2 c2 d2 )2 8abcd. +

+

=

- + - +-.

+

6.

+

+

+

+

+

+

=

1.

10

8. Solve in complex numbers the equation

(x + l)(x + 2)(x + 3) 2 (X + 4)(x + 5) = 360.

9. Solve in real numbers the equation yX + VY

+ 2v'z=2 + v'u + Vv = x + y + z + u + v.

16. Prove that if x, y, z are real numbers such that xS + yS + ZS i= 0, then the

ratio

2xyz - (x + y S+ z) xS + yS + Z equals � if and only if x + y + z = O. 17. Solve in real numbers the equation .vx;:-=-r

10. Find the real solutions to the equation

(x + y) 2 = (X + 1) (y - 1). .

12. Solve the equation

a, b, c

-J x + a + -Jx + b + -J x + c = -J x + a + b - c,

are real parameters. Discuss the equation in terms of the values of the where parameters.

13.

a and b be distinct positive real numbers. Find all pairs of positive real (x, y), solutions to the system of equations { xx42 -- yy42 == axf/a-2 -byb2 .

Let numbers

14. Solve the equation

[

25x4-2] = 13x3+ 4'

[a] denotes the integer part of a real number a. 1 +v1s 15. Prove that If. a � -2-' then 1+ = n, n = O, 1,2, . . .

where

[ [�]l

1

+ 2-J X2 - 4 + . . . + nJXn - n2 = �(Xl 2 + X2 +

. . .

+

xn).

18. Find the real solutions to the system of equations

11. Solve the equation

x + V4X + V16X + J. . + J4nx + 3 - JX = 1.

11

1. 1 . PROBLEMS

ALGEBRA

-x1 +-y1 = 9 1 + -1 1 + -1 1 + -1 - 18 (� ?'Y)( �)( ?'Y) -

19. Solve in real numbers the system of equations

y22 + u22 + v22 + w22 = 4x - 1 x + u + v + w = 4y - 1 x22 + y22 + v22 + w22 = 4u - 1 x2 + y2 + u2 + w2 = 4v - 1 x + y + u + v = 4w - 1 20. Let aI, a2 , as, a4 , a5 be real numbers such that al + a2 + as + a4 + a5 = 0 and max l�i- 109 (a + b + C)2 -

-

_

a, b, c be real numbers such that abc = 1. Prove that at most two of the 2a - -1b ' 2b - c'1 2c--a1 are greater than 1. 23.

Let numbers

12

1.

24. Let a, b,

c,

ALGEBRA

1.1.

d be real numbers. Prove that

- d}, d - a2 ) � �. 25. Let aI, a2 , ' . . ,an be numbers in the interval (0, 1) and let k ;::: 2 be an integer. Find the maximum value of the expression n L yt ai (I - ai+ I), i= l where an+1 a . 26. Let m and n be positive integers. Prove that x mn - 1 > x n - I --- -min(a

=

- b2 , b

2 - c ,

34. Show that for any positive integer n the number

n ; 1) 22n + n : 1) 22n-2 . 3+ . . . + n : 1) 3n C2 C

C

C

-

x

for any positive real number x.

27. Prove that m! ;::: (n!) [�] for all positive integers m and n. 28. Prove that

n ;::: 2.

1 +-+"' 1 +-1 > n � I +-n +2 -1 y'2 v'3 \Iii

29. Prove that

n (1 - 1/ vn) + 1 > 1 + �2 + �3 + . . . + .!.n > n (\In + 1 - 1) for any positive integer n. 30. Let aI, a2 , . . . , an (0, 1 ) and let tn al +naa2a+2 . .. .. .a+n an . Prove that n L loga; tn ;::: (n - I ) n . i=l 31. Prove that between n and 3n there is at least a perfect cube for any integer n ;::: 1. 32. Compute the sum n k. Sn L-(-I) k= l 33. Compute the sums: l a) S n � (k + l) (k + 2) (�); b) T n = � (k + l)(k + 2)(k + 3) (�). E

1

=

=

=

13

is the sum of two consecutive perfect squares.

35. Evaluate the sums:

1

m

for any integer

PROBLEMS

�

Io(k+l) 2

36. Pr.ove that

12 (�) + 32 (�) + 52 (�) + . . . n(n + 1)2n-3 for all integers n ;::: 3. 37. Prove that 2Ln [log kJ = (n - 2)2n + n + 2 k= l 2 for all positive integers n. 38. Let Xn 22n + 1, n 1,2,3, . . . Prove that 2n-1 < ­1 -Xl1 + -X22 + -X223 + . . . +-Xn 3 for all positive integers n. 39. Let f be a function such that f(z)f(iz) = Z2 for all z Prove that f(z) + f( -z) 0 for all z 40. Consider a function f : (0,00) and a real number a > 0 such that f( a) 1. Prove that if f(x)f(y) + f (;) f (�) 2f(xy) for x, y (0, 00), =

=

:

C

=

-+

E C.

C

E C

=

-+ 1R

=

=

all

E

then f is a constant function.

41. Find with proof if the function f·: 1R -+ [- 1, 1), f(x) 42. For all i,j IS(i,j)l·

=

=

sin[x) is periodical.

l,n define S(i,j) = kL=n l ki + . Evaluate the determinant � = i

14

43. Let Xij

=

1.

{a

1. 1 .

ALGEBRA

48. Let P(x) be a polynomial of degree n. If

i if i = j 0 if i i j, i + j i 2n + 1 if i + j = 2n + 1

bi where ai , b i are real numbers. Evaluate the determinant � 2 n

evaluate

44. a) Compute the determinant

=

>

=

0, 1 , . . . , n

n.

for all real numbers x. y

y

x

v

z

z

P(m), where m

k for k P(k) -k+1

49. Find all polynomials P(x) with integral coefficients such that

= IXij l .

x

15

BROBLEMS

v

z

v

x

y

v

y

50.

Consider the polynomials Pi, i = 1 , 2, . . . , n with degrees at least 1. Prove that if the polynomial

z

x

P(x) PI (xn+l ) + XP2 (xn+l ) + . . . + xn- lpn (xn+1 ), i s divisible by xn +x n - l + . . · + x + 1, then all polynomials Pi(x) , i 1, n, are divisible =

abed, bade, cdab, deba are divisible by a prime p, a + b + e + d, a + b - - d, a - b + e - d, a - b - e + d, is divisible by p. 45. Consider the quadratic polynomials t1 (x) x2 + PI X + q r and t2 (x) x2 + P2 X + q�, where Pl , P2 , ql , q 2 are real numbers. Prove that if polynomials tl and t2 have zeros of the same nature, then the polynomial b) Prove that if the numbers then at least one of the numbers

e

=

=

51. Let P be a prime number and let

P(x) aoxn + al xn-1 . + an be a polynomial with integral coefficients such that an =1= 0 (mod p). Prove that if there are n + 1 integers 0' 1 , 0'2 , , an+ 1 such that P (ar) 0 (mod p) for all 1, 2, . . . , n + 1, then there exist i,j with i i j such that a i aj (mod p). 52. Determine all polynomials P with real coefficients such that pn (x) P(xn ) =

46. Let a, b, e be real numbers with a

>

0 such that the quadratic polynomial

ax2 + bex + b3 + e3 - 4abe has nonreal zeros. Prove that exactly one of the polynomials Tl (x) ax2 + bx + e and T2 (x) ax 2 + ex + b has only positive values.

+

.

.

• • •

== ==

r =

for all real numbers x, where n

53. Let

has real zeros.

=

by x - 1.

>

1 is a given integer.

P(x) aoxn + alxn-l + . . . + an , an i 0, =

be a polynomial with complex coefficients such that there is an integer

T(x) =

=

47. Consider the polynomials with complex coefficients

P(x) xn + alxn-l + . . . + an and Q(x) xn + bl xn- l + . . . + bn having zeros Xl, X2 , . . . ,X n and x ? , x� , . . . , x; respectively. Prove that if al + a3 + a5 + . . . and a2 + a4 + a6 + . . . are real numbers, then bl + b2 + . . . + bn is also a real number. =

=

=

m with

P has at least a zero with the absolute value less than 1. 54. Find all polynomials P of degree n having only real zeros Xl , X2 , . . . , Xn such that n 1 2 tt P(x) - Xi XPnI (X) , Prove that the polynomial

=

for all nonzero real numbers x.

55. Consider the polynomial with real coefficients

P(x) aoxn + alxn- l + . . . + an , =

16

1.

an

and

f:. O. Prove that if the equation equation

ALGEBRA

P(x) 0 has all of its roots real and distinct, then the x2 PII (x) 3xP'(x) P(x) 0

has the same property.

=

+

+

=

56. Let R�f and R�) be the sets of polynomials with real coefficients having no multiple zeros and having multiple zeros of order respectively. Prove that if P(x) R�j and P(Q(x)) R�j , then Q'(x) R�]-l). 57. Let P(x) be a polynomial with real coefficients of degree at least 2. Prove

SOLUTIONS

n

E

E

that if there is a real number

E

a such that P(a) plI (a) (P'(a)) 2 ,

ai.

aoxn alxn- 1

X4 - (2 l)x3 m +

+

+ (m

where m is a real parameter.

+ ... +

an

=

0

- l)x2 (2 2 l)x

60. Solve the equation

x2n a1 x2n-1 +

if all of its roots are real.

+ ... +

(l) f:. a1 and f(k) f:. a1 For f(l) and f(k) there are - 1 possibilities of choosing a character from C2 , . .. ,en and for f(i) , 1 < i < k there are such possibilities. Therefore the number of strings f (l) f ( 2) ... f(k - l) f ( k) is Nk ( - 1 )2 k -2 It follows that Nl N2 Nm ( - 1) ( - 1 ) 2 ( ( - 1)2 1 1 ) 2 m-2 n

with real coefficients Prove that if the equation has all of its roots real, then (n 2:: 2n O . Is the reciprocal true?

59. Solve the equation

+

= n

n

58. Consider the equation

a a2

A

C,

- 1. The number that we seek is Nl N2 Nm· {I, 2,...,k}, 1 � k � We need to find out the number of functions

Clearly, Nl

f

P has at least two nonreal zeros.

- l)ar

m.

Ak = m. k -+ A, k = 2,n with the properties

Let

f:

>

then

1. Let Nk be the number of words having exactly k characters from the set

1�k�

+ ... +

+

m +

a2n_2X2 - 2 x n

+

= n

+

+m =

0,

+ ... +

= n

+ n

n

O n + n

n + ... + n -

n

=

(Dorin Andrica) 1

=

0,

2.

Suppose, for the sake of contradiction, that there are two sets A and B such that AU B = {I, 2,...,5n}, An B = 0 and the sets contain together less than n pairs > with the desired property. Let k be a given number, k = 1, n. If k and 2k are in the same set A or B the same can be said about the difference 2k - k = k. The same argument is applied for 4k and 2k. Consider the case when k and 4k are elements of A and 2k is an element of B. If 3k is an element of A, then 4k - 3k = k E A, so let 3k E B. Now if 5k E A, then 5k - 4k = k E A and if 5k E B, then 5k - 3k = 2k E B; so among the numbers k, 2k, 3k,4k,5k there is at least a pair with the desired property. Because k = 1 , 2,... , n, it follows that there are at least n pairs with the desired property. Revista Matematica Timi§oara (RMT) , No. 2(1978), pp. 75, Problem 3698)

(x, y), x y,

-

(Dorin Andrica,

3. Note that (1)

17

1.

18

1.2.

ALGEBRA

and furthermore for all integers ml, m2 2:: 1 . Suppose that for all integers k 2:: 1 we have I[k](x) iBecause there are n! permutations, it follows that for k such that positive integers nl >

x.

n2

ai-, in=2

Let h = n l since numbers or

(2)

l[n2+h](x)

=

n!

x

E

[a, b],

(3)

= (10 l[n2-1])(x), X [a, b]. Because I is injective, we obtain l[n2+h-l] (x) l[n2-1] (x), x [a, b] (10 l[n2+h-l])(x)

and in the same manner

= I(x), l[h](X) = x, x

l[h+l](X)

or

Alternative solution.

E

E

=

x E

E

[a, b]

[a, b].

Let Sn be the symmetric group of order n and Hn the cyclic subgroup generated by a. It is clear that Hn is a finite group and therefore there is integer h such that a[h) is identical permutation. Notice that if = 1, n l I k ](X) = otherwIse

{ axq[k1(i)

x

�i' i =

= x and the solution is complete. (Dorin Andrica, Revista Matematidi Timi§oara (RMT) , No. 2(1978) , pp. 53,

Then I[h](x) Problem 3540)

x + y + z = 0, we obtain x + y -z, y + z -x, z + x -y, or, by squaring and rearranging, x2 + y2 = Z2 2xy, y2 + Z2 = x2 2yz, Z2 + x2 y2 - 2zx. The given equality is equivalent to x2 - 2yz y2 - 2zx x3 y3 Z3 Z2 - 2xy -z--=- + -x + -y -yz + -zx + -, xy 4.

Because

=

_

--

=

=

_

=

=

Z3 -(x +y + z) + 2 ( -Xyz + -yzx + -zxY ) = yz-x3 + zx-y3 + -. xy

2(X2 y2 + y2z2 + 2 X2 ) = x4 + y4 + Z4 . On the other side, from x + y + z = 0 we obtain (x + y + Z) 2 = 0 or x2 + y2 + Z2 = -2( xy + yz + zx). Squaring yields X4 + y4 + Z4 + 2(X2y2 + y2z2 + Z2 X2 ) 4(X2 y2 + y2z 2 + Z2X2 ) + 8xyz(x + y + z) Z

(3)

l[n2](x),

and consequently to

The last equality is equivalent to >

0 and observe that for all k the functions I[k] are injective, 1 , n are distinct. From relation we derive that

>

19

SOLUTIONS

=

or as desired.

(Titu Andreescu,

Revista Matematica Timi§oara (RMT) , No. 3(1971), pp. 25, Problem 483; Gazeta Matematica (GM-B) , No. 12(1977) , pp. 501, Problem 6090)

a, b,

d are different from zero. Consider the equation x4 - (L a) x3 + (L ab) x2 - (L abc) x + abed 0 with roots a, b, e, d. Substituting x with a, b, e and d and simplifying by a, b, e, d i- 0, 5. We assume that numbers

c,

=

after summing up we obtain

Because

If

L

a=

0, it follows that

L a3 = 3 L abe.

one of the numbers is zero, say

a,

then

b+ e + d = 0, or b e = -d. It is left to prove that b3 e3 + d3 = 3bed. Now b3 + e3 + d3 = b3 + e3 - (b + e) 3 -3be(b + ) = 3bed as desired. (Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 1-2(1979) , pp. 47, +

+

=

Problem 3803)

6. Because

a+b+e

a3 + b3 + e3 3abe =

=

a5 + b5 + e5 5abe( a2 + b2 + e2 + ab + be + ea)

0, we obtain

and

c

=

1.

20 The given relation becomes

3abc = 5abc(a2 + b2 + c2 + ab + bc + ca) or a2 + b2 + e2 + ab + be + = "5'3 since a, b, e are nonzero numbers. follows that "21 [(a + b + e)2 + a2 + b2 + e2 ] ="53 and, using again the relation a + b + e = 0, we obtain a2 + b2 + e2 = �5 ' as desired. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1977), pp. 59, Problem 3016) It

ca

7. Consider the equation

x4 - (L:a) x3 + (L: ab) X2 - (L:abc) x + abed = 0, with roots a, b, e, d. Substituting x with a, b, e and d, respectively, we obtain after summation that L:a4 + (L:ab) L:a2 +4abed is divisible by L: a. Taking into account that we deduce that

L: a. Hence 2(a4 + b4 + e4 + d4 ) - (a2 + b2 + e2 + d2 )2 + 8abed is divisible by a + b + e + d, as desired. (Dorin Andrica) is divisible by

8. The equation is equivalent to

1.2 . 21 or y3 + 22y2 + 157y = 0, with solutions Yl = 0, Y2 = -11 + 6i, Y3 = -11 - 6i. Turning back to the substitution, we obtain a first equation, x2 + 6x = 0, with solutions Xl = 0, X2 = -6. The equation x2 + 6x -11 + 6i is equivalent to (x + 3)2 = -2 + 6i. Setting x + 3 = u + iv, u, v we obtain the system { U22uv-=v26 = -2 It follows that (u2 + V2)2 = (u2 - V2)2 + (2UV)2 = 4 + 36 = 40. Therefore { U2u2 +- v2v2 == 2y'lO -2 and u2 = y'lO - 1, v2 = vTO + 1, yielding the solutions X3, 4 = -3 ± V..flO - 1 ± iV..flO + 1 where the signs + and - correspond. The equation x2 + 6x = -11 - 6i can be solved in a similar way and it has the solutions X5 = -3 + VvTO - 1 - iVvTO + 1, X6 = -3 - V.J[O - 1 + iVvTO + 1. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 3(1972), pp. 26, Problem 1255) SOLUTIONS

ALGEBRA

(x2 + 6x + 5)(x2 + 6x + 8)(x2 + 6x + 9) = 360. Setting x2 + 6x = y yields (y + 5)(y + 8)(y + 9) = 360,

=

E lR,

9. The equation is equivalent to

X - ..;x + y

+ - 2vz-=-2 + u - ..jU + v - ..;v = 0, or (VX- D 2 + (VV - D 2 + (vz-=-2 - 1)' + + (VU - D 2 + (vv - D 2 = 0 Because x, y, U, v are real numbers, it follows that 1 ..;x = VY = ..jU = ..;v = "2 and vz-=-2 = 1. Hence X = Y = u = v = 4'1 = 3. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1974), pp. 47, Problem 2002; Gazeta Matematica ( GM-B ) , No. 10(1974), pp. 560, Problem 14536) -

VY

z

Z,

Z

22

1.2.

1. ALGEBRA

10. Setting X = x + 1 and Y = y - 1 yields

(X + y)2 = XY or � [X2 + y2 + (X + y) 2] = o. Hence X = Y = 0, so the solution is x -1 and y = 1. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1(1977), pp. 40, Problem 2811) =

11. The equation is equivalent to

� x + V4X + V16X + l·· + v'4nx + 3 = v'Z + 1

Summing up relations

SOLUTIONS

(1) and (2) we obtain Jx + b Jx + a + b - c,

23

=

a = c. b = c, x = -b. b a c, b c a = c, x = -a 2(1978), pp. 26, (Titu Andreescu, 3017) 13. Because a and b are distinct numbers, x and y are distinct as well. The second

and then To conclude, we have found that (i ) If then the equation has the solution ( ii ) If f:. c and f:. there is no solution. ( iii ) If f:. and then is the only solution. Revista Matematica Timi§oara (RMT ) , No. Problem

equation could be written as

Squaring the equation yields

V4x + V16x + l·· + v'4nx + 3 = 2v'Z +

a b. We have a22 b22 = b22y22 + 2b2Y (X24__ y24 )3 + (X4 _ y4)2 a x = b x + X (X y ) .

1

and the system could be solved in terms of and

Squaring again implies

V16X + l·· + v'4nx + 3 = 4v'Z + 1

Subtracting the first equation from the second yields

Continuing this procedure yields

4nx + 3 = 4nx + 2 · 2nVX + 1 1 and 2 . 2n Vx 2 . Hence x = n ' 4 (Titu Andreescu, Revista Matematica Timi§Oara (RMT) , No. 4-5(1972), pp. 43, Problem 1385) =

12. We distinguish two cases:

1) b = c. The equation becomes Jx + a+ Jx + b = Jx + a, so x:= -b. 2) b f:. c. The equation is equivalent to Jx + b + Jx + c = Jx + a + b - c - Jx + a, or b-c b-c Jx x + a + b - c + Jx + a ' + c Jx + b J so Jx + b - Jx + c = Jx + a + b - c+ Jx + a.

(1)

which reduces to

b b = y3 + 3x2 y a x3 + 3xy2 ) 3 3 2 2 b y - x y a = x - xy ). a = x3 +3xy2 a x(x2 _y22 ) 3b y(y2 _ X2) 3 3 b 3x y + y . a + b = (x + y) a - b = (x - y) . x + y = -Va + b x - y = -Va - b and its unique solution is x = (-Va + b + -Va - b)/2, y = (-Va + b - -Va - b)/2. (Titu Andreescu, Korean Mathematics Competitions, 2001) 13x-+-4 = y, Y It follows that 14. Let 3 3y - 4 x ---u-

Solving the quadratic equation in yields ( and = ( and or = The second alternative is not possible because = and = cannot be both positive. It follows that The system now and = Hence and becomes

E Z.

(2)

=

24

[ �(3Y : 4) - 2] = y,

and the equation is equivalent to

or

[75Y 52- 126] = Y Using that for any real number a, [a] :s; a < [a] + 1, we obtain y:S; 75y 52- 126 < y + 1 , 126 178 or 126 :s; 23y < 178, so 23 :s; y < 23· Note that y Z, therefore y = 6 or y = 7, thus Xl = 1314 and X2 = 1317 ' are the desired solutions. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 3(1972), pp. 25, Problem 1552) 1 15. From a -> 1 +V5 2 we obtam. a2 - a - I -> 0, or a -> -a + 1. We have 1 +-na-2 ] = 1 + na - a, O:S;a < 1. [a a E

---

Hence

That is because and

[ 1 + [�] l = [ l + �:na - a]

= [(1 + � - aH + n] = n, n � O. (1 + -a1 - a) a-1 :s; (a - a)-a1 = 1 - a-a < 1 (1 + a1 ) -a1 > -a12 > 0. -

- Q

(Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1978), pp. 45, 3479) 16. First we consider the case when X + y + z = 0. Then x3 + y3 + Z3 = 3xyz and

Problem

the ratio equals

�, as desired.

1 .2 . 2xyz3 - (x3+ y +3 z) = -,2 x +y +Z 3

25

SOLUTIONS

1. ALGEBRA

Conversely, if

:--�-�---'--:

then and so

2(x3 + y3 + Z3 - 3xyz) + 3(x + y + z) = 0. U sing the formula x3 + y3 + Z3 - 3xyz = (x + y + z)(x2 + y2 + Z2 - xy - yz - zx), we obtain by factorization that

(x + y + z)[2(x2 + y2 + Z2 - xy - yz - zx) + 3] = and so (x + y + z)[(x - y)2 + (y - Z)2 + (z - X) 2 + 3] = 0. Because the second factor is positive, it follows that x + y + z = 0, as desired. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1(1973), pp. 30, Problem 1513) °

17. We write the equation as

Xl - 2VXI - 1 + X2 - 2 · 2JX2 - 22 + . . . + Xn - 2nJxn - n2 = 0, or (VXI - 1 - 1)2 + (.JX2 - 22 - 2) 2 + . . . + ( Jxn - n2 - n) 2 = 0 . Because the numbers Xi , i = 1, n are real, it follows that Xl = 2, X2 = 2 . 22 , . . . ,xn = 2n2 (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1(1977), pp. 14, Problem 2243) 18. Using the identity

(a + b + C) 3 = a3 + b3 + c3 + 3(a + b)(b + c)(c + a) we obtain (� + � + l) " = � + � + 1 + 3 (� + �) (1 + �) (1 + �) = = 9 + 1 + 54 = 64 Hence

1.2.

1. and so 1 +-1 = 3. fIX * The system is now reduced to .!.+.!. x y =9 1 + 1 = 3, fIX *

26

ALGEBRA

and consequently

{

which is a symmetric system, having the solution

x = -,81 y = 1

and

x = 1, Y = -.81

(Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 4-5(1972), pp. 43, 1386)

Problem

19. By summing up the equations of the system we obtain It follows that

(4x2 - 4x + 1) + (4y2 - 4y + 1) + (4u2 - 4u + 1) + +(4v2 - 4v + 1) + (4w2 - 4w + 1) = O.

5 t a� - (t al) " � 50 5 5 Note that I: ai = 0 and so I: ar ::; 10, as claimed. i=l Romanian Mathematica i= l Olympiad - second round 1979; Revista (Titu Andreescu, Matematica Timi§oara (RMT) , No. 1-2(1980), pp. 61, Problem 4094) 21. The inequality (a + b)2 ;::: 4ab yields 1 1 4 a+b a b - -1 1 1 1 4 4 and , sI. mI. larly' - + - -- ' - + b e b+e e a e+a Summing up these inequalities yields 1 1 1 1 1 1 2a + 2b + 2e ;::: a + b + b + e + e + as desired. (Dorin Andrica, Gazeta Matematica (GM-B), No. 8(1977), Problem 5966) -+- >

-

x,y,u,v,w are real numbers, we obtain x = y = u = v = w = 21 (Dorin Andrica, Gazeta Matematica (GM-B), No. 8(1977), pp. 321, Problem 16782)

- > -- .

a

a5 + b5 + e5 = (a + b + e) 5 - 5(a + b)(b + e)(e +

l

ai - aj l ::; lai - ai+l l + . . . + laj-l - j l ::;

2 2 i 1 and (2a - D (2b - D (20 - D > 3. From the relation (1) and using abc = 1 we obtain 3 > 2(a b c) - ( � � � ) a b c +

+

1 ( 2)

( )

+

+

+

+

and so

2

-

-+

which is a contradiction. The proof is complete. Revista Matematica Timi§oara (RMT) , No. ( Problem )

2 1986), pp. 72,

24. Assume by contradiction that all numbers are greater than 1/4. Then

d - a2 > 41 41 41 41 hence 0> -ar G-br G-or G-dr G This is a contradiction so the claim holds. a - b2 b - c2 c +

+

�J2

- a- +

-+-+-+-

+

+

+

(Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1(1985), pp. 59, Problem 5479) 25. Setting ai = sin2 ai for i = 1, n, where aI, a2 , . . . , an are real numbers, the expression becomes n 2 E =

2:: t/sin

i=l

ai cos2 ai+b an+1 a1· =

Using the AM-GM'inequality yields

k -k1 2:: bf ;::: b1 b2 . . . bk , bi > 0, i U. i=l For b1 sint ai, b2 = cost ai+1 and b3 = b4 bk = � we obtain ij2 =

=

= ... =

2

,

1e- 2

-

;; and it is reached if and only if a1 a2 = . . . an 1

Hence the maximum value of E is

=

=

=

2'

Problem

+

(Titu Andreescu, 5982

1e

-

(Dorin Andrica, Revista Matematica Timi§oara (RMT), No. 1(1978), pp. 63, 3266)

2(a b c) ( a1 -1b -c1 ) > 3 +

n.

,

n(k

---

On the other hand, relation ( ) gives +

1 ,2 ...,n yields 2) _l � (n k 2 ) > 2-;;2; n n · 21 t E< = --2 {14 2

Summing up these relations for k =

(3)

.

+

29

SOLUTIONS

26. Because x and

m

are positive, we have to prove that

x(xmn - 1) - m(xn - 1) ;::: 0, or (xn - 1)[(xn) m-1 x (xn) m-2 x x - ] � 0. 1 x (xn ) m-2x x - and note that if x � 1, then Define E (x) = (x n ) m xn ;::: 1 and E (x) � 0, so the inequality holds. In the other case, when x < 1, we have xn < 1 and E (x) < and again the inequality holds, as claimed. + ... +

+

+ .. . +

+

m

m

°

(Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1978), pp. 45,

Problem 3480)

27. For

[:].

m ::; n the inequality is clearly true, so consider m > n and define This implies that m = pn + q with q E {O, ...,n - I} and the inequality can be written as (pn + q ! � (n!) P.

p=

1,

)

We have

=

(1 ·

1

)

(pn + q ! � (pn) !

=

2 ... n) (n + ) ...(2n) ... ((P - l)n +

and we are done.

1) . . . (pn)

;::: (n!) P,

(Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1977), pp. 61,

Problem 3034)

28. We will use the inequality

xmn >- n_m1-1_ (Xl X2 Xn )m , which holds for all positive real numbers X 1 ,X2 , . . . ,X n and all (-00 , 0] [1 ,00) . Xm1 x2m +

+ .. .

+

+

+ .. . +

m

E

U

30

1. ALGEBRA

Xl 1, X2 2, . . . , Xn n and _ .!..n We obtain 1 1 . . +-1 >-1 l [n(n+l)l-� n n{;f;' 1 +-+-+ n+l y'2 V'3 vn 2 n- �

Now set

=

=

Summing up these inequalities yields

m =

=

.

=

-

_

n n (a a a ) n- 1 Llogai ga tn 2: L o l n i=l l 2 . . . n i= l n-l-[n + (logal a2 + loga2 al) + . . . + (logal an + logan al) + . . . + n +(logan an-l + logan_l an )]. 1 Note that a + > a - 2 for all a 0, so n-l n + 2(n - 1) + 2(n - 2) + . . . + 2] (n-l)n, � logai t n 2: --[ � n i= l =

(Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1974), pp. 52, 2035)

Problem

-

29. From the AM-GM inequality we deduce

n 1� � i + 1 > TI i +. 1 v n + l, n i=l i= l n1 1 + n1 � i=l � > \In + 1, n + 1 - 1) ' 1 +-21 + -3l + .. . + -nI > n (\In -

or

.-_

�

-

and so

�

-�

=

as claimed.

(Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1977), pp. 62, 3038)

Problem

-;- -

31. We begin with the following lemma.

or

i 1 , i 1, n,

n-l � 1 1 2 1 +-+ 2 -3 +n· · · +-n > n n _vn_ . =

n (1 - _vn1_) + 1 > 1 + !2 + . .. + .!..n

(Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1977), pp. 62, Problem 3037) 30. Because the numbers aI, a2 ,. " ,an are positive, from the AM-GM inequality al + a2 + . . . + an 2: '\I'al a2 . . . an nl n we deduce that tn � (ala2 ' " an ) � Using that numbers ai are less than 1 we obtain n-l (al a2 . . . ) logai tn 2: --loga. n .

•

an .

Let a > b be two positive integers such that �-�> 1. Then between numbers a and b there is at least a perfect cube. Proof. Suppose, for the sake of contradiction, that there is no perfect cube between a and b. Then there is an integer c such that c3 � b < a � (c + 1) 3 . Lemma.

as desired. Observe that the inequality is strict because the numbers � = are � distinct. In order to prove the first inequality we apply the AM-GM inequality in the form

Therefore

>

=

,.�

_

=

i

_

as desired.

�

31

1.2. SOLUTIONS

This means so which is false. 0 Now we can easily check that for If 2: then

n 10, 11,12, 13,14, 15 the statement holds. n> (2,5)3 (1,4 1- 1) 3 > ({13 1_ 1) 3 ' or � 1 n> {I3-r' Hence � - � 1, and using the above lemma the problem is solved. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1-2(1990), pp. 59, Problem 4080) n 16,

=

=

>

32

1.

1. 2 .

32. Note that the number k(k 2+ 1)

.

= 4p + 1 or k = 4p + 2 and is

= 4p + 3 = 4p, p = 4m, - 1 - 4p- 2 + 4p+ 3 + 4p+4) = 4m. k = �(-4p Sn =�(-1) k =1 p =O ii) if n = 4m + 1, then Sn = 4m - (4m + 1) = -1. iii) if n = 4m + 2, then Sn = 4m - (4m + 1) - (4m + 2) = -(4m + 3). iv) if n = 4m + 3 then Sn = 4m - (4m + 1) - (4m + 2) + (4m + 3) = o. Hence 4m if n = 4m 4m + 1 Sn = �-1(4m + 3l ifif nn == 4m +2 if n = 4m + 3 . (Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 1(1981), pp. 50, Problem 4303) even for k or k where We have the following cases: i) if n then m- l n k(k+l) "'"'

"'"'

2

{

33. a) Summing up the identities

(n +k 2) = (n +(n2 -+ k)2)((nn ++ 1)1 - k) (nk)

= 0 to k = n yields (n +k 2) _ (nn ++ 21) _ (nn ++ 22) ) = Sn - (n + 2)(n1 + 1) (� k =O 1 2 n+2 - (n + 3) [2 n+2 (n + 2) _ 1] . (n + 2)(n + 1) (n + 2)(n + 1) b) Summing up the identities + 2)(n + 1) (n) (n +k 3) - (n + 3(n- +k) 3)(n (n + 2 - k) (n + 1 - k) k for k = 0 to k = n yields 1 n - (n + 3)(n + 2)(n + 1) for k

_

_

_

T.

_

_

SOLUTIONS

n

odd for k is a positive integer. IS

33

. (�( ; 3) -(�:D -(�:!) -(�:�)) = 1 (2n+3 - 1 (n2 + 3n + 2) ) (n + 3)( n + 2)(n + 1)

ALGEBRA

_

2

_

n+ 4 3n + 2) - 2(n2 + -3)(n(n2++2)(n + 1) .

_

(Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1975), pp. 43, 2116) 34. Let Sn be the number in the statement. It is not difficult to see that Sn = 41 [(2 + V3) 2n+l + (2 - V3) 2n+l ] . The required property says: there exists k 0 such that Sn (k - 1) 2 + k2 , or, equi valently, 2k2 - 2k + 1 - Sn = o. The discriminant of this equation is � = 4(2Sn - 1), and, after usual computations, we obtain e +l 1- H r = l + .J3)2n ; ( .J3)2n

Problem

=

>

A

Solving the equation, we find 2 n+1 k=

+ (1 + v'.3)2n++l + (1 - V3)2n+l

2n 2 Therefore, it is sufficient to prove that k is an integer. Let us denote v'3) v'3) where is a positive integer. Clearly, is an integer for all We will prove that Moreover, the numbers divides satisfy the relation

Em

Em (1 + m + (1 - m , [ ] m Em , m = 1,2,3, . . . 2� Emm . Em = 2Em-1 + 2Em-2 •

The property now follows by induction. Romanian IMO Selection Test,

(Dorin Andrica,

1999)

35. Differentiating the identity sin nx yields

= sinn X ((�) cotn-1 X - (;) cotn-3 X + (�) cotn-5 X -

n cos nx where

= n sinn- 1 x cos xP(cot x) - sinn Xsm �X PI (cot X),

• • •

)

1.

34 For

1. 2.

ALGEBRA

x = "4 we obtam v'2) n ( ncosn"47r = 2 (nP(l) - 2P'(1)) . 7r

•

we have

nP(l) = n (�) - n (;) + n (;) - . . . -2P'(1) = -2(n - 1) (�) + 2(n - 3) (;) - 2(n - 5) (;) + . . . ,

nP(1) - 2P'(1) = - [(n - 2) (�) - (n - 6) (;) + (n - lO) (;) - ... J = = -n (G) - (;) + (;) - . . . ) + 28n • To conclude, use that (�) (;) + (;) _ '" = (V2) n sm n; , _

hence

v'2 n )n

Sn - 2 (cos ""4n7r + sm. ""4n7r ) (Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1977), pp. 89, Problem 3200) 36. Differentiating with respect to x the identities (x + l) n = (�) + (�) x + . . . + (:) xn and (x _l) n = (:) xn - (n � l) xn-. + . . . + (-1)" (�) _

(

yields

and

and

and

n(x + l)n-. + n(n - l)x(x + 1)"-2 = (�) + 22 (;) X + . . . + n2 (:) x"- ' n(x - 1)"-' + n(n - l)x(x - 1) n-2 = n2 (:) x"- ' - (n - 1)2 (n � 1) X"-2 + + . . . + (-1)"- ' (�) Setting x = 1 yields 12 (�) + 22 (;) + . . . + n2 (:) = n(n + 1)2"-2 Summing up the last two identities gives

Sn = 12 (n1 ) + 32 (n3 ) + . . . = n(n + 1)2n-3 ,

as desired.

(Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 1(1978), pp. 90, 3438)

Problem

37. Note that

n_12i+1_1 [log k] + [log 2n], 2 2 = k= k= 2i and [log2 k] i for 2 i � k < 2i + l . Hence 2" n-l 2:l �og2 k] = 2: i . 2i + [log2 2n] = (n - 2)2n + n + 2 i=O k= 2>1

2:l [log2 k] = 2: i O 2:

=

as claimed.

(Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1981), pp. 63, 4585; Gazeta Matematica (GM-B ) , No. 2-3(1982), pp. 83, Problem 19113) 38. Let Yn = 22 >1 - 1 for all positive integers n . Then 1 - 2 = 2 1 - 2 +21 = (22)1)2 - 2 .)122" + 1 = Yn Yn+l 2 >1 - 1 2 >1 - 1 (22)1 - 1)(22 +1 - 1) )1 - (22)1 _(221)(22_ )1+11) 2 _ 1) - (222)12 )2--1 1 = 22>1 1+ 1 = x1n and therefore 1 1 2 Problem

n(x - 1} "-' = n (:) x n-. - (n - 1) (n : 1) xn-2 + . . . + ( _ 1) "- ' (�). Multiplying by x gives nx(x+ 1) "-' = (�) x + 2 (;) x2 + . . . + n (:) xn

35

Differentiating again we obtain

Because and

SOLUTIONS

_

_

>1

1.

36

1.2.

ALGEBRA

Consequently,

--

=

-

f(iz)f(-z) =

_ Z2 .

C,

=

(_

)

C

(Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1976), pp. 56, 2583) 40. Setting X = Y = 1 yields f2 (1) +f2 (a) 2f (1 ) and {f(l) - 1 )2 = 0 so f(l) = 1. Substituting Y = 1 gives f(x)f(l)+f (�) f(a) = 2f(x) Problem

x.

or

Y Xa and observe that =

-

f(x)f (�) +f (�) f(x) = 2f(a).

f(x)

f

1 12(1977), 45, 18455) 10(1980), 439,

f

(Titu Andreescu, 2849;

T 0 f(x

+

It follows that

T) =

f(x) or sin[x +T] = sin [x] , for all x E

[x +T] - [x] 2k(x)7r, X =

k

where : 1R -+ Z is a function. Because all E 1R and therefore

x

7r

R

E lR,

is irrational, we deduce that

[x] [x +T] for x all

=

k(x) 0 for =

E 1R

which is false, since the greatest integer function is not periodical. Revista Matematica Timi§oara ( RMT ) , No. Problem

(Dorin Andrica, 3430)

1(1978), pp. 89,

42. Considering the determinant

=

Take now

12 (0) +12 (�) 2/(t)

= and because the left-hand side is positive, it follows that is positive and for all Then is a constant function, as claimed. pp. Revista Matematica Timi§oara ( RMT ) , No. Problem Problem pp. Gazeta Matematica ( GM-B ) , No.

41.

f(iz) = 0 or fez)+f(-z) = o . From the relation f(z)f(iz) = Z2 we deduce that fez) = 0 if and only if z = o. Hence if z f:. 0, then f(iz) f:. 0 and so fez) + f( -z) = 0 and, if z = 0, then fez) + f(-z) = 2f(0) O. Clearly, fez) + f(-z) = 0 for all numbers z E as desired. Remark. A function f : -+ satisfying the relation f(z)f(iz) = Z2 is fez) = V22 +iV22 z. C

> o. = 0, that gives

The function is not periodical. Suppose, by way of contradiction, that there is a number > such that

f(iz)(f(z)+f( -z)) = 0,

so

=

=

39. Substituting z with iz in the relation f(z)f(iz) = Z2

Summing up gives

x Y

--

(Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 1-2(1980), pp. 67, Problem 4135)

yields

f(x)f (�) = 1,

f2 (X) = 1, x

therefore Now set

xn Yn Yn+l Summing up these relations yields 1 + -2 + -22 +. . . + 2n-l 1 - 2n < ­1 xn Yl Yn+l Yl Xl X2 X3 for all positive integers n, as desired.

37

SOLUTIONS

we have

12 22 32 n 2 2 1 n 3 8= I n 2 n 3 n nn 11 122 133 In 2 n = 82 , � = IS(i,j)1 = 8· 2 2 2 n

l n2 n3

n

n

because the second determinant is obtained from 8 by interchanging rows and columns.

1.

38

1.2.

ALGEBRA

On the other hand,

1 1 8 = n!

1 n- 1

1 2

1 3

Because the coefficient of X4 is equal to

1 n

2n-1 3n-1

43. The determinant is al

0 0 �2n = 0 0

b2n

0 a2 0 0 b2n-l 0

0 0

a3 b2n-2

0 0

0 0

0 b2 b3 0 0 a2n-2 0 a2n-l 0 0

b1

0 0 0 0

a2n

y x v z

z v z x y y x

=

a b e d

b a d e

w

=

(

1, we have = 1 and so ..\

x + y + z + v) (x + y - z - v) (x - y + z - v) (x - y - z + v)

b) As shown above, we have

(here we used the known result on Vandermonde determinants). Therefore

(Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 1(1982), pp. 52, Problem 3862)

x y z v

39

SOLUTIONS

�

e d a b

d e b a

= (a b + e + d) (a + b - e - d) (a - b + e - d) (a - b - e + d) +

1000,

100,

On the other hand, multiplying the first column by the second by and adding all these to the fourth, we obtain on the last column the third by the numbers abed, bade, edab, debao Because all those numbers are divisible by the prime number p, it follows that p divides � and therefore p divides at least one of the numbers a b e d, a b - e - d, a - b e - d, a - b - e d.

10

+++ + (Titu Andreescu)

+

+

x

x

Because the quadratic polynomials h ( ) and t2 ( ) have zeros of the same nature, it follows that their discriminants have the same sign, hence

45.

Expanding along the first and then the last row we obtain

� 2n which gives

�2n

= (al a2n - bl b2n)�2n-2 '

= k=lITn (ak a2n-k+l - bk b2n-k+l )

(Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1977), pp. 90, Problem 3201; Gazeta Matematica (GM-B) , No. 8(1977), pp. 325, Problem 16808) 44. a) Adding the last three columns to the first one yields that x + y + z + v

divides the determinant. Adding the first and second columns and subtracting the last two columns implies that divides the determinant. Analogously we can check that and divide the determinant, and taking into account that it has degree 4 in each of the variables, the determinant equals

x +y - z- v

where

..\

x-y+z-v x-y-z+v

..\(x + y + z + v) (x + y - z - v) (x - y + z - v)(x - y - z + v),

is a constant.

Consequently,

(pIP2

+ 4ql q2 )2 - 4(plq2 + P2Ql )2 2:: O.

Note now that the left-hand side of the inequality is the discriminant of the quadratic polynomial and the conclusion follows. Revista Matematica Timi§oara (RMT), No. pp. Problem Gazeta Matematica (GM-B), No. pp. Problem

(Titu Andreescu, 3267;

t

5(1979), 191,

1(1978), 63, 17740)

46. Because the quadratic polynomial T has nonreal zeros, the discriminant � is negative. a bserve that

= b2 e2 - 4a (b3 + e3 - 4abe) =

0,

� ( b2 - 4ae) (e2 - 4ab) < where � l b2 - 4ae and �2 e2 - 4ab are the discriminants of the quadratic polynomials Tl and T2 • Hence exactly one of the numbers � l and � 2 is negative and since a > the conclusion follows .

= = 0, (Titu Andreeseu, Revista Matematica Timi§oara (RMT), No. 1(1977), pp. 40, Problem 2810)

1. 47. Observe that al +a2 + . . · +an and al -a2 . . . + ( _ l) n -l an are real numbers, that is P(l) and P( - 1) are real numbers. Hence (1) P(l) = P(l) and P( -1) = P( -1) Because P(x) = (x - xd . . . (x - x n ), the relations (1) become (1 - xd . . . (1 - xn ) = (1 - xd . . . (1 - xn) and (1 + xd . . . (1 xn ) = (1 xd . . . (1 + xn )

1.2.

ALGEBRA

40

+

+

+

Multiplying these relations yields

(1 - xi) . . . (1 - x� ) = (1 - xi) . . . (1 - x�), or Q(l) = Q(l). Therefore b1 b2 + . . . + bn is a real number. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1(1977), pp. 47, Pro blem 2864) 48. Because P(O) 0, there is a polynomial Q with P(x) xQ(x) . Then Q(k) = +1 l ' = 1,n. Define H(x) = (x + l)Q(x) - 1. It is clear that degH(x) = n and H(k) = 0 for all = !,n, hence H (x) = (x + l)Q(x) - 1 = ao(x - l)(x - 2) . . . (x - n) (1) Setting x = m n in relation (1) yields Q (m) = ao (m - 1) (m -m 2)+ .1. . (m - n) + 1 . On the other hand, setting x = -1 in the same relation implies ao = ((n_ l)+n+I)!l Therefore - 2) . . . ( - n) + -1 Q(m) = ( _ l)n+l (m(n- l)(m l)!(m + 1) m 1 and then - 1) . . . (m - n) m ' P(m) - ( _ l)n+l(nm(m + l)!(m + 1) + -m+l (Dorin Andrica, Gazeta Matematica (GM-B), No. 8(1977), pp. 329, Problem 16833; Revista Matematica Timi§oara (RMT) , No. 1-2(1980), p. 67, Problem 4133) +

=

=

k

k

k

m,

>

m

+

49. We are looking for a polynomial with integral coefficients

P(x) = aoxn + alxn- 1 + . . . + an , ao i= O.

+

4

SOLUTIONS

1

We have

P'(x) = naoxn- 1 + (n - 1)alxn-2 + . . . + an- l and by identifying the coefficient of x(n -l) n in the relation P(P'(x)) = P'(P(x)), we obtain or

aonn- 1 = 1. Hence

ao = nn1- l and since ao is an integer, we deduce that n = 1 and ao = 1. Then P(x) = x aI, P'(x) = 1 and P(P'(x)) = P'(P(x)) yields 1 + al = 1 or al = O. Therefore P(x) = x is the only polynomial with the desired property. (Titu Andreescu, Revista Matematica Timi§oara (RMT), No. 1-2(1979), Problem 3902) 50. Let (}l , (}2 , . . . , (}n be the roots of the equation xn + xn- 1 . . . + x + 1 = O. They are all distinct and Or+1 = 1, i = !,n. Because P(x) is divisible by xn + xn - 1 + . . . + x + 1, it follows that P( Oi ) = 0, i = !,n, hence PI (1) (}1P2 (1) + . . . + O�- lPn (l) = 0 PI (1) + 02 P2(1) + . . . + O�- lPn (l) = 0 +

+

+

The above system of equations has the determinant

1 01 on1 --l1 V = 1 02 0'2 1 On (}nn- l Because all of the numbers 01 , 02 , . . . , On are distinct, it follows that V i= 0 and so the system has only the trivial solution PI (1) = P2 (1) = . . . = Pn (l) = O. This is just another way of saying that x - I divides pi(X) for all i = 1, n. (Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1977), pp. 75, Problem 3120; Gazeta Matematica (GM-B), No. 8(1977), pp. 329, Problem 16834)

1.

42

1. 2.

ALGEBRA

51. Consider the determinant =

an nII+l (ak - a,) 10 , 1 = 1 10 > 1

an-I, the third by an-2 , ... , the last by ao and P(ad P(a2 ) P(an+d a +l al a2 v= an;+l = an nII+l (ak - at) a� a� a�+l af a� On the other hand, P (ar ) 0 (mod p), for all r = 1, n 1 and an 0 (mod p) II< +l (ak - a,) 0 (mod p). Therefore there are at least two numbers implies l r, therefore Q(x) = 0 and P(x) = xm , for n even and P(x) = ±xm , for n odd. Alternative solution. Let degP(x) = m and let P(x) aoxm alxm- l am . If P(x) = x k Q(x) with k a positive integer, then xkn Qn (x) = xkn Q(xn ) or Qn (x) = Q(xn ) Note that Q satisfies the same condition as P. Assume that P(O) f:. O. Setting x = 0 in the initial condition yields a� = am ' Then am = 1 if n is even and a m = ± 1 if n is odd. Differentiating the relations implies npn-l (x)P'(x) = nP'(xn)x n- l . (1) Setting now x = 0 gives P'(O) 0 and so am - l = O. Differentiating again in relation ( 1 ) yields analogously a m -2 = 0 and then am-3 = am-4 = . . . = ao = O. The polynomials are P(x) = xm , if n is even and P(x) = ±xm , if n is odd.

This is impossible if the polynomials are

V=

+

SOLUTIONS

It follows that

+

and by applying the triangle ' s inequality for complex numbers, we deduce that 1

L I xd l x21 .. · I xn-m l > (mn ) . Consider Xo = min l x I , I X 2 1 , . . . , I xn - m l } . Then {

+

so

Xo

<

1, as claimed.

I

1. ALGEBRA

44

1.2. SOLUTIONS

(Titu Andreescu, Revista Matematica Timi§oara (RMT), No. 2(1978), pp. 52, 3531)

Problem

54. We have

PI(X) = -;-n2 . P(x) i= l - X·

�

L...J

p,

Li=l ln I P (x) - xii = n2 In C lx l ,

or

C>

0

IIi=nl IP(x) - xii = In Cn2 Ixl n2 Ill. (F(x) - Xi) I = klxl n' , k > 0 IP (P(x))1 = k l xl n2 . In

Hence

or

Eliminating the modules gives

P(P(x))

2 = AX n ,

A E JR.

P (x) = axn with a E (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1(1977), pp. 47, Problem 2863; Gazeta Matematica (GM-B), No. 1(1977), pp. 22, Problem 17034) 55. Define Q(x) = xP(x). Because an f:. 0, the polynomial Q has distinct real zeros, so the polynomial QI has distinct real zeros as well. Consider H(x) = XQI(X). Again, we deduce that HI has distinct real zeros, and since HI (x) = x2 PI (x) 3xPI (x) P(x) the conclusion follows. (Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1978), pp. 52, Problem 3530) R

Therefore

+

56. Let

m

a of order k. Since P(Q(a)) = 0, we have m ao II(Q(a) - Xi) = 0, i=l and so there is an integer 1 � � such that Q(a) - xp = O. Observe that = Q(a) - xp f:. 0, for all j f:. otherwise Xj xp , which is false. Hence Q(x) - Xj has the multiple zero a of order k and so QI(X) = (Q(x) - xp )' = QI(X) has a multiple zero of order k - 1. This concludes the proof. (Dorin Andrica, Romanian Mathematical Olympiad - final round, 1978; Revista Matematica Timi§oara (RMT) , No. 2(1978), pp. 67, Problem 3614) 57. Assume by way of contradiction that P(x) has less than two nonreal zeros. As a polynomial with real coefficients P(x) cannot have only one nonreal zero, hence all of its are real. Let Xl , X2 , . . . ,X n be the zeros of P(x). Then PI(X) - n _1_ P(x) � X - Xi has a multiple zero

p,

�

Integrating the equation yields n

+

= degP (x) and let

P(x) = ao (x - XI)(X - X2 ) . . . (x - xm ). Because P(x) E R[x], X l , X 2 , . . . , x m are distinct zeros. Now P(Q(x)) = (Q(x) - xt )(Q(x) - X2 ) . . . (Q(x) - xm )

45

p

m,

_

and differentiating we obtain

PII (x)P(x) - [PI(X)] 2 = - n

1 � (x - Xi)2 Setting X = a we reach a contradiction, therefore P(x) has at least two nonreal zeros, as claimed. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1-2(1979), pp. 59, Problem 3883) 58. Let P(x) = aoxn al xn- l . . . an be a polynomial with real coefficients. If all of its zeros are real, then the same is true for the polynomials pI , P", . . . , pCn -2) . Because (n - 2)! P n-2) (x) = -2- [n (n - l) ao x2 2(n - l)alX + 2a2] P2 (X)

+

+

+

C

+

is a quadratic polynomial with real zeros, we have or

(n - l)a� ;::: 2naO a2 . The reciprocal is not always true, as we can see from the following example: 2 + (a + l)x + a, = x3 + (a + with

a

P(x) E ( -00 -1] [2, (0). ,

U

1)x

1.

46

2

( (Dorin Andrica)

ALGEBRA

(

Observe that 2 a + 1) ;::: 2 . 3 a + 1), or (a + l) (a - 2) ;::: 0, so the inequality holds. On the other hand, P (x ) = ( x + a ) ( x2 + x + 1) does not have all zeros real.

59. For m = 0 the equation becomes

X4 - x3 x2 X 0 and has roots X l 0, X2 - 1, X3 X4 1. If m -:f. 0, we will solve the equation in terms of m. We have 2xm2 ( x2 - 2x3 l ) m X4 - x3 - x2 X 0 -

=

=

=

=

=

+

+

+

+

+

=

and

Chapter 2

It follows that

NUMBER THEORY

The initial equation becomes

[

2

x2 - 1 [m - ( x - x )] m - �

Hence and

]

=

0.

x 2 - x - m = 0, with solutions

=

1±� 2

2

=

m±

x - 2mx - 1 = 0,

Xl,2 with solutions X3 , 4

VI m2 . +

(Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1977) ,

Problem 3121)

pp.

75,

60. From the relations between the zeros and the coefficients we obtain L

XIX2 · · · X2n-1

=

2n and

XIX2 · · · X2n 1. =

Hence

2n 1 2n, Xk l k= X2n ' so we have the equality case in the AM-GM inequality. Therefore Xl X2 Since Xl X2 ... X 2n and 2n - > 0 "' ' �X k l = k X2 1. we have Xl X 2 . (Titu Andreescu, Revistan Matematica Timi§oara (RMT) , No. 2(1977) , 52, L

=

=

Problem 2299)

=

=

=

1

. .

-

= .. . =

1

=

=

pp.

PROBLEMS

1. How many 7-digit numbers that do not start nor end with 1 are there? 2. How many integers are among the numbers l·m 2·m -n ' -n-

p·m

where p, m, n are given positive integers?

3. Letn

divides 1 p

p +

2

n

> be a prime number and let n be a positive integer. Prove that n n 2p + . . l) p .

. + (p _

p

4. Prove that for any integer n the number 55

n +1

+ 55 n + 1

is not prime.

5. Let

n

be an odd integer greater than or equal to 5. Prove that

is not a prime number.

6. Prove that

345 + 456

102002 . 7. Find all positive integers n such that [\1111] divides 111. 8. Prove that for any distinct positive integers a and b the number 2a(a2 + 3b2 )

is a product of two integers, each of which is larger than

is not a perfect cube.

9.

Let p be a prime greater than 5. Prove that p of an integer.

10.

(x,y)

-

4 cannot be the fourth power

Find all pairs of nonnegative integers such that simultaneously perfect squares.

49

x2 +3y and y2 +3x are

50

2.

2.1.

NUMBER THEORY

11. Prove that for any positive integer n the number (17 + 120) n _ (17 _ 120) n

is an integer but not a perfect square.

22. Prove that the equation

4v'2

has infinitely many solutions in positive integers such that U and

24. Solve in nonnegative integers the equation

un+

Un

x + y + z + xyz = xy + yz + zx + 2

+5

25. Solve in integers the equation

+ 3n + 3

xy(x2 + y2 ) 2z4 .

15. Let p be a prime. Prove that a product of 2p + 1 positive consecutive numbers

2

cannot be the p + I-power of an integer.

16. Let p be a prime and let a be a positive real number such that pa2 Prove that [n0i - �] = [n0i+ �]

for all integers

<

=

26. Prove that for all positive integers n the equation

1

4'

n [ y'1 -a2avp] + 1.

x2 + y2 + Z2 59n =

has integral solutions.

27. Let n be a positive integer. Prove that the equations

2::

17. Let n be an odd positive integer. Prove that the set 18. Find all positive integers and n such that (:) = 1984. 19. Solve in nonnegative integers the equation m

x2 + 8y2 + 6xy - 3x - 6y

and

xn + y n + zn + u n = vn+ 1

have infinitely many solutions in distinct positive integers.

contains an odd number of odd numbers.

20. Solve in integers the equation

are both primes.

x2 (y - z) + y2 (z - x) + Z2 (x - y) = 2.

n 6 between and 1 there is a perfect square. 13. Prove that for all positive integers n the number n ! is not a perfect square. 14. Prove that if n is a perfect cube then n2 cannot be a perfect cube. 2::

v

23. Find all triples (x, y, z) of integers such that

12. Let (Un) n�l be the Fibonacci sequence: Prove that for all integers

51

PROBLEMS

=

3

(x2 + 1)(y2 + 1) + 2(x - y)(1 - xy) = 4(1 + xy) 21. Let p and q be prime numbers. Find all positive integers x and y such that 1 1 1 - + - = -. x y pq

28. Let n be a positive integer. Solve in rational numbers the equation

x n - 1 + yn - 1 . 29. Find all nonnegative integers x and y such that x(x + 2)(x + 8) = 3Y• xn + y n =

30. Solve in nonnegative integers the equation

(1 + x!)(1 + y!) = (x + y)!.

31. Solve the equation

x! + y! + z! = 2v! .

52

2.

2. 1 .

NUMBER THEORY

32. Find all distinct positive integers X l , X2 , . . . ,Xn such that

Xl + 2XIX2 + . . . + (n - 1)XIX2 . . . Xn-l XIX2 . . . Xn . 33. Prove that for all positive integers n and all integers aI, a2 , . . . , an , bl , b2 , . . . , bn the number n II k=l (a� - b�) 1+

34. Find all integers x, y,

v, t such that X + Y + + v + t xyvt + (x + y)(v + t) xy + + vt xy(v + t) + vt(x + y). 35. Prove that for all nonnegative integers a, b, d such that a and b are relatively prime, the system ax - yz 0 bx - yt + d 0 Z,

=

Z

Z

=

c,

-

=

c

=

has at least a solution in nonnegative integers.

36. Let p be a prime and let Xl , X2 , . . . ,xp be nonnegative integers.

Prove that if

Xl + X + . . . + x 0 (mod p) x� + x�2 + . . . + x�p 0 (mod p) Xp-l l + Xp-2 l + . . . + xp - I 0 (mod p) then there are k, l E {I, 2, . . . , p}, k =I l, such that Xk - Xl 0 (mod p) . 37. Prove that for any odd integers n, aI, a2 , . . . ,an , the greatest com­ mon divisor of numbers aI, a2 , . . . ,an is equal to the greatest common divisor of al -+--a2 ' -a + a an + al -2 2 -2--3 ---2-== ==

P

==

==

3.8 . Let O. 53. Let X l X2 X3 1 and Xn+3 Xn + Xn+I Xn+2 for all positive integers n . Prove that for any positive integer there is an integer k > 0 such that divides Xk · 54. Let (an ) n�O be the sequence defined by ao 0, al 1 and an+l - 3an + an- l ( _ l) n 2 for all integers n O. Prove that an is a perfect square for all n 2:: O. 55. Let al a2 97 and an+l anan-l V(a; - l)(a;_ 1 - 1), n > 1. Prove that a) 2 + 2an is a perfect square. b) 2 J2 2an is a perfect square. 56. Let k � 2 be an integer. Find in closed form for the general term an of the sequence defined by ao 0 and an - a [ �] 1 for all n O. 57. Let ao al 3 and an+l tan - an- l for n 2:: 1. Prove that an - 2 is a perfect square for all n 2:: 1. 58. Let a and {3 be nonnegative integers such that 0'2 + 4{3 is not a perfect square. Define the sequence (x n ) n �O by =

m

m

=

=

=

>

=

=

=

+

+

+

=

=

=

=

=

>

(4n

+

yields all odd positive integers less than or equal to

Prove that

=

=

±1 ± ± ± . . . ±

+

=

55

60. Prove that for different choices of signs + and - the expression

Xn in closed form.

=

2. 1 . for all integers n 2:: 0, where X l and X2 are positive integers. Prove that there is no positive integer no such that X� Xno-IXno+I ' 59. Let n > 1 be an integer. Prove that there is no irrational number a such that \/'a + Ja2 - 1 + \fa - Ja2 - 1 is rational. PROBLEMS

NUMBER THEORY

1),

(2n 1) (4n +

+

1).

SOLUTIONS

1. The problem is equivalent to finding the number of functions

{1,2,3,4,5,6, 7} {0, 1,2, . . . ,9} such that 1(1) f. 0, 1(1) f. 1 and 1(7) f. l. Because 1(1) E {2, 3, . . . , 9}, there are 8 possibilities to define 1(1) . For 1(7) there are 9 possibilities and for 1(2),1(3),1(4),1(5),1(6) there are 10 . To conclude, there are 8 . 9 . 105 72 . 105 numbers with the desired property. (Dorin Andrica, Gazeta Matematica (GM-B) , No. 11(1979), pp. 421, Problem 17999) I:

-t

=

m and n. Hence m ml d and n nl d 1 · ml , 2 · ml p · ml nl nl nl and, since m" are relatively prime, there are [ :, ] integers among them. Because gCd(m, n)p] .mtegers. n n It lo11ows that there are [ nl d gcd(m, n) n d

2.

Let be the greatest common divisor of and n l . for some integers The numbers are

=

ml

--

=

--

n,

=

-

'

=

.r

(Dorin Andrica, Gazeta Matematica (GM-B ) , No. 11(1979), pp. 429, Problem 0:89)

pn and note that k is odd. Then dk (p d) k p[dk-l dk-2 (p d) . . . (p d) k-l ] Summing up the equalities from d 1 to d [�] implies that p divides 1 k 2k (p - l ) k , claimed. 3. Define k

=

+

=

_

_

_

+

+

=

=

as

+ ...+

_

+

(Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 1-2(1979), pp. 49, Problem 3813) 4. Define m 55n • Then 55n +1 55n 1 m5 + m 1 (m2 + m 1)(m3 - m2 + 1) 57 =

+

+

=

+

=

+

2.

58

1, (Titu Andreescu, 2001) 5. Let N = (�) - 5 (�) + 52 (�) - + 5n- ' (�). Then 5N = 1 - 1 + 5 (�) _ 52 (�) + 53 (�) _ . . . + 5n (�) = 1 + (-1 + W. Hence N = � W + l) = � [C2n + 1)2 - (2� n = = � [2n - 2 � + 1] [2n + 2� + 1] = = � [ (2 ";-' - 1r + 2n- ' ] [ (2 ";-' + 1r + 2n-' ] . Because n is greater than or equal to 5, both factors of the numerator are greater than 5. One of them is divisible by 5, call it 5N1 , N1 > 1, the other being N2 . Then N = N1 N2 , where N1 and N2 are both integers greater than 1, and we are done. (Titu Andreescu, Korean Mathematics Competition, 2001) 6. The given number is of the form m4 + � n4 , where m = 344 and n = 4� = 2� .

(Titu Andreescu)

and, since both factors are greater than the conclusion follows. Korean Mathematics Competition, ' "

The conclusion follows from the identity

2m - mn + in1 2 > n ( in1 - m) = 2 �2 (2 562_1 - 344 ) > 61 . �( 54 � 1 ( 56 -1 -5 1

and the inequalities

> 2 2 2 2 - 2 2 4 ) > 2 2 . 25 2 2 2 2 - 1 ) > > 210. � 251 2 > 21 0.54 . 210.50 > 103.54 . 103'50 > 102002 (Titu Andreescu, Korean Mathematics Competition, 2002) 7. The positive divisors of 111nare 1, 3, 37, 111 . So we have the following cases: 1) [yl111] = 1 or 1 $ 111 2 , hence n � 7. 2) [yl111] = 3, or 3n $n 111 4n , son n = 4. 3) [yl111] = 37, or 37 $ 111 38 , impossible. 4) [yl111] = 111, or 111n $ 111 112n , and so n = 1. Therefore n = 1, n = 4 or n � 7 . .

<

<

2. 2.

NUMBER THEORY

<

<

59

SOLUTIONS

8. Note that

2a(a2 + 3b2 ) = (a + b)3 + (a - b)3 The Fermat equation for n = 3 x3 +y3 = z3 has no solution in positive integers (see T. Andreescu, D. Andrica, "An Introduction to Diophantine Equations" , GIL Publishing House, 2002 , pp. 87-93) . Hence there is no integer c such that

if

a > b.

On the other hand, if

b > a then there is no integer

c

such that

This concludes the proof. Revista Matematidi Timi§oara (RMT ) , No. Problem

(Titu Andreescu, 1911)

1(1974) , pp. 24,

4 = q4 for some positive integer q. Then p = q4 + 4 and q > 1 . We obtain p = (q2 2q + 2)(q2 + 2q + 2), a product of two integers greater than 1, contradicting the fact that p is a prime. (Titu Andreescu, Math Path Qualifying Quiz, 2003) 9. Assume that p

-

_

10. The inequalities

0 � x + y + 8, which is x2 +23y (x + 2 y2 + 3x (y + 2)2 is true. Without x + 3y (x + 2)2 . x2 x2 + 3y (x + x + 3y = (x + 1)22 , hence 3y2 = 2x + 1 . x = 3k 1 y = 2k + 1 k 0 y + 3x = 4k + 13k + 4. k > 5, (2k + 3)2 4k2 + 13k + 4 (2k + 4)2 so y 2 + 3x cannot be a square. It is easy to check that for k E {O, 1,2,3,4}, y2 + 3x is not a square but for k = 0, y 2 + 3x = 4 = 2 2 . Therefore the only solution is (x, y) = (1, 1) . (Titu Andreescu) cannot hold simultaneously because summing them up yields false. Hence at least one of 2 ) 2 or < < loss of generality assume that < From < < 2 ) 2 we derive Then + and for some integer � and so If then <

<

2. 11. Note that 17 + 12V2 (V2 + 1) 4 and 17 - 12V2 (V2 _ 1) 4 , so (17 + 12V2) n _ (17 - 12V2) n - (V2 + 1) 4n (V2 _ 1) 4n 4V2 4V2 2n 2 2 2 n n n 1) 1) 1) _ + (V2 (V2 1) (V2 + (V2 + 2 2V2 Define 2n - (V2 _ 1) 2n 2 2 A (V2 + 1) n +2 (V2 _ 1) n and B (V2 + 1) 2V2

60

2. 2.

NUMBER THEORY

=

If

=

n >- 2v'UUn-1+1 >y""n+1 - yUn - v'UUnn+1+1 -+ U..;u;;, n 1 1 U � . n 2y ....Un-1 2v3. fi)v'Un-1 > 1 and so between Un and U n+1 there is a perfect square. (Dorin Andrica)

_

�---=---!....-�-

that

( V2 + 1 ) 2n = X + yV2, (V2 - 1) 2n = X - yV2

Then X=

AB

(V2 + 1) 2n + (V2 - 1) 2n 2

and

and so is as integer, as claimed. Observe that

A B

so and are relatively prime. not a perfect square. We have

It

y such

2n - (V2 _ 1) 2n y - (V2 + 1) 2V2 _

is sufficient to prove that at least one of them is

2n 2 A (V2 + 1) n +2 (V2 - 1) =

[(V2 + 1) nV2 -

=

Since only one of the numbers

(y'2

- 1) n ] 2 + 1

(2)

l) n + (v'2 _ l) n (v'2 + l) n - (v'2 _ 1) n V2 V2 is an integer - depending on the parity of n - from the relations (1) and (2) we derive that A is not a square. This completes the proof. (Dorin Andrica, Revista Matematidi Timi§oara (RMT) , No. 1(1981), pp. 48, Problem 4285) 12. The claim is true for n 6 and n 7, because U6 8 9 U7 13 16 Us 21. =

---

=

=

<

<

=

<

<

If

n � 5,

n! + 5 5(5k + 1) for some integer k and therefore is not a perfect square, as (Dorin Andrica, Gazeta Matematidi (GM-B) , No. 8(1977), pp. 321, Problem 16781; Revista Matematica Timi§oara (RMT) , No. 1(1978), pp. 61, Problem 3254) 14. Suppose by way of contradiction that n2 +3n+3 is a cube. Hence n(n2 +3n+3) is a cube. Note that n (n2 + 3n + 3) = n3 + 3n2 + 3n = (n + 1)3 - 1 and since (n + 1) 3 - 1 is not a cube, we obtain a contradiction. (Dorin Andrica, Gazeta Matematica (GM-B ) , No. 8(1977), pp. 312, Problem E5965; Revista Matematica Timi§oara (RMT) , No. 1-2(1979), pp. 28, Problem 3253) 15. Consider the product of 2p + 1 consecutive numbers P(n) (n + l)(n + 2) . . . (n + 2p + 1) Observe that P(n) > (n + 1) 2p+1 . On the other hand, P(n) [(n + l) + (n + 2)2p++ . 1. . + (n + 2P + l) ] 2P+1 (n +p+ l) 2p+1 from the AM-GM inequality. If P(n) m 2 p+1 , then m {n + 2, ... ,n + pl. Assume by way of contradiction that there is k {2,3, . . . ,p} such that P(n) (n + k) 2 p+1 . Then (n + 1)(n + 2) ... (n + k - 1)(n + k + 1) ... (n + 2p + 1) (n + k)2P. (1) We have two cases: I. k p 1) If n 0 (mod p), then (n + k)2p is divisible by p2p . The left-hand side of the equality (1) is clearly not divisible by p2p , hence we reach a contradiction. 2) n r (mod p), r -::j:. 0, then the left-hand side of the equality (1) is divisible by p2 , because of the factors n + p - r and n + 2p while the right-hand side is not, since (n + k) 2p r2 P. This is a contradiction. II. k {2,3, . . . ,p- l} then desired.

=

<

(v'2 +

=

�

=

(1) and

.

13. If n = 1,2,3 or 4 then n! + 5 = 6,7, 11 or 29, so it is not a square.

.-- �= �--.-:..--=!...-

U sing the binomial expansion formula we obtain positive integers x and

u .�

� =

_

=

n � 8, then

61

SOLUTIONS

=

E

=

E

=

=

=

==

If

==

-

==

E

r,

62

2.

NUMBER THEORY

SOLUTIONS

1) If n -k (mod p), then the right-hand side of (1) is divisible by p2p , but the left-hand side is not. 2) n -q (mod p), q f. k and q {O, 1, . . ,p - I}, then the left-hand side of (1) is divisible by p. On the other hand (n + k)2p (q - k)2 (mod p) � 0 (mod p), because 0 < Iq - kl < p. Both cases end up in contradictions, so the problem is solved. (Dorin Andrica) ==

If

E

==

.

==

16.

[ JI -Q'2Q'vp] + 1.

It suffices to prove that there are no integers in the interval

(nvp - �n ,ny'P + �n ] for n �

Assume by way of contradiction that there is integer

ny'P- -Q'n < k If n � 3, then ( ) � ( ) ' so 1984 � n 6 3 m3 3m 2 + 2m or ( - 30)(m2 + 27m + 812) � -12456 < 0, and so < 30. This ( - 1) ( - n + 1) does not contam· the implies that ( ) f. 1984, because n! n factor 31 of 1984. To conclude, the solutions are 1984, n 1 and 1984, n 1983. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1(1985), pp. 80, Problem 5)

k such that

m m

=

n2p - 1 < k2 < n2p + 1 It follows that k2 pn 2 or Vp � , which is false, since p is prime. n (GM-B) , No. 8(1977), pp. 324, Problem (Dorin Andrica, Gazeta Matematica 16804) 17. For n 1 the claim is clear, so let n � 3. Define Sn (�) + (; ) + + (n�l) . Then =

=

=

=

...

S!,- 2n 1 - 1. Sn Sn ) 2(1984), 71, (Titu Andreescu, 5346) 18. Because (:) ( ': n) ' we can assume that m :5 [iJ . If n 0, then 1 1984, false. If 1, then 1984.

or = Because is odd it follows that the sum contains an odd number of odd terms, as desired. Revista Matematica Timi§oara (RMT , No. pp. Problem =

= n=

= m=

m

m m

m

m

mm

m

-

m

.

m

... m =

m=

m=

=

19. The equation is equivalent to

(x + 2y)(x + 4y) - 3(x + 2y) 3, (x + 2y)(x + 4y - 3) 3. =

or

=

(.) { xx ++ 4y2y - 33 1 , with solution (x, y) (2, -21 )

We have

=

1

so

=

m

-

63

=

=

. (x,y) (-4, 2'5 ) . (ii) { xX ++ 4y2y - 31 3 , with solutIOn Note that there are no solutions in integers, as claimed. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1(1971), pp. 20, Problem 312)

or

=

=

=

20. The equation is equivalent to

x2y2 2xy + 1 + x2 + y2 - 2xy + 2(x - y)(I - xy) 4, or (xy - 1)2 + (x - y) 2 + 2(x - y)(1 - xy) 4. Hence (1 - xy + x - y) 2 4 and, consequently, 1(1 + x)(1 - y)1 2. We have two cases: I. (1 + x)(1 - y) 2. Then a) 1 + x 2, 1 - y 1, so x 1, y O . b) 1 + x -2, 1 - y -1, so x -3, y 2. c ) 1 + x 1, 1 2, so x = 0, y = -1. d ) 1 + x -1, 1 - y -2, so x -2, y 3. II. (1 + x)(1 - y) -2 . Then _

=

=

=

=

=

= =

=

=

=

y

=

=

=

=

=

=

=

=

=

=

2. a) 1 + x = 2, 1 - y = -1, so x = 1, y = 2. b) 1 + x = -2, 1 - y = 1, so x = -3, y = O. c) 1 + x = 1, 1 - y = -2, so x = 0, y = 3. d) 1 + x = -1, 1 - y = 2, so x = -2, y = -l. (Titu Andreescu, Revista Matematica Timi§oara (RMT), No. 4-5(1972), pp. 43, Problem 1383)

64

NUMBER THEORY

21. The equation is equivalent to

65 2. 2 . Observe that ( x - y ) + (y - z ) = x -z . On the other hand, 2 can be written as a product of three distinct integers in the following ways i) 2 = (-1) . (-1) . 2, ii) 2 = 1 . 1 . 2, iii) 2 = (-1) · 1 · (-2). SOLUTIONS

a) We have the cases:

1) x pq = 1, y pq = p2 q2 2 , so X = 1 + y = pq(l + pq). 2) x -pq = p, y pq = pq , so X = p(l + q), y = pq(l + q). 3) x -pq = q,2 y pq = p2 2q, so x = q(l + p), y = pq(l + p) . 4) x pq = p , Y -pq = q , so X = p(p + q), Y = q(p + q). 5) x - pq = pq, y -pq = pq, so x = 2pq, y = 2pq. The equation is symmetric, so we have also: 6) x = pq(l + pq), Y = 1 + pq. 7) x = pq(l + q), Y = p(l + q). 8) x = pq(l + p), Y = q(l + p). 9) x = q(l + q), y = p(p + q). (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1978), pp. 45, Problem 3486) -

pq,

-

-

b)

-

-

l.

Xn+l = 2xn + Yn

and

Yn+l = Xn + 2Yn

for all n � By induction we obtain that

x = xp", y = YP,,' = 3, = y2 = . Alternative solution. p q p 3. pq = 2k + 1, 2k+1 = (pq + 1 pq - 1 ) ( x, y, ) = -- ' --' p, q satIsfy the equatIOn. (Dorin Andrica)2 2

Denote by Pk the k-th prime number. Then U v Pk is a V solution of equation x2 U for any integer k > O Let and be two arbitrary primes, � Then for some positive integer k. Because (k+ 1) 2 - k 2 , it follows that all quadruples

{ x� -� �z =: 21 { y: -= z: =: l=� { xxy --- yzz === l-1l

Since in the first case any two factors do not add up to the third, we only have three POSS b i e so (

x,y, z) = (k + 1, k, k - 1) for some integer k;

so (

x, y,z) = (k - 1,k + 1, k) for some integer k;

so (x, y, z ) = (k, k - 1, k + 1) for some integer k. - z = -2. Revista Matematica Timi§oara (RMT) , No. 1-2(1989), pp. 97, yAndreescu, (Titu Problem 2) c)

24. We have

xyz - (xy + yz + zx) + x + y + z 1 = 1, and, consequently, ( x - 1) ( y - 1) (z - 1) = 1. Because x, y, z are integers, we obtain x - I = y 1 = z - 1 = 1, so x = y = z = 2. (Titu Andreescu, Revista Matematica Timi§oara (RMT), No. 3(1971), pp. 26, Problem 487) 25. Multiplying by 8 yields -

-

•

-

.

u, v

23. The equation is equivalent to (

.

x - y) (x - z) (y - z) = 2.

x + y)4 _ (x _ y)4 = (2z)4 , and so ( x - y ) 4 + (2z) 4 = (x + y ) 4 . This is Fermat's equation for the case = 4 and it is known that this equation has solutions only if x -y = 0 or 2z = 0 (see T. Andreescu, D. Andrica, "An Introduction to Diophantine Equations" , GIL Publishing House, 2002, pp. 85-87). or

(

n

2.

66

Case I.

If

x-

2 . 2.

NUMBER THEORY

Y = 0, then x = Y and Z = ±x. The solutions are x = y = m,

for any integer m. Case II. If z = are

is a solution to the equation

z = ±m

x = O,

y = m,

z=o

x = m,

y = O,

z=o

for any integer m .

(Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1(1978), Problem 2813; Gazeta Matematica (GM-B) , No. 11(1981), pp. 424, Problem 0:264) 26. Consider the sequences (Xn ) n �l' (Yn ) n�l' (Zn ) n�l, defined by Xn+2 = 592xn ' Xl = 1, X2 = 14 Yn+2 = 592Yn ' YI = 3, Y2 = 39 Zn+2 = 592zn , Zl = 7, Z2 = 42 for all � 1. It is easy to check that x� + y� + z; = 59 n , for all integers � 1. (Dorin Andrica, Romanian, Mathematical Olympiad - second round, 1979, Revista Matematica Timi§oara (RMT) , No. 1-2(1980), pp. 58, Problem 4075) n

27. Observe that the equation has infinitely many solutions in distinct nonnegative integers, for example

(1)

Xk = (1 + kn)n -2 , Yk = k(l + kn )n-2 , Zk = (1 + kn)n- l , for any integer k � O . Let (Xk l , Ykl , Zkl ) and (Xk2 ' Yk2 ' Zk2 ) be two solutions of equation (1) with kl =1= k2 . Thtm and mUltiplying yields

(2)

has infinitely many solutions in distinct nonnegative integers, for example

and

(Xkl Xk2 )n + (Xkl Yk2 )n + (Ykl Xk2 )n + (Ykl Yk2 )n = (Zkl Zk2 )n-1

This means that

kl =1= k2

Since are arbitrary positive integers, the conclusion follows. For the second equation, the proof is similar, based on the fact that the equation

0, then (x - y)4 = (x + y)4 and so x = 0 or y = O . The solutions

n

67

SOLUTIONS

Xk = 1 + kn , Yk = k(l + kn), Zk = 1 + kn , for any integer k � O . (Dorin Andrica) 28. It is clear that x = 0, Y = 0 is a solution to the equation xn + yn = xn -l + yn- l Let a =1= -1 be a rational number such that Y = ax. Hence

1 + ann-l ' Y = a 1 + ann-l 1+a 1+a U sing the symmetry of the equation, we also have the solution 1 + ann- l , Y = 1 + ann-l x=a 1+a 1 +a with a =1= -1 rational. a = - 1 and > 1, then again x = Y = O . This concludes the solution. (Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1981), pp. 62, Problem 4578) 29. Let x = 3u, x + 2 = 3v , x + 8 = 3t so + + t = y . Then 3v - 3u = 2 and 3t - 3u = 8. It follows that t 3U (3V -U - 1) = 2 and 3U (3 -u 1) = 8. Hence = 0 and 3v - 1 = 2, 3t - 1 = 8, therefore = 1, t = 2. The solution is x = 1, Y = 3. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1978), pp. 47, Problem 2812; Gazeta Matematica (GM-B), No. 12(1980), pp. 496, Problem 18541) 30. If x, Y � 2 then 1 + xl and 1 + yl are both odd and (x + y) l is even. Hence

so

x=

If

n

U

v

-

u

the equation has no solutions.

v

2. NUMBER THEORY

2.2. SOLUTIONS

1. The equation becomes 2(1 + yl) = (1 + y)l and it is not difficult to notice .the solution y = 2. If y � 3, then 3 divides (1 + y) l but not 2(1 + yl) and y = 1 does not satisfy the equality. Hence x = 1, y = 2 or x = 2, = 1 due to the symmetry of the equation. (Titu Andreescu, Revista Matematidl. Timi§oara (RMT) , No. 2(1977), pp. 60, Problem 3028; Gazeta Matematica (GM-B), No. 2(1980), pp. 75, Problem 0:118)

1, x = 2 or x = 3 and all those cases lead to a contradiction. 3, then xl = 2vl - 8. Then v � 3 and x! = 2 (2VI - - 1) � 2 7. It follows that x � 5 and because x = 5 does not yield a solution and x � 6 implies 23 (2VI -3 - 1) 0 (mod 16), which is false, we do not obtain a solution here. In case II we have found only x = 1, y = 1, z = 2, v = 2,

68

Consider the case x =

Y

31. Without loss of generality we may assume that x � y � z. The equation is

Hence x = If

3

y =

- 1) . . . (z + 1) + y (y - 1) . . . (z + 1) + 1] = 2vl . If z � 3, then the right-hand side is divisible by 3 but the left-hand side is not, so z � 2. We have two cases. I. z = 1. Then we have xl + yl = 2vl - 1, or yl [x(x - 1) . . . ( y + 1) + 1] = 2 vl - 1. y � 2, then the right-hand side is an even number but the left-hand side is odd, so y = 1. Then xl = 2(2VI - 1 - 1). If x � 4, then 2(2VI - 1 - 1) 0 (mod 8), false. It remains to examine the cases x = 1, x = 2, and x = 3. If x = 1, then 1 = 2(2VI - 1 - 1), impossible. If x = 2, then 2 = 2(2VI - 1 - 1) or vI = 2, so v = 2. x = 3, then 2vl - 1 - 1 = 3 or vI = 3, false. Hence the only solution in this case is x = 2, = 1, z = 1, v = 2 II. z = 2. Now we have xl + y! = 2 vl - 2, or y! [x(x - 1) . . . (y + 1) + 1] = 2(2VI - 1 - 1). If y � 4, the 2(2VI - 1 - 1) 0 (mod 8), false. It follows that y = 1, y = 2, or y = 3. y = 1, then xl = 2vl - 3. Since x � 2 implies 2vl - 3 0 (mod 2) false, then x = 1, v = 2. If y = 2, then xl = 2vl - 4. We must have v � 3 so xl = 4(2VI -2 - 1). If x � 3, then 4(2VI -2 - 1) 0 (mod 8), false. If

x=

If

==

==

•

2,

Y

=

1,

z=

1, v = 2

and, due to the symmetry of the equation, we also have

1, x = 1, x=

and

2, y = 1, y=

v=2 z = 2, v = 2. z = 1,

(Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1981), pp. 62, 4576)

Problem

32. The equation is equivalent to or XI [X2 . . . Xn

Hence X l = 1 and X 2 [X3 . " Xn

- (n - l) x - ( - 1) x n

Since X2 f:. Xl , it follows that X3 [X4 . " Xn

Y

==

3

which does not satisfy the condition x � y � z . To conclude, we have the solution - from case I -

If

==

3

==

equivalent to

zl [x (x

69

2 . . . Xn- l

3 . . . Xn - l

=

X2

- (n - 1) x

2 and

4 . . . Xn- l

- ... - 2X2 - 1] = 1. - . . . - 3X3 - 2] = 2. - . . . - 4X4 - 3] = 3. = 3.

Because X3 f:. X2 and X3 f:. Xl , we obtain X3 Continuing with the same procedure we deduce that Xk = k for all k. Turning back to the equation we find the identity

Remark.

1 + I! . 1 + 2 ! . 2 + . . . + ( - 1)!(n - 1) = nIno (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 3(1973), pp. 23, Problem 1509) 33. We proceed by induction on For n = 1 the claim is true. Using the identity (x ) = (xu + ) - ( xv + ) )( n

2

-y

2

n.

2 2 U -v

YV

2

YU

2

2.

70

and the fact that the claim holds for as desired. We have

n we deduce that the property is valid for + 1, n

Alternative solution. n n n Pn = kII=l (a� - b�) = kII= l (ak - bk) kII= l (ak + bk) = where An , B n are integers. Hence

as claimed.

(

Dorin Andrica, Revista Matematidi Timi§oara (RMT) , No. 2(1975), pp. 45, 2239; Gazeta Matematidi (GM-B ) , No. 7(1975), pp. 268, Problem 15212)

Problem

34. Subtracting the equalities yields

(x + y - xy) + (v + y - vt) = (x + y - xy)(v + t - vt) , or [(x + y - xy) 1][(v + t - vt) - 1] = 1, so (1) (1 x)(l - y)(l - v) (l - t) = 1. It follows that 11 - xl = 11 - yl = 1 1 - v i = 1 1 - tl = 1, and using (1) we obtain (x, y, z, t) = (0,0,0,0), (0,0, 2,2), (0,2,0,2), (0,2, 2,0), (2,0,0,2), (2,0,2,0), (2, 2,0,0) and (2,2, 2,2). Turning back t o the system we obtain (x, y, z, v, t) = (0,0,0,0,0), (0,0, -4, 2, 2), (0,2,0,0,2), (0,2,0,2,0), (2,0,0,0, 2), (2,0,0,2,0), (2, 2, -4,0,0) and (2,2, 24,2,2) ( Titu Andreescu, Revista Matematica Timi§oara (RMT ) , No. 2(1978), pp. 46, Problem 3431; Gazeta Matematica ( GM-B ) , No. 5(1981), pp. 216, Problem 18740) -

-

35. We start with a useful lemma.

a and b are relatively prime positive integers, then there are positive integers u and v such that au - bv = 1. Lemma. If

2. 2 .

NUMBER THEORY

71

SOLUTIONS

Proof. Consider the numbers (1) 1 . 2, 2 · a, . . . , (b - 1) . a When divided by b the remainders of these numbers are distinct. Indeed, otherwise we have kl f. k2 E { I, 2, . . . , b - I} such that kla =Plb+ r, k2 a = P2 b+r for some integers PI , P2 ' Hence a b Ikl - k2 1 = (mod b), which is 1 I kl - k2 1 b. (1) is divisible by b. Indeed, if k 2, . k . a = P . b for some integer p. Let d be the greatest common divisor of k and p. Hence k = k l d, P = PI d, for some integers PI ,kl with ged(pI ' kl ) = 1. Then kl a = P I b and since ged( a, b) = 1, we have kl = b, PI = a. This is false, because kl b . It follows that one of the numbers from (1) has the remainder 1 when divided by b so there is u E { I , 2, . . . , b I} such that au = bv + 1 and the lemma is proved. We prove now that the system { axbx -yt - yz - e = O +d= with a, b, e, d nonnegative integers and ged(a, b) = 1 has at least a solution in nonneg­ ative integers. Because ged(a, b) = 1 using the lemma, there are positive integers u and v such that au - bv = 1. Hence x = cu + dv, y = ad + be, z = v, t = u, is a solution to the system. ( Titu Andreescu, Revista Matematica Timi§oara ( RMT ) , No. 2(1977), pp. 60, Problem 3029)

Since and are relatively prime it follows that false because � < On the other hand, none of the numbers listed in so, then there is E {I, . , n I} such that .

°

-

<

-

°

36. Consider the determinant

1 Xl �= Xpl - l

1 X2 Xp-2 l

1 xp X� - l

P

II (Xi - Xj )

i , i=l i>i

72

2.

NUMBER THEORY

SOLUTIONS

Summing up all columns to the first one and applying the hypothesis yields A == 0 p

p), hence II ( ) 0 (mod p) . Because P is a prime number, it follows that there are distinct positive integers k, I E {I, 2, . . . ,p} such that Xk 0 (mod p) . (Dorin Andrica) (mod

',j= 1 i >j

Xi - Xj

==

- Xl ==

37. Let

a = d(al, a2 , . . . , an ) and b = d ( al +2 a2 ' a2 +2 a3 ' . . . , an +2 al ) Then ak = aka, for some integers ak, k = 1,2, . . . , n. It follows that ak + ak+l ak + ak+l a, (1) 2 2 where an+l = al and an+l = al. Since ak are odd numbers, ak are also odd, so ak + ak+ l are mtegers. . 2 ak + ak+1 for all so a divides b. From relation (1) it follows that a divides ak + ak+1 = f3kb, for some 2mtegers . f3k. Then On the other hand, 2 ak + ak+ l 0 (mod 2b) for all k E {I, 2, . . . , n}. Summing up from k = 1 to k n yields 2(al + a2 + . . . + an) 0 (mod 2b) Since n, al, a2 , . . . , an are all odd al + a2 + . . . + an ¢ 0 (mod 2), hence al + a2 + . . . + an 0 (mod b) . (3) Summing up for k = 1,3, . . . , n - 2 implies al + a2 + . . . + an- l 0 (mod 2b) and furthermore al + a2 + . . . + an- I 0 (mod b) . (4) 0 (mod b), then using relation (2) we obtain Subtracting (4) from (3) implies a ak 0 (mod b) for all k. Hence bla andn the proof is complete. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1(1978), pp. 47, Problem 2814) gc

gc

=

==

=

==

==

==

==

==

==

38. It is known that

cp(kl) = cp(k)cp(l) for any relatively prime positive integers k and l .

73 2 . 2. On the other hand, it is easy to see that if p is a prime number, then cp(pl ) = pl pl - I for all positive integers Let n 2 · 3m , where is a positive integer. Then cp(n) = cp(2 . 3m ) cp(2)cp(3m ) = 3m - 3m- 1 = 2 · 3m- 1 �3 for infinitely many values of n, as desired. (Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 1(1978), pp. 61, Problem 3255) 39. For n 1 to n = 6 it is easy to check the claim. For n � 7 note that the number of even positive integers less than n is [�]. Moreover, the number of positive multiples of 3 less than n which are odd is [iJ - [i]. Then 7r (n) < n - [�J - ([iJ - [i]) , n � 7. Since x - I < [x] � x, it follows that ( n) < n - (� - 1) - (� - 1) + � = � + 2 ' 2 3 6 3 as desired. (Dorin Andrica, Revista Matematidi Timi§oara (RMT) , No. 1(1978), pp. 61, Problem 3256) 40. Summing the inequalities Pk+1 - Pk � 2 from k = 1 to k = n - 1 yields Pn - 2 � 2 (n - 1) and so Pn > 2n - 1, n � 1 . Then PI + P2 + ' ' ' +Pn > 2 n(n2+ 1) - n = n2 . The inequality a\" + ar : . . . + a::' ;::: ( a, + a2 : . . . + an ) m holds for any positive real numbers al, a2 , . . . , an ' Hence PmI + Pm2 + . . . + Pmn � n (Pl + P2 +n . . . +Pn ) m > n ( n-;-2 ) m = nm+ , as desired. (Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1978), pp. 45, Problem 3483) 41. Let S = L d and let cp(n) be the number of numbers less than n and relatively prime with n. _

I.

m

=

=

=

=

7r

1

d a(AAl - IA') + b(BBl - IB') + c(CCl - IC' ) ,

which is false. Therefore

1 = H and ABC is an equilateral triangle, as desired. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1981) , pp. 67,

Problem 4616)

15. Using standard notations, we have 64c2 + 49a2 + 9b2 - 112ac - 48bc + 42ab = 6c2 - 6a2 - 6b2 . This is equivalent to

4SABC = 1

14. It is known that if H is the orthocenter of a triangle ABC then

IC + IC' � CCI

or

(AM-GM inequality)

= = (Titu Andreescu,

IB + IB' � BBl ,

are strict. It follows that

c3

abc R' Equality in the first step requires that Y Z be parallel to B C and so on. This occurs if and only if P is the circumcenter of ABC. Equality in the second step requires that a b c. Thus equality holds if and only if ABC is equilateral and P is its center. USA IMO Team Selection Test, 2000)

C

= O. Viewing this as a quadratic equation in b, the condition � � 0 is satisfied. That 441a2 - 1008ac + 576c2 - 825a2 + 1680ac - 870c2 � O. 15b2 + 2b(21a - 24c) + 55a2 - 112ac + 58c2

is

The last relation is equivalent to

-6(64a2 - 112ac + 49c2 ) � 0,

:$ O.

or (8a - 7C) 2 It follows that 8a yields 3a 7b. We obtain

=

= 7c. Substituting back into the given condition ,

106

hence triangle ABC is similar to triangle A' B' C' having sides 7 , 3, 8. In this triangle 3 2 + 82 72 1 cos AI = 2' .3.8

-

2

It follows that A = A7 = 60°. Korean Mathematics Competition, 2002)

(Titu Andreescu, 16. First solution. By the Cauchy-Schwarz Inequality, ..jPB2 + PC2 ..jAC2 + AB2

Applying the (Generalized) Ptolemy's Inequality to quadrilateral ABPC yields

2:: PA . BC.

Because PA is the longest side of an obtuse triangle with side lengths PA, PB, PC, we have PA VPB2 + PC2 , and hence

>

P A . BC

2:: ..jPB2 + PC2 . BC.

AQ 2

2:: AQ2 _ QC2 =

>

=

(Ap2 _ PQ 2 ) _ (Cp 2 _ PQ 2 )

AP2 - PC2

> BP2 2:: DQ2 .

=

l,

l,

o

P o

O�--�O�----�Or---O

A

D

Q

c

+

+

It

=

+

=

z · z

C

=

z -

z

+

=

z ·

+

z

=

0

Applying the Lemma to points A, B , C, P gives

o � AB 2 + Bp2 + PC 2 + CA2 _ AP 2 - BC2 =

>

Let I be the ray AC minus the point A. Note that, since PA PC, Q lies on ray D did not lie on then AQ would be less than or equal to DQ, a contradiction. Thus, D lies on and angle BAC is acute. B

l. If

=

c ,

=

=

2:: BC,

implying that angle BAC is acute. With some careful argument, it can be proved that quadrilateral ABPC is indeed convex. We leave it as an exercise for the reader. Let D and Q be the feet of the perpendiculars from B and P to line AC, respectively. Then DQ � BP. Furthermore, the given conditions imply that Ap2 Bp2 + PC2 , which can be written as Ap2 - PC 2 BP 2 . Hence,

Note. Second solution. >

=

=

z

Combining these three inequalities yields

..jAB2 + AC2

Third solution. Set up a coordinate system on the plane with A (0, 0) , B (a,O), C (b, ) and P (x, y). Without loss of generality, we may assume that a > 0 and that > O. Proving that angle BAC is acute is equivalent to proving that b > O . Since PA2 > PB2 PC2 , x2 + y2 > (x _ a)2 + y2 (x _ b)2 + (y _ C)2 . Hence o > (x - a) 2 - 2bx + b2 (y - C) 2 2:: -2bx . Since P A > P B, we have x > � > O. follows that b > 0, as desired. Fourth solution. We first prove the following Lemma. Lemma. For any four points W, X, Y, Z in the plane, WY2 + X Z2 � WX2 + Xy2 + YZ2 + ZW2 . Proof. Pick an arbitrary origin 0 and let w, x, y, denote the vectors from 0 to W,X,Y, Z, respectively. Then W X 2 + X y 2 + Y Z2 + Z W 2 - W y 2 - X Z2 I w - x l 2 + I x - Yl 2 + I y - Zl 2 I w l 2 - Iw - Yl 2 - I x - Zl 2 w · w + x· x + y . y + - 2(w , x + x · y y . + w - w · y - x · ) I w + y - x - Z1 2 , which is always nonnegative. Equality holds if and only if w y x + which is true if and only if W XY Z is a (possibly degenerate) parallelogram. +

2:: PB · AC + PC · AB.

P B . AC + PC . AB

107

3.2. SOLUTIONS

3. G EOMETRY

(PB2 + PC 2

_

PA2 ) + (AB 2

< 0 + (AB2 + AC2 - BC2 )

=

+

AB 2

=

z,

=

AC2 - BC2 ) < +

AC 2 - BC 2 .

Therefore angle B AC is acute. In this solution, sin - 1 takes on values between 0° and 90° . Note that -p;fjj < 90° , since PB < P A. Applying the Law of Sines to triangle PAB yields - PB - PB sin PAB = sin ABP � PA PA ' It follows that . - 1 PB PAii -< sm PA ' Since P A2 PB 2 + PC 2 , we have similarly

Fifth solution.

>

. _ 1 VPA2 _ PB2 < sm ' - 1 PC < sm P AC _ PA PA

108

the circumcircle of ABPC, arcs AB and AC are bigger than arcs PC and PB, respectively. Thus, IiPC > IfAiJ Because these two angles are supplementary, angle BAC is acute.

Thus -

If () =

-

--

JPA2 - PB2 PB + sin - 1 BAC � BAP + PAC < sin PA PA 1 sin then

��,

sin(90° - ()) Hence

=

1

.

'\.

JPA2 - PB2 . cos () = V'1 - sin2 () = PA

. - 1 JPA2 _ PB2 . - 1 PB + sm BAC < sm PA

PA

and angle B AC is acute.

=

C '\.

1

B

B

As we mentioned at the end of the first solution, the conditions in the problem imply that quadrilateral ABPC is indeed convex. Thus, the diagram on the right-hand side is not possible, but this solution does not depend on this fact.

( \

...-I

I

\

\

"-

I

\

'-...

""

,/

'\.

I

/

17.

A

/

A

All angles will be in degrees. Let x = F0B. Then PiiC Law of Sines (or the trigonometric form of Ceva's Theorem) ,

P

I

,/

'-

\

(Titu Andreescu, USA Mathematical Olympiad, 2001)

P

/'

\

P

90° ,

C

Sixth solution.

109

3.2. SOLUTIONS

3. G EOMETRY

_

/

_

;'

B \

P C

'"

\

P A PB P C PB PC PA

./

I

=

� !!.!!. �

sin sin sin sin PAB sin PBC sin PCA

80 - x. By the

=

4 sin x sin 40 cos 10 sin(80 - x)

sin 20 sin x sin 40 sin 10 sin(80 - x) sin 30 B

\ A

,/

...--

\

=

=

x

/

Note that PA2 > PB 2 + PC2 . Regard P as fixed and A, B, C as free to rotate on circles of radii PA, PB, PC about P, respectively. As A, B, C vary, IfAiJ will be maximized when B and C are on opposite sides of line PA and Jfijp and ;[(j'p are right angles, i.e. , lines AB and AC are tangent to the circles passing through B and C. Without loss of generality, we assume that PA > PB � PC. In this case, ABPC is cyclic and AB 2 = P A2 - PB 2 > PC2 , and similarly AC 2 > PB 2 . Hence on

A

C

The identity 2 sin a cos b = sin( a - b) + sin( a + b) (a consequence of the addition formula) now yields 1=

2 sin x(sin 30 + sin 50) sin(80 - x)

=

sin x(1 + 2 cos 40) , sin(80 - x)

so 2 sin x cos 40

=

sin(80 - x) - sin x = 2 sin(40 - x) cos 40.

This gives x = 40 - x and thus x triangle ABC is isosceles.

=

20. It follows that AcE

=

50

=

IfAiJ, so

110

3 . G EOMETRY

3.2. SOLUTIONS

Alternative solution. Let D be the reflection of A across the line BP. Then triangle D is isosceles with vertex angle APi5 = 2(180 - EPA) = 2(PAii + AiiP) = 2(10 + 20) = 60, and so is equilateral. Also, YSiiA = 2PBA = 40. Since 1f.AC = 50, we have D B AC. AP

.L

B

it follows that

111

r = 4( v. 3 + 1)r sm. "2A sm. 2"B sm. "2C In

or 1 2 ( va + 1) Then 0- 1 -4-

= 2 (sm. 2"A sm. "2C ) sm. "2B '

A-C A + C) . B = (cos - - cos -2- sm 2" 2

= 30° , we obtain J3 - 1 = ( V6 +4 V2 sin �2 ) sin �2 . 4 . B = x Y1e . lds ' sm Lettmg "2 V6 +4 V2x + 04- 1 _- 0 , x - V2 and x = 2 ..J2 ' It follows that "2B = 15 o or 2"B = 45° . whose solutions are x = V6 4 The second solution is not acceptable, because A 2: B. Hence B = 30°, A = 90° and C 60° . Thus triangle ABC is right angled. ( Titu Andreescu, Korean Mathematics Competition, 2002)

and, since A - C x

_

A

C

D

E be the intersection of DB with CPo Then PEi5 = 180 - CEi5 = 180 - (90 - AcE) = 90 + 30 = 120 and so PEi5 + J5Ai5 = 180. We deduce that the quadrilateral APED is cyclic and -' therefore D EA = D PA = 60. Finally, we note that I5EA = 60 = DEC . Since AC .L DE, we deduce that A and C are symmetric across the line DE, which implies that BA = BC, as desired. ( Titu Andreescu, USA Mathematical Olympiad, 1996) 18. Let max{A, B} = A. If triangle ABC is right-angled, then A = 90° , B = 30° R and C = 60° . In order to find - , we may assume that ABC is the triangle with sides = 2, b 1 , = J3. We haverR 1 and va S ABC 2 r = s = 2 + 1 + J3 _- 3 +J3J3 ' Let

a

=

so !!:.

3 + J3 J3

r=

=

c

2

= va + 1.

Conversely, assume that

-r R

=

r=

J3

+ 1. From the identity

. B . C · A sm 4R sm "2 "2 sm "2

2

_

=

19.

Construct in the exterior of triangle ABC three circles equal to the circum­ circle ABC that pass through two vertices of the triangle. By the five-coin theorem the circles will have a common point P, as desired (see Dorin Andrica, Csaba Varga, Daniel Vacare�u, " Selected Topics and Problems in Geometry" , PLUS, Bucharest, 2002, pp. 51-56) . Let H be the orthocenter of triangle ABC. The reflections of H across the sides of the triangle are points of the circumcircle of triangle ABC. Therefore the circum circles of HAB, HBC, HCA are equal to the circumcircle of ABC and for P H the claim holds. ( Revista Matematica Timi§oara (RMT), No. 2(1978) , pp. 74)

Alternative solution. = Dorin Andrica,

20.

Denote the midpoints of A'B', B'C' , C'A' by Co , Ao , Bo, respectively, and the three perpendiculars in question by , Consider the centroid of triangle A'B' C'.

lc, lA lB .

11 2

3.

C

B A C' Since AoG : GA' BoG : GB' CoG : GC' 1 : 2, the dilatation h with center G and coefficient -2 takes Ao, Bo, Co to A', B', C', respectively. Since dilatations carry straight lines into parallel lines, h transforms le into the line through C' perpendicular to AB. But C' is the point of tangency of the incircle and AB, so this line passes through the incenter of triangle ABC. The same applies to the images of lA and lB under h. Since the images of lA , lB , le under h are concurrent, so are lA , IB , le themselves. (Titu Andreescu, Romanian IMO Selection Test, 1986) 21. Because triangle A1 A2 A3 is not isosceles, it is not difficult to see that the circumcenter of the triangles A 1 B 1 I, A2 B2 I, A3 B 3 1 are defined. We start with a sim pIe lemma. Lemma. Let ABC be a triangle with the incenter I. Let T be the circumcenter of the triangle BIC. Then T lies on the internal bisector of the angle A. Proof. Let us draw the external bisectors of the angles B and C as shown in the figure below. B =

=

A

C

The lemma is proved.

..1

113

SOLUTIONS

=

They intersect at the excenter E, which lies on the internal bisector of the angle A. Since BE BI and GE GI, the quadrilateral BEGI is cyclic with the center of the circumscribed circle on IE . This center will be also the circumcenter of BIC. ..1

3.2.

GEOMETRY

i 1, 2, 3 Qi Ai+1 IAi+2 . Oi Ai ' Ti T1 T2T3 I. 0i+1 0i+2 Ti+1 Ti+2 ' Ti+l Ti+2 Bili AlB1 I, A2B2I, A3 B3 I, 101 02 , 102 03 , 103 01 (T1 ' T2 , Q3 ), (T2 ' T3 , Q1 ), (T3 ' TI, Q2 ), 01 T1 . IT2 . 02 Q3 - 1 , IT1 02T2 01 Q3 IT 0 T Q Q 1 03 T33 . IT2 22 . 023 Q1 - 1 , ITI 0 T 0 1 Q OlTl . IT3 33 . 03 Q22 - 1 . Multiplying them all one gets 02 Q3 . 03 Q1 . 01 Q2 1, 0 1 Q 3 0 2 Q 1 03 Q 2 which means that the points Q 1 , Q 2 , Q 3 are collinear. Alternative solution. This proof will be based on inversion. We take the incenter I to be the center of the inversion and the power of the inversion is arbitrary. Using

Let us prove the main statement. For = we denote by the center of the circle Ci and by the circumcenter of the triangle Clearly, lies on the internal bisector of the angle By the lemma, also lies on the same bisector. Thus the triangles and are perspective from the point By Desargues' theorem these triangles are perspective from a line. This is to say that if we denote then and = to be the point of intersection of the lines are collinear. But since is the perpendicular bisector the points of and is the perpendicular bisector of these points are exactly respectively. the circumcenter of the triangles A student not familiar with Desargues' theorem may proceed from the point as follows. Applying Menelaus' theorem to the triangles and to the triples of points respectively, one can, observing usual agreement about the signs, write:

Ti 0 1 02 03 Qi , i 1,2,3, Q 1 , Q , Q3 Ail 0i+120i+2 Remark.

_

=

primes to denote images of points under the inversion we have the following " dual" figure shown below.

3.

11 4

3. 2.

G EOMETRY

SOLUTIONS

11 5

23. Without loss of generality assume that

B'I

PC = max{PA, PB, PC}. The condition in the hypothesis is PB · PC + PA · PC = PA · PB + 1 or

B'2 Indeed, the image of the circle Ci is a straight line Bi+ 1 Bi+2 ' with these lines forming the triangle Bi B�B� . The line AiAi + 1 will be transformed into the circle ri+ 2 , with the side AiAi+ 1 becoming the arc AiAi+ 1 which does not contain I. Note that all these circles have equal radii since the distances from I to the sides of AI A2 A3 were equal. Let us note that if �I ' � 2 ' � 3 are three circles passing through the common point I and no two of them touch, then their centres are collinear if and only if there is another common point f:. I through which all these three circles pass. We will use this observation for �i being the circumcircle of AiBiI. Since the inversion takes :Ei to the line AiB�, the desired result is to show that the lines A� Bi , A� B� , A�B� are concurrent. For this, it suffices to show that the triangles A� A� A� and Bi B�B� are homothetic, which is the same to say that their corresponding sides are parallel. Since the radii of the circles r I , r 2 , r3 are equal, the triangle PI P2 P3 formed by their centre has its sides parallel to the corresponding sides of the triangle Bi B�B�. The homothety of ratio centred at I takes the triangle A� A� A� into the triangle whose vertices are the midpoints of the triangle PI P2 P3 . Therefore the corresponding sides of the triangles A� A�A� and PI P2 P3 are also parallel and the result follows. IMO 1997 Shortlist)

J

PC PA · PB + l · 1 -1- PA . 1 PC . 1 . From the converse of the second theorem of Ptolemy it follows that P ACB is a cyclic quadrilateral. Note that P cannot be A, B or C otherwise the denominator of the right-hand side equals Hence the locus of point P is the circumcircle of triangle ABC without the vertices A, B, C. Revista Matematidl Timi§oara (RMT), No. 1(1985) , Problem C7:3)

(Titu Andreescu,

24. We have

Ja2 b2 - 4S2 + Ja2 c2 - 4S2 = Ja2 b2 - a2 b2 sin2 C+ + Ja2 c2 2- a2c22 sin22 B = ab2 cos C + ac cos B = = ab a +2abb - c +ac a +2acc2 - b2 = a2 , as desired. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 8(1971), pp. 25,

Problem 1006)

25. The relation is equivalent to

Fa + y'rb + Fc = 1 , -

from Van Aubel's theorem, therefore line segment parallel to BC.

(Titu Andreescu)

::, is constant. Hence the locus of point I is a

..jrarbre

or

(Titu Andreescu,

intersection point of lines AI and BC. We have AI AB' AC' = + B' B C' C IA"

+

O.

1/2

22. Let I be the intersection point of lines CB' and C' B and let A' be the

=

On the other hand,

so

(

1 1 1 1 - + - + - = -,

ra rb re r 1 1 1 1 1 1 --+ --- + --- = - + + - . rb re Fay'rb y'rbFc FcFa ra -

so Then

r 1 1 1 1 --= -. + --+ --.;r;.fo vr;.;r;. .;r;.y'rb r

2 2 1 ) - 1 ) 2 + ( 1 1 ) = 0, + ( vr;. - .;rb fo Fc vr; - .;r;. 1

1

ra = rb = re ' It follows that the triangle is equilateral, as desired. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1(1974) , pp. 21,

Problem 1903)

116

3. 26.

If the triangle is equilateral the conclusion is true. To prove the converse, we assume by way of contradiction that the triangle is not Then equilateral and say that

b f:. c. b2 +2 c2 a2 (b + C)42 - a2 =p(p - a) m2a -- - 4 > and likewise m� 2: p(p - b), m� 2: p(p - c). It follows that 1 ) 1 1 1 = + -+ -+ -1 + -1 < -1 ( -m� m� m� p p - a p - b p - c -p2 + ab +2 bc + ca _

-

Problem 3513; Gazeta Matematica ( GM-B ) , No . 11(1981), Problem 0258; No. 2(1988) , pp. 78, Problem 21353)

-�____ _

2"

28. For any positive real numbers x, y, Z we have

(Titu Andreescu,

Setting gives so

A . B . C 1 . < -' sm - sm - sm 8 2 2 2 C B A ' C. ' B sm ' A sm cos "2 cos "2 cos "2 2: sm

xyz ::; (x + y)(y + z)(z + x) x = -a + b + c, y = a - b + c, Z = a + b - c, ( -a + b + c) (a - b + c) (a + b - c) ::; abc, 8

2'

which is a contradiction. Revista Matematidi Timi§oara (RMT ) , No. 2(1977) , pp. 66, Problem 3063)

so

=:=

(Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1978) , pp. 49,

8 -r� + -r� + -r� = -82 [(p - a) 2 + (p _ b)2 + (p _ C)2] = p2 + a282+ b2 + c2 Since b f:. c then ab + bc + ca < a2 + b2 + c2 hence 1 + m12b + m-1e2 < r21a + r1b + r1e m2a -

On the other hand 1 1 1 1

27. We know that

117 3.2. Alternative solution. Let Zl, Z2 , Z3 be the afixes of points A, B, C such that I zd = IZ2 1 = I Z3 1 = 1 . We have a BC = I Z2 - z3 1 , b = AC = IZ3 - zd, c = AB = IZI - z2 1 · U sing the identity Z; (Z2 - Z3 ) + Z� (Z3 - zd + Z� (ZI -Z2 ) = (Zl - Z2 )(Z2 - Z3 )(Z3 - zd and triangle inequality it follows abc = IZI - z2 1 z2 - z311 z3 - zd ::; IZll2 1 z2 - z3 1 + I Z2 12 1 z3 - zll + IZ3 1 2 1 z1 - z2 1 = = I Z2 - z3 1 + IZ3 - zd + I ZI -z2 1 = a + b + c. SOLUTIONS

GEOMETRY

It follows that (1)

On the other hand,

' A + sm' B + sm' C = 4 cos "2A cos "2B cos "2C '

sm

then Hence

3

sin A

+ sin B + sin C 2: 4 sin A sin B sin C.

Since the circumradius is 1, we have

a = 2sinA, b = 2 sin B, c = 2 sin C, and relation (2) yields a + b + c 2: abc, as

claimed.

pabc

2 ' 32

so inequality (1) gives (2) as

desired.

(Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1974) , pp. 51,

Problem 2028)

29. Using the inequality 3(

a2 + b2 + c2 ) � (a + b + C)2 ,

11 8

3.

we obtain

a2 + b2 + C2 � p2 ' 4S2

Hence as

3.2.

GEOMETRY

1

h�

desired.

+

1

h�

3S2

+

1

h�

�

1 3r2 '

as

desired.

(Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1(1973), pp. 43,

Problem 1585)

33. We have

(Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1977), pp. 66,

It follows that

Problem 3062)

30. From the inequality we obtain

1 1 1 1 - > -- + -- + -- . ..jrbrc ..jrcra ..jrarb

r-

Then

hence as

-

-

desired.

(Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1(1984), pp. 67,

Problem 5221)

hence ..jrarb + ..jrbrc + ..jrcra � 9r, as

-) 1 V (P - b)-'-bc(P C-'- < -2 ' (p - b) (P - c) :s; 4bc ' -'---

1 1 1 1 1 1 - + + - > -- + -- + -ra rb rc - ..jrbrc ..jrcra ..jrarb '

119

SOLUTIONS

34. Let 0 1 , O2 , 03 be the centers ofthe three circles and S the area of the common

region .

claimed.

(Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1978), pp. 64,

Pro blem 3277)

31. We have ma =

and likewise

b2 +- - a42 � J(b + C) 2 - a2 = Vp(p - a) J4 2 ,?

It follows that ma m b m c � as

desired.

pVp(p - a)(p - b) (P - c) = pS = rarbrc,

(-Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1 (1978), pp. 64,

The three sectors with centers 0 1 , O2 , 03 which subtend the arcs 02 03 , 0 1 03 ,02 0 1 , respectively, cover the 2 surface of area S and twice more r the surface of triangle 0 1 02 03 ( which is f3 ). On the other hand, the area of these three circular sectors equals the area of a semicircle, which is

Problem 3276)

p3 � 27abc. Hence

r2 J3

1 7fT = S + 2 . -- ' 4 2

-

32. By the AM-GM inequality, so 8

2

�7fT2. Hence

(a + b + c)3 � 27abc, abc · S = 27Rr, 2p2 -> 274S p -

therefore

S

= 21 r2 (1/' - V3).

(Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1978) , pp. 50,

Problem 3522)

1 20

3. G EOMETRY

35. The parallel BD through C meets AB at point E. By Stewart's formula, we

D F N

obtain

AC2 · BE + CE2 . AB - CB 2 · AE = AB · BE · AE Because CE = BD and BE = CD, we deduce AC 2 . CD + BD 2 . AB - BC 2 . (AB + CD) = AB . CD . (AB + CD) D

121

3.2. SOLUTIONS

C

I (1)

C

B

E

M

A Then

AB + CD = EF, 2 which is the length of the midline and hence the length of the altitude. It follows that IM and IN are also altitudes in triangles lAB and ICD therefore lAB and ICD are isosceles. Thus ABCD is isosceles, as claimed. Revista Matematidi Timi§oara (RMT), No. 1 (1978) , pp. 48, Pro blem 28 17) IM + IN =

A

B

E

(Titu Andreescu,

Drawing the parallel to AC through D and using similar computations yields BD 2 . CD + AC2 . AB - AD 2 . (AB + CD) = AB · CD · (AB + CD)

(2)

Subtracting the relation (2) from (1) gives

38. From the Law of Cosines we deduce that

2AB . AC cos IiAC = AB 2 - BC2 + AC2

(AC 2 - BD 2 ) (AB - CD) = (AD 2 - BC2 ) (AB + CD) ,

2DC · AC cos DcA = CD 2 - AD 2 + AC2 2AB . DB cos ISiiA = AB 2 - AD 2 + DB 2

as desired.

(Dorin Andrica, Gazeta Matematidi (GM-B) , No. 9(1977) , Problem 6852; Revista

Matematidi Timi§oara (RMT), No. 1-2(1980) , pp. 64, Problem 4119)

36. Let I be the intersection point of the diagonals AC and BD. Since AB · AC AB · BD IA = and IB = AB + CD AB + CD the condition in the statement becomes Hence AiB = 90° , as desired . Revista Matematidi Timi§oara (RMT) , No. 2(1978) , pp. 59, Problem 3524)

(Dorin Andrica,

37. Let ABCD be the trapezoid. Point I is the intersection of diagonals and

M, N are the midpoints of AB and DC. In a right triangle the length of the median corresponding to the hypothenuse is half of lenght of the the hypothenuse. Hence AB CD IM = and IN = . 2 2

Note that

2DC · DB cos CiJij = CD 2 - BC2 + DB 2

(1) (2) (3) (4)

IiAC = DcA, ISiiA = CiJij,

so dividing relations (1) and (2), (3) and (4) yields AB AB 2 - AD 2 + DB 2 AB 2 - BC2 + AC2 = = DC ' CD2 - AD2 + AC2 CD2 - BC2 + DB2 as desired.

(Dorin Andrica) 39. Let a, b the lengths of the bases, c, d be the lengths of the nonparallel sides

and d 1 , d2 be the lengths of the diagonals. From Euler's theorem for quadrilaterals, it follows that Hence

c

(d1 - d2 ) 2 + 2d1 d2 = ( - d) 2 + 2(ab +

cd) ,

3.

1 22

d1 - d2

d

3. 2.

GEOMETRY

d1 d2 ab cd.

= + = C - implies From Ptolemy's Theorem, we deduce that the trapezoid is cyclic and so isosceles, as claimed. Revista Matematica Timi§oara (RMT) , No. 1(19 78), pp. 48, Problem 281 7) and

(Titu Andreescu,

40. First solution.

Assume the opposite. Then lAC - BD I > lAB - CD I or lAC - BDI > lAD - BCI. Without loss of generality, l AC - BD I > lAB - CDI , otherwise switch B and D. We have AC2 - 2AC . BD + BD 2 > AB 2 - 2AB . CD + CD 2

(1)

and, from Euler's relation, AB 2 + BC2 + CD 2 + AD 2 = AC2 + BD2 + 4MN2 ,

AD 2 + BC2 - 2AC · BD > 4MN2 - 2AB · CD .

lAC - BDI = 2 1 sin(a

+,6) - sinCa + ,) 1 = I Sin a � , cos ( a � , +,6) I .

Since 0 (a + ,)/2 ::; (a + ,)/2 + ,8 ::; 7r/2 (by the assumption ,8 8) and the cosine function is nonnegative and decreasing on [0, 7r /2] , we conclude that l AB _ CD I � l AC - BDI , and similarly lAD - BCI � lAC - BD l . USA Mathematical Olympiad, 1999)

�

�

(Titu Andreescu,'

41.

Let E be the intersection point of the diagonals. Consider AD basis of the trapezoid and AB the altitude. Since

<

BC the

AB 2 = AD . BC, then

( 3)

Let P be the midpoint of AB. Then NP = AD/2, MP = BC/ 2 and since MN � INP - MPI , it follows that 4MN 2 � (AD - BC) 2 .

and

(2)

where M and N are the midpoints of AC and BD, respectively. From (1) and (2) ,

1 23

SOLUTIONS

AB BC AD AB ' so the right triangles ABC and ABD are similar. On the other hand we have

(4)

(3)

and (4) , -2AC · BD > -2AB · CD - 2AD · BC, in contradiction with From Ptolemy's Theorem. We are done. The cyclicity is essential. The inequality fails if ABCD is a parallelogram. Let E be the intersection of AC and BD. Then the triangles ABE and DCE are similar, so if we let x = AE, y = BE, z = AB, then there exists k such that kx = l)E, ky = CE, kz = CD. Now

Note. Second solution.

hence It follows that ill = 90° so the diagonals are perpendicular, as claimed. Revista Matematica Timi§oara (RMT) , No. 2 (19 72) , pp . 28, Problem 1 164)

(Titu Andreescu,

lAB - CDI = Ik - l i z and

lAC - BDI = I (kx + y) - (ky + z ) 1 = I k - 1 1 · Ix - y l · Since Ix - yl ::; z by the triangle inequality, we conclude that l AB - CDI � lAC � BD I , and similarly IBC - DAI � lAC - BD l. These two inequalities imply the desired result. Let 20', 2,8, 2,, 28 be the measures of the arcs subtended by AB, BC, CD , DA, respectively, and take the radius of the circumcircle of ABCD to be 1. Assume without loss of generality that ,8 8. Then a + ,8 + , + 8 = 7r, and (by the Extended Law of Sines)

Third solution.

�

I

l AB - CDI = 2 1 sin a - sin ,1 = Sin

a

� , cos a �' I

42. Let I be the intersection point of the diagonals AF and BE of the rectangle

ABFE. Notice that N I is the median of the right triangle ANF with hypothenuse AF, so AF IN = - = IA 2 Likewise, BE IM = - = IE ' 2

AF IQ = - = IF ' 2

Since IA = IE = IF = IB, it follows that 1M is cyclic, as desired.

=

BE IP = - = IB . 2 IN = IP = IQ. Hence MNPQ

124

3.

A

125 3. 2. 44. Let I be the intersection point of lines BD and AF. The parallel to BD through C meets line AF at point T. First we consider AF .l BD and prove that AB .l CD. F F

GEOMETRY

SOLUTIONS

B P N

Q D E (Titu Andreescu)

< a1 :::; a2 :::; a3 :::; a4 < a1 + a2 + a3 , aI, a2 , a3 , a4 · a, b, c, d, e, j, m AB, BC, CD, DA, AC, BD, MN, b + d 2: a + c. P BC. MP NP are midlines CAB BDC, MP -a2 NP -c.2 Then 2m 2MN < 2MP + 2NP a+c so 2m < a + c < b + d. 43.

By Sturm's theorem, we know that if 0 then there is a cyclic quadrilateral having side lengths Denote by the lengths of the segments respectively. Without loss of generality assume that Let be the midpoint of the side The segments and in triangles and so 1 1 = and = =

=

On the other hand, if 0 is the intersection point of the diagonals, we have

b + d BC + DA < BO + OC + DO + OA AC + BD e + j, hence 2m < a + c < b + d < e + j. It suffices to prove that e + j < 2m + a + c + b + d. Note that e < c + d, j < b + c, e < a + b, j < a + d. =

A �------� D

C

F

=

=

Summing up these inequalities yields

e + j < a + b + c + d < a + b + c+d+ 2m and the proof is complete. (Titu Andreescu, Revista Matematidi Timi§oara (RMT) , No. 1 (1978) , pp. 66, Problem 3288; Gazeta Matematidi ( GM-B ) , No. 10 ( 1981 ) , pp. 402, Problem C148 )

i) Assume that we obtain

D lies on the segment CEo Then ArC

On the other hand

=

90° . Since IfAjJ : BnC,

(1)

CTI I BD so

(2)

EATC

Relations ( 1 ) and imply fiE : EaT , therefore is cyclic. It follows that BEG = 90° , hence as desired. In the right triangle ii ) Assume that lies on the segment median, so

AB .l CD, DE.

C

Because

CTF, TE is the

Ere : EaT .

CTI IB D, we have

(2) (3)

( 4)

Also,

ATEC

(5)

so from ( 3 ) , (4) , (5) , we obtain Ere : Mo. Hence is cyclic, then AEC = ArC = 90° , and .l as desired. Conversely, consider that i) If is on the segment then ME == AcE On the other hand AcE == Aci5 == ABi5 so ME == ABi5 and is cyclic. It follows that JiiF = liEF = 90° , hence as desired. ii ) If is on the segment then

AB CD,

D

C

,

BD .l AF,

AB .l CD. CE, FBIE DE,

.

,

(6)

1 26

11oreover, J.U515 = ACB15,

SO

AcE = ABl.

S

(7)

a, b, c, d

Let be the side lengths of the quadrilateral and let Because the quadrilateral is cyclic, we have

S2 = (p - a)(p - b)(P - c)(P - d). 1

S be its area. (1)

S a � b � c � d. Since S is prime number, p- d=p - c = l md p - a = p- b = � Summing up these equalities yields 4p-2p = 2 + 2S so p = S + 1. Hence a = b = 1 and c = d = S. The required quadrilaterals are either rectangles or kites. (Titu Andreescu, Revista 11atematidi Timi§oara (R11T) , No. 2(1977), pp. 66, Problem 3067)

Without loss of generality assume that from relation we obtain

(1)

a

46. From the A11-G11 inequality it follows that

S

Hence

S

1

1

(6) and (7) we obtain ME = ABl, hence FEB! is cyclic. Note that liEF = 900, so iiiF = 900 and BD ..L AF, as desired. (Titu Andreescu, Revista 11atematica Timi§oara (R11T) , No. 1(1986), pp. 106, Problem C6: 4 ) From

45.

12 7

3.2. SOLUTIONS

3. GEOMETRY

or

abcd -< ( a + b +4 c+d ) 4 = -p41 4 . 16abcd � p4 , 8(ac + bd)2 -p4 � 8(a2 c2 + b2d2 ).

The desired inequality is now obtained from Ptolemy ' s Theorem: N �mbers

a, b, c, d are odd, hence a + b+c+d P = --2--

is an integer. If is odd, then which is false. Hence p is even.

p

p - a, p - b, p - c, p - d are even and so 82 is divisible by 16,

ac+ bd = ef.

(Titu Andreescu, Revista 11atematidi Timi§oara (R11T), No. 3(1973), pp. 36, 1811; Gazeta 11atematidi (G11-B) , No. 8(1980), pp. 364, Problem 18370)

Problem

47.

Let m be the length of the segment determined by the midpoints of the diagonals . From Euler 's Theorem for quadrilaterals we have

(1)

12

3.

8

and from Ptolemy's Theorem,

3. 2.

G EOMETRY

ae + bd = ef.

(1) and (2) we obtain (a - e) 2 + (b - d) 2 = (e - 1) 2 + 4m 2 . Since max{ la - el , I b - dl} � 1, then 2 = 1 + 1 � (a - e)2 + (b - d)2 = (e - 1) 2 + 4m2 � (e - 1) 2 .

129

SOLUTIONS

D

(2)

From relations

Hence

I e - 11 � V2,

AI

as desired .

(Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1978), pp. 51,

Problem 3527)

ABC DAC

AI, MI, BI AIBI, CI DI AC BI CI DIAl . Mil (Dorin Andrica, 3855;

48. The quadrilateral is cyclic so S = J(p - a) (p - b)(P - e) (p - d) 2 Since S = (�) , we have

{j(p - a) (p - b)(P - e) (p - d) = P.2 = (p - a) + (p - b) + (p - e) + (p - d)

51. We have

4 Note that this is the equality case in the AM-GM inequality, hence p - a d, so the quadrilateral is a square. p - It follows that a Revista Matematica Timi§oara (RMT) , No. pp. Problem

p e = d. (Titu Andreescu, 2136)

= p-b = 1(1977), 24,

=b=e=

4�. Observe that SPAB . SPCD = SPBC . SPDA, since both are equal to � PA . PB· PC· PD · sin P. The numbers SPAB , SPCD and SPBC , SPDA have the same sum and the same product, thus SPAB = SPBC and SPCD = SPDA or SPAB = SPDA and SPBC = SPCD, i.e. P is the midpoint of AC or BD, desired. as

We recall the Simpson's theorem: the projections of a point of the circumcircle of a triangle onto the sides of the triangle are collinear. Applying this result to triangles and yields that are collinear and are collinear. Hence the lines and meet at as claimed. and we deduce that (ii) From Simpson's Theorem for triangles and Since is a point of the conclusion follows. is on the lines Revista Matematidi Timi§oara (RMT), No. pp. 54, Problem Romanian Regional Mathematical Contest " Grigore MoisH" ,

(Titu Andreescu, Korean Mathematics Competitions, 2001) 50. (i) Let MI and Mil be the projections of point M onto diagonals AC and BD .

CI, MI, DI MI, ABD BDC AC,

M il

1(1979),

1995)

SABCD � 21 AC . BD,

AC BD. Since 2002 = SABCD � 21 AC . BD � 1 52 · 77 � 2 (AP + PC) . (BP +PD) = -- = 2002, 2 it follows that the diagonals AC and BD are perpendicular and intersect at P. Thus, AB = .J242 + 322 = 40, BC = .J282 + 322 = 4y'fi3, CD = .J282 + 452 = 53, and DA .J452 + 242 = 51. The perimeter of ABCD is therefore 144 + 4 vTi3 = 4(36 + vTi3) . (Titu Andreescu, American Mathematics Contest 12 (AMC 12 - Contest B), 2002, with equality if and only if

.1.

=

Problem 24)

52.

ABCD. Assume without loss of gener­ V2PA = PB +PD.

Let a be the side length of the square ality that max( We have

PA, PC) = PA.

y A(O,a) P(x, y)

a..j2PA = aPB + aPD, hence BD · PA = AD · PB +AB · PD. From the converse of the Ptolemy ' s Theorem it follows that PDAB is a cyclic quadrilateral, therefore P lies on the circumcircle of square ABCD. Conversely, using the Ptolemy's Theorem we deduce that any point of the cir­ cumcircle of square ABC D has the given property. P Then

B ABCD. P Alternative solution. P(x, y) y 0. A, B, C, D PC PA, V2PC = PB + PD Squaring both sides yields 2X2 + 2(y + b) 2 = (x - b) 2 + (x + b) 2 + 2y2 + 2 V(x - b) 2 + y2 + 2 V(x + b) 2 + y2 , then 2by = V(x - b) 2 + y2 + V(x + b)2 + y2 , hence (x2 + y2 + b2 2bx)(x2 + y2 + b2 + 2bx) = 4b2 y2 .

It follows that the locus of point is the circumcircle of square be a point with the given property and assume Let are considered like in the diagram. > Point so > Note that

It

B(-a,O

x

(Titu Andreescu, Romanian IMO Selection Test, 1981; Revista Matematidl. Timi§oara (RMT), No. 2(1981), pp. 87, Problem 4751) 53. Let ABCD be a quadrilateral from the set P and let M, N, P, Q be the midpoints of sides AB, CD, BC, AD, respectively. The Euler's relation for the parallelogram MN PQ is MP2 + PN2 +NQ2 +QM2 = MN2 +PQ2 , or

C

Q

_

follows that

A

M

On the other hand, we have

Thus

P

D(a, 0) C(O, -a)

so

x 2 + y 2 = b2

BD ABCD.

so

A.

Likewise, for that contains and so point lies on the semicircle of diameter that contains and lies on the semicircle of diameter ::; we deduce that finally we obtain the circumcircle of square

y 0

131

3.2. SOLUTIONS

3. G EOMETRY

130

P

BD

C

Hence

DB MP = AC MQ = -, 2 2 AC2 + BD2 = MN2 + PQ2 . 2 AC2 � BD2 , A (P) + A� (P) = �

B

3.

132

3. 2.

GEOMETRY

In order to have equality everywhere we must have

which is, clearly, a constant.

(Dorin Andrica)

The first equality is equivalent to . rr sm

AF = 2R sin �;. 5rr

. sm 12 12 � + -. rr- = 4, sm sin 12 12 1 = 4 sm. -5rr12 sm. -12rr . 1 = 4 cos 12rr sm. 12rr or 1 = 2 sm. '6rr

which is clear. For the second equality, we have

AC2 -- 4R2 . 2 AD 2 - 4R2 sm. 2 12 ' AE2 -- 4R2 . 2 12 ' AF2 -- 4R2 sm. 2 5rr12 ' It reduces to rr 2rr + sm. 2 -3rr + sm. 2 -4rr + sm. 2 -5rr = -5 . 2sin2 - + sm 12 12 12 12 12 2 ' which is also clear. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 6(1971), pp. 27, Problem 821) 55. Let A 1 A2 . . . An be the given polygon and let S be the area of AIA2 . . . An . There is a point M inside the polygon such that n L k= 1 MA% = 2S. We write n S = L SAk MAk+l ' An+l = AI. k=1 Hence n MAk MAk+1 sin AkMAk+1 - � L..;n MAk MAk+ 1 S = L..; 2 2 k=l k= l 1� 1� ::; 4 L..; ( MA2k + MA2k+ 1 ) = 2 L..; MA2k = S. k=l k=l 2rr sm 12 ' 4rr sm

�

3rr

_

<

�

, no

is

-

Furthermore,

Ak�+l = 1 and MAk = MAk+1 , k = 1,2, . . . It follows that M is the circumcenter of a cyclic polygon and all sides subtent arcs of 90°. That is the polygon a square, desired. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1978), pp. 50, Problem 35 28) 56. In a triangle ABC with the altitude AA' and the circumradius R the following equality holds: AB · AC = 2R · AA'. (1) Let R be the circumradius of the polygon A lA2 .. . An and let P}, P2 , . . . , Pn be the projections of a point P on the circumcircle onto the sides A l A2 , A2 A3 , . . . , An Al, respecti vely. Applying (1) for triangles PAi Ai+l, where An+l = AI, yields PAi · PAi+1 = 2R · PPi , i = 1,2, sin

54. From the law of cosines we derive AB = 2R sin ;2 and

or

133

SOLUTIONS

<

as

. . . , n.

Hence

which is a constant,

as

claimed.

(Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 1-2(1980), pp. 65,

Problem 4123)

57. Applying relation (1) from the previous problem to triangles M Al A2 , MA3 A4 , MA5A6 , . . . , MA2n- l A2n , we obtain MAl ' MA2 = 2R · MK}, MA3 · MA4 = 2R· MK3 , . . . , MA2n-1 • MA2n = 2R · MK2n-1 • MUltiplying these equalities yields MK1 · MK3 · · · MK2n - l _- MAl · MA2nR2 n. . . MA2n (2) For the triangles

relation

(1) yields MA2 · MA3 = 2R · MK2 , MA4 · MA5 = 2R · MK4 , . . . ,

134

3.

3.2.

GEOMETRY

Multiplying these equalities gives

(3) M 2 M 4 . . . M 2n = MAl ' M2AnR2 n. . . MA2n Similarly, we obtain • MA n (4) MH1 . MH2 · · · MHn - MAl ' MA 2n R2n 2 by applying relation (1) to triangles MAl An+ l, MA 2 An+ 2 , . . . , MA n A2 n. From equalities (2), (3) and (4) we draw the conclusion. (Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1981), pp. 68, Problem 4622) 58. Let x and y be the measures of the arcs subtended by the sides a and b, respectively. We have nx + ny 27r, or 27r x + y = -. Let R be the circumradius of the polygon. Then . 2x = 2Ra and sm' 2y = 2Rb sm 27r - x so Now y = . 2'y = sm. ( ;;7r - 2X ) = sm. ;;7r cos 2X - cos ;;7r sm. 2x = 2bR' sm hence b + cos -7r SI. n -.X . -7r cos X2 = sm 2R n 2 Squaring b aths sides yields . 2 �7r cos2 2X 4Rb22 + 2 2Rb cos �7r sin 2X + cos2 ;;7r sin2 2X sm or b cos -7r sin -X + cos2 7r- sin2 -X . . 2 -7rn ( 1 sm. 2 -X2 ) = 4Rb22 + 2sm 2R 2 2 x a . � ' a S mce sm 2 = R' we btam 2 sin2 � (4R2 - a2 ) b2 + 2ab cos � + a2 cos2 � . K .

=

n

'

n

n

-

=

-

Therefore

(Dorin Andrica)

--

n

59. We have from the Cauchy-Schwarz Inequality that

K

K

=

n

n

n

2 2 R = �/a 2 sin - + 2ab cos :': + b . n

n

n

135

SOLUTIONS

(tt=l �) (tt=l ai . bi) (tt=l �) 2 �

for any positive numbers Setting n

= 4,

ai , bi , i = 1, 2, . . .

yields

, n.

BCD S 2 3V " SPA where V is the volume of the tetrahedron. Then " SBCD S2 S2 = � L...J PA - 3V rS r ' L...J

>

>

,

=

as desired. The equality occurs if and only if

� = a N. V b;

i = 1, 2,3,4 hence bi = a- I , i 1,2,3,4. Then PAl = PBI = PC1 = PD1, so P is the incenter of tetrahedron AB CD. Remark. The inequality holds for convex polyhedra circumscribed about a sphere. (Titu Andreescu, Romanian IMO Selection Test, 1982; Revista Matematica Timi§oara (RMT) , No. 1(1982), pp. 82, Problem 4910) 60. All summations here range from i = 1 to i = 4. Let be the circumcenter and R2be the 2circumradius of A1 A2A3 A4 . By the Power-of-a-point Theorem, GAi ·GA� = R - OG , for 1 � i � 4. Hence the desired inequalities are equivalent to (1) and (R2 - OG2 ) L G�i � L GAi• Now (1) follows immediately from (3)

for

=

0

136

3. G EOMETRY

by the Arithmetic-Geometric-Mean Inequality. To prove (3) , let P denote the vector from 0 to the point P. Then (4)

(4) vanishes. By Cauchy-Schwarz 4 L GA� � (L GAi) 2 and L GAi L G�i � 16, 41 6 GAi2 6 GAi1 � 1 (6 (GAi ) 2 ) 2 6 G1Ai � 6 GAi. Hence (2) also follows from (3) . (Titu Andreescu, IMO 1995 Shortlist)

This is equivalent to (3), since the last term of Inequality, so

�

�

16

�

�

�

Chapter 4 TRIGON OMETRY

PROBLEMS

1. Prove that 2. Prove that cos3 for all x E

JR.

X

-

3

+ cos3

X

+ 27r

--

3

+ cos3

X

+ 47r

--

3

=

3 cos x 4

-

3. Evaluate the sum n-1

Sn L k= l sin kx cos(n - k)x. =

4. Evaluate the sums

Sl S2

= =

+ sin 2x cos 3y + . . . + sin(n - l)x cos ny, cos x sin 2y + cos 2x sin 3y + . . + cos(n - l)x sin ny. sin x cos 2y

.

5. Evaluate the products 1) PI 2) P2

=

=

(1 - tg1°)(1 - tg2°) . . . (1 - tg8g 0 ) j (1 tg1 °) (1 tg2°) . . . (1 tg44°).

+

6. Prove that

+

+

(4 cos2 go - 3) (4 cos2 27° - 3)

=

7. Let x be a real number such that sec x - tan x 8. Evaluate the product where Ixl

< 2n+7r 2 ' 139

tan g o ,

=

2. Evaluate sec x + tan x.

4.

1 40

4 .1.

TRIGONOMETRY

9. Let a, b, e, d, x be real numbers such that x f:. k7r , k E sin x a

=

sin 2x -b

sin 3x

sin 4x

= -e- = -d- '

.z

and

17. Let a, (3 be real numbers with (3 � 1. Prove that

6 e = and 2 sin a - 6 sin b + 7 sin e - 9 sin d = 0. Prove that 3 cos(a + d) = 7 cos(b + e) . 2 cos a + cos b + 7 cos + 9 cos d

arccos a + arccos b + arccos

then

°

e = 7r,

for all a E

12. Let a, b, e be positive real numbers such that

e e = 1. 1 1 1 arctg - + arctg - + arctg - = 7r . a e b ab + b + a

13. Let x and y be real numbers from the interval (0, i) such that

7r Prove that x + y = 2 '

cos2 (x - y)

= sin 2x sin 2y

(1 + 2 sin2 a)t3 + (1 + 2 cos2 a)t3 � 213+ 1

JR.

18. Let x be a real number, x E [-1,1] . Prove that for all positive integers n.

19. Prove that for all integers

Prove that

-- --

sin3 a cos3 a > sec(a - b) + sin b cos b -

7r for all 0 < a, b < 2"

10. Let a, b, e, d E [O , 7r] such that

11. Prove that if

16. Prove that

14 1

PROBLEMS

n

1_ < x2n + (1 - x2 ) n -< 1 _ 2n - 1 -

+ cosec2nx � 2n+1 � and for all x E (0, �) . sec2n x

°

20. Prove that for all real numbers x.

(1 + sin x) (l + cos x) :::; � + V2

21. Find the maximal value of the expression

E = sin Xl COS X2 + sin x2 cos x3 + . . . + sin xn cos Xl ,

when X l , X 2 , . . . , Xn are real numbers.

22. Find the extreme values of the function f : JR

-t

1R,

= a cos 2x + b cos x + e, where a, b, e are real numbers and a, b > 0. 23. Let ao, aI , . . . , an be numbers from the interval (0, /2) such that tan ( ao - �) + tan (a1 - �) + . . . + tan (an - i) � - 1 . Prove that tan ao tan a1 . . . tan an � n+ f (x)

(0, i ) such that �2 (1 - tga) (l - tg(3) (1 - tg,) = 1 - (tga + tg(3 + tg,). Prove that a + (3 + , = "4' 15. Let a, b E (0, i ) , Prove that ( Si�2 a ) 2 + ( cos2 a ) 2 = 1 sm b cos b if and only if a = b. 14. Consider the numbers a, (3" 7r

E

7r

n

n

1.

24. Find the period of the function f(x) if p , q are positive integers.

=

cospx + cos qx,

x E 1R

142

4.

TRIGONOMETRY

4. 1 .

2

25. Let aD = � + V3 + v'6 and let an+1 = 2 (aann-+52) for n -> O. Prove that an = cot for all n.

(

34. Prove that in any triangle B

A

C

B

C

A

� cos3 -2 sin -2 sm. -2 = cos -2 cos -2 cos 2 � sm. 2 -2 '

2n 3 7r -- ) - 2

�

-

3

-

�

35. Let n be a positive integer. Prove that in any triangle

26. Let n be an odd positive integer. Solve the equation cos nx =

A

1 43

PROBLEMS

2n-1 cos x.

and

27. Solve the equation

L sin nA sin nB sin nC = (_ I) n+1 + cos nA cos nB cos nC L cos nA cos nB sin nC = sin nA sin nB sin nCo

36. Consider a triangle ABC such that

A sin 2 x + B sin 2x + C = 0,

sin A sin B + sin B sin C + sin C sin A = A

where A, B, C are real parameters. and

28. Solve the equation

(1 + sin A) (1 + sin B) (1 + sin C) = 2(A + 1)

. z cos x = '23 . y cos z + sm . x cos y + sm sm

29. Prove that the equation

. .2

4

Prove that triangle AB C has a right angle.

37. Let A > 1 be a real number and let ABC be a triangle such that

3

. 3x sm . x= 4 sm x sm x sm

has no real solutions.

and

30. Solve the system of equations

{

Prove that the triangle is isosceles.

2 sin x + 3 cos y = 3 4 3 sin y + cos x = .

2

38. Prove that the triangle ABC is equilateral if and only if 1 A B C tg "2 + tg 2" + tg = 4S 2"

31. Solve the system of equations

{X

sin y +

VI 7r

x + y = 4'

-

x2 cos Y =

�

2

a cos A + b cos B + ccos C =

39. Let ABC be a triangle such that Prove that triangle ABC has a right angle.

37r x+y+Z= 4 tgx + tgy + tgz = 5 tgx . tgy . tgz = 1.

33. Prove that in any triangle

(a2 + b2 + c2 ).

sin2 B + sin2 C = 1 + 2 sin B sin C cos A.

32. Solve the system of equations

1

aA cos B + bA cos A = cA a2A-1 cos B + b2A-1 cos A = C2A-1•

40. Let ABC be a triangle such that

(cot �r + (2 cot �r + (3 cot �r = ( �; r ,

abc -2R2

where s and r denote its semiperimeter and its inradius, respectively. Prove that triangle ABC is similar to a triangle whose side lengths are all positive integers with no common divisor and determine these integers.

T

1 44

1 45 51. Let ZI,Z2 ,Z3 be complex numbers, not all real, such that I Z11 I Z21 I z l 1 and 2(ZI + Z2 + Z3 ) - 3Z1 Z2Z3 E JR. Prove that 4. 1 .

4. 41. Prove that in any triangle . -A2 cos -B2 cos -C2 -< -89 . � sm TRIGONOMETRY

PROBLEMS

=

L..J

=

a =

42. Prove that in any triangle

A be ea ab >- 4 (sm. 2 -2 + sm. 2 -B2 + sm. 2 -C2 ) .

�

�

�

52.

-+-+-

43. Prove that in any triangle cos A

a3

+

cos B

b3

+

Let n be an even positive integer such that complex roots of unity of order n. Prove that

nII- l (a + bc%) (a � + b � )2 k=O for any complex numbers a and b. 53. Let be an odd positive integer and co, C1 , . .. , cn -l the complex roots of unity of order Prove that nII-1 (a + bc%) an + bn k=O for all complex numbers a and b. 54. Let ZI,Z2 ,Z3 be distinct complex numbers such that I Z11 I Z21 I Z3 1 Prove that

e3 >- � 16p3 .

=

cos C

44. Prove that in any triangle

2 A 2 B 2 C > _9 be ea ab .

sec - sec - sec 2 2 + ___ 2 + ___

__

45. Prove that in any triangle

!!.

-

n

n.

p2

=

> 3V3.

r -

46. Let ABC be a triangle. Prove that

A -2 B + cos B -2-C + cos C 2-A ' 47. Find the number of ordered pairs (a, b) such that (a+bi)2002 a-bi, a, b E 3A + sin 3B + sin 3C

sin 2

2

=

R

48. Find

and

Imz 5 Im5 and the values of z for which the minimum is reached. min

arg

:s arg

Z2 :s . . . :s arg Z2n

:s

7r.

55. Let ZI,Z2 ,Z3 be distinct complex numbers such that I Z11

Z2 f:. Z3 · Prove that

=

IZ2 1 . IZ2n l and = .: =

50. For all positive integers k define

Ak {z E I Zk I}. Prove that for any integers and with 0 < < we have Al U A2 U · · · U Am An-m+1 An-m+2 An . =

O. Consider the equation

k7l' n (2n - 1) � � cot 2 21 ' 3 k= 1 _

n+

sin(2n + l)x with roots

(1)

= 0,

n7l' 2 71' 71' ..., . 2n + 1 ' 2n + 1 ' 2n + 1 Expressing sin(2n + l)x in terms of sin x and cos x, we obtain

n = (2n : 1) cos2n x sin x en : 1) COs2n-2 sin3 + . . = = sin2n+, ( en : 1) cat2n en: 1) cot2n-2 x + . . . ) _

sin(2 + l)x

X

Set x

-

X

X

.

_

k7l' -I- 0 , we have , k 1 , 2 , . . . , n. S"Ince SIn2n+ l x -r 2n + 1 2n 1 1 cat2n X _ cot 2n-2 x + . . . O .

--

-

en: ) ( :) = Substituting y = cot2 x yields en: l) yn en: l) yn- l + . . . = 0, -

n7l' Using the relation between coefwith roots cot2 71' , cot 2 � , . . . , cot 2 2 +1 2n + 1 2n + l ' ficients and roots, we obtain 2n + 1 n k n(2n - 1) 3 7l' � 2. cot2 2n + 1 1 3

n

-

( ) = en: )

= 3, the desired conclusion follows. (Dorin Andrica, Revista Matematidi Timi§oara (RMT), No. 1-2(1979) , pp. 51,

Setting n

Problem 3831)

1 47

148

2. Applying the identity cos for

.[

4.2. SOLUTIONS

4. TRIGONOMETRY

t = x, t = x + 271", t

=

t = 4 cos3 3"t - 3 cos 3"t ' t E

lR

x + 471" and summing up the three relations, we obtain + 271" + 471" 4 cos3 3' + cos3 -- + cos3 -- 3 3

( X X X ) 3 cos x = ( X3 X +3 271" X + 471" ) -3 cos - + cos -- + cos -- . 3

On the other hand, x + 471" x 271" cos - + cos -- + cos 3 3 3

+

X = --

+ = --- + --

2x + 471" 471" 2 cos - cos 6 6 x 21T + 1 cos 0

= (2 cos � )

�

cos

82 - 81

Hence

271"

X

81

3

=

and the desired identity follows. Revista Matematica Timi§oara (RMT) , No. 2(1975), pp. 44, Problem 2124)

=

= = + . . . + sin(n - l)x cos x+ + sin(n - l)x cos x + sin(n - 2)x cos 2x + . . . + cos(n - l)x sin x = = sin nx + sin nx + . . . + sin nx = (n - 1) sin nx, n-1 8n = -- sin nx. (Dorin Andrica, Gazeta Matematica2 (GM-B), No. 8(1977) , pp . 324, Problem 28n 8n + 8n sin x cos(n - l)x + sin 2x cos(n - 2)x

so

1 2

[

sin

.

]

+ (n - 1)h2 ) =

]

� sin [Y + (n - ll'� ] + sin � sin [Y + ( n - l) �] .

n l

1

= h . h2 2 sm sin -f 2 (Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1977), pp. 65, 82

Problem 3056)

5. We have PI = 0 because of the factor 1 - tan 45° = O. On the other hand we have ( cos 1 ° + sin 1 ) . . . (cos 44° + sin 44°) P = _

0

(../2

2

../2 . ) (../2 ( )

cos 1 ° . . . cos 44 °

. 440 y'2 sm ° . . . T COS 440 + T T cos 1° + T sm 1 ../2 �

16803; Revista Matematica Timi§oara (RMT) , No. 2(1978) , pp. 30, Problem 3055)

4. Note that

44

cos 1 ° . . . cos 44 °

)

= sincos46°1 ° . .. ...cossin4489°° (v'2) = 222 . 44

= sin(x + 2y) + sin(2x + 3y) + . . . + sin[(n - l)x + nyJ and 82 - 8 = sin(2y - x) + sin(3y - 2x) + . . . + sin[ny - (n - l)x] . Setting x + y = hI and y - x = h 2

(Titu Andreescu)

81 + 82 1

yields

81 + 82

.

+

[

.

]

hI . n hl sm T sm y + (n - 1) 2 . hI sm 2 h2 ) + sin(y 2 h2 ) + . . . + sin(y . nh2 sm y + (n - 1) h2 sm 2 T h2 . sm 2

. n hl sm y + ( n - 1) hI sm 2 T h1 sm 2

and

(Dorin Andrica,

3. We have

=

1 2

= sin(y +

149

= sin(y + hI ) + sin(y + 2hI ) + . . . + sin(y + (n - l)hI ) =

6. We have cos 3x = 4 cos3 x - 3 cos x, so 4 cos2 X - 3 =

(2k + 1) . 90°, k E Z. Thus

(4 cos2 9 ° - 3) (4 cos2 27° - 3) as

desired.

(Titu Andreescu)

--

cos 3x for all x cos x

sin 9° cos 81 ° -cos 27° . -= coscos819°° = = --- = tan 9 ° , cos 9 ° cos 9 ° cos 27°

f.

15 0

4. 2 .

4. TRIGONOMETRY

7. From the identity 1 + tan2 x = sec2 x it follows that

(Titu Andreescu, Korean Mathematics Competition, 2002)

= sec2 x - tan2 x = (sec x - tan x) (sec x + tan x) = 2(sec x + tan x), so sec x + tan x = 0.5. (Titu Andreescu, American High School Mathematics Examination, 1999, Prob­ 1

11. From the hypothesis it follows that sin( arccos a + arccos b + arccos c)

for all Ixl

<

+

1 tan2 2 k x (1 - tan2 2 k x)2

: it follows that 2 +2 '

=

L cos a cos ,B sin 'Y

cos2 2 k x cos2 2 k+ 1 X

sin(arccos x)

sin(a + ,B

=

+ 'Y) + sin a sin ,B sin 'Y

� and sin(arcsin x)

we obtains ab � + bc � + ca J1=b2 as desired.

9. Let

Problem 3054)

Because sin2 4x =

=

and the formulas

(Dorin Andrica)

= x,

x E [- 1, 1] ,

= V(I - a2 ) (1 - b2) (1 - c2 ) ,

(Titu Andreescu, Revista Matematidi Timi§oara (RMT) , No. 2(1977) , pp. 64,

sin x sin 2x sin 3x sin 4x -- - -- - -- - -- - ;\ b a c d _

_

Then

= O.

Using the identity

lem 15)

8. Since

151

SOLUTIONS

_

_

12. The identity

.

sin 2 4x = 2 sin2 2x(l - sin2 2x). 2 2 sin 2x(1 - sin2 2x) , we obtain

= 2b2 (1 ;\2 b2 ) . On the other hand, sin 3 x = AC, sin x = Aa, and since sin 3x = sin x (3 - 4 sin 2 x), we have c = a(3 - 4A2 a2 ). d2

_

Eliminating A from the relations (1) and (2) yields 2a3 (2b3 - �) = b4 (3a - c),

as desired.

(Dorin Andrica)

10. Rewrite the two equalities as 2 sin a - 9 sin d

+

= 6 sin b - 7 sin c

arctgx + arctgy + arctgz implies

(1)

and the conclusion follows.

=

85

+ 42 cos(b + c) ,

+ + +

1 1 1 ab + bc ca - 1 arctg- arctg- + arctg- = arctg k7r. ab c - ( a + b c) a b c Because ab + bc + ca = 1, we obtain 1 1 1 arctg - + arctg + arctg = k7r, a b c where k is integer. 7r Note that 0 < arctgx < - for all real x > 0, hence 2 1 1 3 7r 1 o < arctg + arctg z; + arctg c < � 2' Therefore k 1 and 1 1 1 arctg- + arctg- + arctg- 7r, a b c as claimed. Revista Matematidi Timi§oara (RMT) , No. 1 (1977), pp. 42, Problem 2827)

+

-

(2)

=

-

=

(Titu Andreescu,

13. The given relation is equivalent to

2 cos a 9 cos d = -6 cos b - 7 cos c. By squaring the two relations and adding them up we obtain 85 + 18 cos(a + d)

= arctg 1 x-+(xyy++zyz- +xyzzx) + k7r

= 4 sin x sin y cos x cos y, (cos x cos y - sin x sin y) 2 = 0

(cos x cos y + sin x sin y) 2 or

152

4. Hence

cos2 (x +

=

y) = 0,

x, y E (o, �), we obtain x + y = �, as desired. Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1(1977) , pp. 42,

and since (

4. 2.

TRIGONOMETRY

Problem 2826)

or tan 2 a tan2 b. Because a, b E 0, we obtain a b. The converse is clear and we are done. From the given relation we deduce that there is a number such that e E 0,

( -i),

Alternative solution.

sin2 a sin b

1 "2 (1 - tan a - tan ,B - tan 1 + tan a tan ,B + tan ,B tan 1+ + tan 1 tan a - tan a tan ,B tan 1) 1 - (tan a + tan ,B tan 1) ,

Hence

=

or

°

<

<

sin 2 a It follows that

From relation (1) we derive 1 - tan a tan ,B - tan ,B tan 1 - tan 1 tan a 1' tan a tan ,B + tan 1 - tan a tan ,B tan 1 therefore cot(a + ,B + 1) 1. Hence a + ,B + 1 as desired. ( Revista Matematica Timi§oara (RMT) , No. 1(1973) , pp. 42, Problem 1582)

=

=

= i, Titu Andreescu,

15. The relation in the statement is equivalent to

(

sin4 a cos4 a . 2 b + cos2 b) (sm -+ .2 b -cos2 b sm

or •

sm4 a It follows that

hence

+

cos2 b . 4 - sm a cos 4 a + -sin2 b

+

) = 1,

sin2 b --- cos4 a cos2 b

Furthermore,

cos b . 2 sm a sin b

= cossin bb cos2 a,

= sin b sin e

and

= cos(b - e)

cos2 a cos b

= cos e cos2 a = cos b cos e.

and

cos 2a

as

Problem 2825; Gazeta Matematica (GM-B), No. 11(1977) , pp. 452, Problem 16934)

=

16.

Multiplying the inequality by sin a sin b + cos a cos b cos(a - b), we obtain the equivalent form Sin3 a cos3 a . . --- (sm a sm b + cos a cos b) 2: 1. . -sm b cos b

(

)

+

But this follows from Cauchy-Schwarz Inequality, because, according to this in­ equality, the left-hand side is greater than or equal to (sin2 a + cos2 a) 2 1. (

=

Titu Andreescu)

17. Using the inequality

x? + xr -> ( 2

Xl

+ X2 2

)m

for m 2: 1 we obtain

= 1.

cos 2 b . 4 sin2 b . 2 a cos 2 a + 1 - 2 sm -- sm a + --- cos4 a . sm2 b cos2 b sin b 2 ) 2 (-COSsm.-bb sm. 2 a - -cos a = 0 . cos b

and

°

(1)

tan a + tan ,B + tan 1 1= tan a tan ,B tan 1.

= sm. e

= cos(b + e) Since a, b, e E (0, i), we have b - e = and 2a = b + e, hence a = b, desired. ( Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1(1977) , pp. 41, 1

7r ,

+

=

( i)

14. Expanding yields

= + tan a + tan ,B + tan 1 - tan a tan ,B tan 1 = = 1 - tan a tan ,B - tan ,B tan 1 - tan 1 tan a. Since a, ,B, 1 E ( 0, i ) , we have a + ,B + 1 hence

1 53

SOLUTIONS

= 1,

(1 as

+ 2 8in2 alP + (1 + 2 C082 aJII � 2 C + 2 8in2 � + 2 cos2 a ) p = 2P+1 ,

desired. (

Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1(1974) , pp. 30,

Problem 1942)

Because X E [- 1, 1J , there is a real number y such that x = sin y . It suffices to prove that 1 > sin2n + cos2n > __ 12n - 1

18.

y

y

� 1 and I cos yl � 1, hence sin2n y + cos2n y � sin2n-2 y + COS2 n -2 y � . . . � sin2 y + cos2

I

Y

= 1,

desired. For the right-hand side we use the inequality x? x� > Xl X2 n 2 2

( + )

+

Hence as

as

'4

2

2

vIn

=

"2

=

"2

(Titu Andreescu, Revista Matematidl Timi§oara (RMT) , No. 2(1978) , pp. 47,

claimed.

Alternative Solution. By setting u X2 and v = 1 - X2 the inequality becomes u,v E [0, 1] and u + v = 1, we have un � vn � v, n- l � uunn++vvnn �� 1. +Because 2implying v 1. Also, by then power mean inequality, 1_ un + vn - 2 ( u+2 V ) = 2 (!2 ) n = _ 2n - l . (Dorin Andrica) =

1

U

U,

=

>

19. We have

+ 1 2:: 2 tan x and cosec2 x = cot2 x + 1 2:: 2 cot x by the AM-GM inequality. It follows that sec 2n x + cosec 2n x 2:: 2 n (tan n x + cot n x). Since tan n x + cotn x 2:: 2, we obtain sec2n x + cosec2n x 2:: 2n +l , sec2 x = tan2 x

as

+ (sin x + cos x) = 3 + V2 sin (x + 7r ) � 3 + 2, desired. Note that equality holds for x � + 2k 7r , where k is integer. Alternative Solution. Expanding the left-hand side, we see that 1 + (sin x + cos x) + � sin 2x = 1 + V2 sin (x + �) + � sin 2x � � 1 + V2 + 1 3 + V2. = 23

For the left-hand side note that sin yl as

155

4.2. SOLUTIONS

4. TRIGONOMETRY

154

desired.

Alternative Solution. Using the AM-GM inequality we obtain sec2 n x

>

+ cosec2n x - 2Jsec2n xcosec2nx = 2 sinn x1cosn x

(Dorin Andrica,

l_ = 2n+ l __ n

>

2n+l sin 2x Gazeta Matematidi (GM-B), No. 3(1975), pp. 104, Problem

-

14900)

20. We have

Problem 3500)

21. We have E = sin Xl COS X2 + sin x2 cos x3 + ... + sin xn cos X l �

� sin2 Xl +2 cos2 X2 + sin2 X2 +2 cos2 X3 + . . . + sin2 Xn +2 cos2 Xl 2 Therefore the maximal value of E is and it is reached, for example, when 2 7r X l = X2 = . . . = X n = '4 ' (Dorin Andrica, Revista Matematidi Timi§oara (RMT), No. 2(1977), pp. 65, n

n

Problem 3058)

22. Because a, b

>

0, it follows that the maximal value of f is Setting y = cos x yields f (x)

=

a(2y 2

a+b+

c.

- 1) + by + c = 2ay2 + by + c - a.

If - � E [- 1, 0) , then the minimal value of f is 4a � 8a(c - a) 8a 8a

-b2 +

- :a E ( -00 -1), then the minimal value of is f ( -1) 2a - b + c - a = a - b + c. (Dorin Andrica, Revista Matematidi Timi§oara (RMT) , No. 1(1981) , pp. 52,

If

,

f

=

Problem 4315)

. )(1 + coo x) - (1 + sin x) 2 + (1 + COS X)2 2

(1 + ffin x _

-

2

<

+ 2(sin x + cos x) + (sin2 x + cos2 x) _

2

=

23. First Solution.

Let hypothesis that for each k,

bk = tan(ak - 7r/4), k = 0, 1, . . . ,n.

-1 bk 1 and 1 + bk 2:: 9¥k5; L n (1 - bl ) . <

It

follows from the

<

0

(1)

156

157

4. TRIGONOMETRY

4.2. SOLUTIONS

Applying the Arithmetic-Geometric-Mean Inequality to the positive numbers 1 bz , = 0, . . . , k - 1, k + we obtain

other by a positive number chosen so that the sum of the pair does not change. Each such change decreases the product of the i'S. It follows that for a given sum of the bi 'S, the minimum product is attained when all of the bi ' S are equal. In this case we > for each so have bi n + 1'

l 1,

1, . . . ,n, l in (2) II n (1 bl) l=lk'5,n (1 - bl) � n 09=1k'5, O'5,L From (1) and (2) it follows that l in !!n (1 + bk ) � nn+1 (Don (1 - bt )n) , and hence that n 1 + bk n+l II -1 - b-k - n . k=O Because 1 + bk -- 1 + tan (ak - �) = tan ( (ak - �) + �) = tan ak , -1 - b k 1 - tan (ak - �) the conclusion follows. Second Solution. We first prove a short lemma: Let W,X,Y,z be real numbers with x + y w + z and Ix - yl < Iw - zl . Then xy. wz

=

=

>

-1 < bk < 1 and (1) tjtk G � ��) G � :: ) 1 + 1 + bj�k 1 bj + b + k First note that because -1 < bk < 1 and bo + b1 + . + bn � n - 1, it follows that bj + bk 0 for all 0 ::; j, k ::; n with j f:. k. Next note that if bj + bk > 0 and bj f:. b k , then it follows from the lemma applied to (1) that the value of tj tk can be Then

=

=

>

.

_

made smaller by replacing bj and bk by two numbers closer together and with the same sum. In particular, if bj < 0, then replacing bj and bk by their average reduces the problem to the case where bi > 0 for all We . may now successively replace the b/s by their arithmetic mean. As long as the bi are not equal, one is greater than the mean and another one is less than the mean. We can replace one of this pair by the arithmetic mean of all the bi'S, and the

i.

=

1 58

4.

4.2.

TRIG ONOMETRY

To establish (2) it is sufficient to show that for 0 � x < 1, I(x) attains its minimum value at x = aJ (n + 1) . Towards this end we differentiate to obtain 1 ! ' (x) = n (a - (n + l)x) ( (l + x) n - 1 (n + 1 - a) n+ - (1 - x) n - 1 (n + 1 + a) n+ 1 ) = = n (a - (n + l)x)g(x) , 1 n where g(x) = (1 + x) - (n + 1 - a) n+ 1 - (1 - x) n -1 (n + 1 + a) n+ 1 . It is clear that I'

(n: 1)

=

0, so we check the second derivative. We find

( : l ) = -n(n + l)g ( n : l ) > 0,

f" n

so I has a local minimum at x = aJ (n + 1). But I' (x) could have another zero, obtained by solving the equation g(x) = Because

O.

t,

-

g' (x) = (n - 1)(1 + x) n- 2 (n + 1 - a) n+ 1 + (n - 1) (1 - x) n 2 (n + 1 + a) n+ 1 is obviously positive for all x E [0, 1), there is at most one solution to the equation g(x) = 0 in this interval. It is easy to check that g(aJ (n + 1)) < 0 and g(l) > O. Thus there is a real number aJ (n + 1) < < 1, with g(t) = For this we have

t,

t

t

O. I" (t) = n(a - (n + l)t)g' (t) < O.

t

Thus, is a local maximum for I, and no other extrema exist on the interval (0, 1). The only thing left is to check that 1 (1) � l(aJ(n + 1)). Note that the case x = 1 is also an extreme case with bo = b1 = . . . = bn - 1 = 1 . This case does not arise in our problem, but we must check to be sure that on the interval 0 � x < 1, I (x) has a minimum at x = aJ (n + 1). We have 1(1) = 2n (1 + a - n) (n + 1 - a) n+ 1 0,

�

since n - 1 � a � n + 1, and l(aJ (n + 1)) = 0 (by design). Thus I(x) indeed attains a unique minimum at x = aJ(n + 1). USA Mathematical Olympiad, 1998, Problem 3)

(Titu Andreescu,

24. Let d be the greatest common divisor of p and q. We prove that T = 27r is d the lowest positive period of the function f. It is clear that I(x + T1 ) = I (x)

for all real x, therefore T is a period of function I. Suppose there are T1 > 0 and an integer A > 0 such that T = AT1 and l (x + T1 ) = I (x) for all real x. Then I( T1 ) = 1(0) = 2, so cos pT1 + cos qT1 = 2,

1 59

SOLUTIONS

therefore

COSpT1 = COS qT1 = 1. It follows that

T1 =

O.

2k l7r

P

=

2k2 7r q

for some integers k1 ' k2 > 27r Since T = AT1 and T1 = Ad ' 1 k1 k2 =q= and so p = k1 ( Ad) , q = k2 ( Ad) . Ad P On the other hand, d = gcd (p , q) , so A = 1, hence T = T1 as desired. Revista Matematidt Timi§oara (RMT) , No. 2(1978), Problem 3695)

(Dorin Andrica, 25. We have

pp.

75,

7r 7r 7r 1 + cos 12 2 cos2 cos 7r 24 24 = -.-7r- = cot 7r . 24 sm sm 2 sin !!..- cos !!..12 24 24 24 \I'2 v'6 l+ + 1 + cos 4 4 7r 7r . sm 3 - "4 4 4 2 4( + ( ) + + 4( v'6 \1'2) v'6 \1'2 v'6 + \1'2) + 8 + 4V3 4 6-2 = 2 + \1'2 + V3 + VB = ao + 2. 2n 3 7r - 2 is true for n = 0. Hence an = cot 2n 3 7r 1. The It suffices to prove that bn = cot , where b n = an + 2, n recursive relation becomes ( b - 2) 2 - 5 bn+ 1 2 - n ' 2bn 2 k -3 7r b2 - 1 . Assuming, inductively, that bk = cot ck , where Ck = -3- ' yields or bn+ 1 = ;bn cot 2 Ck - 1 = cot(2Ck) = cot Ck + 1 , bk + 1 = 2 COt ck and we are done. Korean Mathematics Competition, 2002)

(i �) ( )

_

_

( ; )

_

(; )

_

_

(Titu Andreescu,

26. If n = 1, then all real numbers x are solutions to the equation.

2::

4.

1 60

4. 2.

TRIGONOMETRY

Let n > 1 and note that cos nx

= (�) cosn - (�) cosn-2 x sin2 x + . . . + n + ( - 1) """"2 ( _ ) cos x sin n - 1 n 1 X

n-l

X.

We have two cases: a) x f:. (2k + 1) � for any integer k. Then 2

l (�) cosn- 1 - (�) cosn-3 sin2 + . . . + + ( - 1) n;' ( � ) sin n-1 x l ::; n 1 ::; I cos xl ((�) I cosn - 1 xl + (�) i cosn - 3 x sin 2 xl + . . . + ( � ) I sin n - 1 xl ) < n 1 I cos nxl

=

I cos x '

X

X

hence there are no solutions in this case. b) x (2k + for some integer k. Then

=

l)i {

i

cos x

= 0 and cos nx = 0, }

since n is odd, so (2k + 1) lk integer is the set of solutions.

Alternative Solution. With the substitution x = i - y the equation becomes cos (n � - ny ) = 2n - 1 sin y. (1)

Because n is odd, (1) is equivalent to

= 2n-1 sin y. Taking modules gives (2) I sin nyl = 2n - 1 1 sin yl . But I sin nYI � nl sin yl for all y in hence nl sin yl � 2n-1 I sin yl · If y f:. k7r, k E Z, then n � 2n - 1 , which implies n E {1, 3}. The case n = 1 is clear and for n = 3 the original equation reduces to cos 3x = 4 cos x, that is 4 cos3 - 3 cos x = 4 cos x. Taking into account that cos x f:. 0, this yields cos2 x = �, which i s not possible. It follows that y k7r, which gives the solutions x = (2k + l)i, k E Z. (Titu Andreescu, Gazeta Matematidi (GM-B), No. 7(1978), pp. 304, Problem ± sin ny

27. The equation is equivalent to

(A + G) sin2 x + 2B sin x cos x + G cos2 = 0 We have the following cases: i) A + G = 0 and G f:. O. Then 2B cos x = 0 or cot x = - 0 ' hence 2 (2k ; 1) 71" xE { 1 k E Z } U { arcctg ( - %) + h l k E Z } . ii) A + G f:. 0 and G = O. Then 2B sin x = 0 or tan x = --­ ' A+G hence 2 x E {k71" 1 k E Z} U { arctg ( - : ) + hi k E Z }. A iii) A = B = G = O. Then any real number is a solution. iv) A = G = 0 and B f:. O. Then sin 2x = 0 and so x E {ki l k E Z } . v) A + G f:. 0 and G f:. O. The equation is equivalent to X

c

(A + G) tan2 x + 2B tan x + G = 0,

hence tan x for B 2 + G 2

=

17297; Revista Matematidl Timi§oara (RMT) , No . 1-2(1980), pp. 63, Problem 4107)

= -B ± JB2A +_GAG + G2

� AG. It follows that

Z} U

x E {arctgY1 + k7r 1 k E {arctgY1 + k7r1 k E Z } if B 2 + G2 AG. Otherwise there are no solution. Revista Matematica Timi§oara (RMT), No. 1(1978), pp. 89, Problem 3429)

� (Dorin Andrica,

lR,

X

16 1

SOLUTIONS

28. The equation is equivalent to 2 sin x cos y + 2 sin y cos z + 2 sin z cos x or It follows that

(sin x - cos y) 2 + (sin y - COS Z ) 2 + (sin z - COS X ) 2 = O. sin x

Hence

= 3,

= cos y,

sin y

= cos z ,

sin z = cos x.

1 62

4. Z

4.2.

TRIGONOMETRY

7r + X = (4k3 + 1 ) '2

and

for some integers k1 ' k2 ' k3 and therefore

�

Y

x = [4(k1 - k2 + k3 ) + 1 ] ,

= [4(k1 + k2 - k3 )

�

and

z = [4(-k1 + k2 + k3 ) + 1] ,

+ 1] � ,

k1 , k2 , k3 E z .

4 y = arctan 3 + 2l7r,

for some integers k and l. Revista Matematica Timi§oara (RMT ) , No. 2 ( 1978 ) , pp. 74, Problem 3694)

(Titu Andreescu,

(Titu Andreescu, Gazeta Matematica (GM-B ) , No. 11 ( 1977) , pp. 451, Problem

31. Observe that x E [-1, 1] and

29. Note that

From the first equation we obtain

x = sin ( arcsin x) ,

16931 ; Revista Matematica Timi§oara ( RMT ) , No. 1-2 ( 1979 ) , pp. 52, Problem 3835 )

sin x sin 2x sin 3x sin 4x =

� (cos 3x - cos 5x) (cos x - cos 5x) =

1 = 4 ( cos2 5x - cos 3x cos 5x - cos 5x cos x + cos x cos 3x ) =

1 6 3 = - ( 2 cos2 5x - cos 2x + cos 8x - cos 4x cos 6x ) < - = -, 8 4 8 hence the equation has no solution. Revista Matematica Timi§oara (RMT ) , No. 1 ( 1977 ) , pp. 41, Problem 2923 )

+

(Titu Andreescu,

30. Squaring both equations and summing up yields 4(sin2 x

+ cos2 x) + 9 (cos2

or 13 Hence

Y

+ sin2 y ) + 12 (sin x cos y

+ sin y cos x) = 25,

+ 12 sin(x + y) = 25. sin ( x + y ) = 1,

and so X

7r + y = (4k + 1 ) '2

3

tan x = 4 '

. .j2 cos ( y - arcsm x ) = 2' then y - arcsin x = 7r

. x + y = "4 ' we get U smg for some integer k. Case x + arcsin x = have k = 0, hence

1.

Therefore

� - 2k7r . Because x E [- 1, 1] and arcsin x E [-�, �] , we 7r x + arcsm x = '2 •

7r arcsm x = '2 - x, •

or x = cos x.

For this equation there is only one solution X o E solution 7r Y = "4 - Xo X = Xo ,

(0, �). The system has the

+

arcsin x = -x .

sin y = cos x.

4

tan y = 3 ·

Note that sin x, cos x, sin y, cos y are all positive, therefore 3 x = arctan 4 2k7r

+

±� + 2k7r.

� ± � + 2k7r,

x + arcsin x =

2.

Turning back to the system we obtain . x = cos y = 53 and sm . y = cos x = 54 ' sm hence

� = cos ( arcsin x) .

Case x arcsin x = 2k7r. Using similar arguments, k = 0, so

for some integer k. It follows that sin x = cos y and

1 63

SOLUTIONS

This equation has the unique solution x = ° so the system has the solution x = o,

(Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1 ( 1977) , pp. 41,

Problem 2824)

32. Using the formula t an (x

+ y + z) = 1tan- tanx +xtantanyy+-tantanzy-tantanz x-tantanyztantanzx

--=----------=---

-

164

4.

we have -1 = tan

4. 2 .

TRIGONOMETRY

because the first column is the sum of the other two. Computing the determinant, we obtain A A B O A . B . O - sm - = cos - cos - cos - """ sin2 """ cos3 - sm 2' 2 6 2 2 2 2 2 6 as desired.

5-1 37r = 4 1 - tan x tan y - tan y tan z - tan z tan x

Hence tan x tan y + tan y tan z The equation

+ tan z tan x = 5.

t3 - 5t2 + 5t - 1 = 0

(Dorin Andrica)

has roots tan x, tan y, tan z, from the relations between the roots and the coefficients. On the other hand, the equation has the roots 1, 2 V3, 2 - V3, hence {x, y, z} ==

+

{� + k", �; + h", ;2 + p,,}, for some integers k,l,p.

35. Denote E1 =

L sin nA sin nB cos nO

Observe that

(Titu Andreescu, Revista Matematica Timi§oara (RMT), No. 8(1971), pp. 27,

(cos A

Problem 1018)

33. Using the Extended Law of Sines we obtain

+ sin O cos 0) = R(sin 2A + sin 2B + sin 20) = R(2 sin(A + B) cos(A - B) + sin 20) = A-B + 0 A-B-O = = 2R sin O(cos(A - B) + cos O) = 4R sin O cos cos 2 2 a cos A + b cos B + c cos O = 2R(sin A cos

= -4R sin 0 cos as desired.

(�2 - B) cos (�2 - A)

A

==

+ sin B cos B

4R sin A sin B sin 0

==

==

abc 2R2 '

(Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1977) , pp. 65,

Problem 3060)

A B B O . O . "2 cos "2 = sm cos "2 + sm "2 cos "2 ' A 0 0 A . "2 . "2 cos "2 = sm cos "2 + sm cos "2 ' B

A B B A . "2 . "2 cos "2 = sm cos "2 + sm cos "2' O

0 A sin - cos 2 2

0 cos "2

B sin - cos 2 2

A

==

L cos nA cos nB sin nO.

+ i sin A) (cos B + i sin B) (cos 0 + i sin O) =

= cos(A + B + 0) + i sin(A + B + 0) = cos 7f By de Moivre ' s formula, (cos nA

+ i sin 7r = -1.

==

+ i sin nA) (cos nB + i sin nB) (cos nO + i sin nO) = (_ l) n .

Expanding the brackets yields -E1 + Hence

iE2 + cos nA cos nB cos nO - i sin nA sin nB sin nO = (_ l)n .

E1 = ( _ 1) n+ 1

(Dorin Andrica,

+ cos nA cos nB cos nO

and E2 = sin nA sin nB sin nO. Revista Matematica Timi§oara (RMT), No. 1(1978) , pp. 65,

A 0 sin - cos 2 2 B A sin - cos 2 2

36. Subtracting from the second equality the first multiplied by 2 yields (sin A - 1) (sin B - l) (sin 0 - 1) = O. Hence sin A, sin B or sin 0 is 1, so AB 0 is a right triangle. Revista Matematica Timi§oara (RMT), No. 2(1978), pp. 49, Problem 3514)

(Titu Andreescu, 37. Note that

A . B 0 . 0 B cos - sm "2 cos "2 sm "2 cos 2" 2 B cos "2

and &

Problem 3278)

34. Because A + B + 0 = 7r, we have

Hence

1 65

SO LUTIONS

= 0,

{ aA

a cos B + b cos A - c = O.

The system of linear equations

bA A 2a A- 1 b b2 A- 1 c C2 A- 1 = 0 a cos B + cos A - c = 0 cos B + cos A - = 0 cos B + cos A -

166

4.

4. 2.

TRIGONOMETRY

has the solution (cos B, cos A, - 1) and is homogeneous. Therefore the determinant Ll =

a aAA-1 a2

b c b A C cA b2A-1 2A-1

From the hypothesis we have sin 2 B

+ sin2 C = 1 + 2 sin B sin C cos A,

therefore sin 2 A = 1. It follows that A = �, hence the triangle ABC is right, as 2 desired. Revista Matematica Timi§oara (RMT) , No. 1-2(1977), pp. 52, Problem 3838)

(Dorin Andrica,

is zero. On the other hand, 1

1

1 abc aAA--1 bAA-1-1 CcA-1(a 1 )2 (b ) 2 ( A 1 ) 2 = abc(aA - 1 _ b A -1 )(aA-1 _ cA-1 ) (bA-1 _ cA-1 ) . Therefore a = b, b c or c a, hence the triangle is isosceles. (Dorin Andrica, Revista Matematidi Timi§oara (RMT) , No. 2(1977), pp. 89, Problem 3 1 99 ) Ll

167

SOLUTIONS

=

=

=

2 + 32 + 22 = 72 and A B C r cot -2 + cot -2 + cot -2

40. Because 6

s

- =

1 2 J(P - b)(p - c) + J(P - c)(p - a) + J(P - a)(p - b) = S ( + b2 + c2 ) 4 Jp(P - a) Jp(p - b) c) Jp(p or 1 1 2 S L(P - a)(p - b) S (a + b2 + c2 ). 4 Expanding the brackets yields _p2 + ab + bc + ca = -41 (a2 + b2 + c2 ), a

=

then

4(ab + bc + ca) a2 + b2 + c2 + (a + b + C)2 . It follows that (a - b) 2 + (b - C)2 + (c - a) 2 0, hence b = c. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1972) , pp. 28, =

=

39. By the Extended Law of Sines,

a = 2R sin A, b

=

2R sin B,

c = 2R sin C.

On the other hand,

This means that we have equality in the Cauchy-Schwarz inequality. It follows that C B A cot - 2 cot - 3 cot 2 2 2 2 3 6 A B 7 C 7 Plugging back into (1) gives cot 2 = 7, cot 2" = 4 ' and cot 2" = g ' Hence by 63 28 7 . . . . the Double angle formulas, sm A = , sm B = , and sm C = 1 3 0 Thus the SIde 65 25 lengths of T are 26, 40, and 45. USA Mathematical Olympiad, 2002, Problem 2)

__ -

_

__

-

-

41 . Summing up the formulas ra

A B C = 4R sin 2 cos 2" cos 2 '

rb = 4R sin 2"B cos 2"C cos 2A ' r = 4R sm. 2C cos 2B cos 2A yields B C r + rb + r . A ""'" sm ' 6 2 cos "2 cos "2 4R On the other hand + rb + 4R + hence B C 4R + r . A ""'" sm 6 2 cos "2 cos 2 = --m- = 1 + 4R ' c

ra

sin2 B

+ sin2 C

=

sin2 A

+ 2 sin B sin C cos A.

'

(Titu Andreescu,

=

so

(1)

= (6 cot 2A + 6 cot 2"B + 6 cot 2"C ) 2

a =

Problem 1160)

A B C cot - cot - cot - ' 2 2 2

the given relation is equivalent to

38. The relation is equivalent to ....!.=--;:::::;=: .==:::;:-=::-

=

rc =

r,

c

a

r

168

Because

as

� � r, it follows that

hence

1 9 A B C < 1+- = '" sin - cos - cos - L...J 2 8 8' 2 2

desired.

(Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1978), pp. 49,

42. We have

b e B ecause � + b

�

a2 2 cos A + be . 2, we obtam a2 be

�

=

!!.- + sin2 Q2 ) '

(Dorin Andrica)

B be ea 1 , sec2 "2 = , --A -= p(p - b) P (p - a) cos2 2 so it suffices to prove that 1 9 1 1 -- + -- + -- > . p - a p - b p - e - pSetting x = p - a, Y = P - b, Z = P - e in the inequality

(

(x + Y + Z) .!. + � + � x Y Z P

2be cos A 2ae cos B 2ab cos C + + b2 a2 e2 _

,

) �9

1 1 1 ( p-+ -- + -- ) > 9 , -a p-b p-e -

45. By the AM-GM inequality,

p = (p - a) + (p - b) + (p - e)

Then

_

3+

e)

(Dorin Andrica)

+ a2e2 · 2ae cos B + 1 - ab22 + eb22 b2

2be cos A b2 +1=2 a2 a

2ab cos C a2 b 2 +1+ e2 e2 e2 · Summing up these equalities implies

C ab sec2 "2 = p(p -

and the solution is complete.

_

and

=

yields

43. We have

Likewise,

+)

� "2 (

A sec2 "2

b2 > 4 sin2 !i 2' ae Summing up these inequalities yields a2 b2 e2 > 4 sin2 + sin2 + + be ea ab 2 2 desired.

so

3 cos A cos B cos C > -- . -- + -- + -- a3 b3 e3 2abe By the AM-G M inequality, 3 3 3 3 81 = 16p3 ' 2abe a+b e therefore 81 cos A cos B cos C > + + -as ---,;s � - 16pS · Revista Matematica Timi§oara (RMT) , No. 2(1975), pp. 46, Problem 2134)

44. We have

A 2(1 - cos A) = 4 sin2 "2

and likewise

as

- "2 ' '

(Dorin Andrica,

b e + -. e b

-

( �

be cos A ea cos B ab cos C 3 - + --- > -a-2- + -e2 b2

and moreover

Problem 3510)

or

169

4.2. SOLUTIONS

4. TRIGONOMETRY

p3

( ab22 + ab2 ) +

so p4

2

� 3 \!(P - a) (p - b) (P - e),

� 27(p - a) (p - b) (p - e)

� 27p(p - a) (p - b) (p - e) = 2782 •

� 3V38, and since 8 = !!.r � 3V3, desired. (Titu Andreescu, Revista Matematica Timi§oara (RMT), No. 2(1982), pp. 66,

It follows that p2 Problem 4993)

pr,

as

4.

170

I"V = Q2 , Then 0° < a, (3" < 90° and a + (3 + , = 90° , We have B-C 3A = sin 3a - cos((3 - ,) = sin 3a - sin(a + 2,) = cos sin: 2 2 46. Let a

=

� ' (3 = � , 2 2

I

_

=

2 cos(2a + ,) sin(a - ,)

=

-2 sin(a - (3) sin(a - ,).

In exactly the same way, we can show that C3B _ = -2 sin ((3 - a) sin( (3 - , ) cos sin 2 2

A

and

A

3C -B sin "'2 - cos -2 Hence it suffices to prove that

=

sin(a - (3 ) sin(a - ,) + sin(, - (3 ) [sin(, - a) - sin((3 - a )] ,

0

which is positive as function y = sin x is increasing for ° < x < 90°. We keep the notation of the first solution. We have sin 3a = sin a sin 2a + sin 2a cos ai

= sin(2a + ,) sin 2a cos , + sin , cos 2ai cos((3 - ,) sin(2, + a) = sin 2, cos a + sin a cos 2,i sin 3, = sin , cos 2, + sin 2, cos , . =

=

+

sin 3a + sin 3, - cos((3 - a) - cos((3 - ,)

�

O.

In exactly the same way, we can show that sin 3(3

+ sin 3a - cos(, - (3) - cos(, - a) � 0

and sin 3, + sin 3(3 - cos(a - ,) - cos(a - (3)

�

O.

=

= z·

=

=

=

= z,

=

=

(Titu Andreescu, American Mathematics Contest 12A, 2002, Problem 24) 4 248.2 Let5 a, b be real numbers such that z = a + bi, b -=f:. Then 1m z5 5a b -

10a b + b and

Imz 5 Im 5

=

5

( � ) 4 - 10 ( � ) 2 + 1.

O.

b z b 2 Setting x = (�) yields 1m Z5 - = 5x 2 - lOx + 1 = 5(x - 1) 2 - 4. 1m 5 z The minimum value is -4 and is obtained for x = 1 i.e, for z

=

i), -=f:.

a(1 ± a O . Revista Matematica Timi§oara (RMT) , No. 1(1984) , pp. 67, =

M1 , M2 , . . . ,M2n be the points with the complex coordinates Zl, Z , " " Z and let A , A '"'' A be the midpoints of segments M1 M2n , M2 M2 2n-1 , •2•n" Mn Mn+1 • I 2 n M +l M2 M2n-1n AAn2 Ml M2n Al 49.

=

(sin a - sin , ) (cos 2a - cos 2,) + 2(cos a - cos ,) cos(a ,) sin(a - ,). Note that sin x is increasing and cos x is decreasing for 0 < x < 90° . Since < 90° , each of the two products in the last addition is less than or o < a" , a equal to O . Hence

+,

=

Problem 5221)

=

(sin a - sin ,) (cos 2a - cos 2,) + (cos a - cos ,) (sin 2a - sin 2,)

z=

required conditions.

(Titu Andreescu,

It follows that =

=

z.

=

. (, - . . ( - a ) sm - 2 sm (3) ,

sin 3a + sin 3, - cos((3 - a) - cos((3 - ,)

a + bi, a - bi, and I z l Ja2 + b2 • The given relation becomes 2Z 00247.= LetNotez that = O.

0

=

(Titu Andreescu,

I z l (l z l 200l - 1) Hence I z l 0, and (a, b) (0,0), or I z l 1. In the case I z l = 1, we have Z 2 002 which is equivalent to Z 2003 z I z l2 1. Since the equation Z2003 1 has 2003 distinct solutions, there are altogether 1 + 2003 2004 ordered pairs that meet the

Note that this inequality is symmetric with respect to a, (3 " , we can assume without loss of generality that ° < a < (3 < , < 90°. Then regrouping the terms on the left-hand-side gives

cos((3 - a)

Adding the last three inequalities gives the desired result. USA IMO Team Selection Test, 2002, Problem 1)

from which it follows that

sin(a - (3) sin(a - ,) + sin((3 - a) sin((3 - ,) + sin(, - a) sin(, - (3) � O .

Alternative Solution.

171

4.2. SOLUTIONS

TRIG ONOMETRY

Let

172 Mi, i = 1

The points , 2n lie on the upper semicircle centered in origin and with are radius 1. Moreover, the lengths of the chords in a decreasing order, hence are increasing. Thus

M M , M M , ., M M OAl , OA2 , . . . , OAnl 2n 2 2n-l .. n n+l 1 Z. �Z2n 1 1 Z2 + ;2n-. 1 . . . 1 Zn +2Zn+1 1 and the conclusion follows. Alternative Solution. Consider Zk = r(cos tk +i sin tk), k = 1, 2, . . . , 2n and observe that for any j = 1, 2, . . . , n, we have $

$

$

= r2 [(costj + Cos t2n j+ 2 (sin tj + sin t2n_j+l )2] = = r2 [2 + 2(cos tj cos t2n-j+l + sin tj sin t2n-j+l )] = = 2r2 [1 + COS(t2n-j+l - tj )] = 4r2 cos2 t2n-3'+21 - tj . t 1 - tj and the inequalities Therefore I Zj + Z2 n - j+ ll = 2r cos 2 n -j� IZI + z2n l :::; IZ2 + Z2n-ll :::; :::; IZn + zn+ll are equivalent to t2 n -tl � t2 n - l -t2 � . . . � tn + l -tn · Because :::; tl :::; t2 :::; ... :::; t2n :::; the last inequalities are obviously satisfied. d +

. • .

°

7r,

(Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 1 (1984) , pp. 67, ) 50. Let p = 1, 2, . . . , m and let Z Then zP = 1. Note that n - m + l, n - m + 2, . . . , n are m consecutive integers, and, since p :::; m, there is an integer k {n - m + 1, n - m + 2, . . . , n } such that p divides k. Let k = k p. It follows that z k = (zp) k ' = 1, so Z Ak An - m+l A n - m+2 . . . U An , as claimed. Remark. An alternative solution can be obtained by using the fact that (an -k 1)(an-l -1 1) . . . (an- k+l - 1) (a - 1) (a - 1) ... (a - 1) is an integer for all positive integers a > 1 and n > k . (Dorin Andrica, Romanian Mathematical Regional Contest " Grigore Moisil" , 1997) 51. Let Zk = cos t + i sin tk, k {I, 2, 3}. implies The condition 2 (ZI + Z2 + Z3 ) -3ZlZ2 Z3 2(sin tl + sin t2 sin t3 ) = 3 sin ( tl + t2 + t3 ). (1) Problem 5222

E Ap e

i

E

E

k

k

E

+

-

E

1R

C

U

U

(tl t2 , t3 ) i,[

tl, t2 , t3 < i' Let

Assume by way of contradiction that max < hence 7r 7r . E 0, '6 . The sme functIOn IS concave on 0, '6 , so

t = tl + t32 + t3

. .

( ) 1

From the relations

'

( . tl + sm. t2 + sm. t3 ) sm. tl + t32 + t3 .

'3 sm

(1) and (2) we obtain

then

)

( 2)

:::;

t. It follows that 4 sin3 t - sin t � 0, i.e. sin2 t � �. Hence sin t � �, then t � i, which contradicts that t (0, i) , Therefore max( tl , t2 , t3 ) � i, as desired. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1 ( 1986) , 91, Problem 5862 ) 52. Let n = 2 ( 28 + 1) and b f:. 0, otherwise the claim is obvious. Consider a complex number a such that a2 = � and the polynomial f = xn - 1 = (X - co)(X -cl) . . . (X -cn- l) ' We have f en = G) a (a - iso ) . . . (a -isn- . ) and f (-'n = ( �1 r (a + iso ) . . . (a + ic"n- ,), hence f ( -;:a ) f (- a-;:) = (a2 + co2 ) . . . (a2 + 2n l ) sin

_

173

4 . 2 . SOLUTIONS

4. TRIGONOMETRY

3t

:::; 2 sin

E

pp.

C

-

'

Therefore

nII- l (a+ bc� ) = bn nII-l � +c�) = bn nII- l (a2 +c� ) = ( = bn f en f ( _ IT) = bn [(a2 )2.+. + 1]2 = bn [ m 2.+. + Ij " = b2(2s+1) ( a2s+bl2s++ bl 2S+l ) 2 = (a � + b � )2 . (Dorin Andrica, Romanian Mathematical Olympiad - second round, 2000) 53. If ab = 0, then claim is obvious, so consider the case when f:. and b f:. 0. k=O

k=O

k=O

a

We start with a useful lemma.

°

4.

174

4.2.

TRIGONOMETRY

Lemma. If an odd integer, then

,cn- l are the complex roots of unity of order where is n-IIl (A + B ) = An + Bn , for all complex numbers A and B. Proof. Using the identity xn - 1 = II (x c ) for x = � yields l - ( BAnn + 1) = n!!- ( A + ) , and the conclusion follows. Consider the equation bX2 + a = 0 with roots X l and X . Since n,

CO , C 1 , ' "

Ck

n- l

-

k

k=O

-

-

13

The desired inequality is equivalent to

n

k=O

Ck

0

2

175

SOLUTIONS

-ab1 + -be1 + -ea1 -> ­ R12 abc2 = -48 = a+ b + e -> R R 4Rpr

..

ie or

R � 2r, which is Euler's inequality for a triangle.

(Dorin Andrica, Revista Matematidi Timi§oara (RMT) , No. 2(1985), pp. 82, 5720)

Problem

55. Let AI , A2 , A3 and A be the points of complex coordinates Zl , Z2 , Z3 and let a)z3, a Hence point A lies on the line A A and the triangle ZA =A aA3Z2 +has(1its- circumcenter in the origin of the complex plane. l Point 2 B is the foot of the altitude from A l in the triangle A l A2 A3 . It follows that Al A � Al B, so E

R

2

3

We have we have

nII-l (a + bc� ) = bn nII- l (ck - Xt}(ck -X2 ) = bn nII-l - Xl) nII-l (€k X2 ) Using the lemma for A = -Xl, B = 1 and then for A = -X2 , B = 1 gives nII- l (ck - xt} = (-Xl) n + 1 = 1 X�, nII- l (ck - X2 ) = (-X2 )n + 1 = 1 - x� . Hence nII- l (a + bcT: ) = bn (l - x�)(l - x�) = = bn [l + (XlX2 ) n (x� + x�)] = bn [1 + (�) n] = an + bn , . Xl X2 = a and Xnl + X2n = Xnl + ( -Xl )n = 0. SInce (Dorin Andrica, Romanian Mathematical Olympiad - second round, 2000) 54. Consider the triangle with vertices of complex coordinates Zl , Z3 and the circumcenter in the origin of the complex plane. Then the circumradius R equals IZII = IZ2 1 = IZ3 1 = r and the side lengths are a = IZ2 - z31 , b = IZI - z31 , e = IZI 1 k=O

k=O

k=O

(c k

-

k=O

-

8A1A2A3

therefore as desired.

I ZI - z 11zl - I Z2 -2 z3 1 h - IZI - z2 11zl 2- z3 1 sin Al - 2 h = IZ I - z22r11 zl - z31 '

_

_

_

- z3

2

1

·

IZ2 2r- z31

y

k=O

k=O

k=O

-

o

X A(z)

b

Z2 ,

-

z2 ·

(Dorin Andrica, Romanian Mathematical Olympiad - final round, 1984) 56. Denote I z + 1 /z 1 by r. From the hypothesis,

176

4.

Hence r3 ::; 2 + 3r, which by factorization gives (r - 2) ( r + 1) 2 ::; O. This implies r ::; 2, desired. (Titu Andreescu, Romanian Mathematical Olympiad - first round, 1987; Revista as

Matematidl Timi§oara (RMT) , No. 1(1987) , pp. 75, Problem 6191)

57. Denote t =

ZIZ , t 2

z? - aZlZ2 + z� = 0 is equivalent 2 - 4 ::; 0, hence t a± iJ42 a2 and It I to t2 - at + 1 = O. We have a a2 4 - a2 = 1. f t = cos a + i sin a, then z�n tn cos na + i sin na and we can Jwrite +zfn--4-a zfzg z . + z� n 0, where an 2 cos na [-2, 2] . n Alternative Solution. Because a [-2, 2] ' we can write a = 2 cos a. The relation z? - aZlZ2 + z� = 0 is equivalent to E

1; �

Prove that the sequence is convergent and find its limit.

15. Evaluate

8. Find a sequence (an ) n2:l of positive real numbers such that

(i)

nlim-+oo (an+l - an ) = 00 lim (y'a n+ l �) = O. n-+oo

and

-

9. Let (Xn ) n 2:l be an increasing sequence of positive real numbers such that

n-+oo xn ( k 2:l X l - Xn k lim nk+ k-+oo nk 10. Let (3 be real numbers and let (Xn ) n2:l, (Yn ) n2: l , (Zn ) n2:l be real sequences such that max { x� + O'Yn , y� + (3 xn } :::; Zn for all n ;::: l. 1 a) Prove that Zn ;::: - 8 (0'2 + (32 ) for all n ;::: 1. b) If nl�� Zn = � (O'2 + (32 ), prove that the sequences (Xn ) n 2:b (Yn ) n 2:l are 1.

1m 2 = 0. n Prove that there is a sequence n h of positive integers such that

(ii)

_

convergent and find their limits .

11. The sequences (x ) 2:l and (Yn ) n>l are defined by Xl 2, Yl 1 and Xn+l = x� + 1, Yn+ l XnYnn nfor all n ;::: 1 . a) Prove that Xn / Yn < V7 for all n ;::: 1 . b) Prove that the sequence (Zn ) n 2:l, Zn = xn / Yn , is convergent and lim Zn < V7. n-+oo 12. Let ;::: 0 and a f= 0 be real numbers and let (Xn ) n2:l be an increasing sequence of real numbers such that nlim-+oo nC¥ (xn+ l - xn ) = a. P �ove that the sequence is bounded if and only if > 1. =

a

a

13. Evaluate 14. Evaluate

� k(n - k) ! + (k + 1) nlim-+oo f::o (k + l)!(n - k) ! n (-k � +l lim L n-+oo k= l n2 )

(4n +

�)qn+l ' q > 1.

16. Let (Xn ) n 2:l be an increasing sequence of positive integers such that Xn+2 + Xn > 2Xn+l for all n ;::: 1. Prove that the number 00 1 n=l 10Xn fJ - " - L.J

is irrational.

17. Prove that

= o.

a,

183

PROBLEMS

is irrational for all

An = k=L00l (k!)1 n

n ;::: 1 .

18.

Let k, s be positive integers and let numbers such that

aI, a2 , · . . , ak , bl, b2 , . . . , bs be positive real

y!al + y'a2 + . . . + ifiik = yb; + yb; + . . . + \I'bs

for infinitely many integers n ;::: Prove that k = S; = b2 . . . bs·

2.

1) 2) ala2 . .. ak bl 19. Let (xn ) n 2:l be a sequence with Xl 1 and let X be a real number such that =

Prove that

II00l

n=

( 1 x_n ) = e-x • _

Xn+l

_

A f= ±1 be a real number. Find all functions f : R � R and : (0,00) � R f(ln x + A ln y) = (y'X) + (..fij) for all x, Y E (0, 00). 21. Let f be a continuous real-valued function on the interval [a, b] and let ml, m2 be real numbers such that ml m 2 > O. Prove that the equation m f(x) = � a - x + b -2x has at least a solution in the interval (a, b) . 20.

9

Let such that

9

9

18 4

5.

22.

a b

5 . 1.

MATHEMATICAL ANALYSIS

(0, 1/2),

Let and be real numbers in the interval and let 9 be a continuous real-valued function such that = for all real Prove that ex for some constant e.

g(g(x)) ag(x)+bx

x.

g(x)

=

23. Find all continuous functions f : � � [0,00) such that for all real numbers

[0 ,

x, y.

f2 (X + y) - f2 (X - y) 4f(x)f(y) =

24. (i) Prove that if the continuous functions f : � � (-00,0] and ) have a fixed point, then

f+

9

has a fixed point. (ii) Prove that if the continuous functions : � � a fixed point, then has a fixed point. 00

rp'ljJ

rp

Prove that the limit

f('\x) - f(f-£X) 1mo -"----� x x exists and is finite if and only if f is differentiable at the origin. 31. The sequence (Xn ) n �l is defined by Xl 0, Xn+l eXn - 1, n � 1. Prove that n�oo lim nX n -2. 32. Let Xo (0,1] and Xn+ l Xn - arcsin(sin3 xn ), � 0. Evaluate nl�� Vnxn . 33. Let f : � � � be a twice differentiable function with the second derivative

�

25. Let rp : � � � be a differentiable function at the origin and satisfying rp(O) 0 . Evaluate �� � [rp(x) + rp (�) + . . . + rp (�)] , where n is a positive integer. 26. Let a be a positive real number. Prove that there is a unique positive real number f-£ such that f-£X XIL - aIL-x for all x 0. 27. Let f : [a, b) � � be a twice differentiable function on [a, b) such that f(a) f(b) and f'(a) j'(b). Prove that for any real number ,\ the equation f"(x) - ,\(f'(X)) 2 has at least a solution in the interval (a, b). 28. Find all functions f : [0, 2] � (0, 1] that are differentiable at the origin and satisfies f(2x) 2f2 (x) - 1, x E [0 , 1] >

=

=

=°

=

o

29.

Let such that

,\

be a positive integer. Prove that there is a unique positive real number

for all real number

=

=

E

nonnegati ve. Prove that

34. Let a

=

n

f(x + f'(x)) � f(x), x

=

>

1.

<

g:� [0,1] and 'ljJ : � � [1,00) have

>

30. Let f : � � � be a function continuous at the origin and let '\, f-£ be two

distinct positive real numbers.

E

�.

b be positive real numbers.+ Prove that the equation ( a ; b r Y a" bY has at least a solution in the interval (a, b) . 35. Find with proof if there are differentiable functions rp : � � � such that rp( x) and rp'(x) are integers only if x is integer. 36. Let f : [a, b) � � be a differentiable function. Prove that for any positive integer n there are numbers 01 O2 On in the interval (a, b) such that f(b) - f(a) f'(OI) + f'(02 ) + . . . + f'(On ) n b-a 37. Let f, : � � � be differentiable functions with continuous derivatives such that f (x) + (x) f' (x) - g' (x) for all x E �. Prove that if Xl, X are two consecutive real solutions of the equation f(x) -g(x) 0, then the equation f2(x) + g(x) has at least a solution in the interval (Xl, X2 ). 38. Let f : [-�, �] � (-1, 1) be a differentiable function whose derivative f' is continuous and nonnegative. Prove that there exists Xo in [-�, �] such that (f(XO))2 + (I' (XO)) 2 � 1. <

=

<

< ... <

_

9

9

=

=

=°

x 0.

18 5

PROBLEMS

186

5.

5. 1 .

MATHEMATICAL ANALYSIS

47. Let n > 1 be an integer and let f : [0, 1] �

39. Prove that there are no positive real numbers x and y such that

x2Y + y2-X x + y. ( ) n(n+1) for some integer 40. a) Prove that if x � y �

that

=

yX +

b) Prove that

41. Let

n: 1

n +� � \IY

X l , X 2 , . . . , Xn

be positive real numbers such that X l

Prove that

f

() c

42. Let f :

n � 2, then

+ +...+

= ° for some

�

c

lR

= 1.

lR

lR

�

lR

(x), h(x), .. . , fn (x))dx

45.

46. Let 1 = (0, 00) and let f : I � I be a function with an antiderivative F that

F(x)f G) x, g(x) F(x)F (�) is a constant function and =

x in I. Prove that g : then find f.

1 -+ 1R,

=

t

b).

[ f(x)dx oj O.

a

<

a

<

f3 < b such that :[

b] [c, dJ be a bijective

Let a, c be nonnegative real numbers and let f a , � increasing function. Prove that there is a unique real number f-£ E ( a, such that

50.

51. Let r.p :

Let p be a polynomial of odd degree such that p' has no multiple zero and let f : lR � lR be a function such that f o p is a derivative. Prove that f is a derivative.

R

=

[

44. Evaluate

�

t f(x)dx (b - a.) f ({3) ·

be continuous functions. Prove that

is a derivative and evaluate

satisfies the condition

a,

be a continuous function such that

Prove that there are numbers

E lR.

max( h

for all

has at least a solution in the interval ( a,

be a function with a noninjective antiderivative. Prove that

43. Let h , h, ... , fn :

E

X

=

49. Let f : [a, b] � lR

1

n

=

Xn

be a continuous function such

1

=

9

X2

lR

f(x)dx 1 + -2 + . . . + - . 1 1 Prove that there is a real number o (0, 1), such that f(xo ) 11 -- xgXo 48. Consider the continuous functions f, : [ b] Prove that the equation f(x) /." g(t)dt g(x) l f ( )dt a

+ n +V'x.

n Vn'n'H + n +� l n + 1 >- 2n + 1, n _> 3.

187

PROBLEMS

lR

�

lR

b)

f(t)dt = (IJ

- a)c + (b - 1J)d.

be a continuous function such that

{Jxx+Y

x, y

r.p (t)dt =

X J{x -y

r.p(t)dt,

for all E lR. Prove that r.p is a constant function.

52. Let f :

lR

�

lR

!:.±lL

be a differentiable function such that

1. f(t)dt !!:.±lLy 2

x y.

5

for all real number < Prove that is a nondecreasing function.

f

53. Let f :

lR

�

lR

f (t)dt

2

be an injective and differentiable function.

188

5,

Prove that the function

MATHEMATICAL ANALYSIS

F : (0, ) � lR, F(x) = xl 1a a: f(t)dt 00

-

is monotone,

54. Prove that that

55. Prove that there are no Riemann integrable functions f

for all real numbers that

�

1 lim n 2 1{ x a: + 1 dx = - , n-too 2 0

1(Ya: f(t)dt = f(x) f(y) ,

x =J y.

SOLUTIONS :

Let e =

56. Let f : [0 , 1] � lR be a differentiable function with continuous derivative such Prove that

If(l) - f (O)1 < 1 .

1. We prove that there is an integer n > 1 such that

lR � lR \ {o} such

[[/'(X}]2 dX = 1.

{r

1

1a 1 f(x)(x - f(x))dx = 12

1.

-

Prove that if there i s an integer m 2:: ° such that

[ fm (t}dt = (m : 1}1 '

fa has a fixed point. 5,9 . Let f : [-1, 1] � lR be a differentiable function with nondecreasing derivative. Prove that 21 1(_I1 f(x)dx � f(-I) + f'(I). 60. Let f, g : [a, b] � lR be continuous functions. Prove that there is a real number e E (a, b) such that {1 C f (x)dx .+ (e - a)g(e) = !. g(x)dx + (b - e)f (e). a b

n f3n ,

(Dorin Andriea, 4955)

2. Note that

f3n -

1.

f3,

1 1(1982), 90,

1,

. 1n) 2 (n+1)+1 = (l + v3. 1n) 2n+3 = ( l + v3. 1n) 2n 1 ( 1 + V3) 2 = ( l + v3 = (an + bn V3) (4 + 2V3) = 4an + 6bn + (2an + 4bn )V3.

58. Let fa : [0, 1] � lR be a continuous function and let the sequence ( fn ) n->l be

fn(x) = { fn I (t}dt, x E [0, 1].

+ e}n - an = (�) an- I e + . . . + en > nan-I e > ne,

because a > Take an integer > �e . Then an > n n The interval (a , f3 ) has length greater than hence there is an integer m > such that a < m < or a < rm < as desired. Revista Matematidi Timi§oara (RMT) , No. pp. Problem

- .

then the function

- an = (a

n

57. Find all continuous functions f : [0, 1] � lR such that

defined by

f3 - a. Then

+

On the other hand,

(1 + V3) 2 (n+1) +1 = an+1 + bn+1 V3, and since a n , b n are integers we derive that (i) an+1 = 4an + 6 bn i (ii) bn+1 = 2 an + 4bn . an+1 - 4a

bn = 6 n . Substituting in relation (ii) implies an+2 -6 4an+1 = 2an + 4 an+1 6-4an

From relation (i) we obtain

or

an+2 = 8an+1 -4an . b n+ 1 2- 4bn ' and the first relatI. On gI.ves O n t he ot her hand, an = bn+2 -2 4bn+1 = 4 bn+1 2-4bn + 6bn · 189

190

5.

5.2 .

MATHEMATICAL ANALYSIS

Thus

i) If

bn+2 8bn+1 - 4bn· It follows that the sequences (an ) n �l ' (bn ) n �l are given by =

Xl < 1, then Xn

=

{ XlX2

1

if n = if n is even if n is odd and n >

1 X2 X3 . The equation X2 1 + J2X2 - x� V2 In all other cases the sequence is divergent. . X2 -2 +-2-' has only the solutIOn ii) Xl 1, then odd Xn { 21 ifif isis even and the sequence is divergent. iii) If Xl > 1, then odd Xn { XlX2 ifif isis even. 2 + V2 . It follows that the sequence is convergent if and only if Xl X2 i . e. Xl (Titu Andreescu, Revista Matematidi Timi§oara (RMT) , No. 1-2(1979), pp� 56, Problem 3865) 5. We prove a more general statement: is a positive integer and (Xn ) n->l is a sequence such that . x�P + x�P + . . . + x2np 0, (1) nhm�oo then ... (2) nl'�oo Xl + X2 + + Xn 0 For this, recall the inequality ( Xl + X2 : + Xn rp ::; x�P + x�P : + X� . X3

So the sequence converges if and only if

1. (Dorin Andrica, Revista Matematidi Timi§oara (RMT) , No. 2(1981), pp. 71, Problem 4648) 3. Solving the quadratic equation in Xn+1 and taking into account condition 1 ) yields + 1) Xn+l - Xn + 1 +XnJ-2(x; (1) l ' 0, 1, 2, . . That is Xn+ l I(xn ), 0, 1,2, . . . , where 1 : (1, 00) � is the function given by I (x) x + 1 +xJ-2(xI 2 + 1) . 2 + 0. We It is not difficult to check that 1 is decreasing and 1(2 + O) distinguish three cases: Case 1. If Xo 2 + 0, then Xn 2 + J3 for all Case 2. If Xo E (1, 2 + J3), then from the monotonicity of function 1 it follows that Xo < X2 < X4 < . . , < 2 + V3 < . . . < X5 < X3 < Xl. Case 3. If Xo E (2 + V3, ) then Xl < X3 < X5 < . . . < 2 + V3 < . . . < X4 < X2 < Xo. In all cases the sequence (xn ) n �O is convergent and lim Xn 2 + 0. n�oo Regional Contest (Titu Andreescu and Dorin Andrica, Romanian Mathematical " Grigore Moisil" , 2003) for all

n

2::

_

n =

.

1R

n =

=

=

=

=

n.

00 ,

=

4.

Xn+2 1 + JXn+1 (2 - Xn+l) 1 + J( 1 + J2xn - X�) (1 - J2xn - X�) 1 + J1 - 2xn + x� 1 + /xn - 1 / , hence X n 2:: 1 for all 2:: 2.

•

=

If

=

n

We study three cases.

=

=

n

n

=

n

=

=

If p

=

n

1m

•

It follows that

p

=

= .

n

. . •

. •

I Xl + X2 : Xn I ::; x�p + x�p : . . . + X� • . .

=

n

=

=

=

=

=

Note that

=

191

SOLUTIONS

+

'p

(1),

•

Using the squeeze theorem and the hypothesis the conclusion we obtain the initial problem. The converse is not true. Take = and observe that

1

Xn (-l) n Xl + X2 + . . . + Xn -1 n

=

{o

n

if n is even if n is odd

(2) follows. For

12

5.

9

5 . 2.

MATHEMATICAL ANALYSIS

Hence

... n�oo Xl + X2 +n + Xn = 0 but XI + X� + . . . + X2n = 1 . lim n�oo n (Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1977), pp . 47, Problem 2570) 6. Assume by way of contradiction that there is M > 0 such that an+1 -an M

First,

lim

<

for all n . Summing up these inequalities from 1 to n yields

193

SOLUTIONS

an+ ! - an

= (n + 1) In(n + 1) - n ln n = In(n + 1) + In

so

(1 + �r

Second,

v'an+1 - va;; J(n + 1) In(n + 1) - v'n In n (n + 1) In(n + 1) - n ln n - (n + 1) In(n + 1) + v'n ln n J ln In ( 1 + - ) In(n + 1) n - (n + 1) In(n + 1) + v'n ln n + (n + 1) In(n + 1) + v'n ln n = J J ln In ( 1 + - ) 1 n In(n + 1) + n+1 / n In n (n + 1) In (n + 1) + n In n J 1+ V (n + 1) In (n + 1) Because n = 1, lim In ( 1 + .!. ) n-too n we have In ( l + � r .n�oo = O. hm n ln n J(n + l) ln(n + 1) + Jn1Iln In(n + 1) n ln n . Because lim = 1, It follows that lim n�oo n + 1 0 and n�oo (n + 1) n (n + 1) nlim�oo (v'an+ 1 - va;;) = 0, as desired. Remark. Another such sequence is given by an = ny'n, n � 1. (Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1977), pp. 70, Problem 3087) 9. Assume by way of contradiction that there is no sequence (nkh�l with the desired property. Then there is a > 0 such that xn+1 - xn - a 0 n hence Xn+ l - xn ;::: an, for all n � 1. It follows that Xn - Xl ;::: a(l + 2 + . . . + (n - 1)) = a (n -2 l) n ' =

=

_

_

-

_

or

an+1 a1 + M. (1) n n From an ;::: nbn it follows that a-n+ln ;::: n n+ 1 bn+1 Since the sequence (b n ) n>l is not bounded from above we obtain that the sequence ( an n+ 1 ) n>l is not bou:ded from above, which contradicts (1). (Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 1(1978), pp. 91, <

--

Problem 3441)

. lty X2 = (aXl++1)XI(a ++1a) 2 a. The eft mequa and b ' the right inequality is equivalent to Xl a. Since 0 a we obtain - likewise X3 a and then Xn E (0, a) by inducting on n. On the other hand, Xn - Xn- 1 (aXn+- ll)X+ n(-a1++1a) 2 - Xn- l = Xn-1a2 +- X�(a _+1 1) 0 therefore the sequence is increasing and bounded. It follows that the sequence is convergent . and let l nlim�oo Xn. Then (a + l)l + a2

7. Note that 0

<

<

<

o

<

1

1"

IS 0

VlOUS

< X2 <

<

=

>

,

=

1 =

so 1 = a.

l + (a + 1)

(Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1976) , pp. 53, 2567) 8. We prove that an = n In n, n � 1, satisfies the conditions.

Problem

----;:= : ====;�==

. �. v

.

=

1

>

>

194

5.

then

5. 2.

MATHEMATICAL ANALYSIS

Xn - n - 1 2l a n 2n n 3 for all n � 1. This is in contradiction with l' n�� nn2 = 0, and we are done. (Dorin Andrica, Romanian Mathematical Olympiad - final round, 1984) 10. a) Summing the inequalities x� O'Yn � Zn and y� f3xn � Zn we obtain 0 ::; ( �) 2 (Yn � r ::; 2 (zn + a2 � ,12 ) , for all "= 1 . The conclusion follows immediately. b ) Notice that f3 � and Yn a � J2 ( ---f32 ) � � J2 Zn --- ) 8 I I I f31 ( 8 From the squeeze theorem follows that n-+limoo Xn �2 and n-+1moo Yn = - -a2 . (Dorin Andrica, Romanian Mathematical Regional Contest " Grigore MoisH" , 1997) and 11. By induction on n we obtain Yn = l 2 1 1 1 ... Zn = Xl for all � 1. Since Xn+l = x� - Xn 1 0, the sequence is increasing. From 5

On the other hand,

X 0' 2 > -- + - > ­

Xn+1 Z----; n+l ;: - Yn+l Xn - XXn+lYn - Xn+lYn l nYn+ Yn Xn+l x� 2 1 - 1 �2 > 1 _

+

Xn +

Xn + "2

.

+

+

+

n

0'2 + 2

=

+ "2

Zn +

0'2 +

1.

-

X

+ - + -- + Xl Xl X2

X

. . . X n -l

+

-----

Xl X2 · . . Xn- l

n

X2 =

>

+

Xn

(Xn ) n �l

we obtain

hence It

1 - (5) Zn < 2 "21 -21-. 5 + . . -2-. -51n---2 = 2 "21 --'--'": 1.-: --1 - -5 5 [ 1 ] < 2 + 85 = "821 < ..[7, = 2 8 1 - ( 5) l l n-

+

+

.

+

+

n- l

+

as

desired.

_

X2n

.

-

+ Xn

+

'

Xn

is convergent and (Zn ) n�l is an increasing and bounded sequence. Then 2 1 nlim-+oo Zn � 8 < ..[7, claimed. (Dorin Andrica and $erban Buzeteanu, Romanian Mathematical Regional Contest " Grigore MoisH" , 1992) 12. Let 0 such that o . From lim n C¥ (x n+ l - xn ) = n-+oo it follows that there is an integer nl such that 1 1 (1) nC¥ < Xn+l - Xn < for all � nl. Summing up inequality (1) from n = nl to n - 1, 0 implies (2) L 1 L n 1 < xnl+p - xnl 1 The series f ( l + ) converges if and only if a 1, therefore applying the squeeze theorem to the inequality (2) leads to the conclusion. (Dorin Andrica, Gazeta Matematidi (GM-B ) , No. 11(1979), pp. 422, Problem 18011) 13. Note that n en n 1 1) n � l (n � � n =L L ( ki1 - (k 1 ) + 1 L (k) = nL + = 1 - (n 1 + -2n . Since 1 n-+1moo --(n = n-+1moo n! 0 so

(Zn ) n �l

as

c>

a-c >

a

(a + c) ncx

(a - c)

n

=

p- l

(a - c)

k=O

k

follows that

_

_

_

Xn X nYn

X

+

195

SOLUTIONS

k

k=O (k

I)!

k

n

k + (k + I) !

=

n! n - k)! ! k ! ( k=O

=

>

( cx n l + k)

+ I) !

=

n

+ I)!

k=O

+ I) !

.

k!(n - k) !

n

---

1

P

>

_

I

k=O

c¥

k)! + (k + ( k + ) ! - k)! +

p-l

« a + c)

+ c¥ k=O ( l k) n

nl + P

1.

n!

2

n

-

=

,

n!

k=O

n

196

5.

it follows that

� n�� � k(n(k -+ k)!1)!(n+ (k k)!+ 1) r

_

= 1.

(Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1975), pp. 52, 2281)

Problem

14. Note that Z lim z-+O X

z>O

Indeed, Z lim ",-+0 X

z >O

=

e z 1n z ",lim -+0 ",>0

after applying L'Hospital rule. Then

=

= 1.

!�o ¥ e "'>O z

(-z) z>O = e l� = 1,

x +1 1 . lim z_o x z>o Let e > O . There is an integer n ee) > 0 such that for any integer n � nee), we have ( � +1 1 - e < :2 )k < 1 + e, k = 1, 2, . . . , n. n2 Summing up from k = 1 to k = n and using algebraic manipulations yields n (�) � +1 L 1 - e < k= 1 nn2 k < 1 + e, n � n (e), L n2 k=1 or n -21 - -21 (e - -n1 + -ne ) < k=�l ( -nk2 ) � +1 < -21 + -12 (e+ -n1 + -ne ) ' for any integer n � nee) . Therefore n ( -k � +1 1 lim n-+oo t; n2 ) = -.2 (Dorin Andrica) z

--

=

L....J

15. (i) We have

1 +x+ x + · .. +xn + " ' = I -1 x , I x l < 1. 2

5.2.

MATHEMATICAL ANALYSIS

--

197

SOLUTIONS

Hence

r1/q (1 + x + x2 + . . . )dx = r1 /q dx ' Jo 1 - x Jo so 00 1 q q > 1. In = n I: 1 nq q n=1 (ii ) For any I x l < 1 we have 1 1 + x4 + x 8 + . . . + x4 n + . . . = _ 1 -_x4 . Hence 1 /q (1 + x4 + x8 + . . . )dx = 1 /q dx -1-- x4 ' 1 1 o so 1 4n+l - q1 + 41 ln (1 - 12 ) + '21 arctgq1 ' q � (4n + l)q (Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1977), pp. 70, Problem 3091; Gazeta Matematica ( GM-B ) , No. 1(1981), pp. 40, Problem 18608) _

'

0

00

=

16. The number () has the decimal representation

() = 0 .0 . . . 010'-.;--' . . . 01 . . . 01 . . . 10'-.;--' . . . 010'-.;--'

k1 , k '

where 2 Because

. . . ,

k

n

kn

k2

kl

are the number of zeros between two consecutive ones.

we have

kl < k2 < k3 < . . . < kn < . . . ,

hence () does not have a periodical decimal representation. It follows that () is irrational, as claimed. Revista Matematica Timi§oara (RMT ) , No. Problem

(Dorin Andrica, 4661)

17. Note that

An = 1 + L00 (k!1) > 1 k=2

n

and that

An = -1 + L00 (k!1) n < -1 + L00 k!1 = -1 + e < 2 . Hence 1 < An < 2 for all n � 1. So An is not an integer. k=O

k=O

2(1981), pp. 73,

198

5.

5. 2.

MATHEMATICAL ANALYSIS

n,p, q with q :f 1 A . n q pqn- l (q - 1) 1 = A + (q +1 l )n + (q + l )n1(q + 2)n + . < 1 1 o. Let = ln x + Alny and b = Iny + Alnx. Then Ab- a and y = e�, x = e� hence f ( ) = f(b) = (e2(�t-_al) ) + ( (�2-=-b ) , b E It follows that f is a constant and let f (x) = C. Then for x = y we have g(.jX) = � , so is a constant and g(x) = � . (Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 1-2(1979), 51, Problem 3827) 21. Consider the function F I F(x) = (x - a)(x - b)f(x) + ml (x - b) + m2 (x ) 1

9

s tim es

times

hence

(Dorin Andrica, Romanian Mathematical Regional Contest "Grigore Moisil" , 1999)

A a. - b

a

9

9

e2

l)

R

a,

9

pp.

:

-t lR,

-

a .

200

5.

Note that

5.2.

MATHEMATICAL ANALYSIS

F is continuous and

hence

E

=

9

9

l

E

9

9

9

----+

9

----+

- 00 .

E

9

= =

2001, Problem B-5)

x y f(O) O. For x = y we obtain f2 (2x) = 4f2 (X) and f( ) 2f(x), f(x) � O. f(nx) = nf(x), n � 1. Assume that f(kx) = kf(x) for all k = 1,2, . . . , n. We have f2 ((n + l)x) - f2 ((n - l)x) = 4f(nx)f(x),

Setting = = 0 yields then 2;C = since We prove that

23.

then

=

as desired. It follows that if p, are positive integers then

q

qf (�) f(P) = pf(l), so f (� ) � f(l) and f(r) r f(l) for any positive rational r. Setting x 0 in the initial condition gives =

=

=

=

00 ,

=

9

f((n + l)x) (n + l)f(x), =

c (a, b) such that F(c) = O. It follows that m f(c) � a - c + b -2c and the solution is complete. (Dorin Andrica) 22. Note that g(x) = g(y) implies that g(g(x)) = g(g(y)) and hence x = y from the given equation. That is, is injective. Since is also continuous, is either strictly increasing or strictly decreasing. Moreover, cannot tend to a finite limit L as x or else we'd have g(g(x)) - ag(x) = bx, with the left side bounded and the right side unbounded. Similarly, cannot tend to a finite limit as x Together with monotonicity, this yields that is also surjective. Pick Xo arbitrary, and define Xn for all n Z recursively by Xn+ l = g(xn ) for n > 0, and Xn - l = g - 1 (xn) for n < O. Let rl = (1+Ja 2 + 4b)j2 and r2 = (a-Ja2 + 4b)j2 be the roots of x2 - ax - b = 0, so that rl > 0 > r2 and 1 > I r ll > I r2 1 . Then there exist C , C2 lR such that Xn clr� + c2 r� for all n E Z. Suppose is strictly increasing. If C2 f:. 0 for some choice of xo, then Xn is dominated by r� for n sufficiently negative. But taking Xn and X n+ 2 for n sufficiently negative of the right parity, we get 0 < Xn < Xn+ 2 but g(xn) > g(Xn+ 2 ), contradiction. Thus C2 = O. Since Xo = Cl and Xl Clrl, we have g(x) = rl X for all x. Analogously, if is strictly decreasing, then C2 0 or else Xn is dominated by r� for n sufficiently positive. But taking Xn and Xn+ 2 for n sufficiently positive of the right parity, we get o < X n+ 2 < Xn but g(Xn+ 2 ) < g(xn), contradiction. Thus in that case, g(x) = r2 X for all x. (Titu Andreescu, The " William Lowell Putnam" Mathematical Competition, hence there is

201

SOLUTIONS

then for all real

f(y) f ( -y), f(r) Ir l f (l), =

y, hence r. f(x) I x l f (l) (rn ) n->l

=

for all rational numbers = We prove that for all real numbers Let be an arbitrary real = Because number and let be a sequence of rational numbers with lim

x. x �oo rn x.

f is a continuous function, it follows that nlim�oo f(rn) nlim�oo I rnl f (l) = f (nlim�oo rn) , hence f(x) = f(l)lxl · Note that a = f(l) � 0, therefore the desired functions are f(x) = al x l for some a � O. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1977) , pp. 90,

and

=

Problem 3203)

24. (i) Let a and b be fixed points for f and g, respectively. We have a - f(a) � 0, b = g(b) � 0 so a � b.



205

5.2. SOLUTIONS

so the function f is differentiable at the origin. Conversely, if f is differentiable at the origin, then f( x) f(O ) f (AX) - f(O) z � AX = 1' (0) and zlim-+o l-" J.£-X = f' (O) . Hence f(AX) - f(l-"x) = (A - 1-") f'(O), lim o -+ x z as desired. (Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1978), pp. 76, Problem 3708) 31. Since eZ > 1 + x for all real x ::j::. 0, we have Xn+l - Xn = eZn - 1 - Xn > 0 so the sequence (Xn ) n �l is increasing. By induction on n we obtain Xn < 0 for all n � 1, so the sequence (Xn ) n �l is bounded. Therefore the sequence converges and let l be its limit. The equation el = l + 1 has the unique solution l = 0, hence nlim Xn = O. -+oo Using zlim -+o (�x - � eZ - 1 ) = -21 , it follows that 1 1 1 1 X X eZn n+l n nlim-+oo (n + 1) - n nlim-+oo - 11 Cesaro-Stolz's Theorem implies 1 nlim-+oo nXn hence nlim -+oo nXn = -2, as desired. r

206

5. 2.

5 . MATHEMATICAL ANALYSIS

(Dorin Andrica, Revista Matematidl. Timi§oara (RMT) , No. 2 (1982), pp . 68,

Problem 5004)

32. We first prove by induction that Xn > O. This is true for n = 0 and assuming

Xk > 0 for some positive integer k yields 0 < sin xk < 1, hence sin3 Xk implies Xk+ l > Xk - arcsin(sin xk ) It is not difficult to see that (xn )n> O is convergent and lim X n n �oo

<

sin xk, which

= o.

= O. We have

(vn sin x n ) 2

=+ sin2 X n

and

=

(

=

)

+

(

)

•

+

X�+ l

[

1

_

0,

<

=

e

Problem 6143)

-----------=-----(xn - arcsi n (s in3 xn ) ) 2

1 ]=

<

+ f'(x)) 0, as desired. Let x be a real number such that f' (x) > O. Likewise, f(x + ! ' (x) ) - f(x) ! '(x) !' (c) , for some E (x, x + f' (x)) and f' (e) > f' ex) > O. Hence f(x + f' (x)) � f(x) for all real numbers x, as claimed. (Dorin Andrica, Revista Matematidl Timi§oara (RMT) , No. 1-2(1989), pp. 67,

sin2 Xn

It is clear that the last limit is obtained from (arcsin t3 ) (2 arcsin t - arcsin t3 ) lim lim t-+ o (arcsin t - arcsin t3 ) 2 arcsin2 t t-+ O (arcsin t - arcsin t3) 2 arcsin2 t

! , (x)

f(x) - f(x

x n ) 2 -1 · . -n- hm . ( =2 = nhm-+oo . -1 n-+oo sm Xn

x�

<

and

= J22 . Alt�rnative Solution. We have seen above that (x n ) n >O is convergent, lim Xn = 0 n �oo and X n > 0 for all n = 0, 1,2, . . . We will calculate 1 . n + 1 - n1 = nhm-+. oo ---hm (1) -=-1-----""""1::-- · -+oo 1 -

=

o.

+ f' ex)) = - !, (X)f' (c) ,

! ,(e)

Thus nlim foxn -+oo

n

+

+

. --n1- =

hm (vnX n ) 2 n -+oo

(Titu Andreescu,

for some c E (x f' ex) , x). Because the second derivative is nonnegative, f' is non­ decreasing, hence

+

+

hence

(1)

f(x) - f(x

•

=

t3

is � and the conclusion is obtained via Cesaro-Stoltz It follows that the limit 2 Theorem. Gazeta Matematidl. (GM-B) , No. 10(2002) , pp. 409, Problem C:2557)

+

•

•

---

t

33.

n+1-n 1 = -

(2 arcsin t - t2 --arcsin t3 ) t = tlim-+O ( arcsin t arcsint3 t3 ) 2 = 2. --- - t2

207

Let x be a zero of 1 '. Then f(x f' (x) ) f(x) and the conclusion follows. Let x be a real number such that f' ex) < Applying the Mean Value Theorem on the interval [x f' (x) , x) we obtain

. 2 X + sm2 X sm 1. n n l lim -----:1 ::- ------:;--1m . . 2X + oo sm2 Xn - SIn + nn �oo n l sin2 X n+ l sin 2 Xn . 2 Xn sm2 X n+l sm2 X n sm2 Xn+ l sm lim lim n -+ oo sin(xn - xn+ d sin(xn Xn+ l ) n -+ oo x� sin(xn xn+d . Xn . sm + l 2 lim sm Xn 2 lim X n Xn+ l lim n-+ oo Xn Xn+ l n -+oo sin(x n + Xn + l ) n -+oo Xn 1. 1 1 . 1.1m -1m Xn+ l - -2 · n -+ oo Xn + l n�oo 1 Xn From Cesaro-Stolz Theorem we obtain 1 -2 ' nhm -+oo •

=

t(2 arcsin t - arcsin t3 ) lim t-+ O (arcsin t - arcsin t3 ) 2

SOLUTIONS

34. Applying the Mean Value Theorem a [a, � b] yields a + b - In a 1 In 2b - a x ' 2 --

for the function f(t)

= In t on the interval

( a + b)

x E a ' -- . 2

Hence

=

(1)

208

5. 2.

5. MATHEMATICAL ANALYSIS

Using the same argument for the interval a+b ln b - In -2 b-a 2 hence In From the equalities

1

y'

[ a ; b , b] gives yE

Summing up these equalities implies

( a + b , b) , --

or

2

Y = b - a. (� a + b) 2

(2)

(1) and (2) we obtain ( a + b ) :I: = ( � ) Y 2a a+b

35. We prove that no such function exists. Assume the contrary and let k be an

integer. From the Mean Value Theorem we obtain

cp(k + 1) - cp(k) = (e) , e E (k, k + 1). Since cp(k) and cp(k + 1) are integers, cp(k + 1) - cp(k) is also an integer and so is cp' (e) · On the other hand, e is not an integer, hence cp' (e) is not integer, a contradiction. (Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1978), pp. 67, Problem 3618) 36. Define Xk = a + -k (b - a) , k = 0, 1, and note that b-a , k = 0, 1, Xk+ l - Xk = --;;:'

. . . ,n

. . . , n.

The Mean Value Theorem yields

desired.

(Dorin Andrica, Revista Matematica Timi§oara (RMT), No. 1-2(1979), pp. 58, 3878) 37. Let F lR lR, F(x) f(x) - g(x) and note that F is differentiable. Since F(x ) = F (X2 ) = 0 from Rolle's Theorem there is E (X l , X2 ) such that F'(c) = O. On the other hand, F'(x) = f '(x) - g' ( X ) = f (x) + g(x), and therefore f(c) + g(c) = 0, desired. (Dorin Andrica, Revista Matematid1 Timi§oara (RMT) , No. 2(1977), pp. 74, Problem 3113) 38.- Assume that ( f (X)) + ( f' (X)) 2 1 for all X in [-i, i]' Then f'(x) Jl - (f(X))2 1 for all X E [-�, �] . Integrating from - i to i yields arcsin (i ) - arcsin f (- i ) 2 2 7r. On the other hand, it is clear that arcsin f ( 2"7r ) - arcsin f ( - 2"7r ) � 2"7r + 2"7r = 7r. contradicting the previous inequality. ( Titu Andreescu, Mathematical Horizons, 2000) 39. If X = y, then x = y = 0, which is impossible, because x and y are positive. Assume that there are x f:. y 0 such that x . 2Y + y 2-:1: = X + y, and let y = Xl - X2 , X = X2 - X3 for some Xl X2 X3 O. Then 2:1:1 -:1:2 - 1 1 - 2:1:3- :1:2 or 2:1:1 2:1:2 2:1:2 2:1:3 Xl - X2 X2 - X3 By the Mean Value Theorem there are (} l E (X2 ' xI ) and (}2 E (X3,X 2 ) such that 2:1:1 2:1:2 --- = 2()1 n 2, (} l E (X 2 ,X I) and Xl - X2 I

(Dorin Andrica, Revista Matematica Timi§oara (RMT), No. 2(1978), pp. 54, Problem 3548)

n

n

:

desired.

cp

as

f (b) - f (a) = � t!'((}i ), b-a i=l

Problem

Then as

209

SOLUTIONS

�

=

c

as

2

>

>

f

>

.

>

_

_

1

>

_

>

>

5.

21 0

5.2.

MATHEMATICAL ANALYSIS

(}2 ,

Hence 291 In 2 = 29 2 In 2, that implies (h = a contradiction. Revista Matematidi Timi§oara (RMT), No. 1-2(1980), pp. 70, Problem 4152)

(Dorin Andrica,

40. a) Consider the differentiable function f

:

Consider f(x) = ln x, have

211

SOLUTIONS

Yi = �Xi , YlY2 . · ' Yn and Ai = !!..Yi for i = 1 , 2, . . . , n. We P=

(0, 00) � lR,

f(t) = ifi n+0. We have !,(t) = .!.n t n�l - 1 (t ntn\l) n n+_1 ) ' n(n+ l) then f'(t) 2:: O. Hence f is increasing if t 2:: ( on ) n(n+l ) n :) 1and for x � y � r (n+1) , we have /(x) � /(y). There­ (n: 1 [(n: 1) -

_

'

SO,

, 00

fore as

Hence

yr yr . . . y:£; :s In

desired. b) We prove that

n 2:: 3.

for all The inequality is equivalent to

which is clearly true, because

and since

(n

\Iii

Yi = �Xi , i = 1 ,

(1 + � r $ n,

(1 + �r < e < 3 for all n 2:: 1. Setting x = n n+ l and Y = + l) n , x 2:: Y in the inequality from a) yields + 1 2:: 2n + 1, n 2:: 3 n + n+mn

as

so

_

n

desired.

(Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1981), pp. 74,

Problem 4668)

41. Recall Jensen's inequality for a concave function f:

,

2 . . . n,

(t ) i= l �Yl

i�l -Ji

it follows that

1 X�l X�2 x�n

--,----- :s • • .

thus as desired.

( ) nnXi L i= l

:E Zi

0=1

=

n,

(Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1977) , pp. 74,

Problem

3111)

F Xl < X2 F(xI) F(X2 ) ' [X l,X2 ] f(c) F'(c) (Titu Andreescu)

42.

Let be an antiderivative of the function f. Since F is noninjective, there = are real numbers such that Applying Rolle's Theorem on the interval we obtain = = 0 for some c E as desired.

43. Because max( h (x) , h (x))

=

h (x)

(X l ,X2 ),

+ h (x) + I h (x) 2

-

h (x) 1

212

5.

5 .2.

MATHEMATICAL ANALYSIS

and iI , h are continuous, ma:x.(fI , h) is also continuous. Assume that if iI , are continuous, then ma:x.(iI , is continuous. It follows that ma:x.( ma:x.(iI , . ma:x.( iI , I),

12, ·· ., Ik-l

12, ... ,Ik - I) . . , Ik- Ik)

12, . . . Ik) =

. , Ik)

and according to the 'first step, the function ma:x.(iI , . . is continuous. Hence ma:x.(iI , is continuous and furthermore a derivative function. Note that if n is even, then

. . . , In)

n (-oo,-I) x, .. . ,xn ) = { x1,n , XX EE [-I, x , x E (1,00),I] and xn++l 1 + C, X E (-oo, -I) J max(l, x, ... ,xn)dx xxn++lC,+ X E [-I, I] + C, x E (1,00). n + 1 On the other hand, if n is odd, then x -l, n { x E (-00, -1) n ma:x.(I, x, ... ,x ) = X [-I, I] xn , x EE (1,00), and xn + 1 + C, x E (-00,-1) [- I,I] J max(l, x, . ,xn)dx xxn++lC,+ n + C, XX EE (I,oo). + 1 (RMT), (Dorin Andrica, Revista Matematidin Timi§oara ma:x.(I,

=

{

-n

n

n

1,

..

Problem

5185)

=

1

-n n

No .

2(1983), pp. 62,

We have

earctg:c dx VI x2earctg:c dx = II J VI + x2 J 1 x2 J V I + x2d(earctg:c ) = .)1 + x2earctg:c J VIxearctg:c + x2 dx = =

=

hence

On the other hand,

I

12 = J VIxearctg:cx2 dx J xl +1 x+2 exarctg:c 2 dx = J xV 1 x2 d(earctg:c ) = dx' = x yii+x2earctg:c - I - 2 J xVI2 earctg:c 2 + x so :c dx = xvf1+X2earctg:c - (I 12 ) = J XVI2 earctg 2 2 +x arctg:c 2 e = (x - I) VI 2+ x + C. (Dorin Andrica, Romanian Mathematical Olympiad - final round, 1975; Revista Matematica Timi§oara (RMT), No. 2(1978), pp. 35, Problem 2125)

�

=

+

+

_

+

=

�

+

1

I

Let g, : be functions such that: is a derivative; is differentiable with a continuous derivative. . Then h is a derivative function. Proof. Let H be an antiderivative of h and define u : Then u' (x) = g(x)h(x) + g' (x)H(x), 45.

+

We start with the following lemma. h lR --t lR

Lemma.

1) h 2) 9 9

lR

--t

R,

u(x) g(x)H(x). =

g(x) h(x) = u'(x) -g'(x)H(x). The function u' is a derivative and g' . H is continuous, hence the function g . h is a derivative, as claimed. Applying the lemma for h l o p and g(x) = xk for a nonnegative in­ k teger k, it follows that x (f p)(x) is a derivative, hence p'( 1 p) is a deriva­ tive. Since the degree of p is odd, p( ) = llt Assuming that lim p x = -00, :c-+-oo there are real numbers Xl, x� , x�, ... , xm , x� such that p is increasing on each of the interval ( -00 Xl , [X , X2 ] , ... , [X�_ 1 ,X ],[x �,00) and p(xI) = p (x = Mil P(X2 ) = p(x�) = M2 , ... , p(xm ) = p(x�) m= Mm . Let Fl ,Hl , ... ,Hm be an anti­ derivative of p( 1 p) on the interval (-00, X l ], [x�, X2 ] ' " . , [x�, 00), respectively. It follows that Fl p - l, HI p - l, .. . , Hm p l are antiderivative of I on the intervals or

0

=

0

,

44. Denote

21 3

SOLUTIONS

0

0

J

0

lR

D

�

0

( )

0

-

(-00, Ml], [Ml , M2], . . . , [Mm , 00), respectively, hence I is a derivative function on all llt (Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1985), pp. 76, Problem 2) 46. Substituting x -x1 in the given condition yields F ( ; ) f(x) = � , (1) --t

21 4

5. MATHEMATICAL ANALYSIS

for all x in I. We have g' (x) = F'(x)F

G)

+ F(x)F'

(D ( - :2 )

= f (x)F

1 1

G) : f G) F(x) � -

0

for all x in I, so 9 is constant. Then there is a constant c > such that

(1) we obtain

1

f (x) F(x)

ex '

48. Consider the function F : [a, b) F(x) =

2

= - - -2 · x = o X x

for all x in I, and from

F(x)F (�) = c

x E I.

F

�e ln x + ln d, where d > 0. It follows that F(x) = dx � , for all x in I. The relation F( x)F ( � ) = c becomes tP = c, so d = ..;c. Finally, 1 = 1 x c , x E I, f (x) = 1 ...;c xF ( - ) x where e is any positive real constant . (Titu Andreeseu, Romanian Mathematical Olympiad - final round, 1987; Revista Matematica Timi§oara (RMT) , No. 2(1987), pp. 86, Problem 6307) 47. Consider the function [0 , 1) lR, g(x) = f (x) - (1 + x + . . . + xn - ) 1

and note that

9

11

is continuous. We have

1

1

g(x)dx = =

[

g(x)dx

f (x)dx -

- 11 (1 + . . . + n (1 + � + . . . + � ) = O. x+

x

-1

) dx =

(0, 1) such that g(xo ) = [ g(x)dx = 0, 1 - x� f( Xo ) = l + xo + · · · + x o = -- , 1 - Xo n- l

as desired.

[ t dt l f(t)dt. g( )

[

0

g(t)dt = g(c)

(Titu Andreeseu, Revista Matematica Timi§oara (RMT) , No. 1-2(1979), pp. 33, Problem 3444)

O. Applying Rolle's Theorem,

t f(t)dt,

(Dorin Andriea)

49. Consider the function F : [a, b) F(t) =

1.'

--+

1R,

f(x)dx

-[

and observe that F is continuous and F(a)F(b) that F(O') = so

0

['

f (x)dx =

l

f (x)dx

<

O. Then there is

a

E (a,

b) such

f(x)dx

b) such that l f(x)dx = (b - )f ({3) , therefore there are numbers f3 E (a, b), f3, with { f(x)dx = (b - a)f({3) , From the Mean Value Theorem there is f3 E ( a , a

a,

From the Mean Value Theorem there is Xo E

hence

b)

f t c)

1. - 1

--+

lR,

--+

Note that is differentiable and F(a) = F(b) = we obtain e E (a, such that F' (e) = hence

as desired.

Integrating gives In F(x) =

9 :

215

5.2. SOLUTIONS

a <

as claimed.

(Dorin Andrica, Revista Matematica Timi§oara (RMT), No. 1-2(1979), pp. 61, 3897)

Problem

50.

Since f is an increasing bijective function, f is continuous. Denote 81 =

l

f (x)dx,

and note from the diagram below that

82 = ld r 1 (y)dy,

(1)

216

y . . . . . . . . . . . . . . .(

x, y f (I). (Titu Andreescu, 2349;

c a

=

'"

--+

=

lit

=

=

=

:

R

-+

R,

=

=

x, f 1(1976), 56, 2(1980), 68, 18154)

--+

�,

h(x) xf(x) - 10" f(t)dt. Because f is differentiable, h is differentiable and (1) h(x) = xf'(x), x � O. Since f is injective and continuous, f is either increasing or decreasing, so f' (x) � o for all x or f'(x) � 0 for all real numbers x. Case 1. If f'(x) � 0 for all x, then from (1) we deduce that h'(x) � 0, x � 0, hence h is nondecreasing. It follows that h(x) � h(O) = 0, x � O. Case 2. If f'(x) � 0 for all x, then h'(x) � 0, x � 0, and h is nonincreasing. It follows that h(x) � h(O) 0, x � O. Since F is differentiable and xf(x) - 1(0X f(t)dt h(x) F'(x) = x """'X2 ' we derive that F'(x) � 0 for all x 0 or F'(x) 2:: 0 for all x 0, hence F is a monotonic function, as desired. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1978), pp. 76, Problem 3709; Gazeta Matematica (GM-B), No. 1(1980), pp. 38, Problem 18115) 54. Recall from Problem 14 that =

(c, d) s,uch that { f- l (y)dy (d - c) rl (e). Observe that e is unique and let fJ; = f - 1 (e) . The relation (1) gives t f(t)dt ( - a)e + (b - ",)d, E (a, b) , as desired. (Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 2(1978), pp. 56, Problem 3556) 51. Consider the function F � �, F(s) f There is 0 > such that for all Then for n > "8 we obtam

I

' n

=n

n

2

< 0,

< c.

'

< cn

2

c

218

5. It follows that

5. 2 .

MATHEMATICAL ANALYSIS

57. The relation is equivalent to

* (XX+l - x)dx = 0, r 2 lim n n-+oo io * nlim-+oo n2 iro xx+l dx -21 ,

hence

ri l (xf(x) - f2 (X))dx = irl 4x2 dx. o o Hence [ (f2 (X) - xf(x) + x: ) = 0, or [ (f(x) - � r = O. Because f is continuous, f(x) - i = 0, for all x E [0, 1], so f (x) = �, x E [0, 1]. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1(1978), pp. 72, Problem 3319)

=

as

desired .

dx

Alternative Solution. Consider the function F(t) = f.' x·+1 dx. Then F(O) = 0 and we can write 1 * xx+l dx nlim n2 F ( 1 ) limO F(u)2 limO F'(u) = lim n 2 -+oo n u-+ u u-+ 2u n-+oo 1 uu+ l 1 u-+limo 2u = -2 u-+limo UU = -2 . (Dorin Andrica, Gazeta Matematidi (GM-B) , No. 11(1979), pp. 424, Problem 18025; Revista Matematica Timi§oara (RMT) , No. 1-2(1980), pp. 71, Problem 4160) 55. Assume the contrary and let x =f. y be real numbers. Then rY f(t)dt = f(x)(y) and 1x f(t)dt = f(fexy)) , f Jx hence f (x) f(y) It follows that f 2 (X) + f 2 (y) 0, so f(x) f(y) = 0, which is absurd since f(x) =f. 0 for all x. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1978), pp. 35, Problem 3126) =

0

=

-

=

--

=

dx

--

58. The relation

--

is equivalent to Since

y

=

and

as

desired.

[ (fm (t) - : ) dt = O.

fm is continuous by the Mean Value Theorem there is Xo E (0,1) such that

xo m- l iro (fm- l(t) - (mt I)! ) dt O. Using the same argument, we obtain Xl E (O,xo) such that fm- l (xt } = (mX�-I)!l ' Continuing this procedure, we obtain Xm E (0, x m - d such that fo (xm ) = x m, desired. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1(1978), pp. 72, Problem 3320) 59. From the Mean Value Theorem we deduce that for any x E [-1, 1] there is Cx E (-1, x) such that f (x) f ( -1) (x + 1) f' (cx ) . Since f'( cx ) � f'(I), f(x) - f ( - 1) � ( x + 1) 1'(1), hence L f(x)dx - 2f( -1) � (x � 1) 2 1�J' (1) =

_

=

_

[(I!' (X)I - l)2dx ;::: O. f. (f'(x)) 2 dx - 2 irol I f'(x)ldx + irol dx � 0,

as

I

1 ;::: [ 1 !,(x)l dx ;::: I [ !'(X)dx l ,

(Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1977), pp. 77, 3130)

Problem

[ fm (t)dt = (m : 1)1 '

or

56. It is clear that Hence

219

SOLUTIONS

-

j

J

=

5 . MATHEMATICAL ANALYSIS

220

and the conclusion follows. Revista Matematica Timi§oara (RMT) , No. 2(1981) , Problem 4686)

(Titu Andreescu,

=

�

-

and note that h is differentiable and h' lt)

=

(a (

77,

(a, b] �, (b t) [ f (x)dx + (t - a) [ g(x)dx

60. Consider the function h : h( t)

pp.

-[ f(x)dx

+ (b -

t)f(t) + [ g(x)dx - (t - )g(t) . a

Since h ) = h b) = 0, from Rolle's Theorem it follows that there is a real number b) such that h' (c) = Then cE

(a,

- /.' f(x)dx

O.

-

a g( ) + (b - e)f(e) + t g(x)dx = 0,

(e - )

(Titu Andreescu,

C O M P REHEN SIVE P RO B LEMS

e

and the conclusion follows. Revista Matematica Timi§oara (RMT) , No. 2(1985) , Problem 5628)

Chapter 6

pp.

56,

PROBLEMS

1. Let A be a set with

A

B

elements and let be a subset of with m � 1 elements. � B. Find the number of functions f : --t such that n

A

A A

f(B) elements and let X, Y be subsets of A with p � 1

2. Let be a set with n and q 1 elements respectively. Find the number of functions f : YC

�

f(X).

3.

A

X

Let be a set with n elements and let be a subset of Find the number of functions f : --t such that =

A A

4. Consider the sets A = {I,

2,

...,n

A A such that -+

A with k � 1 elements.

f(X) X. } B = {I, 2, . , m} and let k ,

. .

B having exactly k fixed points. 5. Let aI, a2 , ' . . , an be positive real numbers and let m � 1. Prove that (I al )m (1 a2 )m . . + (I + an ) � . 2 .

Find the number of functions f :

A

--t

+ .

m

}

:::; mini n, m .

m

a2 a3 al 6. Let aI, a2 , . . . ,a n be positive real numbers and let k � O. Prove that +

+

+

n

7. Let a, b, c be positive real numbers such that abc = 1. Prove that 8.

Let numbers x,

(3" be positive real numbers and let [a, b] be an interval. Find the y, z E [a, b) such that E(x , y, z) = o (x - y) 2 + {3(y - Z) 2 + ,(z - X) 2 0,

is maximum.

223

224

6.

6.1 .

CO MPREHENSIVE PROBLEMS

9. Find the maximum number of nonzero terms of the sum

n L If (i) - f (j) 1 i,j= l where f : {I, 2, . .. , n } {a, b, c} is one of the 3n possible functions. 10. Let aI, a2 , . . . , an , bl :::; b2 :::; . . . :::; bn be positive numbers such that al :::; bl , al + a2 :::; b1 + b2 , . . . , al + a2 + . . . + an :::; b1 + b2 + . . . + bn · -+

Prove that

+ ..fo2 + . . . + � ::; .jb; + .jb; + . . . + A· n 11. Define S (n, p) = L (n + 1- 2i ) 2p for all positive integers n and p. Prove that i= l for all positive real numbers ai , i r,n the following inequality holds 4P � p �l� (ai - aj) 2p :::; S (n,p) L....t a2i . 1 �� l an ; n �oo bn ... ) aan+l + bn+l 2 nthat limbn an , o.n - 1 . Prove n�oo bn 39. Let (un ) n 2:: l , be a sequence defined by UI E 1R \ {O, I} and Un U n Un+l Ul U2 U3 0 i) is strictly monotonic and unbounded; ii) there exists lim -

11l

>

=

=

n

If the sequence converges, evaluate

1 lim II (1 + Ul . . . U k ). n-+oo n k=2 40. Let (an ) n2:: 1 and (bn ) n2:: 1 be sequences such that i) 0 b l b2 bn b I + ii) � � k 1, n � 1; n an . iii) there exists n� lim oo . and IS. equa to n�oo . oo abn+n+I1b--n abnn eXIsts Prove that n� hm 1m -abnn -

<

< .. . <

<

< ... ;

>

1

1·

•

41. Let k be a positive integer and let

n�O Prove that

42. Let f

n > 2.

PROBLEMS

:

1 8k2 1R

�

1R

n f(x) k=L1 sinak x, =

230

6.

43. Let f : lR

f(xo ) -I O.

-t

1

CO MPREHENSIVE P ROBLEMS

where a k are real numbers. Prove that if number Xo such that that

.,

lai l -I lajl for i -I j, then there is a real

lim t(x) = 00. f(x) = x-*oo x-*oo Prove that the function : lR lR, g(x) = sinf(x), lim

for any � y > O. ii) Prove that

x

is not periodical.

an.

-t

[0, 1), f(n) = {an } i.e. the fractional

part of the number i) Prove that is injective if and only if a is irrational. ii) If is rational, find the number of elements of the set

f

a

=

f: f f(x) M f(x) = M

Let lR -t lR be a function such that i) has a period T > ii) � for all iii) if and only if kT, for some integer k. Prove that, for any irrational () the function 9 lR -t 1R,

x;

0;

x= : g(x) = f(x) + f(()x),

-t

lR be a continuous function with a period T > O.

[ f(x), max f(X)] there is a

a) Prove that if T is irrational, then for any A E min sequence

(Xn) n� 1 of integers such that lim

n-* oo

xER

f(xn ) = A.

[

xER

]

b) "Prove that if T is rational, then for any A E min f(X) , max f(X) and for any irrational number (), there is a sequence

47.

x,

xER

1

xER

(Xn) n� 1 of integers such that

i) Let y, z , v be distinct positive integers such that that there is no A > such that

d1' . p

(n +enl) n n! (n +en1)n+ , ---

1. 49. Let a, c be nonnegative real numbers and let f : [a, b) [c, d), be a bijective function. i) If f is increasing, prove that /. 1 f(t)dt + ld r' (t)dt = bd - ac. ii) If f is decreasing, prove that /.1 f(t)dt - ld r' (t)dt = bc - ad. 50. i) Let J.L : (0,00) lR be a continuous function such that x-*"oo lim J.L(x) = Prove that lim e - x ifX e t J.L(t)dt = 0 x-*oo o ii) Let f : [0, 00) lR be an n-time differentiable function with the n-derivative continuous such that there exists n k lim L x-*oo k=O C� f ( )(x) = A. Prove that lim f (x) exists and x-*"oo lim f(x) = A. x -*oo 51. Let f : [a, b) [c, d] be continuous function such that a � b /.1 J' (x)dx = cd. Prove that if c + d -I 0, then 1 I.b b - a ( c - d) 2 � -- -° � -f(x)dx c+d 4 c+d 1

<

<

n�

o.

-t

is not periodical.

46. Let f : lR

=

-t

M { f (n)1 n E N} .

45.

b, c, d be distinct positive integers such that

+ bP cP + la - c l + I b - d l �

Prove that

23 1

PROBLEMS

ii) Let p be a prime number and let a,

48. i) Prove that

-t

44. Let a be a real number and let f : N

6. 1 .

aP

lR be a differentiable function with continuous derivative such

9

1

x+

y

= z

-t

-t

+

v.

Prove

a

a

232

6. 1 .

6. COM PREHENSIVE PROBLEMS

52. Let I : [a, b]

lR be a continuous monotonic function and let F : [a, b] lR, (x ) = (x - a) I.' f (t)dt + (x - b) [ f (t)dt.

-t

F

-t

Prove that all values of F have the same sign.

lR and [1, 00) lR such that 1) is differentiable with continuous derivative; 2) I is continuous and the function h : [1,00) lR, h(x) = gl(X) - I(g(x)) is 53. a) Consider the functions I : (0, 00) 9

-t

-t

9:

-t

nonincreasing. Denote

qn - 1 n (X) -- { 'acos(q)�X ,cos X . . cos X , XX = 00 n E W , n is a derivative function for any integer O. 58. 1) Let I : lR lR be a continuous function such that 1 1 I(x)dx = M(/) . lim -

57. Let q be a positive integer. Find the number an (q) such that In : lR f

�

an = kL h(x) . an+1 - h(l) � 1.9(n+1) I (x)dx ::; an , n 1 1 lim L cot = 00.

b) Prove that

2) I : lR lR

n -roo

k=l

M(/)

k

k2

54. Let a be a positive real number and let I : [0 ,1]

function. Evaluate

-t

lR+ be an integrable

() (fa�f k � - n) . 55. Consider the functions In : lR lR, (s�n r) X 0, X k17r ' k E Z* sm ­ X X = k1 ' E Z * lim n

-roo

-t

2

-IT

7r x=o

� O.

=f. k

for all integers n Find the numbers Find the number

1)

2)

56. Let

p,

a ,k such that In is continuous on lR*. that In is a derivative function. annsuch q � 0 be integers. Find the numbers such that I : lR lR, 1 1 f (x) = { sm X cos x- ' ifif xx #= 0O •.•

is a deri vati ve function.

.

p

_

.

q

Cp,q

{

0

G) , g(X) = fM(/), x =O

p,q

-t

= Itllim-roo -lt iot I (x)dx = l iTo I(x)dx

O. Prove that

T

I : lR lR : lR lR, g(x) = I /(x) 1 also a 60. If /l , /2 : lR lR are derivative functions then I = max{/l , /2 } is also a

59.

If -t is a derivative function, then is derivative function? derivative function?

=l

,

Y

I Y I -r oo Y

is a derivative function. Let -t be a continuous function with a period T >

g(l)

lR,

=f.

.

-t

=l

Prove that

-t

n>

Prove that the function

n

233

PROBLEMS

-t

9

-t

SOLUTIONS

m functions h B B. Each of them can be extended in nn-m ways to a function f : A A which satisfies f(B) � B. nn -m . Hence the required number is (Dorin Andrica, Revista Matematidl Timi§oara (RMT) , No. 1-2(1981), 81, Problem Cl: l ) 2. Let Xq be a subset with q elements of X. Since Y has elements, it follows that there are q! bijective functions Xq Y. Each of them can be extended in nn- q ways to a function f A A which satisfies Y � f(X). The number of subsets Xq of X is (:) , hence the requested number is (:) nn - p . Remark. If q consider (:) 1. There are

:

m

---t

---t

mm .

pp.

:

9

---t

> p,

:

q

---t

= O.

(Dorin Andrica, Romanian Mathematical Regional Contest " Grigore MoisH" , 2000)

f(X) X, k!nn - k .

3.

X.

k! such bijections : A A. Hence the

Since = it follows that f is bijective on There are ways to a function f and each of them can be extended in desired number is

nn-k

---t

(Dorin Andrica)

4.

We consider two cases. i) � m. Let There is only one be a subset with elements of the set that has the property h (i) = i for all i E This function can ---t function h : be extended in (m ways to a function ---t B such that f= i for all iE

n

A \ Ak•

k

A Ak Akk n- k - l)

A.

f:A

Ak .

f(i)

Ak of A is (�), hence the desired number is (�) l) n-k . ii) n. Let Bk be a subset with k elements of the set There is only one function h : Bk Bk such that h(i) i for all i Bk . This function can be extended in - l) m- k ways to a function B B such that g(i) f= i for all i \ (m ­

The number of the subsets m �

(m

---t

=

9

:

---t

235

E

B.

E

B

Bk •

23 6

mn-m ways to a function f : A

Moreover, each function 9 can be extended in that clearly has exactly fixed points.

k

The number of the subsets

(7) mn -m (m _ l)m-k .

Bk

of

Therefore the number of functions f

B

is

:A

-4

B with

k fixed points is (�) (m ­

l)n-k if n ::; m and (7) mn-m (m _ l) m-k if m ::; n. (Dorin Andrica, Romanian Mathematical Regional Contest " Marian 'farina,

2001) 5. Using the inequality xr for

+ x� + . . .

+

x�� �

From the previous relation we deduce

B,

ll (k + ail

(7) hence the desired number is

--t

n�- l (Xl + X2 + . . . + Xn ) m

kn + (�) (ll air kn- 1 + (;) (ll a;f kn-2 + n .+ . . + (�) k+

�

Thus

(llf ( � k + � II a; ::; II (k + a; ). n

so

1 Ln ai (k + ai ) ::; k + II ni l i n

(2) (Titu Andreescu, Revista Matematidi Timi§oara (RMT) , No. 2(1977), pp. 63, Problem 3045)

we obtain

First Solution.

On the other hand,

al a a a al a a2 + a23 + . . . + aln >- n a2 . a23 . . . anl = n from the AM-GM inequality. Therefore (l + aa2l ) m + (1 + aa23 ) m + " ' + (l + aaln ) m � � nm (2n)m = n ' 2m, as desired. (Titu Andreescu, Revista Matematidi Timi§oara (RMT) , No. 1(1974), pp. 7, n

I

1564; Gazeta Matematidi (GM-B), No. 2(1976), pp. 65)

6. We have

n + ai) = kn + a kn- l + L a a kn-2 + . . . + II(k l2 Ll i= l

Using the AM-GM inequality gives

(1)

Using again the AM-GM inequality, we obtain

n

Problem

237

6.2. SOLUTIONS

6. COMPREHENSIVE PROBLEMS

a l a2

· · . an ·

=

=l

'

abc = 1,

7. Since this non-homogeneous inequality can be trans­ formed into a homogeneous one by a suitable change of variables. In fact, there exist positive real numbers such that

p, q, r

a = p-q , b = -,qr c = -.pr Rewriting the inequality in terms of p, q, r, we obtain (1) (p - q + r)(q - r + p)(r -p + q) ::; pqr, where p, q, r > o. At most one of the numbers = p-q+r , = q-r+p, = r-p+ q is negative, because any two of them have a positive sum. If exactly one of the numbers is negative, then ::; 0 pqr. If they are all nonnegative, then by the AM-GM inequality, 1 ..;uv ::; "2 ( + ) = p. Likewise, ..;vw ::; q and ..jWu ::; r. Hence ::; pqr, as desired. Second Solution. Expanding out the left-hand side of (1) gives (p-q +r)(q -r +p)(r -p+q) = [p(p -r) + (r -q) (P -q) + q(r - q) +pq][r + (q -p)] = = pr(p - r) + r(r - q)(p - r) + rq(r - q) + pqr + p(p - r)(q -p)+ +(r - q)(p - r)(q -p) + q(r - q)(q -p) + pq(q -p) . u

uvw

v

<

u

v

uvw

W

u, v , W

6.

238

,

Note that pr (p

- r) + rq(r - q) + pq(q - p) + (r - q) (P - r) (q - p)

=

O.

=

+

<

=

=

+

ca .

=

we

=

239

=

(1) is equivalent to o � p (p - q)(p - r) + q (q - r)(q - p) + r(r - p) (r - q), which is a special case of Schur's inequality. Third Solution. Denoting the left-hand side of the desired inequality by L, we have L = abeL = b (a - 1 + D c ( b - l + D a (c- 1 + D = (ab - b + 1)(be - e + 1)(ea - a 1) = L1 • Also, since lIb = ae, lIe = ab, 11a = be, L = (a - 1 + D (b - 1 + D (C- 1 + D = = (a - 1 + ae) (b - 1 + ab) (e - 1 + be) = L2 . If U = a - 1 + lIb � 0, then a 1 and b > 1, implying that v = b - 1 + lIe > 0 and W = e - 1 + 1/a > o . Then L uvw � 0, as desired. Similarly, either u � 0 or v � 0 yields the same result. If u, v, W > 0, then all factors of Ll and L2 are positive. The AM-GM inequality gives J(ab - b + l)(b - 1 + ab) � 21 [(ab - b + 1) + (b - 1 + ab)] abo Likewise, J(be - e + 1) (e - 1 be) � be, J(ea - a + 1) (a - 1 + ae) � Hence L = � � (ab)(be)(ea) = (abe) 2 = 1. Fourth Solution. Using the notations established in the third solution, it is easy to verify the equalities beu + ve = 2, eav + aw = 2, abw + bu 2. As in the third solution, only_ need to consider the case when u, v, W > O . The AM-GM inequality gives 2 2:: 2 e�, 2 2:: 2aJcvw, and 2 2:: 2Jawu, from which uvw � 1 . Fifth Solution. Let Ul = ab-b+ 1, VI = be-e+ 1, WI 1; U2 1-be+c, = 1 -ea + a, and W = 1-ab + b. As in the third solution, we only need to consider V2 2 the case in which U i , Vi, Wi > 0 for i = 1 , 2 . Again, we have Thus

6.2. Let X = a + b + e and Y = ab + be + ca. Then Ul + VI + WI = Y - X + 3 and U2 + V2 + W2 X - Y + 3. Hence either U l + VI + WI � 3 or U2 + V2 + W2 � 3. either case L � 1 follows from the AM-GM inequality. (Titu Andreescu, IMO 2000, Problem 2) 8. Without loss of generality can assume that a � (3 � 'Y. Let x, y , be three arbitrary numbers from the interval [a, b] such that � y � Then E( x , y, z ) - E(x ,z, y ) = ('Y - a)(( z - X) 2 - (y - )) 2:: 0, and E (a, y, b) = a (y - a) 2 + (3 (y - b) 2 + 'Y (b - ) E ( b, y, a) = a (y - b) 2 + f3(y - a) 2 + 'Y (b - a) . We need to find the maximal values of the functions It ( y ) a(y - a) 2 + f3 (y - b) 2 and h (y) = a(y - b) 2 + (3(y - a)2 on the interval [a, b] . Since II (a) = f3 (b - a) 2 2:: a(b - a) II (b ) and the coefficient of y 2 in II is a + f3 2:: 0, it follows that the maximum value of II is obtained for y = a. Likewise, h(b) 2:: h(a) and the maximum value of h is obtained for y b. Therefore max E(a, y, b) = E ( a , a, b) = (f3 + 'Y)(b - a) , and max E( b , y , a) = E (b , b, a) ({3 + 'Y ) (b - a) . It follows that the maximum value of E is (f3 + 'Y)(b - a) 2 and is obtained for x a, y = a, z = b or x = b, y = b, z = a. (Dorin Andrica and loan Ra§a, Revista Matematica Timi§oara (RMT) , No. 1(1983), pp. 66, Problem C5:2) 1- 1 ({b} ) , 9. For a function I : { 1, 2, . . . , n} -+ {a,b, e} let = 1- 1 ({a}) , 1 respec­ = 1- ({e}) and let p, q, r be the number of elements of sets tively. Obviously p + q + r = n and without loss of generality we may assume that p 2:: q 2:: r . A term I /(i) - 1(j ) 1 is different from 0 if the pair (i, j ) is in one of the sets , , , or Hence the number of nonzero terms in the sum L I / ( i ) - l(j) 1 is 2 (pq + qr + rp). SO LUTIONS

CO MPREHENSIVE PROBLEMS

ca - a +

=

In

we

z

x

Z.

x

a

2

2

2

=

2

=

=

2

yE[a , b]

2

=

yE[ a , b]

=

Me

Ma

Mb

Ma

x

Mb Mb

X

Ma Ma

x

Me, Me n

i ,j=1

X

Ma Mb

X

Me

=

Ma, Mb , Me,

Me

X

Mb .

240

6. COMPREHENSIVE PROBLEMS

2 (pq+qT+TP) (Po , qo , TO) 2 (pq+qT+Tp) 1. Po - TO 2 = Po - 1, ql qo , = TO + 1. Then + ql + = n and

p+q+T

The problem reduces to finding the maximal value of when == n and 2:: 0 are integers. Note that if then the absolute is a triplet that maximizes value of any difference of two numbers from this triplet is at most Indeed, assume that 2:: and define

p, q, T

=

PI

PI

,j

241

6.2. SOLUTIONS

{

� ... ( + . . . + A)] � �2 [( � A + A + + yo.;; ) + .Jb;+ 0. an

We have

Tl

Tl

= Poqo + POTo + qoro + Po - ro - 1 > Poqo + POTO + qoTo , which contradicts the maximality of 2(Poqo + qoro + ropo) · We have the following cases. 1) n = 3k . Then Po + qo + TO 3k, Po 2:: qo 2:: ro 2:: Po - 1, hence Po = k and then qo = ro k. In this case the maximal value is 2(k2 + k2 + k2 ) = 6k2 2n3 2) n = 3k + 1. Then Po + qo + ro = 3k + 1, Po 2:: qo 2:: TO 2:: Po 1, so 3po 2:: 3k + 1 2:: 3po - 2. Hence Po k + 1 and then qo = TO = k. In this case the maximal value is 2((k + l)k + (k + l)k + k2 ) = 2(3k2 + 2k) = � (n - 1). 3) n = 3k + 2. Then Po + qo + ro 3k + 2, 2:: qo 2:: TO � Po 1, so 3Po 2:: 3k + 1 2:: 3po - 2. It follows that Po = k + 1 and qo + ro = 2k + 1. Because k + 1 2:: qo 2:: TO � k, qo = k + 1 and ro = k. The maximal value is in this case n2 --1 2[(k + l)(k 1) + (k + l)k + (k + l)k] 2(k + 1)(3k + 1) = 2 · 3 2 2 n 1 n Therefore the requested number is 2 "3 if 3 divides n and 2 -- otherwise. Remark. The problem can be reformulated as follows: Suppose3 that n points in =

=

2

=

.

-

=

2

Po

=

+

-

(Dorin Andrica Pal Dalyai,

10. From the inequality � ( .fiil %I _ 4 fb1 ) + ( f02 2

Y Ul

+ . . . + ( A1 yO;1 ) (b2 + bJ) + yO;1 b .Jb; + 0. + . . . + A, + + . .. + va;; � .Jb; + 0. + . . . + A, __ __ _

hence

_

83,

4917)

'b) ) + . . . + (va; �

4 fb U22 Y

2

_

4 Un Y

2

> - 0,

va;

as claimed .

_-

1

=

.Ja2

(Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1977), pp. 63,

Problem 3046 )

1 1 . By multiplying the numbers aI , a2 , . . . , an with a suitable factor

M = '"'1 2p

-n

� ai

we may reduce the problem to the case when Assume without loss of generality that

=

space are colored by three different colors. Find the maximum number of segments AB such that A and B are different colors. and Romanian IMO Selection Test, 1982; Revista Matematica Timi§oara ( RMT ) , No. 1 ( 1982), pp. Problem

it follows that

1

a

Let a

n

L a;P = 1. i=1

= � � . .. � a2

al

an ·

=� S (n, p) and suppose by way of contradiction that 2

Then hence

L a;P > L[a + a(i - 1)] 2P. n

n

i= 1

i= 1

Consider the function 0, x (0, 1), I is concave up on (0, 1). It follows that equation (2) has at most two solution in (0, 1) therefore the conclusion follows. Remark. The claim that equation (2) has a unique positive solution it follows from 1(0) / (1) = b(b - 1) 0 and from the fact that I (x) = Xk-1 + Xk-2 + . . . + X2 - (k - l)x + b has a unique variation of sign (Descartes). (Dorin Andrica, Revista Matematica Timi§oara (RMT) , No. 1-2(1979), pp. 56, Problem 3866) 38. From the relation iii) we obtain an+l - an + bn+1 - bn - 0 , n � 1, an bn <

E

:

-+

JR,

<

<

(1) Al = an-l + an-2 A2 = a�_ 2 + an-2 an - 3 + a�_ 3 Ak-l = ank-2-k+2 + . . . + an -k+2 ank-_3k+l + ank-2-k+1 ·

.

E

an = k1 (b + an -l + an2 -2 + . . . + ank-l-k+1 ) ,

261

> 0 for n � k, so Note that al = a2 = . . . Al , A2 , . . . , Ak-l � O. On the other hand, 6. 1 = 6.2 = . . , 6.k-1 = 0 and from relation (1) it follows that 6.n > 0 for all n � k . Hence the sequence (an ) n�k is increasing. We prove that an < 1 for all n � 1 . Assume that

-

x � - Xn -l Xn+l -(3 - 6.(3 6. - X�_ l - XnXn -2 _

6.2. SOLUTIONS ak-l = 0 implies an

----

2 62

then Using ii) yields

(1)

=

(2)

It is easy to see that relations (1) and (2) imply an 0, Im n-+oo bn as desired. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 1(1978) , pp. 69, Problem 3304) 39. Consider the determinant 1·

=

X l X2 X l 0 X2 X l X2 0

Xn Xn Xn

Xl X2 X3

0

0

� (Xl , X2 , . . . , X n ) =

Note that

�(O, X 2 , . . . , xn ) = �(XI ' 0, X3 , ' . . , xn ) = . . . = �(XI ' X2 , . · . , Xn -l , 0) = O . Moreover, if Xl + X2 + . . . + Xn = 0, then �(XI ' X2 , . . . , xn ) = 0, therefore �(XI' X2 , · · . , xn) = aXIX2 . . . Xn(XI + X2 + . . . + x n), for some real number a. By identifying the coefficients of XI X2 . . . Xn from both sides we obtain a = 1 .

then Since U2

UI + U2 + . . . + Uk+l . . . Uk = 1 + UI +U.k+l . . + Uk UI + U2 + . . . + Uk UI + U2 + . . . + Un� +l II (1 + UI . . . U k ) = + U UI 2 k=2 I we obtain n 1 UI + . . . + Un+l - II (1 + UI . . . U k ) = nU l ( 1 - UI) n k=2

1 + UI

n

=

-u ,

Let U nlim-+oo Un and note that UI + . . . + Un+l nlim -+oo n It follows that =

_ an -+oo anbn+l+ l -- anbn nlim -+oo ( bn ) = nlim On the other hand, by the Stolz-Cesaro theorem we have lim anbn+l+l -- anbn nlim-+oo anbn n-+oo

Hence We have

26 3

6.2. SOLUTIONS

6. CO MPREHENSIVE PROBLEMS

-

=

U.

(Titu Andreescu, Revista Matematica Timi§oara (RMT), No. 2(1978), pp. 52, Problem 3533) an 40. Let a = lim n-+oo bn and let c > O. There is an integer n (c ) > 0 such that an < a + c -k + 1 for n 2: n (c ) a - c -kk +- 11 < k-1 bn Since bn > 0 it follows that abn - c kk +_ 11 bn < an < abn + c k _+ 11 bn for all n 2: n (c ) . Then k+1 -

-

k

a(bn+l - bn) - c k < a(bn+l

_

1

(b n+l + bn ) < an+l - an <

- bn) + c kk + 11 (bn+ 1 + bn), _

and, dividing by bn+1 - bn > 0, we obtain k+1 k-1

a - c -- '

bn+l + bn < an+l - an bn+ I - bn bn+ I - bn

for all n 2: n (c) . From relation ii) we deduce

< a+c

k + 1 b n+l + bn -' , k + 1 b n+ I - bn

k-1 bn+�l + bn < -bn+l - b n - k + 1 , n 2: 1, +l - an < a + c for all n 2: n (c ) a - c < anbn+l - bn -

--

hence Therefore as desired. (Dorin Andrica, reciproca a teoremei Stolz-Cesaro §i aplicatii ale acesteia", Revista Matematica Timi§oara (RMT), No. 2(1978) , pp. 6-12) 41. Let Xl k + Jk2 + 1 and X 2 k - Jk2 + 1. We have "0

=

=

IX21 = Xl1

-

<

1 1 < 2k - ­2 '

-

264

6.2. SOLUTIONS 265 It follows that a� = ai for some k =I- l i.e. lak I = l ad , which is a contradiction.

6. COMPREHENSIVE PROBLEMS

so

Hence

n x� + x� - 1 < x� + G ) - 1 < !In < x� - G r + 1 1, it follows that f is increasing on (0, �) . Both x, z are in (0, �) , so f(x) f:. f(z), because x f:. z. This implies XA + yA f:. ZA + vA, which is a contradiction. ii) Because is a prime, by Little Fermat's Theorem we have aP - a == bP - b == cP - c == d == 0 (mod p) , hence -(aP - a) + (bP - b) - ( cP c) + (dP - d) 0 (mod p) . From aP + bP = cP + dP, we deduce that (1) a - c + b - d == 0 (mod p) By i) we note that a + b f:. c + d, therefore u

-+

(0 ,

u

=

E

u

u ,

p

at

-

==

-

la - c + b - dl � p,

and then la - c l + I b - d l � p, as desired. (Titu Andreescu, Revista Matematica Timi§oara (RMT) , No. 2(1978), pp. 55, Problem 3550) 48. i) Consider the function