Kol PreRmo_2015_nov_22_2015_u

Pre-RMO 2015 (West Bengal): November 22, 2015 QUESTIONS & ANSWERS

1. Find the sum S =

2015 X

(−1)

k(k+1) 2

k.

k=1

0 2. Suppose in 4ABC, AB =

√

3, BC = 1, CA = 2. Suppose there exists a point P0 in

the plane of 4ABC such that AP0 + BP0 + CP0 ≤ AP + BP + CP for all points P in the plane of 4ABC. Find (AP0 + BP0 + CP0 )2 .

7 3. Consider all the 7-digit numbers containing each of the digits 1, 2, 3, 4, 5, 6, 7 exactly once, and not divisible by 5. Arrange them in decreasing order. What is the 2015-th number (from the beginning) in this list?

4712536 4. Find the largest positive integer n such that 2n divides 34096 − 1.

14 5. Suppose x2 − x + 1 is factor of 2x6 − x5 + ax4 + x3 + bx2 − 4x − 3. Find a − 4b.

6 6. Let P (x) = (x − 3)(x − 4)(x − 5). For how many polynomials Q(x), does there exist a polynomial R(x) of degree 3 such that P (Q(x)) = P (x)R(x)?

22

1

Pre-RMO 2015 (West Bengal): November 22, 2015 QUESTIONS & ANSWERS 7. For positive integers m and n, let gcd(m, n) denote the largest integer that is a factor of both m and n. Find the sum of all possible values of gcd(a − 1, a2 + a + 1) where a is a positive integer.

4 8. For how many pairs of odd positive integers (a, b), both a, b less than 100, does the equation x2 + ax + b = 0 have integer roots?

0 9. Find the sum of all those integers n for which n2 +20n+15 is the square of an integer.

−40 10. Determine the largest 2-digit prime factor of the integer   200 , i.e., 200 C100 . 100

61 11. Suppose a, b, c > 0. What is the minimum value of       1 2 1 2 1 2 + 2b + + 2c + ? 2a + 3b 3c 3a

8 12. Find the sum of all the distinct prime divisors of 2015 2015 X 2015 X r2 , i.e., of r2 ·2015 Cr . r r=0

r=0

61 2

Pre-RMO 2015 (West Bengal): November 22, 2015 QUESTIONS & ANSWERS 13. Let 4ABC be a triangle with base AB. Let D be the mid-point of AB and P be the mid-point of CD. Extend AB in both direction. Assuming A to be on the left of B, let X be a point on BA extended further left such that XA = AD. Similarly, let Y be a point on AB extended further right such that BY = BD. Let P X cut AC at Q and P Y cut BC at R. Let the sides of 4ABC be AC = 13, BC = 14, and AB = 15. What is the area of the pentagon P QABR?

56 14. Suppose we wish to cut four equal circles from a circular piece of wood whose area is equal to 25π square ft. We want these circles (of wood) to be the largest in area that can possibly be cut from the piece of wood. Let R ft. be the radius of each of the four new circles. Find the integer nearest to R.

2 15. Let x3 + ax + 10 = 0 and x3 + bx2 + 50 = 0 have two roots in common. Let P be the product of these common roots. Find the numerical value of P 3 , not involving a, b.

500 16. In right-angled triangle ABC with hypotenuse AB, AC = 12, BC = 35. Let CD be the perpendicular from C to AB. Let Ω be the circle having CD as a diameter. Let I be a point outside 4ABC such that AI and BI are both tangents to the circle Ω. Let the ratio of the perimeter of 4ABI and the length of AI be m/n, where m, n are relatively prime positive integers. Find m + n.

12753

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