XV Asian Pacific Mathematics Olympiad March 2003

Time allowed: 4 hours No calculators are to be used Each question is worth 7 points

Problem 1.

Let a, b, c, d, e, f be real numbers such that the polynomial

p(x) = x^{8} − 4x^{7} + 7x^{6} + ax^{5} + bx^{4} + cx^{3} + dx^{2} + ex + f

factorises into eight linear factors x − x_{i}, with x_{i} > 0 for i = 1, 2, …, 8. Determine all possible values of f.

Problem 2.

Suppose ABCD is a square piece of cardboard with side length a. On a plane are two parallel lines ℓ_{1} and ℓ_{2}, which are also a units apart. The square ABCD is placed on the plane so that sides AB and AD intersect ℓ_{1} at E and F respectively. Also, sides CB and CD intersect ℓ_{2} at G and H respectively. Let the perimeters of △AEF and △CGH be m_{1} and m_{2} respectively. Prove that no matter how the square was placed, m_{1} + m_{2} remains constant.

Problem 3.

Let k ≥ 14 be an integer, and let p_{k} be the largest prime number which is strictly less than k. You may assume that p_{k} ≥ 3k/4. Let n be a composite integer. Prove:

(a) if n = 2p_{k}, then n does not divide (n − k)! ;

(b) if n > 2p_{k}, then n divides (n − k)! .

Problem 4.

Let a, b, c be the sides of a triangle, with a + b + c = 1, and let n ≥ 2 be an integer. Show that

n \sqrt{a^{n} + b^{n}} + ^{n} \sqrt{b^{n} + c^{n}} + ^{n} \sqrt{c^{n} + a^{n}} < 1 + \frac{\sqrt[n]{2}}{2} .

Problem 5.

Given two positive integers m and n, find the smallest positive integer k such that among any k people, either there are 2m of them who form m pairs of mutually acquainted people or there are 2n of them forming n pairs of mutually unacquainted people.