គណិតវិទ្យា​អូឡាំពិច​អាស៊ីប៉ាស៊ីភិច​២០០៣


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 \triangle AEF and \triangle 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 \geq 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 \geq 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 \geq 2 be an integer. Show that
\sqrt[n]{a^n+b^n}+\sqrt[n]{b^n+c^n}+\sqrt[n]{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.

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