## 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.