Asian Pacific Mathematics Olympiad · 2003

Problems

Problem 1Let $p(x)=x^8-4x^7+7x^6+ax^5+bx^4+cx^3+dx^2+ex+f$ factorise into eight linear factors $x-r_i$ , with $r_i>0$ for $i=1,2,\ldots,8$ . Determine all possible values of $f$ .Solutions: 1
Problem 2Suppose $ABCD$ is a square of side length $a$ . Two parallel lines $\ell_1$ and $\ell_2$ in the plane are $a$ units apart. The square is placed so that $AB$ and $AD$ meet $\ell_1$ at $E$ and $F$ , while $CB$ and $CD$ meet $\ell_2$ at $G$ and $H$ . If the perimeters of $\triangle AEF$ and $\triangle CGH$ are $m_1$ and $m_2$ , prove that $m_1+m_2$ is constant, regardless of the placement.Solutions: 1
Problem 3Let $k\ge14$ be an integer, and let $p_k$ be the largest prime strictly less than $k$ . You may assume that $p_k\ge3k/4$ . Let $n$ be composite. Prove: (a) if $n=2p_k$ , then $n$ does not divide $(n-k)!$ ; (b) if $n>2p_k$ , then $n$ divides $(n-k)!$ .Solutions: 1
Problem 4Let $a,b,c$ be the side lengths of a triangle with $a+b+c=1$ , and let $n\ge2$ be an integer. Prove that $\sqrt[n]{a^n+b^n}+\sqrt[n]{b^n+c^n}+\sqrt[n]{c^n+a^n}<1+\frac{2^{1/n}}2$ .Solutions: 1
Problem 5Given positive integers $m$ and $n$ , find the smallest positive integer $k$ such that among any $k$ people, either there are $2m$ people who can be divided into $m$ pairs of mutually acquainted people, or there are $2n$ people who can be divided into $n$ pairs of mutually unacquainted people.Solutions: 1