Asian Pacific Mathematics Olympiad · 2010

Problems

Problem 1Let $ABC$ be a triangle with $\angle BAC\ne90^\circ$ . Let $O$ be the circumcenter of triangle $ABC$ and let $\Gamma$ be the circumcircle of triangle $BOC$ . Suppose that $\Gamma$ intersects segment $AB$ at $P\ne B$ and segment $AC$ at $Q\ne C$ . Let $ON$ be a diameter of $\Gamma$ . Prove that quadrilateral $APNQ$ is a parallelogram.Solutions: 1
Problem 2For a positive integer $k$ , call an integer a pure $k$ -th power if it is $m^k$ for some integer $m$ . Show that for every positive integer $n$ there exist $n$ distinct positive integers whose sum is a pure $2009$ -th power and whose product is a pure $2010$ -th power.Solutions: 1
Problem 3Let $n$ be a positive integer. $n$ people take part in a party. For each pair, either the two people are acquainted or they are not. What is the maximum possible number of pairs that are not acquainted but have a common acquaintance among the participants?Solutions: 1
Problem 4Let $ABC$ be an acute triangle with $AB>BC$ and $AC>BC$ . Let $O,H$ be its circumcenter and orthocenter. The circumcircle of $AHC$ meets line $AB$ again at $M$ , and the circumcircle of $AHB$ meets line $AC$ again at $N$ . Prove that the circumcenter of triangle $MNH$ lies on line $OH$ .Solutions: 1
Problem 5Find all functions $f:\mathbb R\to\mathbb R$ such that for all $x,y,z\in\mathbb R$ , $f(f(x)+f(y)+f(z))=f(f(x)-f(y))+f(2xy+f(z))+2f(xz-yz)$ .Solutions: 1