Asian Pacific Mathematics Olympiad · 2018

Problems

Problem 1Let $H$ be the orthocenter of triangle $ABC$ . Let $M$ and $N$ be the midpoints of sides $AB$ and $AC$ , respectively. Assume that $H$ lies inside quadrilateral $BMNC$ and that the circumcircles of triangles $BMH$ and $CNH$ are tangent to each other. The line through $H$ parallel to $BC$ intersects the circumcircles of triangles $BMH$ and $CNH$ at points $K$ and $L$ , respectively. Let $F$ be the intersection point of $MK$ and $NL$ , and let $J$ be the incenter of triangle $MHN$ . Prove that $FJ = FA$ .Solutions: 1
Problem 2Let $f(x)$ and $g(x)$ be given by $f(x)=\dfrac1x+\dfrac1{x-2}+\dfrac1{x-4}+\cdots+\dfrac1{x-2018}$ and $g(x)=\dfrac1{x-1}+\dfrac1{x-3}+\dfrac1{x-5}+\cdots+\dfrac1{x-2017}$ . Prove that $|f(x)-g(x)|>2$ for any non-integer real number $x$ satisfying $0<x<2018$ .Solutions: 1
Problem 3A collection of $n$ squares on the plane is called tri-connected if the following criteria are satisfied: (i) All the squares are congruent. (ii) If two squares have a point $P$ in common, then $P$ is a vertex of each of the squares. (iii) Each square touches exactly three other squares. How many positive integers $n$ are there with $2018\le n\le 3018$ , such that there exists a collection of $n$ squares that is tri-connected?Solutions: 1
Problem 4Let $ABC$ be an equilateral triangle. From vertex $A$ we draw a ray toward the interior of the triangle so that it reaches one of the sides. When the ray reaches a side it bounces off following the law of reflection, that is, if it arrives with directed angle $\theta$ it leaves with directed angle $180^\circ-\theta$ . After $n$ bounces the ray returns to $A$ without ever landing on either of the other two vertices. Find all possible values of $n$ .Solutions: 1
Problem 5Find all polynomials $P(x)$ with integer coefficients such that for all real numbers $s$ and $t$ , if $P(s)$ and $P(t)$ are both integers, then $P(st)$ is also an integer.Solutions: 1