Asian Pacific Mathematics Olympiad · 2019

Problems

Problem 1Let $\mathbb{Z}^+$ be the set of positive integers. Determine all functions $f:\mathbb{Z}^+\to\mathbb{Z}^+$ such that $a^2+f(a)f(b)$ is divisible by $f(a)+b$ for all positive integers $a$ and $b$ .Solutions: 1
Problem 2Let $m$ be a fixed positive integer. The infinite sequence $\{a_n\}_{n\ge1}$ is defined in the following way: $a_1$ is a positive integer, and for every integer $n\ge1$ , $a_{n+1}=a_n^2+2^m$ if $a_n<2^m$ , and $a_{n+1}=a_n/2$ if $a_n\ge2^m$ . For each $m$ , determine all possible values of $a_1$ such that every term of the sequence is an integer.Solutions: 1
Problem 3Let $ABC$ be a scalene triangle with circumcircle $\Omega$ . Let $M$ be the midpoint of $BC$ . A variable point $P$ is selected on segment $AM$ . The circumcircles of triangles $BPM$ and $CPM$ meet $\Omega$ again at points $D$ and $E$ , respectively. The lines $DP$ and $EP$ meet the circumcircles of triangles $CPM$ and $BPM$ again at points $X$ and $Y$ , respectively. Prove that, as $P$ varies, the circumcircle of triangle $AXY$ passes through a fixed point $T$ distinct from $A$ .Solutions: 1
Problem 4Consider a $2018\times2019$ board with an integer written in each unit square. Two unit squares are called neighbours if they share a common edge. In each turn, some unit squares are chosen; then, for each chosen square, the average of all its neighbours is computed, and finally, after all these computations are done, the number in each chosen square is replaced by its corresponding average. Is it always possible to make the numbers in all squares equal after finitely many turns?Solutions: 1
Problem 5Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x^2+f(y))=f(f(x))+f(y^2)+2f(xy)$ for all real numbers $x$ and $y$ .Solutions: 1