Asian Pacific Mathematics Olympiad · 2022

Problems

Problem 1Find all pairs $(a,b)$ of positive integers such that $a^3$ is a multiple of $b^2$ , and $b-1$ is a multiple of $a-1$ . (Here $n$ is called a multiple of $m$ if $n=km$ for some integer $k$ .)Solutions: 1
Problem 2Let $ABC$ be a right triangle with $\angle B=90^\circ$ . Point $D$ lies on the line $CB$ such that $B$ is between $D$ and $C$ . Let $E$ be the midpoint of $AD$ and let $F$ be the second intersection point of the circumcircle of $\triangle ACD$ and the circumcircle of $\triangle BDE$ . Prove that as $D$ varies, the line $EF$ passes through a fixed point.Solutions: 1
Problem 3Find all positive integers $k<202$ for which there exists a positive integer $n$ such that $\left\{\frac{n}{202}\right\}+\left\{\frac{2n}{202}\right\}+\cdots+\left\{\frac{kn}{202}\right\}=\frac{k}{2},$ where $\{x\}$ denotes the fractional part of $x$ . (Here $\{x\}$ is the real number $r$ with $0\le r<1$ such that $x-r$ is an integer.)Solutions: 1
Problem 4Let $n$ and $k$ be positive integers. Cathy is playing the following game. There are $n$ marbles and $k$ boxes, with the marbles labelled $1$ to $n$ . Initially, all marbles are placed inside one box. Each turn, Cathy chooses a box and then moves the marble with the smallest label, say $i$ , to either any empty box or the box containing marble $i+1$ . Cathy wins if at any point there is a box containing only marble $n$ . Determine all pairs of integers $(n,k)$ such that Cathy can win this game.Solutions: 1
Problem 5Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2+d^2=1$ . Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a,b,c,d)$ such that the minimum value is achieved.Solutions: 1