Asian Pacific Mathematics Olympiad · 2021

Problems

Problem 1Prove that for each real number $r>2$ , there are exactly two or three positive real numbers $x$ satisfying the equation $x^2=r\lfloor x\rfloor$ . (Here $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$ .)Solutions: 1
Problem 2For a polynomial $P$ and a positive integer $n$ , define $P_n$ as the number of positive integer pairs $(a,b)$ such that $a<b\le n$ and $|P(a)|-|P(b)|$ is divisible by $n$ . Determine all polynomials $P$ with integer coefficients such that $P_n\le2021$ for all positive integers $n$ .Solutions: 1
Problem 3Let $ABCD$ be a cyclic convex quadrilateral and $\Gamma$ be its circumcircle. Let $E$ be the intersection of the diagonals $AC$ and $BD$ , let $L$ be the center of the circle tangent to sides $AB$ , $BC$ , and $CD$ , and let $M$ be the midpoint of the arc $BC$ of $\Gamma$ not containing $A$ and $D$ . Prove that the excenter of triangle $BCE$ opposite $E$ lies on the line $LM$ .Solutions: 1
Problem 4Consider a $32\times32$ table. We put a mouse (facing up) in the bottom-left cell and pieces of cheese in several other cells. The mouse then starts moving. It moves forward except that when it reaches a piece of cheese, it eats a part of it, turns right, and continues moving forward. A subset of cells containing cheese is called good if, during this process, the mouse tastes each piece of cheese exactly once and then falls off the table. Show that (a) no good subset consists of $888$ cells; (b) there exists a good subset consisting of at least $666$ cells.Solutions: 1
Problem 5Determine all functions $f:\mathbb{Z}\to\mathbb{Z}$ such that $f(f(a)-b)+b\,f(2a)$ is a perfect square for all integers $a$ and $b$ .Solutions: 1