Asian Pacific Mathematics Olympiad · 2026

Problems

Problem 1Find all quadruples of positive integers $(a, p, m, o)$ with $o \ge 2$ such that $a! + p! = m^o + 26$ .Solutions: 1
Problem 2Let $n \ge 2$ be an integer. Let $A_1, A_2, \dots, A_{2^n}$ be the $2^n$ subsets of an $n$ -element set, listed in some order. Prove that $|A_1\setminus A_2| + |A_2\setminus A_3| + \cdots + |A_{2^n-1}\setminus A_{2^n}| + |A_{2^n}\setminus A_1| \ge 2^{n-2}.$ Solutions: 1
Problem 3Let $\mathbb{R}_+$ denote the set of positive real numbers. Determine all functions $f:\mathbb{R}_+\to\mathbb{R}$ such that for all $x,y,z\in\mathbb{R}_+$ , $|x-y|<|y-z|\ \text{if and only if}\ |f(x)-f(y)|<|f(y)-f(z)|.$ Solutions: 1
Problem 4Let $ABCD$ be a quadrilateral with an incircle $\omega$ of centre $I$ . The diagonals $AC$ and $BD$ intersect at $E$ . Let $J$ be the incentre of triangle $ABD$ . The extension of the ray $EJ$ intersects $\omega$ at $P$ . Prove that $PI\perp BD$ .Solutions: 1
Problem 5Let $n$ be a positive integer. There are $n(n+1)$ rooms arranged in an $(n+1)\times n$ grid. Between every two adjacent rooms there is a door. Find the number of ways to choose a subset of doors and lock them so that there exist two rooms $S$ and $G$ satisfying: (i) $S$ is in the first row and $G$ is in the $(n+1)$ -th row. (ii) $G$ can be reached from $S$ using only unlocked doors.Solutions: 1