APMO 2026 - Asian Pacific Mathematics Olympiad

Find all quadruples of positive integers $(a,p,m,o)$ with $o\ge2$ such that
$$a!+p!=m^o+26.$$

Let $n\ge2$ be an integer. Let $A_1,A_2,\ldots,A_{2^n}$ be all $2^n$ subsets of an $n$-element set, in some order. Prove that
$$|A_1\cap A_2|+|A_2\cap A_3|+\cdots+|A_{2^n-1}\cap A_{2^n}|+|A_{2^n}\cap A_1|\ge2^{n-2}.$$

Let $\mathbb R_+$ denote the positive real numbers. Determine all functions $f:\mathbb R_+\to\mathbb R$ such that for all $x,y,z\in\mathbb R_+$,
$$|x-y|\lt |y-z|\quad\text{if and only if}\quad|f(x)-f(y)|\lt |f(y)-f(z)|.$$

Let $ABCD$ be a quadrilateral with an incircle $\omega$ of center $I$. The diagonals $AC,BD$ intersect at $E$. Let $J$ be the incenter of triangle $ABD$. The extension of ray $EJ$ intersects $\omega$ at $P$. Prove that $PI\perp BD$.

AI-generated configuration using only the given points:

Let $n$ be a positive integer. There are $n(n+1)$ rooms arranged in an $(n+1)\times n$ grid. Every pair of adjacent rooms has a door between them. Find the number of ways to choose a subset of doors to lock such that there exist two rooms $S,G$ satisfying:
1. $S$ is in the first row and $G$ is in the $(n+1)$st row.
2. One can reach $G$ from $S$ using only unlocked doors.