APMO 2026

Find all quadruples of positive integers $(a, p, m, o)$ with $o \geq 2$ such that

$$a! + p! = m^o + 26.$$

Let $n \geq 2$ be an integer. Let $A_1, A_2, \cdots, A_{2^n}$ be the $2^n$ subsets of an $n$-element set in some order. Prove that

$$|A_1 \cap A_2| + |A_2 \cap A_3| + \cdots + |A_{2^n-1} \cap A_{2^n}| + |A_{2^n} \cap A_1| \geq 2^{n-2}.$$

Let $\mathbb{R}_+$ denote the set of positive real numbers. Determine all the functions $f : \mathbb{R}_+ \to \mathbb{R}$ such that for all $x, y, z \in \mathbb{R}_+$,

$$|x - y| < |y - z| \text{ if and only if } |f(x) - f(y)| < |f(y) - f(z)|.$$

Let $ABCD$ 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 $PI \perp BD$.

Let $n$ be a positive integer. There exist $n(n+1)$ rooms arranged in a $(n+1) \times n$ grid. For each two adjacent rooms, there is a door between them. Find the number of ways to choose a subset of doors and lock them such that there are two rooms $S$ and $G$ satisfying the following properties:

(i) $S$ is in the first row and $G$ is in the $(n+1)$-th row.

(ii) One can reach $G$ from $S$ only through unlocked doors.