APMO 2025

Let $A B C$ be an acute triangle inscribed in a circle $\Gamma$ . Let $A_{1}$ be the orthogonal projection of $A$ onto $B C$ so that $A A_{1}$ is an altitude. Let $B_{1}$ and $C_{1}$ be the orthogonal projections of $A_{1}$ onto $A B$ and $A C$ , respectively. Point $P$ is such that quadrilateral $A B_{1}P C_{1}$ is convex and has the same area as triangle $A B C$ . Is it possible that $P$ strictly lies in the interior of circle $\Gamma$ ? Justify your answer.

Let $\alpha$ and $\beta$ be positive real numbers. Emerald makes a trip in the coordinate plane, starting off from the origin $(0,0)$ . Each minute she moves one unit up or one unit to the right, restricting herself to the region $|x - y|< 2025$ , in the coordinate plane. By the time she visits a point $(x,y)$ she writes down the integer $\lfloor x\alpha +y\beta \rfloor$ on it. It turns out that Emerald wrote each non- negative integer exactly once. Find all the possible pairs $(\alpha ,\beta)$ for which such a trip would be possible.

Let $P(x)$ be a non- constant polynomial with integer coefficients such that $P(0) \neq 0$ . Let $a_{1}, a_{2}, a_{3}, \ldots$ be an infinite sequence of integers such that $P(i - j)$ divides $a_{i} - a_{j}$ for all distinct positive integers $i, j$ . Prove that the sequence $a_{1}, a_{2}, a_{3}, \ldots$ must be constant, that is, $a_{n}$ equals a constant $c$ for all $n$ positive integer.

Let $n \geq 3$ be an integer. There are $n$ cells on a circle, and each cell is assigned either 0 or 1. There is a rooster on one of these cells, and it repeats the following operations:

If the rooster is on a cell assigned 0, it changes the assigned number to 1 and moves to the next cell counterclockwise. If the rooster is on a cell assigned 1, it changes the assigned number to 0 and moves to the cell after the next cell counterclockwise.

Prove that the following statement holds true after sufficiently many operations:

If the rooster is on a cell $C$ , then the rooster would go around the circle exactly three times, stopping again at $C$ . Moreover, every cell would be assigned the same number as it was assigned right before the rooster went around the circle 3 times.

Consider an infinite sequence $a_{1},a_{2},\ldots$ of positive integers such that

$$100!(a_{m} + a_{m + 1} + \cdot \cdot \cdot +a_{n}) \mathrm{is~a~multiple~of} a_{n - m + 1}a_{n + m}$$

for all positive integers $m,n$ such that $m\leq n$

Prove that the sequence is either bounded or linear.

Observation: A sequence of positive integers is bounded if there exists a constant $N$ such that $a_{n}< N$ for all $n\in \mathbb{Z}_{>0}$ . A sequence is linear if $a_{n} = n\cdot a_{1}$ for all $n\in \mathbb{Z}_{>0}$