For a positive integer $n$ denote by $P_0(n)$ the product of all non-zero digits of $n$. Let $N_0$ be the set of all positive integers $n$ such that $P_0(n)|n$. Find the largest possible value of $\ell$ such that $N_0$ contains infinitely many strings of $\ell$ consecutive integers. Proposed by Navid Safaei, Iran

Let $ABC$ be an acute angled triangle and let $P, Q$ be points on $AB, AC$ respectively, such that $PQ$ is parallel to $BC$. Points $X, Y$ are given on line segments $BQ, CP$ respectively, such that $\angle AXP = \angle XCB$ and $\angle AYQ = \angle YBC$. Prove that $AX = AY$. Proposed by Ervin Maci$\acute{c},$ Bosnia and Herzegovina

Alice and Bob play the following game on a square grid with $2024 \times 2024$ unit squares. They take turns covering unit squares with stickers including their names. Alice plays the odd-numbered turns, and Bob plays the even-numbered turns. On the $k$-th turn, let $n_k$ be the least integer such that $n_k\geqslant\tfrac{k}{2024}$. If there is at least one square without a sticker, then the player taking the turn: selects at most $n_k$ unit squares on the grid such that at least one of the chosen unit squares does not have a sticker. covers each of the selected unit squares with a sticker that has their name on it. If a selected square already has a sticker on it, then that sticker is removed first. At the end of their turn, a player wins if there exist $123$ unit squares containing stickers with that player's name that are placed on horizontally, vertically, or diagonally consecutive unit squares. We consider the game to be a draw if all of the unit squares are covered but no player has won yet. Does Alice have a winning strategy? Proposed by Erik Paemurru, Estonia

Ana plays a game on a $100\times 100$ chessboard. Initially, there is a white pawn on each square of the bottom row and a black pawn on each square of the top row, and no other pawns anywhere else. Each white pawn moves toward the top row and each black pawn moves toward the bottom row in one of the following ways: it moves to the square directly in front of it if there is no other pawn on it; it captures a pawn on one of the diagonally adjacent squares in the row immediately in front of it if there is a pawn of the opposite color on it. (We say a pawn $P$ captures a pawn $Q$ of the opposite color if we remove $Q$ from the board and move $P$ to the square that $Q$ was previously on.) Ana can move any pawn (not necessarily alternating between black and white) according to those rules. What is the smallest number of pawns that can remain on the board after no more moves can be made? Proposed by José Alejandro Reyes González, Mexico

Let $\mathbb{R}_{>0}$ be the set of all positive real numbers. Find all strictly monotone (increasing or decreasing) functions $f:\mathbb{R}_{>0} \to \mathbb{R}$ such that there exists a two-variable polynomial $P(x, y)$ with real coefficients satisfying $$ f(xy)=P(f(x), f(y)) $$for all $x, y\in\mathbb{R}_{>0}$. Proposed by Navid Safaei, Iran

Let $a\equiv 1\pmod{4}$ be a positive integer. Show that any polynomial $Q\in\mathbb{Z}[X]$ with all positive coefficients such that $$Q(n+1)((a+1)^{Q(n)}-a^{Q(n)})$$is a perfect square for any $n\in\mathbb{N}^{\ast}$ must be a constant polynomial. Proposed by Vlad Matei, Romania