This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 2

2022 Brazil National Olympiad, 4
Tags: number theory , combinatorics , cool problem
Initially, a natural number $n$ is written on the blackboard. Then, at each minute, [i]Neymar[/i] chooses a divisor $d>1$ of $n$, erases $n$, and writes $n+d$. If the initial number on the board is $2022$, what is the largest composite number that [i]Neymar[/i] will never be able to write on the blackboard?

2024 Iranian Geometry Olympiad, 4
Tags: geometry , cool problem
Point $P$ is inside the acute triangle $\bigtriangleup ABC$ such that $\angle BPC=90^{\circ}$ and $\angle BAP=\angle PAC$. Let $D$ be the projection of $P$ onto the side $BC$. Let $M$ and $N$ be the incenters of the triangles $\bigtriangleup ABD$ and $\bigtriangleup ADC$ respectively. Prove that the quadrilateral $BMNC$ is cyclic. [i]Proposed by Hussein Khayou - Syria[/i]