APMO 2021

Prove that for each real number $r>2$, there are exactly two or three positive real numbers $x$ satisfying the equation $x^{2}=r\lfloor x\rfloor$. Note: $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$

For a polynomial $P$ and a positive integer $n$, define $P_{n}$ as the number of positive integer pairs $(a, b)$ such that $a<b \leq n$ and $|P(a)|-|P(b)|$ is divisible by $n$. Determine all polynomial $P$ with integer coefficients such that for all positive integers $n, P_{n} \leq 2021$.

Let $A B C D$ be a cyclic convex quadrilateral and $\Gamma$ be its circumcircle. Let $E$ be the intersection of the diagonals $A C$ and $B D$, let $L$ be the center of the circle tangent to sides $A B, B C$, and $C D$, and let $M$ be the midpoint of the arc $B C$ of $\Gamma$ not containing $A$ and $D$. Prove that the excenter of triangle $B C E$ opposite $E$ lies on the line $L M$.

Given a $32 \times 32$ table, we put a mouse (facing up) at the bottom left cell and a piece of cheese at several other cells. The mouse then starts moving. It moves forward except that when it reaches a piece of cheese, it eats a part of it, turns right, and continues moving forward. We say that a subset of cells containing cheese is good if, during this process, the mouse tastes each piece of cheese exactly once and then falls off the table. Show that: (a) No good subset consists of 888 cells. (b) There exists a good subset consisting of at least 666 cells.

Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f(f(a)-b)+b f(2 a)$ is a perfect square for all integers $a$ and $b$.