APMO 2021 - Asian Pacific Mathematics Olympiad

Prove that for each real number $r\gt 2$, there are exactly two or three positive real numbers $x$ satisfying $x^2=r\lfloor x\rfloor$.

Here $\lfloor x\rfloor$ is the largest integer not exceeding $x$.

For a polynomial $P$ and a positive integer $n$, let $P_n$ be the number of positive integer pairs $(a,b)$ with $a\lt b\le n$ for which $|P(a)|-|P(b)|$ is divisible by $n$. Determine all polynomials $P$ with integer coefficients satisfying $P_n\le2021$ for every positive integer $n$.

Let $ABCD$ be a cyclic convex quadrilateral with circumcircle $\Gamma$. Let $E=AC\cap BD$, let $L$ be the center of the circle tangent to sides $AB,BC,CD$, and let $M$ be the midpoint of the arc $BC$ of $\Gamma$ not containing $A,D$. Prove that the excenter of triangle $BCE$ opposite $E$ lies on $LM$.

On a $32\times32$ board, a mouse starts at the bottom-left cell facing up, and pieces of cheese are placed in some other cells. The mouse moves forward; whenever it reaches cheese, it eats part of it, turns right, and continues. A set of cheese cells is good if the mouse tastes each piece exactly once and then leaves the board. Prove:
(a) no good set has $888$ cells;
(b) some good set has at least $666$ cells.

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