In a scalene triangle $ABC$ with centroid $G$ and circumcircle $\omega$ centred at $O$, the extension of $AG$ meets $\omega$ at $M$, lines $AB$ and $CM$ intersect at $P$, and lines $AC$ and $BM$ intersect at $Q$. Suppose the circumcentre $S$ of the triangle $APQ$ lies on $\omega$ and $A,O,S$ are collinear. Prove that $\angle AGO=90^\circ$.
A grid of cells is tiled with dominoes such that every cell is covered by exactly one domino. A subset $S$ of dominoes is chosen. Is it true that at least one of the following $2$ statements is false?
(1) There are $2022$ more horizontal dominoes than vertical dominoes in $S$.
(2) The cells covered by the dominoes in $S$ can be tiled completely and exactly by the tetrominoes shown below (including rotations & reflections):
Let $n\ge2$ be a positive integer. For any integer $a$, let $Q_a(x)$ denote the polynomial $x^n+ax$. Let $p$ be a prime number and $S_a$ be the set
$$S_a=\{b\mid0\le b\le p-1,\ \exists c\in\mathbb Z,\ Q_a(c)\equiv b\pmod p\}.$$
Show that the expression $\dfrac1{p-1}\sum_{a=1}^{p-1}|S_a|$ is an integer.
Determine all real numbers $x$ between $0$ and $180$ such that it is possible to partition an equilateral triangle into finitely many triangles, each of which has an angle of $x^\circ$.
