Let $n$ be a positive integer. Let $T$ be the set of points $(x,y)$ in the plane where $x$ and $y$ are non-negative integers with $x+y\lt n$. Each point of $T$ is coloured red or blue, subject to the following condition: if a point $(x,y)$ is red, then so are all points $(x',y')$ of $T$ with $x'\leq x$ and $y'\leq y$. Let $A$ be the number of ways to choose $n$ blue points with distinct $x$-coordinates, and let $B$ be the number of ways to choose $n$ blue points with distinct $y$-coordinates. Prove that $A=B$.
Let $BC$ be a diameter of circle $\omega$ with center $O$. Let $A$ be a point of circle $\omega$ such that $0^\circ \lt \angle AOB \lt 120^\circ$. Let $D$ be the midpoint of arc $AB$ not containing $C$. Line $\ell$ passes through $O$ and is parallel to line $AD$. Line $\ell$ intersects line $AC$ at $J$. The perpendicular bisector of segment $OA$ intersects circle $\omega$ at $E$ and $F$. Prove that $J$ is the incenter of triangle $CEF$.
Find all pairs of positive integers $m, n \ge 3$ for which there exist infinitely many positive integers $a$ such that $$
\frac{a^m+a-1}{a^n+a^2-1}
$$ is itself an integer.
Let $n \ge 2$ be a positive integer with divisors $1 = d_1 \lt d_2 \lt \dots \lt d_k = n$. Prove that $d_1d_2 + d_2d_3 + \dots + d_{k-1} d_k$ is always less than $n^2$, and determine when it is a divisor of $n^2$.
Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ such that $$
\left(f(x)+f(z)\right)\left(f(y)+f(t)\right)
= f(xy-zt)+f(xt+yz)
$$ for all real numbers $x$, $y$, $z$, $t$.
Let $n \ge 3$ be a positive integer. Let $C_1$, $C_2$, …, $C_n$ be unit circles in the plane, with centers $O_1$, $O_2$, …, $O_n$ respectively. If no line meets more than two of the circles, prove that $$
\sum_{1 \le i \lt j \le n } \frac{1}{O_i O_j}
\le \frac{(n-1)\pi}{4}.
$$