Let $ABC$ be an acute angled triangle with circumcircle $\Gamma$. The perpendicular from $A$ to $BC$ intersects $\Gamma$ at $D$, and the perpendicular from $B$ to $AC$ intersects $\Gamma$ at $E$. Prove that if $|AB| = |DE|$, then $\angle ACB = 60^o$.
2022 Canadian Junior Mathematical Olympiad
You have an infinite stack of T-shaped tetrominoes (composed of four squares of side length 1), and an n × n board. You are allowed to place some tetrominoes on the board, possibly rotated, as long as no two tetrominoes overlap and no tetrominoes extend off the board. For which values of n can you cover the entire board?
those were also the first CMO problems
Assume that real numbers $a$ and $b$ satisfy $$ab+\sqrt{ab+1}+\sqrt{a^2+b}\sqrt{a+b^2}=0.$$Find, with proof, the value of $$b\sqrt{a^2+b}+a\sqrt{b^2+a}.$$
Let $d(k)$ denote the number of positive integer divisors of $k$. For example, $d(6) = 4$ since $6$ has $4$ positive divisors, namely, $1, 2, 3$, and $6$. Prove that for all positive integers $n$, $$d(1) + d(3) + d(5) +...+ d(2n - 1)\le d(2) + d(4) + d(6) + ... + d(2n).$$
Vishal starts with $n$ copies of the number $1$ written on the board. Every minute, he takes two numbers $a, b$ and replaces them with either $a+b$ or $\min(a^2, b^2)$. After $n-1$ there is $1$ number on the board. Let the maximal possible value of this number be $f(n)$. Prove $2^{n/3}<f(n)\leq 3^{n/3}$.