PMO 2020

A T-tetromino is formed by adjoining three unit squares to form a $1 \times 3$ rectangle, and adjoining on top of the middle square a fourth unit square. Determine the least number of unit squares that must be removed from a $202 \times 202$ grid so that it can be tiled using T-tetrominoes.

Determine all positive integers $k$ for which there exist positive integers $r$ and $s$ that satisfy the equation $$(k^2-6k+11)^{r-1}=(2k-7)^{s}.$$

Define the sequence $\{a_i\}$ by $a_0=1$ , $a_1=4$ , and $a_{n+1}=5a_n-a_{n-1}$ for all $n\geq 1$ . Show that all terms of the sequence are of the form $c^2+3d^2$ for some integers $c$ and $d$ .

Let $\triangle ABC$ be an acute triangle with circumcircle $\Gamma$ and $D$ the foot of the altitude from $A$ . Suppose that $AD=BC$ . Point $M$ is the midpoint of $DC$ , and the bisector of $\angle ADC$ meets $AC$ at $N$ . Point $P$ lies on $\Gamma$ such that lines $BP$ and $AC$ are parallel. Lines $DN$ and $AM$ meet at $F$ , and line $PF$ meets $\Gamma$ again at $Q$ . Line $AC$ meets the circumcircle of $\triangle PNQ$ again at $E$ . Prove that $\angle DQE = 90^{\circ}$ .