PAMO 2021

Let $n$ be an integer greater than $3$ . A square of side length $n$ is divided by lines parallel to each side into $n^2$ squares of length $1$ . Find the number of convex trapezoids which have vertices among the vertices of the $n^2$ squares of side length $1$ , have side lengths less than or equal $3$ and have area equal to $2$ Note: Parallelograms are trapezoids.

Let $\Gamma$ be a circle, $P$ be a point outside it, and $A$ and $B$ the intersection points between $\Gamma$ and the tangents from $P$ to $\Gamma$ . Let $K$ be a point on the line $AB$ , distinct from $A$ and $B$ and let $T$ be the second intersection point of $\Gamma$ and the circumcircle of the triangle $PBK$ .Also, let $P'$ be the reflection of $P$ in point $A$ . Show that $\angle PBT=\angle P'KA$

Let $(a_i)_{i\in \mathbb{N}}$ and $(p_i)_{i\in \mathbb{N}}$ be two sequences of positive integers such that the following conditions hold: $\bullet ~~a_1\ge 2$ . $\bullet~~ p_n$ is the smallest prime divisor of $a_n$ for every integer $n\ge 1$ $\bullet~~ a_{n+1}=a_n+\frac{a_n}{p_n}$ for every integer $n\ge 1$ Prove that there is a positive integer $N$ such that $a_{n+3}=3a_n$ for every integer $n>N$

Find all integers $m$ and $n$ such that $\frac{m^2+n}{n^2-m}$ and $\frac{n^2+m}{m^2-n}$ are both integers.

Find all functions $f$ $:$ $\mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x,y \in \mathbb{R}$ : $$(f(x)+y)(f(y)+x)=f(x^2)+f(y^2)+2f(xy)$$

Let $ABCD$ be a trapezoid which is not a parallelogram, such that $AD$ is parallel to $BC$ . Let $O=BD\cap AC$ and $S$ be the second intersection of the circumcircles of triangles $AOB$ and $DOC$ . Prove that the circumcircles of triangles $ASD$ and $BSC$ are tangent.