Let $p$ be an odd prime number. a) Show that $p$ divides $n2^n + 1$ for infinitely many positive integers n. b) Find all $n$ satisfy condition above when $p = 3$

Let $I $be the incenter of triangle $ABC$and $J$ the excenter of the side $BC$: Let $M$ be the midpoint of $CB$ and $N$ the midpoint of arc $BAC$ of circle $(ABC)$. If $T$ is the symmetric of the point $N$ by the point $A$, prove that the quadrilateral $JMIT$ is cyclic

For $n \ge 3$, it is given an $2n \times 2n$ board with black and white squares. It is known that all border squares are black and no $2 \times 2$ subboard has all four squares of the same color. Prove that there exists a $2 \times 2$ subboard painted like a chessboard, i.e. with two opposite black corners and two opposite white corners.

There are $n$ people with hats present at a party. Each two of them greeted each other exactly once and each greeting consisted of exchanging the hats that the two persons had at the moment. Find all $n \ge 2$ for which the order of greetings can be arranged in such a way that after all of them, each person has their own hat back.

Let sequences of real numbers $(x_n)$ and $(y_n)$ satisfy $x_1 = y_1 = 1$ and $x_{n+1} =\frac{x_n + 2}{x_n + 1}$ and $y_{n+1} = \frac{y_n^2 + 2}{2y_n}$ for $n = 1,2, ...$ Prove that $y_{n+1} = x_{2^n}$ holds for $n =0, 1,2, ... $

The triangle $ABC$ ($AB > BC$) is inscribed in the circle $\Omega$. On the sides $AB$ and $BC$, the points $M$ and $N$ are chosen, respectively, so that $AM = CN$, The lines $MN$ and $AC$ intersect at point $K$. Let $P$ be the center of the inscribed circle of triangle $AMK$, and $Q$ the center of the excircle of the triangle $CNK$ tangent to side $CN$. Prove that the midpoint of the arc $ABC$ of the circle $\Omega$ is equidistant from the $P$ and $Q$.

Let $19$ integer numbers are given. Let Hamza writes on the paper the greatest common divisor for each pair of numbers. It occurs that the difference between the biggest and smallest numbers written on the paper is less than $180$. Prove that not all numbers on the paper are different.

Let $ABCD$ is a trapezoid with $\angle A = \angle B = 90^o$ and let $E$ is a point lying on side $CD$. Let the circle $\omega$ is inscribed to triangle $ABE$ and tangents sides $AB, AE$ and $BE$ at points $P, F$ and $K$ respectively. Let $KF$ intersects segments $BC$ and $AD$ at points $M$ and $N$ respectively, as well as $PM$ and $PN$ intersect $\omega$ at points $H$ and $T$ respectively. Prove that $PH = PT$.

Let $300$ students participate to the Olympiad. Between each $3$ participants there is a pair that are not friends. Hamza enumerates participants in some order and denotes by $x_i$ the number of friends of $i$-th participant. It occurs that $\{x_1,x_2,...,x_{299},x_{300}\} = \{1, 2,..., N - 1,N\}$ Find the biggest possible value for $N$.