$N$ different positive integers are arranged around a circle , in such a way that the sum of every $5$ consecutive numbers in the circle is a multiple of $13$. Let $A $ be the smallest possible sum of the $n$ numbers. Calculate the value of $A$ for $\bullet$ $n = 99$, $\bullet$ $n = 100$.

Let $ABC$ be a triangle such that $M$ and $N$ are the midpoints of $AC$ and $BC$, respectively. Let $I$ be the incenter of $ABC$ and $E$ be the intersection of $MN$ with $Bl$. Let $P$ be a point such that $EP$ is perpendicular to $MN$ and $NP$ parallel to $IA$. Prove that $IP$ is perpendicular to $BC$.

Find the smallest natural number $n$ for which there exists a natural number $x$ such that $$(x+1)^3 + (x + 2)^3 + (x + 3)^3 + (x + 4)^3 = (x + n)^3.$$

We have an infinite sequence of integers $\{x_n\}$, such that $x_1 = 1$, and, for all $n \ge 1$, it holds that $x_n < x_{n+1} \le 2n$. Prove that there are two terms of the sequence,$ x_r$ and $x_s$, such that $x_r - x_s = 2018$.

A $300\times 300$ board is arbitrarily filled with $2\times 1$ dominoes with no overflow, underflow, or overlap. (Tokens can be placed vertically or horizontally.) Decide if it is possible to paint the tiles with three different colors, so that the following conditions are met: $\bullet$ Each token is painted in one and only one of the colors. $\bullet$ The same number of tiles are painted in each color. $\bullet$ No piece is a neighbor of more than two pieces of the same color. Note: Two dominoes are neighbors if they share an edge.

Let $ABC$ be a triangle with $AB < AC$ and $M$ the midpoint of the arc $BC$ containing $A$, plus $T$ the foot of the perpendicular from $M$ on side $AC$. Prove that $AB + AT = TC$.