OFM 2020

Let $ABC$ be a triangle such that $AB <AC$ , $\omega$ its inscribed circle and $\Gamma$ its circumscribed circle. Let also $\omega_b$ be the excircle relative to vertex $B$ , then $B'$ is the point of tangency between $\omega_b$ and $(AC)$ . Similarly, let the circle $\omega_c$ be the excircle exinscribed relative to vertex $C$ , then $C'$ is the point of tangency between $\omega_c$ and $(AB)$ . Finally, let $I$ be the center of $\omega$ and $X$ the point of $\Gamma$ such that $\angle XAI$ is a right angle. Prove that the triangles $XBC'$ and $XCB'$ are congruent.

Emperor Zorg wishes to found a colony on a new planet. Each of the $n$ cities that he will establish there will have to speak exactly one of the Empire's $2020$ official languages. Some towns in the colony will be connected by a direct air link, each link can be taken in both directions. The emperor fixed the cost of the ticket for each connection to $1$ galactic credit. He wishes that, given any two cities speaking the same language, it is always possible to travel from one to the other via these air links, and that the cheapest trip between these two cities costs exactly $2020$ galactic credits. For what values of $n$ can Emperor Zorg fulfill his dream?

Let $n$ be an integer greater than or equal to $1$ . Find, as a function of $n$ , the smallest integer $k\ge 2$ such that, among any $k$ real numbers, there are necessarily two of which the difference, in absolute value, is either strictly less than $1 / n$ , either strictly greater than $n$ .

Find all the integers $x, y$ and $z$ greater than or equal to $0$ such that $2^x + 9 \cdot 7^y = z^3$