We say that a triple of integers $(x, y, z)$ is of jenifer type if $x, y$, and $z$ are positive integers, with $y \ge 2$, and $$x^2 - 3y^2 = z^2 - 3.$$a) Find a triple $(x, y, z)$ of the jenifer type with $x = 5$ and $x = 7$. b) Show that for every $x \ge 5$ and odd there are at least two distinct triples $(x, y_1, z_1)$ and $(x, y_2, z_2)$ of jenifer type. c) Find some triple $(x, y, z)$ of jenifer type with $x$ even.

In a sequence of positive integers, a inversion is a pair of positions, where the number in left is greater than the number in right. For example in the sequence $2, 5, 3, 1, 3$ has $5$ inversions{(5,1),(3,1),(5,3),(2,1),(5,3)}. Find the greatest number of inversions in a sequence where the sum of elements is $n$ a) where $n=7$ b) where $n=2019$

Let $ABC$ be a triangle and $E$ and $F$ two arbitrary points on sides $AB$ and $AC$, respectively. The circumcircle of triangle $AEF$ meets the circumcircle of triangle $ABC$ again at point $M$. The point $D$ is such that $EF$ bisects the segment $MD$ . Finally, $O$ is the circumcenter of triangle $ABC$. Prove that $D$ lies on line $BC$ if and only if $O$ lies on the circumcircle of triangle $AEF$.

Twenty players participated in a chess tournament. Each player faced every other player exactly once and each match ended with either player winning or a draw. In this tournament, it was noticed that for every match that ended in a draw, each of the other $18$ players won at least one of the two players involved in it. We also know that at least two games ended in a draw. Show that it is possible to name the players as $P_1, P_2, ...., P_{20}$ so that player $P_k$ beat player $P_{k+1}$, to each $k \in \{ 1, 2, 3,... , 19\}$.