Let $x,y,z$ be real numbers ( $x \ne y$, $y\ne z$, $x\ne z$) different from $0$. If $\frac{x^2-yz}{x(1-yz)}=\frac{y^2-xz}{y(1-xz)}$, prove that the following relation holds: $$x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$$

$2.$ Let $ABC$ be a triangle and $AD$ the angle bisector ($D\in BC$). The perpendicular from $B$ to $AD$ cuts the circumcircle of triangle $ABD$ at $E$. If $O$ is the center of the circle around $ABC$ , prove $A,O,E$ are collinear. Click to reveal hidden texthttps://artofproblemsolving.com/community/c6h605458p3596629 https://artofproblemsolving.com/community/c6h1294020p6857833

$3.$ Let $S$ be the set of all positive integers from $1$ to $100$ included. Two players play a game. The first player removes any $k$ numbers he wants, from $S$. The second player's goal is to pick $k$ different numbers, such that their sum is $100$. Which player has the winning strategy if : $a)$ $k=9$? $b)$ $k=8$?

$4.$ Let there be a variable positive integer whose last two digits are $3's$. Prove that this number is divisible by a prime greater than $7$.