$x, y, z$ are positive reals such that $x \leq 1$. Prove that $$xy+y+2z \geq 4 \sqrt{xyz}$$

In a school with $101$ students, each student has at least one friend among the other students. Show that for every integer $1<n<101$, a group of $n$ students can be selected from this school in such a way that each selected student has at least one friend among the other selected students.

Let $m, n, a, k$ be positive integers and $k>1$ such that the equality $$5^m+63n+49=a^k$$holds. Find the minimum value of $k$.

In parallellogram $ABCD$, on the arc $BC$ of the circumcircle $(ABC)$, not containing the point $A$, we take a point $P$ and on the $[AC$, we take a point $Q$ such that $\angle PBC= \angle CDQ$. Prove that $(APQ)$ is tangent to $AB$.