If $x,y,z>0$, prove that $(3x+y)(3y+z)(3z+x) \ge 64xyz$. When we have equality;

Let $ABC$ be an acute triangle inscribed in a circle of center $O$. If the altitudes $BD,CE$ intersect at $H$ and the circumcenter of $\triangle BHC$ is $O_1$, prove that $AHO_1O$ is a parallelogram.

Prove that there is not a positive integer $n$ such that numbers $(n+1)2^n, (n+3)2^{n+2}$ are both perfect squares.

Pupils of a school are divided into $112$ groups, of $11$ members each. Any two groups have exactly one common pupil. Prove that: a) there is a pupil that belongs to at least $12$ groups. b) there is a pupil that belongs to all the groups.