Let $P(x)=x^4-x^3-3x^2-x+1.$ Prove that there are infinitely many positive integers $n$ such that $P(3^n)$ is not a prime.

Prove that for each triangle, there exists a vertex, such that with the two sides starting from that vertex and each cevian starting from that vertex, is possible to construct a triangle.

In the Cartesian plane $\mathbb{R}^2,$ each triangle contains a Mediterranean point on its sides or in its interior, even if the triangle is degenerated into a segment or a point. The Mediterranean points have the following properties: (i) If a triangle is symmetric with respect to a line which passes through the origin $(0,0)$, then the Mediterranean point lies on this line. (ii) If the triangle $DEF$ contains the triangle $ABC$ and if the triangle $ABC$ contains the Mediterranean points $M$ of $DEF,$ then $M$ is the Mediterranean point of the triangle $ABC.$ Find all possible positions for the Mediterranean point of the triangle with vertices $(-3,5),\ (12,5),\ (3,11).$

In a mathematical contest, some of the competitors are friends and friendship is mutual. Prove that there is a subset $M$ of the competitors such that each element of $M$ has at most three friends in $M$ and such that each competitor who is not in $M,$ has at least four friends in $M.$