Given a polyhedron $P$. Mikita claims that he can write one integer on each face of $P$ such that not all the written numbers are zeros, and for each vertex $V$ of $P$ the sum of numbers on faces containing $V$ is equals to 0. Matvei claims that he can write one integer in each vertex of $P$ such that not all the written numbers are zeros, and for each face $F$ of $P$ the sum of numbers in vertices belonging to $F$ is equals to 0. Show that if the number of edges of polyhedron $P$ is odd, then at least one of the boys is right.

Given $n \geq 2$ positive real numbers $x_1 \leq x_2 \leq \ldots \leq x_n$ satisfying the equalities $$x_1+x_2+\ldots+x_n=4n$$$$\frac{1}{x_1}+\frac{1}{x_2}+\ldots+\frac{1}{x_n}=n$$Prove that $\frac{x_n}{x_1} \geq 7+4\sqrt{3}$

Let $a>1$ be an integer which is not divisible by four. Prove that there are infinitely many primes $p$ of the form $4k-1$ such that $p | a^d-1$ for some $d<\frac{p-1}{2}$

Prove that the equation $2+x^3y+y^2+z^2=0$ has no solutions in integers.

The circle $\Omega$ centered at $O$ is the circumcircle of the triangle $ABC$. Point $D$ is chosen so that $BD \perp BC$ and points $A$ and $D$ lie in different half-planes with respect to the line $BC$. Let $E$ be a point such that $\angle ADB=\angle BDE$ and $\angle EBD+\angle ACB=90$. Point $P$ is chosen on the line $AD$ so that $OP \perp BC$. Let $Q$ be an arbitrary point on $\Omega$, and $R$ be a point on the line $BQ$ such that $PQ \parallel DR$. Prove that $\angle ARB=\angle BRE$. (All angles are oriented in the same way)

Vika calls some positive integers nice, and it is known that among any ten consecutive positive integers there is at least one nice. Prove that there are infinitely many positive integers $n$ for which $ab-cd=2n^2$ for some pairwise distinct nice numbers $a,b,c,$ and $d$