Let $P(x)=a_{0}+a_{1}x+...+a_{n}x^{n}\in\mathbb{C}[x]$ , where $a_{n}=1$. The roots of $P(x)$ are $b_{1},b_{2},...,b_{n}$, where $|b_{1}|,|b_{2}|,...,|b_{j}|>1$ and $|b_{j+1}|,...,|b_{n}|\leq 1$. Prove that $\prod_{i=1}^{j}|b_{i}|\leq\sqrt{|a_{0}|^{2}+|a_{1}|^{2}+...+|a_{n}|^{2}}$.

Given a sequence of eight integers $x_{1},x_{2},...,x_{8}$ in a single operation one replaces these numbers with $|x_{1}-x_{2}|,|x_{2}-x_{3}|,...,|x_{8}-x_{1}|$. Find all the eight-term sequences of integers which reduce to a sequence with all the terms equal after finitely many single operations.

Suppose that $n$ persons meet in a meeting, and that each of the persons is acquainted to exactly $8$ others. Any two acquainted persons have exactly $4$ common acquaintances, and any two non-acquainted persons have exactly $2$ common acquaintances. Find all possible values of $n$.

Let $m_{1},m_{2},...,m_{n}$ be mutually distinct integers. Prove that there exists a $f(x)\in\mathbb{Z}[x]$ of degree $n$ satisfying the following two conditions: a)$f(m_{i})=-1\forall i=1,2,...,n$. b)$f(x)$ is irreducible.

Let $P$ be a point on the circumcircle of a triangle $A_{1}A_{2}A_{3}$, and let $H$ be the orthocenter of the triangle. The feet $B_{1},B_{2},B_{3}$ of the perpendiculars from $P$ to $A_{2}A_{3},A_{3}A_{1},A_{1}A_{2}$ lie on a line. Prove that this line bisects the segment $PH$.

Let $a,b,c,d$ are integers such that $(a,b)=(c,d)=1$ and $ad-bc=k>0$. Prove that there are exactly $k$ pairs $(x_{1},x_{2})$ of rational numbers with $0\leq x_{1},x_{2}<1$ for which both $ax_{1}+bx_{2},cx_{1}+dx_{2}$ are integers.