Do there exist a sequence $a_{1}, a_{2}, a_{3}, \ldots$ of real numbers and a non-constant polynomial $P(x)$ such that $a_{m}+a_{n}=P(mn)$ for every positive integral $m$ and $n?$ Proposed by A. Golovanov

In the plane are given 100 lines such that no 2 are parallel and no 3 meet in a point. The intersection points are marked. Then all the lines and k of the marked points are erased. Given the remained points of intersection for what max k one can reconstruct the lines? Proposed by A. Golovanov

An acute triangle $ABC$ is inscribed in a circle of radius 1 with centre $O;$ all the angles of $ABC$ are greater than $45^\circ.$ $B_{1}$ is the foot of perpendicular from $B$ to $CO,$ $B_{2}$ is the foot of perpendicular from $B_{1}$ to $AC.$ Similarly, $C_{1}$ is the foot of perpendicular from $C$ to $BO,$ $C_{2}$ is the foot of perpendicular from $C_{1}$ to $AB.$ The lines $B_{1}B_{2}$ and $C_{1}C_{2}$ intersect at $A_{3}.$ The points $B_{3}$ and $C_{3}$ are defined in the same way. Find the circumradius of triangle $A_{3}B_{3}C_{3}.$ Proposed by F.Bakharev, F.Petrov

There are many opposition societies in the city of N. Each society consists of $10$ members. It is known that for every $2004$ societies there is a person belonging to at least $11$ of them. Prove that the government can arrest $2003$ people so that at least one member of each society is arrested. Proposed by V.Dolnikov, D.Karpov

50 knights of King Arthur sat at the Round Table. A glass of white or red wine stood before each of them. It is known that at least one glass of red wine and at least one glass of white wine stood on the table. The king clapped his hands twice. After the first clap every knight with a glass of red wine before him took a glass from his left neighbour. After the second clap every knight with a glass of white wine (and possibly something more) before him gave this glass to the left neughbour of his left neighbour. Prove that some knight was left without wine. Proposed by A. Khrabrov, incorrect translation from Hungarian

The incircle of triangle $ABC$ touches its sides $AB$ and $BC$ at points $P$ and $Q.$ The line $PQ$ meets the circumcircle of triangle $ABC$ at points $X$ and $Y.$ Find $\angle XBY$ if $\angle ABC = 90^\circ.$ Proposed by A. Smirnov

Zeroes and ones are arranged in all the squares of $n\times n$ table. All the squares of the left column are filled by ones, and the sum of numbers in every figure of the form [asy][asy]size(50); draw((2,1)--(0,1)--(0,2)--(2,2)--(2,0)--(1,0)--(1,2));[/asy][/asy] (consisting of a square and its neighbours from left and from below) is even. Prove that no two rows of the table are identical. Proposed by O. Vanyushina

It is known that $m$ and $n$ are positive integers, $m > n^{n-1}$, and all the numbers $m+1$, $m+2$, \dots, $m+n$ are composite. Prove that there exist such different primes $p_1$, $p_2$, \dots, $p_n$ that $p_k$ divides $m+k$ for $k = 1$, 2, \dots, $n$. Proposed by C. A. Grimm