Ten volleyball teams played a tournament; every two teams met exactly once. The winner of the play gets 1 point, the loser gets 0 (there are no draws in volleyball). If the team that scored $n$-th has $x_{n}$ points ($n=1, \dots, 10$), prove that $x_{1}+2x_{2}+\dots+10x_{10}\geq 165$. Proposed by D. Teryoshin

Solve the equation \[(a^{2},b^{2})+(a,bc)+(b,ac)+(c,ab)=199.\] in positive integers. (Here $(x,y)$ denotes the greatest common divisor of $x$ and $y$.) Proposed by S. Berlov

Do there exist quadratic trinomials $P, \ \ Q, \ \ R$ such that for every integers $x$ and $y$ an integer $z$ exists satisfying $P(x)+Q(y)=R(z)?$ Proposed by A. Golovanov

Unit square $ABCD$ is divided into $10^{12}$ smaller squares (not necessarily equal). Prove that the sum of perimeters of all the smaller squares having common points with diagonal $AC$ does not exceed 1500. Proposed by A. Kanel-Belov

All positive integers are distributed among two disjoint sets $N_{1}$ and $N_{2}$ such that no difference of two numbers belonging to the same set is a prime greater than 100. Find all such distributions. Proposed by N. Sedrakyan

Non-zero numbers are arranged in $n \times n$ square ($n>2$). Every number is exactly $k$ times less than the sum of all the other numbers in the same cross (i.e., $2n-2$ numbers written in the same row or column with this number). Find all possible $k$. Proposed by D. Rostovsky, A. Khrabrov, S. Berlov

$ABCD$ is a convex quadrilateral; half-lines $DA$ and $CB$ meet at point $Q$; half-lines $BA$ and $CD$ meet at point $P$. It is known that $\angle AQB=\angle APD$. The bisector of angle $\angle AQB$ meets the sides $AB$ and $CD$ of the quadrilateral at points $X$ and $Y$, respectively; the bisector of angle $\angle APD$ meets the sides $AD$ and $BC$ at points $Z$ and $T$, respectively. The circumcircles of triangles $ZQT$ and $XPY$ meet at point $K$ inside the quadrilateral. Prove that $K$ lies on the diagonal $AC$. Proposed by S. Berlov

Click for solution First we can see that in fact $ AC$ is the radical axis of the circles $ PXY$ and $ QZT$, therefore their intersection in the interior of the quadrilateral will trivially lie on $ AC$. Now, let's prove that $ AC$ is their axis. We will prove only that $ A \in d$, where $ d$ is the radical axis, for $ C$ beeing the same thing. But $ A \in d \iff AZ \cdot AQ = AX \cdot AP$, (the power of $ A$ w.r.t to the circles), but we have that $ AD \cdot AQ = AB \cdot AP$, because the quadrilateral $ BPQD$ is cyclic, therefore we only need to prove that : $ \frac {AD}{AZ} = \frac {AB}{AX}\iff \frac {PD}{AP} = \frac {AQ}{QB}$ $ \iff \frac {PD}{AQ} = \frac {AP}{QB}$, which is obvious because the triangles $ \triangle{PAD}$ and $ \triangle{QAB}$ are similar.

Is it possible to colour all positive real numbers by 10 colours so that every two numbers with decimal representations differing in one place only are of different colours? (We suppose that there is no place in a decimal representations such that all digits starting from that place are 9's.) Proposed by A. Golovanov

$16$ chess players held a tournament among themselves: every two chess players played exactly one game. For victory in the party was given $1$ point, for a draw $0.5$ points, for defeat $0$ points. It turned out that exactly 15 chess players shared the first place. How many points could the sixteenth chess player score?

Is it possible to arrange integers in the cells of the infinite chechered sheet so that every integer appears at least in one cell, and the sum of any $10$ numbers in a row vertically or horizontal, would be divisible by $101$?

Let ABC be an acute isosceles triangle ($AB=BC$) inscribed in a circle with center $O$ . The line through the midpoint of the chord $AB$ and point $O$ intersects the line $AC$ at $L$ and the circle at the point $P$. Let the bisector of angle $BAC$ intersects the circle at point $K$. Lines $AB$ and $PK$ intersect at point $D$. Prove that the points $L,B,D$ and $P$ lie on the same circle.

Natural numbers $1, 2, 3,.., 100$ are contained in the union of $N$ geometric progressions (not necessarily with integer denominations). Prove that $N \ge 31$

On the side $AB$ of an isosceles triangle $AB$ ($AC=BC$) lie points $P$ and $Q$ such that $\angle PCQ \le \frac{1}{2} \angle ACB$. Prove that $PQ \le \frac{1}{2} AB$.

Several rational numbers were written on the blackboard. Dima wrote off their fractional parts on paper. Then all the numbers on the board squared, and Dima wrote off another paper with fractional parts of the resulting numbers. It turned out that on Dima's papers were written the same sets of numbers (maybe in different order). Prove that the original numbers on the board were integers. (The fractional part of a number $x$ is such a number $\{x\}, 0 \le \{x\} <1$, that $x-\{x\}$ is an integer.)

Can three persons, having one double motorcycle, overcome the distance of $70$ km in $3$ hours? Pedestrian speed is $5$ km / h and motorcycle speed is $50$ km / h.