Let $m$ and $n$ be fixed positive integers. Tsvety and Freyja play a game on an infinite grid of unit square cells. Tsvety has secretly written a real number inside of each cell so that the sum of the numbers within every rectangle of size either $m$ by $n$ or $n$ by $m$ is zero. Freyja wants to learn all of these numbers. One by one, Freyja asks Tsvety about some cell in the grid, and Tsvety truthfully reveals what number is written in it. Freyja wins if, at any point, Freyja can simultaneously deduce the number written in every cell of the entire infinite grid (If this never occurs, Freyja has lost the game and Tsvety wins). In terms of $m$ and $n$, find the smallest number of questions that Freyja must ask to win, or show that no finite number of questions suffice. [i]Nikolai Beluhov[/i]

Let $ABCD$ be a convex quadrilateral with $\angle CBD = 2 \angle ADB$, $\angle ABD = 2 \angle CDB$ and $AB = CB$. Prove that $AD = CD$.

Prove that among any seven real numbers there exist two,$ a$ and $b$, such that $\sqrt3|a-b|\le |1+ab|$. Give an example of six real numbers not having this property.

Let $n$ be a positive integer. Prove that \[\sum_{i=1}^nx_i(1-x_i)^2\le\left(1-\frac{1}{n}\right)^2 \] for all nonnegative real numbers $x_1,x_2,\ldots ,x_n$ such that $x_1+x_2+\ldots x_n=1$.

In $\triangle ABC$ with side lengths $AB=13$, $BC=14$, and $CA=15$, let $M$ be the midpoint of $\overline{BC}$. Let $P$ be the point on the circumcircle of $\triangle ABC$ such that $M$ is on $\overline{AP}$. There exists a unique point $Q$ on segment $\overline{AM}$ such that $\angle PBQ = \angle PCQ$. Then $AQ$ can be written as $\frac{m}{\sqrt{n}}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Show that for any triangle $ABC$, the following inequality is true: \[ a^2 + b^2 +c^2 > \sqrt{3} max \{ |a^2 - b^2|, |b^2 -c^2|, |c^2 -a^2| \} . \]

In tetrahedron $ABCD$ with circumradius $2$, $AB = 2$, $CD = \sqrt{7}$, and $\angle ABC = \angle BAD = \frac{\pi}{2}$. Find all possible angles between the planes containing $ABC$ and $ABD$.

Let $n$ be a positive integer. Let $P,Q,A_1,A_2,...,A_n$ be distinct points such that $A_1,A_2,...,A_n$ are not collinear. Suppose that $PA_1 + PA_2 + ...+PA_n$, and $QA_1 + QA_2 +...+ QA_n$, have a common value $s$ for some real number $s$. Prove that there exists a point $R$ such that $$RA_1 + RA_2 +... + RA_n < s.$$

Let $P$ be the point of intersection of the diagonals of a convex quadrilateral $ABCD$.Let $X,Y,Z$ be points on the interior of $AB,BC,CD$ respectively such that $\frac{AX}{XB}=\frac{BY}{YC}=\frac{CZ}{ZD}=2$. Suppose that $XY$ is tangent to the circumcircle of $\triangle CYZ$ and that $Y Z$ is tangent to the circumcircle of $\triangle BXY$.Show that $\angle APD=\angle XYZ$.

The integers $ 1,2,...,n$ are placed in order so that each value is either strictly bigger than all the preceding values or is strictly smaller than all preceding values. In how many ways can this be done?

a) Show that $m^2- m +1$ is an element of the set $\{n^2 + n +1 | n \in N\}$, for any positive integer $ m$. b) Let $p$ be a perfect square, $p> 1$. Prove that there exists positive integers $r$ and $q$ such that $$p^2 + p +1=(r^2 + r + 1)(q^2 + q + 1).$$

Find the real number $k$ such that $a$, $b$, $c$, and $d$ are real numbers that satisfy the system of equations \begin{align*} abcd &= 2007,\\ a &= \sqrt{55 + \sqrt{k+a}},\\ b &= \sqrt{55 - \sqrt{k+b}},\\ c &= \sqrt{55 + \sqrt{k-c}},\\ d &= \sqrt{55 - \sqrt{k-d}}. \end{align*}

Let $\left(a_k\right)_{k=1}^\infty$ be a sequence of real numbers such that \[a_{k-1}+a_{k+1}\ge2a_k\] for all $k>1.$ For $n\ge1$ denote \[A_n=\frac1n\left(a_1+\cdots+a_n\right).\] Show that also the inequality \[A_{n-1}+A_{n+1}\ge2A_n\] holds for every $n>1.$

For real numbers $x$ and $y$ we define $M(x, y)$ to be the maximum of the three numbers $xy$, $(x- 1)(y - 1)$, and $x + y - 2xy$. Determine the smallest possible value of $M(x, y)$ where $x$ and $y$ range over all real numbers satisfying $0 \le x, y \le 1$.

Let $S_r(n)=1^r+2^r+\cdots+n^r$ where $n$ is a positive integer and $r$ is a rational number. If $S_a(n)=(S_b(n))^c$ for all positive integers $n$ where $a, b$ are positive rationals and $c$ is positive integer then we call $(a,b,c)$ as [i]nice triple.[/i] Find all nice triples.

Let $f$ be a polynomial function such that, for all real $x$, \[f(x^2 + 1) = x^4 + 5x^2 + 3.\] For all real $x$, $f(x^2-1)$ is $ \textbf{(A)}\ x^4+5x^2+1\qquad\textbf{(B)}\ x^4+x^2-3\qquad\textbf{(C)}\ x^4-5x^2+1\qquad\textbf{(D)}\ x^4+x^2+3\qquad\textbf{(E)}\ \text{None of these} $

Several (at least three) turtles are crawling along the plane, the velocities of which are constant in magnitude and direction (all are equal in magnitude, but pairwise different in direction). Prove that regardless of the initial location, after some time all the turtles will be at the vertices of some convex polygon.

Points $A, B$ and $P$ lie on the circumference of a circle $\Omega_1$ such that $\angle APB$ is an obtuse angle. Let $Q$ be the foot of the perpendicular from $P$ on $AB$. A second circle $\Omega_2$ is drawn with centre $P$ and radius $PQ$. The tangents from $A$ and $B$ to $\Omega_2$ intersect $\Omega_1$ at $F$ and $H$ respectively. Prove that $FH$ is tangent to $\Omega_2$.

Part of an "$n$-pointed regular star" is shown. It is a simple closed polygon in which all $2n$ edges are congruent, angles $A_{1}$, $A_{2}$, $\ldots$, $A_{n}$ are congruent and angles $B_{1}$, $B_{2}$, $\ldots$, $B_{n}$ are congruent. If the acute angle at $A_{1}$ is $10^{\circ}$ less than the acute angle at $B_{1}$, then $n = $ [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=dir(90-2*36), B=dir(90-36), C=dir(90), D=dir(90+36), E=dir(90+2*36); pair F=2*dir(90-1.5*36), G=2*dir(90-0.5*36), H=2*dir(90+0.5*36), I=2*dir(90+1.5*36); draw(A--F--B--G--C--H--D--I--E); label("$B_2$", B, -0.3*dir(B)); label("$B_1$", C, -0.3*dir(C)); label("$B_n$", D, -0.3*dir(D)); label("$A_3$", F, dir(F)); label("$A_2$", G, dir(G)); label("$A_1$", H, dir(H)); label("$A_n$", I, dir(I)); [/asy] $ \textbf{(A)}\ 12\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 36\qquad\textbf{(E)}\ 60 $

For every positive integer $m$ prove the inquality $|\{\sqrt{m}\} - \frac{1}{2}| \geq \frac{1}{8(\sqrt m+1)} $ (The integer part $[x]$ of the number $x$ is the largest integer not exceeding $x$. The fractional part of the number $x$ is a number $\{x\}$ such that $[x]+\{x\}=x$.) A. Golovanov

Let $x,y,z$ be real numbers such that $(x-z)(y-z)=x+y+z-3.$ Prove that $x^2+y^2+z^2\ge3.$

Let there be $131$ given distinct natural numbers, each having prime divisors not exceeding $42$. Prove that one can choose four of them whose product is a perfect square.

For each integer $n\geqslant2$, determine the largest real constant $C_n$ such that for all positive real numbers $a_1, \ldots, a_n$ we have \[\frac{a_1^2+\ldots+a_n^2}{n}\geqslant\left(\frac{a_1+\ldots+a_n}{n}\right)^2+C_n\cdot(a_1-a_n)^2\mbox{.}\] [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 2)[/i]

A point $O$ lies inside $\triangle ABC$ so that $\angle BOC=90-\angle BAC$. Let $BO, CO$ meet $AC, AB$ at $K, L$. Points $K_1, L_1$ lie on the segments $CL, BK$ so that $K_1B=K_1K$ and $L_1C=L_1L$. If $M$ is the midpoint of $BC$, then prove that $\angle K_1ML_1=90^{o}$. [i]Proposed by Anton Trygub[/i]

Given a convex polygon. You can put no triangle with area $1$ inside it. Prove that you can put the polygon inside a triangle with the area $4$.