4 tokens are placed in the plane. If the tokens are now at the vertices of a convex quadrilateral $P$, then the following move could be performed: choose one of the tokens and shift it in the direction perpendicular to the diagonal of $P$ not containing this token; while shifting tokens it is prohibited to get three collinear tokens. Suppose that initially tokens were at the vertices of a rectangle $\Pi$, and after a number of moves tokens were at the vertices of one another rectangle $\Pi'$ such that $\Pi'$ is similar to $\Pi$ but not equal to $\Pi $. Prove that $\Pi$ is a square.

Let $C_0$ be the midpoint of hypotenuse $AB$ of triangle $ABC$; $AA_1, BB_1$ the bisectors of this triangle; $I$ its incenter. Prove that the lines $C_0I$ and $A_1B_1$ meet on the altitude from $C$. [i]Proposed by A.Zaslavsky[/i]

$f(x)=ax^2+bx+c;a,b,c$ are reals. $M=\{f(2n)|n \text{ is integer}\},N=\{f(2n+1)|n \text{ is integer}\}$ Prove that $M=N$ or $M \cap N = \O $

Let $n \ge 2$ be an integer. Prove that if $k^2 + k + n$ is prime for all integers $k$ such that $0 \leq k \leq \sqrt{\frac{n}{3}}$, then $k^2 +k + n$ is prime for all integers $k$ such that $0 \leq k \leq n - 2$.

$64$ distinct points are positioned in the plane so that they determine exactly $2003$ different lines. Prove that among the $64$ points there are at least $4$ collinear points.

A game is played by $n$ girls ($n \geq 2$), everybody having a ball. Each of the $\binom{n}{2}$ pairs of players, is an arbitrary order, exchange the balls they have at the moment. The game is called nice [b]nice[/b] if at the end nobody has her own ball and it is called [b]tiresome[/b] if at the end everybody has her initial ball. Determine the values of $n$ for which there exists a nice game and those for which there exists a tiresome game.

Cindy was asked by her teacher to subtract $ 3$ from a certain number and then divide the result by $ 9$. Instead, she subtracted $ 9$ and then divided the result by $ 3$, giving an answer of $ 43$. What would her answer have been had she worked the problem correctly? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 34 \qquad \textbf{(C)}\ 43 \qquad \textbf{(D)}\ 51 \qquad \textbf{(E)}\ 138$

Let $a_1<a_2 \dots <a_n$ be positive integers, with $n\geq 2$. An invisible frog lies on the real line, at a positive integer point. Initially, the hunter chooses a number $k$, and then, once every minute, he can check if the frog currently lies in one of $k$ points of his choosing, after which the frog goes from its point $x$ to one of the points $x+a_1, x+a_2 \dots x+a_n$. Based on the values of $a_1, a_2 \dots a_n$, what is the smallest value of $k$ such that the hunter can guarantee to find the frog within a finite number of minutes, no matter where it initially started? [i]Proposed by David Anghel[/i]

A farmer bought $749$ sheeps. He sold $700$ of them for the price paid for the $749$ sheep. The remaining $49$ sheep were sold at the same price per head as the other $700$. Based on the cost, the percent gain on the entire transaction is: ${{ \textbf{(A)}\ 6.5 \qquad\textbf{(B)}\ 6.75 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 7.5 }\qquad\textbf{(E)}\ 8 } $

Let $S$ denote the set of $2$ by $2$ matrices with integer entries and determinant $1$, and let $T$ denote those matrices of $S$ which are congruent to the identity matrix $I\pmod3$ (so $\begin{pmatrix}a&b\\c&d\end{pmatrix}\in T$ means that $a,b,c,d\in\mathbb Z,ad-bc=1,$ and $3$ divides $b,c,a-1,d-1$). (a) Let $f:T\to\mathbb R$ be a function such that for every $X,Y\in T$ with $Y\ne I$, either $f(XY)>f(X)$ or $f(XY^{-1})>f(X)$. Show that given two finite nonempty subsets $A,B$ of $T$, there are matrices $a\in A$ and $b\in B$ such that if $a'\in A$, $b'\in B$ and $a'b'=ab$, then $a'=a$ and $b'=b$. (b) Show that there is no $f:S\to\mathbb R$ such that for every $X,Y\in S$ with $Y\ne\pm I$, either $f(XY)>f(X)$ or $f(XY^{-1})>f(X)$.

Prove that there are no perfect squares in the array below: \[\begin{array}{cccc}11&111&1111&...\\22&222&2222&...\\33&333&3333&...\\44&444&4444&...\\55&555&5555&... \\66&666&6666&...\\77&777&7777&...\\88&888&8888&...\\99&999&9999&...\end{array}\]

Let $n$ be an integer greater than $1$. Define \[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\] where $[z]$ denotes the largest integer less than or equal to $z$. Prove that \[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]

Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$. The lines $AB$ and $CD$ meet at $P$, the lines $AD$ and $BC$ meet at $Q$, and the diagonals $AC$ and $BD$ meet at $R$. Let $M$ be the midpoint of the segment $PQ$, and let $K$ be the common point of the segment $MR$ and the circle $\omega$. Prove that the circumcircle of the triangle $KPQ$ and $\omega$ are tangent to one another.

Let $f : R \to R$ be a function satisfying $f(f(x)) = 4x + 1$ for all real number $x$. Prove that the equation $f(x) = x$ has a unique solution.

Let \(\mathbb R_{>0}\) denote the set of positive real numbers. Find all functions \(f:\mathbb R_{>0}\to\mathbb R_{>0}\) such that for all positive real numbers \(x\) and \(y\), \[f(xy+1)=f(x)f\left(\frac1x+f\left(\frac1y\right)\right).\] [i]Proposed by Luke Robitaille[/i]

Let $x,y,z\in \mathbb{R}^+$ be such that $xy+yz+zx=x+y+z$. Prove the following inequality: $\frac{1}{x^2+y+1}+\frac{1}{y^2+z+1}+\frac{1}{z^2+x+1}\leq 1$.