In a group of interpreters each one speaks one of several foreign languages, 24 of them speak Japanese, 24 Malaysian, 24 Farsi. Prove that it is possible to select a subgroup in which exactly 12 interpreters speak Japanese, exactly 12 speak Malaysian and exactly 12 speak Farsi.

What is the square root of the sum of the first 2006 positive odd integers?

A positive integer $m$ is called a [i]beautiful [/i] integer if that there exists a positive integer $n$ such that $m = n^2+ n + 1$. Prove that there are infinitely many [i]beautiful [/i] integers with square factors, and the square factors of different beautiful integers are relative prime.

A billiard table. (see picture) A white ball is on $p_1$ and a red ball is on $p_2$. The white ball is shot towards the red ball as shown on the pic, hitting 3 sides first. Find the minimal distance the ball must travel. [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=280[/img]

Let $a_1, a_2, \ldots , a_{11}$ be 11 pairwise distinct positive integer with sum less than 2007. Let S be the sequence of $1,2, \ldots ,2007$. Define an [b]operation[/b] to be 22 consecutive applications of the following steps on the sequence $S$: on $i$-th step, choose a number from the sequense $S$ at random, say $x$. If $1 \leq i \leq 11$, replace $x$ with $x+a_i$ ; if $12 \leq i \leq 22$, replace $x$ with $x-a_{i-11}$ . If the result of [b]operation[/b] on the sequence $S$ is an odd permutation of $\{1, 2, \ldots , 2007\}$, it is an [b]odd operation[/b]; if the result of [b]operation[/b] on the sequence $S$ is an even permutation of $\{1, 2, \ldots , 2007\}$, it is an [b]even operation[/b]. Which is larger, the number of odd operation or the number of even permutation? And by how many? Here $\{x_1, x_2, \ldots , x_{2007}\}$ is an even permutation of $\{1, 2, \ldots ,2007\}$ if the product $\prod_{i > j} (x_i - x_j)$ is positive, and an odd one otherwise.

Let $ABCD$ be a square and let $S$ be the point of intersection of its diagonals $AC$ and $BD$. Two circles $k,k'$ go through $A,C$ and $B,D$; respectively. Furthermore, $k$ and $k'$ intersect in exactly two different points $P$ and $Q$. Prove that $S$ lies on $PQ$.

In a triangle $ABC$, the circle through $C$ touching $AB$ at $A$ and the circle through $B$ touching $AC$ at $A$ have different radii and meet again at $D$. Let $E$ be the point on the ray $AB$ such that $AB = BE$. The circle through $A$, $D$, $E$ intersect the ray $CA$ again at $F$ . Prove that $AF = AC$.

Given positive integers $a,b,c$ which are pairwise coprime. Let $f(n)$ denotes the number of the non-negative integer solution $(x,y,z)$ to the equation $$ax+by+cz=n.$$ Prove that there exists constants $\alpha, \beta, \gamma \in \mathbb{R}$ such that for any non-negative integer $n$, $$|f(n)- \left( \alpha n^2+ \beta n + \gamma \right) | < \frac{1}{12} \left( a+b+c \right).$$

There is a table with $n$ rows and $18$ columns. Each of its cells contains a $0$ or a $1$. The table satisfies the following properties: [list=1] [*]Every two rows are different. [*]Each row contains exactly $6$ cells that contain $1$. [*]For every three rows, there exists a column so that the intersection of the column with the three rows (the three cells) all contain $0$. [/list] What is the greatest possible value of $n$?

Each of the $15$ coaches ranked the $50$ selected football players on the places from $1$ to $50$. For each football player, the highest and lowest obtained ranks differ by at most $5$. For each of the players, the sum of the ranks he obtained is computed, and the sums are denoted by $S_1\le S_2\le\ldots\le S_{50}$. Find the largest possible value of $S_1$.

Eminem is trying to find the real Slim Shady in a row of $2025$ indistinguishable Slim Shady clones, one of which is the real Slim Shady. Eminem randomly guesses, and if he guesses wrong, a new clone joins the row and all the clones randomly rearrange themselves. He keeps guessing as more identical clones are added, trying to find the real Slim Shady. Find the probability that he will eventually find him within $15$ guesses.

An equilateral triangle of side length $ 10$ is completely filled in by non-overlapping equilateral triangles of side length $ 1$. How many small triangles are required? $ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 100 \qquad \textbf{(D)}\ 250 \qquad \textbf{(E)}\ 1000$

Each edge of a complete graph on $30$ vertices is coloured either red or blue. It is allowed to choose a non-monochromatic triangle and change the colour of the two edges of the same colour to make the triangle monochromatic. Prove that by using this operation repeatedly it is possible to make the entire graph monochromatic. (A complete graph is a graph where any two vertices are connected by an edge.)

Solve in natural numbers the equation $x^2-y^2=105$.

Determine the maximal size of the set $S$ such that: i) all elements of $S$ are natural numbers not exceeding $100$; ii) for any two elements $a,b$ in $S$, there exists $c$ in $S$ such that $(a,c)=(b,c)=1$; iii) for any two elements $a,b$ in $S$, there exists $d$ in $S$ such that $(a,d)>1,(b,d)>1$. [i]Yao Jiangang[/i]

Let $ f(x) \equal{} x^2 \plus{} 2007x \plus{} 1$. Prove that for every positive integer $ n$, the equation $ \underbrace{f(f(\ldots(f}_{n\ {\rm times}}(x))\ldots)) \equal{} 0$ has at least one real solution.

Let $\omega$ be a semicircle with $A B$ as the bounding diameter and let $C D$ be a variable chord of the semicircle of constant length such that $C, D$ lie in the interior of the arc $A B$. Let $E$ be a point on the diameter $A B$ such that $C E$ and $D E$ are equally inclined to the line $A B$. Prove that (a) the measure of $\angle C E D$ is a constant; (b) the circumcircle of triangle $C E D$ passes through a fixed point.

Let $n,p>1$ be positive integers and $p$ be prime. We know that $n|p-1$ and $p|n^3-1$. Prove that $4p-3$ is a perfect square.

Five different points $O,A,B,C,D$ are given in plane such that \[OA\le OB\le OC\le OD.\] Show that for area $P$ of any convex quadrilateral with vertices $A,B,C,D$ (not necessarily in this order) the inequality \[P\le \frac12(OA+OD)(OB+OC)\] holds and determine when equality occurs.

Let $ABC$ be an acute-angled triangle with $AB =4$ and $CD$ be the altitude through $C$ with $CD = 3$. Find the distance between the midpoints of $AD$ and $BC$

Does there exist an infinite non-constant arithmetic progression, each term of which is of the form $a^b$, where $a$ and $b$ are positive integers with $b\ge 2$?

Let $a_1, a_2, . . . , a_n$ be positive integers, with $n \ge 2$, such that $$ \lfloor \sqrt{a_1 \cdot a_2\cdot\cdot\cdot a_n} \rfloor = \lfloor \sqrt{a_1} \rfloor \cdot \lfloor \sqrt{a_2} \rfloor \cdot\cdot\cdot \lfloor \sqrt{a_n} \rfloor.$$ Prove that at least $n - 1$ of these numbers are perfect squares. Clarification: Given a real number $x$, $\lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$. For example $\lfloor \sqrt2\rfloor$ and $\lfloor 3\rfloor =3$.

Let $AXYZB$ be a regular pentagon with area $5$ inscribed in a circle with center $O$. Let $Y'$ denote the reflection of $Y$ over $\overline{AB}$ and suppose $C$ is the center of a circle passing through $A$, $Y'$ and $B$. Compute the area of triangle $ABC$. [i]Proposed by Evan Chen[/i]

What is the smallest positive integer that is divisible by $225$ and that has ony the numbers one and zero as digits?

Let a triangle $ABC$ be given. For any point $X$ of the triangle denote $m(X)=\min\{XA,XB,XC\}.$ Find all points $X$ (of triangle $ABC$) such that $m(X)$ is maximal.