A [i]hypergraph[/i] consists of a set of vertices $V$ and a set of subsets of those vertices, each of which is called an edge. (Intuitively, it's a graph in which each edge can contain multiple vertices). Suppose that in some hypergraph, no two edges have exactly one vertex in common. Prove that one can color this hypergraph's vertices such that every edge contains both colors of vertices.

The unit cells of a 5 x 5 board are painted with 5 colors in a way that every cell is painted by exactly one color and each color is used in 5 cells. Show that exists at least one line or one column of the board in which at least 3 colors were used.

A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that atleast $15$ balls of a single colour will be drawn?

For a positive integer $n$, a [i]sum-friendly odd partition[/i] of $n$ is a sequence $(a_1, a_2, \ldots, a_k)$ of odd positive integers with $a_1 \le a_2 \le \cdots \le a_k$ and $a_1 + a_2 + \cdots + a_k = n$ such that for all positive integers $m \le n$, $m$ can be [b]uniquely[/b] written as a subsum $m = a_{i_1} + a_{i_2} + \cdots + a_{i_r}$. (Two subsums $a_{i_1} + a_{i_2} + \cdots + a_{i_r}$ and $a_{j_1} + a_{j_2} + \cdots + a_{j_s}$ with $i_1 < i_2 < \cdots < i_r$ and $j_1 < j_2 < \cdots < j_s$ are considered the same if $r = s$ and $a_{i_l} = a_{j_l}$ for $1 \le l \le r$.) For example, $(1, 1, 3, 3)$ is a sum-friendly odd partition of $8$. Find the number of sum-friendly odd partitions of $9999$.

In any set of $181$ square integers, prove that one can always find a subset of $19$ numbers, sum of whose elements is divisible by $19$.

$162$ pluses and $144$ minuses are placed in a $30\times 30$ table in such a way that each row and each column contains at most $17$ signs. (No cell contains more than one sign.) For every plus we count the number of minuses in its row and for every minus we count the number of pluses in its column. Find the maximum of the sum of these numbers.

Each cell of the checkered plane is colored in one of $n^2$ colors so that in any square of $n \times n$ cells all colors occur. It is known that in some line all the colors occur. Prove that there exists a column colored in exactly $n$ colors.

Minimal distance of a finite set of different points in space is length of the shortest segment, whose both ends belong to this set and segment has length greater than $0$. a) Prove there exist set of $8$ points on sphere with radius $R$, whose minimal distance is greater than $1,15R$. b) Does there exist set of $8$ points on sphere with radius $R$, whose minimal distance is greater than $1,2R$?

For covering the floor of a rectangular room rectangular tiles of sizes $2 \times 2$ and $4 \times 1$ were used. Show that it's not possible to cover the floor if there is one plate less of one type and one more of the other type.

In the Cartesian plane, each point with integer coordinates is colored either red, green, or blue. It is possible to form right isosceles triangles ($45^\circ - 90^\circ - 45^\circ$) using colored points as vertices. Prove that regardless of how the coloring is done, there always exists a right isosceles triangle such that all its vertices are either the same color or all different colors.

Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ having integer coefficients. On each move, Vasya pays him a ruble, and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation $P(x)=a$. The game continues until Petya is forced to repeat an answer. What minimal amount of rubles must Vasya pay in order to win? [i](Anant Mudgal)[/i] (Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])

A set $S$ of $100$ points, no four in a plane, is given in space. Prove that there are no more than $4 .101^2$ tetrahedra with the vertices in $S$, such that any two of them have at most two vertices in common.

There are some checkers in $n\cdot n$ size chess board.Known that for all numbers $1\le i,j\le n$ if checkwork in the intersection of $i$ th row and $j$ th column is empty,so the number of checkers that are in this row and column is at least $n$.Prove that there are at least $\frac{n^2}{2}$ checkers in chess board.

Let $A$ be a set with $2n$ elements, and let $A_1, A_2...,A_m$ be subsets of $A$e ach one with n elements. Find the greatest possible m, such that it is possible to select these $m$ subsets in such a way that the intersection of any 3 of them has at most one element.

Given a positive integer $k,$ prove that for any integer $n \geq 20k,$ there exist $n - k$ pairwise distinct positive integers whose squares add up to $n(n + 1)(2n + 1)/6.$ [i]The Problem Selection Committee[/i]

Mr. Ganzgenau would like to take his tea mug out of the microwave right at the front. But Mr. Ganzgenau's microwave doesn't really want to be very precise play along. To be precise, the two of them play the following game: Let $n$ be a positive integer. The turntable of the microwave makes one in $n$ seconds full turn. Each time the microwave is switched on, an integer number of seconds turned either clockwise or counterclockwise so that there are n possible positions in which the tea mug can remain. One of these positions is right up front. At the beginning, the microwave turns the tea mug to one of the $n$ possible positions. After that Mr. Ganzgenau enters an integer number of seconds in each move, and the microwave decides either clockwise or counterclockwise this number of spin for seconds. For which $n$ can Mr. Ganzgenau force the tea cup after a finite number of puffs to be able to take it out of the microwave right up front? (Birgit Vera Schmidt) [hide=original wording, in case it doesn't make much sense]Herr Ganzgenau möchte sein Teehäferl ganz genau vorne aus der Mikrowelle herausnehmen. Die Mikrowelle von Herrn Ganzgenau möchte da aber so ganz genau gar nicht mitspielen. Ganz genau gesagt spielen die beiden das folgende Spiel: Sei n eine positive ganze Zahl. In n Sekunden macht der Drehteller der Mikrowelle eine vollständige Umdrehung. Bei jedem Einschalten der Mikrowelle wird eine ganzzahlige Anzahl von Sekunden entweder im oder gegen den Uhrzeigersinn gedreht, sodass es n mögliche Positionen gibt, auf denen das Teehäferl stehen bleiben kann. Eine dieser Positionen ist ganz genau vorne. Zu Beginn dreht die Mikrowelle das Teehäferl auf eine der n möglichen Positionen. Danach gibt Herr Ganzgenau in jedem Zug eine ganzzahlige Anzahl von Sekunden ein, und die Mikrowelle entscheidet, entweder im oder gegen den Uhrzeigersinn diese Anzahl von Sekunden lang zu drehen. Für welche n kann Herr Ganzgenau erzwingen, das Teehäferl nach endlich vielen Zügen ganz genau vorne aus der Mikrowelle nehmen zu können? (Birgit Vera Schmidt) [/hide]

A segment of length $2$ is divided into $n$, $n\ge 2$, subintervals. A square is then constructed on each subinterval. Assume that the sum of the areas of all such squares is greater than $1$. Show that under this assumption one can always choose two subintervals with total length greater than $1$.

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

Let $M$ be a finite set of real numbers with the following property: From three different elements of $M$ can always be chosen two whose sum is located in $M$. How many elements can $M$ have at most?

Let $n \ge 3$ be an integer, and consider a circle with $n + 1$ equally spaced points marked on it. Consider all labellings of these points with the numbers $0, 1, ... , n$ such that each label is used exactly once; two such labellings are considered to be the same if one can be obtained from the other by a rotation of the circle. A labelling is called [i]beautiful[/i] if, for any four labels $a < b < c < d$ with $a + d = b + c$, the chord joining the points labelled $a$ and $d$ does not intersect the chord joining the points labelled $b$ and $c$. Let $M$ be the number of beautiful labelings, and let N be the number of ordered pairs $(x, y)$ of positive integers such that $x + y \le n$ and $\gcd(x, y) = 1$. Prove that $$M = N + 1.$$

A square $ n \times n $, ($ n> 2 $) contains nonzero real numbers. It is known that every number is exactly $ k $ times smaller than the sum of all the numbers in its row or sum of all number in its column. For which real numbers $ k $ is this possible?

Prove that the squares with sides $\frac{1}{1}, \frac{1}{2}, \frac{1}{3},\ldots$ may be put into the square with side $\frac{3}{2} $ in such a way that no two of them have any interior point in common.

The numbers $1, 2,\ldots ,50$ are written on a blackboard. Ana performs the following operations: she chooses any three numbers $a, b$ and $c$ from the board and replaces them with their sum $a + b + c$ and writes the number $(a + b) (b + c) (c + a)$ in the notebook. Ana performs these operations until there are only two numbers left on the board ($24$ operations in total). Then, she calculates the sum of the numbers written down in her notebook. Let $M$ and $m$ be the maximum and minimum possible of the sums obtained by Ana. Find the value of $\frac{M}{m}$.

Let $ n > 1, n \in \mathbb{Z}$ and $ B \equal{}\{1,2,\ldots, 2^n\}.$ A subset $ A$ of $ B$ is called weird if it contains exactly one of the distinct elements $ x,y \in B$ such that the sum of $ x$ and $ y$ is a power of two. How many weird subsets does $ B$ have?

A convex $1994$-gon $M$ is given in the plane. A closed polygonal line consists of $997$ of its diagonals. Every vertex is adjacent to exactly one diagonal. Each diagonal divides $M$ into two sides, and the smaller of the numbers of edges on the two sides of $M$ is defined to be the length of the diagonal. Is it posible to have (a) $991$ diagonals of length $3$ and $6$ of length $2$? (b) $985$ diagonals of length $6, 4$ of length $8$, and $8$ of length $3$?