$ a)$ There are $ 25$ participants in a mathematics contest having four problems. Each problem is considered solved or not solved (that is, partial solutions are not possible). Show that either there are four contestants having solved the same problems (or not having solved any of them), or two contestants, one of which has solved exactly the problems that the other did not solve. $ b)$ There are $ k$ sport clubs for the students of a secondary school. The school has $ 100$ students, and for any selection of three of them, there exists a club having at least one of them, but not all, as a member. What is the least possible value of $ k$?

A competition consists of $25$ sports, each awarding one gold medal to a winner. $25$ athletes participate, each in all $25$ sports. There are also $25$ experts, each of whom must predict the number of gold medals each athlete will win. In each prediction, the medal counts must be non-negative integers summing to $25$. An expert is called competent if they correctly guess the number of gold medals for at least one athlete. What is the maximum number \( k \) such that the experts can make their predictions so that at least \( k \) of them are guaranteed to be competent regardless of the outcome? \\

Let $a, b, c, d$ be integers. Show that the product \[(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)\] is divisible by $12$.

Let us consider the sequence $1,2, 3, \ldots , 2002$. Somebody choses $1002$ numbers from the sequence. Prove that there are two of the chosen numbers which are relatively prime (i.e. do not have any common divisors except $1$).

A positive integer is [i]dapper[/i] if at least one of its multiples begins with $ 2008$. For example, $ 7$ is dapper because $ 200858$ is a multiple of $ 7$ and begins with $ 2008$. Observe that $ 200858 \equal{} 28694\times 7$. Prove that every positive integer is dapper.

An [i]$ n\minus{}m$ society[/i] is a group of $ n$ girls and $ m$ boys. Prove that there exists numbers $ n_0$ and $ m_0$ such that every [i]$ n_0\minus{}m_0$ society[/i] contains a subgroup of five boys and five girls with the following property: either all of the boys know all of the girls or none of the boys knows none of the girls.

A drawer in a darkened room contains $100$ red socks, $80$ green socks, $60$ blue socks and $40$ black socks. A youngster selects socks one at a time from the drawer but is unable to see the color of the socks drawn. What is the smallest number of socks that must be selected to guarantee that the selection contains at least $10$ pairs? (A pair of socks is two socks of the same color. No sock may be counted in more than one pair.) $ \textbf{(A)}\ 21\qquad\textbf{(B)}\ 23\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 50$

$ f(x)$ and $ g(x)$ are two polynomials with nonzero degrees and integer coefficients, such that $ g(x)$ is a divisor of $ f(x)$ and the polynomial $ f(x)\plus{}2009$ has $ 50$ integer roots. Prove that the degree of $ g(x)$ is at least $ 5$.

Let $A$ be a subset of $\mathbb{Z}/n\mathbb{Z}$ with at most $\frac{\ln(n)}{100}$ elements. Define $f(r)=\sum_{s\in A} e^{\dfrac{2 \pi i r s}{n}}$. Show that for some $r \ne 0$ we have $|f(r)| \geq \frac{|A|}{2}$.

All the vertices of a convex pentagon are on lattice points. Prove that the area of the pentagon is at least $\frac{5}{2}$. [i]Bogdan Enescu[/i]

Determine all natural numbers $n$ for which there is a partition of $\{1,2,...,3n\}$ in $n$ pairwise disjoint subsets of the form $\{a,b,c\}$, such that numbers $b-a$ and $c-b$ are different numbers from the set $\{n-1, n, n+1\}$.

There are 2010 boxes labeled $B_1,B_2,\dots,B_{2010},$ and $2010n$ balls have been distributed among them, for some positive integer $n.$ You may redistribute the balls by a sequence of moves, each of which consists of choosing an $i$ and moving [i]exactly[/i] $i$ balls from box $B_i$ into any one other box. For which values of $n$ is it possible to reach the distribution with exactly $n$ balls in each box, regardless of the initial distribution of balls?

Of the thirteen members of the volunteer group, Hannah selects herself, Tom Morris, Jerry Hsu, Thelma Paterson, and Louise Bueller to teach the September classes. When she is done, she decides that it's not necessary to balance the number of female and male teachers with the proportions of girls and boys at the hospital $\textit{every}$ month, and having half the women work while only $2$ of the $7$ men work on some months means that some of the women risk getting burned out. After all, nearly all the members of the volunteer group have other jobs. Hannah comes up with a plan that the committee likes. Beginning in October, the comittee of five volunteer teachers will consist of any five members of the volunteer group, so long as there is at least one woman and at least one man teaching each month. Under this new plan, what is the least number of months that $\textit{must}$ go by (including October when the first set of five teachers is selected, but not September) such that some five-member comittee $\textit{must have}$ taught together twice (all five members are the same during two different months)?

Each point of a circle is colored either red or blue. [b](a)[/b] Prove that there always exists an isosceles triangle inscribed in this circle such that all its vertices are colored the same. [b](b)[/b] Does there always exist an equilateral triangle inscribed in this circle such that all its vertices are colored the same?

Let $A$ be an $m\times n$ matrix with rational entries. Suppose that there are at least $m+n$ distinct prime numbers among the absolute values of the entries of $A.$ Show that the rank of $A$ is at least $2.$

Find the smallest integer $n$ satisfying the following condition: regardless of how one colour the vertices of a regular $n$-gon with either red, yellow or blue, one can always find an isosceles trapezoid whose vertices are of the same colour.

In a group of $m$ girls and $n$ boys, any two persons either know each other or do not know each other. For any two boys and any two girls, there are at least one boy and one girl among them,who do not know each other. Prove that the number of unordered pairs of (boy, girl) who know each other does not exceed $m+\frac{n(n-1)}{2}$.

In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$, two distinct points $A$, $ B\in S$ will be called $ k$-[i]friends[/i] if there is a point $ C\in S$ such that the area of the triangle $ ABC$ is equal to $ k$. A set $ T\subset S$ will be called $ k$-[i]clique[/i] if every two points in $ T$ are $ k$-friends. Find the least positive integer $ k$ for which there exits a $ k$-clique with more than 200 elements. [i]Proposed by Jorge Tipe, Peru[/i]

Let $A$ and $B$ be two subsets of $S = \{1, 2, . . . , 2000\}$ with $|A| \cdot |B| \geq 3999$. For a set $X$ , let $X-X$ denotes the set $\{s-t | s, t \in X, s \not = t\}$. Prove that $(A-A) \cap (B-B)$ is nonempty.

Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible. [i]Carl Schildkraut, USA[/i]

We consider graphs with vertices colored black or white. "Switching" a vertex means: coloring it black if it was formerly white, and coloring it white if it was formerly black. Consider a finite graph with all vertices colored white. Now, we can do the following operation: Switch a vertex and simultaneously switch all of its neighbours (i. e. all vertices connected to this vertex by an edge). Can we, just by performing this operation several times, obtain a graph with all vertices colored black? [It is assumed that our graph has no loops (a [i]loop[/i] means an edge connecting one vertex with itself) and no multiple edges (a [i]multiple edge[/i] means a pair of vertices connected by more than one edge).]

Prove that among $81$ natural numbers whose prime divisors are in the set $\{2, 3, 5\}$ there exist four numbers whose product is the fourth power of an integer.

Let $k$ be an odd number that is greater than or equal to $3$. Prove that there exists a $k^{th}$-degree integer-valued polynomial with non-integer-coefficients that has the following properties: (1) $f(0)=0$ and $f(1)=1$; and. (2) There exist infinitely many positive integers $n$ so that if the following equation: \[ n= f(x_1)+\cdots+f(x_s), \] has integer solutions $x_1, x_2, \dots, x_s$, then $s \geq 2^k-1$.

Ten girls, numbered from 1 to 10, sit at a round table, in a random order. Each girl then receives a new number, namely the sum of her own number and those of her two neighbours. Prove that some girl receives a new number greater than 17.

Let $u_{jk}$ be a real number for each $j=1,2,3$ and each $k=1,2$ and let $N$ be an integer such that \[\max_{1\le k \le 2} \sum_{j=1}^3 |u_{jk}| \leq N\] Let $M$ and $l$ be positive integers such that $l^2 <(M+1)^3$. Prove that there exist integers $\xi_1,\xi_2,\xi_3$ not all zero, such that \[\max_{1\le j \le 3}\xi_j \le M\ \ \ \ \text{and} \ \ \ \left|\sum_{j=1}^3 u_{jk}\xi_k\right| \le \frac{MN}{l} \ \ \ \ \text{for k=1,2}\]