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Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that

- each contestant solved at most six problems, and
- for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy.

Show that there is a problem that was solved by at least three girls and at least three boys.

Consider that we have a $5\times 5$ board. In each square, there is a flea. At the same time all the fleas jump and move to a square which is adjacent to their current square. Is it possible that after they jump that all the squares have one flea on them?

If every pair of people are acquaintances or are strangers. Show that in any group of 6 people that there are 3 who all know each other or 3 who are all strangers.

At a party, everyone shakes hands with some other people at the party (possibly 0 people). Show that there are 2 people at the party who had the same number of handshakes.

In a $2\times 2$ square, show that for any 5 points, there is a pair which are at most $\sqrt{2}$ apart.

If $ABCDE$ is a convex pentagon with all its vertices on lattice points, prove that there exists a lattice point strictly in the interior of $ABCDE$.

If $n+1$ numbers are selected from the set $\{1,2,\dots 2n\}$ show that there are two of them such that one divides the other.

Let $n$ be odd and $\pi$ a permutation of $\{1,2,\dots, n\}$. Prove that

$$(1+\pi(1))(2+\pi(2))\dots(n+\pi(n))$$

is even.

Given two disks, one smaller than the other. Each disk is divided into 200 congruent sectors. In the larger disk 100 sectors are chosen arbitrarily and painted red; the other 100 sectors are painted blue. In the smaller disk each sector is painted either red or blue with no stipulation on the number of red and blue sectors. The smaller disk is placed on the larger disk so that the centers and sectors coincide. Show that it is possible to align the two disks so that the number of sectors of the smaller disk whose color matches the corresponding sector of the larger disk is at least 100.

Show that for any sequence of $mn+1$ distinct real numbers. Show there is either an increasing sequence of length $m$ or a decreasing sequence of length $n$.

A $10\times 10$ table is filled with positive integers such that any 2 adjacent squares differ by at most 5. Show that there exists two identical integers.

If every pair of people are friends, acquaintances, or strangers. Show that in any group of 17 people that there are 3 who are all friends, or 3 who are all acquaintances or 3 who are all strangers.

There are 2009 clubs, each with 45 members, any two of which have exactly one person in common. Show that there is one person who is a member of all the clubs.

Find the smallest positive integer $n$ such that if $n$ squares of a $1000\times 1000$ chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board.

In a $10\times 10$ table are written natural numbers not exceeding 10. Any two numbers that appear in adjacent or diagonally adjacent spaces are relatively prime. Prove that some number appears in the table at least 17 times.

A chess master who has 11 weeks to prepare for a tournament decides to play at least one game every day but, in order to not tire himself out, he decides to play at most 12 games in a given calendar week. Show that there exists a succession of consecutive days during which the chess master will have played exactly 21 games.

For $i=1,2,\dots 7$, $a_i$ and $b_i$ are nonnegative numbers such that $a_i+b_i \leq 2$. Prove that there exist distinct indices $i,j$ such that $|a_i-a_j|+|b_i-b_j|\leq 1$.

Each cell of a $50\times 50$ square is colored in one of four colors. Show that there exists a square which has cells of the same color as it directly above, directly below, directly to the left, and directly to the right of it (though not necessarily adjacent to it).

Show that in every convex finite three dimensional polyhedron there exist two faces with the same number of sides.

Let $a,b,c$ be positive integers. Define the sequence $x_0=c$ and $x_{n+1}=ax_n+b$ for $n\geq 0$. Prove that the set of terms of the sequence which are composite is infinite.

Let $x_1, x_2,\dots x_{19}$ be positive integers each of which is less than or equal to 93. Let $y_1, y_2\dots y_{93}$ be positive integers each of which is less than or equal to 19. Prove that there exists a (nonempty) sum of some $x_i$'s equal to a sum of some $y_j$'s

A circle is divided into $432$ congruent arcs by $432$ points. The points are colored in four colors such that some $108$ points are colored Red, some $108$ points are colored Green, some $108$ points are colored Blue, and the remaining $108$ points are colored Yellow. Prove that one can choose three points of each color in such a way that the four triangles formed by the chosen points of the same color are congruent.

For a prime $p$ and a given integer $n$ let $\nu_p(n)$ denote the exponent of $p$ in the prime factorisation of $n!$. Given $d \in \mathbb{N}$ and $\{p_1,p_2,\ldots,p_k\}$ a set of $k$ primes, show that there are infinitely many positive integers $n$ such that $d\mid \nu_{p_i}(n)$ for all $1 \leq i \leq k$.