In a convex quadrilateral. eight segments are drawn, each of them connects a vertex with the midpoint of some opposite side. Seven of these segments have the same length $ a$. Prove that the eight one is also of length $ a$.

Pete chose a positive integer $ n$. For each (unordered) pair of its decimal digits, he wrote their difference on the blackboard. After that, he erased some of these differences, and the remaining ones are $ 2,0,0,7$. Find the smallest number $ n$ for which this situation is possible.

Determine if there exist prime numbers $ p_{1},p_{2},...,p_{2007}$ such that $ p_{2}|p_{1}^{2}-1,p_{3}|p_{2}^{2}-1,...,p_{1}|p_{2007}^{2}-1$.

On the chessboard, $ 32$ black pawns and $ 32$ white pawns are arranged. In every move, a pawn can capture another pawn of the opposite color, moving diagonally to an adjacent square where the captured one stands. White pawns move only in upper-left or upper-right directions, while black ones can move in down-left or in down-right directions only; the captured pawn is removed from the board. A pawn cannot move without capturing an opposite pawn. Find the least possible number of pawns which can stay on the chessboard.

There are $ 11$ coins, which are indistinguishable by sight. Nevertheless, among them there are $ 10$ geniune coins ( of weight $ 20$ g each) and one counterfeit (of weight $ 21$ g). You have a two-pan scale which is blanced when the weight in the left-hand pan is twice as much as the weight in the right-hand one. Using this scale only, find the false coin by three weighings.

A number $ B$ is obtained from a positive integer number $ A$ by permuting its decimal digits. The number $ A-B=11...1$ ($ n$ of $ 1's$). Find the smallest possible positive value of $ n$.

Given an isosceles triangle $ ABC$ with $ AB = BC$. A point $ M$ is chosen inside $ ABC$ such that $ \angle AMC = 2\angle ABC$ . A point $ K$ lies on segment $ AM$ such that $ \angle BKM =\angle ABC$. Prove that $ BK = KM+MC$.

In the class, there are $ 15$ boys and $ 15$ girls. On March $ 8$, some boys made phone calls to some girls to congratulate them on the holiday ( each boy made no more than one call to each girl). It appears that there is a unique way to split the class in $ 15$ pairs (each consisting of a boy and a girl) such that in every pair the boy has phoned the girl. Find the maximal possible number of calls.

Pete chooses $ 1004$ monic quadratic polynomial $ f_{1},\cdots,f_{1004}$, such that each integer from $ 0$ to $ 2007$ is a root of at least one of them. Vasya considers all equations of the form $ f_{i}=f_{j}(i\not =j)$ and computes their roots; for each such root , Pete has to pay to Vasya $ 1$ ruble . Find the least possible value of Vasya's income.

$ 25$ boys and some girls came to the party and discovered an interesting property of their company. Take an arbitrary group of $ \geq 10$ boys and all the girls which are acquainted with at least one of them. Then in the joint group, the number of girls is by one greater than the number of boys. Prove that there exists a girl who is acquainted with at least $ 16$ boys.

Two triangles have equal longest sides and equal smallest angles. A new triangle is constructed, such that its sides are the sum of the longest sides, the sum of the shortest sides, and the sum of the middle sides of the initial triangles. Prove that the area of the new triangle is at least twice as much as the sum of the areas of the initial ones.

Given a triangle. A variable poin $ D$ is chosen on side $ BC$. Points $ K$ and $ L$ are the incenters of triangles $ ABD$ and $ ACD$, respectively. Prove that the second intersection point of the circumcircles of triangles $ BKD$ and $ CLD$ moves along on a fixed circle (while $ D$ moves along segment $ BC$).

An infinite increasing arithmetical progression consists of positive integers and contains a perfect cube. Prove that this progression also contains a term which is a perfect cube but not a perfect square.

A set contains $ 372$ integers from $ 1,2,...,1200$ . For every element $ a\in S$, the numbers $ a+4,a+5,a+9$ don't belong to $ S$. Prove that $ 600\in S$.

Prove that the inequality $ (x^{k}-y^{k})^{n}<(x^{n}-y^{n})^{k}$ holds forall reals $ x>y>0$ and positive integers $ n>k$.

Given a triangle $ ABC$. A circle passes through vertices $ B$ and $ C$ and intersects sides $ AB$ and $ AC$ at points $ D$ and $ E$, respectively. Segments $ CD$ and $ BE$ intersect at point $ O$. Denote the incenters of triangles $ ADE$ and $ ODE$ by $ M$ and $ N$, respectiely. Prove that the midpoint of the smaller arc $ DE$ lies on line $ MN$.

A point $ D$ is chosen on side $ BC$ of a triangle $ ABC$ such that the inradii of triangles $ ABD$ and $ ACD$ are equal. Consider in these triangles the excircles touching sides $ BD$ and $ CD$, respectively. Prove that their radii are also equal.

Given an integer $ n>6$. Consider those integers $ k\in (n(n-1),n^{2})$ which are coprime with $ n$. Prove that the greatest common divisor of the considered numbers is $ 1$.

Two quadratic polynomials $ f_{1},f_{2}$ satisfy $ f_{1}'(x)f_{2}'(x)\geq |f_{1}(x)|+|f_{2}(x)|\forall x\in\mathbb{R}$ . Prove that $ f_{1}\cdot f_{2}= g^{2}$ for some $ g\in\mathbb{R}[x]$.

Find all positive integers $ n$ for which there exist integers $ a,b,c$ such that $ a+b+c=0$ and the number $ a^{n}+b^{n}+c^{n}$ is prime.

Prove that $ \prod_{i=1}^{n}(1+x_{1}+x_{2}+...+x_{i})\geq\sqrt{(n+1)^{n+1}x_{1}x_{2}...x_{n}}\forall x_{1},...,x_{n}> 0$.