Let $ a,b,c,d$ be positive real numbers with $ a+b+c+d = 4$. Prove that \[ a^{2}bc+b^{2}cd+c^{2}da+d^{2}ab\leq 4.\]

A set of balls contains $ n$ balls which are labeled with numbers $ 1,2,3,\ldots,n.$ We are given $ k > 1$ such sets. We want to colour the balls with two colours, black and white in such a way, that (a) the balls labeled with the same number are of the same colour, (b) any subset of $ k+1$ balls with (not necessarily different) labels $ a_{1},a_{2},\ldots,a_{k+1}$ satisfying the condition $ a_{1}+a_{2}+\ldots+a_{k}= a_{k+1}$, contains at least one ball of each colour. Find, depending on $ k$ the greatest possible number $ n$ which admits such a colouring.

Let $ k$ be a circle and $ k_{1},k_{2},k_{3},k_{4}$ four smaller circles with their centres $ O_{1},O_{2},O_{3},O_{4}$ respectively, on $ k$. For $ i = 1,2,3,4$ and $ k_{5}= k_{1}$ the circles $ k_{i}$ and $ k_{i+1}$ meet at $ A_{i}$ and $ B_{i}$ such that $ A_{i}$ lies on $ k$. The points $ O_{1},A_{1},O_{2},A_{2},O_{3},A_{3},O_{4},A_{4}$ lie in that order on $ k$ and are pairwise different. Prove that $ B_{1}B_{2}B_{3}B_{4}$ is a rectangle.

Determine all pairs $ (x,y)$ of positive integers satisfying the equation \[ x!+y!=x^{y}.\]

Let $ a,b,c,d$ be real numbers which satisfy $ \frac{1}{2}\leq a,b,c,d\leq 2$ and $ abcd=1$. Find the maximum value of \[ \left(a+\frac{1}{b}\right)\left(b+\frac{1}{c}\right)\left(c+\frac{1}{d}\right)\left(d+\frac{1}{a}\right).\]

For a set $ P$ of five points in the plane, no three of them being collinear, let $ s(P)$ be the numbers of acute triangles formed by vertices in $ P$. Find the maximum value of $ s(P)$ over all such sets $ P$.

A tetrahedron is called a MEMO-tetrahedron if all six sidelengths are different positive integers where one of them is $ 2$ and one of them is $ 3$. Let $ l(T)$ be the sum of the sidelengths of the tetrahedron $ T$. (a) Find all positive integers $ n$ so that there exists a MEMO-Tetrahedron $ T$ with $ l(T)=n$. (b) How many pairwise non-congruent MEMO-tetrahedrons $ T$ satisfying $ l(T)=2007$ exist? Two tetrahedrons are said to be non-congruent if one cannot be obtained from the other by a composition of reflections in planes, translations and rotations. (It is not neccessary to prove that the tetrahedrons are not degenerate, i.e. that they have a positive volume).

Find all positive integers $ k$ with the following property: There exists an integer $ a$ so that $ (a+k)^{3}-a^{3}$ is a multiple of $ 2007$.