MEMO 2007

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 $2007$ .