AUSTRIAN-POLISH 2002

Let $P_{1}P_{2}\dots P_{2n}$ be a convex polygon with an even number of corners. Prove that there exists a diagonal $P_{i}P_{j}$ which is not parallel to any side of the polygon.

Let $ABCD$ be a tetrahedron and let $S$ be its center of gravity. A line through $S$ intersects the surface of $ABCD$ in the points $K$ and $L$ . Prove that $$\frac{1}{3}\leq \frac{KS}{LS}\leq 3$$

For each positive integer $n$ find the largest subset $M(n)$ of real numbers possessing the property: $$n+\sum_{i=1}^{n}x_{i}^{n+1}\geq n \prod_{i=1}^{n}x_{i}+\sum_{i=1}^{n}x_{i} \text{for all} x_{1},x_{2},\cdots,x_{n}\in M(n)$$ When does the inequality become an equality ?

Let $A$ be the set $\{2,7,11,13\}$ . A polynomial $f$ with integer coefficients possesses the following property: for each integer $n$ there exists $p \in A$ such that $p|f(n)$ . Prove that there exists $p \in A$ such that $p|f(n)$ for all integers $n$ .

The diagonals of a convex quadrilateral $ABCD$ intersect in the point $E$ . Let $U$ be the circumcenter of the triangle $ABE$ and $H$ be its orthocenter. Similarly, let $V$ be the circumcenter of the triangle $CDE$ and $K$ be its orthocenter. Prove that $E$ lies on the line $UK$ if and only if it lies on the line $VH$ .

Find all real functions $f$ definited on positive integers and satisying:

(a) $f(x+22)=f(x)$ ,

(b) $f\left(x^{2}y\right)=\left(f(x)\right)^{2}f(y)$ for all positive integers $x$ and $y$ .

Determine the number of real solutions of the system $$\left\{ \begin{aligned}\cos x_{1}&= x_{2} &\cdots \cos x_{n-1}&= x_{n} \cos x_{n}&= x_{1} \end{aligned}\right.$$

A set $P$ of $2002$ persons is given. The family of subsets of $P$ containing exactly $1001$ persons has the property that the number of acquaintance pairs in each such subset is the same. (It is assumed that the acquaintance relation is symmetric). Find the best lower estimation of the acquaintance pairs in the set $P$ .

For all real number $x$ consider the family $F(x)$ of all sequences $(a_{n})_{n\geq 0}$ satisfying the equation $$a_{n+1}=x-\frac{1}{a_{n}} (n\geq 0).$$ A positive integer $p$ is called a minimal period of the family $F(x)$ if

(a) each sequence $\left(a_{n}\right)\in F(x)$ is periodic with the period $p$ ,

(b) for each $0<q<p$ there exists $\left(a_{n}\right)\in F(x)$ such that $q$ is not a period of $\left(a_{n}\right)$ .

Prove or disprove that for each positive integer $P$ there exists a real number $x=x(P)$ such that the family $F(x)$ has the minimal period $p>P$ .