AUSTRIAN-POLISH 2000

Find all polynomials $P(x)$ with real coefficients having the following property: There exists a positive integer n such that the equality $$\sum_{k=1}^{2n+1}(-1)^k \left[\frac{k}{2}\right] P(x + k)=0$$ holds for infinitely many real numbers $x$ .

In a unit cube, $CG$ is the edge perpendicular to the face $ABCD$ . Let $O_1$ be the incircle of square $ABCD$ and $O_2$ be the circumcircle of triangle $BDG$ . Determine min $\{XY|X\in O_1,Y\in O_2\}$ .

For each integer $n \ge 3$ solve in real numbers the system of equations: $$\begin{cases} x_1^3 = x_2 + x_3 + 1 ...x_{n-1}^3 = x_n+ x_1 + 1x_{n}^3 = x_1+ x_2 + 1 \end{cases}$$

For which integers $n \ge 5$ is it possible to color the vertices of a regular $n$ -gon using at most $6$ colors in such a way that any $5$ consecutive vertices have different colors?

Consider the solid $Q$ obtained by attaching unit cubes $Q_1...Q_6$ at the six faces of a unit cube $Q$ . Prove or disprove that the space can be filled up with such solids so that no two of them have a common interior point.

In the plane are given $27$ points, no three of which are collinear. Four of this points are vertices of a unit square, while the others lie inside the square. Prove that there are three points in this set forming a triangle with area not exceeding $1/48$ .

If three nonnegative reals $a$ , $b$ , $c$ satisfy $a+b+c=1$ , prove that

$2 \leq \left(1-a^{2}\right)^{2}+\left(1-b^{2}\right)^{2}+\left(1-c^{2}\right)^{2}\leq \left(1+a\right)\left(1+b\right)\left(1+c\right)$ .

The plan of the castle in Baranow Sandomierski can be presented as the graph with $16$ vertices on the picture. A night guard plans a closed round along the edges of this graph. (a) How many rounds passing through each vertex exactly once are there? The directions are irrelevant. (b) How many non-selfintersecting rounds (taking directions into account) containing each edge of the graph exactly once are there?