AUSTRIAN-POLISH 2005

For a convex $n$ -gon $P_n$ , we say that a convex quadrangle $Q$ is a diagonal-quadrangle of $P_n$ , if its vertices are vertices of $P_n$ and its sides are diagonals of $P_n$ . Let $d_n$ be the number of diagonal-quadrangles of a convex $n$ -gon. Determine $d_n$ for all $n\geq 8$ .

Determine all polynomials $P$ with integer coefficients satisfying $$P(P(P(P(P(x)))))=x^{28}\cdot P(P(x)) \forall x\in\mathbb{R}$$

Let $a_0, a_1, a_2, ... , a_n$ be real numbers, which fulfill the following two conditions: a) $0 = a_0 \leq a_1 \leq a_2 \leq ... \leq a_n$ . b) For all $0 \leq i < j \leq n$ holds: $a_j - a_i \leq j-i$ .

Prove that $$\left( \sum_{i=0}^n a_i \right)^2 \geq \sum_{i=0}^n a_i^3.$$

Determine the smallest natural number $a\geq 2$ for which there exists a prime number $p$ and a natural number $b\geq 2$ such that $$\frac{a^p - a}{p}=b^2.$$

Given is a convex quadrilateral $ABCD$ with $AB=CD$ . Draw the triangles $ABE$ and $CDF$ outside $ABCD$ so that $\angle{ABE} = \angle{DCF}$ and $\angle{BAE}=\angle{FDC}$ . Prove that the midpoints of $\overline{AD}$ , $\overline{BC}$ and $\overline{EF}$ are collinear.

Determine all monotone functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ , so that for all $x, y \in \mathbb{Z}$ $$f(x^{2005} + y^{2005}) = (f(x))^{2005} + (f(y))^{2005}$$

For each natural number $n\geq 2$ , solve the following system of equations in the integers $x_1, x_2, ..., x_n$ : $$(n^2-n)x_i+\left(\prod_{j\neq i}x_j\right)S=n^3-n^2, \forall 1\le i\le n$$ where $$S=x_1^2+x_2^2+\dots+x_n^2.$$

Given the sets $R_{mn} = \{ (x,y) \mid x=0,1,\dots,m; y=0,1,\dots,n \}$ , consider functions $f:R_{mn}\to \{-1,0,1\}$ with the following property: for each quadruple of points $A_1,A_2,A_3,A_4\in R_{mn}$ which form a square with side length $0<s<3$ , we have $$f(A_1)+f(A_2)+f(A_3)+f(A_4)=0.$$ For each pair $(m,n)$ of positive integers, determine $F(m,n)$ , the number of such functions $f$ on $R_{mn}$ .

Consider the equation $x^3 + y^3 + z^3 = 2$ . a) Prove that it has infinitely many integer solutions $x,y,z$ . b) Determine all integer solutions $x, y, z$ with $|x|, |y|, |z| \leq 28$ .

Determine all pairs $(k,n)$ of non-negative integers such that the following inequality holds $\forall x,y>0$ : $$1+ \frac{y^n}{x^k} \geq \frac{(1+y)^n}{(1+x)^k}.$$