AUSTRIAN-POLISH 1986

A non-right triangle $A_1A_2A_3$ is given. Circles $C_1$ and $C_2$ are tangent at $A_3, C_2$ and $C_3$ are tangent at $A_1$ , and $C_3$ and $C_1$ are tangent at $A_2$ . Points $O_1,O_2,O_3$ are the centers of $C_1, C_2, C_3$ , respectively. Supposing that the triangles $A_1A_2A_3$ and $O_1O_2O_3$ are similar, determine their angles.

The monic polynomial $P(x) = x^n + a_{n-1}x^{n-1} +...+ a_0$ of degree $n > 1$ has $n$ distinct negative roots. Prove that $a_1P(1) > 2n^2a_o$

Each point in space is colored either blue or red. Show that there exists a unit square having exactly $0, 1$ or $4$ blue vertices.

Find all triples (m,n,N) of positive integers numbers m,n and N such that $m^N-n^N=2^{100}$ with N>1

Find all real solutions of the system of equations $$\begin{cases} x^2 + y^2 + u^2 + v^2 = 4 xu + yv + xv + yu = 0 xyu + yuv + uvx + vxy = - 2 xyuv = -1 \end{cases}$$

Let $M$ be the set of all tetrahedra whose inscribed and circumscribed spheres are concentric. If the radii of these spheres are denoted by $r$ and $R$ respectively, find the possible values of $R/r$ over all tetrahedra from $M$ .

Let $k$ and $n$ be integers with $0 < k < n^2/4$ such that k has no prime divisor greater than $n$ . Prove that $k$ divides $n!$ .

Pairwise distinct real numbers are arranged into an $m \times n$ rectangular array. In each row the entries are arranged increasingly from left to right. Each column is then rearranged in decreasing order from top to bottom. Prove that in the reorganized array, the rows remain arranged increasingly.

Find all continuous monotonic functions $f : R \to R$ that satisfy $f (1) = 1$ and $f(f (x)) = f (x)^2$ for all $x \in R$ .