AUSTRIAN-POLISH 1991

Show that there are infinitely many integers $m \ge 2$ such that $m \choose 2$ $= 3$ $n \choose 4$ holds for some integer $n \ge 4$ . Give the general form of all such $m$ .

Find all solutions $(x,y,z)$ to the system $$\begin{cases}(x^2 - 6x + 13)y = 20 (y^2 - 6y + 13)z = 20 (z^2 - 6z + 13)x = 20 \end{cases}$$

Given two distinct points $A_1,A_2$ in the plane, determine all possible positions of a point $A_3$ with the following property: There exists an array of (not necessarily distinct) points $P_1,P_2,...,P_n$ for some $n \ge 3$ such that the segments $P_1P_2,P_2P_3,...,P_nP_1$ have equal lengths and their midpoints are $A_1, A_2, A_3, A_1, A_2, A_3, ...$ in this order.

Let $P(x)$ be a real polynomial with $P(x) \ge 0$ for $0 \le x \le 1$ . Show that there exist polynomials $P_i (x) (i = 0, 1,2)$ with $P_i (x) \ge 0$ for all real x such that $P (x) = P_0 (x) + xP_1 (x)( 1- x)P_2 (x)$ .

If $x,y, z$ are arbitrary positive numbers with $xyz = 1$ , prove the inequality $$x^2+y^2+z^2 + xy+yz + zx \ge 2(\sqrt{x} +\sqrt{y}+ \sqrt{z})$$ .

Suppose that there is a point $P$ inside a convex quadrilateral $ABCD$ such that the triangles $PAB$ , $PBC$ , $PCD$ , $PDA$ have equal areas. Prove that one of the diagonals bisects the area of $ABCD$ .

For a given positive integer $n$ determine the maximum value of the function $f (x) = \frac{x + x^2 +...+ x^{2n-1}}{(1 + x^n)^2}$ over all $x \ge 0$ and find all positive $x$ for which the maximum is attained.

Consider the system of congruences $$\begin{cases} xy \equiv - 1 \, (mod z) yz \equiv 1 (mod x) zx \equiv 1 (mod y)\end {cases}$$ Find the number of triples $(x,y, z)$ of distinct positive integers satisfying this system such that one of the numbers $x,y, z$ equals $19$ .

For a positive integer $n$ denote $A = \{1,2,..., n\}$ . Suppose that $g : A\to A$ is a fixed function with $g(k) \ne k$ and $g(g(k)) = k$ for $k \in A$ . How many functions $f: A \to A$ are there such that $f(k)\ne g(k)$ and $f(f(f(k))= g(k)$ for $k \in A$ ?