AUSTRIAN-POLISH 1984

Prove that if the feet of the altitudes of a tetrahedron are the incenters of the corresponding faces, then the tetrahedron is regular.

Let $A$ be the set of four-digit natural numbers having exactly two distinct digits, none of which is zero. Interchanging the two digits of $n\in A$ yields a number $f (n) \in A$ (for instance, $f (3111) = 1333$ ). Find those $n \in A$ with $n > f (n)$ for which $gcd(n, f (n))$ is the largest possible.

Show that for $n>1$ and any positive real numbers $k,x_{1},x_{2},...,x_{n}$ then $$\frac{f(x_{1}-x_{2})}{x_{1}+x_{2}}+\frac{f(x_{2}-x_{3})}{x_{2}+x_{3}}+...+\frac{f(x_{n}-x_{1})}{x_{n}+x_{1}}\geq \frac{n^2}{2(x_{1}+x_{2}+...+x_{n})}$$ Where $f(x)=k^x$ . When does equality hold.

A regular heptagon $A_1A_2... A_7$ is inscribed in circle $C$ . Point $P$ is taken on the shorter arc $A_7A_1$ . Prove that $PA_1+PA_3+PA_5+PA_7 = PA_2+PA_4+PA_6$ .

Given $n > 2$ nonnegative distinct integers $a_1,...,a_n$ , find all nonnegative integers $y$ and $x_1,...,x_n$ satisfying $gcd(x_1,...,x_n) = 1$ and $$\begin{cases} a_1x_1 + a_2x_2 +...+ a_nx_n = yx_1 a_2x_1 + a_3x_2 +...+ a_1x_n = yx_2 ... a_nx_1 + a_1x_2 +...+ a_{n-1}x_n = yx_n \end{cases}$$

In a dancing hall, there are $n$ girls standing in one row and $n$ boys in the other row across them (so that all $2n$ dancers form a $2 \times n$ board). Each dancer gives her / his left hand to a neighboring person standing to the left, across, or diagonally to the left. The analogous rule applies for right hands. No dancer gives both hands to the same person. In how many ways can the dancers do this?

A $m\times n$ matrix $(a_{ij})$ of real numbers satisfies $|a_{ij}| <1$ and $\sum_{i=1}^m a_{ij}= 0$ for all $j$ . Show that one can permute the entries in each column in such a way that the obtained matrix $(b_{ij})$ satisfies $\sum_{j=1}^n b_{ij} < 2$ for all $i$ .

The functions $f_0,f_1 : (1,\infty) \to (1,\infty)$ are given by $f_0(x) = 2x$ and $f_1(x) =\frac{x}{x-1}$ . Show that for any real numbers $a, b$ with $1 \le a < b$ there exist a positive integer $k$ and indices $i_1,i_2,...,i_k \in \{0,1\}$ such that $a <f_{i_k}(f_{i_{k-1}}(...(f_{i_j}(2))...))< b$ .

Find all functions $f: Q \to R$ satisfying $f (x + y) = f (x)f (y) - f(xy) + 1$ for all $x,y \in Q$