CZ-SK-MO 1993

Find all natural numbers $n$ for which $7^n -1$ is divisible by $6^n -1$

In fields of a $19 \times 19$ table are written integers so that any two lying on neighboring fields differ at most by $2$ (two fields are neighboring if they share a side). Find the greatest possible number of mutually different integers in such a table.

Let $AKL$ be a triangle such that $\angle ALK > 90^o +\angle LAK$ . Construct an isosceles trapezoid $ABCD$ with $AB \parallel CD$ such that $K$ lies on the side $BC, L$ on the diagonal $AC$ and the lines $AK$ and $BL$ intersect at the circumcenter of the trapezoid.

The sequence ( $a_n$ ) of natural numbers is defined by $a_1 = 2$ and $a_{n+1}$ equals the sum of tenth powers of the decimal digits of $a_n$ for all $n \ge 1$ . Are there numbers which appear twice in the sequence ( $a_n$ )?

Find all functions $f : Z \to Z$ such that $f(-1) = f(1)$ and $f(x)+ f(y) = f(x+2xy)+ f(y-2xy)$ for all $x,y \in Z$

Show that there exists a tetrahedron which can be partitioned into eight congruent tetrahedra, each of which is similar to the original one.