ÖMO 1989

Consider the set $S_n$ of all the $2^n$ numbers of the type $2\pm \sqrt{2 \pm \sqrt {2 \pm ...}},$ where number $2$ appears $n+1$ times. $(a)$ Show that all members of $S_n$ are real. $(b)$ Find the product $P_n$ of the elements of $S_n$ .

Find all triples $(a,b,c)$ of integers with $abc=1989$ and $a+b-c=89$ .

Show that it is possible to situate eight parallel planes at equal distances such that each plane contains precisely one vertex of a given cube. How many such configurations of planes are there?

We are given a circle $k$ and nonparallel tangents $t_1,t_2$ at points $P_1,P_2$ on $k$ , respectively. Lines $t_1$ and $t_2$ meet at $A_0$ . For a point $A_3$ on the smaller arc $P_1 P_2,$ the tangent $t_3$ to $k$ at $P_3$ meets $t_1$ at $A_1$ and $t_2$ at $A_2$ . How must $P_3$ be chosen so that the triangle $A_0 A_1 A_2$ has maximum area?

Find all real solutions of the system:

$x^2+2yz=x,$ $y^2+2zx=y,$ $z^2+2xy=z.$

Determine all functions $f: \mathbb{N}_0 \rightarrow \mathbb{N}_0$ such that $f(f(n))+f(n)=2n+6$ for all $n \in \mathbb{N}_0$ .