SE-SKOLORNAS 1989

Let $n$ be a positive integer. Prove that the numbers $n^2(n^2 + 2)^2$ and $n^4(n^2 + 2)^2$ are written in base $n^2 +1$ with the same digits but in opposite order.

Find all continuous functions $f$ such that $f(x)+ f(x^2) = 0$ for all real numbers $x$ .

Find all positive integers $n$ such that $n^3 - 18n^2 + 115n - 391$ is the cube of a positive intege

Let $ABCD$ be a regular tetrahedron. Find the positions of point $P$ on the edge $BD$ such that the edge $CD$ is tangent to the sphere with diameter $AP$ .

Assume $x_1,x_2,..,x_5$ are positive numbers such that $x_1 < x_2$ and $x_3,x_4, x_5$ are all greater than $x_2$ . Prove that if $a > 0$ , then $$\frac{1}{(x_1 +x_3)^a}+ \frac{1}{(x_2 +x_4)^a}+ \frac{1}{(x_2 +x_5)^a} <\frac{1}{(x_1 +x_2)^a}+ \frac{1}{(x_2 +x_3)^a}+ \frac{1}{(x_4 +x_5)^a}$$

On a circle $4n$ points are chosen ( $n \ge 1$ ). The points are alternately colored yellow and blue. The yellow points are divided into $n$ pairs and the points in each pair are connected with a yellow line segment. In the same manner the blue points are divided into $n$ pairs and the points in each pair are connected with a blue segment. Assume that no three of the segments pass through a single point. Show that there are at least $n$ intersection points of blue and yellow segments.