SE-SKOLORNAS 1984

Let $A$ and $B$ be two points inside a circle $C$ . Show that there exists a circle that contains $A$ and $B$ and lies completely inside $C$ .

The squares in a $3\times 7$ grid are colored either blue or yellow. Consider all $m\times n$ rectangles in this grid, where $m \in \{2,3\}$ , $n \in \{2,...,7\}$ . Prove that at least one of these rectangles has all four corner squares the same color.

Prove that if $a,b$ are positive numbers, then $$\left( \frac{a+1}{b+1}\right)^{b+1} \ge \left( \frac{a}{b}\right)^{b}$$

Find all positive integers $p$ and $q$ such that all the roots of the polynomial $(x^2 - px+q)(x^2 -qx+ p)$ are positive integers.

Solve in natural numbers $a,b,c$ the system $$\left\{ \begin{array}{l}a^3 -b^3 -c^3 = 3abc a^2 = 2(a+b+c) \end{array} \right.$$

Assume $a_1,a_2,...,a_{14}$ are positive integers such that $\sum_{i=1}^{14}3^{a_i} = 6558$ . Prove that the numbers $a_1,a_2,...,a_{14}$ consist of the numbers $1,...,7$ , each taken twice.