OBM 1993

The sequence $(a_n)_{n \in\mathbb{N}}$ is defined by $a_1 = 8, a_2 = 18, a_{n+2} = a_{n+1}a_{n}$ . Find all terms which are perfect squares.

A real number with absolute value less than $1$ is written in each cell of an $n\times n$ array, so that the sum of the numbers in each $2\times 2$ square is zero. Show that for odd $n$ the sum of all the numbers is less than $n$ .

Given a circle and its center $O$ , a point $A$ inside the circle and a distance $h$ , construct a triangle $BAC$ with $\angle BAC = 90^\circ$ , $B$ and $C$ on the circle and the altitude from $A$ length $h$ .

$ABCD$ is a convex quadrilateral with $$\angle BAC = 30^\circ$$$$\angle CAD = 20^\circ$$$$\angle ABD = 50^\circ$$$$\angle DBC = 30^\circ$$ If the diagonals intersect at $P$ , show that $PC = PD$ .

Find at least one function $f: \mathbb R \rightarrow \mathbb R$ such that $f(0)=0$ and $f(2x+1) = 3f(x) + 5$ for any real $x$ .