ZA-SAMO 2013

Determine all pairs of polynomials $f$ and $g$ with real coefficients such that $$x^2 \cdot g(x) = f(g(x)).$$

Some coins are placed on a $20 \times 13$ board. Two coins are called neighbours if they are in the same row or column and no other coins lie between them. What is the largest number of coins that can be placed on the board if no coin is allowed to have more than two neighbours?

Let $ABC$ be an acute-angled triangle with $AC \neq BC$ , and let $O$ be the circumcentre and $F$ the foot of the altitude through $C$ . Furthermore, let $X$ and $Y$ be the feet of the perpendiculars dropped from $A$ and $B$ respectively to (the extension of) $CO$ . The line $FO$ intersects the circumcircle of $FXY$ a second time at $P$ . Prove that $OP<OF$ .