ISL 1980

The function $f$ is defined on the set $\mathbb{Q}$ of all rational numbers and has values in $\mathbb{Q}$ . It satisfies the conditions $f(1) = 2$ and $f(xy) = f(x)f(y) - f(x+y) + 1$ for all $x,y \in \mathbb{Q}$ . Determine $f$ .

Three points $A,B,C$ are such that $B \in ]AC[$ . On the side of $AC$ we draw the three semicircles with diameters $[AB], [BC]$ and $[AC]$ . The common interior tangent at $B$ to the first two semi-circles meets the third circle in $E$ . Let $U$ and $V$ be the points of contact of the common exterior tangent to the first two semi-circles. Denote the area of the triangle $ABC$ as $S(ABC)$ . Evaluate the ratio $R=\frac{S(EUV)}{S(EAC)}$ as a function of $r_1 = \frac{AB}{2}$ and $r_2 = \frac{BC}{2}$ .

Two circles $C_{1}$ and $C_{2}$ are (externally or internally) tangent at a point $P$ . The straight line $D$ is tangent at $A$ to one of the circles and cuts the other circle at the points $B$ and $C$ . Prove that the straight line $PA$ is an interior or exterior bisector of the angle $\angle BPC$ .

Ten gamblers started playing with the same amount of money. Each turn they cast (threw) five dice. At each stage the gambler who had thrown paid to each of his 9 opponents $\frac{1}{n}$ times the amount which that opponent owned at that moment. They threw and paid one after the other. At the 10th round (i.e. when each gambler has cast the five dice once), the dice showed a total of 12, and after payment it turned out that every player had exactly the same sum as he had at the beginning. Is it possible to determine the total shown by the dice at the nine former rounds ?