BALKAN 1995

For all real numbers $x,y$ define $x\star y = \frac{ x+y}{ 1+xy}$ . Evaluate the expression $$( \cdots (((2 \star 3) \star 4) \star 5) \star \cdots ) \star 1995.$$

Macedonia

The circles $\mathcal C_1(O_1, r_1)$ and $\mathcal C_2(O_2, r_2)$ , $r_2 > r_1$ , intersect at $A$ and $B$ such that $\angle O_1AO_2 = 90^\circ$ . The line $O_1O_2$ meets $\mathcal C_1$ at $C$ and $D$ , and $\mathcal C_2$ at $E$ and $F$ (in the order $C$ , $E$ , $D$ , $F$ ). The line $BE$ meets $\mathcal C_1$ at $K$ and $AC$ at $M$ , and the line $BD$ meets $\mathcal C_2$ at $L$ and $AF$ at $N$ . Prove that $$\frac{ r_2}{r_1} = \frac{KE}{KM} \cdot \frac{LN}{LD} .$$ Greece

Let $a$ and $b$ be natural numbers with $a > b$ and having the same parity. Prove that the solutions of the equation $$x^2 - (a^2 - a + 1)(x - b^2 - 1) - (b^2 + 1)^2 = 0$$ are natural numbers, none of which is a perfect square.

Albania

Let $n$ be a positive integer and $\mathcal S$ be the set of points $(x, y)$ with $x, y \in \{1, 2, \ldots , n\}$ . Let $\mathcal T$ be the set of all squares with vertices in the set $\mathcal S$ . We denote by $a_k$ ( $k \geq 0$ ) the number of (unordered) pairs of points for which there are exactly $k$ squares in $\mathcal T$ having these two points as vertices. Prove that $a_0 = a_2 + 2a_3$ .

Yugoslavia