IT-CESENATICO 1986

Two circles $\alpha$ and $\beta$ intersect at points $P$ and $Q$ . The lines connecting a point $R$ on $\beta$ with $P$ and $Q$ intersect $\alpha$ again at $S$ and $T$ respectively. Prove that $ST$ is parallel to the line tangent to $\beta$ at $R$ .

Determine the general term of the sequence ( $a_n$ ) given by $a_0 =\alpha > 0$ and $a_{n+1} =\frac{a_n}{1+a_n}$ .

Two numbers are randomly selected from interval $I = [0, 1]$ . Given $\alpha \in I$ , what is the probability that the smaller of the two numbers does not exceed $\alpha$ ?

Is the answer $(100 \alpha)$ %, it just seems too easy. :|

Prove that a circle centered at point $(\sqrt{2},\sqrt{3})$ in the cartesian plane passes through at most one point with integer coordinates.

I tried to prove that any circle with center at $(0,0)$ has at most one point with coordinates $(a-\sqrt{2},b-\sqrt{3})$ ; $a,b \in \mathbb{Z}$ . So that when we translate the center to $(\sqrt{2},\sqrt{3})$ we have what we wanted to show.

Given an acute triangle $T$ with sides $a,b,c$ , find the tetrahedra with base $T$ whose all faces are acute triangles of the same area.

Show that for any positive integer $n$ there exists an integer $m > 1$ such that $(\sqrt2-1)^n=\sqrt{m}-\sqrt{m-1}$ .