IT-CESENATICO 1987

Show that $3x^5 +5x^3 -8x$ is divisible by $120$ for any integer $x$

A tetrahedron has the property that the three segments connecting the pairs of midpoints of opposite edges are equal and mutually orthogonal. Prove that this tetrahedron is regular.

Show how to construct (by a ruler and a compass) a right-angled triangle, given its inradius and circumradius.

Given $I_0 = \{-1,1\}$ , define $I_n$ recurrently as the set of solutions $x$ of the equations $x^2 -2xy+y^2- 4^n = 0$ , where $y$ ranges over all elements of $I_{n-1}$ . Determine the union of the sets $I_n$ over all nonnegative integers $n$ .

Let $a_1,a_2,...$ and $b_1,b_2,..$ . be two arbitrary infinite sequences of natural numbers. Prove that there exist different indices $r$ and $s$ such that $a_r \ge a_s$ and $b_r \ge b_s$ .

There are three balls of distinct colors in a bag. We repeatedly draw out the balls one by one, the balls are put back into the bag after each drawing. What is the probability that, after $n$ drawings, (a) exactly one color occured? (b) exactly two oclors occured? (c) all three colors occured?