IT-CESENATICO 1995

Determine for which values of $n$ it is possible to tile a square of side $n$ with figures of the type shown in the picture

No two of $20$ students in a class have the same scores on both written and oral examinations in mathematics. We say that student $A$ is better than $B$ if his two scores are greater than or equal to the corresponding scores of $B$ . The scores are integers between $1$ and $10$ . (a) Show that there exist three students $A,B,C$ such that $A$ is better than $B$ and $B$ is better than $C$ . (b) Would the same be true for a class of $19$ students?

In a town there are four pubs, $A,B,C,D$ , and any two of them are connected to each other except $A$ and $D$ . A drunkard wanders about the pubs starting with $A$ and, after having a drink, goes to any of the pubs directly connected, with equal probability. (a) What is the probability that the drunkard is at $C$ at its fifth drink? (b) Where is the drunkard most likely to be after $n$ drinks ( $n > 5$ )?

An acute-angled triangle $ABC$ is inscribed in a circle with center $O$ . The bisector of $\angle A$ meets $BC$ at $D$ , and the perpendicular to $AO$ through $D$ meets the segment $AC$ in a point $P$ . Show that $AB = AP$ .

Two non-coplanar circles in space are tangent at a point and have the same tangents at this point. Show that both circles lie on some sphere.

Find all pairs of positive integers $x,y$ such that $x^2 +615 = 2^y$