IT-CESENATICO 1988

Players $A$ and $B$ play the following game: $A$ tosses a coin $n$ times, and $B$ does $n+1$ times. The player who obtains more "heads" wins; or in the case of equal balances, $A$ is assigned victory. Find the values of $n$ for which this game is fair (i.e. both players have equal chances for victory).

In a basketball tournament any two of the $n$ teams $S_1,S_2,...,S_n$ play one match (no draws). Denote by $v_i$ and $p_i$ the number of victories and defeats of team $S_i$ ( $i = 1,2,...,n$ ), respectively. Prove that $v^2_1 +v^2_2 +...+v^2_n = p^2_1 +p^2_2 +...+p^2_n$

A regular pentagon of side length $1$ is given. Determine the smallest $r$ for which the pentagon can be covered by five discs of radius $r$ and justify your answer.

Show that all terms of the sequence $1,11,111,1111,...$ in base $9$ are triangular numbers, i.e. of the form $\frac{m(m+1)}{2}$ for an integer $m$

Given four non-coplanar points, is it always possible to find a plane such that the orthogonal projections of the points onto the plane are the vertices of a parallelogram? How many such planes are there in general?

The edge lengths of the base of a tetrahedron are $a,b,c$ , and the lateral edge lengths are $x,y,z$ . If $d$ is the distance from the top vertex to the centroid of the base, prove that $x+y+z \le a+b+c+3d$ .