IT-CESENATICO 1989

Determine whether the equation $x^2 +xy+y^2 = 2$ has a solution $(x,y)$ in rational numbers.

There are $30$ men with their $30$ wives sitting at a round table. Show that there always exist two men who are on the same distance from their wives. (The seats are arranged at vertices of a regular polygon.)

Prove that, for every tetrahedron $ABCD$ , there exists a unique point $P$ in the interior of the tetrahedron such that the tetrahedra $PABC,PABD,PACD,PBCD$ have equal volumes.

Points $A,M,B,C,D$ are given on a circle in this order such that $A$ and $B$ are equidistant from $M$ . Lines $MD$ and $AC$ intersect at $E$ and lines $MC$ and $BD$ intersect at $F$ . Prove that the quadrilateral $CDEF$ is inscridable in a circle.

A fair coin is repeatedly tossed. We receive one marker for every "head" and two markers for every "tail". We win the game if, at some moment, we possess exactly $100$ markers. Is the probability of winning the game greater than, equal to, or less than $2/3$ ?

Given a real number $\alpha$ , a function $f$ is defined on pairs of nonnegative integers by $f(0,0) = 1, f(m,0) = f(0,m) = 0$ for $m > 0$ , $f(m,n) = \alpha f(m,n-1)+(1- \alpha)f(m -1,n-1)$ for $m,n > 0$ . Find the values of $\alpha$ such that $| f(m,n)| < 1989$ holds for any integers $m,n \ge 0$ .