RIOPLATENSE 1997

Find all positive integers $n$ with the following property: there exists a polynomial $P_n(x)$ of degree $n$ , with integer coefficients, such that $P_n(0)=0$ and $P_n(x)=n$ for $n$ distinct integer solutions.

Consider a prism, not necessarily right, whose base is a rhombus $ABCD$ with side $AB = 5$ and diagonal $AC = 8$ . A sphere of radius $r$ is tangent to the plane $ABCD$ at $C$ and tangent to the edges $AA_1$ , $BB _1$ and $DD_ 1$ of the prism. Calculate $r$ .

Prove that there are infinitely many positive integers $n$ such that the number of positive divisors in $2^n-1$ is greater than $n$ .

Circles $c_1$ and $c_2$ are tangent internally to circle $c$ at points $A$ and $B$ , respectively, as seen in the figure. The inner tangent common to $c_1$ and $c_2$ touches these circles in $P$ and $Q$ , respectively. Show that the $AP$ and $BQ$ lines intersect the circle $c$ at diametrically opposite points.

Let $x_1, x_2, ... , x_n$ be non-negative numbers $n\ge3$ such that $x_1 + x_2 + ... + x_n = 1$ . Determine the maximum possible value of the expression $x_1x_2 + x_2x_3 + ... + x_{n-1}x_n$ .

Let $N$ be the set of positive integers. Determine if there is a function $f: N\to N$ such that $f(f(n))=2n$ , for all $n$ belongs to $N$ .