FI-MO 2014

Determine the value of the expression $x^2 + y^2 + z^2$ ,

if $x + y + z = 13$ , $xyz= 72$ and $\frac1x + \frac1y + \frac1z = \frac34$ .

The center of the circumcircle of the acute triangle $ABC$ is $M$ , and the circumcircle of $ABM$ meets $BC$ and $AC$ at $P$ and $Q$ ( $P\ne B$ ). Show that the extension of the line segment $CM$ is perpendicular to $PQ$ .

The points $P = (a, b)$ and $Q = (c, d)$ are in the first quadrant of the $xy$ plane, and $a, b, c$ and $d$ are integers satisfying $a < b, a < c, b < d$ and $c < d$ . A route from point $P$ to point $Q$ is a broken line consisting of unit steps in the directions of the positive coordinate axes. An allowed route is a route not touching the line $x = y$ . Tetermine the number of allowed routes.

The radius $r$ of a circle with center at the origin is an odd integer. There is a point ( $p^m, q^n$ ) on the circle, with $p,q$ prime numbers and $m,n$ positive integers. Determine $r$ .

Determine the smallest number $n \in Z_+$ , which can be written as $n = \Sigma_{a\in A}a^2$ , where $A$ is a finite set of positive integers and $\Sigma_{a\in A}a= 2014$ . In other words: what is the smallest positive number which can be written as a sum of squares of different positive integers summing to $2014$ ?