PMO 2013

Determine, with proof, the least positive integer $n$ for which there exist $n$ distinct positive integers, $\left(1-\frac{1}{x_1}\right)\left(1-\frac{1}{x_2}\right)......\left(1-\frac{1}{x_n}\right)=\frac{15}{2013}$

Let P be a point in the interior of triangle ABC . Extend AP, BP, and CP to meet BC, AC, and AB at D, E, and F, respectively. If triangle APF, triangle BPD and triangle CPE have equal areas, prove that P is the centroid of triangle ABC .

Let n be a positive integer. The numbers 1, 2, 3,....., 2n are randomly assigned to 2n distinct points on a circle. To each chord joining two of these points, a value is assigned equal to the absolute value of the difference between the assigned numbers at its endpoints. Show that one can choose n pairwise non-intersecting chords such that the sum of the values assigned to them is $n^2$ .

Let $a$ , $p$ and $q$ be positive integers with $p \le q$ . Prove that if one of the numbers $a^p$ and $a^q$ is divisible by $p$ , then the other number must also be divisible by $p$ .

Let $r$ and $s$ be positive real numbers such that $(r+s-rs)(r+s+rs)=rs$ . Find the minimum value of $r+s-rs$ and $r+s+rs$