Let the diagonals of the square $ABCD$ intersect at $S$ and let $P$ be the midpoint of $AB$. Let $M$ be the intersection of $AC$ and $PD$ and $N$ the intersection of $BD$ and $PC$. A circle is inscribed in the quadrilateral $PMSN$. Prove that the radius of the circle is $MP-MS$.
Let $a_1,a_2,\ldots,a_n$ be positive integers, not necessarily distinct but with at least five distinct values. Suppose that for any $1\le i\lt j\le n$, there exist $k,l$, both different from $i$ and $j$, such that $a_i+a_j=a_k+a_l$. What is the smallest possible value of $n$?
A student divides an integer $m$ by a positive integer $n$, where $n\le100$, and claims that $$\frac mn=0.167a_1a_2\ldots.$$ Show that the student must be wrong.
The numbers $\frac11,\frac12,\ldots,\frac1{2010}$ are written on a blackboard. A student chooses any two of the numbers, say $x,y$, erases them and then writes down $x+y+xy$. He continues to do this until only one number is left on the blackboard. What is this number?