International Competitions 2002

$S$ is the set of all $(h,k)$ with $h,k$ non-negative integers such that $h+k<n$. Each element of $S$ is colored red or blue, so that if $(h,k)$ is red and $h'\leq h,k'\leq k$, then $(h',k')$ is also red. A type 1 subset of $S$ has $n$ blue elements with different first member and a type 2 subset of $S$ has $n$ blue elements with different second member. Show that there are the same number of type 1 and type 2 subsets.

$BC$ is a diameter of a circle center $O$. $A$ is any point on the circle with $\angle AOC>60^{\circ}$. $EF$ is the chord which is the perpendicular bisector of $AO$. $D$ is the midpoint of the minor arc $AB$. The line through $O$ parallel to $AD$ meets $AC$ at $J$. Show that $J$ is the incenter of triangle $CEF$.

Find all pairs of integers $m>2,n>2$ such that there are infinitely many positive integers $k$ for which $k^{n}+k^{2}-1$ divides $k^{m}+k-1$.

The positive divisors of the integer $n>1$ are $d_{1}<d_{2}<\ldots<d_{k}$, so that $d_{1}=1,d_{k}=n$. Let $d=d_{1}d_{2}+d_{2}d_{3}+\cdots+d_{k-1}d_{k}$. Show that $d<n^{2}$ and find all $n$ for which $d$ divides $n^{2}$.

Find all real-valued functions on the reals such that $(f(x)+f(y))(f(u)+f(v))=f(xu-yv)+f(xv+yu)$ for all $x,y,u,v$.

$n>2$ circles of radius 1 are drawn in the plane so that no line meets more than two of the circles. Their centers are $O_{1},O_{2},\cdots,O_{n}$. Show that $\sum_{i<j}1/O_{i}O_{j}\leq(n-1)\pi/4$.