IMO 2002

Problem 1. Let $n$ be a positive integer. Let $T$ be the set of points $(x,y)$ in the plane where $x$ and $y$ are non-negative integers and $x+y<n$. Each point of $T$ is colored red or blue. If a point $(x,y)$ is red, then so are all points $(x^{\prime },y^{\prime })$ of $T$ with both $x^{\prime }\le x$ and $y^{\prime }\le y$. Define an $X$-set to be a set of $n$ blue points having distinct $x$-coordinates, and a $Y$-set to be a set of $n$ blue points having distinct $y$-coordinates. Prove that the number of $X$-sets is equal to the number of $Y$-sets.

Problem 2. Let $BC$ be a diameter of circle $\omega$ with center $O$. Let $A$ be a point of circle $\omega$ such that $0^{\circ }<\angle AOB<120^{\circ }$. Let $D$ be the midpoint of arc $AB$ not containing $C$. Line $\ell$ passes through $O$ and is parallel to line $AD$. Line $\ell$ intersects line $AC$ at $J$. The perpendicular bisector of segment $OA$ intersects circle $\omega$ at $E$ and $F$. Prove that $J$ is the incenter of triangle $CEF$.

Problem 3. Find all pairs of integers $m,n\ge 3$ such that there exist infinitely many positive integers $a$ for which

$\frac{a^m+a-1}{a^n+a^2-1}$

is an integer.

Problem 4. Let $n$ be an integer greater than 1. The positive divisors of $n$ are $d_1,d_2,\dots ,d_k$ where $1=d_1<d_2<\cdots <d_k=n$. Define $D=d_1{d}_2+d_2{d}_3+\cdots +d_{k-1}{d}_k$.

(a) Prove that $D<n^2$.

(b) Determine all $n$ for which $D$ is a divisor of $n^2$.

Problem 5. Find all functions $f$ from the set $\mathbb{R}$ of real numbers to itself such that

$(f(x)+f(z))(f(y)+f(t))=f(xy-zt)+f(xt+yz)$

for all $x,y,z,t$ in $\mathbb{R}$.

Problem 6. Let ${\Gamma }_1,{\Gamma }_2,\dots ,{\Gamma }_n$ be circles of radius 1 in the plane, where $n\ge 3$. Denote their centers by $O_1,O_2,\dots ,O_n$ respectively. Suppose that no line meets more than two of the circles. Prove that

${\sum }_{1\le i<j\le n}\frac{1}{O_i{O}_j}\le \frac{(n-1)\pi }{4}.$