International Mathematical Olympiad · 2002

Problems

Problem 1Let $n$ be a positive integer. Let $T$ be the set of points $(x,y)$ in the plane where $x$ and $y$ are non-negative integers with $x+y<n$ . Each point of $T$ is coloured red or blue, subject to the following condition: if a point $(x,y)$ is red, then all points of $T$ with first coordinate at most $x$ and second coordinate at most $y$ are also red. Let $A$ be the number of ways to choose $n$ blue points with distinct $x$ -coordinates, and let $B$ be the number of ways to choose $n$ blue points with distinct $y$ -coordinates. Prove that $A=B$ .Solutions: 1
Problem 2Let $BC$ be a diameter of circle $\omega$ with center $O$ . Let $A$ be a point of $\omega$ such that $0^\circ<\angle AOB<120^\circ$ . Let $D$ be the midpoint of arc $AB$ not containing $C$ . The 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 $CEF$ .Solutions: 1
Problem 3Find all pairs of positive integers $m,n\ge3$ for which there exist infinitely many positive integers $a$ such that $\frac{a^m+a-1}{a^n+a^2-1}$ is itself an integer.Solutions: 1
Problem 4Let $n\ge2$ be a positive integer with divisors $1=d_1<d_2<\cdots<d_k=n$ . Prove that $d_1d_2+d_2d_3+\cdots+d_{k-1}d_k$ is always less than $n^2$ , and determine when it is a divisor of $n^2$ .Solutions: 1
Problem 5Find all functions $f:\mathbb R\to\mathbb R$ such that $(f(x)+f(z))(f(y)+f(t))=f(xy-zt)+f(xt+yz)$ for all real numbers $x,y,z,t$ .Solutions: 1
Problem 6Let $n\ge3$ be a positive integer. Let $C_1,C_2,\ldots,C_n$ be unit circles in the plane, with centers $O_1,O_2,\ldots,O_n$ respectively. If no line meets more than two of the circles, prove that $\sum_{1\le i<j\le n}\frac1{O_iO_j}\le\frac{(n-1)\pi}{4}$ .Solutions: 1