IMO 2002

Official IMO 2002 problems  ·  6/6 solved.

Problem 1   solved · 8m44s · Solution →

$S$ 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 $h' \le h,k' \le k$, 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.

Problem 2   solved · 14m01s · Solution →

$BC$ is a diameter of a circle center $O$. $A$ is any point on the circle with $\angle AOC \not\le 60^\circ$. $EF$ 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 $CEF$.

Problem 3   solved · 15m09s · Solution →

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

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

is itself an integer.

Problem 4   solved · 14m10s · Solution →

Let $n>1$ be an integer and let $1=d_{1}<d_{2}<d_{3} \cdots <d_{r}=n$ be all of its positive divisors in increasing order. Show that $$ d=d_1d_2+d_2d_3+ \cdots +d_{r-1}d_r <n^2. $$

Problem 5   solved · 14m11s · Solution →

Find all functions $f:\Bbb{R}\to \Bbb{R}$ such that

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

for all real numbers $x,y,z,t$.

Problem 6   solved · 14m29s · Solution →

Let $n \ge 3$ be a positive integer. Let $C_1,C_2,...,C_n$ be unit circles in the plane, with centers $O_1,O_2,...,O_n$ respectively. If no line meets more than two of the circles, prove that

$$ \sum_{1\le i< j \le n}^{}\frac{1}{O_iO_j}\le\frac{(n-1)\pi}{4} $$