2002 IMO Problems

Problems of the 2002 IMO.

Day I

Problem 1

$S$ 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.

Solution

Problem 2

$BC$ 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 $CEF$ .

Solution

Problem 3

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.

Solution

Day II

Problem 4

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 $d=d_1d_2+d_2d_3+ \cdots +d_{r-1}d_r <n^2.$

Solution

Problem 5

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 $x,y,z,t$ .

Solution

Problem 6

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

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

Solution

2002 IMO (Problems) • Resources
Preceded by 2001 IMO	1 • 2 • 3 • 4 • 5 • 6	Followed by 2003 IMO
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2002 IMO Problems

Contents

Day I

Problem 1

Problem 2

Problem 3

Day II

Problem 4

Problem 5

Problem 6

See Also