1998 IMO Problems

Problems of the 1998 IMO.

Day I

Problem 1

In the convex quadrilateral $ABCD$ , the diagonals $AC$ and $BD$ are perpendicular and the opposite sides $AB$ and $DC$ are not parallel. Suppose that the point $P$ , where the perpendicular bisectors of $AB$ and $DC$ meet, is inside $ABCD$ . Prove that $ABCD$ is a cyclic quadrilateral if and only if the triangles $ABP$ and $CDP$ have equal areas.

Solution

Problem 2

In a competition, there are $a$ contestants and $b$ judges, where $b\ge3$ is an odd integer. Each judge rates each contestant as either “pass” or “fail”. Suppose $k$ is a number such that, for any two judges, their ratings coincide for at most $k$ contestants. Prove that $\frac{k}{a}\ge\frac{b-1}{2b}$ .

Solution

Problem 3

For any positive integer $n$ , let $d(n)$ denote the number of positive divisors of $n$ (including 1 and $n$ itself). Determine all positive integers $k$ such that $\frac{d(n^2)}{d(n)} = k$ for some $n$ .

Solution

Day II

Problem 4

Determine all pairs $(a, b)$ of positive integers such that $ab^{2} + b + 7$ divides $a^{2}b + a + b$ .

Solution

Problem 5

Let $I$ be the incenter of triangle $ABC$ . Let the incircle of $ABC$ touch the sides $BC$ , $CA$ , and $AB$ at $K$ , $L$ , and $M$ , respectively. The line through $B$ parallel to $MK$ meets the lines $LM$ and $LK$ at $R$ and $S$ , respectively. Prove that angle $RIS$ is acute.

Solution

Problem 6

Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$ ,

$f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}.$

Solution

1998 IMO (Problems) • Resources
Preceded by 1997 IMO	1 • 2 • 3 • 4 • 5 • 6	Followed by 1999 IMO
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1998 IMO Problems

Contents

Day I

Problem 1

Problem 2

Problem 3

Day II

Problem 4

Problem 5

Problem 6

See Also