1991 IMO Problems

Problems of the 1991 IMO.

Day I

Problem 1

Given a triangle $\,ABC,\,$ let $\,I\,$ be the center of its inscribed circle. The internal bisectors of the angles $\,A,B,C\,$ meet the opposite sides in $\,A^{\prime },B^{\prime },C^{\prime }\,$ respectively. Prove that \[\frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}.\]


Problem 2

Let $\,n > 6\,$ be an integer and $\,a_{1},a_{2},\cdots ,a_{k}\,$ be all the natural numbers less than $n$ and relatively prime to $n$. If \[a_{2} - a_{1} = a_{3} - a_{2} = \cdots = a_{k} - a_{k - 1} > 0,\] prove that $\,n\,$ must be either a prime number or a power of $\,2$.


Problem 3

Let $S = \{1,2,3,\cdots ,280\}$. Find the smallest integer $n$ such that each $n$-element subset of $S$ contains five numbers which are pairwise relatively prime.


Day II

Problem 4

Suppose $\,G\,$ is a connected graph with $\,k\,$ edges. Prove that it is possible to label the edges $1,2,\ldots ,k\,$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1.


Problem 5

Let $\,ABC\,$ be a triangle and $\,P\,$ an interior point of $\,ABC\,$. Show that at least one of the angles $\,\angle PAB,\;\angle PBC,\;\angle PCA\,$ is less than or equal to $30^{\circ }$.


Problem 6

An infinite sequence $\,x_{0},x_{1},x_{2},\ldots \,$ of real numbers is said to be bounded if there is a constant $\,C\,$ such that $\, \vert x_{i} \vert \leq C\,$ for every $\,i\geq 0$. Given any real number $\,a > 1,\,$ construct a bounded infinite sequence $x_{0},x_{1},x_{2},\ldots \,$ such that \[\vert x_{i} - x_{j} \vert \vert i - j \vert^{a}\geq 1\] for every pair of distinct nonnegative integers $i, j$.


1991 IMO (Problems) • Resources
Preceded by
1990 IMO
1 2 3 4 5 6 Followed by
1992 IMO
All IMO Problems and Solutions