International Mathematical Olympiad  – IMO 1991  Problems

Problem 1

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 2

Let $n geq 3$ and consider a set $E$ of $2n – 1$ distinct points on a circle. Suppose that exactly $k$ of these points are to be colored black. Such a coloring is good if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly $n$ points from $E$. Find the smallest value of $k$ so that every such coloring of $k$ points of $E$ is good.

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.

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 $i, j$.