IMO 1991

Official IMO 1991 problems  ·  4/6 solved.

Problem 1   solved · 18m14s · Solution →

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   solved · 31m33s · Solution →

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   solved · 31m43s · Solution →

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   solved · 11m04s · Solution →

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 $i, j$.