IMO 1991

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

Problem 2. Let $n>6$, be an integer and $a_1,a_2,\dots ,a_k$, 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 $S=\{1,2,3,\dots ,280\}$. 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 $1,2,\dots ,k$, 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.

[A graph/ consists of a set of points, called vertices/, together with a set of edges/ joining certain pairs of distinct vertices. Each pair of vertices $u,v$, belongs to at most one edge. The graph $G$ is connected/ if for each pair of distinct vertices $x,y$, there is some sequence of vertices $x=v_0,v_1,v_2,\dots ,v_m=y$, such that each pair $v_i,v_{i+1}(0\le i<m)$, is joined by an edge of $G$.]

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 $x_0,x_1,x_2,\dots$ of real numbers is said to be bounded/ if there is a constant $C$, such that $|x_i|\le C$, for every $i\ge 0$.

Given any real number $a>1,$, construct a bounded infinite sequence $x_0,x_1,x_2,\dots$ such that

$|x_i-x_j||i-j{|}^a\ge 1$

for every pair of distinct nonnegative integers $i,j$.