International Mathematical Olympiad 1991

Problem 1Given a triangle ABC, let I be the incenter. The internal bisectors of angles A, B, C meet the opposite sides in A', B', C' respectively. Prove that

1/4<AI\cdot BI\cdot CI/(AA'\cdot BB'\cdot CC')\le8/27

.Solutions: 1

Problem 2Let n>6 be an integer and let

a_1,a_2,\ldots,a_k

be all positive integers less than n and relatively prime to n, in increasing order. If

a_2-a_1=a_3-a_2=\cdots=a_k-a_{k-1}>0

, prove that n is either a prime or a power of 2.Solutions: 1

Problem 3Let

S=\{1,2,3,\ldots,280\}

. Find the smallest integer n such that each n-element subset of S contains five numbers which are pairwise relatively prime.Solutions: 1

Problem 4Suppose 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 1.Solutions: 1

Problem 5Let ABC be a triangle and X an interior point of ABC. Show that at least one of the angles XAB, XBC, XCA is less than or equal to 30 degrees.Solutions: 1

Problem 6Given any real number a>1, construct a bounded infinite sequence

x_0,x_1,x_2,\ldots

such that

|x_n-x_m|\,|n-m|^a\ge1

for every pair of distinct n,m.Solutions: 1