International Mathematical Olympiad · 1985

Problems

Problem 1A circle has its center on the side $AB$ of the cyclic quadrilateral $ABCD$ . The other three sides are tangent to the circle. Prove that $AD + BC = AB$ .Solutions: 1
Problem 2Let $n$ and $k$ be relatively prime positive integers with $k < n$ . The set $M = \{1, 2, \ldots, n-1\}$ is colored so that each number is either blue or white, subject to: (i) for each $i \in M$ , the numbers $i$ and $n-i$ have the same color; and (ii) for each $i \in M$ with $i \ne k$ , the numbers $i$ and $|i-k|$ have the same color. Prove that all the numbers in $M$ must have the same color.Solutions: 1
Problem 3For an integer-coefficient polynomial $P(x)=a_0+a_1x+\cdots+a_kx^k$ , let $w(P)$ be the number of odd coefficients. Put $Q_i(x)=(1+x)^i$ for $i=0,1,2,\ldots$ . Prove that if $0\le i_1<i_2<\cdots<i_n$ , then $w(Q_{i_1}+Q_{i_2}+\cdots+Q_{i_n})\ge w(Q_{i_1})$ .Solutions: 1
Problem 4Given a set $M$ of $1985$ distinct positive integers, none of which has a prime divisor greater than $23$ , prove that $M$ contains four distinct elements whose product is the fourth power of an integer.Solutions: 1
Problem 5A circle with center $O$ passes through vertices $A$ and $C$ of triangle $ABC$ and intersects segments $AB$ and $BC$ again at distinct points $K$ and $N$ , respectively. The circumcircles of $ABC$ and $KBN$ meet at exactly two distinct points $B$ and $M$ . Prove that $\angle OMB$ is a right angle.Solutions: 1
Problem 6For every real number $x_1$ , define a sequence by $x_{n+1}=x_n\left(x_n+\frac1n\right)$ . Prove that there exists exactly one value of $x_1$ for which $0<x_n<x_{n+1}<1$ for all $n$ .Solutions: 1