IMO 1985

Problem 1. A circle has center on the side $AB$ of the cyclic quadrilateral $ABCD.$ The other three sides are tangent to the circle. Prove that $AD+BC=AB.$

Problem 2. Let $n$ and $k$ be given relatively prime natural numbers, $k<n$. Each number in the set $M=\{1,2,...,n-1\}$ is colored either blue or white. It is given that

(i) for each $i\in M$, both $i$ and $n-i$ have the same color;

(ii) for each $i\in M,i\ne k$, both $i$ and $\left|i-k\right|$ have the same color. Prove that all numbers in $M$ must have the same color.

Problem 3. For any polynomial $P(x)=a_0+a_{1}x+\cdots +a_k{x}^k$ with integer coefficients, the number of coefficients which are odd is denoted by $w(P)$. For $i=0,1,...$, let $Q_i(x)=(1+x{)}^i$. Prove that if $i_1{i}_2,...,i_n$ are integers such that $0\le i_1<i_2<\cdots <i_n,$ then

$w(Q_{i_1}+Q_{i_2},++Q_{i_n})\ge w(Q_{i_1}).$

Problem 4. Given a set $M$ of $1985$ distinct positive integers, none of which has a prime divisor greater than $26$. Prove that $M$ contains at least one subset of four distinct elements whose product is the fourth power of an integer.

Problem 5. A circle with center $O$ passes through the vertices $A$ and $C$ of triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$, respectively. The circumscribed circles of the triangles $ABC$ and $EBN$ intersect at exactly two distinct points $B$ and $M$. Prove that angle $OMB$ is a right angle.

Problem 6. For every real number $x_1$, construct the sequence $x_1,x_2,...$ by setting

$x_{n+1}=x_n\left(x_n+\frac{1}{n}\right)\text{for each }n\ge 1.$

Prove that there exists exactly one value of $x_1$ for which

$0<x_n<x_{n+1}<1$

for every $n$.