International Mathematical Olympiad  – IMO 1985  Problems

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 $M = \{ 1,2, \ldots , n-1 \}$ 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 $i \in M, i \neq k$, both $i$ and $|i-k|$ have the same color.

Prove that all the numbers in $M$ 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 $i = 0, 1, \ldots$, let $Q_i (x) = (1+x)^i$. Prove that if $i_1, i_2, \ldots , i_n$ are integers such that $0 \leq i_1 < i_2 < \cdots < i_n$, then

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

Problem 4

Given a set $M$ of $1985$ distinct positive integers, none of which has a prime divisor greater than $23$, prove that $M$ contains a subset of $4$ elements whose product is the $4$th power of an integer.

Problem 5

A circle with center $O$ passes through the vertices $A$ and $C$ of the triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$ respectively. Let $M$ be the point of intersection of the circumcircles of triangles $ABC$ and $KBN$ (apart from $B$). Prove that $\angle OMB = 90^{\circ}$.

Problem 6

For every real number $x_1$, construct the sequence $x_1,x_2,\ldots$ by setting:

$x_{n + 1} = x_n(x_n + {1\over n}).$

Prove that there exists exactly one value of $x_1$ which gives $0 < x_n < x_{n + 1} < 1$ for all $n$.