1985 IMO Problems

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Problems of the 26th IMO Finland.

Day I

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 $AD + BC = AB$ .

Solution

Problem 2

Let $n$ and $k$ be given relatively prime natural numbers, $n < k$ . 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 $|i-j|$ have the same color.

Prove that all number in $M$ have the same color.

Solution

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

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

Solution

Day II

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.

Solution

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 $\angle OMB = 90^{\circ}$ .