International Competitions 1985

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

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 numbers in $M$ must have the same color.

For any polynomial $P(x) = a_0 + a_1x + \cdots + a_kx^k$ 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 $$w(Q_{i_1} + Q_{i_2} + \cdots + Q_{i_n}) \geq w(Q_{i_1}).$$

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.

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.

For every real number $x_1$, construct the sequence $x_1, x_2, \ldots$ by setting $$x_{n+1} = x_n\left(x_n + \frac{1}{n}\right) \text{ for each } n \geq 1.$$ Prove that there exists exactly one value of $x_1$ for which $$0 < x_n < x_{n+1} < 1$$ for every $n$.