1985 IMO Problems

Problems of the 26th IMO Finland.

Day I

Problem 1

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

Solution

Problem 2

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

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

(ii) for each $i \in M, i \neq k$ , both $\displaystyle i$ and $\displaystyle |i-j|$ have the same color.

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