IMO 1985 — Official

A circle whose center is on the side $ED$ of the cyclic quadrilateral $BCDE$ touches the other three sides. Prove that $EB+CD=ED$.

Each of the numbers in the set $N=\{1,2,3,\ldots,n-1\}$, where $n\ge3$, is colored with one of two colors, say red or black, so that:
(i) $i$ and $n-i$ always receive the same color, and
(ii) for some $j\in N$ relatively prime to $n$, $i$ and $|j-i|$ receive the same color for all $i\in N$, $i\ne j$.
Prove that all numbers in $N$ must receive the same color.

The weight $w(p)$ of a polynomial $p$, $p(x)=\sum_{i=0}^n a_ix^i$, with integer coefficients $a_i$ is defined as the number of its odd coefficients. For $i=0,1,2,\ldots$, let $q_i(x)=(1+x)^i$. Prove that for any finite sequence $0\le i_1\lt i_2\lt \cdots\lt i_n$ the inequality
$$w(q_{i_1}+\cdots+q_{i_n})\ge w(q_{i_1})$$
holds.

Given a set $M$ of 1985 positive integers, none of which has a prime divisor larger than 26, prove that $M$ has four distinct elements whose geometric mean is an integer.

A circle with center $O$ passes through points $A$ and $C$ and intersects the sides $AB$ and $BC$ of the triangle $ABC$ at points $K$ and $N$, respectively. The circumscribed circles of the triangles $ABC$ and $KBN$ intersect at two distinct points $B$ and $M$. Prove that $\angle OMB=90^\circ$.

The sequence $f_1,f_2,\ldots,f_n,\ldots$ of functions is defined for $x>0$ recursively by
$$f_1(x)=x,\qquad f_{n+1}(x)=f_n(x)\left(f_n(x)+\frac1n\right).$$
Prove that there exists one and only one positive number $a$ such that $0\lt f_n(a)\lt f_{n+1}(a)\lt 1$ for all integers $n\ge1$.