IMO 1984

Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$ , where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$ .

Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$ , but $(a+b)^7-a^7-b^7$ is divisible by $7^7$ .

Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$ , where $\angle AOX$ is measured in radians in the range $[0,2\pi)$ . Prove that we can find a point $X$ , not on $OA$ , such that its color appears on the circumference of the circle $C(X)$ .

Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$ . Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.

Let $d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $n$ vertices (where $n>3$ ). Let $p$ be its perimeter. Prove that: $$n-3<{2d\over p}<\Bigl[{n\over2}\Bigr]\cdot\Bigl[{n+1\over 2}\Bigr]-2,$$ where $[x]$ denotes the greatest integer not exceeding $x$ .

Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$ . Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$ , then $a=1$ .