Question bank IMO 1984

Let $x,y,z$ be nonnegative real numbers with $x+y+z=1$. Show that
$$0\le xy+yz+zx-2xyz\le\frac7{27}.$$

Find two positive integers $a,b$ such that none of the numbers $a,b,a+b$ is divisible by $7$ and $(a+b)^7-a^7-b^7$ is divisible by $7^7$.

In a plane two different points $O$ and $A$ are given. For each point $X\ne O$ of the plane denote by $\alpha(X)$ the angle $AOX$ measured in radians $(0\le\alpha(X)\lt 2\pi)$ and by $C(X)$ the circle with center $O$ and radius $OX+\frac{\alpha(X)}{OX}$. Suppose each point of the plane is colored by one of a finite number of colors. Show that there exists a point $X$ with $\alpha(X)\gt 0$ such that its color appears somewhere on the circle $C(X)$.

Let $ABCD$ be a convex quadrilateral for which the circle of diameter $AB$ is tangent to the line $CD$. Show that the circle of diameter $CD$ is tangent to the line $AB$ if and only if the lines $BC$ and $AD$ are parallel.

Let $d$ be the sum of the lengths of all diagonals of a convex polygon of $n$ $(n\gt 3)$ vertices, and let $p$ be its perimeter. Prove that
$$\frac{n-3}{2}\lt \frac{d}{p}\lt \frac12\left(\left[\frac n2\right]\left[\frac{n+1}{2}\right]-2\right).$$

Let $a,b,c,d$ be odd positive integers such that $a\lt b\lt c\lt d$, $ad=bc$, and $a+d=2^k$, $b+c=2^m$ for some integers $k$ and $m$. Prove that $a=1$.