IMO 1984

Problem 1. Prove that $0\le yz+zx+xy-2xyz\le 7/27$, where $x,y$ and $z$ are non-negative real numbers for which $x+y+z=1$.

Problem 2. Find one pair of positive integers $a$ and $b$ such that:

(i) $ab(a+b)$ is not divisible by $7$;

(ii) $(a+b{)}^7-a^7-b^7$ is divisible by $7^7$ .

Justify your answer.

Problem 3. In the plane two different points $O$ and $A$ are given. For each point $X$ of the plane, other than $O$, denote by $a(X)$ the measure of the angle between $OA$ and $OX$ in radians, counterclockwise from $OA(0\le a(X)<2\pi ).$ Let $C(X)$ be the circle with center $O$ and radius of length $OX+a(X)/OX.$ Each point of the plane is colored by one of a finite number of colors. Prove that there exists a point $Y$ for which $a(Y)>0$ such that its color appears on the circumference of the circle $C(Y)$.

Problem 4. Let $ABCD$ be a convex quadrilateral such that the line $CD$ is a tangent to the circle on $AB$ as diameter. Prove that the line $AB$ is a tangent to the circle on $CD$ as diameter if and only if the lines $BC$ and $AD$ are parallel.

Problem 5. Let $d$ be the sum of the lengths of all the diagonals of a plane convex polygon with $n$ vertices $(n>3)$, and let $p$ be its perimeter. Prove that

$n-3<\frac{2d}{p}<\left[\frac{n}{2}\right]\left[\frac{n+1}{2}\right]-2,$

where $[x]$ denotes the greatest integer not exceeding $x$.

Problem 6. Let $a,b,c$ and $d$ 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 $a=1$.