International Competitions 1984

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

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.

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 $OA$ $(0 \leq a(X) < 2\pi)$. 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 $C(Y)$.

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.

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

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

where $\lfloor x \rfloor$ denotes the greatest integer not exceeding $x$.

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 $a = 1$.