International Mathematical Olympiad · 1984

Problems

Problem 1Let $x,y,z$ be nonnegative real numbers with $x+y+z=1$ . Show that $0\le xy+yz+zx-2xyz\le\frac{7}{27}$ .Solutions: 1
Problem 2Find 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 $7^7$ .Solutions: 1
Problem 3Given points $O$ and $A$ in the plane, every point is colored with one of a finite number of colors. For a point $X$ , let $C(X)$ be the circle centered at $O$ with radius $OX+\frac{\angle AOX}{OX}$ , where $\angle AOX$ is measured in radians in $\left[0,2\pi\right)$ . Prove that there is a point $X$ , not on $OA$ , such that the color of $X$ appears on the circumference of $C(X)$ .Solutions: 1
Problem 4Let ABCD be a convex quadrilateral such that CD is tangent to the circle with diameter AB. Prove that AB is tangent to the circle with diameter CD if and only if $BC\parallel AD$ .Solutions: 1
Problem 5Let d be the sum of lengths of all diagonals of a convex n-gon, n>3, and p its perimeter. Prove $n-3<\frac{2d}{p}<\left\lfloor\frac n2\right\rfloor\left\lfloor\frac{n+1}{2}\right\rfloor-2$ .Solutions: 1
Problem 6Let a,b,c,d be odd integers with $0<a<b<c<d$ and $ad=bc$ . Prove that if $a+d=2^k$ and $b+c=2^m$ , then $a=1$ .Solutions: 1