International Mathematical Olympiad · 1988

Problems

Problem 1Two coplanar circles with common center have radii $R>r$ . Let P be fixed on the smaller circle and B vary on the larger circle. Line BP meets the larger circle again at C; the perpendicular to BP at P meets the smaller circle again at A (if tangent, A=P). (i) Find the set of values of $AB^2+BC^2+CA^2$ . (ii) Find the locus of the midpoint of BC.Solutions: 1
Problem 2Let $n$ be positive and let $A_1,\ldots,A_{2n+1}$ be subsets of $B$ . Suppose each $A_i$ has exactly $2n$ elements, every two distinct $A_i$ have exactly one common element, and every element of $B$ belongs to at least two $A_i$ . For which $n$ can one label each element 0 or 1 so that each $A_i$ has 0 on exactly $n$ elements?Solutions: 1
Problem 3A function $f$ on positive integers is defined by $f(1)=1$ , $f(3)=3$ , $f(2n)=f(n)$ , $f(4n+1)=2f(2n+1)-f(n)$ , and $f(4n+3)=3f(2n+1)-2f(n)$ . Determine the number of positive integers $n\le1988$ for which $f(n)=n$ .Solutions: 1
Problem 4Show that the solution set of the inequality $\sum_{k=1}^{70}\frac{k}{x-k}\ge\frac{5}{4}$ is a union of disjoint intervals whose total length is $1988$ .Solutions: 1
Problem 5 $ABC$ is a right-angled triangle with right angle at $A$ , and $AD$ is the altitude to the hypotenuse. The line joining the incenters of $ABD$ and $ACD$ meets $AB,AC$ at $K,L$ . If $S$ and $T$ are the areas of $ABC$ and $AKL$ , prove that $S\ge2T$ .Solutions: 1
Problem 6Let $a,b$ be positive integers such that $ab+1$ divides $a^2+b^2$ . Show that $(a^2+b^2)/(ab+1)$ is the square of an integer.Solutions: 1