IMO 1988

Problem 1. Consider two coplanar circles of radii $R$ and $r$ ($R>r$) with the same center. Let $P$ be a fixed point on the smaller circle and $B$ a variable point on the larger circle. The line $BP$ meets the larger circle again at $C$. The perpendicular $l$ to $BP$ at $P$ meets the smaller circle again at $A$. (If $l$ is tangent to the circle at $P$ then $A=P$.)

(i) Find the set of values of $B{C}^2+C{A}^2+A{B}^2$.

(ii) Find the locus of the midpoint of $BC$.

Problem 2. Let $n$ be a positive integer and let $A_1$, $A_2$, ..., $A_{2n+1}$ be subsets of a set $B$. Suppose that

(a) Each $A_i$ has exactly $2n$ elements,

(b) Each $A_i\cap A_j$ ($1\le i<j\le 2n+1$) contains exactly one element, and

(c) Every element of $B$ belongs to at least two of the $A_i$.

For which values of $n$ can one assign to every element of $B$ one of the numbers 0 and 1 in such a way that $A_i$ has 0 assigned to exactly $n$ of its elements?

Problem 3. A function $f$ is defined on the positive integers by

for all positive integers $n$.

Determine the number of positive integers $n$, less than or equal to 1988, for which $f(n)=n$.

Problem 4. Show that set of real numbers $x$ which satisfy the inequality ${\sum }_{k=1}^{70}\frac{k}{x-k}\ge \frac{5}{4}$ is a union of disjoint intervals, the sum of whose lengths is 1988.

Problem 5. $ABC$ is a triangle right-angled at $A$, and $D$ is the foot of the altitude from $A$. The straight line joining the incenters of the triangles $ABD$, $ACD$ intersects the sides $AB$, $AC$ at the points $K$, $L$ respectively. $S$ and $T$ denote the areas of the triangles $ABC$ and $AKL$ respectively. Show that $S\ge 2T$.

Problem 6. Let $a$ and $b$ be positive integers such that $ab+1$ divides $a^2+b^2$. Show that $\frac{a^2+b^2}{ab+1}$ is the square of an integer.