\nopagenumbers
\noindent {\bf 29th IMO 1988}
\vskip 25pt
\noindent {\bf A1}. Consider two coplanar circles of radii $R>r$ 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 to $BP$ at $P$ meets the smaller circle again at $A$ (if it is tangent to the circle at $P$, then $A=P$).

(i) Find the set of values of $AB^2+BC^2+CA^2$.

(ii) Find the locus of the midpoint of $BC$.
\vskip 12pt
\noindent {\bf A2}. Let $n$ be a positive integer and let $A_1,A_2,\ldots,A_{2n+1}$ be subsets of a set $B$. Suppose that:

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

(ii) The intersection of every two distinct $A_i$ contains exactly one element, and

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

\noindent 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 each $A_i$ has $0$ assigned to exactly $n$ of its elements?
\vskip 12pt
\noindent {\bf A3}. A function $f$ is defined on the positive integers by: $f(1)=1, f(3)=3, f(2n)=f(n),f(4n+1)=2f(2n+1)-f(n)$, and $f(4n+3)=3f(2n+1)-2f(n)$ for all positive integers $n$. Determine the number of positive integers $n\le1988$ for which $f(n)=n$.
\vskip 12pt
\noindent {\bf B1}. Show that the set of real numbers $x$ which satisfy the inequality: $${1\over x-1}+{2\over x-2}+{3\over x-3}+\ldots+{70\over x-70}\ge{5\over4}$$ is a union of disjoint intervals, the sum of whose lengths is $1988$.
\vskip 12pt
\noindent {\bf B2}. $ABC$ 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 $ABD$ and $ACD$ intersects the sides $AB,AC$ at $K,L$ respectively. Show that the area of the triangle $ABC$ is at least twice the area of the triangle $AKL$.
\vskip 12pt
\noindent {\bf B3}. Let $a$ and $b$ be positive integers such that $ab+1$ divides $a^2+b^2$. Show that $a^2+b^2\over ab+1$ is a perfect square.
\vskip 20pt
\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

\bye
