\nopagenumbers
\noindent {\bf 39th IMO 1998}
\vskip 25pt
\noindent {\bf A1}. In the convex quadrilateral $ABCD$, the diagonals $AC$ and $BD$ are perpendicular and the opposite sides $AB$ and $DC$ are not parallel. The point $P$, where the perpendicular bisectors of $AB$ and $CD$ meet, is inside $ABCD$. Prove that $ABCD$ is cyclic iff the triangles $ABP$ and $CDP$ have equal areas.
\vskip 12pt
\noindent {\bf A2}. In a competition there are $a$ contestants and $b$ judges, where $b \ge 3$ is an odd integer. Each judge rates each contestant as either ``pass" or ``fail". Suppose k is a number such that for any two judges their ratings coincide for at most k contestants. Prove $k/a \ge (b-1)/2b$.
\vskip 12pt
\noindent {\bf A3}.  For any positive integer $n$, let $d(n)$ denote the number of positive divisors of $n$ (including $1$ and $n$). Determine all positive integers $k$ such that $d(n^2) = k d(n)$ for some $n$.
\vskip 12pt
\noindent {\bf B1}. Determine all pairs $(a, b)$ of positive integers such that $ab^2 + b + 7$ divides $a^2b+a+b$.
\vskip 12pt
\noindent {\bf B2}. Let $I$ be the incenter of the triangle $ABC$. Let the incircle of $ABC$ touch the sides $BC, CA, AB$ at $K, L, M$ respectively. The line through $B$ parallel to $MK$ meets the lines $LM$ and $LK$ at $R$ and $S$ respectively. Prove that $\angle RIS$ is acute.
\vskip 12pt
\noindent {\bf B3}. Consider all functions $f: N \to N$ on the positive integers satisfying $f(t^2f(s)) = sf(t)^2$ for all $s$ and $t$. Determine the least possible value of $f(1998)$.
\vskip 20pt
\noindent \copyright John Scholes

\noindent jscholes@kalva.demon.co.uk

\noindent 19 August 2003

\bye
