\documentstyle[12pt]{article}
\def\dg{^\circ}
\begin{document}
\begin{center}
${\bf 21}^{\mbox{\bf st}}$ {\bf USA
Mathematical
Olympiad}
\\[.1in]
{\bf April 30, 1992}\\
{\bf Time Limit: 3}${{\bf 1}\over{\bf 2}}$ {\bf
hours}
\end{center}
\begin{enumerate}
\item
Find, as a function of $\, n, \,$ the sum of the digits of
\[
9 \times 99 \times 9999 \times \cdots \times \left( 10^{2^n} - 1
\right),
\]
where each factor has twice as many digits as the
previous
one.
\item
Prove
\[
\frac{1}{\cos 0\dg \cos 1\dg} +
\frac{1}{\cos 1\dg \cos 2\dg} + \cdots +
\frac{1}{\cos 88\dg \cos 89\dg} =
\frac{\cos 1\dg}{\sin^2 1\dg}.
\]
\item
For a nonempty set $\, S \,$ of integers, let $\, \sigma(S) \,$
be the sum of the elements of $\, S$. Suppose that
$\, A = \{a_1, a_2, \ldots, a_{11} \} \,$ is a set of
positive integers with $\, a_1 < a_2 < \cdots < a_{11} \,$
and that, for each positive integer $\, n\leq 1500, \,$
there is a subset $\, S \,$ of $\, A \,$ for which
$\, \sigma(S) = n$. What is the smallest possible
value of $\, a_{10}$?
\item
Chords $\, \overline{AA'}, \, \overline{BB'}, \, \overline{CC'}
\,$ of a sphere meet at an interior point $\, P \,$ but are not
contained in a plane. The sphere through $\, A,B,C,P \,$
is tangent to the sphere through $\, A', B', C', P$.
Prove that $\, AA' = BB' = CC'$.
\item
Let $\, P(z) \,$ be a polynomial with complex coefficients which
is of degree $\, 1992 \,$ and has distinct zeros. Prove that
there exist complex numbers $\, a_1, a_2, \ldots, a_{1992} \,$
such that $\, P(z) \, $ divides the polynomial
\[
\left( \cdots \left( (z-a_1)^2 - a_2 \right)^2 \cdots - a_{1991}
\right)^2 - a_{1992}.
\]
\end{enumerate}
\end{document}