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\begin{center}
${\bf 18}^{\mbox{\bf th}}$ {\bf USA
Mathematical
Olympiad}
\\[.1in]
{\bf April 25, 1989}\\
{\bf Time Limit: 3}${{\bf 1}\over{\bf 2}}$ {\bf
hours}
\end{center}
\begin{enumerate}
\item
For each positive integer $n$, let
\begin{eqnarray*}
S_n &=& 1 + \frac 12 + \frac 13 + \cdots + \frac 1n, \\
T_n &=& S_1 + S_2 + S_3 + \cdots + S_n, \\
U_n &=& \frac{T_1}{2} + \frac{T_2}{3} + \frac{T_3}{4} + \cdots +
\frac{T_n}{n+1}.
\end{eqnarray*}
Find, with proof, integers $0 < a, b,c, d < 1000000$ such that $T_{1988}
= a S_{1989} - b$ and $U_{1988} = c S_{1989} - d$.
\item
The 20 members of a local tennis club have scheduled exactly 14
two-person games among themselves, with each member playing in at least
one game. Prove that within this schedule there must be a set of 6 games
with 12 distinct players.
\item
Let $P(z)= z^n + c_1 z^{n-1} + c_2 z^{n-2} + \cdots + c_n$ be a
polynomial in the complex variable $z$, with real coefficients $c_k$.
Suppose that $|P(i)| < 1$. Prove that there exist real numbers $a$ and
$b$ such that $P(a + bi) = 0$ and $(a^2 + b^2 + 1)^2 < 4 b^2 + 1$.
\item
Let $ABC$ be an acute-angled triangle whose side lengths satisfy the
inequalities $AB < AC < BC$. If point $I$ is the center of the inscribed
circle of triangle $ABC$ and point $O$ is the center of the circumscribed
circle, prove that line $IO$ intersects segments $AB$ and $BC$.
\item
Let $u$ and $v$ be real numbers such that
\[
(u + u^2 + u^3 + \cdots + u^8) + 10u^9 =
(v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8.
\]
Determine, with proof, which of the two numbers, $u$ or $v$, is larger.
\end{enumerate}
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