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${\bf 19}^{\mbox{\bf th}}$ {\bf USA
Mathematical
Olympiad}
\\[.1in]
{\bf April 24, 1990}\\
{\bf Time Limit: 3}${{\bf 1}\over{\bf 2}}$ {\bf
hours}
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\begin{enumerate}
\item
A certain state issues license plates consisting of six digits (from 0
through 9). The state requires that any two plates differ in at least two
places. (Thus the plates \fbox{027592} and \fbox{020592} cannot both be
used.) Determine, with proof, the maximum number of distinct license
plates that the state can use.
\item
A sequence of functions $\, \{f_n(x) \} \,$ is defined recursively as follows:
\begin{eqnarray*}
f_1(x) &=& \sqrt{x^2 + 48}, \quad \mbox{and} \\
f_{n+1}(x) &=& \sqrt{x^2 + 6f_n(x)} \quad \mbox{for $n \geq 1$.}
\end{eqnarray*}
(Recall that $\sqrt{\makebox[5mm]{}}$
is understood to represent the positive square root.)
For each positive integer $n$, find all real solutions of the equation
$\, f_n(x) = 2x \,$.
\item
Suppose that necklace $\, A \,$ has 14 beads and necklace $\, B \,$ has
19. Prove that for any odd integer $n \geq 1$, there is a way to number
each of the 33 beads with an integer from the sequence
\[
\{ n, n+1, n+2, \dots, n+32 \}
\]
so that each integer is used once, and adjacent beads correspond to
relatively prime integers. (Here a ``necklace'' is viewed as a circle in
which each bead is adjacent to two other beads.)
\item
Find, with proof, the number of positive integers whose base-$n$
representation consists of distinct digits with the property that, except
for the leftmost digit, every digit differs by $\pm 1$ from some digit
further to the left.
(Your answer should be an explicit function of $n$ in simplest form.)
\item
An acute-angled triangle $ABC$ is given in the plane. The circle with
diameter $\, AB \,$ intersects altitude $\, CC' \,$ and its extension at
points $\, M \,$ and $\, N \,$, and the circle with diameter $\, AC \,$
intersects altitude $\, BB' \,$ and its extensions at $\, P \,$ and $\, Q
\,$. Prove that the points $\, M, N, P, Q \,$ lie on a common circle.
\end{enumerate}
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