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\begin{center}
The Fifty-Ninth Annual William Lowell Putnam Mathematical 
Competition \\
Saturday, December 5, 1998
\end{center}

\begin{itemize}
\item[A--1]
A right circular cone has base of radius 1 and height 3.  A cube is
inscribed in the cone so that one face of the cube is contained in the
base of the cone.  What is the side-length of the cube?

\item[A--2]
Let $s$ be any arc of the unit circle lying entirely in the first
quadrant.  Let $A$ be the area of the region lying below $s$ and above
the $x$-axis and let $B$ be the area of the region lying to the right of
the $y$-axis and to the left of $s$.  Prove that $A+B$ depends only on
the arc length, and not on the position, of $s$. 

\item[A--3]
Let $f$ be a real function on the real line with continuous third
derivative.  Prove that there exists a point $a$ such that 
\[
f(a) \cdot f'(a) \cdot f''(a) \cdot f'''(a)\geq 0. 
\]

\item[A--4]
Let $A_{1}=0$ and $A_{2}=1$.  For $n>2$, the number $A_{n}$ is defined
by concatenating the decimal expansions of $A_{n-1}$ and $A_{n-2}$ from
left to right.  For example $A_{3}=A_{2}A_{1}=10$,
$A_{4}=A_{3}A_{2}=101$, $A_{5}=A_{4}A_{3}=10110$, and so forth.
Determine all $n$ such that 11 divides $A_{n}$. 

\item[A--5]
Let $\mathcal{F}$ be a finite collection of open discs in
$\mathbf{R}^{2}$ whose union contains a set $E\subseteq
\mathbf{R}^{2}$.  Show that there is a pairwise disjoint subcollection
$D_{1},\dots ,D_{n}$ in $\mathcal{F}$ such that 
\[
\bigcup _{j=1}^{n} 3D_{j} \supseteq E.
\]
Here, if $D$ is the disc of radius $r$ and center $P$, then $3D$ is the
disc of radius $3r$ and center $P$. 

\item[A--6]
Let $A$, $B$, $C$ denote distinct points with integer coordinates in
$\mathbf{R}^{2}$.  Prove that if 
\[
(|AB| + |BC|)^{2} < 8 \cdot [ABC] + 1
\]
then $A$, $B$, $C$ are three vertices of a square.  Here $|XY|$ is the
length of segment $XY$ and $[ABC]$ is the area of triangle $ABC$. 

\pagebreak

\item[B--1]
Find the minimum value of 
\[
\frac{(x+1/x)^{6}-(x^{6}+1/x^{6})-2}{(x+1/x)^{3}+(x^{3}+1/x^{3})}
\]
for $x>0$. 

\item[B--2]
Given a point $(a,b)$ with $0<b<a$, determine the minimum perimeter of a
triangle with one vertex at $(a,b)$, one on the $x$-axis, and one on the
line $y=x$.  You may assume that a triangle of minimum perimeter
exists. 

\item[B--3]
Let $H$ be the unit hemisphere $\{(x,y,z): x^{2}+y^{2}+z^{2}\leq 1,
z\geq 0 \}$, $C$ the unit circle $\{(x,y,0):x^{2}+y^{2}=1 \}$, and $P$ a
regular pentagon inscribed in $C$.  Determine the surface area of that
portion of $H$ lying over the planar region inside $P$, and write your
answer in the form $A \sin \alpha + B \cos \beta $, where $A$, $B$,
$\alpha $, and $\beta $ are real numbers. 

\item[B--4]
Find necessary and sufficient conditions on positive integers $m$ and
$n$ so that 
\[
\sum _{i=0}^{mn-1} (-1)^{\lfloor i/m \rfloor + \lfloor i/n \rfloor} =
0. 
\]

\item[B--5]
Let $N$ be the positive integer with 1998 decimal digits, all of them
1.  Find the thousandth digit after the decimal point of $\sqrt{N}$. 

\item[B--6]
Prove that, for any integers $a$, $b$, $c$, there exists a positive
integer $n$ such that $\sqrt{n^{3}+an^{2}+bn+c}$ is not an integer. 
\end{itemize}

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