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\title{The 59th William Lowell Putnam Mathematical Competition \\
Saturday, December 5, 1998}
\maketitle
\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 $\mathbb R^2$
whose union contains a set $E\subseteq \mathbb R^2$. Show that there
is a pairwise disjoint subcollection $D_1,\ldots, D_n$ in $\mathcal F$
such that
\[E\subseteq \cup_{j=1}^n 3D_j.\]
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 $\mathbb
R^2$. 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 $ABC$.
\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