\documentclass[amssymb,twocolumn,pra,10pt,aps]{revtex4-1}
\usepackage{mathptmx,amsmath, multirow}
\begin{document}
\title{The 78th William Lowell Putnam Mathematical Competition \\
Saturday, December 2, 2017}
\maketitle
\begin{itemize}
\item[A1]
Let $S$ be the smallest set of positive integers such that
\begin{enumerate}
\item[(a)]
$2$ is in $S$,
\item[(b)]
$n$ is in $S$ whenever $n^2$ is in $S$, and
\item[(c)]
$(n+5)^2$ is in $S$ whenever $n$ is in $S$.
\end{enumerate}
Which positive integers are not in $S$?
(The set $S$ is ``smallest'' in the sense that $S$ is contained in any other such set.)
\item[A2]
Let $Q_0(x) = 1$, $Q_1(x) = x$, and
\[
Q_n(x) = \frac{(Q_{n-1}(x))^2 - 1}{Q_{n-2}(x)}
\]
for all $n \geq 2$. Show that, whenever $n$ is a positive integer, $Q_n(x)$ is equal to a polynomial with integer coefficients.
\item[A3]
Let $a$ and $b$ be real numbers with $a**1$. Considering all positive integers $N$ with this property,
what is the smallest positive integer $a$ that occurs in any of these expressions?
\item[B3]
Suppose that $f(x) = \sum_{i=0}^\infty c_i x^i$ is a power series for which each coefficient $c_i$ is $0$ or $1$.
Show that if $f(2/3) = 3/2$, then $f(1/2)$ must be irrational.
\item[B4]
Evaluate the sum
\begin{gather*}
\sum_{k=0}^\infty \left( 3 \cdot \frac{\ln(4k+2)}{4k+2} - \frac{\ln(4k+3)}{4k+3} - \frac{\ln(4k+4)}{4k+4} - \frac{\ln(4k+5)}{4k+5} \right) \\
= 3 \cdot \frac{\ln 2}{2} - \frac{\ln 3}{3} - \frac{\ln 4}{4} - \frac{\ln 5}{5}
+ 3 \cdot \frac{\ln 6}{6} - \frac{\ln 7}{7} \\ - \frac{\ln 8}{8} - \frac{\ln 9}{9}
+ 3 \cdot \frac{\ln 10}{10} - \cdots .
\end{gather*}
(As usual, $\ln x$ denotes the natural logarithm of $x$.)
\item[B5]
A line in the plane of a triangle $T$ is called an \emph{equalizer} if it divides $T$ into two regions having equal area and equal perimeter. Find positive integers $a>b>c$, with $a$ as small as possible, such that there exists a triangle with side lengths $a, b, c$ that has exactly two distinct equalizers.
\item[B6]
Find the number of ordered $64$-tuples $(x_0,x_1,\dots,x_{63})$ such that $x_0,x_1,\dots,x_{63}$ are distinct elements of $\{1,2,\dots,2017\}$ and
\[
x_0 + x_1 + 2x_2 + 3x_3 + \cdots + 63 x_{63}
\]
is divisible by 2017.
\end{itemize}
\end{document}
**