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\begin{document}

\begin{center}
The Sixtieth Annual William Lowell Putnam Mathematical 
Competition \\
Saturday, December 4, 1999
\end{center}

\begin{itemize}
\item[A--1]
Find polynomials $f(x)$, $g(x)$, and $h(x)$, if they exist, such that,
for all $x$, 
\[
|f(x)|-|g(x)| + h(x) = 
\begin{cases}
-1 & \text{if } x < -1 \\
3x+2 & \text{if } -1\leq x\leq 0 \\
-2x+2 & \text{if } x>0. 
\end{cases} \]

\item[A--2]
Let $p(x)$ be a polynomial that is non-negative for all $x$.  Prove
that, for some $k$, there are polynomials $f_{1}(x),\dots ,f_{k}(x)$
such that 
\[
p(x) = \sum _{j=1}^{k} (f_{j}(x))^{2}. 
\]

\item[A--3]
Consider the power series expansion 
\[
\frac{1}{1-2x-x^{2}} = \sum _{n=0}^{\infty } a_{n}x^{n}. 
\]
Prove that, for each integer $n\geq 0$, there is an integer $m$ such
that 
\[
a_{n}^{2} +a_{n+1}^{2} = a_{m}. 
\]

\item[A--4]
Sum the series 
\[
\sum _{m=1}^{\infty } \sum _{n=1}^{\infty }
\frac{m^{2}n}{3^{m}(n3^{m}+m3^{n})}. 
\]

\item[A--5]
Prove that there is a constant $C$ such that, if $p(x)$ is a polynomial
of degree 1999, then 
\[
|p(0)| \leq C \int _{-1}^{1} |p(x)| \, dx.  
\]

\item[A--6]
The sequence $(a_{n})_{n\geq 1}$ is defined by $a_{1}=1$, $a_{2}=2$,
$a_{3}=24$, and, for $n\geq 4$, 
\[
a_{n} = \frac{6a_{n-1}^{2}a_{n-3}-8a_{n-1}a_{n-2}^{2}}{a_{n-2}a_{n-3}}. 
\]
Show that, for all $n$, $a_{n}$ is an integer multiple of $n$.  

\pagebreak

\item[B--1]
Right triangle $ABC$ has right angle at $C$ and $\angle BAC =\theta $;
the point $D$ is chosen on $AB$ so that $|AC|=|AD|=1$; the point $E$ is
chosen on $BC$ so that $\angle CDE = \theta $.  The perpendicular to
$BC$ at $E$ meets $AB$ at $F$.  Evaluate $\lim_{\theta \xrightarrow{}0}
|EF|$.  [Here, $|PQ|$ denotes the length of the line segment $PQ$.] 

\item[B--2]
Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=Q(x)P''(x)$,
where $Q(x)$ is a quadratic polynomial and $P''(x)$ is the second
derivative of $P(x)$.  Show that if $P(x)$ has at least two distinct
roots then it must have $n$ distinct roots.  [The roots may be either
real or complex.]

\item[B--3]
Let $A=\{(x,y):0\leq x,y <1 \}$.  For $(x,y)\in A$, let 
\[
S(x,y) = \sum _{\frac{1}{2}\leq \frac{m}{n}\leq 2} x^{m}y^{n},
\]
where the sum ranges over all pairs $(m,n)$ of positive integers
satisfying the indicated inequalities.  Evaluate the limit (for
$(x,y)\in A$) 
\[
\lim _{(x,y)\xrightarrow{}(1,1)} (1-xy^{2})(1-x^{2}y)S(x,y).  
\]

\item[B--4]
Let $f$ be a real function with continuous third derivative such that
$f(x)$, $f'(x)$, $f''(x)$, $f'''(x)$ are positive for all $x$.  Suppose
that $f'''(x)\leq f(x)$ for all $x$.  Show that $f'(x)<2f(x)$ for all
$x$.  

\item[B--5]
For an integer $n\geq 3$, let $\theta =2\pi /n$.  Evaluate the
determinant of the $n\times n$ matrix $I+A$, where $I$ is the $n\times
n$ identity matrix and $A=(a_{jk})$ has entries $a_{jk}=\cos (j\theta
+k\theta )$ for all $j,k$.  

\item[B--6]
Let $S$ be a finite set of integers, each greater than $1$.  Suppose
that for each integer $n$ there is some $s\in S$ such that $\gcd
(s,n)=1$ or $\gcd (s,n)=s$.  Show that there exist $s,t\in S$ such that
$\gcd (s,t)$ is prime.  [Here, $\gcd (a,b)$ denote the greatest common
divisor of $a$ and $b$.]

\end{itemize}

\end{document}