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\begin{document}
\title{The 55th William Lowell Putnam Mathematical Competition \\
Saturday, December 3, 1994}
\maketitle
\begin{itemize}
\item[A--1]
Suppose that a sequence $a_1, a_2, a_3, \dots$ satisfies
$0 < a_n \leq a_{2n} + a_{2n+1}$ for all $n \geq 1$. Prove that the series
$\sum_{n=1}^{\infty} a_n$ diverges.
\item[A--2]
Let $A$ be the area of the region in the first quadrant bounded by the
line $y = \frac{1}{2} x$, the $x$-axis, and the ellipse $\frac{1}{9} x^2
+ y^2 = 1$. Find the positive number $m$ such that $A$ is equal to the
area of the region in the first quadrant bounded by the line $y = mx$,
the $y$-axis, and the ellipse $\frac{1}{9} x^2 + y^2 = 1$.
\item[A--3]
Show that if the points of an isosceles right triangle of side length
1 are each colored with one of four colors, then there must be two points
of the same color whch are at least a distance $2 - \sqrt{2}$ apart.
\item[A--4]
Let $A$ and $B$ be $2 \times 2$ matrices with integer entries such
that $A, A+B, A+2B, A+3B$, and $A+4B$ are all invertible matrices whose
inverses have integer entries. Show that $A+5B$ is invertible and that
its inverse has integer entries.
\item[A--5]
Let $(r_n)_{n \geq 0}$ be a sequence of positive real numbers such that
$\lim_{n \to \infty} r_n = 0$. Let $S$ be the set of numbers representable
as a sum
\[
r_{i_1} + r_{i_2} + \cdots + r_{i_{1994}},
\]
with $i_1 < i_2 < \cdots < i_{1994}$. Show that every nonempty interval
$(a,b)$ contains a nonempty subinterval $(c,d)$ that does not intersect $S$.
\item[A--6]
Let $f_1, \dots, f_{10}$ be bijections of the set of integers such that for
each integer $n$, there is some composition $f_{i_1} \circ f_{i_2}
\circ \cdots \circ f_{i_m}$ of these functions (allowing repetitions)
which maps 0 to $n$. Consider the set of 1024 functions
\[
\mathcal{F} = \{f_1^{e_1} \circ f_2^{e_2} \circ \cdots \circ f_{10}^{e_{10}}\},
\]
$e_i = 0$ or 1 for $1 \leq i \leq 10$. ($f_i^0$ is the identity function
and $f_i^1 = f_i$.) Show that if $A$ is any nonempty finite set of
integers, then at most 512 of the functions in $\mathcal{F}$ map $A$ to
itself.
\item[B--1]
Find all positive integers $n$ that are within 250 of exactly 15 perfect
squares.
\item[B--2]
For which real numbers $c$ is there a straight line that intersects the curve
\[ x^4 + 9x^3 + cx^2 + 9x + 4
\]
in four distinct points?
\item[B--3]
Find the set of all real numbers $k$ with the following property: For any
positive, differentiable function $f$ that satisfies $f'(x) > f(x)$
for all $x$, there is some number $N$ such that
$f(x) > e^{kx}$ for all $x > N$.
\item[B--4]
For $n \geq 1$, let $d_n$ be the greatest common divisor of the entries of
$A^n - I$, where
\[
A = \begin{pmatrix} 3 & 2 \\ 4 & 3 \end{pmatrix}
\quad \mbox{ and } \quad
I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}.
\]
Show that $\lim_{n \to \infty} d_n = \infty$.
\item[B--5]
For any real number $\alpha$, define the function $f_{\alpha}(x)
= \lfloor \alpha x \rfloor$. Let $n$ be a positive integer. Show that
there exists an $\alpha$ such that for $1 \leq k \leq n$,
\[
f_\alpha^k(n^2) = n^2 - k = f_{\alpha^k}(n^2).
\]
\item[B--6]
For any integer $n$, set
\[
n_a = 101a - 100\cdot 2^a.
\]
Show that for $0 \leq a,b,c,d \leq 99$, $n_a + n_b \equiv
n_c + n_d \pmod{10100}$ implies $\{a,b\} = \{c,d\}$.
\end{itemize}
\end{document}