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\begin{document}
\title{The Forty-Sixth Annual William Lowell Putnam Competition \\
Saturday, December 7, 1985}
\maketitle
\begin{itemize}
\item[A--1]
Determine, with proof, the number of ordered triples $(A_1, A_2, A_3)$
of sets which have the property that
\begin{enumerate}
\item[(i)] $A_1 \cup A_2 \cup A_3 = \{1,2,3,4,5,6,7,8,9,10\}$, and
\item[(ii)] $A_1 \cap A_2 \cap A_3 = \emptyset$.
\end{enumerate}
Express your answer in the form $2^a 3^b 5^c 7^d$, where $a,b,c,d$
are nonnegative integers.
\item[A--2]
Let $T$ be an acute triangle. Inscribe a rectangle $R$ in $T$ with one
side along a side of $T$. Then inscribe a rectangle $S$ in the triangle
formed by the side of $R$ opposite the side on the boundary of $T$,
and the other two sides of $T$, with one side along the side of
$R$. For any polygon $X$, let $A(X)$ denote the area of $X$. Find the
maximum value, or show that no maximum exists, of
$\frac{A(R)+A(S)}{A(T)}$, where $T$ ranges over all triangles and
$R,S$ over all rectangles as above.
\item[A--3]
Let $d$ be a real number. For each integer $m \geq 0$, define a
sequence $\{a_m(j)\}$, $j=0,1,2,\dots$ by the condition
\begin{align*}
a_m(0) &= d/2^m, \\
a_m(j+1) &= (a_m(j))^2 + 2a_m(j), \qquad j \geq 0.
\end{align*}
Evaluate $\lim_{n \to \infty} a_n(n)$.
\item[A--4]
Define a sequence $\{a_i\}$ by $a_1=3$ and $a_{i+1}=3^{a_i}$ for
$i\geq 1$. Which integers between 00 and 99 inclusive occur as the
last two digits in the decimal expansion of infinitely many $a_i$?
\item[A--5]
Let $I_m = \int_0^{2\pi} \cos(x)\cos(2x)\cdots \cos(mx)\,dx$. For
which integers $m$, $1 \leq m \leq 10$ is $I_m \neq 0$?
\item[A--6]
If $p(x)= a_0 + a_1 x + \cdots + a_m x^m$ is a polynomial with real
coefficients $a_i$, then set
\[
\Gamma(p(x)) = a_0^2 + a_1^2 + \cdots + a_m^2.
\]
Let $F(x) = 3x^2+7x+2$. Find, with proof, a polynomial $g(x)$ with
real coefficients such that
\begin{enumerate}
\item[(i)] $g(0)=1$, and
\item[(ii)] $\Gamma(f(x)^n) = \Gamma(g(x)^n)$
\end{enumerate}
for every integer $n \geq 1$.
\item[B--1]
Let $k$ be the smallest positive integer for which there exist
distinct integers $m_1, m_2, m_3, m_4, m_5$ such that the polynomial
\[
p(x) = (x-m_1)(x-m_2)(x-m_3)(x-m_4)(x-m_5)
\]
has exactly $k$ nonzero coefficients. Find, with proof, a set of
integers $m_1, m_2, m_3, m_4, m_5$ for which this minimum $k$ is achieved.
\item[B--2]
Define polynomials $f_n(x)$ for $n \geq 0$ by $f_0(x)=1$, $f_n(0)=0$
for $n \geq 1$, and
\[
\frac{d}{dx} f_{n+1}(x) = (n+1)f_n(x+1)
\]
for $n \geq 0$. Find, with proof, the explicit factorization of
$f_{100}(1)$ into powers of distinct primes.
\item[B--3]
Let
\[
\begin{array}{cccc} a_{1,1} & a_{1,2} & a_{1,3} & \dots \\
a_{2,1} & a_{2,2} & a_{2,3} & \dots \\
a_{3,1} & a_{3,2} & a_{3,3} & \dots \\
\vdots & \vdots & \vdots & \ddots
\end{array}
\]
be a doubly infinite array of positive integers, and suppose each
positive integer appears exactly eight times in the array. Prove that
$a_{m,n} > mn$ for some pair of positive integers $(m,n)$.
\item[B--4]
Let $C$ be the unit circle $x^2+y^2=1$. A point $p$ is chosen randomly
on the circumference $C$ and another point $q$ is chosen randomly from
the interior of $C$ (these points are chosen independently and
uniformly over their domains). Let $R$ be the rectangle with sides
parallel to the $x$ and $y$-axes with diagonal $pq$. What is the
probability that no point of $R$ lies outside of $C$?
\item[B--5]
Evaluate $\int_0^\infty t^{-1/2}e^{-1985(t+t^{-1})}\,dt$. You may
assume that $\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$.
\item[B--6]
Let $G$ be a finite set of real $n\times n$ matrices $\{M_i\}$, $1
\leq i \leq r$, which form a group under matrix
multiplication. Suppose that $\sum_{i=1}^r \mathrm{tr}(M_i)=0$, where
$\mathrm{tr}(A)$
denotes the trace of the matrix $A$. Prove that $\sum_{i=1}^r M_i$ is
the $n \times n$ zero matrix.
\end{itemize}
\end{document}