\input amstex
\input epsfx
\hyphenation{Mass-achusetts Central Missouri Wis-con-sin Muthuvel}
\raggedbottom
\loadmsam
\loadmsbm
\loadeufm
\def\R{\Bbb R}
\def\Z{\Bbb Z}
\def\Q{\Bbb Q}
\def\C{\Bbb C}
\def\N{\Bbb N}
\tolerance=1600
\hsize=33pc
\vsize=42pc
\baselineskip=12pt
\lineskip=12pt
\centerline{\bf 2010 Missouri Collegiate Mathematics Competition}
\vskip 6pt
\centerline{\bf Session I}
\bigskip
1. For the parabola having equation $y = -x^2$ let $a<0$ and $b>0$ with points $P : (a,-a^2)$
and $Q : (b,-b^2)$. Let $M$ be the midpoint of $PQ$ and let $R$ be the intersection of the
vertical line through $M$ with the parabola. Show that the area of the region bounded
by the parabola and the line segment $PQ$ is ${4 \over 3}$ of the area of triangle $PQR$.
(Archimedes, 3rd century B.C.)
\bigskip
2. Suppose that the length of the larger base of an isosceles trapezoid equals the length of a
diagonal and
the length of the smaller base equals the altitude. Find the ratio of the length of the
larger base to the length of the smaller base.
\bigskip
3. Consider the Diophantine equation $x(2x^2 + 3x + 3) = y^3 -1$. Prove that it
does not have a solution $(x,y)$ in positive integers, or find such a solution if it does.
\bigskip
4. The Fibonacci and Lucas numbers are sequences defined by the following initial
conditions and second order recurrence relations. Let $F_0 = 0$, $F_1 = 1$, and for
$n \ge 2$ define $F_n = F_{n-1} + F_{n-2}$; let $L_0 = 2$, $L_1 = 1$, and for $n \ge 2$
define $L_n = L_{n-1} + L_{n-2}$.
Let $\alpha = (1 + \sqrt 5)/2$ and $\beta = (1-\sqrt 5)/2$. The Binet form of the
Fibonacci and Lucas numbers are given by
$$F_n = {\alpha^n - \beta^n \over \sqrt 5} \ \ \text{and}\ \ L_n = \alpha^n + \beta^n ,$$
where $n$ is any nonnegative integer.
Let $k$ be a fixed positive integer and define
$$U_n = {F_{kn} \over L_k} ,$$
where $n$ is any nonnegative integer. Find $U_0$ and $U_1$ and find a second order
recurrence relation involving $L_k$
satisfied by the sequence $\{ U_n \}_{n=0}^\infty$.
\medskip
5. If your calculator is set to radian mode, and you enter any number and then repeatedly
push the ``cosine'' button, the displayed value will converge to $0.73908 \ldots$.
Call this number $d$. If $f(x) = x - \cos x$, then $f(d) = 0$. The number $d$ can be
expressed as a series in odd powers of $\pi$:
$$d = \sum_{n=0}^\infty a_n \pi^{2n+1} .$$
Find $a_0$ and $a_1$.
\vfill\eject
\centerline{\bf 2010 Missouri Collegiate Mathematics Competition}
\vskip 6pt
\centerline{\bf Session II}
\bigskip
1. Let $S$ be the collection of ordered pairs $(x,y)$ in $[0,1] \times [0,1]$ such that
either $x$ or $y$ is irrational. Prove or disprove that for any two distinct ordered pairs
in $S$, we can find a path in $S$ connecting the two points.
\bigskip
2. If $a, b, c$ are positive real numbers, find the value of $x$ that minimizes the function
$$f(x) = \sqrt {a^2 + x^2} + \sqrt {(b-x)^2 + c^2} .$$
(Hint: Think geometrically.)
\bigskip
3. A sequence of $2 \times 2$ matrices, $\{ M_n \}_{n=1}^\infty$, is defined as follows:
$$M_n = \pmatrix
m_{11} = {1 \over (2n+1)!} & m_{12} = {1 \over (2n+2)!} \\
m_{21} = \sum_{k=0}^n {(2n+2)! \over (2k+2)!} & m_{22} = \sum_{k=0}^n {(2n+1)! \over (2k+1)!}
\endpmatrix .$$ For each $n$, let $\det M_n$ denote the determinant of $M_n$. Determine the
value of
$$\lim_{n \to \infty} \det M_n .$$
\bigskip
4. Evaluate the integral
$$I = \int_{1 \over 2}^2 {\ln x \over 1+x^2} dx .$$
\bigskip
5. A function $f$ has the following properties:
\medskip
\item{(a)} For all $x \ge 1$, $f(x)$ is a positive, differentiable, decreasing function;
\item{(b)} Whenever $x$ equals a natural number $k$, we set $f(k) = f_k$, an element of a
numerical sequence;
\item{(c)} The series $\sum_{k=1}^\infty f_k$ diverges to $\infty$;
\item{(d)} $F$ is an arbitrary antiderivative of $f$, but with a fixed constant of integration,
and is defined for all $x \ge 1$;
\item{(e)} $\lim_{x \to \infty} F(x) = \infty$.
\medskip
Prove that
$$\lim_{n \to \infty} \left( \sum_{k=1}^n f_k - F(n) \right)$$
is finite.
\bye