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\begin{center}
2014 Missouri Collegiate Mathematics Competition \\ Session I
\end{center}

\noindent 1.  Let $0 \le a \le 1$ be given.  Determine all nonnegative continuous functions
$f$ on $\lbrack 0,1 \rbrack$ (or prove there are none) which satisfy the following three
conditions.
\begin{align*}
\int_0^1 f(x) \, dx &= 1 \\
\int_0^1 xf(x) \, dx &= a \\
\int_0^1 x^2 f(x) \, dx &= a^2 .
\end{align*}
\ \\ \ \\
\noindent 2. Let $T_0$ be an isosceles triangle with base $b$ and base angles $\alpha$.  Define
a sequence $\{ T_n \} _{n=0}^\infty$ of triangles, recursively, as follows.  The base of each
triangle $T_n$ is $b$, and the base angles of triangle $T_{n+1}$ have half the measure of the base
angles of triangle $T_n$.  Evaluate
$$\lim_{n \to \infty} \frac{2^n \cdot \textrm{area} (T_n)}{\textrm{area} (T_0)} .$$
\ \\ \ \\
\noindent 3. Find the sum of the series $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{6}
+ \frac{1}{8} + \frac{1}{9} + \frac{1}{12} + \cdots$, where the terms are the reciprocals of
the positive integers whose only prime factors are twos and threes.
\ \\ \ \\
\noindent 4. Let $Q$ be an arbitrary convex quadrilateral with vertices $A_0$, $A_1$, $A_2$, and
$A_3$ ordered counterclockwise.  Let $M_i$ be the midpoint of side $A_i A_{i+1}$ interpreting the
subscripts mod 4.  Then draw the medians $A_i M_{i+2}$ (again interpreting the subscripts mod 4).
Prove that the quadrilaterals $A_0 M_0 A_2 M_2$ and $A_1 M_1 A_3 M_3$ have equal areas.
\ \\ \ \\
\noindent 5. For which positive integers $a$ does there exist a right triangle with integer
sides, at least one of which is $a$?
\vfill\eject
\begin{center}
2014 Missouri Collegiate Mathematics Competition \\ Session II
\end{center}

\noindent 1. 101 marbles are numbered from 1 to 101.  The marbles are divided between two baskets A and B.
The marble numbered 40 is in basket A.  This marble is removed from basket A and put in basket B.  The
average of all the
numbers on the marbles in A increases by $\frac{1}{4}$.  The average of all the numbers on the marbles
in B increases by $\frac{1}{4}$ too.  How many marbles were there originally in basket A?
\ \\ \ \\
\noindent 2. If $f(x) = \sin ^6 \left( \frac{x}{4} \right) + \cos ^6 \left( \frac{x}{4} \right)$,
find $f^{(2014)} (0)$.  (Note that $f^{(n)} (x)$ refers to the $n$th derivative of $f$ evaluated
at $x$.)
\ \\ \ \\
\noindent 3. A polynomial $P(x)$ is known to be of the form $P(x) = x^{15} - 9x^{14} + \cdots - 7$,
where the ellipsis ($\cdots$) represent unknown intermediate terms.  It is also known that all
roots of $P(x)$ are integers.  Find the roots (including multiplicities) of $P(x)$.
\ \\ \ \\
\noindent 4. Find all real solutions to the equation $4x^2 - 40 \lfloor x \rfloor + 51 = 0$.  (Note
that $\lfloor x \rfloor$ denotes the floor function, the greatest integer less than or equal to
$x$.)
\ \\ \ \\
\noindent 5. Given the fact that $\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt {\pi}$,
evaluate the integral
$$I = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x^2 + (y-x)^2 + y^2)} \, dx \, dy .$$

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