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\begin{center}
2018 Missouri Collegiate Mathematics Competition \\ Session I
\end{center}

\noindent 1.  Let $f$ be a real-valued function such that for any real $x$,
\begin{itemize}
\item[(a)] $f(10+x) = f(10-x)$.
\item[(b)] $f(20+x) = -f(20-x)$.
\end{itemize}
Prove that $f$ is an odd and periodic function.
\ \\ \ \\
\noindent 2. Let $a>0$. Suppose $f$ is a continuous function on
$\left[  0,a\right]  $ such that $f\left(  x\right)  f\left(  a-x\right)  =1$.
Evaluate%
\[
\int_{0}^{a}\frac{dx}{1+f\left(  x\right)  }.
\]
\ \\ \ \\
\noindent 3. In a theoretical version of the Canadian lottery ``Lotto 6--49'', a ticket consists of six distinct integers chosen by the player from 1 to 49 (inclusive).  A $t$-prize is awarded for any ticket having $t$ or more numbers in common with a designated ``winning'' ticket.  Denote by $f(t)$ the smallest number of tickets required to be certain of winning a $t$-prize.  Clearly $f(1) = 8$ and $f(6) = {49 \choose 6}$.  Show that $f(2) \le 19$.
\ \\ \ \\
\noindent 4. Find all real numbers $x$ that satisfy $6^x + 1 = 8^x - 27^{x-1}$.  Prove your solution set contains all solutions.
\ \\ \ \\
\noindent 5. Let $P_1$, $P_2$, and $P_3$ be three points on the parabola $y = x^2$, and let $\ell_1$, $\ell_2$, and $\ell_3$ be the tangent lines to the parabola at these points.  The tangent lines intersect pairwise in three points.  Denote the intersection of $\ell_1$ and $\ell_2$ by $Q_{12}$, the intersection of $\ell_1$ and $\ell_3$ by $Q_{13}$, and the intersection of $\ell_2$ and $\ell_3$ by $Q_{23}$.  Find the ratio of the area of triangle $P_1P_2P_3$ to the area of triangle $Q_{12}Q_{13}Q_{23}$.
\vfill\eject
\begin{center}
2018 Missouri Collegiate Mathematics Competition \\ Session II
\end{center}

\noindent 1. For each positive integer $n$, let $a_n=\sum_{k=0}^n \frac{\pi^{4k}}{(4k+1)!}$ and $b_n=\sum_{k=0}^n\frac{\pi^{4k}}{(4k+3)!}$. Evaluate $\lim_{n\to \infty} \frac{a_n}{b_n}$.
\ \\ \ \\
\noindent 2. Triangle $ABC$ is a right triangle with right angle at $C$. Set $a=BC$, $%
b=CA $, and $c=AB$. Inscribe a circle inside triangle $ABC$, and inside the
circle inscribe a triangle similar to triangle $ABC$. Repeat the process,
obtaining an infinite sequence of circles and similar triangles. Find in
terms of $a$, $b$, $c$ the sum of the areas of the circles.
\ \\ \ \\
\noindent 3.  What is the coefficient of $x^{2018}$ in
\begin{align*}
&(x+1)^3(x^2+1)^4(x^4+1)^5 \\
&\ \ \ \ \ \times (x^8+1)(x^{16}+1)(x^{32}+1)(x^{64}+1)(x^{128}+1)(x^{256}+1)
(x^{512}+1)(x^{1024}+1)?
\end{align*}
\ \\ \ \\
\noindent 4. Let $P_{2}$ denote the set of real polynomials of
degree less than or equal to 2. Define the map $J:P_{2}\rightarrow\mathbb{R}$
by
\[
J\left(  f\right)  =\int_{0}^{1}\left[  f\left(  x\right)  \right]  ^{2}\,dx.
\]
Let $Q=\left\{  f\in P_{2}:f\left(  1\right)  =1\right\}  $. Show that $J$
attains a minimum value on $Q$, and determine where the minimum occurs.
\ \\ \ \\
\noindent 5. If $A$, $B$, $C$ are angles of an acute triangle, prove that
$$(\tan A + \tan B + \tan C)^2 \ge (\sec A + 1)^2 + (\sec B + 1)^2 + (\sec C + 1)^2 .$$

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