\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 2005 Missouri Collegiate Mathematics Competition}
\vskip 6pt
\centerline{\bf Session I}
\bigskip
1. Find the point in the first quadrant on the graph of $y = 7 - x^2$ such that the distance between the $x$- and $y$- intercepts of the tangent line at the point is minimum.
\bigskip
2. Let $M(n,k)$ be the number of mappings from a set $X$ of $n$ distinct objects onto a set $Y$ of $k$ distinct objects, and let $P(n,k)$ denote the number of partitions of a set of $n$ distinct objects into $k$ nonempty subsets. Determine the relationship between $M(n,k)$ and $P(n,k)$ and use it to show that $M(n,k)$ is a multiple of $24$ whenever $k>3$.
\bigskip
3. Suppose that $f$ is a polynomial of positive degree $n$ with integer coefficients. Prove that there are infinitely many integers $x$ for which $f(x)$ is composite. (Here, composite means those integers, positive or negative, whose absolute value is not $1$ or a prime; thus, $-4$ and $6$ are composite, while $1$ and $-2$ are not.)
\bigskip
4. Determine the value of the integral
$$I( \theta ) = \int_{-1}^1 {\sin \theta \, dx \over 1 - 2x \cos \theta + x^2} ,$$
and locate those points $0 \le \theta \le 2\pi$, where $I( \theta )$ is discontinuous.
\bigskip
5. Prove that in the MacLaurin series for $\tan \theta$, $-\pi / 2 < \theta < \pi /2$, every coefficient is non-negative.
\vfill\eject
\centerline{\bf 2005 Missouri Collegiate Mathematics Competition}
\vskip 6pt
\centerline{\bf Session II}
\bigskip
1. If $a < b < c$, $f' (x)$ is strictly increasing on $(a,c)$, and $f(x)$ is continuous on $[a,c]$, then show that
$$(b-a) f(c) + (c-b) f(a) > (c-a) f(b) .$$
\bigskip
2. Find all integer solutions $(x,y)$ to the equation $xy = 5x + 11y$.
\bigskip
\comment
3. Define a circle of radius $r$ and center $P$ on a sphere to be the locus of points on the surface of the sphere that are a distance $r$ from $P$, where distance is the usual Euclidean distance in $\R ^3$, assuming that $r$ is less than the diameter of the sphere (otherwise, the locus is empty). Thus, a circle divides the sphere into two spherical segments. The area of the circle is the surface area of the segment containing the center point, $P$. Show that this area is $\pi r^2$.
\endcomment
3. Define a circle of radius $r$ and center $P$ on a sphere to be the locus of points on the surface of the sphere that are a distance $r$ from $P$, where distance is the usual Euclidean distance in $\R ^3$. When $r$ is less than the diameter of the sphere, this circle divides the sphere into two spherical segments, or ``caps''. Show that the area of the cap containing the center point $P$ is $\pi r^2$.
\bigskip
4. Let $p > 2$ be a prime. Prove or disprove that all prime divisors of $2^p - 1$ have the form $2kp + 1$.
\bigskip
5. Suppose that $f \colon [0, \infty ) \to [0, \infty )$ is a differentiable function with the property that the area under the curve $y=f(x)$ from $x=a$ to $x=b$ is equal to the arclength of the curve $y=f(x)$ from $x=a$ to $x=b$. Given that $f(0) = 5/4$, and that $f(x)$ has a minimum value on the interval $(0, \infty )$, find that minimum value.
\bye