%Format: AMS TeX
\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 1998 Missouri MAA Collegiate Mathematics Competition}
\vskip 6pt
\centerline{\bf Session I}
\bigskip
1. Let $P \ne (0,0)$ be a point on the parabola $y=x^2$. The normal
line to the parabola at $P$ will intersect the parabola at another point,
say $Q$. Find the coordinates of $P$ so that the length of segment $PQ$
is a minimum.
\bigskip
2. Let $0 < a_1 < a_2 < \cdots < a_n$ and let $e_i = \pm 1$. Prove that
$\sum_{i=1}^n e_i a_i$ assumes at least ${n+1 \choose 2}$ distinct values
as the $e_i$'s range over the $2^n$ possible combinations of signs.
\bigskip
3. If $m$ and $n$ are positive integers and $a < b$, find a formula for
$$\int_a^b {(b-x)^m \over m!} {(x-a)^n \over n!} dx$$
and use your formula to evaluate
$$\int_0^1 (1-x^2)^n dx .$$
\bigskip
4. Describe how to fold a rectangular sheet of paper so that the lower
right corner touches the left edge and the length of the crease is a
minimum. Discuss how the dimensions of the rectangle affect the result.
\bigskip
5. Let $n$ be a positive integer. Prove that there exists a number
divisible by $5^n$ that does not contain a single zero in its decimal
notation.
\vfill\eject
\centerline{\bf 1998 Missouri MAA Collegiate Mathematics Competition}
\vskip 6pt
\centerline{\bf Session II}
\bigskip
1. Let $I$ be the $n \times n$ identity matrix. Prove that $AB - BA \ne
I$ for any $n \times n$ matrices $A$ and $B$.
\bigskip
2. Let $a_1, \ldots , a_n$ be positive real numbers and let $s$ denote
their sum. Show that
$$(1+a_1) (1+a_2) \cdots (1+a_n) \le 1 + {s \over 1!} + {s^2 \over 2!} +
\cdots + {s^n \over n!} .$$
\bigskip
3. Sum the series
$$\sum_{i=1}^\infty {36 i^2 + 1 \over (36 i^2 - 1)^2 } .$$
\bigskip
4. Circle $O$ has a diameter of $3$; let $AOD$ be a diameter. A second
circle, of radius 1, is inscribed in circle $O$ so that its center lies
along $AOD$ and such that this circle is tangent to circle $O$ at point
$D$. A third circle, of radius $1/2$, is next inscribed in circle $O$ so
that its center also lies along $AOD$ and such that it is tangent to
circle $O$ at point $A$. Determine the radius of a fourth circle that
could be constructed inside circle $O$ and which would be simultaneously
tangent to all three circles.
\bigskip
5. A sequence of polynomials $\{ P_i (x) \} _{i=0}^\infty$ is defined by
the generating function
$${2 e^{tx} \over e^t + 1} = \sum_{i=0}^\infty P_i (x) {t^i \over i!} .$$
Show that 1 is a zero of $P_i (x)$ for all even $i>0$, and $1/2$ is a
zero of $P_i (x)$ for all odd $i>0$.
\bye