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\hyphenation{Mass-achusetts Central Missouri Wis-con-sin Muthuvel}
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\centerline{\bf 2001 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 area bounded by the
normal line and the parabola is a minimum.
\bigskip
2. Let $\{ x_i \}$ denote any finite sequence with the
following properties:
\medskip
\item{(a)} $x_i \in \{ -2, 1, 2 \}$ for each $x_i$,
\item{(b)} $\sum_i x_i = 29$,
\item{(c)} $\sum_i x_i^2 = 59$.
\medskip
In considering the family of all such sequences, let $M = \max \{
\sum_i x_i^3 \}$ and $m = \min \{ \sum_i x_i^3 \}$. Determine $M/m$.
\bigskip
3. Let $a$, $b$, and $c$ be the sides of a triangle with perimeter 2.
Prove that
$$3/2 < a^2 + b^2 + c^2 + 2abc < 2.$$
\bigskip
4. Find the sum
$$S = \sum_{k=1}^\infty {k^2 \over 3^k} .$$
\bigskip
5. A set of five cubical dice has the following properties:
\medskip
\item{(a)} On each face of each die is a 3-digit integer. No two
integers on a given face are the same.
\item{(b)} Every integer has a nonzero hundred's digit.
\item{(c)} The sum, when the dice are rolled, of the five integers is a
4-digit integer.
\item{(d)} Whenever the dice are rolled, their sum $S$ can be found
quickly as follows: the sum of the unit's digits of the five dice is the
last two digits of $S$, and the first two digits of $S$ are 50 minus the
sum of the unit's digits.
\medskip
For example, if the dice come up 189, 256, 275, 845, and 168, the
sum of the unit's digits is $9+6+5+5+8=33$, so the value of $S$ is 1733,
since $50-33=17$.
Explain, justifying your statements, how such a set of dice can be
constructed.
\vfill\eject
\centerline{\bf 2001 Missouri MAA Collegiate Mathematics Competition}
\vskip 6pt
\centerline{\bf Session II}
\bigskip
1. Circle $B$ lies wholly in the interior of circle $A$. Find the
loci of points equidistant from the two circles?
\bigskip
2. Show that if $x$, $y$, and $z$ are positive reals such that
$x+y+z=1$, then
$$\biggl( {1 \over x} - 1 \biggr) \biggl( {1 \over y} - 1 \biggr)
\biggl( {1 \over z} - 1 \biggr) \ge 8.$$
\bigskip
3. A convex decagon and all of its diagonals are drawn. How many {\it
interior} points of intersection of the diagonals are there, if it is
assumed that no 3 diagonals share a common {\it interior} point?
\bigskip
4. No matter what $n$ real numbers $x_1$, $x_2$, $\ldots$, $x_n$ may be
selected in the closed unit interval $[0,1]$, prove that there always
exists a real number $x$ in this interval such that the average unsigned
distance from $x$ to the $x_i$'s is exactly $1/2$.
\bigskip
5. Consider the polynomial
$$f(x) = x^n + a_1 x^{n-1} + a_2 x^{n-2} + \cdots + a_{n-1} x + 1,$$
where $a_i \ge 0$. If the equation $f(x) = 0$ happens to have $n$ real roots, is it not
remarkable that the value of $f(2)$ must then be at least $3^n$? Prove
this unlikely consequence: $f(2) \ge 3^n$.
\bye