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\hyphenation{Mass-achusetts Central Missouri Wis-con-sin Muthuvel}
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\centerline{\bf 2002 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 average of the
$y$-coordinates of $P$ and $Q$ is a minimum.
\bigskip
2. A tetrahedron is called {\it isosceles} if the members of each
pair of opposite edges are equal. This means, for tetrahedron
$ABCD$, that $AB = CD$, $BC = AD$, and $AC = BD$.
\medskip
\item{(a)} Prove that all four faces of an isosceles tetrahedron
are congruent.
\item{(b)} Prove that if all four faces of a tetrahedron have the
same perimeter, then the tetrahedron is isosceles.
\item{(c)} Prove that a tetrahedron is isosceles if and only if
the sum of the face angles at each vertex is $180^\circ$.
\bigskip
3. Let $\{ x_n \}$ be the following sequence involving alternating
square roots of 5 and 13:
$$x_1 = \sqrt 5,\ \ x_2 = \sqrt {5 + \sqrt {13}},\ \ x_3 = \sqrt {5 +
\sqrt {13 + \sqrt 5}},\ \ x_4 = \sqrt {5 + \sqrt {13 + \sqrt {5 +
\sqrt {13}}}},$$
and so on. Prove that $\lim_{n \to \infty} x_n$ exists and determine its value.
\bigskip
4. Does the set $X = \{ 1, 2, \ldots , 3000 \}$ contain a subset $A$
of 2000 integers in which no member of $A$ is twice another member of
$A$?
\bigskip
5. Two right circular cylinders of radius $r$ intersect at right
angles to form a solid. This solid has four curved faces. Imagine
one of these faces ``rolled out flat''. Find equations of the
boundary curves of this flattened face and also find its area.
\vfill\eject
\centerline{\bf 2002 Missouri MAA Collegiate Mathematics Competition}
\vskip 6pt
\centerline{\bf Session II}
\bigskip
1. Seven golf balls, labeled 1 through 7, are correctly placed in
corresponding boxes (one to a box), also labeled 1 through 7. The
balls are now removed and then randomly returned to the boxes, one
ball to a box. What is the probability that no ball will find its
correct box?
\bigskip
2.
\item{(a)} Prove that, for any positive integer $n$,
$$\sin n\theta = {n \choose 1} \sin \theta \cos ^{n-1} \theta - {n
\choose 3} \sin ^3 \theta \cos ^{n-3} \theta + {n \choose 5} \sin ^5
\theta \cos ^{n-5} \theta - \cdots$$
and
$$\cos n\theta = \cos ^n \theta - {n \choose 2} \sin ^2 \theta \cos
^{n-2} \theta + {n \choose 4} \sin ^4 \theta \cos ^{n-4} \theta -
\cdots .$$
\item{(b)} Prove that, for all $x$ in the interval $[-1,1]$ and for any positive integer $n$, the function
$$T_n (x) = \cos (n \cos ^{-1} x)$$
is a polynomial in $x$ of degree $n$ and leading coefficient $2^{n-1}$.
\bigskip
3. Suppose three equal circles, each of radius $r$, pass through a
common point $O$ and have three other pairwise intersections at $P_1$,
$P_2$, and $P_3$. Prove that the circle containing $P_1$, $P_2$, and
$P_3$ also has radius $r$.
\bigskip
4. $ABCDE$ is a regular pentagon of side $s$, and $P$ is any point in
the interior of $ABCDE$. Line segments are drawn from $P$
perpendicular to each of the five sides. Denote the sum of the
lengths of these five perpendiculars by $S$. Prove that $S$ is
independent of the location of $P$, and find $S$ in terms of $s$.
\bigskip
5. Let $p(n)$ denote the product of the (decimal) digits of the
positive integer $n$. Consider the sequences, beginning at any
arbitrary positive integer, in which succeeding terms are obtained by
adding to the previous term the product of its digits:
$$n_0 = n,\ \ \text{and for}\ \ r \ge 0,\ \ n_{r+1} = n_r + p(n_r).$$
Is there an initial integer $n$ for which the sequence continues to
increase indefinitely?
\bye