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\hyphenation{Mass-achusetts Central Missouri Wis-con-sin Muthuvel}
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\centerline{\bf 1997 Missouri MAA Collegiate Mathematics Competition}
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\centerline{\bf Session I}
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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 $y$-coordinate of $Q$
is a minimum.
\bigskip
2. Prove that from any row of $n$ integers one may always select a block
of adjacent integers whose sum is divisible by $n$.
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3. Find conditions on the parameters $a$, $b$, $c$, and $d$ so that
$$f(x,y) = a \sin (x+y) + b \cos (x+y) + c \sin (x-y) + d \cos (x-y)$$
can be written as $f(x,y) = g(x) h(y)$.
\bigskip
4. A point $P$ is in the interior of a circle of radius $r$. Place
the vertex of a right angle at $P$ and denote by $A$ and $B$ the
points where the sides of the right angle intersect the circle. Let
$Q$ be the point which completes the rectangle $PAQB$. What is the
locus of $Q$?
\bigskip
5. Let $\{ L_n \}_{n=0}^\infty $ be the sequence of Lucas numbers:
$L_0 = 2$, $L_1 = 1$, $L_n = L_{n-1} + L_{n-2}$ for $n \ge 2$. Let
$DR(N)$ denote the digital root of a positive integer $N$, defined as
the sum of the digits of $N$, composed enough times until a value
between 1 and 9 is obtained. For example, $DR(667) = DR(19) = DR(10)
= 1$. Show that there is a smallest positive integer $k$ such that
$DR(L_{n+k}) = DR(L_n)$ for all integers $n \ge 0$.
\vfill\eject
\centerline{\bf 1997 Missouri MAA Collegiate Mathematics Competition}
\vskip 6pt
\centerline{\bf Session II}
1. Find positive integers $n$ and $a_1, a_2, \ldots , a_n$ such that
$$a_1 + a_2 + \cdots + a_n = 1997$$
and the product $a_1 a_2 \cdots a_n$ is as large as possible.
\bigskip
2. Let $a$, $b$, $c$, $d$ be positive numbers with $abcd = 1$. Prove that
$$a^2 + b^2 + c^2 + d^2 + ab + ac + ad + bc + bd + cd \ge 10.$$
\bigskip
3. A wooden cube of edge 3 is formed by gluing together 27 small cubes
of edge 1. A termite, beginning with any one of the outer small
cubes, begins to eat its way through the large cube, always moving
perpendicular to a face (i.e., no diagonal movements are allowed -
don't ask why, who knows the mind of a termite?) Is it possible for
the termite to follow a path entirely within the large cube (emerging
and crawling on the outside is also not allowed) which passes through
each small cube exactly once and ends in the center cube? Generalize
the problem to the case where the large cube has edge $n$, an odd
integer.
\bigskip
4. Define a family of curves by
$$S_n = \{ (x,y) : y = {1 \over n} \sin (n^2 x),\ 0 \le x \le \pi \},$$
where $n$ is a positive integer. What is the limit of the length of
$S_n$ as $n \to \infty$?
\bigskip
5. Consider the infinite sequences $\{ x_n \}$ of positive real
numbers with the following properties:
$$x_0 = 1,\ \ \text{and for all } i \ge 0,\ \ x_{i+1} \le x_i .$$
(a) Prove that for every such sequence, there is an $n \ge 1$ such
that
$${x_0^2 \over x_1} + {x_1^2 \over x_2} + \cdots + {x_{n-1}^2 \over
x_n} \ge 3.999 .$$
(b) Find such a sequence for which
$${x_0^2 \over x_1} + {x_1^2 \over x_2} + \cdots + {x_{n-1}^2 \over
x_n} < 4\ \ \text{for all } n.$$
\bye