\input amstex
\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 2000 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 distance between 
the $x$-coordinates of $P$ and $Q$ is a minimum. 
\bigskip
2.  If $xyz = (1-x)(1-y)(1-z)$ where $0 \le x, y, z \le 1$, show that
$$x(1-z) + y(1-x) + z(1-y) \ge {3 \over 4} .$$   
\bigskip
3.  Let $n \ge 3$ points be given in the plane.  Prove that three of 
them form an angle which is at most $\pi / n$.  
\bigskip
4.  Justify as far as you can, the equality
$$\int_0^1 x^x dx = 1 - {1 \over 2^2} + {1 \over 3^3} - {1 \over 4^4} + 
{1 \over 5^5} - \cdots .$$ 
\bigskip
5.  Show that a polynomial in $x$ with real coefficients which takes 
rational values for rational arguments and (real) irrational values for 
(real) irrational arguments must be linear. 
\vfill\eject
\centerline{\bf 2000 Missouri MAA Collegiate Mathematics Competition}
\vskip 6pt
\centerline{\bf Session II}
\bigskip
1.  Two points are chosen at random (with a uniform distribution) from 
the unit interval $[0,1]$.  What is the probability that the points will 
be within a distance of $\epsilon$ of each other? 
\bigskip
2.  Write Pascal's Triangle as an infinite array as follows:
$$\matrix 1&1&1&1&1&\cdots \cr
          1&2&3&4&5&\cdots \cr
          1&3&6&10&15&\cdots \cr
          1&4&10&20&35&\cdots \cr
          1&5&15&35&70&\cdots \cr
          \vdots&\vdots&\vdots&\vdots&\vdots&\ddots \cr \endmatrix $$
where the first row and first column consist entirely of ones and every 
other entry is formed by taking the sum of the entry to the left and the 
entry above.  For each positive integer $n$, let $D_n$ denote the $n$ by 
$n$ matrix formed by the first $n$ rows and first $n$ columns of the 
array.  What is the determinant of $D_n$?  Prove your answer.
\bigskip
3.  Given an $n \times n$ checkerboard with the four corners removed, 
characterize for which $n$ this deleted board can be covered with $3 
\times 1$ rectangles.
\bigskip
4.  For $n \ge 2$, let $x_1, \ldots , x_n$ be non-zero real numbers whose 
sum is zero.  Show that there are $i,j$ with $1 \le i < j \le n$ such 
that
$$1/2 \le \vert x_i / x_j \vert \le 2 .$$ 
\bigskip
5.  This problem concerns sequences $x_1 x_2 \cdots x_n$ in which each 
$x_i$ is either $a$, $b$, or $c$.  Determine the number of those
sequences which have length $n$, begin and end with the letter $a$, and
in which adjacent terms are always different letters. 
\bye