\input amstex
\input epsfx
\hyphenation{Mass-achusetts Central Missouri Wis-con-sin Muthuvel}
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\def\R{\Bbb R}
\def\Z{\Bbb Z}
\def\Q{\Bbb Q}
\def\C{\Bbb C}
\def\N{\Bbb N}
\tolerance=1600
\hsize=33pc
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\centerline{\bf 2006 Missouri Collegiate Mathematics Competition}
\vskip 6pt
\centerline{\bf Session I}
\bigskip
1. Find
$$\lim_{n \to \infty} \biggl( {n \over n^2 + 1} + {n \over n^2 + 4} + {n \over n^2 + 9} + \cdots + {1 \over 2n} \biggr) .$$
\bigskip
2. You have a supply of unit squares, $1 \times 2$ rectangles and $1 \times 3$ rectangles. With these you wish to ``tile'' a rectangular strip of dimensions $1 \times n$ ($n$ is a positive integer). For example, if $W_n$ is the number of ways a $1 \times n$ strip can be tiled, then $W_3 = 4$ since
$$3 = 2+1 = 1+2 = 1+1+1 .$$
\comment
$$\epsfxsize=15pc \epsfbox{tiling.eps}$$
\endcomment
Determine how many tiling patterns $W_n$ exist when $n=16$ and prove your answer.
\bigskip
3. Let $\{ a_n \}$ be the sequence defined recursively by
$$a_0 = 1,\ a_1 = 0,\ \text{and}\ a_n = {1 \over n} a_{n-2} \ \ \text{for}\ \ n \ge 2 .$$
If the function $f$ is defined by
$$f(x) = \sum_{n=0}^\infty a_n x^n ,$$
find the exact value of $f(2)$.
\bigskip
4. Find all positive integers $c$ such that $n(n+c)$ is never a perfect square for any positive integer $n$.
\bigskip
5. Let $f(t)$ and $f'(t)$ be differentiable on $[a,x]$ and for each $x$ suppose there is a number $c_x$ such that $a < c_x < x$ and
$$\int_a^x f(t) \, dt = f(c_x) (x-a) .$$
Assume that $f' (a) \ne 0$. Then prove that
$$\lim_{x \to a} {c_x - a \over x - a} = {1 \over 2} .$$
\vfill\eject
\centerline{\bf 2006 Missouri Collegiate Mathematics Competition}
\vskip 6pt
\centerline{\bf Session II}
\bigskip
1. The array below is called a magic square because the sum of the three numbers along any row, any column, or the two diagonals, is the same (namely, $15$).
$$\vbox{\tabskip=0pt \offinterlineskip
\def\tablerule{\noalign{\hrule}}
\halign to125pt{\strut#& \vrule#\tabskip=1em plus2em& \hfil#\hfil& \vrule#& \hfil#\hfil& \vrule#& \hfil#\hfil& \vrule#\tabskip=0pt\cr\tablerule
&&\omit&&\omit&&\omit&\cr
&&\hfil$8$\hfil&&\hfil$1$\hfil&&\hfil$6$\hfil&\cr
&&\omit&&\omit&&\omit&\cr
\tablerule
&&\omit&&\omit&&\omit&\cr
&&\hfil$3$\hfil&&\hfil$5$\hfil&&\hfil$7$\hfil&\cr
&&\omit&&\omit&&\omit&\cr
\tablerule
&&\omit&&\omit&&\omit&\cr
&&\hfil$4$\hfil&&\hfil$9$\hfil&&\hfil$2$\hfil&\cr
&&\omit&&\omit&&\omit&\cr
\tablerule \cr}}$$
\item{(a)} Construct a $3 \times 3$ multi-magic square, that is, a $3 \times 3$ array of $9$ distinct integers such that the PRODUCT of the three numbers along any row, any column, or the two diagonals, is the same.
\item{(b)} Show that no multi-magic square can be constructed with nine {\it consecutive} integers.
\bigskip
2. Evaluate the limit:
$$\lim_{n \to \infty} {1 \over n^4} \sum_{k=3}^n {k \choose 3} .$$
\bigskip
3. Define a sequence of positive integers $\{ x_n \ \vert \ n=1, 2, 3, \ldots \}$ to be {\it Dence} if every positive integer can be expressed as a sum of distinct members of the sequence. Now consider the sequence in which $x_1 = 1$, $x_2 = 2$, $x_3 = 4$, $x_4 = 7$, $x_5 = 15$, and $x_{k+1} = 2x_k - 7$ for all $k \ge 5$. Prove that this sequence is Dence.
\bigskip
4. For a convex polygon with $n$ sides, let $T$ be a point in the interior of the polygon. Triangulate the polygon by drawing line segments from $T$ to each vertex. Denote the distance from $T$ to side $s_i$ of the polygon by $r_i$ and the area of the corresponding triangle by $A_i$. Let $A$ be the total area of the polygon. Show that the number $r$, defined by
$${1 \over r} = \sum_{i=1}^n \biggl( {A_i \over A} \biggr) \biggl( {1 \over r_i} \biggr) ,$$
is independent of the position (inside the polygon) of $T$.
\bigskip
5. Let
$$\{ a_n \}_{n=1}^\infty \ \ \text{and}\ \ \{ \Delta a_n \} _{n=1}^\infty = \{ a_n - a_{n+1} \} _{n=1}^\infty $$
be two decreasing sequences of positive numbers that converge to $0$. Prove that the magnitude of the error, $\vert R_n \vert$, made in approximating the sum of the series
$$\sum_{k=1}^\infty (-1)^{k-1} a_k$$
by its $n$ partial sum is bounded as follows:
$${a_{n+1} \over 2} < \vert R_n \vert < {a_n \over 2} .$$
\bye