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\hyphenation{Mass-achusetts Central Missouri Wis-con-sin Muthuvel}
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\centerline{\bf 2007 Missouri Collegiate Mathematics Competition}
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\centerline{\bf Session I}
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1. Seemingly unimportant changes in the terms of a series can have a dramatic effect on the nature of the sum. Thus, the series
$$\sum_{n=2}^\infty {1 \over n^2}$$
sums to a transcendental number, but if the $n^2$ is replaced by $n^2 + 3n - 4$, the new series sums to a rational number. Find it.
\bigskip
2. A bitwin chain of length one consists of two pairs of twin primes with the property that they are related by being of the form:
$$(n-1, n+1)\ \ \text{and}\ \ (2n-1, 2n+1) .$$ Prove or disprove that $30 \ \vert \ n$ for $n \ge 7$.
\bigskip
3. Let $f$ be the complex function
$$f(z) = e^{z^2} .$$
Find the set of all points $z$ in the complex plane for which $f(z)$ is a real number. Give a geometric description of this set.
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4. A triangle is Pythagorean if it is a right triangle and the lengths of all of its sides are integers. Suppose that $\triangle ABC$ is Pythagorean; for concreteness assume that the lengths of the three sides satisfy $c > a > b$. The median and the altitude are now drawn from $C$ to the hypotenuse, where they meet the latter at $P$, $Q$, respectively. Determine simple conditions upon $a$, $b$, $c$ so that $\triangle CQP$ will also be Pythagorean.
\vfil\eject
5. Associated with the Fibonacci numbers $\{ F_n \}_{n=1}^\infty$, are the Fibonacci polynomials, $\{ U_n (x) \} _{n=1}^\infty$ ,
$$U_n (x) = \cases
1, &\text{$n=1$;}\\
x, &\text{$n=2$;}\\
x U_{n-1} (x) + U_{n-2} (x), &\text{$n>2$}.
\endcases$$
\medskip
\item{(a)} Find a function $F(x,y)$ such that (formally)
$$F(x,y) = \sum_{n=1}^\infty U_n (x) y^n .$$
Such a function is called a generating function for the $U_n (x)$'s.
\item{(b)} Establish that for each $n$, $U_n (1) = F_n$.
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\centerline{\bf 2007 Missouri Collegiate Mathematics Competition}
\vskip 6pt
\centerline{\bf Session II}
\bigskip
1. Suppose that
$${2x+3 \over x^2 - 2x + 2}$$
has the Taylor series
$$\sum_{k=0}^\infty a_k x^k .$$
Find
$$\sum_{k=0}^\infty a_k .$$
\bigskip
2. Find the point on a given line such that the sum of its distances from two fixed points is a minimum. Assume the two fixed points and the given line are in the same plane.
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3. Assume the point $(h,k)$ lies ``outside'' the circle $x^2 + y^2 = r^2$, $0 < r < \sqrt {h^2 + k^2}$. Find an expression for the $x$-intercepts of the tangent lines to the circle from the point $(h,k)$ in terms of $h$, $k$, and $r$.
\bigskip
4. Let
$$S = \{ 5a + 503b\ :\ \text{$a$ and $b$ are nonnegative integers} \} .$$
What is the largest integer which does NOT belong to $S$?
\bigskip
5. Does there exist a real valued continuous function $f$ with domain the set of all real numbers (usual topology on the domain and range) such that if $x$ is rational then $f(x)$ is irrational and if $x$ is irrational then $f(x)$ is rational? Prove your answer.
\bye