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\hyphenation{Mass-achusetts Central Missouri Wis-con-sin Muthuvel}
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\centerline{\bf 2004 Missouri 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 $x$-axis at a point, say $Q$. Let $O=(0,0)$ and form the triangle $OPQ$. Let $OQ$ be the base of this triangle. Find the minimum ratio of the length of the base of $\triangle OPQ$ to its height.
\comment
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 area under the parabola between $P$ and $Q$ is a minimum.
\endcomment
\bigskip
2. The numbers $\pm 1, \pm 2, \ldots \pm 2004$ are written on a blackboard. You decide to pick two numbers $x$ and $y$ at random, erase them, and write their product, $xy$, on the board. You continue this process until only one number remains. Prove that the last number is positive.
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3. A chess position possesses the following property: On every vertical column and on every horizontal row, there is an odd number of pieces. Prove that there is an even number of pieces on black squares.
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4. At a point $P$ on the curve
$$\biggl( {x \over a} \biggr) ^{2/3} + \biggl( {y \over b} \biggr) ^{2/3} = 1 ,$$
the tangent to the curve meets the $x$-axis at $(h,0)$ and the $y$-axis at $(0,k)$. As $P$ moves on the given curve, find the locus of points $Q(h,k)$.
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5. Let $a$, $b$, $c$, and $d$ be integers. Suppose that each of the three quadratics $ax^2 + bx +c$, $ax^2 + bx + (c+d)$, and $ax^2 + bx + (c+2d)$ factors over the integers, i.e. has rational roots. Let $S = ad>0$. Show that $S$ represents the area of some Pythagorean triangle (integer-sided right triangle).
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\centerline{\bf 2004 Missouri Collegiate Mathematics Competition}
\vskip 6pt
\centerline{\bf Session II}
\bigskip
1. Suppose $f$ is a continuous real-valued function on the interval $[0,1]$. Show that
$$\int _0^1 x^2 f(x) \, dx = {1 \over 3} f( \xi )$$
for some $\xi \in [0,1]$.
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2. Prove that $2(3n-1)^n \ge (3n+1)^n$ for all nonnegative integers $n$.
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3. Equations of two ellipses $E_1$ and $E_2$ are
$${x^2 \over a^2} + {y^2 \over b^2} - {2x \over c} = 0 \text{ and } {x^2 \over b^2} + {y^2 \over a^2} + {2x \over c} = 0 ,$$
respectively. $AB$ is a common tangent, meeting $E_1$ at $A$ and $E_2$ at $B$. Prove that when $A$ and $B$ are joined to the origin $O$, angle $AOB$ is a right angle.
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4. For each positive integer $n$, let $s(n)$ denote the sum of the digits of $n$ (when $n$ is written in base 10). Prove that for every positive integer $n$
$$s(2n) \le 2s(n) \le 10 s(2n) .$$
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5. The Fibonacci numbers are defined by $F_0 = 0$, $F_1 = 1$, and $F_n = F_{n-1} + F_{n-2}$, if $n \ge 2$. Use the Fibonacci numbers to express the number $K_n$ of $n$-tuples $(x_1, x_2, \ldots , x_n)$ of 0's, 1's, and 2's such that 0 is never followed by 1.
\bye