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\hyphenation{Mass-achusetts Central Missouri Wis-con-sin Muthuvel}
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\centerline{\bf 1996 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 area bounded by the
normal line and the parabola is a minimum.
\bigskip
2. If
$$\eqalign{
&u = 1 + {x^3 \over 3!} + {x^6 \over 6!} + \cdots ,\cr
&v = {x \over 1!} + {x^4 \over 4!} + {x^7 \over 7!} + \cdots ,\cr
&w = {x^2 \over 2!} + {x^5 \over 5!} + {x^8 \over 8!} + \cdots ,\cr}$$
prove that
$$u^3 + v^3 + w^3 - 3uvw = 1.$$
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3. Each of the numbers $x_1, x_2, \ldots , x_n$ can be 1, 0, or -1.
What is the minimum possible value of the sum of all products of pairs of
these numbers?
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4. A $6 \times 6$ board is tiled with $2 \times 1$
dominos. Prove that the board can be cut into two parts by a straight
line that does not cut dominos.
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5. Let $a, b, c, d, e$ be integers such that $1 \le a < b < c < d < e$.
Prove that
$${1 \over [a,b]} + {1 \over [b,c]} + {1 \over [c,d]} + {1 \over [d,e]}
\le {15 \over 16},$$
where $[m,n]$ denotes the least common multiple of $m$ and $n$ (e.g.
$[4,6] = 12$).
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\centerline{\bf Session II}
1. Evaluate the definite integrals
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(a) $$\int_1^3 {dx \over \sqrt {(x-1)(3-x)}} ,$$
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(b) $$\int_1^\infty {dx \over e^{x+1} + e^{3-x}} .$$
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2. Let $x, y, z$ be three different integers. Prove that
$$(x-y)^5 + (y-z)^5 + (z-x)^5$$
is divisible by $5(x-y)(y-z)(z-x)$.
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3. What is the probability of an odd number of sixes turning up in a
random toss of $n$ fair dice?
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4. A swimmer stands at one corner of a square swimming pool and wishes to
reach the diagonally opposite corner. If $w$ is the swimmer's walking
speed and $s$ is the swimmer's swimming speed ($s \sqrt 2$.]
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5. In a finite sequence of real numbers, the sum of any seven successive
terms is negative and the sum of any eleven successive terms is positive.
Determine the maximum number of terms in the sequence.
\bye