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\begin{document}
\input{header} Team Competition \\ October 27, 2007
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\testitem Let $A$ and $C$ be points in the plane with $AC=1$. Find the area of the set of all points $B$ so that $\angle ABC\geq45^\circ$.
\testitem Fermat's little theorem states that whenever $p$ is prime and $a$ is not a multiple of $p$, $a^{p-1}$ leaves a remainder of $1$ when divided by $p$. Given that $21^5-1$ is not divisible by 11, what is the remainder when $21^5$ is divided by 11?
{\parpic[r]{\includegraphics[width=5cm]{images/doorway.eps}}\testitem A certain rectangular doorway has dimensions $7'\times3'$. A table has a square top that is $5'\times5'$, and it is connected via a $3\frac{1}{2}'$ pole protruding from the center of the square to a circular base that is $3'$ in diameter. A picture of the doorway and table are shown to the right with the table turned on its side. Determine whether it is possible to pass the table through the doorway, and either explain how to do it or prove that it is impossible. Assume that the walls, the table top, the pole, and the base are all negligibly thin.
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\testitem Given that $x_1$, $x_2$, $x_3$, $\ldots$, $x_{100}$ are all integers and that
\[
x_1x_2x_3\cdots x_{99}x_{100}=-1,
\]
how many different possibilities are there for the ordered 100-tuple $(x_1,x_2,\ldots x_{100})$? Express your answer as $p^n$, where $p$ is a prime number and $n$ is a positive integer.
{\parpic[r]{\includegraphics[width=2.6cm]{images/squares.eps}} \testitem A square is inscribed in a unit circle, and four smaller squares are constructed that have two vertices on a side of the larger square and the other two vertices on the circle. Find the area of one of these smaller squares.\picskip{0}}
\testitem Five very small marks are made on the surface of a solid, opaque cube. Prove that no matter how the marks are made, there must be a perspective from which one may view the cube so that at least three of the marks are visible.
\testitem
Find all real solutions of the equation \[(3x-63)(x-47)=x^5-21x^4+13x-273.\]
\testitem Forty-five distinct playing cards are placed in a rectangular array of five columns and nine rows. An observer chooses a card secretly and tells the dealer which of the five columns the card is in. The dealer then puts the cards into a stack, picking up one column at a time. The column specified by the observer is picked up third, and the cards from each column are picked up in order from top to bottom. The cards are again dealt in the rectangular array, the first row being dealt in order from left to right, then the second row, and so on. The observer again specifies the column that contains his card, and the dealer again picks up the columns one at a time, the column specified being picked up third. This process is repeated one more time. Prove that the observer's card will be the twenty-third card in the deck.
\testitem In terms of $n$, find a formula for the determinant of an $n\times n$ matrix whose diagonal entries are all 0 and the rest of whose entries are 1.
\testitem Prove that \[\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{99}+\frac{1}{100}>\pi.\]
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