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\begin{enumerate}
 \item A lattice point is a point that has integer coordinates.  Thus, $(-7,41)$ is a lattice point on $35x+72y=2707$.  Find a lattice point in the 1st quadrant that lies on this line.
 
 \item Determine the length from $O$ to $P$ as the minimum length along the grid lines from $O$ to $P$.  In the diagram, the path length from $O$ to $P$ is 7.  Find the number of different points $P$ with integer coordinates that have a path length from $O$ of 37.
 
 \item If the whole number $60k$ has 60 factors, find the smallest value for $k$.
 
 \item For all $x$ and $y$, given that $2^x={16y \over 3}$ and $3^x = 27y$.
 
 \item Compute all real solutions to the equation $$\sqrt{x^2-x-12} - \sqrt{x^2-6x+8} = \sqrt{x-4}.$$
 
 \item There are four unequal, positive integers $a$, $b$, $c$, and $N$ such that $N=5a+3b+5c$.  It is also true that $N=4a+5b+4c$ and $N$ is between 131 and 150.  Find the value of $a+b+c$.
 
 \item In the diagram (not provided here), angle $P$ has a measure of 15 degrees.  Points $Q$, $R$, $S$, $T$, \ldots, alternate from one side of the angle to the other, each located farther away from point $P$ than the point before.  If $PQ=QR=RS=\cdots$, then find the maximum number of isosceles triangles with equal sides that can be formed.
 
 \item The digits 1, 2, 3, 4 can be arranged to form twenty-four different four-digit numbers.  If these 24 numbers are listed from smallest to largest, in what position is 3142?
 
 \item Compute the product $xyz$ if $x$, $y$, and $z$ are real and $$\matrix{x&+&y&+&z&=&6\cr
                                                                             x^2&+&y^2&+&z^2&=&14\cr
									     x^2& &   &+&z^2&=&10}$$
									     
 \item Consider the integers that are between 250 and 300 and that do not have a zero as a digit.  How many of these integers have three digits whose product is a perfect square?
 
 \item In triangle $ABC$, $AC=3$, $BC=4$, and $AB=5$.  A circle with center on side $AC$ is tangent to sides $AB$ and $BC$.  Find the area of the circle.
 
 \item Trapezoid $ABCD$ is inscribed in a circle with diameter $AB$.  Triangle $ABC$ has area 150 and triangle $ACD$ has area 120.  Find length $BC$.
 
 \end{enumerate}
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