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Mathematics is filled with shapes that are kaleidoscopic in variety. Wielded since ancient times, the power of geometry helps us examine and measure these shapes. See more

Let \(ABC\) an equilateral triangle, \(P\) a point inside the triangle such as \(AP=5, BP=12\) and \(CP=13\). Find the area of \(\triangle ABC\).

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Evaluate

\[\frac{241}{2-\cot^{2} (9^{\circ})}+\frac{241}{2-\cot^{2} (27^{\circ})}+\frac{241}{2-\cot^{2} (45^{\circ})}\\ +\frac{241}{2-\cot^{2} (63^{\circ})}+\frac{241}{2-\cot^{2} (81^{\circ})} . \]

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\[ (x_1 - x_2)^2 + \left ( \sqrt{2-x_1 ^2} - \frac {9}{x_2} \right )^2 \]

What is the minimum value of the above expression where \(x_1 \in (0, \sqrt{2}) \) and \( x_2 \in R^+ \)

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A person is bored waiting in line. He draws 1000 congruent circles in the plane, all passing through a fixed point, P. What is the largest number of regions into which these circles
can split the plane? (Include the region outside the circles in your count)
###### adapted from the Mandlebrot competition

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In the interval \( [ 0, 2\pi ] \), how many solutions are there to

\[ \cos^2 x + \cos^2 2x + \cos^2 3x = 1? \]

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