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# Geometry Warmups

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.

Evaluate

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

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

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

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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