Geometry
# Geometry Warmups

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 \( x_2 \in R^+ \)

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)

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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