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Problem 1222

AIME late
Geometry Difficulty 5.2 Prove it Philippine Mathematical Olympiad · Philippines

A vertical pole has a cross section of a regular hexagon with a side length of 11 foot. If a dog on a leash is tied onto one vertex, and the leash is 33 feet long, determine the total area that the dog can walk on.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

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Official solution

Solution:
The total area is indicated by the lightly shaded region below.

Figure 1

240360π(3)2+2(60360)π(2)2+2(60360)π(1)2=23(9π)+13(4π)+13(π)=233π \begin{aligned} & \frac{240}{360} \pi (3)^2 + 2\left(\frac{60}{360}\right) \pi (2)^2 + 2\left(\frac{60}{360}\right) \pi (1)^2 \\ = & \frac{2}{3}(9\pi) + \frac{1}{3}(4\pi) + \frac{1}{3}(\pi) = \frac{23}{3} \pi \end{aligned}

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