Question

A hexagon that is inscribed in a circle has side lengths $22,$ $22,$ $20,$ $22,$ $22,$ and $20$ in that order. The radius of the circle can be written as $p + \sqrt{q},$ where $p$ and $q$ are positive integers. Find $p + q.$

Answer

Answer: 272

Concepts:chord trigonometry law of cosines quadratic

Difficulty rating: 2560

Solution:

Half a $20$ -chord gives $\sin\frac{\beta}{2} = \frac{10}{r},$ and the law of cosines on the isosceles triangle with legs $r$ and base $22$ gives $22^2 = 2r^2(1 - \cos\alpha),$ so $\cos\alpha = 1 - \frac{242}{r^2}.$ Equating, $\begin{aligned} 1 - \frac{242}{r^2} &= \frac{10}{r} \\ &\quad\Longrightarrow\quad r^2 - 10r - 242 \\ &= 0, \end{aligned}$ so $r = 5 + \sqrt{267}$ (taking the positive root).

Therefore $p + q = 5 + 267 = 272.$

2013 AIME II Problem 8

Answer: 272

Difficulty rating: 2560

Solution:

Problem 8 in Other Years