AIME I 2007 Problem 12 pt. 1

Let’s try something new. I’m going to split this into two parts, because 1) It’s 11:43 PM right now and I want to sleep, and 2) It’s a problem 12, so it’s probably going to be difficult. We’ll see.

Table of Contents:

Part 1

Part 2

Problem:

In isosceles triangle $ABC$, $A$ is located at the origin and $B$ is located at (20,0). Point $C$ is in the first quadrant with $AC = BC$ and angle $BAC = 75^{\circ}$. If triangle $ABC$ is rotated counterclockwise about point $A$ until the image of $C$ lies on the positive $y$-axis, the area of the region common to the original and the rotated triangle is in the form $p\sqrt{2} + q\sqrt{3} + r\sqrt{6} + s$, where $p,q,r,s$ are integers. Find $(p-q+r-s)/2$.

My Attempt:

I tried finding a lot of areas and side lengths, and ended up with one big glop. I’ll try anew tomorrow.

And the time is .

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