2014 AIME I Problems/Problem 10
A disk with radius is externally tangent to a disk with radius . Let be the point where the disks are tangent, be the center of the smaller disk, and be the center of the larger disk. While the larger disk remains fixed, the smaller disk is allowed to roll along the outside of the larger disk until the smaller disk has turned through an angle of . That is, if the center of the smaller disk has moved to the point , and the point on the smaller disk that began at has now moved to point , then is parallel to . Then , where and are relatively prime positive integers. Find .
Let be the new tangency point of the two disks. The smaller disk rolled along minor arc on the larger disk. Let , in radians. The smaller disk must then have rolled along an arc of length , since it has a radius of . Since all of the points on major arc on the smaller disk have come into contact with the larger disk at some point during the rolling, and none of the other points on the smaller disk did, the length of major arc equals the length of minor arc , or . Since , , so the angles of minor arc and minor arc are equal, so minor arc has an angle of . Since the smaller disk has a radius of , the length of minor arc is . This means that equals the circumference of the smaller disk, so , or .
Now, to find , we construct . Also, drop a perpendicular from to , and call this point . Since , is right, and , and . Now drop a perpendicular from to , and call this point . Since , , and . Thus, we know that , and by using the Pythagorean Theorem on , we get that . Thus, , so , and our answer is .
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