Problem:
A unicorn is tethered by a -foot silver rope to the base of a magician's cylindrical tower whose radius is feet. The rope is attached to the tower at ground level and to the unicorn at a height of feet. The unicorn has pulled the rope taut, the end of the rope is feet from its nearest point on the tower, and the length of rope that is touching the tower is feet, where , and are positive integers, and is prime. Find .
Solution:
Suppose that the rope is attached to the ground at point , last touches the tower at point , and attaches to the unicorn at point . Let and be on the ground directly below and , respectively. Let be on the axis of the tower, and let be directly below so that the plane of is horizontal. Then is a radius of the tower, so , and, because is feet from the tower, is too, so . Also, is a right angle, so . If the tower wall were spread flat in the plane of , and , then right triangles and would be similar. Because
where is the rope's length between and and is the length of the projection of the rope onto the ground, . Then the length of rope touching the tower is . Thus .
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The problems on this page are the property of the MAA's American Mathematics Competitions