Problem:
Circle with radius has diameter . Circle is internally tangent to circle at . Circle is internally tangent to circle , externally tangent to circle , and tangent to . The radius of circle is three times the radius of circle and can be written in the form , where and are positive integers. Find .
Solution:
Let circles , and have centers , and , respectively. Let circle be tangent to at . Let circle have radius , and circle have radius . Then , and . The Pythagorean Theorem applied to gives and to gives . Because , it follows that . Squaring and simplifying twice reduces this equation to , which has solutions . Thus , and the requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions