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USAMO 2002 Problem 2

Source:

September 30, 2005
AMCUSA(J)MOUSAMOgeometryinradiusinequalities

Problem Statement

Let ABCABC be a triangle such that (cot⁡A2)2+(2cot⁡B2)2+(3cot⁡C2)2=(6s7r)2, \left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2, where ss and rr denote its semiperimeter and its inradius, respectively. Prove that triangle ABCABC is similar to a triangle TT whose side lengths are all positive integers with no common divisors and determine these integers.
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