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Existence of triangle satisfies trigonometric conditions

Source: Vietnam TST 1985 Problem 3

February 5, 2010
trigonometrygeometrygeometry proposed

Problem Statement

Does there exist a triangle ABC ABC satisfying the following two conditions: (a) sin⁡2A+sin⁡2B+sin⁡2C=cot⁡A+cot⁡B+cot⁡C{ \sin^2A + \sin^2B + \sin^2C = \cot A + \cot B + \cot C} (b) S≥a2−(b−c)2 S\ge a^2 - (b - c)^2 where S S is the area of the triangle ABC ABC.
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