1997 JBMO Problems/Problem 4
Determine the triangle with sides and circumradius for which .
Solving for yields . We can substitute into the area formula to get We also know that , where is the angle between sides and Substituting this yields Since is inside a triangle, . Substitution yields Note that , so multiplying both sides by that value would not change the inequality sign. This means However, by the AM-GM Inequality, . Thus, the equality case must hold, so where . When plugging , the inequality holds, so the value truly satisfies all conditions.
That means so That means the only truangle that satisfies all the conditions is a 45-45-90 triangle where is the longest side. In other words, for all positive
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