2005 Canadian MO Problems/Problem 4

Problem

Let $ABC$ be a triangle with circumradius $R$ , perimeter $P$ and area $K$ . Determine the maximum value of $KP/R^3$ .

Solution Outline

Use the formula $K=\dfrac{abc}{4R}$ to get $KP/R^3=\dfrac{abc(a+b+c)}{4R^4}$ . Then use the extended sine law to get something in terms of sines, and use AM-GM and Jensen's to finish. (Jensen's is used for $\sin A+\sin B+\sin C \le \dfrac{3\sqrt3}{2}$ .

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2005 Canadian MO Problems/Problem 4

Problem

Solution Outline