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You have $ R=\frac{a}{2sinA}=\frac{b}{2sinB}=\frac{c}{2sinC}=\frac{P}{2(sinA+sinB+sinC)}$ and $K=\frac{abc}{4R}$
Let $MNO$ be a triangle such that $R, P$ and $K$ are the circumradius, perimeter and the area of the triangle. The maximum possible value of $\frac {KP}{R^3}$ can be written as $\frac {a}{b}$, where $a, b$ are coprimes. Find $a+b$.
You have $ R=\frac{a}{2sinA}=\frac{b}{2sinB}=\frac{c}{2sinC}=\frac{P}{2(sinA+sinB+sinC)}$ and $K=\frac{abc}{4R}$