2023 SSMO Relay Round 3 Problems/Problem 2

Problem

Let $T=$ TNYWR. In triangle $ABC$ with circumradius and inradius having lengths $R$ and $r,$ respectively. Given that $\sin\angle{A}+\sin\angle{B}+\sin\angle{C}=\left\{\sqrt{N}\right\}$ the maximum value of $8\sin\angle{A}\sin\angle{B}\sin\angle{C}$ is $b+c\sqrt{a},$ for squarefree $a,$ find $|a+b+c|.$ (Note that $\left\{x\right\} = x - \lfloor x \rfloor$ )

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2023 SSMO Relay Round 3 Problems/Problem 2

Problem

Solution