2025 SSMO Accuracy Round Problems/Problem 7

Problem

There is a unique ordered triple of positive reals $(a,b,c)$ satisfying the system of equations $\begin{align*} a^2 + 9 &= (b-8\sqrt{3})^2 + 4 \\ b^2 + 4 &= (c-8\sqrt{3})^2 + 49 \\ c^2 + 49 &= (a-8\sqrt{3})^2 + 9. \end{align*}$ The value of $100a+10b+c$ can be expressed as $m\sqrt{n},$ where $m$ and $n$ are positive integers such that $n$ is square-free. Find $m+n$ .

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2025 SSMO Accuracy Round Problems/Problem 7

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