# Problem 27

Algebra Level 5

$\left\{\begin{matrix} \frac{1}{\sqrt{2}}\leq z\leq \frac{1}{2} \min(x\sqrt{2},y\sqrt{3}) & & \\ x+z\sqrt{3}\geq \sqrt{6} & & \\ y\sqrt{3}+z\sqrt{10}\geq 2\sqrt{5} & & \end{matrix}\right.$

Let $$x,y,z$$ be positive real numbers satisfying the conditions above.

The maximum value of the expression $$Q=\frac{1}{x^2}+\frac{2}{y^2}+\frac{3}{z^2}$$ can be expressed as $$\large \frac{R}{S},$$ with equality achieved if and only if $$x = \sqrt{ \frac{A}{B} }, y = \sqrt { \frac{C}{D} } , z = \sqrt{ \frac{ E}{F} }$$, where $$R, S, A, B, C, D, E, F$$ are positive integers with $$\gcd (R, S) = \gcd(A, B) = \gcd(C,D) = \gcd(E, F) = 1$$.

Compute $$R+S+A+B+C+D+E+F$$.

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