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 \(R+S+A+B+C+D+E+F\).


Not an original problem.This had been used for practice before IMO a few years back.
This problem is in this Set.
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