# Problem 27

Algebra Level 5

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

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

If the maximum value of the expression $$Q$$ can be expressed as $$\large \frac{L}{C}$$ and the equality achieved when $$\large x=\frac{\sqrt{H}}{K};y=\frac{\sqrt{A}}{N};z=\frac{\sqrt{M}}{S}$$ where $$L,C,H,K,A,N,M,S$$ are coprime integers and $$H,A,M$$ are free-square numbers and also $$gcd(L;C)=gcd(H;K)=gcd(A;N)=gcd(M;S)=1$$.

$$Q=\frac{1}{x^2}+\frac{2}{y^2}+\frac{3}{z^2}$$.

Compute $$L+C+H+K+A+N+M+S$$.

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