# 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 \(L+C+H+K+A+N+M+S\).

###### Not an original problem.This had been used for practice before IMO in a few year back.

###### This problem is in this Set.

###### Don't asked me why I put \(H,K,A\)....

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