Maximum sum power of cubic

Algebra Level 5

\[ \large{ \begin{cases} a+b+c+d+e+f=6 \\ a^2+b^2+c^2+d^2+e^2+f^2=\frac{36}5 \end{cases}} \]

Let \(a,b,c,d,e\) and \(f\) be positive real numbers such that the system of equations above are fulfilled. If the maximum value of

\[ a^3+b^3+c^3+d^3+e^3 + f^3 \]

can be expressed as \( \dfrac xy\) for coprime positive integers \(x\) and \(y\), find the value of \(\sqrt{x+y} \).


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