Playing with the Glome

Algebra Level 5

\[2xy+4yz+6zw\]

Let \(M\) be the maximum of the expression above if \(x^2+y^2+z^2+w^2=1\), where \(x,y,z\) and \(w\) are real numbers. Write \(M=\sqrt{p}+\sqrt{q}\) for two positive integers \(p\) and \(q\), and enter \(p+q\).

Bonus question: For \(axy+byz+czw\), with the same constraint, express the maximum \(M\) in terms of \(a,b\) and \(c\).

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