\section*{Problem 18}

Find the minimum value of \(x y / z+y z / x+z x / y\) for positive reals \(x, y, z\) with \(x^{2}+y^{2}+z^{2}=1\).


\section*{Solution}

Answer: \(\min \sqrt{3}\) when all equal.

Let us consider \(z\) to be fixed and focus on \(x\) and \(y\). Put \(f(x, y, z)=x y / z+y z / x+z x / y\). We have \(f(x, y, z)=p / z+z\left(1-z^{2}\right) / p=\left(p+k^{2} / p\right) / z\), where \(p=x y\), and \(k=z \sqrt{ }\left(1-z^{2}\right)\). Now \(p\) can take any value in the range \(0\sqrt{ } 3\).

If \(z>1 / \sqrt{5}\), then the minimum of \(f(x, y, z)\) occurs at \(x=y\) and is \(x^{2} / z+z+z=\left(1-z^{2}\right) /(2 z)+2 z\) \(=3 z / 2+1 /(2 z)=(\sqrt{3}) / 2(z \sqrt{3}+1 /(z \sqrt{3}) \geq \sqrt{3}\) with equality at \(z=1 / \sqrt{3}(\) and hence \(x=y=1 / \sqrt{3}\) also).

