\section*{Problem 3 - 141033}

Given a positive real number \(a\), where \(a \neq 1\).

Determine all real numbers \(x\) that satisfy the equation \(x^{\log _{a} x}=a^{2} x\)!
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For \(\log _{a} x\) to be defined, \(x\) must be positive. By applying \(\log _{a}\) to both sides, we obtain the equivalent equation \(\left(\log _{a} x\right)^{2}=2+\log _{a} x\).

This is a quadratic equation in \(\log _{a} x\) with solutions \(\log _{a} x=-1\) and \(\log _{a} x=2\).

Thus, \(x\) satisfies the equation if and only if \(x=\frac{1}{a}\) or \(x=a^{2}\).