 Solution

The joint likelihood ratio in favor of the alternative isIt is increasing in the product, so the Neyman-Pearson lemma rejects for large product, equivalently small $S=-{\sum }_i\log x_i$. Under $H_0$, the transformation $Z_i=-\log X_i$ givesTherefore the $Z_i$ are independent exponential random variables of rate two, $S\sim \mathrm{Gamma}(n,\text{rate }2)$, and $4S\sim {\chi }_{2n}^2$. An exact size-$\alpha$ critical region isEquivalently, reject when the geometric mean satisfies $g\ge e^{-c_{\alpha }/n}$. The boundary distribution is continuous, so no randomization is needed here.