= Solution
The joint <likelihood ratio> in favor of the alternative is
$$
\frac{L_3}{L_2}=\left(\frac32\right)^n\prod_i x_i.
$$
It 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 $Z_i=-\log X_i$ gives
$$
\mathbb P_0(Z_i>z)=\mathbb P_0(X_i<e^{-z})=e^{-2z},\qquad z\geq0.
$$
Therefore the $Z_i$ are <independent> <exponential random variables> of rate two, $S\sim\operatorname{Gamma}(n,\text{rate }2)$, and $4S\sim\chi_{2n}^2$. An exact size-$\alpha$ <critical region> is
$$
\boxed{S\leq c_\alpha},\qquad\boxed{F_{\chi_{2n}^2}(4c_\alpha)=\alpha}.
$$
Equivalently, reject when the <geometric mean> satisfies $g\geq e^{-c_\alpha/n}$. The boundary distribution is continuous, so no randomization is needed here.
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