= Solution
The <Neyman-Pearson lemma> states that a test which rejects where the joint <likelihood ratio> $L_1/L_0$ is largest is a <most powerful test> of its size: its power is at least that of any test with no larger size. More precisely, choose a threshold $k$ and reject if $L_1/L_0>k$, accept if it is smaller, and if necessary randomize on equality to attain the specified size.
For an independent and identically distributed <statistical sample>, take
$$
Y_i=\log\frac{f_0(X_i)}{f_1(X_i)}.
$$
The $Y_i$ are <independent and identically distributed random variables> under either simple hypothesis, and the log joint ratio is their sum. Thus the nonrandomized threshold form can be written
$$
\boxed{\sum_{i=1}^nY_i\leq c}
$$
with $c=-\log k$, using either strict or non-strict inequality when the boundary has probability zero. The usual finite-log expression assumes positive common support; zero densities can instead be handled by the extended likelihood ratio on the relevant supports. Continuous observations alone do not ensure a continuous likelihood-ratio statistic: ties may require the boundary randomization in the lemma. This qualification preserves the exact-size formulation.
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