 Solution

For any fixed $p>1/4$ below 1, the ratio to the null isIt increases with $y$. Its ordering is therefore the same for every such alternative, while the size is determined only by the null. The very same rule, reject when $Y\ge 4$, is most powerful against each alternative and hence is a uniformly most powerful test. At $p=1$ its power is 1, also optimal.

More generally, for $0<p_0<1$, the binomial likelihood ratio against any $p>p_0$ increases with $y$. Choose an integer cutoff $c$ so thatand reject for $Y>c$, randomize with probabilitywhen $Y=c$, and accept below. This gives exact size $\alpha$ and is a uniformly most powerful test for every positive integer $n$. Exact prescribed size may require this boundary randomization; a deterministic cutoff alone need not attain it. For $p_0=0$, reject all $Y>0$ and reject at $Y=0$ with probability $\alpha$, which is likewise optimal. At $p_0=1$ the stated one-sided alternative is empty. The cutoff generally depends on $p_0,n,\alpha$; there is no claim that one fixed cutoff serves every null parameter.