 Solution

For any fixed $p>1/4$, the alternative-to-null likelihood ratio iswhich is strictly increasing in $S$ for $1/4<p<1$. Consequently the Neyman-Pearson lemma selects the same upper-tail ordering for every such alternative hypothesis. At the null hypothesis value,Thereforeis a uniformly most powerful test of the required exact test size. Its power function is $5p^4(1-p)+p^5=5p^4-4p^5$. At $p=1$ its statistical power is one, so including that boundary alternative preserves the conclusion.