Let's call a subsequence of $5$ consecutive letters a group. Let the score of a group be the number of letters $A$ in it. We need to find a group with a score of $3$. Divide this sequence into $n$ groups. We know that the sum of scores of these $n$ groups is $3n$. If one of these groups has a score of $3$, then we are done. Assume that none of the groups have a score of $3$. Then, there is a group, say $G_1$, with a score greater than $3$ and a group, say $G_2$, with a score less than $3$. Now, see that the difference of scores of consecutive groups (groups that contains exactly four common letters) is $-1, 0$ or $1$. Group $G_1$ has a score greater than $3$ and group $G_2$ has a score less than $3$. Hence, between $G_1$ and $G_2$, there must be a group with a score of $3$.