Answer is No.

Counterexample

My problem! A few years ago I revisited 2013 IMOSL C5 and thought it was natural to generalize $r$ to a function of $m$ and $s$ which resulted in this problem. (Some other students mentioned similarities to 2020 Putnam B6.) My original construction $a_n=\lfloor \phi (n+1) \rfloor - \lfloor \phi n\rfloor$, where $phi$ is the golden ratio is essentially the same to #3. This is motivated from the fact that these are the easiest non-periodic sequences one can come up with. Hopefully these work, and this is slightly sharper then the above construction in the sense that $\phi$ is more 'badly approximable'. For a sketch of the solution, note that $a_m+a_{m+1}+\cdots+a_{m+s}= \lfloor \phi (m+s+1) \rfloor - \lfloor \phi m \rfloor \in \{\lfloor \phi (s+1) \rfloor, \lfloor \phi (s+1) \rfloor +1\}$. Now from $\{ \phi s\}>\frac{1}{\sqrt5 s}$, it follows that $a_m+a_{m+1}+\cdots+a_{m+s}=a_n+a_{n+1}+\cdots+a_{n+s}$ for some $m<n\le m+5(s+1)$. Simpler solutions took $\{a_{2i-1},a_{2i}\}=\{0,1\}$ and carefully controlled the $i$s for which $a_{2i}=1$. I don't think it's easy to guess the answer, but after you're determined to find a counterexample it shouldn't be too hard. Remark. Trying $r(m,s)=O(m)$ (this is easier than $O(s)$ imho) or other functions is also fun