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Let $a_1,\ldots,a_p$ be (not necessarily distinct) residues modulo $p$, such that there exists some $r$ so that if $S\subseteq [p]$ is non-empty and \[\sum_{i\in S}a_i\equiv 0\pmod{p}\] then $\lvert S\rvert=r$. Must there be at most two distinct residues amongst the $a_i$?
A question of Graham.