Let $a_1,\dots ,a_p$ be (not necessarily distinct) residues modulo $p$, such that there exists some $r$ so that if $S\subseteq [p]$ is non-empty and$$\sum _{i\in S}{a}_i\equiv 0(\mathrm{mod}p)$$then $|S|=r$. Must there be at most two distinct residues amongst the $a_i$?

AMS 11 · solved (literature)

theorem erdos_541 : answer(True) ↔ (∀ p, Fact p.Prime → ∀ (a : Fin p → ZMod p),
    (∃ r, ∀ (S : Finset (Fin p)), S ≠ ∅ → ∑ i ∈ S, a i = 0 → S.card = r) →
      (Set.range a).ncard ≤ 2)