Prove that a sequence of $11$ numbers always contains six numbers summing up to a multiple of $6$.
This is a problem from a selection to IMO 2014.
2026-03-30 15:15:54.1774883754
Prove that a sequence of $11$ numbers always contains six numbers summing up to a multiple of $6$.
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This is a special case of the Erdos Ginzburg Ziv theorem (namely with $n=6$).
The theorem in general says that if you take $2n-1$ integers, then some subset of size $n$ must have sum that is a multiple of $n$. You can find several proofs here.