Problem
COM-B1-M05-P013 General Partial Sum Version
#13
★★★☆☆ Level 3 of 5
Prove that among any \(n\) integers, there is a nonempty consecutive block whose sum is divisible by \(n\).
Repeat the proof with \(n\) partial sums.
Consider \(s_1,\ldots,s_n\). If some \(s_i\equiv0\pmod n\), the first \(i\) numbers work. Otherwise all \(s_i\) have one of \(n-1\) nonzero residues. Two sums have the same residue; their difference is the sum of a nonempty consecutive block and is divisible by \(n\).
Key olympiad lemma.