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Partial Sums of Sequence

Source: 2013 CMO #2

March 31, 2013
modular arithmeticnumber theory unsolvednumber theory

Problem Statement

The sequence a1,a2,…,ana_1, a_2, \dots, a_n consists of the numbers 1,2,…,n1, 2, \dots, n in some order. For which positive integers nn is it possible that the n+1n+1 numbers 0,a1,a1+a2,a1+a2+a3,…,a1+a2+⋯+an0, a_1, a_1+a_2, a_1+a_2+a_3,\dots, a_1 + a_2 +\cdots + a_n all have di fferent remainders when divided by n+1n + 1?
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