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permutation of of (1, 2,3,..., n) such that a_1+a2+... +a_k is divisible by k

Source: 2016 Saudi Arabia GMO TST level 4, I p3

August 1, 2020
number theorypermutationdivisible

Problem Statement

Find all positive integer nn such that there exists a permutation (a1,a2,...,an)(a_1, a_2,...,a_n) of (1,2,3,...,n)(1, 2,3,..., n) satisfying the condition: a1+a2+...+aka_1 + a_2 +... + a_k is divisible by kk for each k=1,2,3,...,nk = 1, 2,3,..., n.
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