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Part of 2015 Bosnia and Herzegovina Junior BMO TST

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Bosnia and Herzegovina JBMO TST 2015 Problem 4

Source: Bosnia and Herzegovina Junior Balkan Mathematical Olympiad TST 2015

9/16/2018
Let nn be a positive integer and let a1a_1, a2a_2,..., ana_n be positive integers from set {1,2,...,n}\{1, 2,..., n\} such that every number from this set occurs exactly once. Is it possible that numbers a1a_1, a1+a2,...,a1+a2+...+ana_1 + a_2 ,..., a_1 + a_2 + ... + a_n all have different remainders upon division by nn, if: a)a) n=7n=7 b)b) n=8n=8
combinatoricspositive integersumsremainder