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quadratic permutations

Source: Iran TST 2002 (aka: iranian olympiad/3'rd round/2002)

December 29, 2003
quadraticsinductiongroup theorycombinatorics proposedcombinatorics

Problem Statement

We call a permutation (a1,a2,...,an) \left(a_1, a_2, ..., a_n\right) of (1,2,...,n) \left(1, 2, ..., n\right) quadratic if there exists at least a perfect square among the numbers a1 a_1, a_1 \plus{} a_2, ... ..., a_1 \plus{} a_2 \plus{} ... \plus{} a_n. Find all natural numbers n n such that all permutations in Sn S_n are quadratic. Remark. Sn S_{n} denotes the n n-th symmetric group, the group of permutations on n n elements.
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