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(a_1 + b_1, a_2 +b_2,..., a_n + b_n) consecutive naturals for permutations

Source: 2023 Austrian Regional Competition For Advanced Students p3

March 26, 2024
combinatoricspermutations

Problem Statement

Determine all natural numbers n≥2n \ge 2 with the property that there are two permutations (a1,a2,...,an)(a_1, a_2,... , a_n) and (b1,b2,...,bn)(b_1, b_2,... , b_n) of the numbers 1,2,...,n1, 2,..., n such that (a1+b1,a2+b2,...,an+bn)(a_1 + b_1, a_2 +b_2,..., a_n + b_n) are consecutive natural numbers.
(Walther Janous)
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