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exists permutation a_i of i=1,1000 such that |a_i - i| = k

Source: 2018 Saudi Arabia IMO TST IV p3

July 28, 2020
permutationcombinatorics

Problem Statement

Find all positive integers kk such that there exists some permutation of (1,2,...,1000)(1, 2,...,1000) namely (a1,a2,...,a1000)(a_1, a_2,..., a_{1000}) and satisfy ∣ai−i∣=k|a_i - i| = k for all i=1,1000i = 1,1000.
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