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Sorting permutation. Who wins?

Source: IMEO 2020 Problem 4

July 15, 2020
IMEOgamepermutations

Problem Statement

Anna and Ben are playing with a permutation pp of length 20202020, initially pi=2021−ip_i = 2021 - i for 1≤i≤20201\le i \le 2020. Anna has power AA, and Ben has power BB. Players are moving in turns, with Anna moving first.
In his turn player with power PP can choose any PP elements of the permutation and rearrange them in the way he/she wants.
Ben wants to sort the permutation, and Anna wants to not let this happen. Determine if Ben can make sure that the permutation will be sorted (of form pi=ip_i = i for 1≤i≤20201\le i \le 2020) in finitely many turns, if
a) A=1000,B=1000A = 1000, B = 1000
b) A=1000,B=1001A = 1000, B = 1001
c) A=1000,B=1002A = 1000, B = 1002
Anton Trygub
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