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Is it true that k_i<k_j ? combo game with n2^{n-1} questions for 2^n integers

Source: Balkan BMO Shortlist 2015 C2

August 5, 2019
combinatoricsgamegame strategy

Problem Statement

Isaak and Jeremy play the following game. Isaak says to Jeremy that he thinks a few 2n2^n integers k1,..,k2nk_1,..,k_{2^n}. Jeremy asks questions of the form: ''Is it true that ki<kjk_i<k_j ?'' in which Isaak answers by always telling the truth. After n2n1n2^{n-1} questions, Jeramy must decide whether numbers of Isaak are all distinct each other or not. Prove that Jeremy has bo way to be ''sure'' for his final decision.
(UK)