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Kazakhstan National Olympiad 2017, Final Round, 11-Grade, P5

Source: Kazakhstan National Olympiad 2017, Final Round, 11-Grade, P5

March 18, 2017
logicSetscombinatorics

Problem Statement

Consider all possible sets of natural numbers (x1,x2,...,x100)(x_1, x_2, ..., x_{100}) such that 1≤xi≤20171\leq x_i \leq 2017 for every i=1,2,...,100i = 1,2, ..., 100. We say that the set (y1,y2,...,y100)(y_1, y_2, ..., y_{100}) is greater than the set (z1,z2,...,z100)(z_1, z_2, ..., z_{100}) if yi>ziy_i> z_i for every i=1,2,...,100i = 1,2, ..., 100. What is the largest number of sets that can be written on the board, so that any set is not more than the other set?
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