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Pretty much the same as KMO P7

Source: 2015 Korean Junior MO P8

November 1, 2015
combinatoricsSets

Problem Statement

A positive integer nn is given. If there exist sets F1,F2,⋯FmF_1, F_2, \cdots F_m satisfying the following, prove that m≤nm \le n. (For sets A,BA, B, ∣A∣|A| is the number of elements in AA. A−BA-B is the set of elements that are in AA but not BB)
(i): For all 1≤i≤m1 \le i \le m, Fi⊆{1,2,⋯n}F_i \subseteq \{1,2,\cdots n\}
(ii): ∣F1∣≤∣F2∣≤⋯≤∣Fm∣|F_1| \le |F_2| \le \cdots \le |F_m|
(iii): For all 1≤i<j≤m1 \le i < j \le m, ∣Fi−Fj∣=1|F_i-F_j|=1.
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