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Find all sets that satisfy sum and cardinality condition

Source: Mexico National Olympiad 2021 Problem 6

November 10, 2021
combinatoricsalgebranumber theoryMexico

Problem Statement

Determine all non empty sets C1,C2,C3,⋯C_1, C_2, C_3, \cdots such that each one of them has a finite number of elements, all their elements are positive integers, and they satisfy the following property: For any positive integers nn and mm, the number of elements in the set CnC_n plus the number of elements in the set CmC_m equals the sum of the elements in the set Cm+nC_{m + n}.
Note: We denote ∣Cn∣\lvert C_n \lvert the number of elements in the set CnC_n, and SkS_k as the sum of the elements in the set CnC_n so the problem's condition is that for every nn and mm: ∣Cn∣+∣Cm∣=Sn+m\lvert C_n \lvert + \lvert C_m \lvert = S_{n + m} is satisfied.
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