MathDB
Subsets of a set form a complete residue system mod 2^n

Source: Mongolia national olympiad 2024

April 10, 2024
combinatoricsnumber theorycomplete residue system

Problem Statement

A set XX consisting of nn positive integers is called <spanclass=latexitalic>good</span><span class='latex-italic'>good</span> if the following condition holds: For any two different subsets of XX, say AA and BB, the number s(A)s(B)s(A) - s(B) is not divisible by 2n2^n. (Here, for a set AA, s(A)s(A) denotes the sum of the elements of AA) Given nn, find the number of good sets of size nn, all of whose elements is strictly less than 2n2^n.