Problem

Source: XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade

Tags: combinatorics, set theory



Let $x_1,\dots ,x_n$ be real numbers. We look at all the $2^{n-1}$ possible sums between some of the numbers. If the number of different sums is at least $1.8^n$, prove that the number of sums equal to $2022$ is no more than $1.67^n$.