IMC 2019 · Problem 8

Let $x_1, \ldots, x_n$ be real numbers. For any set $I \subset \{1, 2, \ldots, n\}$ let $s(I) = \sum_{i \in I} x_i$. Assume that the function $I \mapsto s(I)$ takes on at least $1.8^n$ values where $I$ runs over all $2^n$ subsets of $\{1, 2, \ldots, n\}$. Prove that the number of sets $I \subset \{1, 2, \ldots, n\}$ for which $s(I) = 2019$ does not exceed $1.7^n$.