# International Mathematics Competition for University Students 2019

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IMC 2019
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## IMC2019: Day 2, Problem 8

Problem 8. Let $\displaystyle x_1,\ldots,x_n$ be real numbers. For any set $\displaystyle I\subset \{1,2,\ldots,n\}$ let $\displaystyle s(I)=\sum\limits_{i\in I} x_i$. Assume that the function $\displaystyle I\mapsto s(I)$ takes on at least $\displaystyle 1.8^n$ values where $\displaystyle I$ runs over all $\displaystyle 2^n$ subsets of $\displaystyle \{1,2,\ldots,n\}$. Prove that the number of sets $\displaystyle I\subset \{1,2,\ldots,n\}$ for which $\displaystyle s(I)=2019$ does not exceed $\displaystyle 1.7^n$.

Proposed by Fedor Part and Fedor Petrov, St. Petersburg State University

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