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2022 USAMO Problems/Problem 5

Revision as of 13:59, 27 March 2022 by Shihan (talk | contribs) (Created page with "==Problem== A function <math>f: \mathbb{R}\to \mathbb{R}</math> is <math>\textit{essentially increasing}</math> if <math>f(s)\leq f(t)</math> holds whenever <math>s\leq t</mat...")
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Problem

A function $f: \mathbb{R}\to \mathbb{R}$ is $\textit{essentially increasing}$ if $f(s)\leq f(t)$ holds whenever $s\leq t$ are real numbers such that $f(s)\neq 0$ and $f(t)\neq 0$.

Find the smallest integer $k$ such that for any 2022 real numbers $x_1,x_2,\ldots , x_{2022},$ there exist $k$ essentially increasing functions $f_1,\ldots, f_k$ such that\[f_1(n) + f_2(n) + \cdots + f_k(n) = x_n\qquad \text{for every } n= 1,2,\ldots 2022.\]

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