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Omni-MATH / A function f:R→R is textit{essentially increasing} if f(s)≤f(t) holds whenever s≤t are real numbers such that f(s)≠0 and f(t)≠0 . Find the smallest integer k such that for any 2022…

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problem

A function f:R→Rf: \mathbb{R}\to \mathbb{R} is $\textit{essentially increasing}$ if f(s)≤f(t)f(s)\leq f(t) holds whenever s≤ts\leq t are real numbers such that f(s)≠0f(s)\neq 0 and f(t)≠0f(t)\neq 0 . Find the smallest integer kk such that for any 2022 real numbers x1,x2,…,x2022,x_1,x_2,\ldots , x_{2022}, there exist kk essentially increasing functions f1,…,fkf_1,\ldots, f_k such that f1(n)+f2(n)+⋯+fk(n)=xnfor every n=1,2,…2022.f_1(n) + f_2(n) + \cdots + f_k(n) = x_n\qquad \text{for every } n= 1,2,\ldots 2022.
Plain-text mathematical notation (without MathML)
A function f:R→R is $\textit{essentially increasing}$ if f(s)≤f(t) holds whenever s≤t are real numbers such that f(s)≠0 and f(t)≠0 .
Find the smallest integer k such that for any 2022 real numbers x₁,x₂,…,x₂₀₂₂, there exist k essentially increasing functions f₁,…,f_(k) such that f₁(n)+f₂(n)+⋯+f_(k)(n)=x_(n) for every n=1,2,…2022.
Original LaTeX notation
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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Source and history

task_id
20e3558a-d196-587f-94d9-6ddd038502a3
task_key
test--20e3558a-d196-587f-94d9-6ddd038502a3
task_revision_id
1
config / split
/ test

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