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Difference between revisions of "2022 USAMO Problems/Problem 5"

(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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Find the smallest integer <math>k</math> such that for any 2022 real numbers <math>x_1,x_2,\ldots , x_{2022},</math> there exist <math>k</math> essentially increasing functions <math>f_1,\ldots, f_k</math> such that<cmath>f_1(n) + f_2(n) + \cdots + f_k(n) = x_n\qquad \text{for every } n= 1,2,\ldots 2022.</cmath>
 
Find the smallest integer <math>k</math> such that for any 2022 real numbers <math>x_1,x_2,\ldots , x_{2022},</math> there exist <math>k</math> essentially increasing functions <math>f_1,\ldots, f_k</math> such that<cmath>f_1(n) + f_2(n) + \cdots + f_k(n) = x_n\qquad \text{for every } n= 1,2,\ldots 2022.</cmath>
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==Solution==
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Coming soon.
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==See also==
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{{USAMO newbox|year=2022|num-b=4|num-a=6}}
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{{MAA Notice}}

Latest revision as of 14:00, 27 March 2022

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\[f_1(n) + f_2(n) + \cdots + f_k(n) = x_n\qquad \text{for every } n= 1,2,\ldots 2022.\]

Solution

Coming soon.

See also

2022 USAMO (Problems • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions

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