Difference between revisions of "2022 USAMO Problems/Problem 5"
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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> | ||
| + | |||
| + | ==Solution== | ||
| + | Coming soon. | ||
| + | |||
| + | ==See also== | ||
| + | {{USAMO newbox|year=2022|num-b=4|num-a=6}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 14:00, 27 March 2022
Problem
A function 
is 
if 
holds whenever 
are real numbers such that 
and 
.
Find the smallest integer 
such that for any 2022 real numbers 
there exist essentially increasing functions 
such that![\[f_1(n) + f_2(n) + \cdots + f_k(n) = x_n\qquad \text{for every } n= 1,2,\ldots 2022.\]](http://latex.artofproblemsolving.com/f/f/1/ff18b495d0bfd0406f3c2471654f87b5d8be1544.png)
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 | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
