Difference between revisions of "2023 AMC 12B Problems/Problem 22"
(Created page with " ==Problem== A real-valued function <math>f</math> has the property that for all real numbers <math>a</math> and <math>b,</math> <cmath>f(a + b) + f(a - b) = 2f(a) f(b).</cm...") |
|||
| Line 21: | Line 21: | ||
{{AMC12 box|ab=B|year=2023|num-b=21|num-a=23}} | {{AMC12 box|ab=B|year=2023|num-b=21|num-a=23}} | ||
| − | [[Category:Functional | + | [[Category:Functional equation]] |
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 18:33, 15 November 2023
Problem
A real-valued function 
has the property that for all real numbers 
and 
![\[f(a + b) + f(a - b) = 2f(a) f(b).\]](http://latex.artofproblemsolving.com/7/b/d/7bd7b5dc604674f392920ae5bd0a29c5ebbf73d7.png)
Which one of the following cannot be the value of 
Solution 1
Substituting 
we get
![\[f(2a) + f(0) = 2f(a)^2\]](http://latex.artofproblemsolving.com/2/8/c/28cdad33e067510298b8f671157d89556ea3aadb.png)
Substituting 
we find ![\[2f(0) = 2f(0)^2 \implies f(0) \in \{0, 1\}.\]](http://latex.artofproblemsolving.com/0/3/d/03d4c990a866c07ed0ed78c64bfa5e4a9209ca14.png)
This gives ![\[f(2a) = 2f(a)^2 - f(0) \geq 0-1\]](http://latex.artofproblemsolving.com/6/c/5/6c51450ee6d0ae690f78ca2ff96eb73d9dee0be4.png)
Plugging in 
implies 
, so answer choice 
is impossible.
~AtharvNaphade
See also
| 2023 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 21 |
Followed by Problem 23 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
