Difference between revisions of "1968 IMO Problems/Problem 5"
(Created page with "==Problem 5== Let <math>f</math> be a real-valued function defined for all real numbers <math>x</math> such that, for some positive constant <math>a</math>, the equation <cmath>...") |
|||
| (One intermediate revision by one other user not shown) | |||
| Line 20: | Line 20: | ||
(b) <math>f(x) = 1</math> when <math>2n\le x < 2n+1</math> for some integer <math>n</math>, and <math>f(x)=\frac{1}{2}</math> when <math>2n+1\le x < 2n+2</math> for some integer <math>n</math>. | (b) <math>f(x) = 1</math> when <math>2n\le x < 2n+1</math> for some integer <math>n</math>, and <math>f(x)=\frac{1}{2}</math> when <math>2n+1\le x < 2n+2</math> for some integer <math>n</math>. | ||
| + | |||
| + | == See Also == {{IMO box|year=1968|num-b=4|num-a=6}} | ||
| + | |||
| + | [[Category:Olympiad Algebra Problems]] | ||
| + | [[Category:Functional Equation Problems]] | ||
Latest revision as of 12:33, 29 January 2021
Problem 5
Let 
be a real-valued function defined for all real numbers 
such that, for some positive constant 
, the equation
![\[f(x + a) = \frac{1}{2} + \sqrt{f(x) - (f(x))^2}\]](http://latex.artofproblemsolving.com/0/c/8/0c8b3bbc8a56354c4a9940dfe4076f5dad29e31a.png)
holds for all .
(a) Prove that the function is periodic (i.e., there exists a positive number 
such that 
for all ).
(b) For 
, give an example of a non-constant function with the required properties.
Solution
(a) Since
![\[f(x+a) \ge \frac{1}{2}\]](http://latex.artofproblemsolving.com/b/3/c/b3c0d8b1936802188854dee709b4b361158d36e2.png)
is true for any , and
![\[f(x+a)(1-f(x+a)) = \frac{1}{4} - (f(x)-(f(x))^2) = (\frac{1}{2}-f(x))^2\]](http://latex.artofproblemsolving.com/9/b/d/9bdd0bc38caa5ed7be000fb06296b00db4fd02da.png)
We have:
![\[f(x+2a) = \frac{1}{2} + \sqrt{(\frac{1}{2}-f(x))^2} = \frac{1}{2} + (f(x) - \frac{1}{2}) = f(x)\]](http://latex.artofproblemsolving.com/4/1/b/41b8de7a868b77d7f5d148e86e515659aa49cfb7.png)
Therefore is periodic, with 
as a period.
(b) 
when 
for some integer 
, and 
when 
for some integer .
See Also
| 1968 IMO (Problems) • Resources | ||
| Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
| All IMO Problems and Solutions | ||
