Difference between revisions of "1968 IMO Problems/Problem 5"
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(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>. | ||
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+ | [[Category:Olympiad Algebra Problems]] | ||
+ | [[Category:Functional Equation Problems]] |
Revision as of 08:37, 19 July 2016
Problem 5
Let be a real-valued function defined for all real numbers
such that, for some positive constant
, the equation
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
is true for any
, and
We have:
Therefore
is periodic, with
as a period.
(b) when
for some integer
, and
when
for some integer
.