1968 IMO Problems/Problem 5
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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
.
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 |