1975 Canadian MO Problems/Problem 7
Problem 7
A function is
if there is a positive integer such that
for all
. For example,
is periodic with period
. Is the function
periodic? Prove your assertion.
Solution
To prove that is periodic, we need to check if there exists a positive number
such that
for all
. Using the trigonometric property
(where
), this implies
for some integer
. Expanding and simplifying,
, so the equation becomes
. Rewriting,
. For this equation to hold for all
, the term
must vanish, which is only possible if
. However, since
is required for periodicity, no such
exists, meaning
is not periodic. [Intuitively, the argument
of
grows faster than linearly as
increases, causing the values of
to fail to repeat in a regular pattern. Therefore,
is not periodic.]
~sitar .
1975 Canadian MO (Problems) | ||
Preceded by Problem 6 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • | Followed by Problem 8 |