Difference between revisions of "2012 USAMO Problems/Problem 3"
m (→Solution) |
m |
||
Line 6: | Line 6: | ||
==Solution== | ==Solution== | ||
For <math>n</math> equal to any odd prime <math>p</math>, the sequence <math>\left\{a_i = \left(\frac{1-n}{2}\right)^{m_p\left(i\right)}\right\}</math>, where <math>p^{m_p\left(i\right)}</math> is the greatest power of <math>p</math> that divides <math>i</math>, gives a valid sequence. Therefore, the set of possible values for <math>n</math> is at least the set of odd primes. | For <math>n</math> equal to any odd prime <math>p</math>, the sequence <math>\left\{a_i = \left(\frac{1-n}{2}\right)^{m_p\left(i\right)}\right\}</math>, where <math>p^{m_p\left(i\right)}</math> is the greatest power of <math>p</math> that divides <math>i</math>, gives a valid sequence. Therefore, the set of possible values for <math>n</math> is at least the set of odd primes. | ||
+ | |||
+ | {{solution}} | ||
==See Also== | ==See Also== |
Revision as of 09:53, 25 April 2012
Problem
Determine which integers have the property that there exists an infinite sequence
,
,
,
of nonzero integers such that the equality
holds for every positive integer
.
Solution
For equal to any odd prime
, the sequence
, where
is the greatest power of
that divides
, gives a valid sequence. Therefore, the set of possible values for
is at least the set of odd primes.
This problem needs a solution. If you have a solution for it, please help us out by adding it.
See Also
2012 USAMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |