2011 AIME I Problems/Problem 7
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Problem 7
Find the number of positive integers 
for which there exist nonnegative integers 
, 
, 
, 
such that
![\[m^{x_0} = \sum_{k = 1}^{2011} m^{x_k}.\]](http://latex.artofproblemsolving.com/3/7/c/37c3a4d8143a7bc88f260f55b43ac3f3fbb7678e.png)
Solution
NOTE: This solution is incomplete. Please help make it better.
This formula only works if is exactly 1 more than a factor of 2010. (Someone else should insert a convincing proof here; I forgot exactly what I did. However, it involved starting with each 
equal to an unspecified large number 
, and then decreasing the powers of a certain number of the terms [equal to to certain powers] by 1 repeatedly in certain ways until the expression becomes 
• 
for some integer p.)
Since 2010 factors as 
, there are 
such factors.
See also
| 2011 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
