Difference between revisions of "2025 USAMO Problems/Problem 1"
(→Solution) |
|||
| Line 4: | Line 4: | ||
== Problem == | == Problem == | ||
Let <math>k</math> and <math>d</math> be positive integers. Prove that there exists a positive integer <math>N</math> such that for every odd integer <math>n>N</math>, the digits in the base-<math>2n</math> representation of <math>n^k</math> are all greater than <math>d</math>. | Let <math>k</math> and <math>d</math> be positive integers. Prove that there exists a positive integer <math>N</math> such that for every odd integer <math>n>N</math>, the digits in the base-<math>2n</math> representation of <math>n^k</math> are all greater than <math>d</math>. | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
==See Also== | ==See Also== | ||
Revision as of 05:37, 23 March 2025
- The following problem is from both the 2025 USAMO #1 and 2025 USAJMO #2, so both problems redirect to this page.
Contents
Problem
Let 
and 
be positive integers. Prove that there exists a positive integer 
such that for every odd integer 
, the digits in the base-
representation of 
are all greater than .
See Also
| 2025 USAMO (Problems • Resources) | ||
| Preceded by First Question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAMO Problems and Solutions | ||
| 2025 USAJMO (Problems • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAJMO Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
