2002 AMC 12A Problems/Problem 6
- The following problem is from both the 2002 AMC 12A #6 and 2002 AMC 10A #4, so both problems redirect to this page.
Problem
For how many positive integers does there exist at least one positive integer n such that ?
infinitely many
Solution
Solution 1
For any we can pick , we get , therefore the answer is .
Solution 2
Another solution, slightly similar to this first one would be using Simon's Favorite Factoring Trick.
Let , then
This means that there are infinitely many numbers that can satisfy the inequality. So the answer is .
Solution 3
If we subtract from both sides of the equation, we get . Factor the left side to get . Divide both sides by and we get . The fraction if . There is an infinite amount of integers greater than 1, therefore the answer is .
See Also
2002 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 5 |
Followed by Problem 7 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
2002 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.