Mock AIME 4 2006-2007 Problems/Problem 3
Problem
Find the largest prime factor of the smallest positive integer such that are distinct integers such that the polynomial has exactly nonzero coefficients.
Solution
- The following solution is non-rigorous.
We would normally expect 2007 terms after multiplying out all of the binomials, but our goal is minimize the number of non-zero terms. We could get rid of some terms by applying repeated difference of squares. In other words, we let . Then our polynomial reduces to . This is the product of binomials, which gives us terms (with nonzero coefficients). Since (assuming the constant term counts as a coefficient), our answer is .
Note that we could not apply difference of cubes etc, since that would require complex roots.
See also
Mock AIME 4 2006-2007 (Problems, Source) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |