1975 USAMO Problems/Problem 3
Contents
Problem
If denotes a polynomial of degree such that for , determine .
Solution 1
Let , and clearly, has a degree of .
Then, for , .
Thus, are the roots of .
Since these are all of the roots of the degree polynomial, by the Factor Theorem, we can write as where is a constant.
Thus,
We plug in to cancel the and find :
Finally, plugging in to find gives:
If is even, this simplifies to . If is odd, this simplifies to .
~Edits by BakedPotato66
Solution 2
It is fairly natural to use Lagrange's Interpolation Formula on this problem:
through usage of the Binomial Theorem.
~lpieleanu (minor editing and reformatting)
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
1975 USAMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.