1975 USAMO Problems/Problem 3
Contents
Problem
If denotes a polynomial of degree such that for , determine .
Solution
Let . Clearly, has a degree of .
Then, for , .
Thus, are the roots of .
Since these are all of the roots, we can write as: where is a constant.
Thus,
Plugging in gives:
Finally, plugging in gives:
If is even, this simplifies to . If is odd, this simplifies to .
Solution 2
It is fairly natural to use Lagrange's Interpolation Formula on this problem:
through usage of the Binomial Theorem.
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.