1975 USAMO Problems/Problem 3
If denotes a polynomial of degree such that for , determine .
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, we can write as where is a constant.
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
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.
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