Difference between revisions of "1975 USAMO Problems/Problem 3"
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==Problem== | ==Problem== | ||
− | If <math>P(x)</math> denotes a polynomial of degree <math>n</math> such that < | + | If <math>P(x)</math> denotes a polynomial of degree <math>n</math> such that <cmath>P(k)=k/(k+1)</cmath> for <math>k=0,1,2,\ldots,n</math>, determine <math>P(n+1)</math>. |
==Solution== | ==Solution== |
Revision as of 01:42, 21 May 2015
Contents
[hide]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.