Difference between revisions of "1975 USAMO Problems/Problem 3"
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<math>P(n+1) = \dfrac{(-1)^{n+1} + (n+1)}{n+2}</math> | <math>P(n+1) = \dfrac{(-1)^{n+1} + (n+1)}{n+2}</math> | ||
− | If <math>n</math> is even, this simplifies to <math>P(n+1) = \dfrac{n}{n+2}</math>. If <math>n</math> is odd, this simplifies to <math>P(n+1) = 1</math>. | + | If <math>n</math> is even, this simplifies to <math>P(n+1) = \dfrac{n}{n+2}</math>. If <math>n</math> is odd, this simplifies to <math>P(n+1) = 1</math>. |
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+ | {{alternate solutions}} | ||
==See Also== | ==See Also== |
Revision as of 15:09, 17 September 2012
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
.
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 |