1975 USAMO Problems/Problem 3
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
.
See Also
1975 USAMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |