Difference between revisions of "2004 IMO Problems/Problem 2"
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[[Category:Olympiad Algebra Problems]] | [[Category:Olympiad Algebra Problems]] |
Latest revision as of 00:52, 19 November 2023
Problem
Find all polynomials with real coefficients such that for all reals
such that
we have the following relations
Solution
Solution 1
From , we have
, so
is even, and all the degrees all of its terms are even. Let
Let *; then we have
. Comparing lead coefficients, we have
, which cannot be true for
. Hence, we have
. We can easily verify by expanding that all such polynomials work.
- The substitution arises from writing
.
Solution 2
Let ,
, and
. Then it is easy to check that
, so
for all . Hence, for the coefficient of
to be nonzero, we must have
.
This does not hold for , and if
is odd and
, then the LHS is irrational and the RHS is a positive integer, so
must be even.
Let . Then
, so
. This holds for
and
, and
, so
for
. Therefore,
must be of the form
.
See also
- <url>viewtopic.php?p=99448#99448 AoPS/MathLinks discussion</url>
2004 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |