2020 INMO Problems/Problem 2
PROBLEM
Suppose is a polynomial with real coefficients, satisfying the condition
, for every real
. Prove that
can be expressed in the form
for some real numbers
and non-negative integer
.
SOLUTION(1)
Assume to the contrary. Suppose satisfies
for all real
, and is of minimal degree and not of the prescribed form.
For some
, we have
.
Note that
. Set
. Then
as the latter vanishes at both
. Now let
for some
.
Then holds for all
, by plugging
in the original equation, since we have the identities
and
.
(Subtlety for beginners: while the equation in only holds for
away from roots of
, since these form a discrete subset of
, the equation extends to these as
is continuous.)
In particular, plugging we get
and
so
, hence
. Thus,
as desired.
Finally, we see that and
so
has the prescribed form. But then
also has the prescribed form, and our result follows.
~anantmdgal09