1975 Canadian MO Problems/Problem 8
Problem 8
Let be a positive integer. Find all polynomials
where the
are real, which satisfy the equation
.
1975 Canadian MO (Problems) | ||
Preceded by Problem 7 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • | Followed by Last Question |
Solution
Let be the degree of polynomial
. We begin by noting that
. This is because the degree of the LHS is
and the RHS is
. Now we split
into two cases.
In the first case, is a constant. This means that
or
if
is even.
In the second case, is non-constant with coefficients of
. If we divide by
on both sides, then we have that
. This can only be achieved if
. This is because if we factor out a
, then clearly these terms are not constant. Thus,
and our second solution is
.
~bigbrain123 ~minor edits by clever14710owl