2013 Canadian MO Problems/Problem 1
Problem
Determine all polynomials with real coefficients such that is a constant polynomial.
Solution
Let
In order for the new polynomial to be a constant, all the coefficients in front of for need to be zero.
So we start by looking at the coefficient in front of :
Since ,
We then evaluate the term of the sum when :
Note that since , , and are not present in the expression before , they can be anything and the coefficient in front of is still zero because the expression before also starts with , and the expression before also starts with and so on...
In fact in matrix form when solving for for for all , it will look something like this:
This matrix is singular because the 3nd row and the 4rd row also start at the 3rd column, then all coefficients for need to be zero so that the coefficient in front of is zero.
That is, .
So now we just need to find and , for that we look at the coefficient in front of in :
Since =0 for :
Therefore , thus satisfies the condition for to be a constant polynomial.
So we can set and , and all the polynomials will be in the form:
where
~Tomas Diaz. orders@tomasdiaz.com
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.