Problem A. 574. (November 2012)
A. 574. Let n2 and let be a polynomial with real coefficients. Prove that if for some positive integer k the polynomial (x-1)k+1 divides p(x) then .
CIIM 2012, Guanajuato, Mexico
Deadline expired on 10 December 2012.
Solution. For convenience, define the leading coefficient an=1 also.
Lemma. For every polynomial q(y) with degree at most k, we have .
Proof. Let 0(y)=1 and let for . By (x-1)k|p(x), for 0k we have
Every polynomial with degree at most k can be repesented as a linear combination of the polynomials , so with some real numbers . Then
To prove the problem statement, let Tk be the kth Chebyshev-polynomial, and choose
Then , and
(In the last step we applied the inequality .)
By applying the lemma,