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(x-1)^{k} | P(x)

Source: Ukrainian TST 2007 problem 3

September 19, 2007

algebrapolynomiallogarithmsalgebra proposed

Problem Statement

It is known that

k

and

n

are positive integers and k \plus{} 1\leq\sqrt {\frac {n \plus{} 1}{\ln(n \plus{} 1)}}. Prove that there exists a polynomial

P(x)

of degree

n

with coefficients in the set \{0,1, \minus{} 1\} such that (x \minus{} 1)^{k} divides

P(x)