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Polynomials written on a board

Source: Germany 2010 - Problem 5

December 7, 2022
algebrapolynomialcalculusderivativecombinatorics

Problem Statement

The polynomial x8+x7x^8 +x^7 is written on a blackboard. In a move, Peter can erase the polynomial P(x)P(x) and write down (x+1)P(x)(x+1)P(x) or its derivative P′(x).P'(x). After a while, the linear polynomial ax+bax+b with a≠0a\ne 0 is written on the board. Prove that a−ba-b is divisible by 49.49.
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