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polynomial with coefficients in {1,-1}

Source: Polish MO 2006

April 8, 2006
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Problem Statement

Find all pairs of integers a,ba,b for which there exists a polynomial P(x)∈Z[X]P(x) \in \mathbb{Z}[X] such that product (x2+ax+b)⋅P(x)(x^2+ax+b)\cdot P(x) is a polynomial of a form xn+cn−1xn−1+⋯+c1x+c0 x^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0 where each of c0,c1,…,cn−1c_0,c_1,\ldots,c_{n-1} is equal to 11 or −1-1.
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