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p(x)=1 and p(x)=3 have integer solutions, can p(x)=2 have 2?

Source: Baltic Way 1994

December 22, 2011
algebrapolynomialalgebra proposed

Problem Statement

Let p(x)p(x) be a polynomial with integer coefficients such that both equations p(x)=1p(x)=1 and p(x)=3p(x)=3 have integer solutions. Can the equation p(x)=2p(x)=2 have two different integer solutions?
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