IMO 1989 LL IRE53

Let f(x) = (x −a1)(x −a2) \cdot \cdot \cdot (x −an) −2, where n \geq3

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IMO 1989 LL IRE53

Origin: IRE

Problem

Let f(x) = (x −a1)(x −a2) \cdot \cdot \cdot (x −an) −2, where n \geq3 and a1, a2, . . . , an are distinct integers. Suppose that f(x) = g(x)h(x), where g(x), h(x) are both nonconstant polynomials with integer coeffi- cients. Prove that n = 3.