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P(a_1) = P(a_2) = P(a_3) = P(a_4) = 1, P(n) = 12 impossible, P(n) = 1998

Source: ITAMO 1998 p5

January 25, 2020
polynomialInteger Polynomialalgebra

Problem Statement

Suppose a1,a2,a3,a4a_1,a_2,a_3,a_4 are distinct integers and P(x)P(x) is a polynomial with integer coefficients satisfying P(a1)=P(a2)=P(a3)=P(a4)=1P(a_1) = P(a_2) = P(a_3) = P(a_4) = 1. (a) Prove that there is no integer nn such that P(n)=12P(n) = 12. (b) Do there exist such a polynomial and ana_n integer nn such that P(n)=1998P(n) = 1998?
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