MathDB
Polynomial up to a Translation

Source: 2021 China TST, Test 1, Day 2 P4

March 17, 2021
algebrapolynomialnumber theorymodular arithmetic

Problem Statement

Let f(x),g(x)f(x),g(x) be two polynomials with integer coefficients. It is known that for infinitely many prime pp, there exist integer mpm_p such that f(a)≡g(a+mp)(modp)f(a) \equiv g(a+m_p) \pmod p holds for all a∈Z.a \in \mathbb{Z}. Prove that there exists a rational number rr such that f(x)=g(x+r).f(x)=g(x+r).
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