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Polynomial such that |P(10)-P(0)|<1000

Source: Dutch IMO TST II Problem 5

July 17, 2014

algebrapolynomialcalculusintegrationratioarithmetic sequencealgebra unsolved

Problem Statement

Let

P(x)

be a polynomial of degree

n \le 10

with integral coefficients such that for every

k \in \{1, 2, \dots, 10\}

there is an integer

m

with

P(m) = k

. Furthermore, it is given that

|P(10) - P(0)| < 1000

. Prove that for every integer

k

there is an integer

m

such that

P(m) = k.