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Polynomial taking downhill values

Source: Nikolai Beluhov, USAMTS 2017

May 2, 2018
algebrapolynomialUSAMTS

Problem Statement

A positive integer is called downhill if the digits in its decimal representation form a nonstrictly decreasing sequence from left to right. Suppose that a polynomial P(x)P(x) with rational coefficients takes on an integer value for each downhill positive integer xx. Is it necessarily true that P(x)P(x) takes on an integer value for each integer xx?
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