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Characterisation of special polynomials

Source: All-Russian MO 2023 Final stage 11.7

April 23, 2023
algebrapolynomial

Problem Statement

We call a polynomial P(x)P(x) good if the numbers P(k)P(k) and P′(k)P'(k) are integers for all integers kk. Let P(x)P(x) be a good polynomial of degree dd, and let NdN_d be the product of all composite numbers not exceeding dd. Prove that the leading coefficient of the polynomial Nd⋅P(x)N_d \cdot P(x) is integer.
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