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Actually, estimating the size of A is harder than the problem

Source: 2023 KMO P3

November 4, 2023
algebrapolynomial

Problem Statement

For a given positive integer n(≥2)n(\ge 2), find maximum positive integer AA such that there exists P∈Z[x]P \in \mathbb{Z}[x] with degree nn that satisfies the following two conditions.
[*] For any 1≤k≤A1 \le k \le A, it satisfies that A∣P(k)A \mid P(k), and [*] P(0)=0P(0)= 0 and the coefficient of the first term of PP is 11, which means that P(x)P(x) is in the following form where c2,c3,⋯ ,cnc_2, c_3, \cdots, c_n are all integers and cn≠0c_n \neq 0. P(x)=cnxn+cn−1xn−1+⋯+c2x2+xP(x) = c_nx^n + c_{n-1}x^{n-1}+\dots+c_2x^2+x
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