MathDB

2

Part of 2021 APMO

Problems(1)

Bounding Divisibility

Source: 2021 APMO P2

6/9/2021
For a polynomial PP and a positive integer nn, define PnP_n as the number of positive integer pairs (a,b)(a,b) such that a<b≤na<b \leq n and ∣P(a)∣−∣P(b)∣|P(a)|-|P(b)| is divisible by nn. Determine all polynomial PP with integer coefficients such that Pn≤2021P_n \leq 2021 for all positive integers nn.
algebrapolynomial