Problem 2

For a polynomial

P

and a positive integer

n

, define

P_n

as the number of positive integer pairs

(a,b)

such that

a<b\le n

and

|P(a)|-|P(b)|

is divisible by

n

. Determine all polynomials

P

with integer coefficients such that

P_n\le2021

for all positive integers

n

Step 1 of 6: Answer

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P(x)=x+d\ (d\ge-2022)\quad\text{or}\quad P(x)=-x+d\ (d\le2022)

Detailed analysis

The answer consists of the two families $P(x)=x+d$ with $d\ge-2022$ and $P(x)=-x+d$ with $d\le2022$ . The later steps prove necessity from a residue-distinctness lemma and then verify these bounds by counting the pairs with equal absolute values.