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 6 of 6: Count the exact solutions and match the bound; the case c=-1 mirrors it

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\sup_n P_n = \max(-d-1,0)\ \le 2021 \iff d\ge-2022

Detailed analysis

For $d\ge0$ no pair satisfies $a+b=2m=-2d$ (as $m\le0$ ), so $P_n=0$ for all $n$ . For $d=-m<0$ , the pairs with $a<b\le n$ , $a+b=2m$ , $a<m\le b$ number exactly $\min(m-1,\,n-m)$ when this is nonnegative, which equals $m-1$ once $n\ge2m-1$ ; so $\sup_nP_n=m-1$ . Requiring $m-1\le2021$ gives $m\le2022$ , i.e. $d\ge-2022$ , and this bound is also sufficient. Replacing $P$ by $-P$ leaves every $P_n$ unchanged (since $|{-P}|=|P|$ ), so the family $P(x)=-x+d$ works exactly when $-d\ge-2022$ , i.e. $d\le2022$ , completing the classification.