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 5 of 6: For c=1, only exact equality of |P(a)| and |P(b)| matters

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|P(a)|-|P(b)|<n\ \text{always, so equality }|a+d|=|b+d|\text{ is forced}

Detailed analysis

Take $P(x)=x+d$ . If $a+d,b+d$ have the same sign, $|P(a)|-|P(b)|=\pm(a-b)$ with $0<|a-b|<n$ (as $1\le a<b\le n$ ), so $n$ never divides it, giving no pairs. If they have opposite signs, say $a<-d\le b$ (requiring $-d=:m>0$ ), then $|P(a)|-|P(b)|=-(a+b+2d)$ , and since $1\le a\le m-1$ , $m\le b\le n$ , one checks $|a+b-2m|<m\le n$ ; so again the difference has absolute value less than $n$ , forcing $a+b=2m$ exactly for $n$ to divide it.