MathDB

Let

N

denote the number of ordered 2011-tuples of positive integers

(a_1,a_2,\ldots,a_{2011})

with

1\le a_1,a_2,\ldots,a_{2011} \le 2011^2

such that there exists a polynomial

f

of degree

4019

satisfying the following three properties: [*]

f(n)

is an integer for every integer

n

; [*]

2011^2 \mid f(i) - a_i

for

i=1,2,\ldots,2011

; [*]

2011^2 \mid f(n+2011) - f(n)

for every integer

n

. Find the remainder when

N

is divided by

1000

.
Victor Wang

Problem Statement