MathDB

Determine whether there exists a polynomial

f(x_1, x_2)

with two variables, with integer coefficients, and two points

A=(a_1, a_2)

and

B=(b_1, b_2)

in the plane, satisfying the following conditions:
(i)

A

is an integer point (i.e

a_1

and

a_2

are integers);
(ii)

|a_1-b_1|+|a_2-b_2|=2010

;
(iii)

f(n_1, n_2)>f(a_1, a_2)

for all integer points

(n_1, n_2)

in the plane other than

A

;
(iv)

f(x_1, x_2)>f(b_1, b_2)

for all integer points

(x_1, x_2)

in the plane other than

B

.
Massimo Gobbino, Italy

Problem Statement