Question 195
Determine whether there exist a polynomial $f(x_1, x_2)$ in two variables, with integer coefficients, and two points $A = (a_1, a_2)$ and $B = (b_1, b_2)$ in the plane, satisfying all 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) $f(n_1, n_2) \gt f(a_1, a_2)$, for all integer points $(n_1, n_2)$ in the plane other than $A$;
(iv) $f(x_1, x_2) \gt f(b_1, b_2)$, for all points $(x_1, x_2)$ in the plane other than $B$.
(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) $f(n_1, n_2) \gt f(a_1, a_2)$, for all integer points $(n_1, n_2)$ in the plane other than $A$;
(iv) $f(x_1, x_2) \gt f(b_1, b_2)$, for all points $(x_1, x_2)$ in the plane other than $B$.