RMM 2010 Problem 4

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