Ответ. Does not exist.

Решение. Let $F$ and $G$ be an important pair of polynomials. Consider pairs of residues modulo 100 of numbers $F(a,b)$ and $G(a,b)$, where $a,b$ range over all integer pairs from 0 to 99. According to the problem's condition, all such residue pairs are distinct. Since there are $100^2$ possible number pairs, each residue pair modulo 100 occurs exactly once. Therefore, all 4 possible parity combinations of $F(a,b)$ and $G(a,b)$ are achieved.
Since the parity of a polynomial's value with integer coefficients at point $(a,b)$ depends only on the parity of $a$ and $b$, we conclude that the value pairs $(F(0,0); G(0,0))$, $(F(1,0); G(1,0))$, $(F(0,1); G(0,1))$, and $(F(1,1); G(1,1))$ must give all four possible parity combinations.
However, observe that for both polynomial pairs $F = P, G = Q$ and $F(x,y) = P(x,y) - xy, G(x,y) = Q(x,y) + xy$, the first three parity pairs are identical, while the fourth pair differs. Consequently, both such polynomial pairs cannot be important.