I have the distinct feeling that the problem is related to the following inequality:
x^2 + y^2 >= 2xy
and to the fact that the distance from the origin to point (x, y) is sqrt(x^2 + y^2).
However, I'm a bit stuck in calculations (or maybe not), so, since I'm not sure about my answer, I won't give an explanation, just post my answer and wait for someone else's.
My answer is E

I couldn't determine a formula approaching quickly, so I plugged in values:

(1) m n - p q = 12
Let's assume, (m,n) is (10,2)
m*n = 12 + p*q
10*2 = 12 + p*q
20 = 12 + p*q
8 = p*q
For all values p and q, the distance from the origin for (m,n) is greater than (p,q).

Let's assume, (m,n) = (-2,2)
m*n = 12 + p*q
-4 = 12 + p*q
8 = p*q
For all values p and q, the distance from the origin for (m,n) is less than (p,q).

This conflicts. So (1) is insufficient.

(2) p + q = 12
This tells us nothing about (m,n), so (2) is insufficient.

(1) and (2) together?
Suppose (p,q) is (14,-2) such that 14 + -2 = 12.
mn = 12 - 28
mn = -16
If (m,n) is (-16,1), then the distance from the origin to (m,n) is greater than (p,q).
If (m,n) is (-8,2), then the distance from the origin to (m,n) is less than (p,q).

This conflicts. So together (1) and (2) are insufficient.

Answer E.

PS: I'm really interested in a non-plug method.