I found this question nice so i would like to bring my two cents
First lets simplify the inequality: 2xy<x^2+y^2 ----> 0<x^2 +y^2-2xy ----> 0<(x-y)^2. So in fact the question asks whether (x-y) are not equal to 0?
statement 1: xy<0. This statement tells us that x and y have different signs. Thus (x-y) necessarily couldn't equal to 0 because positive number - negative number = positive number and negative number - positive number = negative number. Sufficient.
statement 2: x+y=5. (x-y) could be zero just if both x and y equal to each other. Since x and y are integers we can conclude that x and y are not equal to each other an thus that (x - y) couldn't be equal to 0. Sufficient.
Answer: D