Answer
At least one pair of mid point of line joining 5 different pairs of integers have integer values.
Work Step by Step
Let us take 2 pair of integers ($x_{1}$,$y_{1}$) and ($x_{2}$,$y_{2}$).
Their midpoint is given by ($\frac{x_{1}+x_{2}}{2}$ , $\frac{y_{1}+y_{2}}{2}$)
The mid point will be an integer if both line joining integer pair has same parity i.e. (odd,odd), (odd,even), (even,odd), (even,even).
As we have 5 different pair of integers and 4 possible parity , by pigeonhole principle we can say that at least one pair of mid point has integer coordinates .
