An integer point has integer coordinates. Also call point (x,y) different if all the distances from (x,y) to integer points are different.
Show that (3/4, 2/5) is not different
So you want to show that there exist points $\displaystyle (x_1, y_1)$ and $\displaystyle (x_2, y_2)$, either $\displaystyle x_1\ne x_2$ or $\displaystyle y_2\ne y_1$, such that $\displaystyle \sqrt{(x_1- 3/4)^2+ (y_1- 2/5)^2}= \sqrt{(x_2- 3/4)^2+ (y_2- 2/5)^2}$.
I recommend starting by squaring both sides. You can then write the equation in terms of $\displaystyle x_1- x_2$ and $\displaystyle y_1- y_2$. Obviously one solution will be $\displaystyle x_1= x_2$ and $\displaystyle y_1= y_2$ but there is another solution. Find it.