2017 AMC 10A Problem 17

The integer-coordinate points on $x^2+y^2=25$ are $(\pm5,0)$, $(0,\pm5)$, $(\pm3,\pm4)$, and $(\pm4,\pm3)$. For $PQ$ and $RS$ to be irrational, the squared distance must not be a perfect square. To maximize the ratio, make $PQ$ as large as possible and $RS$ as small as possible under that condition. The largest possible irrational distance is between $(-4,3)$ and $(3,-4)$, giving $PQ=\sqrt{7^2+7^2}=\sqrt{98}$. The smallest possible irrational distance is between $(3,4)$ and $(4,3)$, giving $RS=\sqrt{1^2+1^2}=\sqrt2$. The greatest possible ratio is $\dfrac{\sqrt{98}}{\sqrt2}=7$. Thus, D is the correct answer.