Question

Distinct points $P,$ $Q,$ $R,$ $S$ lie on the circle $x^{2}+y^{2}=25$ and have integer coordinates. The distances $PQ$ and $RS$ are irrational numbers.

What is the greatest possible value of the ratio $\dfrac{PQ}{RS}?$

$3$

$5$

$3\sqrt{5}$

$7$

$5\sqrt{2}$

Answer: D

Concepts:lattice point circle distance formula optimization

Difficulty rating: 1790

Solution:

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.

2017 AMC 10A Problem 17

Answer: D

Difficulty rating: 1790

Solution:

Problem 17 in Other Years