Problem:
Distinct points P,Q,R, and S lie on the circle x2+y2=25 and have integer coordinates. The distances PQ and RS are irrational numbers. What is the greatest possible value of the ratio RSPQ?
Answer Choices:
A. 3
B. 5
C. 35
D. 7
E. 52
Solution:
The ratio RSPQ has its greatest value when PQ is as large as possible and RS is as small as possible. Points P,Q,R, and S have coordinates among (±5,0),(±4,±3),(±4,∓3),(±3,±4), (±3,∓4), and (0,±5). In order for the distance between two of these points to be irrational, the two points must not form a diameter, and they must not have the same x-coordinate or y-coordinate. If R=(a,b) and S=(a′,b′), then ∣a−a′∣≥1 and ∣b−b′∣≥1. Because (3,4) and (4,3) achieve this, they are as close as two points can be, 2 units apart. If P=(a,b) and Q=(a′,b′), then PQ is maximized when the distance from (a′,b′) to (−a,−b) is minimized. Because ∣a+a′∣≥1 and ∣b+b′∣≥1, the points (3,−4) and (−4,3) are as far apart as possible, 98 units. Therefore the greatest possible ratio is 298=49=(D)7.