06/07/2020 (2011: National Sprint Round, Problem 22)

Q: Circle O has a radius of 10 units . Point P is on radius OQ and OP = 6 units. How many different chords containing P, including the diameter, have integer lengths?

A: 8

Because the radius of circle O is 10, and OP is 6, that means the chord perpendicular to OQ passing through P is 2 * √(10² - 6²) = 2 * √(100 - 36) = 2 * √(64) = 2 * 8 = 16. That is the shortest chord to pass through P.

The longest chord is the diameter, which is 10 * 2 = 20 inches. That means there are (20 - 16) - 2 = 3 integer lengths between 20 and 16 (non-inclusive). There are 2 of each of these chords, making there be 3 * 2 = 6 chords.

We add the 2 chords left over (20 & 16 ) and get 6 + 2 = 8.