MathDB

22

Part of 2016 Harvard-MIT Mathematics Tournament

Problems(1)

2016 Guts #22

Source:

12/24/2016
On the Cartesian plane R2\mathbb{R}^2, a circle is said to be nice if its center is at the origin (0,0)(0,0) and it passes through at least one lattice point (i.e. a point with integer coordinates). Define the points A=(20,15)A = (20,15) and B=(20,16)B = (20,16). How many nice circles intersect the open segment ABAB?
For reference, the numbers 601601, 607607, 613613, 617617, 619619, 631631, 641641, 643643, 647647, 653653, 659659, 661661, 673673, 677677, 683683, 691691 are the only prime numbers between 600600 and 700700.