Problem 1387. Points on a circle.

Created by James

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This problem is related to 1283, Points on a Sphere. In this case, instead of a sphere, you have a circle. Given a radius R, calculate the number of points on the circumference of the circle that have two integer coordinates. For a circle of radius 5, you would have 12 points:

(0, 5) and (0, -5)
(5, 0) and (-5, 0)
(4, 3) and (4, -3)
(-4, 3) and (-4, -3)
(3, 4) and (3, -4)
(-3, 4) and (-3, -4)

Some radii are quite large, so watch out. Good luck!

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Last Solution submitted on Nov 26, 2025

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Chris Cleveland on 14 Jul 2017

Great problem! Learned a bunch from others solutions as well.

HH on 3 Dec 2017

In the problem statement, I think you meant '3' and '4', rather than '1' and '2', in the respective +/- combinations.

James on 4 Dec 2017

Thanks for catching my stupidity on that one, HH. It's fixed now.

Rafael S.T. Vieira on 23 Jun 2020

The tip for this question is the sum of squares function, which I use in my solution. There are many possible solutions to this problem, but the huge circle radii limit what we can employ, so be careful.

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Problem 1387. Points on a circle.

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Problem 1387. Points on a circle.

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Problem Recent Solvers24

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