If a circle is centered on the origin and has radius r, it may pass through some points with integer coordinates. In fact, by symmetry, this will be a multiple of 4.

Find the smallest radius that will pass through 4n integer points where n=1,2,3,4,5.

Feel free to go further.

Playing around with Excel, I came up with a radius of 25*sqrt(2) = about 35.35534. It passes through (5,35), (17,31), (25,25),(31,17) and (35,3).

5^2+35^2 = 17^2+31^2 = 25^2 + 25^2 = 1250. I didn't see a lower value available, but maybe I missed something (plus, I was only looking for multiples of sqrt(2).

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No, wait! I wrote too soon. 25 works.

It passes through (0,25), (15,20),(20,15),(7,24) and (24,7).

25^2 = 15^2+20^2 = 7^2 + 24^2 = 625.

And now I really don't see a minimum.

I guess I'll tackle n = 4 next.