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.
Without thinking got the 1st four (1,2,3,4)==> (1, sqrt(5), 5, sqrt(65))
time allowing, might try later