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Imagine the points (Posted on 2021-09-22) Difficulty: 3 of 5
A circle with radius 4410001/2 is drawn at on a Cartesian plane with the origin as its centre.

How many lattice points (coordinate with integer coordinates) does it pass?

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution computer solution | Comment 1 of 3
clc, clearvars
intCt=0;
r=sqrt(sym(441000));
for x0=0:floor(r)
    x=sym(x0);
    y=sqrt(r*r-x*x);
    if y==round(y)
       disp([x round(y)]) 
    else
       intCt=intCt+floor(y);
    end
end
disp(intCt*4+1)

finds points

[210, 630]
[378, 546]
[546, 378]
[630, 210]

that the circle passes through

That's just the first quadrant, so there are 16 lattice points exactly on the circumference of the circle.

While it was at it, not asked for in the puzzle, the program counted 1,378,381 lattice points strictly within the circle. Well actually it counted  344,595 points on the positive y axis and strictly in the first quadrant not on any axis or circumference, which I multiplied by 4 and added 1 for the origin.

  Posted by Charlie on 2021-09-22 11:06:03
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