Problem:
Centered at each lattice point in the coordinate plane are a circle radius and a square with sides of length whose sides are parallel to the coordinate axes. The line segment from to intersects of the squares and of the circles. Find .
Solution:
Let and . Between the points and can intersect the square at lattice point only if it passes between the upper-left and lower-right corners of the square. That is,
This implies . The only lattice points with which satisfy this requirement are , and . In the cases of and passes through the center of the square centered at that point, so it also intersects the circle centered at that point. In the cases of and passes through the lower-right and upper-left corners of the square centered at that point, respectively, so it does not interest the circle centered at that point. Altogether the segment joining to intersects squares and circles. The same conclusion applies to the segment joining to for each with . Because the points ) belong to two of these segments for intersects of the squares and of the circles. The requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions