Problem 295Published on Saturday, 5th June 2010, 01:00 pm; Solved by 331; Difficulty rating: 75%
We call the convex area enclosed by two circles a lenticular hole if:
- The centres of both circles are on lattice points.
- The two circles intersect at two distinct lattice points.
- The interior of the convex area enclosed by both circles does not contain any lattice points.
Consider the circles:
The circles C0, C1 and C2 are drawn in the picture below.
C0 and C1 form a lenticular hole, as well as C0 and C2.
We call an ordered pair of positive real numbers (r1, r2) a lenticular pair if there exist two circles with radii r1 and r2 that form a lenticular hole. We can verify that (1, 5) and (5, √65) are the lenticular pairs of the example above.
Let L(N) be the number of distinct lenticular pairs (r1, r2) for which 0 < r1 ≤ r2 ≤ N.
We can verify that L(10) = 30 and L(100) = 3442.
Find L(100 000).