yaworsw/euler-manager

---
:id: 295
:name: Lenticular holes
:url: https://projecteuler.net/problem=295
:content: "We call the convex area enclosed by two circles a _lenticular hole_ if:\n\n-
  The centres of both circles are on lattice points.\n- The two circles intersect
  at two distinct lattice points.\n- The interior of the convex area enclosed by both
  circles does not contain any lattice points.\n\nConsider the circles:  \nC<sub>0</sub>:
  <var>x</var><sup>2</sup>+<var>y</var><sup>2</sup>=25  \nC<sub>1</sub>: (<var>x</var>+4)<sup>2</sup>+(<var>y</var>-4)<sup>2</sup>=1
  \ \nC<sub>2</sub>: (<var>x</var>-12)<sup>2</sup>+(<var>y</var>-4)<sup>2</sup>=65\n\nThe
  circles C<sub>0</sub>, C<sub>1</sub> and C<sub>2</sub> are drawn in the picture
  below.\n\n ![p295_lenticular.gif]({{ images_dir }}/p295_lenticular.gif)\n\nC<sub>0</sub>
  and C<sub>1</sub> form a lenticular hole, as well as C<sub>0</sub> and C<sub>2</sub>.\n\nWe
  call an ordered pair of positive real numbers (r<sub>1</sub>, r<sub>2</sub>) a _lenticular
  pair_ if there exist two circles with radii r<sub>1</sub> and r<sub>2</sub> that
  form a lenticular hole. We can verify that (1, 5) and (5, √65) are the lenticular
  pairs of the example above.\n\nLet L(N) be the number of **distinct** lenticular
  pairs (r<sub>1</sub>, r<sub>2</sub>) for which 0 \\< r<sub>1</sub> ≤ r<sub>2</sub>
  ≤ N.  \nWe can verify that L(10) = 30 and L(100) = 3442.\n\nFind L(100 000).\n\n"