57 lines
1.3 KiB
Markdown
57 lines
1.3 KiB
Markdown
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---
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id: 5900f4931000cf542c50ffa6
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title: 'Problem 295: Lenticular holes'
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challengeType: 5
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forumTopicId: 301947
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dashedName: problem-295-lenticular-holes
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---
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# --description--
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We call the convex area enclosed by two circles a lenticular hole if:
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The centres of both circles are on lattice points.
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The two circles intersect at two distinct lattice points.
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The interior of the convex area enclosed by both circles does not contain any lattice points.
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Consider the circles: C0: x2+y2=25 C1: (x+4)2+(y-4)2=1 C2: (x-12)2+(y-4)2=65
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The circles C0, C1 and C2 are drawn in the picture below.
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C0 and C1 form a lenticular hole, as well as C0 and C2.
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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.
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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.
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Find L(100 000).
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# --hints--
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`euler295()` should return 4884650818.
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```js
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assert.strictEqual(euler295(), 4884650818);
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```
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# --seed--
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## --seed-contents--
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```js
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function euler295() {
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return true;
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}
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euler295();
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```
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# --solutions--
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```js
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// solution required
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```
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