63 lines
1.7 KiB
Markdown
63 lines
1.7 KiB
Markdown
---
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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:
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$$\begin{align}
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& C_0: x^2 + y^2 = 25 \\\\
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& C_1: {(x + 4)}^2 + {(y - 4)}^2 = 1 \\\\
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& C_2: {(x - 12)}^2 + {(y - 4)}^2 = 65
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\end{align}$$
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The circles $C_0$, $C_1$ and $C_2$ are drawn in the picture below.
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<img class="img-responsive center-block" alt="C_0, C_1 and C_2 circles" src="https://cdn.freecodecamp.org/curriculum/project-euler/lenticular-holes.gif" style="background-color: white; padding: 10px;">
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$C_0$ and $C_1$ form a lenticular hole, as well as $C_0$ and $C_2$.
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We call an ordered pair of positive real numbers ($r_1$, $r_2$) a lenticular pair if there exist two circles with radii $r_1$ and $r_2$ that form a lenticular hole. We can verify that ($1$, $5$) and ($5$, $\sqrt{65}$) are the lenticular pairs of the example above.
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Let $L(N)$ be the number of distinct lenticular pairs ($r_1$, $r_2$) for which $0 < r_1 ≤ r_2 ≤ N$. 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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`lenticularHoles()` should return `4884650818`.
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```js
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assert.strictEqual(lenticularHoles(), 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 lenticularHoles() {
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return true;
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}
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lenticularHoles();
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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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