55 lines
1.8 KiB
Markdown
55 lines
1.8 KiB
Markdown
---
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id: 5900f53d1000cf542c510050
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title: 'Problem 465: Polar polygons'
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challengeType: 5
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forumTopicId: 302140
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dashedName: problem-465-polar-polygons
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---
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# --description--
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The kernel of a polygon is defined by the set of points from which the entire polygon's boundary is visible. We define a polar polygon as a polygon for which the origin is strictly contained inside its kernel.
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For this problem, a polygon can have collinear consecutive vertices. However, a polygon still cannot have self-intersection and cannot have zero area.
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For example, only the first of the following is a polar polygon (the kernels of the second, third, and fourth do not strictly contain the origin, and the fifth does not have a kernel at all):
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<img class="img-responsive center-block" alt="five example polygons" src="https://cdn.freecodecamp.org/curriculum/project-euler/polar-polygons.png" style="background-color: white; padding: 10px;">
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Notice that the first polygon has three consecutive collinear vertices.
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Let $P(n)$ be the number of polar polygons such that the vertices $(x, y)$ have integer coordinates whose absolute values are not greater than $n$.
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Note that polygons should be counted as different if they have different set of edges, even if they enclose the same area. For example, the polygon with vertices [(0,0), (0,3), (1,1), (3,0)] is distinct from the polygon with vertices [(0,0), (0,3), (1,1), (3,0), (1,0)].
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For example, $P(1) = 131$, $P(2) = 1\\,648\\,531$, $P(3) = 1\\,099\\,461\\,296\\,175$ and $P(343)\bmod 1\\,000\\,000\\,007 = 937\\,293\\,740$.
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Find $P(7^{13})\bmod 1\\,000\\,000\\,007$.
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# --hints--
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`polarPolygons()` should return `585965659`.
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```js
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assert.strictEqual(polarPolygons(), 585965659);
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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 polarPolygons() {
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return true;
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}
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polarPolygons();
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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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