53 lines
1.6 KiB
Markdown
53 lines
1.6 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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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) = 1648531, P(3) = 1099461296175 and P(343) mod 1 000 000 007 = 937293740.
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Find P(713) mod 1 000 000 007.
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# --hints--
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`euler465()` should return 585965659.
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```js
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assert.strictEqual(euler465(), 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 euler465() {
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return true;
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}
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euler465();
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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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