51 lines
1.1 KiB
Markdown
51 lines
1.1 KiB
Markdown
---
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id: 5900f5351000cf542c510047
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title: 'Problem 456: Triangles containing the origin II'
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challengeType: 5
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forumTopicId: 302130
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dashedName: problem-456-triangles-containing-the-origin-ii
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---
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# --description--
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Define:
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$$\begin{align} & x_n = ({1248}^n\bmod 32323) - 16161 \\\\ & y_n = ({8421}^n\bmod 30103) - 15051 \\\\ & P_n = \\{(x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\\} \end{align}$$
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For example, $$P_8 = \\{(-14913, -6630), (-10161, 5625), (5226, 11896), (8340, -10778), (15852, -5203), (-15165, 11295), (-1427, -14495), (12407, 1060)\\}$$
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Let $C(n)$ be the number of triangles whose vertices are in $P_n$ which contain the origin in the interior.
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Examples:
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$$\begin{align} & C(8) = 20 \\\\ & C(600) = 8\\,950\\,634 \\\\ & C(40\\,000) = 2\\,666\\,610\\,948\\,988 \end{align}$$
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Find $C(2\\,000\\,000)$.
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# --hints--
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`trianglesContainingOriginTwo()` should return `333333208685971500`.
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```js
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assert.strictEqual(trianglesContainingOriginTwo(), 333333208685971500);
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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 trianglesContainingOriginTwo() {
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return true;
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}
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trianglesContainingOriginTwo();
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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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