86 lines
2.1 KiB
Markdown
86 lines
2.1 KiB
Markdown
---
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id: 5900f52e1000cf542c510041
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challengeType: 5
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title: 'Problem 450: Hypocycloid and Lattice points'
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forumTopicId: 302123
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---
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## Description
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<section id='description'>
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A hypocycloid is the curve drawn by a point on a small circle rolling inside a larger circle. The parametric equations of a hypocycloid centered at the origin, and starting at the right most point is given by:
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$x(t) = (R - r) \cos(t) + r \cos(\frac {R - r} r t)$
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$y(t) = (R - r) \sin(t) - r \sin(\frac {R - r} r t)$
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Where R is the radius of the large circle and r the radius of the small circle.
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Let $C(R, r)$ be the set of distinct points with integer coordinates on the hypocycloid with radius R and r and for which there is a corresponding value of t such that $\sin(t)$ and $\cos(t)$ are rational numbers.
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Let $S(R, r) = \sum_{(x,y) \in C(R, r)} |x| + |y|$ be the sum of the absolute values of the x and y coordinates of the points in $C(R, r)$.
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Let $T(N) = \sum_{R = 3}^N \sum_{r=1}^{\lfloor \frac {R - 1} 2 \rfloor} S(R, r)$ be the sum of $S(R, r)$ for R and r positive integers, $R\leq N$ and $2r < R$.
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You are given:C(3, 1) =
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{(3, 0), (-1, 2), (-1,0), (-1,-2)}
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C(2500, 1000) =
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{(2500, 0), (772, 2376), (772, -2376), (516, 1792),
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(516, -1792), (500, 0), (68, 504), (68, -504),(-1356, 1088), (-1356, -1088), (-1500, 1000), (-1500, -1000)}
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Note: (-625, 0) is not an element of C(2500, 1000) because $\sin(t)$ is not a rational number for the corresponding values of t.
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S(3, 1) = (|3| + |0|) + (|-1| + |2|) + (|-1| + |0|) + (|-1| + |-2|) = 10
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T(3) = 10; T(10) = 524 ;T(100) = 580442; T(103) = 583108600.
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Find T(106).
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</section>
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## Instructions
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<section id='instructions'>
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</section>
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## Tests
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<section id='tests'>
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```yml
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tests:
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- text: <code>euler450()</code> should return 583333163984220900.
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testString: assert.strictEqual(euler450(), 583333163984220900);
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```
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</section>
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## Challenge Seed
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<section id='challengeSeed'>
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<div id='js-seed'>
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```js
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function euler450() {
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// Good luck!
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return true;
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}
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euler450();
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```
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</div>
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</section>
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## Solution
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<section id='solution'>
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```js
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// solution required
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```
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</section>
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