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67 lines
2.1 KiB
Markdown
67 lines
2.1 KiB
Markdown
---
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id: 5900f52e1000cf542c510041
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title: 'Problem 450: Hypocycloid and Lattice points'
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challengeType: 5
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forumTopicId: 302123
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dashedName: problem-450-hypocycloid-and-lattice-points
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---
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# --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 $T(N) = \sum_{R = 3}^N \sum_{r=1}^{\left\lfloor \frac{R - 1}{2} \right\rfloor} S(R, r)$ 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:
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$$\begin{align}
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C(3, 1) = & \\{(3, 0), (-1, 2), (-1,0), (-1,-2)\\} \\\\
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C(2500, 1000) = & \\{(2500, 0), (772, 2376), (772, -2376), (516, 1792), (516, -1792), (500, 0), (68, 504), \\\\
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&(68, -504),(-1356, 1088), (-1356, -1088), (-1500, 1000), (-1500, -1000)\\}
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\end{align}$$
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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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$T(3) = 10$; $T(10) = 524$; $T(100) = 580\\,442$; $T({10}^3) = 583\\,108\\,600$.
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Find $T({10}^6)$.
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# --hints--
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`hypocycloidAndLatticePoints()` should return `583333163984220900`.
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```js
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assert.strictEqual(hypocycloidAndLatticePoints(), 583333163984220900);
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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 hypocycloidAndLatticePoints() {
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return true;
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}
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hypocycloidAndLatticePoints();
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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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