79d9012432
* fix(curriculum): tests quotes * fix(curriculum): fill seed-teardown * fix(curriculum): fix tests and remove unneeded seed-teardown
84 lines
2.1 KiB
Markdown
84 lines
2.1 KiB
Markdown
---
|
|
id: 5900f52e1000cf542c510041
|
|
challengeType: 5
|
|
title: 'Problem 450: Hypocycloid and Lattice points'
|
|
---
|
|
|
|
## Description
|
|
<section id='description'>
|
|
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:
|
|
$x(t) = (R - r) \cos(t) + r \cos(\frac {R - r} r t)$
|
|
$y(t) = (R - r) \sin(t) - r \sin(\frac {R - r} r t)$
|
|
Where R is the radius of the large circle and r the radius of the small circle.
|
|
|
|
|
|
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.
|
|
|
|
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 $C(R, r)$.
|
|
|
|
|
|
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 $2r < R$.
|
|
|
|
|
|
You are given:C(3, 1) =
|
|
{(3, 0), (-1, 2), (-1,0), (-1,-2)}
|
|
C(2500, 1000) =
|
|
{(2500, 0), (772, 2376), (772, -2376), (516, 1792),
|
|
(516, -1792), (500, 0), (68, 504), (68, -504),(-1356, 1088), (-1356, -1088), (-1500, 1000), (-1500, -1000)}
|
|
|
|
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.
|
|
|
|
|
|
S(3, 1) = (|3| + |0|) + (|-1| + |2|) + (|-1| + |0|) + (|-1| + |-2|) = 10
|
|
|
|
T(3) = 10; T(10) = 524 ;T(100) = 580442; T(103) = 583108600.
|
|
|
|
|
|
Find T(106).
|
|
</section>
|
|
|
|
## Instructions
|
|
<section id='instructions'>
|
|
|
|
</section>
|
|
|
|
## Tests
|
|
<section id='tests'>
|
|
|
|
```yml
|
|
tests:
|
|
- text: <code>euler450()</code> should return 583333163984220900.
|
|
testString: assert.strictEqual(euler450(), 583333163984220900, '<code>euler450()</code> should return 583333163984220900.');
|
|
|
|
```
|
|
|
|
</section>
|
|
|
|
## Challenge Seed
|
|
<section id='challengeSeed'>
|
|
|
|
<div id='js-seed'>
|
|
|
|
```js
|
|
function euler450() {
|
|
// Good luck!
|
|
return true;
|
|
}
|
|
|
|
euler450();
|
|
```
|
|
|
|
</div>
|
|
|
|
|
|
|
|
</section>
|
|
|
|
## Solution
|
|
<section id='solution'>
|
|
|
|
```js
|
|
// solution required
|
|
```
|
|
</section>
|