2.1 KiB
| id | title | challengeType | forumTopicId | dashedName |
|---|---|---|---|---|
| 5900f52e1000cf542c510041 | Problem 450: Hypocycloid and Lattice points | 5 | 302123 | problem-450-hypocycloid-and-lattice-points |
--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}^{\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.
You are given:
\begin{align}
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)\\}
\end{align}$$
**Note:** (-625, 0) is not an element of $C(2500, 1000)$ because $\sin(t)$ is not a rational number for the corresponding values of $t$.
$S(3, 1) = (|3| + |0|) + (|-1| + |2|) + (|-1| + |0|) + (|-1| + |-2|) = 10$
$T(3) = 10$; $T(10) = 524$; $T(100) = 580\\,442$; $T({10}^3) = 583\\,108\\,600$.
Find $T({10}^6)$.
# --hints--
`hypocycloidAndLatticePoints()` should return `583333163984220900`.
```js
assert.strictEqual(hypocycloidAndLatticePoints(), 583333163984220900);
```
# --seed--
## --seed-contents--
```js
function hypocycloidAndLatticePoints() {
return true;
}
hypocycloidAndLatticePoints();
```
# --solutions--
```js
// solution required
```