1.9 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}^{\\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).
--hints--
euler450()
should return 583333163984220900.
assert.strictEqual(euler450(), 583333163984220900);
--seed--
--seed-contents--
function euler450() {
return true;
}
euler450();
--solutions--
// solution required