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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

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