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# Problem 450

Hypocycloid and Lattice points

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, $R \le N$ and $2r \lt R$.

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).

$$x(t)=(R-r)\cos(t)+r \cos(\frac{R-r}{r}t)$$ $$y(t)=(R-r)\sin(t)-r \sin(\frac{R-r}{r}t)$$

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)}