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Problem 450

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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 $2r<R$.

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).

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圆内旋轮线与格点

在大圆内部有一小圆与其相切并不断滚动，由小圆上一点所构成的曲线称为圆内旋轮线。以原点为圆心，以最右侧的点为起点的圆内旋轮线，其参数方程如下所示：

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

其中R是大圆的半径，而r是小圆的半径。

记$C(R,r)$是大小圆半径分别为R和r时落在圆内旋轮线上的格点的集合，且此时对应的t值满足$\sin (t)$和$\cos (t)$都是有理数。

记$S(R,r)=\sum _{(x,y)\in C(R,r)}|x|+|y|$是$C(R,r)$中所有点的坐标绝对值之和。

记$T(N)=\sum _{R=3}^N\sum _{r=1}^{\lfloor \frac{R-1}{2}\rfloor }S(R,r)$是满足R和r均为正整数，$R\le N$以及$2r<R$的所有$S(R,r)$的和。

已知：
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)}

注意：(-625, 0)不是C(2500, 1000)中的元素，此时对应的t值不满足$\sin (t)$为有理数。

因此S(3, 1) = (|3| + |0|) + (|-1| + |2|) + (|-1| + |0|) + (|-1| + |-2|) = 10

且已知T(3) = 10；T(10) = 524；T(100) = 580442；T(10³) = 583108600。

求T(10⁶)。

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