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Solve the following system of equations for \(x\) and \(y.\)

$$\left\{ \begin{array}{ll} \log x + \log y = \log 2\\ x^{2}+y^{2} = 5 \end{array} \right.$$

Enter your answers below:

\(x = \)

\(y = \)

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:

$$x(t) = (R - r)\cos(t) + r\cos\left( \frac{R-r}{r}t \right)$$

$$y(t) = (R - r)\sin(t) - r\sin\left( \frac{R - r}{r}t \right)$$

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 \(\sum_{(x, y)\in C(R, r)} |x|+|y|\) 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:

$$C(3, 1) = \{(3, 0), (-1, 2), (-1,0), (-1,-2)\}$$

$$C(2500, 1000) = \{(2500, 0), (772, 2376), (772, -2376),$$

$$(516, 1792), (516, -1792), (500, 0), (68, 504),$$

$$(68, -504), (-1356, 1088), (-1356, -1088),$$

$$(-1500, 1000), (-1500, -1000)\}$$

Note: \((-625, 0)\) is not an element of \(C(2500, 1000)\) because \(\sin(t)\) is not a rational number for the corresponding values of \(t\).

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

$$(|-1| + |-2|) = 10$$

$$T(3) = 10; T(10) = 524 ;T(100) = 580442;$$

$$T\left(10^{3}\right) = 583108600.$$

Find \(T\left(10^{6}\right).\)

\(T\left(10^{6}\right)\) =

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