In the \(xyz\)-coordinate system, a thin solid square of side length \(S\) lies in the \(xy\)-plane with its center at the origin. An isotropic light source is positioned at \((x,y,z) = \left(0,0,\frac{S}{2}\right)\).

What percentage of the light energy is incident on the square \((\)to one decimal place; i.e., \(66\frac{2}{3}\%\) would be entered as 66.7\()?\)

\[ m^3-n^3+mn(m-n)=315^9\]

Find the number of solutions \((m, n)\) to the equation above, where \(m\) and \(n\) are both positive rational numbers and their difference is an integer.

In the pattern lock system below,

- all of the 9 nodes have to be connected without any cycle, and
- any connection using one or more diagonal edges is not allowed.

For example, pattern #1 below is good.

Pattern #2 is invalid because it includes a cycle, as indicated by the red square. Pattern #3 is invalid because it contains an implicit cycle, i.e. the red segment must be traveled twice to go from the lower left corner to the lower right corner. Pattern #4 is invalid because a diagonal edge is used to connect 2 nodes.

So, how many different patterns can be generated for this system?

**Bonus**: What if using diagonal edges is allowed while intersections between 2 or more diagonal edges are prohibited. Can you find the number of different patterns in this case?

An infinity of circles covers part of the plane in a fractal spiral. As the figure below shows, there is one largest circle, which we'll call the first one, followed by others whose radii are in geometric progression. The \(9^\text{th}\) and \(10^\text{th}\) circles are tangential to the first circle.

Let \(A\) be the total area of all the circles, and \(B\) the total area of the interstitial spaces formed by these circles. What is \( \left \lfloor 1000 \times \frac{ A } { A+B} \right \rfloor\)?

**Bonus Question:** Does "circle area density on the infinite plane" always has a meaning? For a regular array of circles on the infinite plane, it does. When does it fail to have a meaning? Consider the case of "mass density" of the universe, for example. Does that always necessarily have a meaning?

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