If two point charges \(+q\) and \(−q\) are separated by distance \(d,\) then according to Coulomb's law, the attractive force between them is

\[F_e=k_e\frac{q^2}{d^2},\]

where \(k_e\) is Coulomb's constant. But what if the charges were distributed on spheres instead of at points? Two charged conducting spheres with charges \(+q\) and \(−q\) are separated by center-to-center distance \(d.\)

The attractive force between them is \(\text{__________}.\)

\(n\) | \(\hspace{10mm} 2^n\) | Concatenation of the powered numbers | Divisibility checked |

0 | \(\hspace{5mm} 2^0=1\) | \(\hspace{25mm} 1\) | \(\hspace{8mm} 1\, \big|\, 1\) |

1 | \(\hspace{5mm} 2^1=2\) | \(\hspace{25mm} 12\) | \(\hspace{8mm} 2\, \big|\, 12\) |

2 | \(\hspace{5mm} 2^2=4\) | \(\hspace{25mm} 124\) | \(\hspace{8mm} 4\, \big|\, 124\) |

3 | \(\hspace{5mm} 2^3=8\) | \(\hspace{25mm} 1248\) | \(\hspace{8mm} 8\, \big|\, 1248\) |

4 | \(\hspace{5mm} 2^4=16\) | \(\hspace{25mm} 124816\) | \(\hspace{8mm} 16\, \big|\, 124816\) |

As we get greater and greater numbers in column 3 of the table by concatenation (i.e. 12481632, 1248163264, ...) for \(n>4,\) will the divisibility in the last column still hold?

However, it's not common for someone to know the shape of the graph \[ \big\lfloor x^2 \big\rfloor + \big\lfloor y^2 \big\rfloor = 1,\] where \(\lfloor \cdot \rfloor\) is the floor function. To three decimal places, what is the area of the region on the coordinate plane that satisfies the equation \( \big\lfloor x^2 \big\rfloor + \big\lfloor y^2 \big\rfloor = 1?\)

\(\)

**Note:** Try to draw a hand sketch of this curve and provide analysis instead of using any software!

Three identical 30-60-90 right triangles are arranged as shown in the figure.

What is the ratio of the \(\color{red}\text{red segment's}\) length to the \(\color{blue}\text{blue segment's}\) length?

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