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# 2018-01-08 Intermediate 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? It's common knowledge that the graph of $$x^2 + y^2 \leq 1$$ is a unit disk, and the area of the region is $$\pi.$$

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!

Does there exist a function $$f: \mathbb{R} \rightarrow \mathbb{R}$$ such that for every two distinct real numbers $$a$$ and $$b,$$ $$f(a)$$ and $$f(b)$$ differ by at least $$1?$$ 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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