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 numbersDivisibility 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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