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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 $\text{\_\_\_\_\_\_\_\_\_\_}.$
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$n$ | $\hspace{10mm} 2^n$ | 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?
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$x^2 + y^2 \leq 1$ is a unit disk, and the area of the region is $\pi.$
It's common knowledge that the graph ofHowever, 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!
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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 $1?$
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Three identical 30-60-90 right triangles are arranged as shown in the figure.
What is the ratio of the $\color{#D61F06}\text{red segment's}$ length to the $\color{#3D99F6}\text{blue segment's}$ length?
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