Excel in math and science
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Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, interactive explorations.
Used and loved by over 7 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
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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 \(\text{__________}.\)
by
A Brilliant Member
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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?
by
Michael Huang
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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!
by
Rupen Kohli
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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 \(1?\)
by
George Ivanchyk
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Three identical 30-60-90 right triangles are arranged as shown in the figure.
What is the ratio of the \(\color{red}\text{red segment's}\) length to the \(\color{blue}\text{blue segment's}\) length?
by
Ronald Chén
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