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2018-01-08 Intermediate

         

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

Fe=keq2d2,F_e=k_e\frac{q^2}{d^2},

where kek_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+q and q-q are separated by center-to-center distance d.d.

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

nn2n\hspace{10mm} 2^nConcatenation of the powered numbersDivisibility checked
020=1\hspace{5mm} 2^0=11\hspace{25mm} 111\hspace{8mm} 1\, \big|\, 1
121=2\hspace{5mm} 2^1=212\hspace{25mm} 12212\hspace{8mm} 2\, \big|\, 12
222=4\hspace{5mm} 2^2=4124\hspace{25mm} 1244124\hspace{8mm} 4\, \big|\, 124
323=8\hspace{5mm} 2^3=81248\hspace{25mm} 124881248\hspace{8mm} 8\, \big|\, 1248
424=16\hspace{5mm} 2^4=16124816\hspace{25mm} 12481616124816\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,n>4, will the divisibility in the last column still hold?

It's common knowledge that the graph of x2+y21x^2 + y^2 \leq 1 is a unit disk, and the area of the region is π.\pi.

However, it's not common for someone to know the shape of the graph x2+y2=1, \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 x2+y2=1? \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:RRf: \mathbb{R} \rightarrow \mathbb{R} such that for every two distinct real numbers aa and b,b, f(a)f(a) and f(b)f(b) differ by at least 1?1?

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

What is the ratio of the red segment’s\color{#D61F06}\text{red segment's} length to the blue segment’s\color{#3D99F6}\text{blue segment's} length?

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