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# Problems of the Week

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A uniform solid right-circular cone of mass $$M$$ and radius $$R$$ is kept on a rough horizontal floor (coefficient of friction $$\mu$$) on its circular base. It is spun with initial angular velocity $$\omega_{0}$$ about its symmetry axis. Neglecting toppling effects (if any), find the time after which it stops spinning.


Details and Assumptions:

• Take the moment of inertia of the cone about its symmetry axis as $$I$$.

###### If you are looking for more such twisted questions, Twisted problems for JEE aspirants is for you!

$$x,y,z,t$$ are all integers satisfying the following system of equations: \begin{cases} \begin{align} xz-2yt&=3 \\ xt+yz&=1. \end{align} \end{cases} Find $$x^2+y^2+z^2+ t^2$$.

A uniform wire of resistance $$R$$ is cut into four circular rings of equal radius. Rings are then connected such that their centers lie on the vertices of a square (as illustrated in the figure above).

If the equivalent resistance between $$A$$ and $$B$$ can be expressed as $$\frac{aR}{b}$$, where $$a$$ and $$b$$ are coprime positive integers, determine the value of $$a+b$$.

$S= \frac{1^{2}}{10}+\frac{2 \times 1^{2}+2^{2}}{10^{2}}+\frac{3 \times 1^{2}+2 \times 2^{2}+3^{2}}{10^{3}}+\cdots$

The sum $$S$$ defined above is an infinite sum whose $$n^\text{th}$$ term is $\dfrac{n \times 1^{2}+(n-1) \times 2^{2}+(n-2) \times 3^{2} + \cdots + n^{2}}{10^{n}}$ for $$n=1,2,\ldots$$.

If $$S$$ can be expressed in the form $$\frac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers, find $$a+b$$.

What is the maximum number of regions a plane may be cut into by 2017 ellipses?

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