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.
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Details and Assumptions:
\(x,y,z,t\) 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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