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Find the set of integers \(n\) such that \[ n^6 + 24n^3 + 192 \] is a perfect cube. If there are \(N\) distinct solutions \(n_1,n_2,\ldots ,n_N\), submit ...

Find all solutions to the equation \[ 2n^3 + 6n^2 + 102n + 98 \; = \; m^3 \hspace{2cm} m,n \in \mathbb{Z} \;. \] If the answers are ...

A uniform solid in the shape of a quarter segment of a sphere of radius \(R\) is released from rest with its diameter vertical, and the center a height of ...

Triangle \(OAB\) has vertices with coordinates \(O=(0,0), A=(4,0), B=(0,2)\). As shown in the diagram, there are \(9\) points with integer coordinates that lie either ...

Evaluate the limit \[\large \lim_{n \to \infty} n^2\sum_{j=1}^n \frac{(-1)^{n-j} j^{n-1} \ln j}{j! (n-j)!}. \]

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