Excel in math, science, and engineering

Players A and B are playing a game:

2 circles intersect at A and B. The tangent to the first circle at A intersects the second circle at C. The tangent to the second circle at A intersects ...

Let \(0<x<1\) and \(0<y<1\).

Find the number of pairs \((x,y) \) such that \(53x + 11y\) and \(43x + 47y\) are both integers.

A celebrity among a group of \(n\) people is the one who knows nobody but is known by everyone else.

You go to a party and wonder if there is ...

True or False?

\[ \large \sum_{i=0}^n \dfrac1{ {n \choose i} } =\frac{n+1}{2^{n+1}} \sum_{i=1}^{n+1} \frac{2^i}i \]

Notation: ...

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