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Who is the liar?

Zack: It is possible to construct an 18-sided irregular polygon whose angles are equal to each other and whose side lengths are all rational.

Cody: It ...

The greatest lower bound of an ordered set \(S \subseteq U\) is the largest number \( x \) in \(U\) such that \( x \leq y\) for all \(y\) in \(S\).

For example ...

\[\frac{1}{ \sqrt[7]{531441} + \sqrt[7]{472392} + \sqrt[7]{419904} + \sqrt[7]{373248} + \sqrt[7]{331776} + \sqrt[7]{294912} + \sqrt[7]{262144} } \]

The expression above simplifies to ...

Find the number of positive integer factors of the determinant of the \( 2017 \times 2017\) matrix whose diagonal elements are all 2017 and off-diagonal elements are all 1.

Notes:

\[ \large\displaystyle \lim_{n\to\infty} \left(\dfrac {n^{n+1}e}{(n+1)^n} - n\right) = \, ? \] Clarification: \(e\) denotes Euler's number, \(e \approx 2.71828\).

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