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Discrete Mathematics Level 5

\(A\) is the number mentioned in this problem.

Suppose that there is a rational number sequence \(s_1\leq s_2\leq\cdots\leq s_{30}\) sastifying:

\[\left\{\begin{matrix} As_1+As_2+As_3+\cdots+As_{30}=k \quad (k\in \mathbb{Q}) \\ \mid s_1\mid +\mid s_2\mid +\cdots+\mid s_{30}\mid =2016 & & \end{matrix}\right.\]

If the minimum value of \(s_{30}-s_1\) can be expressed as \( \dfrac de \), where \(d\) and \(e\) are coprime positive integers, find \(d+e\).

Notation: \(\mathbb Q \) denotes the set of rational numbers.


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