Algebra
# Floor and Ceiling Functions

Find the number of positive integers \(x\) which satisfiy

\[\left\lfloor \dfrac {x}{99} \right\rfloor = \left\lfloor \dfrac {x}{101} \right\rfloor . \]

Given that \(1\le x \le k,\) there are 2017 solutions to the equation \[\large x^3-\left\lfloor x^3 \right\rfloor = \left( x - \lfloor x \rfloor \right)^3.\] The minimum value of \(k\) is \(a+\frac{b\sqrt{c}}{d},\) where \(a, b, c, d\) are positive integers.

Given that \(c\) is square-free and \(b\) and \(d\) are coprime, find the value of \(a+b+c+d.\)

\(\)

**Notation:** \( \lfloor \cdot \rfloor \) denotes the floor function.

Find

\[ \displaystyle\sum_{k=1}^{1995}\dfrac 1{f(k)}, \]

where \(f(n)\) is the integer **closest** to \(\sqrt[4]{n}.\)

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