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Floor and Ceiling Functions

Floor and Ceiling Functions: Level 5 Challenges


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

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

\(\left\lceil 1 \right\rceil + \left\lceil 1.7 \right\rceil + \left\lceil 2.4 \right\rceil + \left\lceil 3.1 \right\rceil+\ldots+\left\lceil 999.9 \right\rceil=?\)

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.


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

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

If the range of positive \(x\) satisfy the equation \( \lceil x \lfloor x \rfloor \rceil + \lfloor x \lceil x \rceil \rfloor = 111 \) is \( \alpha \leq x \leq \beta \). What is the value of \(8\alpha + 7 \beta\)?


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