Algebra

Floor and Ceiling Functions

Floor and Ceiling Functions: Level 5 Challenges

         

Find the number of positive integers xx which satisfiy

x99=x101.\left\lfloor \dfrac {x}{99} \right\rfloor = \left\lfloor \dfrac {x}{101} \right\rfloor .

1+1.7+2.4+3.1++999.9=?\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 1xk,1\le x \le k, there are 2017 solutions to the equation x3x3=(xx)3.\large x^3-\left\lfloor x^3 \right\rfloor = \left( x - \lfloor x \rfloor \right)^3. The minimum value of kk is a+bcd,a+\frac{b\sqrt{c}}{d}, where a,b,c,da, b, c, d are positive integers.

Given that cc is square-free and bb and dd are coprime, find the value of a+b+c+d.a+b+c+d.


Notation: \lfloor \cdot \rfloor denotes the floor function.

Find

k=119951f(k), \displaystyle\sum_{k=1}^{1995}\dfrac 1{f(k)},

where f(n)f(n) is the integer closest to n4.\sqrt[4]{n}.

If the range of positive xx satisfy the equation xx+xx=111 \lceil x \lfloor x \rfloor \rceil + \lfloor x \lceil x \rceil \rfloor = 111 is αxβ \alpha \leq x \leq \beta . What is the value of 8α+7β8\alpha + 7 \beta?

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