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Algebra

# Floor and Ceiling Functions: Level 5 Challenges

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 $$8\alpha + 7 \beta$$?

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

Define a function $$\displaystyle f(n) = n + \left \lfloor \sqrt{n} \right \rfloor$$.

Find the smallest value of $$k$$ such that the composite function,
$$f^{k} (2017) = \underbrace{f\circ f \circ f \circ \cdots \circ f}_{k \text{ times}} (2017)$$ can be expressed as $$m^2$$ for some positive integer $$m$$.

Submit your answer as $$m-k$$.

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

###### This problem is inspired from a Putnam problem.

Find the number of positive integers $$x$$ which satisfiy

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

$\large x^2 - 6\left\lfloor x\right\rfloor + 5 = 0$

Find the sum of squares of the solutions to the above equation.

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