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Floor and Ceiling Functions

Floor functions map a real number x to the largest integer less than or equal to x. You can probably guess what the ceiling function does.

Level 5

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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