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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.

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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.

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

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

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

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

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