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Limit of a sum of greatest integer terms

This type of question, asked 2 times: 2021 Paper 8, 29 Jan 2025. All 2 questions below, oldest first.

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  1. Question 1

    2021 Paper 8

    65
    The value of lim⁡n→∞[r]+[2r]+…..+[nr]n2\lim_{n\rightarrow \infty} \frac{\lbrack r\rbrack + \lbrack 2r\rbrack + \ldots.. + \lbrack nr\rbrack}{n^{2}}, where rr is non-zero real number and [r] denotes the greatest integer less than or equal to rr, is equal to :

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  2. Question 2

    29 Jan 2025

    73
    Let [t]\lbrack t\rbrack be the greatest integer less than or equal to tt. Then the least value of p∈Np \in N for which
    lim⁡x→0+(x([1x]+[2x]+…..+[px])−x2([1x2]+[22x2]+….+[92x2]))≥1\underset{x \rightarrow0^{+}}{\lim} \left( x\left( \left\lbrack \frac{1}{x} \right\rbrack+\left\lbrack \frac{2}{x} \right\rbrack+ \ldots.. +\left\lbrack \frac{p}{x} \right\rbrack \right)-x^{2}\left( \left\lbrack \frac{1}{x^{2}} \right\rbrack+\left\lbrack \frac{2^{2}}{x^{2}} \right\rbrack+ \ldots. +\left\lbrack \frac{9^{2}}{x^{2}} \right\rbrack \right) \right)\geq 1
    is equal to