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2023 IOQM/Problem 1

Revision as of 23:48, 25 September 2023 by Sansgankrsngupta (talk | contribs) (Created page with "==Problem== Let <math>n</math> be a positive integer such that <math>1 \leq n \leq 1000</math>. Let <math>M_n</math> be the number of integers in the set <math>...")
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Problem

Let $n$ be a positive integer such that $1 \leq n \leq 1000$. Let $M_n$ be the number of integers in the set

$X_n = \left\{\sqrt{4n + 1}, \sqrt{4n + 2}, \ldots, \sqrt{4n + 1000}\right\}$. Let $a = \max\{M_n : 1 \leq n \leq 1000\}$, and $b = \min\{M_n : 1 \leq n \leq 1000\}$.

Find $a - b$.

Solution

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