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Junior Balkan Mathematical Olympiad 2021- P1

Source: JBMO 2021

July 1, 2021
JuniorBalkanfloor functionequationalgebra

Problem Statement

Let nn (n≥1n \ge 1) be an integer. Consider the equation
2⋅⌊12x⌋−n+1=(n+1)(1−nx)2\cdot \lfloor{\frac{1}{2x}}\rfloor - n + 1 = (n + 1)(1 - nx),
where xx is the unknown real variable.
(a) Solve the equation for n=8n = 8. (b) Prove that there exists an integer nn for which the equation has at least 20212021 solutions. (For any real number yy by ⌊y⌋\lfloor{y} \rfloor we denote the largest integer mm such that m≤ym \le y.)
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