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Find all Real Numbers such that $\lfloor x\lfloor x\rfloor\rfloor=2024$

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I just came across an Olympiad problem that says:$$\lfloor x\lfloor x\rfloor\rfloor=2024$$I have never encountered floor functions in my life and so I have tried this approach to this:$$x^2=2024\Rightarrow x=44.98$$And therefore, $x$ can take a value around this. Now the maximum value of $x\lfloor x\rfloor$ needs to be less than $2025$ which leads to:$$2024<x\lfloor x\rfloor<2025\Rightarrow \frac{2024}{x}<\lfloor x\rfloor <\frac{2025}{x}$$and then I am completely confused. Now can I lead further?


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