HMMT 2012 A5

Algebra
Answer

Find all ordered triples $(a, b, c)$ of positive reals that satisfy: $\lfloor a\rfloor b c=3, a\lfloor b\rfloor c=4$ , and $a b\lfloor c\rfloor=5$ , where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$ .

Dataset text NuminaMath olympiads. Quoted with source.

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\left(\frac{\sqrt{30}}{3}, \frac{\sqrt{30}}{4}, \frac{2 \sqrt{30}}{5}\right),\left(\frac{\sqrt{30}}{3}, \frac{\sqrt{30}}{2}, \frac{\sqrt{30}}{5}\right)

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