iflugkiwmu wrote: Let $a,b,c$ be real number greater than 1 satisfying $$\lfloor a\rfloor b = \lfloor b \rfloor c = \lfloor c\rfloor a.$$Prove that $a=b=c$ (Here, $\lfloor x \rfloor$ denotes the laregst integer that is less than or equal to $x$.) Note that $a,b,c> 1$ $\implies$ $\lfloor a\rfloor,\lfloor b\rfloor,\lfloor c\rfloor >0$ Then $\frac a{\lfloor b\rfloor}=\frac c{\lfloor c\rfloor}\ge 1$ $\implies$ $a\ge \lfloor b\rfloor$ $\implies$ $\lfloor a\rfloor\ge \lfloor b\rfloor$ And same (circulat permutation) gives $\lfloor a\rfloor\ge \lfloor b\rfloor\ge \lfloor c\rfloor\ge \lfloor a\rfloor$ And so $\lfloor a\rfloor=\lfloor b\rfloor=\lfloor c\rfloor\ge 1$ And initial property becomes $a=b=c$