RU-ALLRUSSIAN 2016 Problem 91

A carpet dealer,who has a lot of carpets in the market,is available to exchange a carpet of dimensions $a\cdot b$ either with a carpet with dimensions $\frac{1}{a}\cdot \frac{1}{b}$ or with two carpets with dimensions $c\cdot b$ and $\frac{a}{c}\cdot b$ (the customer can select the number $c$).The dealer supports that,at the beginning he had a carpet with dimensions greater than $1$ and,after some exchanges like the ones we described above,he ended up with a set of carpets,each one having one dimension greater than $1$ and one smaller than $1$. Is this possible?

Note:The customer can demand from the dealer to consider a carpet of dimensions $a\cdot b$ as one with dimensions $b\cdot a$.