Czech-Slovak-Polish Match

Without loss of generality we can suppose $a=\max \{a,b,c\}$. From the first equation of the given system we obtain $$c(\sqrt{b}-1)\le a(\sqrt{b}-1)=c,\text{i.e.}b\le 4.$$ Similarly, from the second equation of the given system we have $$b(\sqrt{c}-1)=a\ge b,\text{thus}c\ge 4.$$ Using these results together with the third equation we get $$4\le c\le c(\sqrt{c}-1)\le c(\sqrt{a}-1)=b\le 4$$ which yields $a=b=c=4$.

Conclusion. The unique solution of the given system of equations is therefore $(a,b,c)=(4,4,4)$.