Don't come too close to the border with Georgia. Relations between Georgia and Russia are not good and there is a no-go security zone enforced for foreigners 10km of the Georgian border. The Russian army is present and will check you. It can easily become an administrative hassle. Evidently this border zone puts the highest summits Dykh Tau, Koshtan Tau, Katyn Tau, Shkara, etc... out of reach. Too bad, you will only be able to look at them from a distance.
