The mountains would have to be something of an "in between zone" but... it really depends on if you want this to be an earth-like planet, or if it can be more exotic. The mountains would likely have to be east-west and you would need some extreme measures to achieve the effect in an earth-like situation, whereas on an alien planet the rotation of the world could create a daylight and dark side, with the country being on the edge of the two zones, and the mountain range blocking more sun in order to keep that side cold but habitable. A more earth like scenario would require something like, say, a great Sahara desert that is hot as hell with prevailing winds taking that heat northward to mitigate otherwise temperate climates while the heat stops at the mountain range, leaving the other side cold. Volcanic and other tectonic conditions could also lead to a warmer side, while also, a constant sun blocking cloud of volcanic ash above the cold side (with some rationale for that!) could theoretically keep one side of the mountains cold...

Its a reach! but those are off the top of my head, LOL.