Why do the MiG-15 and MiG-17 have a split air intake in their noses?
The advent of (turbo)jet engines resulted in an important change in design of aircraft- the location of the engine, which can either be mounted on the fuselage  or in the wings (as in Me 262).

Also, the engine was shifted from the front (as in the case of a majority of piston engined fighters) to the rear in jet aircraft (as the exhaust has to be routed below the fuselage otherwise). In this case, the intakes can either be in the nose or in the sides (like in Hawker Hunter). 

For the aircraft with nose intakes (a design which was eliminated by the ever increasing size of Radar equipment), the problem was how to get the air to the engine. In case of Mig-15, the Russians split the air in a central splitter, which then passed through two narrow air ducts passing through either side of the cockpit, avionics bay, fuel tanks and the nose landing gear bay. This inlet was quite complex, not to metion it made for a cramped cockpit, but on the other had, it was compact. Similar air intakes were used in aircraft like the Mig-21.

Another method was to put to simply put the cockpit above the air inlet duct, like the F-86 Sabre, which resulted in the raised stance of the cockpit (and better visibility).