Just what the title states.
Since the Wright brothers, aviation technology for fixed-wing craft has advanced by an order or more. Rotary wing craft on the other hand (I know little about aviation; please correct me!) have advanced significantly by way of lift capacity, and maneuverability. Yet velocity has achieved nothing like the kind of advance seen in case of fixed-wing craft.
Currently there is no rotary wing craft capable of supersonic flight.
Combined with the forward motion through the air, the rotating blades attack the air on one side and retreat backwards on the other. As the aircraft moves faster this poses 2 problems:
There are ideas of new blade actuators that can change the blade for 1) reverse airspeed and 2) supersonic aerodynamics. As you have noticed, nothing has panned out so far...
The Republic XF-84H "Thunderscreech" was a USAF jet fighter modified with a turboshaft engine and a propeller designed to operate at supersonic blade speeds. It first flew in 1955, and the result was a noise that was literally deafening. During ground engine runs, "the prototypes could reportedly be heard 25 miles (40 km) away."
Whereas there may be some advantages of a supersonic propeller, the side effects (the noise) prohibit the use of a supersonic propeller even for military applications, let alone civilian ones.
The question proposes a helicopter with supersonic blades, at least on the blade advancing side of the helicopter. There is no research that I know of that would indicate a supersonic blade on the main rotor of a helicopter will act differently than the supersonic blade on a propeller on an airplane. The assumption is therefore that the blades would experience a very significant increase in drag, and would also produce a very significant amount of noise.
Since the question seems about aircraft theory, I'll answer with theory. The short answer is that anything can, and will fly given enough time and effort.
One of the biggest limitations for making a conventional configuration rotary aircraft fly at supersonic speeds is materials. When a rotary blade is inline with the path of travel, it has to withstand significant compression forces along its long axis from the shock. There are three solutions for a conventional configuration - bigger & stronger blade (heavier) new materials, or shorter blades (less lift).
If you don't mind deviating from conventional design, then introducing a shroud around the blades will allow you to remove these forces from consideration by allowing your shroud to absorb the forces of the created shock. You could probably reduce the weight of each individual blade by bracing it against the shroud, (think - beam braced at one end vs both ends) Since the air between a shock wave and expansion waves is sub-sonic. If you're lucky, and physics doesn't hate you, then you may get expansion waves behind the shroud, so your rotor blades will be in a sub-sonic medium. By pumping air from above the shrouded rotor to the underside, you cause lower pressure on top, higher pressure on the bottom, causing a pressure gradient and generating lift.
This is all speculation based on my limited aerodynamics knowledge. In real life physics usually hates you, and you'll get some really messed up expansion waves and shock waves along the inside of the shroud, coupled with some fugly shocks being formed by the rotor blades themselves. Since prototyping has become super expensive, and wind tunnels seem to be a dying race nowadays, this would have to be simulated with CFD, and as far as I know, CFD isn't that fast and accurate yet, which could explain why this hasn't been tried yet.
A rotor on an aircraft/rotorcraft is basically an airfoil section twisted along its length. This twist is also known as the pitch, and in most cases it is variable. On approaching Mach 1 or the speed of sound, shock waves begin to form on an aircraft (eg. on the wings) and there is a sharp rise in drag (known as wave drag). This is due to the fact that at high speeds, the airflow can no longer be treated as incompressible flow. The rotors face the same compressibility effects that an aircraft wing faces near the speed of sound, hence it isn't able to overcome this drag.
