Q:

Is it possible to fly with counterbalanced single blade engines? 

Refer the image for counterbalanced single blade (Source http://www.aviastar.org/helicopters_eng/bo-103.php). 
In ground school, they took us for aircraft inspection at end of our ground school classes.  The instructor told us that "counterbalanced single blade engines" are efficient. 
I have 2 questions here

Why development under these fields are stopped (Bo-102/103)? 
How Single blade will be efficient than 2 blades rotor-craft engines? 

A:

Both Bo-102 and 103 were experimental aircraft, with Bo-102 more of a jig than an actual aircraft.  They were used a testbed for various technologies and the single blade rotor is only one of them , with the main thrust being in the development of (hinge-less) rigid rotor and (glass fiber reinforced) composite blades.  
The single blade rotor offers some advantages over the (normal) multi-blade configurations like,

The blade passes through 'clean' air and is thus more efficient compared to the multi-blade configurations, where the (following) blade passes through the turbulent (wake) air from the previous blade, at-least in hover. 
The single blade propeller is compact and for this reason, is used in some self launching gliders which fold their propellers inside their fuselage,like the Alisport Silent 2 Targa. 

 
However, there are significant problems associated with this design,which makes it (mostly) unsuitable for widespread use. 

For reducing vibrations, the blade has to be counterbalanced by a mass, which is practically dead weight.  While this may be acceptable for small and lightweight propellers used in gliders, this is a major problem in helicopter blades, which are significantly heavier.  For example, the Mi-17 Main Rotor Blade weighs ~140 kg. 
If the mass is  kept low, the lever arm is huge and this increases drag. 
Though the blade may be  balanced at the pitch axis (to reduce vibrations), torque induced in the rotor blade (due to lift) will cause significant vibrations.  As the torque is increased, if the weight is fixed, the rotor will tend to rotate the engine and rip it off.  The only way to prevent this is to balance the blade, both aerodynamically and centrifugally, which is easier said than done. 
The same problem arises as the drag of the blade is different from that of the mass arm, with a torque being created parallel to the axis of rotation.  
The rotor blades usually 'cone' in flight; this means that the lift vector is slightly tilted towards the center, not concentric with the rotor plane. 
The blade has aerodynamic forces acting on it while the balance weight does not.  Due to this, the precess differently, which causes unbalance and vibration.  

A:

The number of blades determine the solidity ratio of a propeller.  If you need to limit tip speed and propeller diameter, the propeller will have a high solidity ratio by using many, short blades.  Take a ship's propeller: It needs to work in the space between the draft of the ship and the waterline, so it will have many stubby blades. 
If space allows, a bigger propeller has a better efficiency because it  can use a larger mass of air for lift creation.  If the propeller has a low advance ratio, the wake of one blade might interfere with the next, so packing more blades will increase the risk of interference.  The fewer blades you need for lift (or thrust) creation, the better.  Efficiency generally runs inverse to the number of blades. 
A single bladed propeller or rotor needs a counterbalance, which in effect is a dead weight.  When the interference between two blades is low enough and the disc loading of the rotor is not too low, a two-bladed rotor will be the better choice.  Disc loading is the lift relative to the area of the rotor disc and dictates the chord of the rotor blade(s) once their number is defined.  Only when structural limits will make the chord of an ideal two-bladed rotor too short will a single-bladed rotor be attractive.