Why weren't the rotor tips of the Djinn directly fed from the turbine's exhaust? The Sud-Ouest Djinn seems to be pretty much the only tip-jet helicopter to have any sort of commercial success. It used a turbine to drive a compressor, which itself fed slightly warm air to pipes in the propeller blades, which itself exited at the tips, making the propeller turn.

This patent from the 90's says that warm air would be more efficient, as exhaust efficiency is limited by the speed of sound, and hot air has a higher speed of sound. As such, the patent is about putting a heat exchanger between the turbine exhaust and the air compressor exhaust.

But then, why didn't they simply plug both exhausts together to go drive the tips, similarly to a high-bypass turbofan? Why simply evacuate the turbine exhaust instead?
The soot in the burned exhaust will eventually clog the ducting. The higher temperature will deform the rotor, or tougher heavier materials will be needed.

The stream of a  high BPR turbofan (ignoring the size) when channeled into narrow ducts will raise the stream velocity, but lower its pressure, which will be an overkill in terms of thrust needed.

Therefore "slightly warm" clean air is best.