Notice traces of oil in the car radiator

You don't mention the type of car and engine, sounds like maybe 3.1, could be bad head or intake gasket.

Head gasket is likely leaking, due to overheating. Frequently, the head warps, valves get damaged, and the head gasket leaks, allowing oil to mix with coolant. Repairs frequently run $2,500+, if it's a 4-cylinder car.

I'll third that diagnosis...

I found sealup to stop this from occurring on my 1996 probe v6 gt, it had a slight amount of what appeared to be a small oil trail, I added some seal up a few days ago and the coolant is now clear.