White smoke coming from exhaust

Those are all signs of a leaking cylinder head gasket. Different things can happen depending on where the leak is. The least common symptom is oil in the coolant. The most common symptom is white smoke from the tail pipe. That is a sign of burning coolant. Combustion gases can also go the other way and push coolant into the reservoir.

Your mechanic can perform a chemical test at the radiator to check for a leaking head gasket. That involves drawing air from the radiator, while the engine is running, through a glass cylinder with two chambers partially-filled with a special dark blue liquid. If combustion gases are present, the liquid will turn bright yellow.