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These symptoms are consistent with a failed thermostat. If the engine is overheating usually within about 5 minuets of starting cold and the coolant level is NOT low, the thermostat has likely failed and no longer will open.
If no coolant is leaking out (I assume the coolant is blown out of relief valve when it overheats) I suggest you replace the thermostat and retest.
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Jake "The BMW Doctor"
BMW Technician
Over 30 years experience, 13 years with BMW, former BMW Shop Foreman, Service Manager