I need some help on a perplexing issue with our 2000 Odyssey.

After many trouble-free miles, it has begun losing oil in dramatic
fashion. We've added a 1/2 quart twice in the last 3 days of light
driving alone.

Here's the mystery:

Absolutely no oil on garage floor. Not a drop.
The coolant levels are perfect.
The engine is not overheating or running warm.
Engine acceleration and power are normal.

Besides losing oil, the only other possible symptom is that the oil on
the dipstick is very dirty.

Any suggestions or ideas? Any help is greatly appreciated.