duplicates problem

Hi all, I have a problem with duplicates (surprise!): I keyframe an object, duplicate it, move the duplicate, then keyframe it (object 2). So at frame 1 I have two objects, in two different locations, with their loc keyframed. Yet as soon as I move through the timeline, the objects get in the same location, one object going where it is not supposed to be.
If I first duplicate then keyframe, this does not happen. But what if I want to duplicate an object that I have already give keyframes to?
How do I fix this?

duplicate problem.blend (59.9 KB)