I am really confused about what is going on.
I think there are three options, one of which I am unsure of:
Your pi is broken.
The AC mains supply to your location is poor -- I am not sure how bad this has to be to affect an AC/DC converter regulator, or what form the effect would take.
Your power supply is insufficient.
If the problem is #2, which is the least likely IMO, there is nothing you can do.
If the problem is #1, which is the next least likely, you will have to return the pi and try a new one.
If the problem is #3, which is the most likely, you need to try another power supply. Personally, I have not found a supply stamped 1+ amps that would not power up any kind of pi. However, it could be that all of mine are decent quality, since there have been people with problems similar to yours and certainly the #1 most successful fix for them reported around here is "I tried another power supply and it worked!".