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Yea, they shorten alot of things, just like in any langauge.

тебе - те
мне - ме (or sometime sounds like не ?) it's hard to tell sometimes they are like 'даймежвачку'
тебя - тя etc.

Also...

I think it IS really formal....I know I'll have problems understanding people in Moscow when I fly there....

a Russian friend of mine told me that they say "тя" instead of "тебя"..etc...
I'M...

Our teacher told us that there is another way of asking this question:

Как называется СЛОВО по-русски?