The audio for this one I found a little tricky. If I had not seen it written, I would never have answered it correctly. "Te egy" seems to get merged together as though it were one word (incomprehensible a word though it may be). Just shows how important it is to listen intently to the audio in order to get your ear attuned to the language.
Te egy - here the e sounds a bit longer. The Hungarian language makes a big difference between long or short sounds, almoust 3:1. E is usually short while é is long. If you listen more carefully you will find that "te egy" has a longer e sound. E.g. o = 1 while ó = 2-3. Has (s [=sh] = 1), hass (s = 2-3). Maybe thos helps you.
Yes, Quasi, I agree with you about it possibly being a contraction. To my ear it sounds like "tég", (as in the way we say 'tag' in English) so I totally missed the 'gy' sound. It just illustrates to me that, with familiarity of the language, our ears will become more accustomed to ill-pronounced or abbreviated words. Keep persevering!