Yes, you are correct. The pronunciation of 'the' doesn't always change before vowels. There are usually "Grammar Book" answers and "real life" answers to questions like this. Or to put it another way, what do grammar teachers say to do, and what do native English speakers actually say. Some English speakers insert a consonant between 'the' and 'Olympics' (or whatever word it may be) in order to avoid saying 'thee'. The consonant is a glottal stop /ʔ/, as in "the/ʔ/Olympics."
Something else you may be hearing is the connection of the words 'thee' and 'eastern'. 'the' can end with the phoneme /i/ and eastern begins with the phoneme /i/. These are connected and said as one sound (the eastern --> theastern). This happens any time a word begins with the same sound as the last word. (past time --> pastime, first time --> firstime).