You should be able to identify the /θ/ sound in "the" (the "th"). This would not be dropped by native speakers. In my accent (London), "win a US open" would sound like "winner US open", but in American accents, you should be able to hear the /r/ at the end of "winner".

http://dictionary.cambridge.org/us/pronunciation/english/win http://dictionary.cambridge.org/us/pronunciation/english/the These will show you how it is pronounced