Google and search engines in general will treat the title as what it sees when indexing the page. There is nothing wrong with changing titles (people do it all the time for testing and optimization) however it isn't ideal to have one page with a dynamic title. By having one page with a dynamic title you are really just confusing search engines, they will have a hard time trying to serve their result page (SERP).
The way us web builders handle the situation you are in is by using parameters. Example:
examplesite.com?state=OH
examplesite.com?state=CO
examplesite.com?state=AZ
These are in essence 3 different pages which would be treated as such for indexing purposes, content wise it may all be the same but changes with parameters on your index.html / asp / php page. Since you must be using parameters to serve a dynamic title why not include some dynamic content and rank for 3 pages?
Hope it helps,
Don