Short business trip

I am an American citizen and will be flying into Hong Kong, then to Taipei, to Beijing, to Xiamen City, back to Beijing, and then back to the USA. I will be staying for less than 72 hours in each location, except for Xiamen City, where I will be for 6 days. Will I need a visa, and will I be able to apply for one on-location, provided I have an invitation letter/passport/photo/application? If I shorten the stay in Xiamen City to under 72hrs, will that eliminate the need for a visa? Thank you.