If you weren’t actively planning your Rio 2016 Olympics trip in the spring of 2015, you must buy your tickets second-hand. I just spent several hours researching the process and here is what I found:
https://www.cosport.com/ is the official vendor for the US. They have been the vendor for the Olympics since Sydney 2000. The tickets are mostly sold out, but they do have some available. There was a lottery that started in spring of 2015 and ended in the summer of 2015.
E-Bay. Folks in the states are receiving their tickets this summer (2016) from the lottery and some are selling them on E-Bay. I do not believe that the tickets are attached to an individual and require ID. I am looking for more verification on this.
This doesn’t help much, but at least you can direct your search.
Forbes: Rio 2016: How To Plan A Trip To The Next Summer Olympics (written May 2015)
Nick Huggin’s Travel Blog (written April 2015)