Therefore: Adam was created on the sixth day, and Rosh Hashanah marks the beginning of the year from the time Adam was created.
Why would we measure the year from the creation of Adam instead of from the very beginning of Hashem's creating? According to some of the rabbis (and a position later adopted by philosophers and physicists), time cannot exist without someone to perceive it. Without going into the nerdy theoretical physics/philosophy, that is how many orthodox Jews can accept both modern scientific findings and Torah, particularly about the age of the earth. The days in Bereishis/Genesis can be different than the human concept of days precisely because there was no human present to perceive time. Of course, there are people who believe that the earth is only 6,000 years old. 5771 years, to be precise.
On a related note, if you ever get the chance to hear a lecture from Dr. Gerald Schroeder, do so! Well, if you're as nerdy as I am. I heard these ideas for years and they made sense to me. If you've studied the theory of special relativity, you already understand the basic concept. But Dr. Schroeder really fleshes out the details in ways I had never imagined. And his analogy to the Hubble telescope and the universe's expansion will simply blow your mind. My understanding is that he is based at Aish in Jerusalem, but I heard him when he was visiting American communities, unrelated to Aish.