When the United States dropped atomic bombs on Japan, it triggered the start of a race between the United States and the Soviet Union. This was a key issue during the Cold War.

This is true. After the first atomic bombs dropped, both countries began developing Nuclear energy based weaponry which was indeed a key issue in the cold war. The Cold war is called that way because it wasn't fought at all, yet the countries relationships were extremely strained and a real war almost broke out a couple of times.