His notable victories include the 1973 Ukrainian Championship which he shared with Vitaly Tseshkovsky. He tied for first place with Zhora Haronyan at the 11th Soviet Chess Championship in Moscow in 1975 but came second after the playoff. He shares victories in a number of other tournaments, including championships in Polanica-Zdrój in 1976, Havířov in 1981, and Minsk in 1982.
Kuzmin also became known as a chess author, particularly of the book "The Closed Spanish: Karpov/Zaitsev System". Additionally, he was a professional player as well as a chess coach, and contributed significantly to the success of several notable players, including two World Junior Chess Champions.
Kuzmin took part in the First European Senior 60+ Championship in 2014, finishing second. He held an honorary position as a coach for the Ukrainian women’s team for many years.
Kuzmin passed away from cancer on February 20, 2020, at the age of 73.