Amar Ramasar was 11 when he first saw ballet. It was a video of the New York City Ballet dancers Heather Watts and Mel Tomlinson, in Balanchine’s radical 1957 ballet “Agon,” the dancers clad in minimal leotards and tights, performing spiky, extraordinary movements to Stravinsky’s astringent score. Mr. Ramasar, an outgoing, talkative boy from the Bronx, knew right away what he thought. “That’s the ballet I want to dance, and that’s the company I’m going to get into,” he said.