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WASHINGTON — A 16-year-old California boy won a premier high-school science competition Monday for his innovative approach to an old math problem that could help in the design of airplane wings.

Michael Viscardi, a senior from San Diego, won a $100,000 college scholarship, the top individual prize in the Siemens Westinghouse Competition in Math, Science and Technology.

Viscardi said he has been home-schooled since fifth grade, although he does take math classes at the University of California at San Diego three days a week. His father is a software engineer and his mother, who stays at home, has a Ph.D. in neuroscience, he said.

"It's unbelievable," Viscardi said of his win. "It's so incredible that I'm in shock right now."

Viscardi tackled a 19th-century math problem known as the Dirichlet problem, formulated by the mathematician Lejeune Dirichlet. The theorem Viscardi created to solve it has potential applications in the fields of engineering and physics, including airplane wing design. He said he worked on it for about six months with a professor at UCSD.

"It was almost impossible for our judges to figure out the limits of his understanding during our questioning, and he's only 16 years old," said lead judge Constance Atwell, a consultant and former research director at the National Institutes of Health.

Anne Lee, 17, a senior at Phoenix Country Day School in Paradise Valley, Ariz., and Albert Shieh, 16, a junior at Chaparral High School in Scottsdale, Ariz., shared the $100,000 top prize in the team category. They improved computer technology that could help locate the genetic roots of some inherited diseases such as Alzheimer's, autism and bipolar disorder.