In general, there are only a handful of public high schools that are considered elite. They often tend to be magnet schools that require application and sometimes a lottery as well and therefore residency the year prior to matriculation. If you look up the high school teams that clean up at national/international competitions, that might get you a decent list of high schools with strong programs.

Off the top of my head for math/science and with soccer/tennis availability:

-Montgomery Blair in Silver Springs, MD
-Stuyvesant in New York, NY
-TJHSST (Thomas Jefferson) in Alexandria, VA