Some third-party lens makers sell mirror or, as they’re more accurately known, catadioptric lenses. Such lenses use a pair of mirrors to fold the light path in half, in addition to containing regular glass elements. The advantages are that mirror lenses can be made fairly cheaply and will also be shorter and lighter than all-glass (refracting) lenses of the same focal length. And mirror lenses with focal lengths of 500mm to 1000mm are not uncommon. Russian makers produce a lot of these lenses.

Unfortunately, this answer is going to be somewhat like the previous one. The challenges of sports and other action photography are twofold. First, by its very nature, sports photography tends to involve rapid motion - fast-moving players or cars or whatever. Second, usually there’s some distance between the action and the camera.

Solving the first problem requires lenses which can let in plenty of light, the use of flash or faster film or high ISO settings on a digital camera. Each of these solutions has drawbacks, however. Fast lenses are large, heavy and expensive. Fast film or high ISO settings result in higher grain or noise and thus lower picture quality. And flash may be inadequate to illuminate the subject effectively, particularly if the subject is some distance away.