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Like many things in photography it all depends how close you can get to the action, you've probably seen the huge white lenses on the sidelines at football matches, normally 400mm f/2.8 and cost about £7K. At least that gives you the kind of focal length you need although you're going to need to compromise on the aperture.
I'm afraid that there's no way to do this on the cheap, light is at a premium most of the time and fast long lenses are expensive. A slightly shorter lens would be the 120 - 300mm f/2.8 Sigma which stretches to a 420mm f/4 with a 1.4X TC however that will cost you around £2500, there's also the 100 - 300mm f/4 Sigma which loses a stop (half the light) but you can pick this lens up second hand on Ebay for around £500 or less.
The cheaper lenses are normally around f/5.6 (only 1/4 the light of the f/2.8) are so compromised in terms of focus speed and shutter speeds they can allow that I'd recommend you avoid them as they would return good results (unless the sun shines very brightly).
An extra flash will not help you unless you manage to install a speedotron system which again is a bit on the expensive side.
In terms of mountain biking you can get a lot closer (don't put yourself in danger) and a second slave flash might be useful. Again fast lenses are best so depending on you close you are several of Sigmas f/2.8 zooms would suit & wouldn't break the bank, 24 -70mm 50 - 150mm & 70 - 200mm
As for landscape a stable tripod and some stitching software !
The problem with the areas you want to photograph is that they need completely different lenses! sports need the long expensive fast glass and landscape the wider angle, slower apertures will do. Perhaps you need to have a think about where you want your photography to go first?
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