Your Flikr photos are set to private so I can't get into them to read the EXIF info so this is going to be just an educated guess on what's going on.
The lighting in your gym photos aren't ideal, so you should adjust your ISO to match. I don't know what your ISO is set to in those photos, but I'm guessing that it's not high enough. IIRC, the Exilims only go so high in ISO in auto mode. You'd have to actually choose a higher ISO if you wanted to go higher.
So if your ISO is 100, 200 or even 400, I'm gonna guess that it's still not high enough and so your shutterspeed has to compensate which means that your shutter will stay open longer. This then has a side effect. The longer the shutter is open, the more susceptible it is to camera shake or motion blur. That's why your pictures aren't coming out sharp. Also, the more you zoom, the more noticeable motion blur or camera shake will be.
Your best option here is to increase ISO in order to quicken your shutterspeed. But in doing so, you're inherently going to get a lot more noise in your picture (grainy colored specks). This can be taken care of with post processing whereas motion blur can't be done quite as easily (sometimes impossible).
General advice would be that if you're in the stands taking pictures zoomed all the way in, try to at least get your shutterspeed up to 1/125 (if you have steady hands) or faster if you want to avoid motion blur and/or camera shake.
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