Hi wanataka –
I wouldn't buy a DSLR for sports or events (no video viewfinder, unless you spend the extra money for a loupe or EVF; limited to 30 minutes of coontinuous recording – so no speeches, plays or sermons; and poor in-camera stabilization).
The one interchangeable lens camera brand that can do this job in your price range is Panasonic, with their mirrorless DSLMs. These cameras, unlike DSLRs, have built-in EVFs, essentially unlimited continuous recording, and unlike most DSLRs (with the exception of the Nikon D3300 and D5300) records at frame rates up to 1080/60p for smooth action and slow motion.
This is what the $678 Panasonic G6 (on sale with the kit lens as of this post) can do on the ski slopes:
I am a Panasonic GH3 shooter. It adds a headphone jack, an all-metal splashproof body and a choice of codecs and higher bit rates to the G6's terrific image quality.
Here is what this $1000 camera can do:
If you prefer the camcorder form factor and you want to shoot skiing, you need world-class stabilization. Sony camcorders are almost unequaled in this regard.
Here is a comparison of GoPro video and the Sony CX730 (the consumer version of the $1849 pro NX30 – please watch at 1080p):
Here is what the NX30 can do on a vehicle mount:
Hope this is helpful,