Then, there's the amazing Cyber Dojo site for practicing code katas in a browser without any setup or somesuch.
Then, we also have another online great site, the Coders Dojo for reading up on and trying out code katas.
Then, there's this site, full of Architectural Katas that look so tempting I can barely wait to try some of them.
Then, there is the original collection of Code Katas from Dave 'PragProg' Thomas that advises on How to Become a Better Developer.
Then, there's the veritable and extensive collection of Code Katas on softwarecraftsmanship.org with screencasts.
As to programming cometition problems, the UVa Online Judge site contains hundreds of problems for practice.
The CodinGame is where you can practice writing games, get it reviewed and possibly even get yourselves hired.
A genuine treasure trove is the problem archive of ACM ICPC aka ACM International Collegiate Programming Contest.
Did I miss any? Do let me know!