USACO
Climb from Bronze to Platinum in the USA Computing Olympiad's online contests
USACO runs online programming contests each winter. In each contest you get a few hours to solve three algorithm problems in your division. A good score promotes you from Bronze to Silver, Gold and then Platinum, and top US students are invited to a training camp that picks the IOI team.
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About USACO
It trains algorithmic problem solving from basic simulation up to advanced data structures and dynamic programming. It suits high school students most, but anyone can take part, and the Bronze division works for beginners.
Contests run on the USACO website within a weekend window, and you pick when to start your timed attempt. It is free, with an account. Past problems with analysis are online for practice, and the season ends with the US Open in March.
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