I've read about "math circles" in the past but only recently observed them in my local community. Young people (K-12) with a penchant for math attend math circles (usually once-per-week) to be challenged by more enriched experiences/problems (not tutored) than they likely get from their classroom experience.
Math circles have been around for decades in the U.S. (having started overseas) and may take on slightly different styles or formats. I feel safe in saying the overall goal is to give young people greater practice in mathematical thinking, although competitions and computational techniques can sometimes be a part of the process. And many now have long waiting lists to attend, such is their popularity.
I've been impressed watching the joy, enthusiasm, and thought (from both students AND teachers) of math-circle participants in my local area.
Read more about them here from Wikipedia:
https://en.wikipedia.org/wiki/Math_circle
...and here from an official website:
https://www.mathcircles.org/
(there's also a Facebook page for math circles: https://www.facebook.com/MathCircles/info/ )
Check below to see if there are any established math circles in your area:
https://www.mathcircles.org/Wiki_ExistingMathCirclePrograms_view
Lastly, by coincidence, Cathy O'Neil recently ran a guest post on the Chicago math circles at her Mathbabe blog:
http://mathbabe.org/2016/02/12/guest-post-a-math-circle-thats-breaking-the-mold/
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