Math contests inspire a wide range of reactions in the math world. Many people do not like them. I’ll take Cathy O’Neil’s post Math contests kind of suck as a nice example from that camp.
On the other end of the spectrum there’s the super-star status that some of the top competitors achieve. A good representative article here is Zach Wener-Fligner’s piece: The Last Contest: Hanging With the Big Dogs at the 2013 Putnam Math Competition
I grew up in math in the world of math contests, so I’m probably pretty biased toward liking them. The reason that I like using them with my kids has nothing to do with competition, though. I like them because the wide variety of questions and topics on the contests help me see where they have misconceptions.
With no school today I had the boys spend a little bit of time on some problem from the 2014 AMC 8:
The 2014 AMC 8 hosted at Art of Problem Solving
My younger son had some questions on problem #18 and that led to a fun conversation about probability:
My older son wanted to talk about question #20. Talking through this problem led to a nice conversation about approximating 
So, a fun morning talking through a few old AMC problems. I’m happy that these problems are available online now – they are a great resource.
Mike's Math Page 