You can find middle school math competition questions online through sites like MathCounts, AMC 8, or previous Math Kangaroo and Math League tests. Beestar also offers practice problems with rankings, which could help your child get used to the competition format and track their progress at home.
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Well i know many specific sites with math team and various national contests on them, but i don't know if i can say without advertising.... just try googling the name of the math competition.. Many links will pop up and usually they are very valuable and filled with old tests and practice problems.