Math Kangaroo is a nice math competition with some parallelism to the AMC layout, but stratified into age-appropriate levels as low as the first grade, so that there is something suitable for most students near their grade level.

SDMC used to administer Math Kangaroo locally at UCSD, but MK developed an individual registration track through their national website that left not much value for SDMC to add.  Since then, we do recommend Math Kangroo as  worthwhile event - especially for younger students - but we do not host it.  Instead, we direct families to Math Kangaroo's own website to use their "individual registration" track:

www.mathkangaroo.org