An implication of this new paper is that girls (and boys) who score well on the AMC but not well enough to advance to the next round (the AIME) should be encouraged to take the AMC in future years. I think some selective schools such as MIT do look at AMC scores.

I don't see where the full paper is freely available.

http://www.nber.org/papers/w24910
Dynamics of the Gender Gap in High Math Achievement
by Glenn Ellison, Ashley Swanson - #24910 (ED)

Abstract:
This paper examines the dynamics of the gender gap in high math
achievement over the high school years using data from the
American Mathematics Competition. A clear gender gap is already
present by 9th grade and the gender gap widens over the high
school years. High-achieving students must substantially improve
their performance from year to year to maintain their
within-cohort rank, but there is nonetheless a great deal of
persistence in the rankings. Several gender-related differences
in the dynamics contribute to the widening of the gender gap,
including differences in dropout rates and in the mean and
variance of year-to-year improvements among continuing students.
A decomposition indicates that the most important difference is
that fewer girls make large enough gains to move up substantially
in the rankings. An analysis of students on the margin of
qualifying for a prestigious second stage exam provides evidence
of a discouragement effect: some react to falling just short by
dropping out of participating in future years, and this reaction
is more common among girls.