You're pretty much right on the math competitions front, but the first few problems on the AMC are pretty simple IMO(pardon the pun). They are barely above the level of mechanical problem solving. It does get harder, though, and I agree that that is more daunting.
You've reminded me, however --
I remembered thinking that the Phillips Exeter Academy model for math education (the "Harkness" method) is amazing.
http://en.wikipedia.org/wiki/Phillips_Exeter_Academy#Harknes...
> Exeter does not teach math with traditional textbooks. Instead,
> math teachers assign problems from workbooks that have been
> written collectively by the Academy's math department. From these
> custom workbooks, students are assigned word problems as
> homework. In class, students then present their solutions at the
> blackboard. This means that in math class at Exeter, students are not
> given theorems, model problems, or principles beforehand. Instead,
> theorems and principles emerge more organically, as students work
> through the word problems."*
This is what we want. Discovering math makes it fun for everyone. Honesty and actual learning with that method(my whole would be not be too difficult, because everyone would inevitably have their own personal spin on the origins of the theorems et cetera. Plus, it'd be embarrassing to go to the blackboard and say that you haven't done the work.
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By the way, expect an email regarding what you said about the IMO.