<p>The math for each one is quite different, but perhaps people don’t view them as equally challenging? </p>
<p>Most of my son’s friends at the local math circle have not gone as deeply into college level math as he has (dif. equations, upper division dif equ., dynamical systems, real analysis); yet, he does not spend time studying math competition math (combinatorics, number theory, stats and probability, etc) and consequently some of them have been to USAMO whereas my son has only gotten AIME 3X. He’s capable of being at their level (given the fact that in 7th grade, he was one of the top 8 students in Mathcounts in our large city along with some of these kids who have been USAMO participants), but he’s chosen a different route in math. We didn’t know about AMCs back then. He dropped out of math competitions for several years and only picked them up again in 10th.</p>
<p>Math competition math is wonderful because it’s a way for exceptionally gifted (or exceptionally disciplined and hard working) students to stay in high school with their peers while being challenged in math. </p>
<p>Math competitions aren’t for every kid, though. Not every kid is competitive like that. One of the most math gifted kids I know never did competitions but researched a particular math topic in a new way while still in middle school.</p>