<p>This is such an interesting discussion and one that I can’t really contribute to except to use my son as an anecdotal example. In his case, qualifying for AIME came after his 800 in math because he never attempted to qualify before 10th grade since he didn’t do math competitions in 8th and 9th grade. He is not tops in our city by any stretch of the imagination, but he was, in 7th grade, 8th in our large city in MathCounts but with no one to help him, he placed only about 70th in Ca. in the state tournament. After that year, though, he was turned off by math competitions and didn’t do anything for two years. Now, as a junior, he’s definitely <em>not</em> 8th best in math competitions by a long shot. He came in at the bottom and worked his way up to mid-level competitor. His math competition math skills are rusty; qualifying for AIME last year and this year was a goal for him. Perhaps this year, he might have as a goal USAMO but given that his AIME was only 100.5, it’s highly unlikely that just taking one AoPS AIME class will give him a 10-11 on the AIME. He will be happy with a 6-8 score, I am guessing.</p>
<p>As far as SAT I math, I think practice in taking the real test was probably most helpful. He took a free SAT I test in 6th grade to see if he could sit still long enough to finish. His math was 580. In 7th grade, he took it again to qualify for SET. He watched some borrowed math videos to prepare and scored a 700 (barely made it in!). In 8th grade, I don’t think he prepped because he’d had so much more classroom math (had finished Calc I by that time) and the 800 came easily. He took the SAT I for the last time this year and again got an 800. </p>
<p>So, my observation is that SAT math is your typical high school math-if you take good math classes, you’ll do well. Math competition math requires a specific set of math skills that students have to be very honed into in order to do well. Math competitors who do well spend hours upon hours honing said skills. My son has the opportunity to work with an amazing local math tutor who focused on olympiad level math but after one session, my son knew that’s not what he wanted to do at all. It was way too intense.</p>