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<p>More qualifiers means a lower cognitive threshold for reaching AIME, since the intelligence distribution in the population is relatively stable. The number of qualifiers can certainly be increased (while keeping the required performance level and the number of competitors the same) by methods such as improved math instruction, free coaching for all students who register for the contests, or a three-month schoolwork exemption prior to the exam. These would raise the number of competitors who reach the bar of qualification but also reduce the performance of the average qualifier later on, by drawing on a weaker talent pool. </p>
<p>The “prestige” of AIME or any other credential attained relatively early in life is mostly in the predictive power. Solving competition problems has little direct value on its own except as a hobby. It is a signal of what can be achieved later, and in this respect raising the number of qualifers weakens the signal.</p>
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<p>The self-selection means that the mathematically higher AMC threshold is not necessarily higher as a psychometric threshold. If the generalists who get 800 on the SAT entered math competitions and prepared for the AMC/AIME, they would probably qualify at similar rates. Getting an 800 math score is unimpressive for a math competition hobbyist because it comes after years of preparation. An 800 math score from a specialist in English literature or politics indicates high ability likely to have reached the AIME qualification had it been applied to mathematics.</p>