## Professor Chen Education Palace

# 陈 教 授 教 育 学 院

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# AMC 10 vs AMC 12

Advantages of taking AMC 12 not AMC 10 for those students who are eligible for AMC 10

1. AMC 12 is NOT much harder to prepare for than AMC 10.

The additional topics tested in AMC 12 but not in AMC 10 are: logarithmic functions, complex numbers, trigonometric functions, general polynomial functions, trigonometric approach to solve geometric problems, advanced 2-dimensional geometry. It is true that a student who plans to take AMC 10 has a lighter learning burden of not having to learn these materials. However, this is NOT a good strategy to prepare for AMC 10. Reasons are as follows.

(1) Although these additional materials are not directly tested in AMC 10, they are quite useful in solving AMC 10 problems.

For instance, in some AMC 10 geometric problems, it is necessary to add auxiliary lines to facilitate the analysis. But adding auxiliary lines is generally non-trivial and requires lots of creative thinking. No student can guarantee that s/he can always find appropriate auxiliary lines to add in an AMC 10 contest in a very limited time. However, many such problems can be easily solved in a more straightforward way by using trigonometry without adding any auxiliary line. Therefore, learning additional materials enables a student to own more powerful tools to easily solve some AMC 10 problems that are challenging without these tools.

(2) These materials are not hard to learn.

A student can quickly and easily learn these materials.

2. It may NOT be easier to advance to AIME by taking AMC 10 compared to taking AMC 12.

(1) The percent of examinees who take AMC 10 and advance to AIME is only half of the percent of those who take AMC 12.

This number is only 2.5% for AMC 10. By contrast, it is doubled to 5% for AMC 12.

(2) The cutoff score for AMC 10 examinees who are eligible to advance to AIME is almost 20% higher than the cutoff score for those who take AMC 12. It is relatively easier to meet AMC 12 cutoff score than AMC 10 cutoff score by taking into account the fact that they have 40%-60% overlapping questions (equivalently, 10-15 overlapping questions) and the difficulty levels of a majority of these questions are easy.

For instance, in 2020 Version A, the cutoff score for AMC 10 was 103.5, 19% higher than the cutoff score for AMC 12, 87. Suppose a student’s answer was correct for each question that s/he answered. This means that this student was qualified for AIME by taking AMC 10 if s/he got 15 correct answers out of 25 questions. By contrast, if this student took AMC 12, then s/he was qualified for AIME by only correctly answering 11 out of 25 questions. Note that AMC 10 and AMC 12 typically have 10-15 overlapping questions. The difficulty level of the majority of their overlapping questions are easy or medium. This entails that beyond correctly answering most of their overlapping questions, a student who takes AMC 12 only needs to correctly answer very few non-overlapping questions. By contrast, a student who takes AMC 10 needs to correctly answer much more than that.

3. If a student aims at advancing to AIME and receiving a good score in it, then it is more efficient and effective to take AMC 12, not AMC 10.

(1) The additional materials covered in AMC 12 but not AMC 10 may appear in AIME.

For example, in 2019 AIME I, 4 out of 15 (27%) problems involve these additional materials: Problem 7 involves logarithmic functions, Problem 8 involves trigonometric functions, Problem 10 involves complex numbers and polynomial functions, Problem 12 involves complex numbers. Note that the break between AMC 10/12 and AIME is about only one month. Hence, it is a big rush to learn these materials in such a short time. Therefore, if a student’s goal is to receive a good score in AIME (example: for the purpose of being admitted to a top college or being qualified for USAMO), a good strategy is to learn these materials with sufficient time before AMC and directly register AMC 12.

(2) The increment of difficulty level from AMC 10 to AIME is much steeper than from AMC 12 to AIME.

Approximately 20% of AIME problems (Problems 13-15) are more challenging than all AMC 12 problems. However, 67% of AIME problems (Problems 6-15) are more challenging than all AMC 10 problems. Therefore, if a student is comfortable with solving most difficult problems in AMC 12, then s/he is ready for most AIME problems. However, if a student is only comfortable with solving most difficult problems in AMC 10, but not AMC 12, then s/he still has a long way to go to be ready for AIME.

4. AMC 12 is more valuable for a student’s college application than AMC 10.

(1) College admission committees weight more on AMC 12 score than AMC 10 score.

Reasons are two-fold. First, AMC 12 covers the entire high school math curriculum (except calculus). But AMC 10 covers only some of them. Second, AMC 12 has a higher difficulty level than AMC 10.

(2) Performing well in AMC 10 but bad in AMC 12 may hurt a student’s college application.

When a college admission committee reviews one application, the committee cares not only this applicant’s past performance, but more about her/his potential. Suppose there was a student who performed exceptionally well in AMC 10 (example: received a distinguished honor and was invited to AIME) but less well in AMC 12 (example: the score was much lower than the AIME cutoff score) after two years. Then this would signal the college that this student might have trouble in learning advanced math topics and might not have a good potential in math or other subjects that use math tools.

(3) Taking AMC 12 while being eligible to take AMC 10 gives a student more chance of getting a good AMC 12 score.

Note that the AMC 10/12 competition time is January-February. Many college application deadlines are before that. So a student’s college application does not get too much help from AMC 12 that s/he takes in Grade 12. Therefore, in the college application, a student has to mainly use her/his scores of AMC 12 that s/he takes in or below Grade 11. Suppose a student takes AMC 12 only if s/he is no longer eligible to take AMC 10 (in Grade 11 only). Then this student has only two chances of taking AMC 12 (It is offered twice per year) before applying to a college. By contrast, if s/he starts to take AMC 12 as early as s/he is eligible to take AMC 10, say, in Grade 9, then s/he has six chances of taking AMC 12 before applying to a college. This 200% increment of the number of times of taking AMC 12 brings this student the following values.

First, each time a student takes AMC 12, s/he gains some experience. This experience is quite valuable for her/him to attend the same contest in future. Thus, it increases the chance that s/he will perform better while taking the contest again.

Second, the difficult problems in different year’s or version’s AMC 12 contests may focus on different topics. Suppose a student is relatively weak on some topic, say, geometry. If this student only took AMC 12 once and unluckily most challenging problems were on geometry, then s/he would not perform well. However, suppose this student started to take AMC 12 as early as s/he was eligible for AMC 10. Then it would be very likely that among all AMC 12s that s/he took, there was one, such that the most challenging problems were on the topics that s/he was comfortable with, rather than the topic that s/he was weak on, geometry. As a result, it is very likely that s/he would perform well in this particular test.