Top 3 Mistakes Students Make on the AIME

The AIME, the second tier of the AMC pathway, is one of the most prestigious high school math competitions worldwide. After over 30,000 people take the AMC, the top 7% are invited to take the AIME, making it a substantial landmark in a child’s math olympiad journey. As an AIME qualifier and a coach who has guided many throughout their AMC and AIME journey, I’ve seen the challenges students face at this level. After qualifying for the AIME, the change in difficulty from the AMC to the AIME can be quite intimidating. Yet despite their preparation, I’ve seen even the strongest students fall short because of three recurring mistakes that cost them valuable points and lower their scores.

Mistake #1: Not Checking Your Work

This is the most disastrous yet preventable flaw on this list. Students tend to keep attempting problems beyond their capability when unsatisfied with their current progress. At this point, the chances of solving a harder problem are significantly lower than catching a mistake in your work! Making computational mistakes on problems is extremely common, especially with the complex and long solutions typical for problems on the AIME. Focusing on fixing these can help raise your score drastically.

Let’s jump into an example.

2019 AIME II/4: A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Without using advanced techniques of generating functions or a multivariable roots of unity filter, all other methods of approaching this problem involve splitting the set of numbers rolled into many cases. With so many cases to consider, computational errors are typical. Forgetting just one can lead to “off-by-one” mistakes or similar issues.

However, considering the early placement of this problem, problem difficulty in the test will continue to increase as we progress through the problems. To maximize your score, it is necessary to get all the “easy” points. That’s why, after solving a casework-heavy problem, it’s important to check your work to ensure you didn’t miss anything.

Mistake #2: Aiming Too High

Similar to the previous mistake, aiming too high is another common mistake top students make on the AIME. Often motivated by an easier test, strong students tend to aim for significantly higher than what they receive on practice tests, causing them to rush through earlier problems. While this occasionally leads to better performance, it primarily creates careless errors on easy problems.

To understand the downfall of this mistake, we can take a look at 2025 AIME I/4.

Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-100$ and $100$ inclusive, such that $12x^2 - xy - 6y^2 = 0$.

This problem was probably the most missed problem on the AIME this year. After a student either factors or uses the discriminant to obtain the equations $y = \frac{4}{3} x$ and $y = \frac{-3}{2} x$, they are so worried about not achieving their far-fetched goal that they forget what they are trying to do with these equations. This is even more exemplified by the problem appearing early on the test, leading students to want to finish it fast. Thus, most students found the solutions to both equations given the problem’s conditions of $-100 \le x,y \le 100$, leading to their answer of 118.

However, rushing this quickly through problems leads to mistakes: in this problem, they overcounted the solution $(x,y) = (0,0)$ which appears in both equations. Unsurprisingly, so many students encountered this mistake that the title of the AoPS post for this problem was “do NOT double count (0,0).”

Mistake #3: Jumping Into Brute-Force

This mistake is all too common for newer test-takers. The problem writers for the AIME intentionally write problems in a way that remains accessible to beginning mathletes in brute-force techniques, yet the intended solution requires far less computation and more insight. Despite the longer exam time provided on the AIME, executing “bash” methods on early problems can quickly drain the given time, taking away time from later problems.

An excellent example of this is 2001 AIME I/6:

A fair die is rolled four times. The probability that each of the final three rolls is at least as large as the roll preceding it may be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

To the new problem solver, casework often feels like the only option. You might try writing all the 4-tuples where $a_1 \leq a_2 \leq a_3 \leq a_4$, and maybe think of doing recursion or messy enumeration. But then you realize: this isn’t small. Even just listing the sequences starting with 1 gives you 84 sequences already, and there are six starting values!

However, if one takes time to think through this problem, they can realize that it is just an application of stars and bars, a tool all test-takers have under their belt. If you were to take your four rolls, unordered, and place them into six boxes labeled 1,2,3,4,5,6, you would realize that this counts exactly what we want: the number of 1’s rolled, the number of 2’s, etc. Using stars and bars, this is simply $\binom{4+6-1}{6-1} = \binom{9}{5} = 126$, taking care of the majority of the problem. Recognizing this problem as stars and bars effectively turns this nasty-looking bash into a one-liner.

Sadly, I have seen too many students succumb to these mistakes, missing USA(J)MO qualification by a tiny margin. But those who understand the roots of these mistakes are able to qualify for the USA(J)MO and beyond!