American Invitational Mathematics Examination

The American Invitational Mathematics Examination (AIME) is a selective 15-question, 3-hour test given since 1983 to students who score sufficiently well on the AMC 12 or the AMC 10 mathematics examinations. The AIME is the second of two tests used to determine qualification for the prestigious United States of America Mathematical Olympiad (USAMO or USAJMO), the first being the AMC or the additional pathway of the USAMTS. The AIME has been held annually since 1983, and the rules and qualification thresholds have changed substantially since then. The competition is administered by the Mathematical Association of America (MAA).

History

The AIME began in 1983, organized by a committee chaired by Hungarian-American mathematician George Berzsenyi (1938–2026), under the auspices of the MAA. The exam was initially given once per year, on a Tuesday or Thursday in late March or early April. Between 2000 and 2026, two AIME exams were offered annually: AIME I and AIME II, and qualifying students were only allowed to officially take one of these two exams. The majority of the students who qualify for the AIME took AIME I.

In 2020, the COVID-19 pandemic led to cancellation of the AIME II for that year. Instead, qualifying students were able to take the American Online Invitational Mathematics Examination, which contained the problems that were originally going to be on the AIME II. The 2021 AIME I and II were also moved online. The AIME resumed fully in-person in 2023.

Changes for the 2027 AIME

In August 2026, the MAA announced several significant changes for the 2027 AIME in order to protect the integrity of the competition. The exam will be administered at specific Pearson testing centers and will only be available to students aged 13 and over. The AIME will be held over a 2-day window, with no alternate AIME II testing date. The 3-hour competition will be split into two 90-minute halves, with 8 questions in the first half and 7 questions in the second half.

Historical results

The table below lists the mean and median scores for recent AIME examinations.

Contest	Mean score	Median score
2026 I	6.26	6
2026 II	6.55	6
2025 I	6.16	6
2025 II	6.24	6
2024 I	5.89	5
2024 II	5.45	5
2023 I	4.28	4
2023 II	4.40	4
2022 I	4.82	4
2022 II	4.40	4
2021 I	5.44	5
2021 II	5.42	5
2020 I	5.70	6
2020 II	6.13	6
2019 I	5.88	6
2019 II	6.47	6
2018 I	5.09	5
2018 II	5.48	5
2017 I	5.69	5
2017 II	5.64	5
2016 I	5.83	6
2016 II	4.43	4
2015 I	5.29	5
2015 II	6.63	6
2014 I	4.88	5
2014 II	5.49	5

Rules and scoring

The 3-hour competition consists of 15 questions of roughly increasing difficulty, where each answer is an integer between 000 and 999 inclusive. Thus the competition effectively removes the element of chance afforded by a multiple-choice test while preserving the ease of automated grading; answers are entered onto an OMR sheet, similar to the way grid-in math questions are answered on the SAT. Leading zeros must be filled in on the OMR sheet; for example, answers of 7 and 43 must be recorded as 007 and 043. Calculators, rulers, protractors, and other aids are not allowed.

Concepts typically covered in the competition include topics in elementary algebra, geometry, trigonometry, as well as number theory, probability, and combinatorics. Many of these concepts are not directly covered in typical high school mathematics courses; thus, participants often turn to supplementary resources to prepare for the competition.

One point is earned for each correct answer, and no points are deducted for incorrect answers. No partial credit is given, so AIME scores are integers from 0 to 15 inclusive.

USA(J)MO qualification

Prior to the 2026-2027 academic year, USA(J)MO qualification was based on a combination of a student's AMC 10 or 12 score and their AIME score. Historically, a student's AMC score would be added to 10 times their AIME score (20 times in the 2025-2026 academic year) to determine their USA(J)MO index. Since 2017, the USA(J)MO qualification cutoffs have been split between the AMC 10/12 A and B examinations, as well as the AIME I and II. Therefore, there have been up to 8 published USA(J)MO qualification cutoffs per year, and a student could have up to two indices depending on whether they took both an AMC 10A or AMC 12A, and an AMC 10B or AMC 12B.

For the 2026-2027 academic year, AMC 10 and AMC 12 scores will no longer affect qualification for the USA(J)MO.

Sample problems

• Given that

${\frac {((3!)!)!}{3!}}=k\cdot n!,$

where $k$ and $n$ are positive integers and $n$ is as large as possible, find $k+n.$ (2003 AIME I #1)

Answer: 839

• Find the number of ordered pairs of integers $(a,b)$ such that the sequence

$3,4,5,a,b,30,40,50$

is strictly increasing and no set of four (not necessarily consecutive) terms forms an arithmetic progression. (2022 AIME I #6)

Answer: 228

• If the integer $k$ is added to each of the numbers $36$, $300$, and $596$, one obtains the squares of three consecutive terms of an arithmetic series. Find $k$. (1989 AIME #7)

Answer: 925

• Complex numbers $a$, $b$ and $c$ are the zeros of a polynomial $P(z)=z^{3}+qz+r$, and $|a|^{2}+|b|^{2}+|c|^{2}=250$. The points corresponding to $a$, $b$, and $c$ in the complex plane are the vertices of a right triangle with hypotenuse $h$. Find $h^{2}$. (2012 AIME I #14)

Answer: 375

The MAA published a book presenting the first six years of AIME problems and solutions (1983–1988) together with the AHSME qualifiers.

See also

• United States of America Mathematical Olympiad (USAMO)
• American Mathematics Competitions
• List of mathematics competitions
• Mandelbrot Competition

External links

• Official website
• AIME Problems and Solutions