The American Invitational Mathematics Examination (AIME) is a selective 15-question, 3-hour test given since 1983 to those who rank in the top 5% on the AMC 12 high school mathematics examination (formerly known as the AHSME), and starting in 2010, those who rank in the top 2.5% on the AMC 10. Since 2022, the AIME competition has invited those who rank around the top 13-15% on the AMC 12 to qualify for the AIME and invites those who rank around the top 6-8% on the AMC 10. Two different versions of the test are administered, the AIME I and AIME II. All domestic students are automatically scheduled for the AIME I and can reschedule for AIME II, while international students are scheduled for the AIME II. Qualifying students can only take one of these two competitions. The majority of the students who qualify for the AIME take AIME I. Additionally, another pathway to qualify for the AIME is through the USAMTS, a free proof-based math contest. Students who score at least a 68 out of 75 qualify for the AIME. The AIME is the second of two tests used to determine qualification for the USA Mathematical Olympiad (USAMO), the first being the AMC or the additional pathway of the USAMTS. If a student qualifies from both competitions, the AMC pathway is used over the USAMTS when calculating the index scores for the cutoff. The use of calculators is not allowed on the AIME, with only pencils, erasers, rulers, and compasses permitted.
Format and scoring The competition consists of 15 questions of 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. 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. Some historical results are:
A student's score on the AIME is used in combination with their score on the AMC to determine eligibility for the USAMO or USAJMO. Before the 2025-2026 competition cycle, a student's AMC score would be added to 10 times their AIME score to compute the USAMO or USAJMO index. Now, a student's AMC score would be added to 20 times their AIME score to compute the USAMO or USAJMO index. Since 2017, the USAMO and USAJMO qualification cutoff has been split between the AMC A and B, as well as the AIME I and II. Hence, there will be a total of 8 published USAMO and USAJMO qualification cutoffs per year, and a student can have up to 2 USAMO/USAJMO indices (via participating in both AMC contests). The student only needs to reach one qualification cutoff to take the USAMO or USAJMO. During the 1990s, fewer than 2,000 students typically qualified for the AIME. However, in 1994, an unprecedented 99 students achieved perfect scores on the AHSME, causing delays in result distribution. The usual pamphlets were replaced by thick newspaper bundles.
History The AIME began in 1983, organized by a committee chaired by Hungarian-American mathematician George Berzsenyi (1938–2026), under the auspices of the Mathematical Association of America (MAA). The exam was initially given once per year, on a Tuesday or Thursday in late March or early April. Beginning in 2000, the AIME is administered twice per year. The second date, typically a week or two after the first, serves as an alternate (commonly called "AIME II" or "AIME 2") for students who miss the first due to conflicts such as spring break or illness. 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. 2021's AIME I and II were also moved online. 2022's AIME I and II were administered both online and in-person, and from 2023 onward, all AIME contests were administered in-person. For the 2025–2026 competition cycle, AIME I and II were held on February 5 and 11, 2026, respectively.
Sample problems Given that
where k {\displaystyle k} and n {\displaystyle n} are positive integers and n {\displaystyle n} is as large as possible, find k + n . {\displaystyle k+n.} (2003 AIME I #1)
Answer: 839 Find the number of ordered pairs of integers ( a , b ) {\displaystyle (a,b)} such that the sequence
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 {\displaystyle k} is added to each of the numbers 36 {\displaystyle 36} , 300 {\displaystyle 300} , and 596 {\displaystyle 596} , one obtains the squares of three consecutive terms of an arithmetic series. Find k {\displaystyle k} . (1989 AIME #7) Answer: 925 Complex numbers a {\displaystyle a} , b {\displaystyle b} and c {\displaystyle c} are the zeros of a polynomial P ( z ) = z 3 + q z + r {\displaystyle P(z)=z^{3}+qz+r} , and | a | 2 + | b | 2 + | c | 2 = 250 {\displaystyle |a|^{2}+|b|^{2}+|c|^{2}=250} . The points corresponding to a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} in the complex plane are the vertices of a right triangle with hypotenuse h {\displaystyle h} . Find h 2 {\displaystyle 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.
Note
See also United States of America Mathematical Olympiad (USAMO) American Mathematics Competitions List of mathematics competitions Mandelbrot Competition
References
External links Official website AIME Problems and Solutions