The American Invitational Mathematics Examination (AIME) is a highly selective and prestigious 15-question 3-hour test given since 1983 to those who rank in the top 5% on the
AMC 12 The American Mathematics Competitions (AMC) are the first of a series of competitions in secondary school mathematics that determine the United States team for the International Mathematical Olympiad (IMO). The selection process takes place over th ...
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 The American Mathematics Competitions (AMC) are the first of a series of competitions in secondary school mathematics that determine the United States team for the International Mathematical Olympiad (IMO). The selection process takes place over the ...
. Two different versions of the test are administered, the AIME I and AIME II. However, qualifying students can only take one of these two competitions.
The AIME is the second of two tests used to determine
qualification
Qualification is either the process of qualifying for an achievement, or a credential attesting to that achievement, and may refer to:
* Professional qualification, attributes developed by obtaining academic degrees or through professional exper ...
for the
United States Mathematical Olympiad
United may refer to:
Places
* United, Pennsylvania, an unincorporated community
* United, West Virginia, an unincorporated community
Arts and entertainment Films
* ''United'' (2003 film), a Norwegian film
* ''United'' (2011 film), a BBC Two f ...
(USAMO), the first being the
AMC
AMC may refer to:
Film and television
* AMC Theatres, an American movie theater chain
* AMC Networks, an American entertainment company
** AMC (TV channel)
** AMC+, streaming service
** AMC Networks International, an entertainment company
*** AM ...
.
The use of calculators is not allowed on the test.
Format and scoring
The competition consists of 15 questions of increasing difficulty, where each answer is an integer between 0 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
The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and scoring have changed several times; originally called the Scholastic Aptitude Test, it was later called the Schol ...
. Leading zeros must be gridded in; for example, answers of 7 and 43 must be written and gridded in as 007 and 043, respectively.
Concepts typically covered in the competition include topics in
elementary algebra
Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values).
This use of variables entai ...
,
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
,
trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
, as well as
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
,
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
, and
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
. Many of these concepts are not directly covered in typical
high school
A secondary school describes an institution that provides secondary education and also usually includes the building where this takes place. Some secondary schools provide both '' lower secondary education'' (ages 11 to 14) and ''upper seconda ...
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. Thus 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
AMC may refer to:
Film and television
* AMC Theatres, an American movie theater chain
* AMC Networks, an American entertainment company
** AMC (TV channel)
** AMC+, streaming service
** AMC Networks International, an entertainment company
*** AM ...
to determine eligibility for the
USAMO
The United States of America Mathematical Olympiad (USAMO) is a highly selective high school mathematics competition held annually in the United States. Since its debut in 1972, it has served as the final round of the American Mathematics Compet ...
. A student's score on the AMC is added to 10 times their score on the AIME. In 2006, the cutoff for eligibility in the USAMO was 217 combined points.
During the 1990s it was not uncommon for fewer than 2,000 students to qualify for the AIME, although 1994 was a notable exception where 99 students achieved perfect scores on the
AHSME The American Mathematics Competitions (AMC) are the first of a series of competitions in secondary school mathematics that determine the United States team for the International Mathematical Olympiad (IMO). The selection process takes place over the ...
and the list of high scorers, which usually was distributed in small pamphlets, had to be distributed several months late in thick newspaper bundles.
History
The AIME began in 1983. It was given once per year on a Tuesday or Thursday in late March or early April. Beginning in 2000, the AIME is given twice per year, the second date being an "alternate" test given to accommodate those students who are unable to sit for the first test because of spring break, illness, or any other reason. However, under no circumstances may a student officially participate both competitions. The alternate competition, commonly called the "AIME2" or "AIME-II," is usually given exactly two weeks after the first test, on a Tuesday in early April. However, like the AMC, the AIME recently has been given on a Tuesday in early March, and on the Wednesday 15 days later, e.g. March 13 and 20, 2019. In 2020, the rapid spread of the
COVID-19 pandemic
The COVID-19 pandemic, also known as the coronavirus pandemic, is an ongoing global pandemic of coronavirus disease 2019 (COVID-19) caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The novel virus was first identif ...
led to the 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.
Sample problems
*Given that
where
and
are positive integers and
is as large as possible, find
(''2003 AIME I #1'')
:''Answer: 839''
*Find the number of ordered pairs of integers
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
is added to each of the numbers
,
, and
, one obtains the squares of three consecutive terms of an arithmetic series. Find
. (''1989 AIME #7'')
:''Answer: 925''
*Complex numbers
,
and
are the zeros of a polynomial
, and
. The points corresponding to
,
, and
in the complex plane are the vertices of a right triangle with hypotenuse
. Find
. (''2012 AIME I #14'')
:''Answer: 375''
:
Note
See also
*
American Mathematics Competitions The American Mathematics Competitions (AMC) are the first of a series of competitions in secondary school mathematics that determine the United States team for the International Mathematical Olympiad (IMO). The selection process takes place over th ...
*
List of mathematics competitions
Mathematics competitions or mathematical olympiads are competitive events where participants complete a math test. These tests may require multiple choice or numeric answers, or a detailed written solution or proof.
International mathematics compe ...
*
Mandelbrot Competition
Named in honor of Benoit Mandelbrot, the Mandelbrot Competition was a mathematics competition founded by Sam Vandervelde, Richard Rusczyk and Sandor Lehoczky that operated from 1990 to 2019. It allowed high school students to compete individually ...
References
External links
The Official AMC Home PageComplete Archive of AIME Problems and Solutions{{American mathematics
Mathematics competitions
Recurring events established in 1983