American Invitational Mathematics Examination
Encyclopedia
The American Invitational Mathematics Examination (AIME) is a 15-question 3-hour test given since 1983 to those who rank in the top 5% (or score at least 100) on the AMC 12 high school mathematics contest (formerly known as the AHSME), and starting in 2010, those who rank in the top 2.5% (or score at least 120) on the AMC 10.
The AIME is the second of two tests used to determine qualification for the United States of America Mathematical Olympiad
(USAMO), the first being the AMC
.
The use of calculators is not allowed on the test.
sheet, similar to the way grid-in math questions are answered on the SAT
.
Concepts typically covered on the exam 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 exam.
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 recent 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. A student's score on the AMC is added to 10 times his/her 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 and the list of high scorers, which usually was distributed in small pamphlets, had to be distributed several months late in thick newspaper bundles.
where
and 
are positive integers and 
is as large as possible, find 
(2003 AIME I #1)
The AIME is the second of two tests used to determine qualification for the United States of America Mathematical Olympiad
United States of America Mathematical Olympiad
The United States of America Mathematical Olympiad is a high school mathematics competition held annually in the United States. Since its debut in 1972, it has served as the final round of the AMC series of contests...
(USAMO), the first being the AMC
American Mathematics Competitions
The American Mathematics Competitions are the first of a series of competitions in high school mathematics that determine the United States team for the International Mathematical Olympiad . This team, consisting of six high school students, competes in the IMO and has traditionally performed well...
.
The use of calculators is not allowed on the test.
Format and scoring
The exam consists of 15 questions, where each answer is an integer between 0 and 999 inclusive. Thus the test effectively removes the element of chance afforded by a multiple-choice test while preserving the ease of automated grading; answers are entered onto an OMROptical mark recognition
Optical Mark Recognition is the process of capturing human-marked data from document forms such as surveys and tests.-OMR background:...
sheet, similar to the way grid-in math questions are answered on the SAT
SAT
The SAT Reasoning Test is a standardized test for college admissions in the United States. The SAT is owned, published, and developed by the College Board, a nonprofit organization in the United States. It was formerly developed, published, and scored by the Educational Testing Service which still...
.
Concepts typically covered on the exam include topics in elementary algebra
Elementary algebra
Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. It is typically taught in secondary school under the term algebra. The major difference between algebra and...
, geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...
, as well as number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
, and combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
. Many of these concepts are not directly covered in typical high school
High school
High school is a term used in parts of the English speaking world to describe institutions which provide all or part of secondary education. The term is often incorporated into the name of such institutions....
mathematics courses; thus, participants often turn to supplementary resources to prepare for the exam.
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 recent results are:
| Year | Students sitting |
Mean score |
Median score |
Students with perfect scores |
| 2006 | 22764 | 2.741 | - | 4 |
| 2005 | 12476 | 2.717 | 2 | 1 |
| 2004 | 11945 | 2.195 | 2 | 4 |
| 2003 | 13444 | 3.059 | 3 | 3 |
A student's score on the AIME is used in combination with their score on the AMC
American Mathematics Competitions
The American Mathematics Competitions are the first of a series of competitions in high school mathematics that determine the United States team for the International Mathematical Olympiad . This team, consisting of six high school students, competes in the IMO and has traditionally performed well...
to determine eligibility for the USAMO. A student's score on the AMC is added to 10 times his/her 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 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 take both exams. The alternate test, 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 7 and 22, 2006.Sample Problems
- Given that
where
and are positive integers and is as large as possible, find (2003 AIME I #1)
- Solution: 839
- 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)
- Solution: 925
See also
- American Mathematics CompetitionsAmerican Mathematics CompetitionsThe American Mathematics Competitions are the first of a series of competitions in high school mathematics that determine the United States team for the International Mathematical Olympiad . This team, consisting of six high school students, competes in the IMO and has traditionally performed well...
- List of mathematics competitions
- Mandelbrot CompetitionMandelbrot CompetitionNamed in honor of the Mandelbrot set, the Mandelbrot Competition is a mathematics competition founded by Sam Vandervelde, Richard Rusczyk and Sandor Lehoczky that allows high school students to compete individually and in four-person teams.-Competition:...