American Invitational Mathematics Examination
The American Invitational Mathematics Examination (AIME) is a 15-question 3-hour test given since 1983 to those who rank in the top 5% 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% on the AMC 10.
The use of calculators is not allowed on the test.
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Format and scoring
The exam consists of 15 questions of increasing difficulty, 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 OMR sheet, similar to the way grid-in math questions are answered on the SAT. 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 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 his score on the AMC to determine eligibility for the USAMO. A student's score on the AMC is added to 10 times his 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.
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.
- 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
- 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)
- Solution: 375