United States of America Mathematical Olympiad
Encyclopedia
The United States of America Mathematical Olympiad (USAMO) 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. Top scorers on the USAMO usually represent the United States at the International Mathematical Olympiad
.
or a legal resident of the United States or Canada
. Only U.S. residents and citizens may join the American IMO
team. In addition, all participants, regardless of geographic location, must meet qualification indices determined by previous rounds of the AMC
contests. Entry to the USAMO is by invitation only.
1.U.S. citizens and students residing in the United States and Canada (with qualifying scores) are eligible to take the USAMO and USAJMO.
2.Selection to the USAMO will be based on the USAMO index which is defined as AMC 12 Score + 10 * AIME Score. Selection to the USAJMO will be based on the USAJMO index which is defined as AMC 10 Score + 10 * AIME Score.
3.Only AMC 12 A or AMC 12 B takers who are U.S. citizens and students residing in the United States and Canada will be eligible for the USAMO.
4.Only AMC 10 A or AMC 10 B takers who are U.S. citizens and students residing in the United States and Canada will be eligible for the USAJMO. This automatically limits Junior Math Olympiad participation to 10th graders and below. Students who take ONLY the AMC 10 test, whether AMC 10 A or AMC 10 B or both, will NOT be eligible for the USAMO regardless of their score on the AMC 10 or the AIME.
5.The approximately 260-270 individual students with the top AMC 12 based USAMO indices will be invited to take the USAMO. These indices will be selected from the pool of AMC 12 takers with an AIME score.
6.The approximately 230-240 individual students with the top AMC 10 based USAMO indices will be invited to take the USAJMO. These indices will be selected from the pool of AMC 10 takers with an AIME score after removing students who also took an AMC 12 test and qualified for the USAMO in rule 5. This means young students MUST take the USAMO if they qualify through an AMC 12 index.
7.We will select the student with the numerically largest index, whether AMC 10 based USAJMO index or AMC 12 based USAMO index, from each US state not already represented in either the USAMO or the USAJMO. The student will be invited to the USAMO if the numerically highest index in the state is AMC 12 based, and invited to the USAJMO if the index is AMC 10 based.
1.Selection to the USAMO and JMO will be based on the USAMO index which is defined as
AMC score + 10 * AIME score.
2.Only AMC 12A or AMC 12B takers are eligible for the USAMO (with the slight exception mentioned in item 5 below).
3.Only AMC 10A and AMC 10B takers are eligible for the JMO. (This automatically limits Junior Math Olympiad participation to 10th graders and below.)
4.Approximately the top 260 AMC12 based USAMO indices will be invited to the USAMO.
5.In order to find unrecognized young talent, AMC 10 takers who score 11 or more on the AIME will be invited to the USAMO. (In 2008 and 2009 this was 5 or 6 students).
6.Select the top index from any state not already represented in the USAMO.
7.Approximately the top 220-230 students with AMC10 based USAMO indices and not already selected to the USAMO via an AMC12 based index will be invited to the JMO.
Source: http://www.unl.edu/amc/e-exams/e0-usajmo/usajmo.shtml
1. The goal is to select about 500 of the top scorers from this
years’s AIME and AMC 12A, AMC 12B, AMC 10A and AMC
10B contests to participate in the USAMO.
2. Selection will be based on the USAMO index which is defined
as 10 times the student’s AIME score plus the student’s score
on the AMC 12 or the AMC 10.
3. The first selection will be the approximately 330 highest
USAMO indices of students taking the AMC 12A or AMC
12B contest.
4. The lowest AIME score among those 330 first selected will
determine a floor value. The second selection of approximately
160 USAMO participants will be among students in
the 10th grade and below who received an AIME score at
least as high as the floor value. If there are more than 160
young students with a score above the floor value, then approximately
160 students will be selected from this group by
using the USAMO index.
5 The student with the highest USAMO index from each state,
territory, or U.S. possession not already represented in the
selection of the first and second groups will be invited to
take the USAMO.
6. To adjust for variations in contest difficulty, the number of
students selected from A & B contests will be proportional
to the number of students who took the A & B Contests.
7. In advising young students (in grade 10 or below) who desire
to be selected for the USAMO whether to take the AMC 12
contest or the AMC 10 contest, please be aware of the following
facts:
a. In 2007, among 506 students invited to take the USAMO,
229 were in 10th grade and below. Those students had
scored 6 or greater on the AIME.
b. Among those 229 students, 87 had their AIME qualifying
high score based on the AMC 12 and 142 had their AIME
qualifying high score based on the AMC 10.
c. In 2007, among 8,312 students who took the AIME, 2,696
were in grades 10 and below. Of those, 998 qualified
for the AIME from the AMC 12 and 1,698 qualified from
the AMC 10.
Source: American Mathematics Competitions
Source: American Mathematics Competitions
Source: American Mathematics Competitions
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examination spread out over two days. (The IMO
uses the same format.) On each day, four and a half hours are given for three questions.
Each question is graded on a scale from 0 to 7, with a score of 7 representing a proof that is mathematically sound. Thus, a perfect score is 42 points. The number of perfect papers each year has varied depending on test difficulty. Regardless, the top 12 scorers are all named contest winners.
The scale of 0 to 7 goes as follows:
In 2002, the Akamai Foundation
, as a major sponsor of the American Mathematics Competitions, invited all USAMO participants to take the test at a central event at MIT in Cambridge, Massachusetts
, all expenses paid. In addition, Akamai invited all 2002 USAMO participants who were not high school seniors (approximately 160 students) to take part in an enlarged Mathematical Olympiad Program
(also known as "MOP") program. Since holding this central event every year would be prohibitively expensive, it has been discontinued. In 2004 and 2005, however, funding was found to send 30 rising freshmen to MOP as well, in a program popularly called "Red MOP."
Each year, the top 12 scorers on the USAMO are considered for selection to the IMO
team for the United States. The students are trained at the MOP in Lincoln, Nebraska
, and then six are selected to the team. The next approximately 18 high scorers, usually excluding high school seniors, are also invited to MOP.
2011:
2010:
2009:
2008:
2007:
2006:
2005:
2004:
2003:
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
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
competition
Competition
Competition is a contest between individuals, groups, animals, etc. for territory, a niche, or a location of resources. It arises whenever two and only two strive for a goal which cannot be shared. Competition occurs naturally between living organisms which co-exist in the same environment. For...
held annually in the United States
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
. Since its debut in 1972, it has served as the final round of 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...
series of contests. Top scorers on the USAMO usually represent the United States at the International Mathematical Olympiad
International Mathematical Olympiad
The International Mathematical Olympiad is an annual six-problem, 42-point mathematical olympiad for pre-collegiate students and is the oldest of the International Science Olympiads. The first IMO was held in Romania in 1959. It has since been held annually, except in 1980...
.
Eligibility
In order to be eligible to take the USAMO, a participant must be either a U.S. citizenUnited States nationality law
Article I, section 8, clause 4 of the United States Constitution expressly gives the United States Congress the power to establish a uniform rule of naturalization. The Immigration and Naturalization Act sets forth the legal requirements for the acquisition of, and divestiture from, citizenship of...
or a legal resident of the United States or Canada
Canada
Canada is a North American country consisting of ten provinces and three territories. Located in the northern part of the continent, it extends from the Atlantic Ocean in the east to the Pacific Ocean in the west, and northward into the Arctic Ocean...
. Only U.S. residents and citizens may join the American IMO
International Mathematical Olympiad
The International Mathematical Olympiad is an annual six-problem, 42-point mathematical olympiad for pre-collegiate students and is the oldest of the International Science Olympiads. The first IMO was held in Romania in 1959. It has since been held annually, except in 1980...
team. In addition, all participants, regardless of geographic location, must meet qualification indices determined by previous rounds of 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...
contests. Entry to the USAMO is by invitation only.
Participant selection process
The USAMO (and the USAJMO since 2010) is restricted to approximately 500 (250 prior to 2006) participants combined each year. To keep this quota constant, the AMC Committee uses a selection process, which has seen a number of revisions in the exam's history.2011
In 2011 we will have slightly revised qualification rules for the USA Mathematical Olympiad and USA Junior Mathematical Olympiad. The goal is to select approximately 500 students total for the two Olympiads, split approximately 270 for the USAMO and 230 for the USAJMO respectively. Selection for the 2011 USA Mathematical Olympiad (USAMO) and 2011 USA Junior Mathematical Olympiad (USAJMO) will be made according to the following rules:1.U.S. citizens and students residing in the United States and Canada (with qualifying scores) are eligible to take the USAMO and USAJMO.
2.Selection to the USAMO will be based on the USAMO index which is defined as AMC 12 Score + 10 * AIME Score. Selection to the USAJMO will be based on the USAJMO index which is defined as AMC 10 Score + 10 * AIME Score.
3.Only AMC 12 A or AMC 12 B takers who are U.S. citizens and students residing in the United States and Canada will be eligible for the USAMO.
4.Only AMC 10 A or AMC 10 B takers who are U.S. citizens and students residing in the United States and Canada will be eligible for the USAJMO. This automatically limits Junior Math Olympiad participation to 10th graders and below. Students who take ONLY the AMC 10 test, whether AMC 10 A or AMC 10 B or both, will NOT be eligible for the USAMO regardless of their score on the AMC 10 or the AIME.
5.The approximately 260-270 individual students with the top AMC 12 based USAMO indices will be invited to take the USAMO. These indices will be selected from the pool of AMC 12 takers with an AIME score.
6.The approximately 230-240 individual students with the top AMC 10 based USAMO indices will be invited to take the USAJMO. These indices will be selected from the pool of AMC 10 takers with an AIME score after removing students who also took an AMC 12 test and qualified for the USAMO in rule 5. This means young students MUST take the USAMO if they qualify through an AMC 12 index.
7.We will select the student with the numerically largest index, whether AMC 10 based USAJMO index or AMC 12 based USAMO index, from each US state not already represented in either the USAMO or the USAJMO. The student will be invited to the USAMO if the numerically highest index in the state is AMC 12 based, and invited to the USAJMO if the index is AMC 10 based.
2010
Starting in 2010, the USA Mathematical Olympiad is split into two parts. The USA Mathematical Olympiad will be administered to approximately 270 students, mostly selected from top ranking AMC12 participants. The AMC10 only participants will take part in USA Junior Mathematical Olympiad.1.Selection to the USAMO and JMO will be based on the USAMO index which is defined as
AMC score + 10 * AIME score.
2.Only AMC 12A or AMC 12B takers are eligible for the USAMO (with the slight exception mentioned in item 5 below).
3.Only AMC 10A and AMC 10B takers are eligible for the JMO. (This automatically limits Junior Math Olympiad participation to 10th graders and below.)
4.Approximately the top 260 AMC12 based USAMO indices will be invited to the USAMO.
5.In order to find unrecognized young talent, AMC 10 takers who score 11 or more on the AIME will be invited to the USAMO. (In 2008 and 2009 this was 5 or 6 students).
6.Select the top index from any state not already represented in the USAMO.
7.Approximately the top 220-230 students with AMC10 based USAMO indices and not already selected to the USAMO via an AMC12 based index will be invited to the JMO.
Source: http://www.unl.edu/amc/e-exams/e0-usajmo/usajmo.shtml
2008
Selection for the USAMO will be made according to the following rules:1. The goal is to select about 500 of the top scorers from this
years’s AIME and AMC 12A, AMC 12B, AMC 10A and AMC
10B contests to participate in the USAMO.
2. Selection will be based on the USAMO index which is defined
as 10 times the student’s AIME score plus the student’s score
on the AMC 12 or the AMC 10.
3. The first selection will be the approximately 330 highest
USAMO indices of students taking the AMC 12A or AMC
12B contest.
4. The lowest AIME score among those 330 first selected will
determine a floor value. The second selection of approximately
160 USAMO participants will be among students in
the 10th grade and below who received an AIME score at
least as high as the floor value. If there are more than 160
young students with a score above the floor value, then approximately
160 students will be selected from this group by
using the USAMO index.
5 The student with the highest USAMO index from each state,
territory, or U.S. possession not already represented in the
selection of the first and second groups will be invited to
take the USAMO.
6. To adjust for variations in contest difficulty, the number of
students selected from A & B contests will be proportional
to the number of students who took the A & B Contests.
7. In advising young students (in grade 10 or below) who desire
to be selected for the USAMO whether to take the AMC 12
contest or the AMC 10 contest, please be aware of the following
facts:
a. In 2007, among 506 students invited to take the USAMO,
229 were in 10th grade and below. Those students had
scored 6 or greater on the AIME.
b. Among those 229 students, 87 had their AIME qualifying
high score based on the AMC 12 and 142 had their AIME
qualifying high score based on the AMC 10.
c. In 2007, among 8,312 students who took the AIME, 2,696
were in grades 10 and below. Of those, 998 qualified
for the AIME from the AMC 12 and 1,698 qualified from
the AMC 10.
2006-2007
Beginning in 2006, the USAMO was expanded to include approximately 500 students (around 430 were actually invited, read below) due to a proposal and sponsorship from the Art of Problem Solving website:- The goal is to select about 500 of the top scorers from this years’s AIME and AMC 12A, AMC 12B, AMC 10A and AMC 10B contests to participate in the USAMO.
- Selection will be based on the USAMO index which is defined as 10 times the student’s AIME score plus the student’s score on the AMC 12 or the AMC 10.
- The first selection will be the approximately 240 highest USAMO indices of students taking the AMC 12A or AMC 12B contest.
- The lowest AIME score among those 240 first selected will determine a floor value. The second selection of approximately 120 USAMO participants will be among students in the 10th grade and below who received an AIME score at least as high as the floor value. If there are more than 120 young students with a score above the floor value, then approximately 120 students will be selected from this group by using the USAMO index.
- The student with the highest USAMO index from each state, territory, or U.S. possession not already represented in the selection of the first and second groups will be invited to take the USAMO.
- To adjust for variations in contest difficulty, the number of students selected from A & B contests will be proportional to the number of students who took the A & B Contests.
- The selection process is designed to favor students who take the more mathematically comprehensive AMC 12A and AMC 12B contests.*
Source: American Mathematics Competitions
- Statement 7 above (quoted from the AMC website) has recently come under controversy. During the selection for the 2006 USAMO, students who qualified by the floor value (in grades seven through ten) were qualified based on AMC scores as well (see * below) as their AIME scores, yet no distinction was made between the AMC 12 contest and the generally easier AMC 10 contest, giving those who took the AMC 10 an advantage over those who took the AMC 12. Students in grades seven through ten who were in the first selection of qualifiers (see 3. above) would still have qualified even if they had taken the AMC 10, except in the rare case that they set the floor themselves, making the AMC 12 still non-advantageous.
2002-2005
Since 2002, the following set of guidelines have been adopted for use in determining each year's USAMO participants:- The goal is to select about 250 of the top scorers from the prior AIME and AMC 12A, AMC 12B, AMC 10A and AMC 10B contests to participate in the USAMO.
- Selection will be based on the USAMO index which is defined as 10 times the student’s AIME score plus the student’s score on the AMC 12 or the AMC 10.
- The first selection (consisting of participants from all grade levels) will be the approximately 160 highest USAMO indices of students taking the AMC 12A or AMC 12B contest.
- The lowest AIME score among those 160 first selected will determine a floor value. The second selection of USAMO participants will be from the highest USAMO indices among students in grades seven through ten who got an AIME score at least as high as the floor value. To note, during 2002-2005 period, this included all students in grades seven through ten.
- The student with the highest USAMO index from each state, territory, or U.S. possession not already represented in the selection of the first and second groups will be invited to take the USAMO.
- To adjust for variations in contest difficulty, the number of students selected from A & B contests will be proportional to the number of students who took the (A & B) Contests.
- The selection process is designed to favor students who take the more mathematically comprehensive AMC 12A and AMC 12B contests.
Source: American Mathematics Competitions
2001 and earlier
Prior to 2001, the following guidelines were used:- First Group: The top 120 students.
- Second Group: The next 20 students in grades 11 and below.
- Third Group: The next 20 students in grades 10 or below.
- Fourth Group: The next 20 students in grades 9 or below.
- Fifth Group: One student from each state, one student from the combined U.S.A. Territories, and one student from the APO/FPO schools- if not represented in the first four groups.
Source: American Mathematics Competitions
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Recent qualification indices
| Year | AMC 12 | AMC 10 | Total number of qualifiers |
|---|---|---|---|
| 2011 | 188.0 (AIME I); 215.5 (AIME II) for USAMO | 179.0 (AIME I); 196.5 (AIME II) for USAJMO | 282 USAMO; 222 USAJMO |
| 2010 | 208.5 (USAMO); 204.5 (USAMO -- 11th and 12th) | 11/15 on AIME (USAMO) OR 188.5+ on index (USAJMO) | 328 USAMO; 235 USAJMO |
| 2009 | 201.0 | 7/15 on AIME AND 215.0+ on index | 514 |
| 2008 | 204.0 | 6/15 on AIME AND 202.5+ on index | 503 |
| 2007 | 197.5 | 6/15 on AIME AND 181.0+ on index | 505 |
| 2006 | 217 | 8/15 on AIME | 432 |
| 2005 | 233 (AIME I); 220.5 (AIME II) | 9/15 on AIME | 259 |
| 2004 | 210 | 7/15 on AIME | 261 |
| 2003 | 226 | 8/15 on AIME | 250 |
| 2002 | 210 | 6/15 on AIME | 326 |
| 2001 | 213 | 7/15 on AIME | 268 |
| 2000 | 212 (12th); 204 (11th) | 9th grade: 7/15 on AIME AND 164+ on index; 10th grade: 8/15 on AIME AND 174+ on index | 239 |
Post-2002
Since 2002, the USAMO has been a six-question, nine-hour mathematical proofMathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
examination spread out over two days. (The IMO
International Mathematical Olympiad
The International Mathematical Olympiad is an annual six-problem, 42-point mathematical olympiad for pre-collegiate students and is the oldest of the International Science Olympiads. The first IMO was held in Romania in 1959. It has since been held annually, except in 1980...
uses the same format.) On each day, four and a half hours are given for three questions.
Each question is graded on a scale from 0 to 7, with a score of 7 representing a proof that is mathematically sound. Thus, a perfect score is 42 points. The number of perfect papers each year has varied depending on test difficulty. Regardless, the top 12 scorers are all named contest winners.
The scale of 0 to 7 goes as follows:
- 0 - No work, or completely trivial work
- 1-2 - Progress on the problem, but not completely solved
- 3-4 - All steps are present, but may lack clarity. (These scores are very rare.)
- 5-6 - Complete solution with minor errors
- 7 - Perfect solution
1996 to 2001
The test consisted of two three-problem sets. Three hours were given for each set; one set was given in the morning (9:00-12:00), and the other in the afternoon (1:00-4:00).1995 and earlier
The test consisted of five problems to be solved in three and a half hours (earlier, three hours). Each problem was worth 20 points, for a perfect score of 100.Test procedures
In most years, students have taken the USAMO at their respective high schools. Prior to 2002, the problems were mailed to the schools in sealed envelopes, not to be opened before the appointed time on the test day. Since 2002, test problems have been posted on the AMC website (see links below) fifteen minutes prior to the official start of the test. Student responses are then faxed back to the AMC office at the end of the testing period.In 2002, the Akamai Foundation
Akamai Foundation
The Akamai Foundation is an American non-profit organization that sponsors mathematics competitions. It was founded and it is supported financially by Akamai Technologies....
, as a major sponsor of the American Mathematics Competitions, invited all USAMO participants to take the test at a central event at MIT in Cambridge, Massachusetts
Cambridge, Massachusetts
Cambridge is a city in Middlesex County, Massachusetts, United States, in the Greater Boston area. It was named in honor of the University of Cambridge in England, an important center of the Puritan theology embraced by the town's founders. Cambridge is home to two of the world's most prominent...
, all expenses paid. In addition, Akamai invited all 2002 USAMO participants who were not high school seniors (approximately 160 students) to take part in an enlarged Mathematical Olympiad Program
Mathematical Olympiad Program
The Mathematical Olympiad Program or MOP is an intensive summer program held at University of Nebraska-Lincoln. The main purpose of MOP, held since 1974, is to select and train the six members of the U.S. team for the International Mathematical Olympiad . Students qualify for the program by taking...
(also known as "MOP") program. Since holding this central event every year would be prohibitively expensive, it has been discontinued. In 2004 and 2005, however, funding was found to send 30 rising freshmen to MOP as well, in a program popularly called "Red MOP."
Each year, the top 12 scorers on the USAMO are considered for selection to the IMO
International Mathematical Olympiad
The International Mathematical Olympiad is an annual six-problem, 42-point mathematical olympiad for pre-collegiate students and is the oldest of the International Science Olympiads. The first IMO was held in Romania in 1959. It has since been held annually, except in 1980...
team for the United States. The students are trained at the MOP in Lincoln, Nebraska
Lincoln, Nebraska
The City of Lincoln is the capital and the second-most populous city of the US state of Nebraska. Lincoln is also the county seat of Lancaster County and the home of the University of Nebraska. Lincoln's 2010 Census population was 258,379....
, and then six are selected to the team. The next approximately 18 high scorers, usually excluding high school seniors, are also invited to MOP.
Exam content
Here are the subjects on the test in different years by problem number (i.e. what subject each problem was from):2011:
- AlgebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
/Inequalities - CombinatoricsCombinatoricsCombinatorics 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 ,...
- GeometryGeometryGeometry 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 ....
- Number TheoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
- GeometryGeometryGeometry 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 ....
- CombinatoricsCombinatoricsCombinatorics 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 ,...
2010:
- GeometryGeometryGeometry 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 ....
- CombinatoricsCombinatoricsCombinatorics 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 ,...
- AlgebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
- GeometryGeometryGeometry 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 ....
/Number TheoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well... - AlgebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
/Number TheoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well... - CombinatoricsCombinatoricsCombinatorics 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 ,...
2009:
- GeometryGeometryGeometry 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 ....
- CombinatoricsCombinatoricsCombinatorics 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 ,...
- CombinatoricsCombinatoricsCombinatorics 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 ,...
/Graph TheoryGraph theoryIn mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of... - AlgebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
- GeometryGeometryGeometry 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 ....
- Number TheoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
2008:
- Number TheoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
- GeometryGeometryGeometry 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 ....
- CombinatoricsCombinatoricsCombinatorics 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 ,...
- CombinatoricsCombinatoricsCombinatorics 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 ,...
- Number TheoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
/CombinatoricsCombinatoricsCombinatorics 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 ,... - Graph TheoryGraph theoryIn mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
/Linear AlgebraLinear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
2007:
- Number TheoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
/CombinatoricsCombinatoricsCombinatorics 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 ,... - GeometryGeometryGeometry 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 ....
- CombinatoricsCombinatoricsCombinatorics 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 ,...
- Graph TheoryGraph theoryIn mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
- Number TheoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
- GeometryGeometryGeometry 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 ....
2006:
- Number TheoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
- AlgebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
/CombinatoricsCombinatoricsCombinatorics 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 ,... - Number TheoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
/AlgebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures... - AlgebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
- AlgebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
/CombinatoricsCombinatoricsCombinatorics 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 ,... - GeometryGeometryGeometry 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 ....
2005:
- Number TheoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
/Graph TheoryGraph theoryIn mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of... - Number TheoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
- GeometryGeometryGeometry 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 ....
- GeometryGeometryGeometry 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 ....
/AlgebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures... - CombinatoricsCombinatoricsCombinatorics 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 ,...
- AlgebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
2004:
- GeometryGeometryGeometry 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 ....
/Inequalities - AlgebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
- CombinatoricsCombinatoricsCombinatorics 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 ,...
/GeometryGeometryGeometry 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 .... - CombinatoricsCombinatoricsCombinatorics 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 ,...
- Inequalities
- GeometryGeometryGeometry 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 ....
2003:
- Number TheoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
- GeometryGeometryGeometry 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 ....
/AlgebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures... - AlgebraAlgebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
- GeometryGeometryGeometry 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 ....
- Inequalities
- CombinatoricsCombinatoricsCombinatorics 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 ,...
External links
- The Official AMC Home Page - Contains rules, problems, qualifiers, and winners for each year since 1999
- The IMO Compendium - huge collection of problems from mathematical competitions, and the most complete collection of IMO shortlists and longlists.
- MathLinks - a large community of Olympiad problems solvers from around the globe.
- How to Write a Proof
- An archive of USAMO Problems