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Simon Lee
2016-08-31
Version –eiπ
A Guide to Math Competitions
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About Me
Simon Lee
• Competed in Math Competitions in Canada
• Top 5 Finisher in Canadian Mathematics Olympiad
and Euclid Math Contest
• Individual Mention, Putnam Competition
• Studied Pure Mathematics as undergraduate and in
graduate school
• Executive in Data Science
• Coach middle school and high school math team
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Agenda
• The Contests
• Description
• Their Difficulty
• Why Do Math Contests?
• How to Prepare
• FAQ
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Mathematics
“Mathematics takes us still further
from what is human, into the region
of absolute necessity, to which not
only the world, but every possible
world, must conform.”
― Bertrand Russell
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AMCs to AMO and IMO
The American Mathematics Competition Program
• The major mathematics competitions in the United States for Middle and
High School students
• Consists of
• American Mathematics Competition 8 (AMC 8)
• American Mathematics Competition 10 A and B (AMC 10)
• American Mathematics Competition 12 A and B (AMC 12)
• American Invitational Mathematics Examination I and II (AIME)
• USA Junior Math Olympiad (USAJMO)
• USA Math Olympiad (AMO)
• Success in the AMO leads to an invitation to Math Olympiad Summer
Program (MOSP) from which a team is chosen to represent the United
States at the very prestigious international competition, the International
Math Olympiad (IMO)
See http://www.maa.org/math-competitions
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AMC 8
American Mathematics Competition 8
Eligibility All students 8th grade and below
Dates November 15, 2016
Location Middle schools, some homeschool organizations and colleges
Format 40 minute, 25 multiple choice questions
Calculators Not permitted
Scoring One point per question correct, no penalty for incorrect
Participants ~ 110,000 (US)
95th Percentile 18/25
99th Percentile 23/25
Qualifies For N/A
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AMC 10
American Mathematics Competition 10
Eligibility All students 10th grade and below
Dates February 7, 2017 (A) and February 15, 2017 (B)
Location High school, some homeschool organizations and colleges
Format 75 minute, 25 multiple choice questions
Calculators Not permitted
Scoring Out of 150; 6 points per correct, 1.5 point penalty per incorrect
Participants ~ 50,000 (A), ~25,000 (B) in US
95th Percentile ~100 / 150
99th Percentile ~120 / 150
Qualifies For ~120 / 150 typically qualifies for AIME
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AMC 12
American Mathematics Competition 12
Eligibility All students before college
Dates February 7, 2017 (A) and February 15, 2017 (B)
Location High school, some homeschool organizations and colleges
Format 75 minute, 25 multiple choice questions
Calculators Not permitted
Scoring Out of 150; 6 points per correct, 1.5 point penalty per incorrect
Participants ~ 45,000 (A), ~25,000 (B) in US
95th Percentile ~100 / 150
99th Percentile ~120 / 150
Qualifies For ~100 / 150 typically qualifies for AIME
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AMC 10 and 12
Additional Rules
• One may only take one of the 10A or 12A
• One may only take one of the 10B or 12B
• Those in 11th and 12th grade may only take the 12A and / or 12 B
• Those below 11th grade may take any combination of the (10A or12A) and
(10B or 12B)
• Only the best score is considered for qualification for the AIME
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AIME
American Invitational Mathematics Examination
Eligibility Qualification through AMC 10 or AMC 12 (or USAMTS)
Dates March 7, 2017 (I) or March 22, 2017 (II)
Location High school, some homeschool organizations and colleges
Format 3 hour, 15 questions, three digit answers
Calculators Not permitted
Scoring Out of 15; 1 point per correct, no penalty per incorrect
Participants ~ 4,000 (I), ~1,000 (II) in US
95th Percentile ~11 / 15
99th Percentile ~13 / 15
Qualifies For USAJMO (below 11th grade) and USAMO
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USAJMO / AMO
Qualifications
USAJMO
• Only individuals below 11th grade may qualify for the USAJMO
• The composite score for qualification is: 10 x AIME score + best AMC 10
score
• The top ~270 scorers qualify for the USAJMO
USAMO
• Any individual may qualify for the USAMO
• The composite score for qualification is: 10 x AIME score + best AMC 12
score
• The top ~230 scorers qualify for the USAMO
• Qualifiers to both the USAJMO and USAMO may only take the USAMO
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USAJMO
American Junior Mathematics Olympiad
Eligibility Qualification through AMC 10 and AIME
Dates April 19-20, 2017
Location Proctored by educator
Format 9 hour, 6 questions, written proofs over 2 days
Calculators Not permitted
Scoring Out of 42; 7 points per question
Participants ~ 270
95th Percentile Varies
99th Percentile Varies
Qualifies For MOSP (Lowest Group)
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USAMO
American Mathematics Olympiad
Eligibility Qualification through AMC 12 and AIME
Dates April 19-20, 2017
Location Proctored by educator
Format 9 hour, 6 questions, written proofs over 2 days
Calculators Not permitted
Scoring Out of 42, 7 points per question
Participants ~ 230
95th Percentile Varies
99th Percentile Varies
Qualifies For MOSP
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IMO
Success!!
• The United States won the International Math Olympiad in 2015 and 2016!!
• The United States hadn’t won since 1986 in the competition dominated by China
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AMC Competitions at a Glance
8th Grade and Below
AMC 8
AMC 10A AMC 12A
AMC 10B AMC 12B
or
or
AIME
USAJMO USAMO
MOSP
IMO
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AMC Competitions at a Glance
10th Grade and Below
AMC 8
AMC 10A AMC 12A
AMC 10B AMC 12B
or
or
AIME
USAJMO USAMO
MOSP
IMO
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AMC Competitions at a Glance
11th and 12th Grade
AMC 8
AMC 10A AMC 12A
AMC 10B AMC 12B
or
or
AIME
USAJMO USAMO
MOSP
IMO
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Other Competitions
Team / Individual Competitions
Competition Level Team /
Individual
Location Comment
MathCounts Middle
School
Both; Multi-
Round
Local initially;
Single location
nationally
Actually four progressive
competitions: School, Chapter, State
and Nationals
Harvard MIT
Math
Tournament
High
School
Both; Multi-
Round
Either Harvard
or MIT
Held on one day in November (easier;
short answer) and February (harder;
proofs)
Princeton
University
Mathematics
Competition
High
School
Both; Multi-
Round
Princeton
University
Held in November. One part is proof
based done over a week. Other parts
are done at Princeton University over
one day.
American
Regions Math
League
High
School
Both; Multi-
Round
Penn State and
three other
locations
Held in June. Short answer and proof
based.
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Competition Difficulty
Difficulty Over Time
• Math competitions have gotten much more difficult over time
• Resources are much better
• Students are much more competitive
Scripps Spelling Bee Winning Words
1959 and onCatamaran
Sycophant
Eczema
Chihuahua
Abalone
2010sStromuhr
Cymotrichous
Guetapens
Knaidel
Feuilleton
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Competition Difficulty
Difficulty Over Time
IMO 1959 #1
Prove that the fraction (21n+4) / (14n+3) is irreducible for every natural number n.
IMO 2010 #1
Determine all functions f:R → R such that the equality
f(g(x)y) = f(x)g(f(y)) holds for all real x and y where g is the floor function.
Competition winners are really “mini-professionals”
Extreme talent and hard work are required
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Competition Difficulty
Relative Difficulty
AMC 8 1 to 1.5
AMC 10 1 to 3
AMC 12 2 to 4
AIME 3 to 6
USAJMO 6 to 8
USAMO 7 to 9
IMO 6.5 to 9.5
MathCounts 0.5 to 3
HMMT
(February)
5 to 8
ARML 2 to 6
See https://www.artofproblemsolving.com/wiki/index.php?title=AoPSWiki:Competition_ratings
Easy
Challenging
Very
Difficult
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Why Compete?
The Goals
Very few will actually achieve the top levels; so why compete?
• Hone problem solving skills that will serve one well beyond high school
• Be able to tackle problems that are difficult and time consuming
• Build confidence
• Strengthen quantitative skills
• Gain an appreciation for mathematics
• Help with college admissions??
• Best reason: Because the student wants to do more mathematics
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How to Prepare
American Education in Mathematics
• Lacking in many ways
• Insufficient progress in covering material
• Conditions students to think that all math problems should be doable
quickly
• Poor in developing mathematical thought
• Very little emphasis on proofs / mathematical writing
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How to Prepare
The Subject Matter
• Algebra
• Geometry
• Number Theory
• Discrete Math (Combinatorics / Graph Theory)
• Only the first two are covered in the standard high school curriculum
• But not to the required depth
• Maturity in the first two topics and in Number Theory and Discrete Math
will required outside work
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How to Prepare
Mathematical Writing / Proofs
• There are a number of students who do quite well on the AIME and qualify
for the USAMO, but don’t do well on the AMO
• One big cause is the inability to think and write mathematical arguments
(proofs)
• In some respects, this is the separation between those who can use
mathematics and those who can produce mathematics
• The ways to improve
• Start with basic proof techniques (induction, etc.)
• Read math proofs
• Do old contests that have proofs (ARML, HMMT, USAMO)
• Try the USA Math Talent Search
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How to Prepare
References
Books and Courses
• Art of Problem Solving
Former Contests
• Available for all competitions mentioned
• Note, though that the earlier years are much easier
Camps
• MathPath
• Canada/USA MathCamp
• PROMYS
• AwesomeMath
• http://www.ams.org/programs/students/emp-mathcamps
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How to Prepare
The Secret …
• There is no secret; it’s just hard work
• Do LOTS and LOTS of problems that are uncomfortably difficult but not
impossible
• Don’t give up on problems too easily but don’t spend too much time on a
problem for the sake of efficiency
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Questions?
FAQ
Q: Will a certain level of mathematics in the standard curriculum prepare a
student adequately for competition math? Precalc? AP Calc BC?
A: No, the material covered in those courses do not train a student because
• Much of the material in competition math is not in the standard
curriculum
• The material is covered in a shallow manner
• Students are not trained to struggle over problems
Q: How well would a student need to do to get into Harvard, Princeton, Yale,
etc.?
A: There is no guarantee to get into those institutions. Their admission criteria
are opaque; extremely strong students are often rejected and very weak one
students can be admitted.
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Questions?
FAQ
Q: When should a student start taking math contests?
A: Ideally, as soon as the student wants to. While the student’s school may not
offer a contest, there are other venues that can be found with some
investigation.
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Thank You