a guide to math competitions - storming robots · pdf fileversion -eiπ page 5 amcs to amo...

Simon Lee

2016-08-31

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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


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



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


Qualifies For ~120 / 150 typically qualifies for AIME

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AMC 12


Eligibility All students before college

Dates February 7, 2017 (A) and February 15, 2017 (B)




Scoring Out of 150; 6 points per correct, 1.5 point penalty per incorrect

Participants ~ 45,000 (A), ~25,000 (B) in US



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)


Format 3 hour, 15 questions, three digit answers


Scoring Out of 15; 1 point per correct, no penalty per incorrect

Participants ~ 4,000 (I), ~1,000 (II) in US



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


Scoring Out of 42; 7 points per question

Participants ~ 270

95th 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


Scoring Out of 42, 7 points per question

Participants ~ 230



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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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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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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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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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

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