High School Math...

High School Math Contests

Blake Madill

University of Toronto Mississauga

[email protected]

August 28, 2017

Blake Madill (UTM) Math Contests August 28, 2017 1 / 19

Outline

1 Benefits of ParticipationFor the StudentsFor the Teachers

2 Available Contests

3 Preparation

4 Other Opportunities

Benefits for the Students

High school math contests...

provide a fun, low-pressure atmosphere for students to practice theirmathematics

expose students to new mathematics not regularly covered in the highschool curriculum

better prepare students for university proof-based courses (eg.MAT102 at UTM)

offer students scholarship opportunities

(in special cases) increase competitiveness in a student’s universityapplication

realize mathematics as both an individual and team sport

Benefits for the Teachers

Besides the obvious benefits for your students...

You get to solve the problems too! Many of the contests containhighly non-trivial problems.

I strongly recommend that you volunteer to mark some of thecontests. (If interested, please feel free to speak with me afterwards)

You may also wish to apply to be a part of the question committees.

I will not be speaking about:

contests not available to Ontario students

Canadian Mathematical Olympiad (CMO)

International Mathematical Olympiad (IMO)

the Putnam

Canadian Open Mathematics Challenge (COMC)

“The COMC is Canada’s premier national mathematics competitionthat is open to any student with an interest in and grasp of highschool mathematics.”

The top 50 students from the COMC will be invited to write theCMO.

contest date: Thursday, November 2, 2017

registration begins: September 1, 2017

French or English

2.5 hour on-paper contest

French or English

Eligibility

A participating student must be under 19 years old as of June 30,2017.

They must be attending school (elementary or secondary) full-timesince September 15, 2017.

They must not have written the Putnam.

Prizes etc.

A prize pool of $2000 will be divided among the top students inCanada.

A group of students in grade 8, 9, or 10 will be invited to the CMSNational Canada Math Camp at the University of Toronto.

Competition results are shared with participating universities.

Eligibility

A participating student must be under 19 years old as of June 30,2017.

They must be attending school (elementary or secondary) full-timesince September 15, 2017.

They must not have written the Putnam.

Prizes etc.

A prize pool of $2000 will be divided among the top students inCanada.

A group of students in grade 8, 9, or 10 will be invited to the CMSNational Canada Math Camp at the University of Toronto.

Competition results are shared with participating universities.

Canadian Senior/ Intermediate Mathematics Contests

“The CSMC and CIMC are two contests designed to give students theopportunity to have fun and develop their mathematical problemsolving ability.”

made by the Centre for Education in Mathematics and Computing(CEMC) at the University of Waterloo

contest date: November 22, 2017

registration deadline: November 3, 2017

French or English

2 hour on-paper contests

The CSMC is a great warm-up for Grade 12 students planning towrite the Euclid!

French or English

Fryer, Galois, and Hypatia Contests

Fryer - ≤ Grade 9

Galois - ≤ Grade 10

Hypatia - ≤ Grade 11

made by the Centre for Education in Mathematics and Computing(CEMC) at the University of Waterloo.

contest date: April 12, 2018

registration deadline: March 23, 2018

French or English

75 minute on-paper contests

held the day after the Euclid

Fryer - ≤ Grade 9

French or English

Fryer - ≤ Grade 9

French or English

Fryer - ≤ Grade 9

French or English

Fryer - ≤ Grade 9

French or English

Fryer - ≤ Grade 9

French or English

Fryer - ≤ Grade 9

French or English

Fryer - ≤ Grade 9

French or English

Euclid Contest

usually written by grade 12 students

French or English

2.5 hour on-paper contests

the top five contestants receive a plaque and a prize of $500

the next top ten students receive a prize of $200

the Euclid score is considered for admission to the University ofWaterloo

Euclid Contest

French or English

Euclid Contest

French or English

Euclid Contest

French or English

Euclid Contest

French or English

Euclid Contest

French or English

Euclid Contest

French or English

Preparation

When preparing for a Grade 12 contest (Euclid or COMC), the followingtopics should be studied and practiced.

1 counting and patterns

2 basic algebra

3 geometry

4 trigonometry

5 functions and graphs

6 logarithms and exponents

7 polynomials

8 divisibility and gcd’s

9 sequences

Counting and Patterns

Example (Easier, COMC 2015)

What is the smallest palindrome larger than 2015?

Example (Harder, COMC 2015)

A polynomial x2 + px + q, where p and q are real numbers, is said to be adouble-up polynomial if it has two real roots, one of which is twice theother one. Determine all double-up polynomials such that p + q = 6.

Counting and Patterns

What is the smallest palindrome larger than 2015?

A polynomial x2 + px + q, where p and q are real numbers, is said to be adouble-up polynomial if it has two real roots, one of which is twice theother one. Determine all double-up polynomials such that p + q = 6.

Algebra

Solve the following system of equations:

Example (Easier, Euclid 2016)

If x2 = 2016 then what is the value of (x + 2)(x − 2)?

Algebra

Solve the following system of equations:

If x2 = 2016 then what is the value of (x + 2)(x − 2)?

Geometry

A rectangle has an area of 180 units2 and a perimeter of 54 units. If eachside of the rectangle is increased by 6 units, what is the area of the newrectangle?

Let A = (0, a),C = (c , 0), and B = (c , b), where a, b, and c are positiveintegers. Suppose P = (p, 0) is the point on OC that minimizes AP + PB.Set X = AP + PB. If a, b, p, and X are positive integers, prove that thereexists a positive integer n ≥ 3 such that n divides both a and b.

Geometry

A rectangle has an area of 180 units2 and a perimeter of 54 units. If eachside of the rectangle is increased by 6 units, what is the area of the newrectangle?

Let A = (0, a),C = (c , 0), and B = (c , b), where a, b, and c are positiveintegers. Suppose P = (p, 0) is the point on OC that minimizes AP + PB.Set X = AP + PB. If a, b, p, and X are positive integers, prove that thereexists a positive integer n ≥ 3 such that n divides both a and b.

If cos θ = tan θ then what are the possible exact values for sin θ?

Example (Very Hard, Euclid 2014)

Let f (n) denote the number of positive integers who have exactly n digitsand whose digits sum to 4. How many of f (1), f (x), . . . , f (2014) have aunits digit of 1?

If cos θ = tan θ then what are the possible exact values for sin θ?

Example (Very Hard, Euclid 2014)

Let f (n) denote the number of positive integers who have exactly n digitsand whose digits sum to 4. How many of f (1), f (x), . . . , f (2014) have aunits digit of 1?

Others

Example (Very Easy, COMC 2012)

Which positive integer satisfies 84 = 4n?

Example (Harder, Euclid 2013)

Solve for x :log2(2x−1 + 3x+1) = 2x − log2(3x)

Others

Example (Very Easy, COMC 2012)

Which positive integer satisfies 84 = 4n?

Example (Harder, Euclid 2013)

Solve for x :log2(2x−1 + 3x+1) = 2x − log2(3x)

A Sample Year

1 COMC - Nov. 2

2 CIMC/CSMC - Nov. 22

3 Fryer, Galois, Hypatia - April 12

4 Euclid - April 11

Math Circles

a high school enrichment program here at UTM

a fun way to practice problem solving skills and prepare for contests

grades 10 to 12

math battles

“Fields trips”

registration is limited to 20 students

contact Marina Tvalavadze for details: [email protected]

Math Circles

grades 10 to 12

math battles

“Fields trips”

Math Circles

grades 10 to 12

math battles

“Fields trips”

Math Circles

grades 10 to 12

math battles

“Fields trips”

Math Circles

grades 10 to 12

math battles

“Fields trips”

Math Circles

grades 10 to 12

math battles

“Fields trips”

Math Circles

grades 10 to 12

math battles

“Fields trips”

Thank You!

High School Math...

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