Post on 13-Mar-2020
High School Math Contests
Blake Madill
University of Toronto Mississauga
b.madill@utoronto.ca
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
Blake Madill (UTM) Math Contests August 28, 2017 2 / 19
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
Blake Madill (UTM) Math Contests August 28, 2017 3 / 19
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
Blake Madill (UTM) Math Contests August 28, 2017 3 / 19
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
Blake Madill (UTM) Math Contests August 28, 2017 3 / 19
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
Blake Madill (UTM) Math Contests August 28, 2017 3 / 19
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
Blake Madill (UTM) Math Contests August 28, 2017 3 / 19
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
Blake Madill (UTM) Math Contests August 28, 2017 3 / 19
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.
Blake Madill (UTM) Math Contests August 28, 2017 4 / 19
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.
Blake Madill (UTM) Math Contests August 28, 2017 4 / 19
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.
Blake Madill (UTM) Math Contests August 28, 2017 4 / 19
I will not be speaking about:
contests not available to Ontario students
Canadian Mathematical Olympiad (CMO)
International Mathematical Olympiad (IMO)
the Putnam
Blake Madill (UTM) Math Contests August 28, 2017 5 / 19
I will not be speaking about:
contests not available to Ontario students
Canadian Mathematical Olympiad (CMO)
International Mathematical Olympiad (IMO)
the Putnam
Blake Madill (UTM) Math Contests August 28, 2017 5 / 19
I will not be speaking about:
contests not available to Ontario students
Canadian Mathematical Olympiad (CMO)
International Mathematical Olympiad (IMO)
the Putnam
Blake Madill (UTM) Math Contests August 28, 2017 5 / 19
I will not be speaking about:
contests not available to Ontario students
Canadian Mathematical Olympiad (CMO)
International Mathematical Olympiad (IMO)
the Putnam
Blake Madill (UTM) Math Contests August 28, 2017 5 / 19
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
Blake Madill (UTM) Math Contests August 28, 2017 6 / 19
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
Blake Madill (UTM) Math Contests August 28, 2017 6 / 19
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
Blake Madill (UTM) Math Contests August 28, 2017 6 / 19
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
Blake Madill (UTM) Math Contests August 28, 2017 6 / 19
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
Blake Madill (UTM) Math Contests August 28, 2017 6 / 19
Canadian Open Mathematics Challenge (COMC)
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.
Blake Madill (UTM) Math Contests August 28, 2017 7 / 19
Canadian Open Mathematics Challenge (COMC)
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.
Blake Madill (UTM) Math Contests August 28, 2017 7 / 19
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!
Blake Madill (UTM) Math Contests August 28, 2017 8 / 19
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!
Blake Madill (UTM) Math Contests August 28, 2017 8 / 19
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!
Blake Madill (UTM) Math Contests August 28, 2017 8 / 19
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!
Blake Madill (UTM) Math Contests August 28, 2017 8 / 19
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!
Blake Madill (UTM) Math Contests August 28, 2017 8 / 19
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!
Blake Madill (UTM) Math Contests August 28, 2017 8 / 19
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
Blake Madill (UTM) Math Contests August 28, 2017 9 / 19
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
Blake Madill (UTM) Math Contests August 28, 2017 9 / 19
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
Blake Madill (UTM) Math Contests August 28, 2017 9 / 19
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
Blake Madill (UTM) Math Contests August 28, 2017 9 / 19
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
Blake Madill (UTM) Math Contests August 28, 2017 9 / 19
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
Blake Madill (UTM) Math Contests August 28, 2017 9 / 19
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
Blake Madill (UTM) Math Contests August 28, 2017 9 / 19
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
Blake Madill (UTM) Math Contests August 28, 2017 9 / 19
Euclid Contest
usually written by grade 12 students
made by the Centre for Education in Mathematics and Computing(CEMC) at the University of Waterloo
contest date: April 11, 2018
registration deadline: March 22, 2018
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
Blake Madill (UTM) Math Contests August 28, 2017 10 / 19
Euclid Contest
usually written by grade 12 students
made by the Centre for Education in Mathematics and Computing(CEMC) at the University of Waterloo
contest date: April 11, 2018
registration deadline: March 22, 2018
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
Blake Madill (UTM) Math Contests August 28, 2017 10 / 19
Euclid Contest
usually written by grade 12 students
made by the Centre for Education in Mathematics and Computing(CEMC) at the University of Waterloo
contest date: April 11, 2018
registration deadline: March 22, 2018
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
Blake Madill (UTM) Math Contests August 28, 2017 10 / 19
Euclid Contest
usually written by grade 12 students
made by the Centre for Education in Mathematics and Computing(CEMC) at the University of Waterloo
contest date: April 11, 2018
registration deadline: March 22, 2018
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
Blake Madill (UTM) Math Contests August 28, 2017 10 / 19
Euclid Contest
usually written by grade 12 students
made by the Centre for Education in Mathematics and Computing(CEMC) at the University of Waterloo
contest date: April 11, 2018
registration deadline: March 22, 2018
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
Blake Madill (UTM) Math Contests August 28, 2017 10 / 19
Euclid Contest
usually written by grade 12 students
made by the Centre for Education in Mathematics and Computing(CEMC) at the University of Waterloo
contest date: April 11, 2018
registration deadline: March 22, 2018
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
Blake Madill (UTM) Math Contests August 28, 2017 10 / 19
Euclid Contest
usually written by grade 12 students
made by the Centre for Education in Mathematics and Computing(CEMC) at the University of Waterloo
contest date: April 11, 2018
registration deadline: March 22, 2018
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
Blake Madill (UTM) Math Contests August 28, 2017 10 / 19
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
Blake Madill (UTM) Math Contests August 28, 2017 11 / 19
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.
Blake Madill (UTM) Math Contests August 28, 2017 12 / 19
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.
Blake Madill (UTM) Math Contests August 28, 2017 12 / 19
Algebra
Example (Easier, COMC 2016)
Solve the following system of equations:
1
x+
1
y= 1,
2
x+
3
y= 4.
Example (Easier, Euclid 2016)
If x2 = 2016 then what is the value of (x + 2)(x − 2)?
Blake Madill (UTM) Math Contests August 28, 2017 13 / 19
Algebra
Example (Easier, COMC 2016)
Solve the following system of equations:
1
x+
1
y= 1,
2
x+
3
y= 4.
Example (Easier, Euclid 2016)
If x2 = 2016 then what is the value of (x + 2)(x − 2)?
Blake Madill (UTM) Math Contests August 28, 2017 13 / 19
Geometry
Example (Easier, COMC 2015)
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?
Example (Harder, COMC 2016)
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.
Blake Madill (UTM) Math Contests August 28, 2017 14 / 19
Geometry
Example (Easier, COMC 2015)
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?
Example (Harder, COMC 2016)
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.
Blake Madill (UTM) Math Contests August 28, 2017 14 / 19
Other
Example (Easier, Euclid 2017)
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?
Blake Madill (UTM) Math Contests August 28, 2017 15 / 19
Other
Example (Easier, Euclid 2017)
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?
Blake Madill (UTM) Math Contests August 28, 2017 15 / 19
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)
Blake Madill (UTM) Math Contests August 28, 2017 16 / 19
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)
Blake Madill (UTM) Math Contests August 28, 2017 16 / 19
A Sample Year
1 COMC - Nov. 2
2 CIMC/CSMC - Nov. 22
3 Fryer, Galois, Hypatia - April 12
4 Euclid - April 11
Blake Madill (UTM) Math Contests August 28, 2017 17 / 19
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: marina.tvalavadze@utoronto.ca
Blake Madill (UTM) Math Contests August 28, 2017 18 / 19
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: marina.tvalavadze@utoronto.ca
Blake Madill (UTM) Math Contests August 28, 2017 18 / 19
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: marina.tvalavadze@utoronto.ca
Blake Madill (UTM) Math Contests August 28, 2017 18 / 19
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: marina.tvalavadze@utoronto.ca
Blake Madill (UTM) Math Contests August 28, 2017 18 / 19
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: marina.tvalavadze@utoronto.ca
Blake Madill (UTM) Math Contests August 28, 2017 18 / 19
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: marina.tvalavadze@utoronto.ca
Blake Madill (UTM) Math Contests August 28, 2017 18 / 19
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: marina.tvalavadze@utoronto.ca
Blake Madill (UTM) Math Contests August 28, 2017 18 / 19