Canadian Open Mathematics Challenge (COMC) — Questions and Answers

Question 1: If f(x) = x² + 1 and a sequence is defined by a₁ = 0, aₙ₊₁ = f(aₙ), what is a₃?

1
5
4
2 (Correct answer)

Correct answer: 2

a₂ = f(0) = 0²+1 = 1; a₃ = f(1) = 1²+1 = 2.

Question 2: What quality assurance measure supports applied methods and techniques?

Annual review is sufficient
Regular self-assessment, peer review, and adherence to established standards (Correct answer)
Quality checks are unnecessary for experienced professionals
Quality only matters for new practitioners

Correct answer: Regular self-assessment, peer review, and adherence to established standards

Ongoing quality assurance through self-assessment, peer review, and standards adherence ensures continuous improvement.

Question 3: What quality assurance measure supports communication and documentation?

Regular self-assessment, peer review, and adherence to established standards (Correct answer)
Quality checks are unnecessary for experienced professionals
Quality only matters for new practitioners
Annual review is sufficient

Correct answer: Regular self-assessment, peer review, and adherence to established standards

Ongoing quality assurance through self-assessment, peer review, and standards adherence ensures continuous improvement.

Question 4: A sequence satisfies a₁ = 2 and aₙ₊₁ = aₙ / (1 + aₙ). What is a₄?

2/7 (Correct answer)
2/9
1/4
1/3

Correct answer: 2/7

a₂ = 2/3, a₃ = (2/3)/(5/3) = 2/5, a₄ = (2/5)/(7/5) = 2/7.

Question 5: What is the foundational principle of core concepts and principles in the Canadian Open Mathematics Challenge field?

Avoiding all challenging situations
Maintaining competence, integrity, and service to stakeholders (Correct answer)
Following the easiest path available
Maximizing personal advancement

Correct answer: Maintaining competence, integrity, and service to stakeholders

The foundational principles of core concepts and principles in Canadian Open Mathematics Challenge center on maintaining competence, integrity, and quality service.

Question 6: What quality assurance measure supports core concepts and principles?

Quality checks are unnecessary for experienced professionals
Annual review is sufficient
Regular self-assessment, peer review, and adherence to established standards (Correct answer)
Quality only matters for new practitioners

Correct answer: Regular self-assessment, peer review, and adherence to established standards

Ongoing quality assurance through self-assessment, peer review, and standards adherence ensures continuous improvement.

Question 7: In how many ways can 8 people be split into two unlabeled groups of 4?

28
56
35 (Correct answer)
70

Correct answer: 35

Choose 4 people in C(8,4) = 70 ways, then divide by 2 because swapping the two groups gives the same split: 70/2 = 35.

Question 8: How should challenges in communication and documentation be addressed?

Delegate all challenges to supervisors
Ignore challenges until they resolve themselves
Apply systematic problem-solving, seek expert guidance when needed, and document decisions (Correct answer)
Avoid challenges and stick to familiar tasks

Correct answer: Apply systematic problem-solving, seek expert guidance when needed, and document decisions

Systematic problem-solving combined with appropriate consultation and documentation ensures challenges are addressed effectively.

Question 9: What is the foundational principle of professional standards and ethics in the Canadian Open Mathematics Challenge field?

Avoiding all challenging situations
Maximizing personal advancement
Following the easiest path available
Maintaining competence, integrity, and service to stakeholders (Correct answer)

Correct answer: Maintaining competence, integrity, and service to stakeholders

The foundational principles of professional standards and ethics in Canadian Open Mathematics Challenge center on maintaining competence, integrity, and quality service.

Question 10: Two angles of a triangle measure 45° and 75°. What is the measure of the third angle?

80°
70°
50°
60° (Correct answer)

Correct answer: 60°

The interior angles of a triangle sum to 180°, so the third angle is 180° − 45° − 75° = 60°.

Question 11: A circle is inscribed in a square with side length 4. What is the area of the circle?

8π
16π
2π
4π (Correct answer)

Correct answer: 4π

The inscribed circle's diameter equals the side of the square, so r = 2 and area = π(2²) = 4π.

Question 12: The Fibonacci sequence is F₁=1, F₂=1, Fₙ=Fₙ₋₁+Fₙ₋₂ for n ≥ 3. What is F₁₀?

34
64
55 (Correct answer)
44

Correct answer: 55

The sequence begins 1,1,2,3,5,8,13,21,34,55, so F₁₀ = 55.

Question 13: In a right triangle, one leg is 7 and the hypotenuse is 25. What is the length of the other leg?

16
24 (Correct answer)
18
20

Correct answer: 24

By the Pythagorean theorem, the other leg is √(25² − 7²) = √(625 − 49) = √576 = 24.

Question 14: How should applied methods and techniques knowledge be maintained and updated?

Through continuous professional development, current literature review, and professional networking (Correct answer)
Initial training provides lifelong competence
Knowledge updates are only needed every five years
Learning stops after certification

Correct answer: Through continuous professional development, current literature review, and professional networking

Professional competence requires ongoing development through education, literature review, and engagement with the professional community.

Question 15: In a geometric sequence with first term 2 and common ratio 3, what is the 5th term?

54
18
486
162 (Correct answer)

Correct answer: 162

Using aₙ = a₁ × r^(n−1): a₅ = 2 × 3⁴ = 2 × 81 = 162.

Question 16: How many ways can a committee of 3 people be chosen from a group of 12?

132
360
66
220 (Correct answer)

Correct answer: 220

C(12,3) = 12!/(3!×9!) = (12×11×10)/(3×2×1) = 220.

Question 17: What award is given to the highest-scoring student in the COMC?

A medal only
National Champion title with a certificate and prize (Correct answer)
Certificate of participation
A scholarship to any university

Correct answer: National Champion title with a certificate and prize

The COMC aims to recognize and celebrate the highest level of mathematical achievement in Canada. The top-scoring student is awarded the prestigious National Champion title, accompanied by a certificate and a prize, signifying their outstanding performance and encouraging future mathematical pursuits.

Question 18: How should core concepts and principles knowledge be maintained and updated?

Knowledge updates are only needed every five years
Learning stops after certification
Initial training provides lifelong competence
Through continuous professional development, current literature review, and professional networking (Correct answer)

Correct answer: Through continuous professional development, current literature review, and professional networking

Professional competence requires ongoing development through education, literature review, and engagement with the professional community.

Question 19: The 3rd term of a geometric sequence is 12 and the 6th term is 96. What is the common ratio?

3
2.5
1.5
2 (Correct answer)

Correct answer: 2

t₆ / t₃ = r³ = 96/12 = 8, so r = 2.

Question 20: How should challenges in safety and compliance be addressed?

Apply systematic problem-solving, seek expert guidance when needed, and document decisions (Correct answer)
Delegate all challenges to supervisors
Avoid challenges and stick to familiar tasks
Ignore challenges until they resolve themselves

Correct answer: Apply systematic problem-solving, seek expert guidance when needed, and document decisions

Systematic problem-solving combined with appropriate consultation and documentation ensures challenges are addressed effectively.

Question 21: How should professionals apply applied methods and techniques in daily practice?

Apply principles selectively based on convenience
Consistently integrate best practices into every aspect of professional work (Correct answer)
Follow standards only for complex tasks
Only when being evaluated

Correct answer: Consistently integrate best practices into every aspect of professional work

Consistent application of professional standards ensures quality outcomes and builds professional credibility.

Question 22: What is the area of a regular hexagon with side length 2?

12
8√3
6√3 (Correct answer)
4√3

Correct answer: 6√3

A regular hexagon with side s has area (3√3/2)s²; with s = 2 the area is (3√3/2)(4) = 6√3.

Question 23: How are points awarded for multiple-choice questions in the COMC?

Negative points for incorrect answers
Partial credit for close answers
Points deducted for unanswered questions
Full points for correct answers only (Correct answer)

Correct answer: Full points for correct answers only

Multiple-choice questions in the COMC are typically scored dichotomously, meaning an answer is either entirely correct or entirely incorrect. This straightforward scoring method ensures objectivity and rewards precise knowledge, without partial credit for nearly correct selections.

Question 24: If P(x) = x³ − 6x² + 11x − 6, what is P(2)?

−2
0 (Correct answer)
2
4

Correct answer: 0

P(2) = 8 − 24 + 22 − 6 = 0, confirming that x = 2 is a root of the polynomial.

Question 25: If a/b = 3/4 and a + b = 28, what is the value of a − b?

−4 (Correct answer)
7
4
−7

Correct answer: −4

Setting a = 3k and b = 4k gives 7k = 28, so k = 4, a = 12, b = 16, and a − b = −4.

Question 26: How should industry best practices knowledge be maintained and updated?

Initial training provides lifelong competence
Through continuous professional development, current literature review, and professional networking (Correct answer)
Learning stops after certification
Knowledge updates are only needed every five years

Correct answer: Through continuous professional development, current literature review, and professional networking

Professional competence requires ongoing development through education, literature review, and engagement with the professional community.

Question 27: How many paths are there from one corner to the opposite corner of a 3×3 grid, moving only right or up?

15
20 (Correct answer)
12
24

Correct answer: 20

Each path uses exactly 3 right and 3 up steps; the number of orderings is C(6,3) = 20.

Question 28: Two similar triangles have corresponding sides in the ratio 3:5. What is the ratio of their areas?

3:5
27:125
6:10
9:25 (Correct answer)

Correct answer: 9:25

Areas of similar figures scale as the square of the linear ratio: (3/5)² = 9/25.

Question 29: What is the total surface area of a cube with edge length 3?

36
72
27
54 (Correct answer)

Correct answer: 54

A cube has 6 square faces each of area 3² = 9, so total surface area = 6 × 9 = 54.

Question 30: How should professionals apply assessment and evaluation in daily practice?

Follow standards only for complex tasks
Only when being evaluated
Consistently integrate best practices into every aspect of professional work (Correct answer)
Apply principles selectively based on convenience

Correct answer: Consistently integrate best practices into every aspect of professional work

Consistent application of professional standards ensures quality outcomes and builds professional credibility.

Canadian Open Mathematics Challenge (COMC)

The Canadian Open Mathematics Challenge (COMC) is an annual national mathematics competition for Canadian students in grades 7–12, organized by the Canadian Mathematical Society. It tests problem-solving skills in algebra, geometry, number theory, and combinatorics through three progressively challenging parts.

Questions

75 min

Time Limit

70%

To Pass

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