The CENTRE for EDUCATION in MATHEMATICS and COMPUTING Le CENTRE d’ÉDUCATION en MATHÉMATIQUES et en INFORMATIQUE www.cemc.uwaterloo.ca 2019 2019 Results Résultats Canadian Senior and Intermediate Mathematics Contests Concours canadiens de mathématiques de niveau supérieur et intermédiaire c 2019 University of Waterloo Competition Organization Organisation du Concours Centre for Education in Mathematics and Computing Faculty and Staff / Personnel du Concours canadien de mathématiques Ed Anderson Conrad Hewitt Jeff Anderson Valentina Hideg Terry Bae Angie Hildebrand Jacquelene Bailey Carrie Knoll Shane Bauman Christine Ko Jenn Brewster Judith Koeller Ersal Cahit Laura Kreuzer Diana Castañeda Santos Bev Marshman Sarah Chan Paul McGrath Ashely Congi Jen Nelson Serge D’Alessio Ian Payne Fiona Dunbar J.P. Pretti Mike Eden Alexandra Rideout Sandy Emms Nick Rollick Barry Ferguson Kim Schnarr Steve Furino Carolyn Sedore John Galbraith Ashley Sorensen Lucie Galinon Ian VanderBurgh Robert Garbary Troy Vasiga Rob Gleeson Heather Vo Sandy Graham Bonnie Yi Problems Committees / Comités des problèmes Canadian Senior Mathematics Contest / Concours canadien de niveau supérieur Stephen New (Chair / président), University of Waterloo, Waterloo, ON Diana Castañeda Santos, University of Waterloo, Waterloo, ON Lee Fryer-Davis, Cameron Heights C.I., Kitchener, ON Kee Ip, Crescent School, Toronto, ON Denes Jakob, Sir Allan MacNab S.S., Hamilton, ON Paul Leistra, Guido de Bres Christian H.S., Hamilton, ON Ian Payne, University of Waterloo, Waterloo, ON Jocelyn Procopio, Citadel H.S., Halifax, ON Joe West, University of Waterloo, Waterloo, ON Canadian Intermediate Mathematics Contest / Concours canadien de niveau intermédiaire Carrie Knoll (Chair / présidente), University of Waterloo, Waterloo, ON Dean Murray (Chair / président), University of Waterloo, Waterloo, ON Ed Barbeau, Toronto, ON Robert Garbary, University of Waterloo, Waterloo, ON Adam Gregson, University of Toronto Schools, Toronto, ON Brian McBain, North Lambton S.S., Forest, ON Ginger Moorey, Abbey Park H.S., Oakville, ON Karen Sullivan, Blenheim D.H.S., Blenheim, ON Fredric Suresh, Mayfield S.S., Brampton, ON Marcel te Bokkel, York Region District School Board, Newmarket, ON 2 Foreword Avant-Propos The Centre for Education in Mathematics and Computing is pleased to announce the results of the 2019 Canadian Senior and Intermediate Mathematics Contests. Our congratulations go to all who participated in this year’s CSMC and CIMC. This year’s Contests were re- sounding successes, with averages of 31.3 and 29.1, respectively. As always, we would like to thank the hard-working Problems Committees. Many of the members of these Com- mittees are active secondary school teachers who volunteer their time and contribute dozens of hours of expertise. Without their intriguing and sometimes amusing problems, these Contests would not be possible. We would also like to thank all participants, both teachers and students. We hope that the papers provided you with some interesting mathematics to think about and play with. Thank you for your support! Please con- tinue to encourage your colleagues and fellow students to become involved in our activities. Le Centre d’éducation en mathématiques et en informatique à d’annoncer les résultats du Conours canadiens de mathématiques de niveau supérieur et intermédiaire 2019. Nos félicitations vont à tous les participants du CCMS et du CCMI de cette année. Les Concours de cette année retentissaient de succès, avec des moyennes de 31,3 et 29,1, respectivement. Nous aimerions surtout remercier les Comités de problèmes pour leur dur travail. Plusieurs membres de ces Comités sont des enseignant(e)s d’école secondaire actifs qui offre leur temps et contribuent des douzaines d’heures d’expertise. Sans leurs problèmes perspicaces et amusants, ces Concours ne seraient pas possibles. Nous aimerions remercier aussi tous les participants incluant les enseignants et les étudiants. Nous espérons que les concours vous ont offert des mathématiques intéressantes qui vous ont amusées et portées à réfléchir. Merci pour votre soutien continué! 3 Comments on the Contests CANADIAN SENIOR MATHEMATICS CONTEST Part A 1. Very well done. When errors were made, they were usually subtraction errors or reporting the incorrect person’s age. Average: 4.9 2. Well done. There were many errors in the formula for the circumference of a circle that was used by contestants, and thus many variations in the final answers given. The other major source of error was the unnecessary substitution of an approximation for π. If a decimal approximation is needed for further computation, the substitution of an approximation for π should be made at the latest possible step in the calculation. Since the length of the side of the triangle was asked for, there was no need to approximate the answer. Average: 4.0 3. Most students were able to correctly determine the product of powers. Determining the sum of powers proved more difficult. A common mistake was to say that 2403 + 2403 = 4403. Another common mistake was to drop the bases of 2 before simplifying the sum, resulting in an incorrect answer of 242. Average: 3.6 4. Most students understood how to approach this question and were able to find values of x that satisfy x2 ≤ x + 6. Most mistakes were related to miscounting the number of possible values of y that correspond with each value of x. Some students graphed y = x2 and y = x + 6 and then counted the number of “lattice points” in the appropriate region. Average: 2.7 5. Of the students who solved this problem, many recognized that 605 = 11 × 55 and scaled the relatively unknown Pythagorean Triple (48; 55; 73) to get 48 × 11 = 528. (Note that this approach does not show that 528 is the largest possibility less than 605, but since the correct answer of 528 was in the box, this still would have gotten 5 marks.) A common incorrect answer was 604; in solutions with this answer, there often was not much written down and we suspect students simply misinterpreted the question. Average: 1.0 6. There were various common mistakes we saw in Question 6. Some students switched the roles of k and 4 − k 4 and ended up with k = 3 , this usually earned 4 of the available 5 marks. If the correct answer was not in the box, we required justification for the formula for the area of 4ARS. The question does not state that k must be an integer, but some students assumed that it must be and simply tested possible integer values for k. Average: 0.7 Part B 1. This question was very well done. In part (b), a few students found values for t and r where t was not in the required range. In part (c), a few students missed that a, b and c were positive integers, and concluded that the minimum possible value of c was 0. Average: 8.8 2. Part (a) was generally well done. When asked for the value of w, that is what should be given rather than a 1 1 value of . In part (b), many contestants tested several values and made the conclusion that x = , which w 2 was the correct value of x. However, having explored and found what appears to be the value of x, it must 1 now be shown using general mathematical methods that the value of x is, in fact, always . 2 Solutions for part (c) required the solution of a cubic equation and some indication of why two of the three roots were inadmissible. Frequent errors were dividing out a linear factor (such as (r − 1)) without giving 4 Comments on the Contests reasons why it was allowable to do this. In either approach, the condition of the question forced r 6= 1, and thus r − 1 6= 0, and the division was allowed. These things were rarely explained leading to less than fully correct solutions. Average: 4.4 3. Part (a) was fairly well done and we saw many different solutions. Most students exploited known facts about 30◦-60◦-90◦ triangles to find concise solutions. There were also many ways to solve part (b). Most methods started with a simple construction followed by a few calculations to arrive at an equation equivalent to r2 = 4r. Very few students attempted part (c). Of those who got started, most got lost in the algebra. Average: 1.2 5 Comments on the Contests CANADIAN INTERMEDIATE MATHEMATICS CONTEST Part A 1. Well done. Some students who had the incorrect answer could have received some part marks, but did not since all of their calculations were just numbers that were not attached to any part of the labelled diagram. Average: 4.5 2. Well done. Some students calculated how many quarters were equal to 20 dimes rather than how many additional quarters were required to match Binh’s total. Average: 4.4 3. Students struggled with this problem. If they were unable to determine the prime factorization, it was difficult for them to proceed. Common errors included mismatching the exponents on 2, 3 and 5, or miscalculating 3a+4b+6c with those exponents. Some students attempted to factor 36 000 into powers of 3, 4 and 6. Other students multiplied 2 × 3 × 5 = 30 and tried to write 36 000 as a power of 30. Average: 3.1 4. Well done. Many students either got a subset or a superset of f14; 15; 16g. Some students had a table showing scenarios, but that was not labelled well enough to earn significant part marks. Average: 3.8 5. Many students correctly tried to use case work to solve this problem and recognized that the values of a and b had to be between −10 and 10. The problem many students faced was that their cases double-counted some solutions or weren’t exhaustive.