2011 Results 2011 Résultats Canadian Senior and Intermediate
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The CENTRE for EDUCATION in MATHEMATICS and COMPUTING Le CENTRE d'EDUCATION´ en MATHEMATIQUES´ et en INFORMATIQUE www.cemc.uwaterloo.ca 2011 2011 Results R´esultats Canadian Senior and Intermediate Mathematics Contests Concours canadiens de math´ematiques de niveau sup´erieuret interm´ediaire c 2012 University of Waterloo Competition Organization Organisation du Concours Centre for Education in Mathematics and Computing Faculty and Staff / Personnel du Concours canadien de math´ematiques Ed Anderson Terry Bae Steve Brown Ersal Cahit Karen Cole Serge D'Alessio Frank DeMaio Jennifer Doucet Fiona Dunbar Mike Eden Barry Ferguson Barb Forrest Judy Fox Steve Furino John Galbraith Sandy Graham Angie Hildebrand Judith Koeller Joanne Kursikowski Bev Marshman Dean Murray Jen Nissen J.P. Pretti Linda Schmidt Kim Schnarr Jim Schurter Carolyn Sedore Ian VanderBurgh Troy Vasiga Problems Committees / Comit´esdes probl`emes Canadian Senior Mathematics Contest / Concours canadien de niveau sup´erieur Mike Eden (Chair / pr´esident), University of Waterloo, Waterloo, ON Kee Ip, Crescent School, Toronto, ON Paul Leistra, Guido de Bres Christian H.S., Hamilton, ON Daryl Tingley, University of New Brunswick, Fredericton, NB Joe West, University of Waterloo, Waterloo, ON Bruce White, Windsor, ON Canadian Intermediate Mathematics Contest / Concours canadien de niveau interm´ediaire John Galbraith (Chair / pr´esident), University of Waterloo, Waterloo, ON Ed Barbeau, Toronto, ON Alison Cornthwaite, Lo-Ellen Park S.S., Sudbury, ON Brian McBain, North Lambton S.S., Forest, ON Ginger Moorey, Abbey Park H.S., Oakville, ON Dean Murray, University of Waterloo, Waterloo, ON 2 Foreword Avant-Propos The Centre for Education in Mathematics and Computing is pleased to announce the results of the 2011 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 32.4 and 32.7, respectively. As always, we would like to thank the hard-working Problems Committees. Many of the members of these Committees are active secondary school teachers who volunteer their time and contribute dozens of hours of ex- pertise. 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'´educationen math´ematiqueset en informatique `ad'annoncer les r´esultatsdu Conours canadiens de math´ematiquesde niveau sup´erieuret interm´ediaire2011. Nos f´elicitationsvont `atous les participants du CCMS et du CCMI de cette ann´ee. Les Concours de cette ann´eeretentissaient de succ`es,avec des moyennes de 32,4 et 32,7, respectivement. Nous aimerions surtout remercier les Comit´esde probl`emespour leur dur travail. Plusieurs membres de ces Comit´essont des enseignant(e)s d'´ecolesecondaire actifs qui offre leur temps et contribuent des douzaines d'heures d'expertise. Sans leurs probl`emesperspicaces et amusants, ces Concours ne seraient pas possibles. Nous aimerions remercier aussi tous les participants incluant les enseignants et les ´etudiants. Nous esp´erons que les concours vous ont offert des math´ematiquesint´eressantes qui vous ont amus´eeset port´ees`ar´efl´echir. Merci pour votre soutien continu´e! 3 Comments on the Contests CANADIAN SENIOR MATHEMATICS CONTEST Part A 31 1. Very well done. A common mistake was giving 16 as the final answer and forgetting to multiply by 16. Average: 4.7 2. Well done. Quite a few students had trouble formulating the correct equations from the given information. Also, some students needed to take more care when defining their variables to specify whether the ages were now, in the past, or in the future. Average: 3.5 3. Well done. Common errors included assuming that each of the possible sums 2 through 12 is equally likely, miscounting the number of results giving a perfect square, and drawing only the top or bottom half of the table to obtain a denominator of 21. Average: 3.5 4. Many students did well on this problem. The most common error was trying to list the divisors by trial and error, and then missing one or more of them. Average: 2.7 5. Generally very well done. Many students got the answer of 5 without sound mathematical reasoning, perhaps by having seen the numbers 3 and 4 and a right-angled triangle. A significant portion of students failed to divide the chord lengths in half when working with half of the chords. Average: 3.0 6. This was a difficult problem in which it was very easy to go in circles. Many students wrote one equation comparing two sums from rows/columns/diagonals but nothing more. Average: 1.0 Part B 1. This question was very well done. In general, it would be helpful to see slightly more in the way of explication or justification of students' work. Average: 9.2 2. Part (a) was attempted by most students. While most students gave a correct answer, some students gave non-integer values for x and y, while others forgot to give a particular solution even after obtaining x = 7y. While parts (b) and (c) were difficult, a good number of students presented complete solutions to these. In parts (b) and (c), many students worked with specific numbers rather than working in a more general setting. An alternate correct solution that appeared a number of times for (b) started with the inequality ad < bc, multiplied both sides by y to obtain ady < bcy, added abx to both sides to obtain abx + ady < abx + bcy, a ax + cy factored to obtain a(bx + dy) < b(ax + cy), and then divided to obtain < , which is part of the b bx + dy desired inequality . Average: 3.3 3. Part (a) was generally well done { most students wrote out the cases and counted correctly. Part (b) was attempted by very few students, though a small percentage noticed without much work that if p = m, the result is true. Average: 1.5 4 Comments on the Contests CANADIAN INTERMEDIATE MATHEMATICS CONTEST Part A 1. Very well done. Average: 4.6 2. Very well done. Quite a few students incorrectly used \guess and check". Also, a number of students subtracted the two given equations to obtain 0 = 4 − c instead of 0 = 4 − 2c. Average: 4.5 3. This question was fairly well done. The most common mistake was the answer 6, from giving the number of weeks at $100 per week, rather than the total number of weeks. The majority of students who solved this problem used an algebraic solution. Average: 4.1 4. Many students did well on this problem. Many students were able to determine that there were 36 equally likely outcomes, but some would then get stuck on either the definition of a perfect square or had too many or too few possibilities. Common mistakes included double counting (2; 2) or not counting rolls like (1; 3) and (3; 1) as separate possibilities. Average: 2.9 5. In this problem, students either obtained the correct answer or made very little progress, with not much in between. Average: 2.0 6. This was a very difficult problem. Most students gave the prime factorization of 616 but were unable to proceed further. Average: 0.3 Part B 1. This question was well done. A number of students used the expression 2πr for the area of a circle with radius r instead of πr2. (Since r = 2 in this problem, the answer turns out to be the same.) Most students understood the concept of subtraction of areas, even if they were unable to obtain the correct answer. Average: 7.8 2. Parts (a) and (b) were well done and most students who attempted part (c) did quite well. Most students solved part (a) by drawing out Figure 4 and adding up the side lengths. Most students solved part (b) by calculating the Ink Lengths of Figures 8 and 9 and then subtracting those two numbers. We required that students provide some justification/explanation of their approach for parts (b) and (c). Average: 5.1 3. This was a difficult problem to solve, but even more so to write a good solution. Many students used diagrams without any numerical work to attempt the problem. Some students did not read the question carefully to get the meaning of the word cross for the two swimmers and thought that a cross occurred when one of the swimmers reached one of the ends of the pool. Often students gave an answer of one more than the correct answer because they misread the question. Average: 1.4 5 Commentaires sur les concours CONCOURS CANADIEN DE NIVEAU SUPERIEUR´ Partie A Partie A 31 1. Ce probl`emea ´et´etr`esbien r´eussi. L'erreur la plus commune ´etaitde r´epondre 16 , soit la valeur de la parenth`ese,en oubliant de la multiplier par 16. Moyenne: 4,7 2. Ce probl`emea ´et´ebien r´eussi.Bon nombre d'´el`eves ont ´eprouv´ede la difficult´e`arepr´esenter les renseigne- ments par des ´equations. De plus, certains ´el`eves auraient eu avantage `ad´efinirles inconnues avec plus de soin et `asp´ecifiersi une inconnue repr´esentait l'^agepass´e,l'^agepr´esent ou l'^agedans cinq ans. Moyenne: 3,5 3. Ce probl`eme a ´et´ebien r´eussi. Certains ´el`eves ont suppos´eque les sommes possibles, de 2 `a12, ´etaient ´equiprobables.D'autres ont mal compt´eles r´esultatsqui donnaient un carr´eparfait.