2013 Results Résultats Canadian Senior and cemc.math.· Résultats Canadian Senior and Intermediate

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    Canadian Senior and IntermediateMathematics Contests

    Concours canadiens de mathmatiquesde niveau suprieur et intermdiaire

    c2013 University of Waterloo

  • Competition Organization Organisation du Concours

    Centre for Education in Mathematics and Computing Faculty and Staff /Personnel du Concours canadien de mathmatiques

    Ed Anderson Sandy GrahamJeff Anderson Angie HildebrandTerry Bae Judith KoellerSteve Brown Joanne KursikowskiErsal Cahit Bev MarshmanAlison Cornthwaite Mike MiniouHeather Culham Dean MurraySerge DAlessio Jen NissenFrank DeMaio J.P. PrettiJennifer Doucet Kim SchnarrFiona Dunbar Carolyn SedoreMike Eden Ian VanderBurghBarry Ferguson Troy VasigaJudy Fox JoAnn VincentSteve Furino Tim ZhouJohn Galbraith

    Problems Committees / Comits des problmes

    Canadian Senior Mathematics Contest / Concours canadien de niveau suprieur

    Mike Eden (Chair / prsident), University of Waterloo, Waterloo, ONKee Ip, Crescent School, Toronto, ONPaul Leistra, Guido de Bres Christian H.S., Hamilton, ONDaryl Tingley, University of New Brunswick, Fredericton, NBJoe West, University of Waterloo, Waterloo, ONBruce White, Windsor, ON

    Canadian Intermediate Mathematics Contest / Concours canadien de niveau intermdiaire

    John Galbraith (Chair / prsident), University of Waterloo, Waterloo, ONEd Barbeau, Toronto, ONAlison Cornthwaite, University of Waterloo, Waterloo, ONBrian McBain, North Lambton S.S., Forest, ONGinger Moorey, Abbey Park H.S., Oakville, ONDean Murray, University of Waterloo, Waterloo, ON


  • Foreword Avant-Propos

    The Centre for Education in Mathematics and Computing is pleased to announce the results of the 2013 CanadianSenior and Intermediate Mathematics Contests.

    Our congratulations go to all who participated in this years CSMC and CIMC. This years Contests were re-sounding successes, with averages of 34.9 and 36.9, 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 providedyou 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 dducation en mathmatiques et en informatique dannoncer les rsultats du Conours canadiensde mathmatiques de niveau suprieur et intermdiaire 2013.

    Nos flicitations vont tous les participants du CCMS et du CCMI de cette anne. Les Concours de cetteanne retentissaient de succs, avec des moyennes de 34,9 et 36,9, respectivement.

    Nous aimerions surtout remercier les Comits de problmes pour leur dur travail. Plusieurs membres de cesComits sont des enseignant(e)s dcole secondaire actifs qui offre leur temps et contribuent des douzaines dheuresdexpertise. Sans leurs problmes perspicaces et amusants, ces Concours ne seraient pas possibles.

    Nous aimerions remercier aussi tous les participants incluant les enseignants et les tudiants. Nous espronsque les concours vous ont offert des mathmatiques intressantes qui vous ont amuses et portes rflchir. Mercipour votre soutien continu!


  • Comments on the Contests


    1. Very well done. There were few trends among incorrect solutions.Average: 4.9

    2. Well done. Most students organized their solution by looking first at the possible combinations for Ben orfor Riley and Wendy. A common mistake was the incorrect conclusion that when Ben gets the number 4,then Sara always gets the number 1 (she could get 1 or 3).Average: 4.5

    3. Fairly well done. Common mistakes included incorrectly factoring 99! in the denominator or incorrectlycancelling the 99! in one term in the denominator.Average: 3.6

    4. Well done by most students. Some students found the length of AP but did not calculate the area of thetriangle. Many students rounded answers found using a calculator as opposed to working with exact values.Average: 4.2

    5. Many students did some counting, but missed an important detail. For example, a number of students forgotto remove the cases where the four-digit number started with 0 (ie. the number 0123 is not a valid four-digitnumber). Some students listed out all 66 possible integers, or attempted to and missed several cases.Average: 2.3

    6. Students found this difficult to approach. Many students tried several pairs (x, y), but did not generalize toa range of pairs. Other students found a range of pairs but either did not count them at all or counted themincorrectly.Average: 1.1

    Part B

    1. Well done by most students. Many students presented clear, numerical or algebraic reasoning in their solu-tions. That being said, a good number of students needed to more clearly justify their statements.Average: 7.8

    2. Part (a) was attempted by most students and was very well done. In part (b), most students saw theconnection to part (a) and used this to factor the cubic equation. The most common mistake was notconsidering negative values for . Most students who solved part (c) used a table. We also saw solutionswhere students sketched the cubic function and determined the values of that make the function negativefrom the sketch.Average: 5.2

    3. Part (a) was generally well done. The vast majority of students who attempted the problem found the foursequences. In part (b), many students made an assumption about x1 = A without considering the case ofx1 = B. Carefully reasoning through how and why a contradiction is reached was quite often missed. Inpart (c), few students constructed a useful sequence and proved the desired results. Some students had lessprecise statements about stretching" the sequence or multiplying the sequence by r" which needed to beformalized. Part (d) saw very few students make progress.Average: 1.3


  • Comments on the Contests


    1. Very well done. Some students did not know that ABC and CBE are supplementary angles that add to180. Some students determined the value of BCE instead of the value of x.Average: 4.6

    2. Very well done. Many students did not realize this question was asking for the lowest common multiple ofthe four given integers and that they need only look at multiples of both 6 and 8.Average: 4.0

    3. Well done, except for some mechanical mistakes. Some students had algebraic difficulties with solving theequation after x = 3 and y = 7 were substituted. Other students had a few problems with substitution,especially substituting 8 for the second a when substituting 8 for x and 3 for a.Average: 4.3

    4. Fairly well done. Students who understood that the shaded area was to be bisected using a line through theorigin (as opposed to a horizontal line) generally had no difficulties with this problem.Average: 3.9

    5. In this problem, many students were awarded part marks for an almost complete list of palindromes, or forcounting too many possibilities. One frequent error was to begin with a palindrome and then multiply it by6. This does not in general result in a palindrome and also misses some palindromes that are divisible by 6.Average: 2.9

    6. This was a hard problem with a short solution that required a lot of insight. Part marks were awarded forincorrect triples if students demonstrated some ability to manipulate fractions.Average: 0.3

    Part B

    1. In part (a), many students unfortunately did not read to the end and thus found the reflections on thediagram but did not find the coordinates of these points. In part (b), students who knew what the questionwas asking did quite well. Some made mistakes by looking at an incorrect diagram with angled line segmentsor by looking at the diagram from part (c) and using some partial circumferences. In part (c), many studentswere able to find the correct volume. Finding the correct surface area was harder and many students didnot understand what the two pieces of the formula represented and struggled with adding or subtracting toomany pieces of circles.Average: 7.2

    2. Part (a) was attempted successfully by many students. Part (b) was well done, although some students onlygave a list of the cups that receive balls. Part (c) was also well done. Some students created lists and missedone of the numbers. In part (d), we saw a common error in solutions when students double-counted the firstball and calculated that the last ball goes 338 8 = 2366 cups further along the circle from cup 1 instead of337 7 = 2359 cups further along the circle from cup 1.Average: 6.5

    3. Many students were able to do parts (a) and (b). The most common error here was to consider only differencesof adjacent pairs of numbers rather than differences for all pairs of numbers. Parts (c) and (d) were moredifficult and required more care.Average: 3.2


  • Commentaires sur les concours


    1. Ce problme a t trs bien russi. Il y avait peu de tendances parmi les solutions errones.Moyenne: 4,9

    2. Ce problme a t bien russi. La plupart des lves ont prsent leur solution en considrant dabord lescartes possibles que Benot pouvait recevoir ou encore celles que Ren et Vianna pourraient recevoir. Bonnombre dlves ont conclu que si Benot recevait la carte numro 4, alors Sara devait recevoir la carte numro1 (elle pouva