The CENTRE for EDUCATION in MATHEMATICS and COMPUTING Le CENTRE d’EDUCATION´ en MATHEMATIQUES´ et en INFORMATIQUE www.cemc.uwaterloo.ca 2018 2018 Results R´esultats Euclid Contest Concours Euclide c 2018 Centre for Education in Mathematics and Computing Competition Organization Organisation du Concours Centre for Education in Mathematics and Computing Faculty and Staff / Personnel du Centre d’´education en math´ematiques et informatique Ed Anderson Angie Hildebrand Jeff Anderson Carrie Knoll Terry Bae Judith Koeller Jacquelene Bailey Laura Kreuzer Grace Bauman Bev Marshman Shane Bauman Mike Miniou Ersal Cahit Dean Murray Serge D’Alessio Jen Nelson Rich Dlin J.P. Pretti Jennifer Doucet Kim Schnarr Fiona Dunbar Carolyn Sedore Mike Eden Kevin Shonk Barry Ferguson Ashley Sorensen Judy Fox Ian VanderBurgh Steve Furino Troy Vasiga John Galbraith Christine Vender Robert Garbary Heather Vo Rob Gleeson Jessica Won Sandy Graham Tim Zhou Conrad Hewitt Problems Committee / Comit´e des probl`emes Fiona Dunbar (Chair / pr´esidente), University of Waterloo, Waterloo, ON Steve Brown, University of Waterloo, Waterloo, ON Serge D’Alessio, University of Waterloo, Waterloo, ON Charlotte Danard, Toronto, ON Garry Kiziak, Burlington, ON Jeremy Klassen, Ross Shepherd H.S., Edmonton, AB Darren Luoma, Bear Creek S.S., Barrie, ON Alex Pintilie, Crescent School, Toronto, ON David Pritchard, Los Angeles, CA Laurissa Werhun, Parkside C.I., Toronto, ON Peter Wood, University of Waterloo, Waterloo, ON 2 Comments on the Paper Commentaires sur les ´epreuves Overall Comments Congratulations to all of the participants in the 2018 Euclid Contest. The average score in 2018 was 54.6. We were very pleased that almost all students achieved some success on the early parts of the paper. At the same time, the later parts of these problems managed to challenge the top students even more than last year’s problems. Special congratulations go to the 45 official contestants who achieved scores of 90 and higher this year. We at the Centre for Education in Mathematics and Computing believe strongly that it is very important for students to both learn to solve mathematics problems and learn to write good solutions to these problems. Many students do a reasonable job of writing solutions, while others still include no explanation whatsoever. Special thanks go to the Euclid Committee that annually sets the Contest problems and manages to achieve a very difficult balancing act of providing both accessible and challenging problems on the same paper. To the students who wrote, the parents who supported them, and the teachers who helped them along the way, thank you for your continuing participation and support. We hope that you enjoyed the Contest and relished the challenges that it provided. We hope that mathematics contests continue to feed your love for and interest in mathematics. Specific Comments 1. Average: 9.5 Very well done. In parts (a) and (b), the most common errors were arithmetical or were introduced by incorrectly copying the equation. In part (c), the most common error was in the relationship between the cost of gum and the cost of the chocolate. 2. Average: 9.4 Very well done. In (a), the most frequent error was not using all five of the required digits. In (b), some errors in the use of the Pythagorean Theorem were found. In part (c), most errors involved not getting the point of intersection as (3, 6). 3. Average: 9.2 Well done. In parts (a) and (b), some students struggled with squaring a binomial. In part (c), some students gave point P back as their answer for point Q. 4. Average: 7.2 Students struggled with (a) and did not show much structure in their approach. Part (b) was well done, although some students tried to incorporate the given example (f(34)) in their solution. 5. Average: 6.7 Many students did well on part (a). Most students followed the Alternate Solution and computed the areas of the triangle and hexagon. Part (b) was also generally well done. Most students solved this problem by drop- 1 ping a perpendicular from P to OA. Many students used the formula area 4OP A = 2 (OP )(OA) sin (\P OA) and were able to solve this problem in a few lines. 6. Average: 6.3 In part (a), students who did not get the correct answer often factored correctly and then did not know how to proceed. Most students who attempted (b) were able to calculate t1 = 27. There were many ways to calculate t2. A common error was finding a value for t2, but forgetting to add it to the time already elapsed. Another common error was working backwards to calculate the second time, but forgetting to subtract from 99. We saw some nice solutions where students visualized the problem by writing Karuna’s and Jorge’s speeds as piecewise functions and graphing these. 3 Comments on the Paper Commentaires sur les ´epreuves 7. Average: 2.3 Students found part (a) to be challenging. In respectable attempts, a common error was to fix the order of the canoes, but inconsistently. In incorrect solutions, we often saw some insightful counting that would still get a mark. For part (b), we saw many solutions similar to both Solutions 1 and 2. A common incorrect assumption was to assume the sides of the square are horizontal and vertical. 8. Average: 1.6 In part (a), we were pleased to see a lot of students successfully simplify the given equation using rules for logarithms and the prime factorizations of 48 and 162. In order to arrive at a correct exact solution, students often followed the approach in the posted solutions. However, we also saw some a large number of slick solutions using the definition of logarithm√ to eventually determine that both sides of the equation equal 8 3 from which one can determine that x = 6. Part (b) was challenging. As with many geometry problems, there are many reasonable things to try. For this problem, similar triangles or something equivalent is the key to finding a solution. Particularly clever answers managed to avoid some of the algebra in the posted solutions by using trigonometry, or by constructing a perpendicular from P to DC to find congruent triangles and/or rectangles. 9. Average: 1.5 Many students correctly demonstrated a tiling in part (a) and an understanding of how to extend small tilings into infinitely many larger tilings. We expected careful rigorous justification to earn full marks in parts (b) and (c). In particular, it does not seem possible to fully answer part (c) without providing an explicit tiling of the 5 × 9 (or 9 × 5) case. 10. Average: 0.8 We saw many correct values for A4 in the first part of this last problem. With a little bit of extra persistence, many students also provided a correct proof in part (b). Also in (b), an additional group of students incorrectly assumed that the result was true and used it to derive true statements, rather than deriving the result. As expected, the last part of this last question was very challenging and we did not see many students make significant progress. Please visit our website at cemc.uwaterloo.ca to download the 2018 Euclid Contest, plus full solu- tions. 4 Comments on the Paper Commentaires sur les ´epreuves Commentaires G´en´eraux F´elicitations`atous les participants du Concours Euclide 2018. La note moyenne ´etait54,6. Nous avons eu le plaisir de constater que presque tous les ´el`eves ont eu du succ`esdans les premi`eresquestions. De plus, les derni`eres parties de ces probl`emespr´esentaient un plus grand d´efipour les meilleurs ´etudiants que l’ann´eederni`ere. Des f´elicitationssp´ecialesvont aux 45 concurrents officiaux qui ont atteint une note sup`erieureou ´egale`a90 sur 100. Le Centre d’´educationen math´ematiqueset en informatique croit fortement qu’il est tr`esimportant pour les ´etudiant(e)s d’apprendre `ar´esoudredes probl`emesde math´ematiquesainsi que d’apprendre `a´ecrirede bonnes solutions `aces probl`emes.Plusieurs ´etudiants d´eveloppent leurs solutions r´esonablement bien, tandis que d’autres n’inclus aucune explication avec leurs r´eponses. Un grand merci va au comit´edu concours Euclide qui rassemble annuellement les probl`emesdu concours et r´eussit`aaccomplir la tˆache difficile de concevoir des probl`emesaccessibles et stimulants sur le mˆeme examen. Un grand merci aux ´etudiant(e)s qui ont ´ecrit,aux parents qui les ont soutenus et aux enseignant(e)s qui les ont aid´espour votre participation continue et votre soutien. Nous esp´eronsque vous avez appr´eci´ele concours et savour´eles d´efisqu’il a pr´esent´es. Nous esp´eronsque les concours de math´ematiquescontinuent `anourrir votre amour et int´erˆetpour les math´ematiques. Remarques particuli`eres 1. Moyenne: 9,5 Cette question a ´et´etr`esbien r´eussie. Dans les parties (a) et (b), les erreurs ´etaient surtout de nature arithm´etique.De plus, certains ´el`eves ont mal copi´el’´equation.Dans la partie (c), l’erreur la plus commune d´ecoulaitde la relation entre le cout d’une palette de chocolat et celui d’un paquet de gomme. 2. Moyenne: 9,4 Cette question a ´et´etr`esbien r´eussie. Dans la partie (a), bon nombre d’´el`eves n’ont pas utilis´etous les cinq chiffres de l’entier. Dans la partie (b), des erreurs ont ´et´ecommises dans l’utilisation du th´eor`emede Pythagore. Dans la partie (c), bon nombre d’´el`eves n’ont pas d´etermin´ecorrectement les coordonn´ees(3, 6) du point d’intersection. 3. Moyenne: 9,2 Cette question a ´et´ebien r´eussie.