Canadian Mathematics Competition An activity of the Centre for Education in Mathematics and Computing, University of Waterloo, Waterloo, Ontario Concours canadien de math´ ematiques Une activit´ e du Centre d’´ education en math´ ematiques et en informatique, Universit´ e de Waterloo, Waterloo, Ontario 2008 Results Pascal Contest (Grade 9) Cayley Contest (Grade 10) Fermat Contest (Grade 11) 2008 R´ esultats Concours Pascal (9 e ann´ ee – Sec. III) Concours Cayley (10 e ann´ ee – Sec. IV) Concours Fermat (11 e ann´ ee – Sec. V) C.M.C. Sponsors Chartered Accountants C.M.C. Supporter c 2008 Centre for Education in Mathematics and Computing
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CanadianMathematicsCompetition
An activity of the Centre forEducation in Mathematics and Computing,University of Waterloo, Waterloo, Ontario
Concourscanadien demathematiquesUne activite du Centre d’educationen mathematiques et en informatique,Universite de Waterloo, Waterloo, Ontario
Wally Webster (Chair / president), Exeter, ONLloyd Auckland, Waterloo, ONJanet Christ, Walter Murray C.I., Saskatoon, SKNerissa Coronel, St. Marguerite d’Youville C.S.S., Brampton, ONJP Pretti, University of Waterloo, Waterloo, ONJim Schurter, University of Waterloo, Waterloo, ONTyler Somer, Kitchener, ONAnna Spanik, Halifax West H.S., Halifax, NS
Cayley Contest / Concours Cayley
Jeff Anderson (Chair / president), Forest Heights C.I., Kitchener, ONTom Barker, Stratford Central S.S., Stratford, ONCharlotte Danard, Branksome Hall, Toronto, ONShawn Godin, Cairine Wilson S.S., Orleans, ONCatherine Haines, Western Technical-Commerical School, Toronto, ONChristopher Ing, Vincent Massey S.S., Windsor, ONJack Koenka, Havergal College, Toronto, ON
Fermat Contest / Concours Fermat
Paul Fennell (Chair / president), London, ONFiona Dunbar, University of Waterloo, Waterloo, ONCliff Lelievre, Oakville Trafalgar H.S., Oakville, ONMichelle McGinty, Waterloo C.I., Waterloo, ONDean Murray, University of Waterloo, Waterloo, ONAlexandru Pintilie, Crescent School, Toronto, ONDan Schnabel, Vaughan S.S., Thornhill, ONDavid Shepherd, Collingwood C.I., Collingwood, ON
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Foreword Avant-Propos
The Centre for Education in Mathematics and Computing is proud to once again be the major sponsor of theCanadian Mathematics Competition this year. The year 2008 marks the 46th year of Contests produced by theCanadian Mathematics Competition.
Our congratulations go to all who participated in this year’s Pascal, Cayley and Fermat Contests. This year’sContests were resounding successes, with average of 87.3, 90.1 and 86.4 across the three Contests. The numberof students with perfect papers on the three Contests were 42, 12 and 6, including International and Unofficialcompetitors. We are very pleased with these results. The overall number of students writing this year’s papersshowed a small decrease over that from 2007.
This year, we tried for the first time having individual school results available online in “real time” as we scannedand marked forms. As a result, we were able to have some students’ scores available within three days of theContests. We were quite happy with how this worked, and will continue to experiment with this.
As always, we would like to thank the hard-working Problems Committees. Most of the members of these Com-mittees are active secondary school teachers from across Canada who volunteers their time and contribute dozensof hours of expertise. Without their intriguing and sometimes amusing problems, these Contests would not bepossible.
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 d’education en mathematiques et en informatique a encore une fois le plaisir d’etre un majeur sup-porter du Concours canadien de mathematiques. L’annee 2008 marque la 46e annee de concours produit par leConcours canadien de mathematiques.
Nos felicitations vont a tous les participants du Concours Pascal, Cayley et Fermat de cette annee. Les Con-cours de cette annee retentissaient de succes, avec des moyennes de 87,3, 90,1 et 86,4 a travers les trois Concours.Le nombre d’etudiants possedant un concours parfait etait de 42, 12 et 6, en incluant les concurrents interna-tionaux et non officiels. Nous sommes satisfait de ces resultats. L’ensemble du nombre de participants cette anneea montre une petite diminution comparee a 2007.
Cette annee, nous avons essaye pour la premiere fois d’avoir les resultats d’ecole disponible en ligne au fur eta mesure que nous avions balaye et corrige les formulaires de reponses. Par consequent, nous avions les resultatspour beaucoup des etudiants disponibles trois jours apres les Concours. Nous etions tout a fait heureux avec lederoulement et nous continuerons a experimenter avec ceci.
Nous aimerions surtout remercier les Comites de problemes pour leur dur travail. La plupart des membres de cesComites sont des enseignant(e)s d’ecole secondaire actifs d’un bout a l’autre du Canada qui offre leur temps etcontribuent des douzaines d’heures d’expertise. Sans leurs problemes perspicaces et amusants, ces Concours neseraient pas possibles.
Nous aimerions remercier aussi tous les participants incluant les enseignants et les etudiants. Nous esperonsque les concours vous ont offert des mathematiques interessantes qui vous ont amusees et portees a reflechir.Merci pour votre soutien continue!
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Enrollment Inscription
Number of schools registered by province /Nombre d’ecoles inscrit par province
Total number of students enrolled / Nombre d’inscriptions individuelles: 82894
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Champions – 2008 Champions – 2008
2008 STUDENT CHAMPIONS AND CHAMPIONSHIP TEAMSPALMARES INDIVIDUEL ET PALMARES PAR EQUIPE 2008
PASCAL CONTEST / CONCOURS PASCAL
STUDENTS /ELEVESStudents are listed in alphabetical order / Les eleves sont nommes en ordre alphabetique
Gold Medals School Location GradeMedailles d’or Ecole Endroit NiveauSEUNGJIN AN St. Andrew’s College Aurora, ON 8AMALIA CONG Windfields J.H.S. North York, ON 9SEAN DEBRONI St. Michael’s University School Victoria, BC 9JACK DILLON Athens D.H.S. Athens, ON 8JASMINE DING William Lyon Mackenzie C.I. North York, ON 9BENOIT FOURNIER College Jean de Brebeuf Montreal, QC 9JIAYUE GAO Stratford Hall Vancouver, BC 9ANDY HAN Gordon Graydon Memorial S.S. Mississauga, ON 9JIMMY HE University of Toronto Schools Toronto, ON 9ALICE HU West Vancouver S.S. West Vancouver, BC 9ALEX JI Marc Garneau C.I. North York, ON 9HEINRICH JIANG Vincent Massey S.S. Windsor, ON 9ERIC KIM Windfields J.H.S. North York, ON 9TIAN LAN Northern S.S. Toronto, ON 9CHIEH MING LIU Fraser Heights S.S. Surrey, BC 9LARRY LIU Point Grey S.S. Vancouver, BC 9RICHARD LIU University of Toronto Schools Toronto, ON 9BRENT LOWRY Riverbend J.H.S. Edmonton, AB 9XIN NING Zion Heights J.H.S. North York, ON 9SUZIE PAN Hugh McRoberts S.S. Richmond, BC 9SUNGHWAN PARK Fredericton H.S. Fredericton, NB 9SUSAN SUN West Vancouver S.S. West Vancouver, BC 8SHELDON TUNG University of Toronto Schools Toronto, ON 9YAO WANG St. Michael’s University School Victoria, BC 9MATTHEW WARREN The Woodlands School Mississauga, ON 9AMY WU Burnaby South S.S. Burnaby, BC 9PHILIP XIAO Sir Winston Churchill S.S. Vancouver, BC 9YUNG LIN YANG Northern S.S. Toronto, ON 9FAN YIN Vincent Massey S.S. Windsor, ON 9JOE ZENG Zion Heights J.H.S. North York, ON 9CYRIL ZHANG Zion Heights J.H.S. North York, ON 9MENG ZHAO Queen Elizabeth H.S. Calgary, AB 9BRYAN ZHONG Victoria Park C.I. North York, ON 9KAIVEN ZHOU Vernon Barford School Edmonton, AB 9KEVIN ZHOU North York, ON 8
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Champions – 2008 Champions – 2008
TEAMS / EQUIPES
School Location ScoreEcole Endroit Note
Champion / Premiere University of Toronto Schools Toronto, ON 450Zion Heights J.S.S. North York, ON 450
Third / Troisieme Northern S.S. Toronto, ON 444West Vancouver S.S. West Vancouver, BC 444Windfields J.H.S. North York, ON 444
2008 STUDENT CHAMPIONS AND CHAMPIONSHIP TEAMSPALMARES INDIVIDUEL ET PALMARES PAR EQUIPE 2008
CAYLEY CONTEST / CONCOURS CAYLEY
STUDENTS /ELEVESStudents are listed in alphabetical order / Les eleves sont nommes en ordre alphabetique
Gold Medals School Location GradeMedailles d’or Ecole Endroit NiveauTYSON ANDRE Richmond Hill H.S. Richmond Hill, ON 10DUNCAN DAUVERGNE University Hill S.S. Vancouver, BC 10ROBERT LEE Thornlea S.S. Thornhill, ON 10SIMON QUACH Gordon Graydon Memorial S.S. Mississauga, ON 10MARIYA SARDARLI McKernan J.H.S. Edmonton, AB 8RICHARD WANG Sir Winston Churchill S.S. Vancouver, BC 10DAVID XU Sir Winston Churchill S.S. Vancouver, BC 10JENNIFER YU McKernan J.H.S. Edmonton, AB 9XINGYU ZHOU Walter Murray C.I. Saskatoon, BC 10
TEAMS / EQUIPES
School Location ScoreEcole Endroit Note
Champion / Premiere Sir Winston Churchill S.S. Vancouver, BC 442Second / Deuxieme Richmond Hill S.S. Richmond Hill, ON 430Third / Troisieme University of Toronto Schools Toronto, ON 428Fourth / Quatrieme Gordon Graydon Memorial S.S. Mississauga, ON 426
St. John’s-Ravenscourt School Winnipeg, MB 426
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Champions – 2008 Champions – 2008
2008 STUDENT CHAMPIONS AND CHAMPIONSHIP TEAMSPALMARES INDIVIDUEL ET PALMARES PAR EQUIPE 2008
FERMAT CONTEST / CONCOURS FERMAT
STUDENTS /ELEVESStudents are listed in alphabetical order / Les eleves sont nommes en ordre alphabetique
Gold Medals School Location GradeMedailles d’or Ecole Endroit NiveauJUNGHOO KIM University Hill S.S. Vancouver, BC 10ZHIQIANG LIU Don Mills C.I. North York, ON 10JULIAN SUN Sir Winston Churchill S.S. Vancouver, BC 11SIMON YIN O’Neill C.V.I. Oshawa, ON 11NOBLE ZHAI Western Canada H.S. Calgary, AB 11
TEAMS / EQUIPES
School Location ScoreEcole Endroit Note
Champion / Premiere Western Canada H.S. Calgary, AB 436Second / Deuxieme University Hill S.S. Vancouver, BC 428Third / Troisieme Sir Winston Churchill S.S. Vancouver, BC 424Fourth / Quatrieme Vincent Massey S.S. Windsor, ON 418Fifth / Cinquieme Point Grey S.S. Vancouver, BC 416
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Champions – 2008 Champions – 2008
2008 FERMAT CONTEST PROVINCIAL CHAMPIONSCHAMPIONS PROVINCIAUX AU CONCOURS FERMAT 2008
Each student below will receive a $200 prize / Chaque etudiant ci-dessous recevrait un prix de 200 $
Province Student School LocationProvince Etudiant Ecole EndroitNL ANTHONY PAYNE Pasadena Academy PasadenaNS TSE AN CHEN King’s-Edgehill School WindsorNB YOONGOO NAM St. Malachy’s H.S. Saint JohnPE SAMUEL CHEN Colonel Gray S.H.S. CharlottetownQC ILIA POLOTSKII College Jean de Brebeuf MontrealON Nord YUHAN CHEN Sir Winston Churchill C.V.I. Thunder BayON Est SIMON YIN O’Neill C.V.I. OshawaON Metro ZHIQIANG LIU Don Mills C.I. North YorkON Ouest CHEN SUN A.B. Lucas S.S. LondonON Central KULJOT ANAND Turner Fenton S.S. BramptonMB CLEMENT KWOKEN St. John’s-Ravenscourt School WinnipegMB GHAEUN LEE Balmoral Hall School WinnipegSK XINDI HU Bedford Road C.I. SaskatoonAB NOBLE ZHAI Western Canada H.S. CalgaryBC JUNGHOO KIM University Hill S.S. VancouverBC JULIAN SUN Sir Winston Churchill S.S. Vancouver
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Correct Answers Bonnes Rponses
# Pascal Cayley Fermat1 E E D2 A B A3 B C C4 C D E5 C C A6 A D E7 D A B8 A C D9 E A D
10 D C A11 A B D12 B E A13 A B B14 D A B15 E A B16 C B C17 B A A18 B C A19 D D E20 D E C21 A A A22 E D E23 D E C24 D E B25 C D D
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Pascal Contest Concours Pascal
Number of Contestants / Nombre de concurrents: 26478Average mark / Note moyenne: 87.32
Percentage of contestants selecting each option / Pourcentage de concurrents par choix:
Correct % not % correctAnswer/ attempted/ of attemptedBonne % sans % tentatives
Reponse reponse reussies# A B C D E1 E 2.05 0.50 0.85 1.16 94.82 0.56 95.412 A 89.61 3.61 4.11 0.85 0.18 1.58 91.103 B 8.90 87.79 0.82 1.18 0.57 0.67 88.444 C 0.70 8.27 86.40 2.05 1.26 1.25 87.565 C 1.85 1.90 77.68 14.73 2.53 1.25 78.716 A 86.04 2.81 0.95 3.93 3.64 2.56 88.367 D 2.21 1.20 3.27 91.07 0.82 1.36 92.398 A 92.61 0.87 2.44 1.06 1.63 1.32 93.929 E 15.24 5.61 5.19 3.35 65.13 5.42 68.91
10 D 13.02 12.74 3.20 53.64 5.47 11.86 60.9111 A 50.22 10.40 4.15 21.16 11.26 2.74 51.6712 B 5.44 66.84 4.95 3.27 7.00 12.44 76.3913 A 59.44 1.84 14.53 19.38 2.48 2.27 60.8614 D 3.12 1.72 3.63 74.73 6.28 10.45 83.5215 E 8.87 3.61 3.21 5.61 65.44 13.20 75.4416 C 18.08 10.19 29.76 4.69 2.85 34.36 45.3917 B 36.32 37.27 3.63 2.87 1.80 18.05 45.5118 B 3.77 58.71 16.89 10.63 2.94 7.00 63.1719 D 5.90 4.67 19.79 39.34 7.58 22.66 50.9120 D 5.45 7.29 7.18 53.94 3.49 22.60 69.7321 A 21.91 6.76 6.76 7.78 12.42 44.29 39.3922 E 10.87 7.89 12.65 8.78 16.66 43.08 29.3123 D 34.10 8.32 6.08 11.88 7.85 31.70 17.4124 D 18.74 11.36 6.44 16.22 9.37 37.82 26.1125 C 10.94 6.28 10.89 5.20 7.15 59.46 26.92
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Cayley Contest Concours Cayley
Number of Contestants / Nombre de concurrents: 19991Average mark / Note moyenne: 90.11
Percentage of contestants selecting each option / Pourcentage de concurrents par choix:
Correct % not % correctAnswer/ attempted/ of attemptedBonne % sans % tentatives
Reponse reponse reussies# A B C D E1 E 0.26 0.11 0.14 0.21 99.15 0.10 99.282 B 2.17 93.98 1.48 0.65 0.22 1.46 95.413 C 0.88 8.09 89.13 0.80 0.82 0.25 89.384 D 1.48 0.78 2.21 94.12 0.58 0.79 94.915 C 1.02 1.44 91.93 3.27 0.94 1.36 93.246 D 0.22 3.95 0.44 94.35 0.50 0.51 94.867 A 73.65 6.05 5.87 2.66 7.36 4.35 77.058 C 3.84 4.05 76.92 6.54 1.84 6.77 82.549 A 50.87 7.26 15.94 5.90 9.98 10.01 56.55
10 C 3.06 0.98 92.26 1.82 0.67 1.18 93.3911 B 3.96 80.82 4.56 2.19 1.18 7.25 87.1812 E 1.74 1.36 1.41 2.26 86.35 6.85 92.7313 B 5.85 70.22 5.13 2.97 13.07 2.73 72.2114 A 67.49 2.56 4.06 10.76 1.53 13.55 78.1115 A 59.12 15.83 3.73 2.29 3.76 15.23 69.7716 B 11.75 37.09 3.93 8.61 15.15 23.43 48.4617 A 28.67 5.23 4.92 6.91 16.82 37.40 45.8418 C 1.35 3.04 72.58 8.03 2.17 12.80 83.2619 D 30.35 8.79 2.78 26.65 7.25 24.13 35.1520 E 8.45 6.69 4.17 5.54 22.98 52.12 48.0521 A 23.13 12.00 7.68 8.82 5.32 43.00 40.6122 D 5.82 11.64 10.44 13.98 28.71 29.37 19.8023 E 5.37 21.20 13.83 6.52 6.31 46.74 11.8524 E 9.04 10.72 9.11 13.39 9.89 47.80 18.9625 D 4.81 3.94 10.38 11.19 4.56 65.07 32.08
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Fermat Contest Concours Fermat
Number of Contestants / Nombre de concurrents: 16354Average mark / Note moyenne: 86.40
Percentage of contestants selecting each option / Pourcentage de concurrents par choix:
Correct % not % correctAnswer/ attempted/ of attemptedBonne % sans % tentatives
Reponse reponse reussies# A B C D E1 D 0.68 0.20 0.75 98.06 0.13 0.15 98.242 A 87.45 1.91 9.70 0.29 0.26 0.37 87.793 C 0.20 0.10 98.60 0.42 0.46 0.20 98.824 E 0.18 1.94 0.23 0.43 97.08 0.13 97.225 A 88.29 0.84 0.41 1.93 7.99 0.52 88.776 E 0.78 1.85 1.28 8.21 86.57 1.30 87.727 B 2.15 88.26 2.56 1.16 3.71 2.15 90.218 D 1.94 0.65 2.36 83.75 4.50 6.79 89.869 D 2.16 2.67 2.22 85.27 3.56 4.10 88.93
10 A 48.75 11.41 5.80 7.61 2.32 24.10 64.2411 D 6.57 3.38 2.75 80.10 2.21 4.98 84.3112 A 69.29 3.02 9.78 9.48 2.49 5.93 73.6713 B 5.00 60.09 12.47 4.15 3.82 14.46 70.2614 B 7.20 49.17 15.03 12.37 6.00 10.21 54.7715 B 2.85 20.56 61.97 5.80 2.23 6.58 22.0116 C 6.71 5.52 47.45 7.75 9.93 22.62 61.3417 A 67.94 4.82 2.77 3.19 2.12 19.15 84.0418 A 26.68 7.31 4.82 6.77 4.56 49.84 53.2119 E 9.59 4.49 3.53 5.08 40.86 36.44 64.3020 C 4.94 7.86 33.04 9.22 5.56 39.37 54.5021 A 15.46 10.44 9.77 5.95 31.55 26.82 21.1322 E 8.88 13.01 7.46 7.86 6.90 55.88 15.6423 C 5.15 9.31 9.69 7.23 7.99 60.60 24.6124 B 11.67 6.42 5.52 6.82 2.75 66.81 19.3525 D 19.84 12.15 9.74 5.41 2.43 50.42 10.91
Number of three student teams / Nombre d’equipes de trois elevesPascal Contest / Concours Pascal : 1105Cayley Contest / Concours Cayley : 1022Fermat Contest / Concours Fermat : 1017
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CanadianMathematicsCompetition
An activity of the Centre forEducation in Mathematics and Computing,University of Waterloo, Waterloo, Ontario
Concourscanadien demathematiquesUne activite du Centre d’educationen mathematiques et en informatique,Universite de Waterloo, Waterloo, Ontario
2008Results
Pascal Contest(Grade 9)
Cayley Contest(Grade 10)
Fermat Contest(Grade 11)
2008Resultats
Concours Pascal(9e annee – Sec. III)
Concours Cayley(10e annee – Sec. IV)
Concours Fermat(11e annee – Sec. V)
Fermat Contest Honour RollsPalmares pour le concours Fermat
ScoreCity/Ville
Rank/Rang
School/École
436Calgary1 Western Canada H.S.
428Vancouver2 University Hill S.S.
424Vancouver3 Sir Winston Churchill S.S.
418Windsor4 Vincent Massey S.S.
416Vancouver5 Point Grey Secondary School
414Richmond6 Richmond S.S.
412Waterloo7 Waterloo C.I.
408North York8 Don Mills C.I.
408Calgary8 Sir Winston Churchill H. S.
406Montreal10 E.S. St. Luc
406Coquitlam10 Pinetree S.S.
406Scarborough10 Sir John A. Macdonald C.I.
402Burnaby13 Burnaby North S.S.
402Toronto13 University of Toronto Schools
402Toronto13 Upper Can Col-Upper School
400Burnaby16 Moscrop Secondary School
400Vancouver16 St. George's School
398Vancouver18 Lord Byng S.S.
398Delta18 Seaquam S.S.
398Brampton18 Turner Fenton S.S.
398West Vancouver18 West Vancouver S.S.
396Vancouver22 Prince of Wales S.S.
396West Vancouver22 Rockridge S.S.
394Scarborough24 Agincourt C.I.
392North York25 A.Y. Jackson S.S.
392Montreal25 College Jean de Brebeuf
391Scarborough27 Woburn Collegiate Institute
390Burnaby28 Burnaby South S.S.
390West Vancouver28 Mulgrave School
390Port Moody28 Port Moody S.S.S.
388North York31 Marc Garneau C.I.
388Mississauga31 St. Francis Xavier Secondary S
386London33 A.B. Lucas S.S.
386Mississauga33 Glenforest S.S.
386Surrey33 Semiahmoo S.S.
386Hamilton33 Westdale S.S.
384Toronto37 Northern S.S.
384Edmonton37 Old Scona Academic H.S.
384Mississauga37 The Woodlands S.
384North York37 Victoria Park C.I.
383Calgary41 Queen Elizabeth H.S.
382Gloucester42 Colonel By S. S.
382Ottawa42 Lisgar C.I.
382Langley42 R.E. Mountain S.S.
381Toronto45 The Abelard School
380Surrey46 Elgin Park S.S.
380Surrey46 Fraser Heights S.S.
379Oshawa48 O'Neill C.V.I.
379Mississauga48 Streetsville S.S.
379Vancouver48 Templeton S.S.
378Oakville51 Appleby College
378Montreal51 Ecole Villa Maria
2008Fermat Contest/Concours Fermat
Team List/Palmarès d'équipes
Grade/Niveau
Location/Endroit
Name/Nom School/École
Group I/Groupe I Scores/Notes 150 - 145KIM JUNGHOO 10VancouverUniversity Hill S.S.
LIU ZHIQIANG 10North YorkDon Mills C.I.
SUN JULIAN 11VancouverSir Winston Churchill S.S.
YIN SIMON 11OshawaO'Neill C.V.I.
ZHAI NOBLE 11CalgaryWestern Canada H.S.
Group II/Groupe II Scores/Notes 144 - 139BramptonTurner Fenton S.S.ANAND KULJOT 11