DISINI sol2016-draft
✂✁☎✄✝✆✟✞✂✠☛✡✌☞✎✍✏✞☎✑✒✡✌✓✕✔✖✆✗✡✌✁✙✘✝✚✜✛✣✢✥✤✧✦✩★✫✪✥✬✮✭✯✁✰✑✎☞✱✘✳✲✳✴✵✘✝✚✜✛✶✢
✷✹✸✻✺✽✼✿✾❁❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊●✷✹❍❋■✶❍❋❏✶❍❇❑❇❑❇❑☎❍❋■✹▲▼✷❖◆✙P✹❉❋✾❁◗✽❉❘❀❙❄❅❄❅✾❖❂❯❚❱❂❲P❨❳✳❚❱P✹❉❘❩☛✸✽❬✵❚❱❭❫❪❇P❈❂❫❪❋✼▼❚✶❚❱❊❅✾
P❈❂✣❴❯❄❵◗❛❚❯❂✶❭✿❜❁❳❝✾❇❉❋❊❞❚❱❂❡❄❢✼▼✾●❳❝❚❱P✹❉❋❩❡P❈❂✿❩❫❉❢✾❇❣✝❤✐P❈❪❥✾✧❄❢✼✿✾❖❜❦◗❧❀❙❄❢✼❫❄❢✼▼✾❖❀❙❉♠P♦♥❈✾❇❉❋P✹❆❈✾❈✸
♣ ❚❈❉q✾❥rsP❈❜t❣✌❤❙✾❈✉☛❴❈❚❱❭✈❪❇P❈❂❡❉❢✾❇❣✌❤❃P❈❪❥✾✂✷tP❈❂✿❩✈■✮◗❧❀❙❄❢✼✇✷✹❑②①✶✉✏❚❈❉q❴❈❚❱❭❡❪❇P❈❂❡❉❢✾❇❣✝❤❃P❈❪❥✾③✷
P❈❂✿❩❫❏t◗✻❀❙❄❢✼④P✮❊❅✾❖❪❥❚❱❂✿❩❲❪❥❚❈❣✶❴③❚❈⑤❛■✶✸⑦⑥❧⑤✒❄❅✾❇❉⑦■✹▲▼✷❖①✮❉❋✾❇❣✝❤❃P❈❪❥✾❖❜t✾❖❂✣❄❢❊❞❚❈⑤⑧❄❢✼✿❀❃❊✻⑨✶❀❃❂✝❩☛✉
❄❢✼▼✾♠❳❝❚❱P✹❉❋❩③◗❧❀❃❤❃❤✳✼✿P❖♥❈✾❁❚❱❂✿❤❙❴❨❚❱❂▼✾q❂✶❭✿❜q❳❝✾❇❉✽❤❙✾❇⑤✒❄✱❚❱❂③❀❙❄❇✸
⑩ P❷❶✂❸❹❉❢❚♦♥❈✾✻❄❢✼✿P✹❄✵❄❢✼▼✾❇❉❢✾❞❀❃❊✵P❺❊❢✾❖❻✣❭▼✾❖❂✝❪❥✾✽❚❈⑤☛❉❢✾❇❣✝❤❃P❈❪❥✾❖❜✖✾❖❂❷❄❢❊❹❄❢✼✿P✹❄✵◗❧❀❃❤✐❤▼❜✖P✹⑨❈✾❧❄❢✼▼✾
❼ ❂✝P❈❤☛❂✣❭✿❜❁❳❝✾❇❉✱✾❖❻✣❭✝P❈❤☛❄❅❚❨■✶✸
⑩ ❳✳❶✙❸❹❉❢❚♦♥❈✾✻❄❢✼✿P✹❄✵❄❢✼▼✾❇❉❢✾❞❀❃❊✵P❺❊❢✾❖❻✣❭▼✾❖❂✝❪❥✾✽❚❈⑤☛❉❢✾❇❣✝❤❃P❈❪❥✾❖❜✖✾❖❂❷❄❢❊❹❄❢✼✿P✹❄✵◗❧❀❃❤✐❤▼❜✖P✹⑨❈✾❧❄❢✼▼✾
❼ ❂✝P❈❤☛❂✣❭✿❜❁❳❝✾❇❉✱✾❖❻✣❭✝P❈❤☛❄❅❚❯✷❇▲❈▲❈▲s✸
✌❽ ❾❈❿➁➀▼➂✎➃➄❾✹➅✌➆ ⑩ P❷❶ ♣ ❀❃❉❋❊❅❄➇❉❋✾❇❣✝❤❃P❈❪❥✾❲■✹▲▼✷❥➈✥P❈❂✝❩➉■✹▲▼✷❖◆➊◗❧❀❙❄❢✼➋■✹▲▼✷❖①✶✉✻P❈❂✝❩➌❄❢✼▼✾❖❂➉❉❢✾❥➍
❣✝❤✐P❈❪❥✾t❄❢✼▼✾✖❄✗◗➎❚④❪❥❚❈❣✌❀❙✾❖❊❺❚❈⑤✱■✹▲▼✷❖①③◗✻❀❙❄❢✼❡P③❊❢❀❃❂✿❆❱❤❙✾✖❪❥❚❈❣✣❴❈✸✙✺✽✼✿❀✐❊❞❤❃✾❖P❖♥❈✾❖❊✧❭✿❊♠◗❧❀❙❄❢✼
➏ ✷✹❍❋■✶❍❇❑❇❑❇❑✰❍❋■✹▲▼✷❖❏✶❍❋■✹▲▼✷❖①✣➐❱✸ ♣ ❉❢❚❱❜➑✼▼✾❇❉❢✾❈✉✳◗➎✾✧❪❇P❈❂④❉❢✾❇❣✝❤❃P❈❪❥✾●■✹▲▼✷❖❏✮P❈❂✿❩④■✹▲▼✷❖①❨◗❧❀❙❄❢✼
■✹▲▼✷❥➈t❄❅❚●❆❈✾❇❄ ➏ ✷✹❍❋■✶❍❇❑❇❑❇❑✰❍❋■✹▲▼✷❖■✶❍❋■✹▲▼✷❥➈s➐❱✸✱➒❡✾❺❪❇P❈❂✙❄❢✼▼✾❖❂✮❉❢✾❇❣✝❤❃P❈❪❥✾❺■✹▲▼✷❖■●P❈❂✿❩✂■✹▲▼✷❥➈
◗❧❀❃❄❢✼❲■✹▲▼✷❖❏✶✉✏P❈❂✝❩❲❊❢❚✮❚❱❂✜✉❝❭✝❂❷❄❢❀❃❤✕◗➎✾✧✾❇♥❈✾❖❂✣❄❢❭✿P❈❤❃❤❃❴③❆❈✾❇❄♠❄❅❚ ➏ ✷✹❍❋❏✣➐❱✸q➒❡✾ ❼ ❂✿❀✐❊❢✼③❳✶❴
❉❢✾❇❣✌❤❃P❈❪❇❀❃❂▼❆✂✷♠P❈❂✿❩❯❏t◗❧❀❙❄❢✼③■✶✸
⑩ ❳✳❶✻➓❞❊❢❀❃❂▼❆✮❄❢✼▼✾✖❊❢P❈❜✖✾✖❪❥❚❱❂✿❊❅❄❅❉❋❭✿❪❥❄❢❀❃❚❱❂❲P❈❊q❀❃❂ ⑩ P❷❶❘✉✏◗➎✾●❪❇P❈❂ ❼ ❂✿❩❫P✙❊❅✾❖❻✣❭✿✾❖❂✿❪❥✾t❚❈⑤
❉❢✾❇❣✌❤❃P❈❪❥✾❖❜t✾❖❂✣❄❢❊❧❄❢✼✿P✹❄✻❉❢✾❖❩✿❭✿❪❥✾❖❊ ➏♦➔ ❍ ➔q→ ✷✹❍❇❑❇❑❇❑♦❍❋➣♦➐●❄❅❚♠↔❢❭✿❊❅❄ ➏♦➔q→ ✷✰➐❱✸✻↕s❀❃❜✖❀❃❤✐P✹❉❋❤❙❴❈✉
❪❇P❈❂✂P❈❤❃❊❢❚ ❼ ❂✿❩✙Pt❊❅✾❖❻✣❭✿✾❖❂✿❪❥✾⑦❚❈⑤✕❉❢✾❇❣✌❤❃P❈❪❥✾❖❜t✾❖❂✣❄❢❊➙❄❢✼✿P✹❄✱❉❢✾❖❩✿❭✿❪❥✾❖❊ ➏♦➔ ❍ ➔❞→ ✷✹❍❇❑❇❑❇❑♦❍❋➣♦➐
❄❅❚⑦↔❢❭✿❊❅❄ ➏ ➣➎➛✥✷✰➐❱✸
➜ ❂⑦❣✝P✹❉❢❄❢❀❃❪❇❭✝❤❃P✹❉❖✉❖◗➎✾❹❪❇P❈❂ ❼ ❂✝❩♠❊❅✾❖❻✣❭✿✾❖❂✿❪❥✾❖❊✜❚❈⑤✶❉❢✾❇❣✝❤❃P❈❪❥✾❖❜✖✾❖❂❷❄❢❊✏❄❢✼✿P✹❄☛❉❢✾❖❩✿❭✿❪❥✾ ➏ ✷✹❍❋■✶❍❇❑❇❑❇❑✹❍❋➝❈➝❈➝✣➐
❄❅❚❁↔❢❭✿❊❅❄ ➏ ➝❈➝❈➞✣➐❱✉✏P❈❂✿❩❲❄❢✼✿P✹❄⑦❉❢✾❖❩✿❭✿❪❥✾ ➏ ✷❇▲❈▲▼✷✹❍❖✷❇▲❈▲❱■✶❍❇❑❇❑❇❑❱❍❋■✹▲▼✷❖◆✣➐➇❄❅❚❁↔❢❭✿❊❅❄ ➏ ✷❇▲❈▲❱■✣➐❱✸
✺✽✼✝❀❃❊♠❤❙✾❖P❖♥❈✾❖❊●❭✿❊q◗❧❀❙❄❢✼ ➏ ➝❈➝❈➞✶❍❖✷❇▲❈▲❈▲s❍❖✷❇▲❈▲❱■✣➐❱✸✈➒❡✾✮❪❇P❈❂➊❉❢✾❇❣✝❤✐P❈❪❥✾❨➝❈➝❈➞❯P❈❂✝❩➌✷❇▲❈▲❱■
◗❧❀❃❄❢✼❲P✂❊❢✾❖❪❥❚❱❂✿❩✈❪❥❚❈❣✶❴❲❚❈⑤❞✷❇▲❈▲❈▲s✉✯P❈❂✝❩❡❄❢✼▼✾❖❂❫❉❢✾❇❣✝❤✐P❈❪❥✾t❄❢✼▼✾t❄✗◗➎❚❯❪❥❚❈❣✝❀❙✾❖❊⑦❚❈⑤⑦✷❇▲❈▲❈▲
◗❧❀❃❄❢✼✂Pt❊❢❀❃❂▼❆❱❤❙✾♠❪❥❚❈❣✶❴✂❄❅❚➇❪❥❚❱❜t❣✝❤❃✾❇❄❅✾❺❄❢✼▼✾❺❪❥❚❱❂✿❊❅❄❅❉❘❭✿❪❥❄❢❀❙❚❱❂☛✸
■✶✸⑦➟➎❚❱❂✝❊❢❀❃❩▼✾❇❉✂❄❢✼▼✾➠⑤➡❚❱❤❃❤❃❚♦◗❧❀❃❂✿❆✥❊❅❴✶❊❢❄❅✾❖❜➢❚❈⑤✂✷❇▲✥✾❖❻✶❭✿P✹❄❢❀❙❚❱❂✿❊④❀❃❂➤✷❇▲➌❉❢✾❖P❈❤♠♥✹P✹❉❋❀❃P✹❳✝❤❙✾❖❊
➥✶➦ ❍❇❑❇❑❇❑♦❍ ➥✶➦✗➧✹➨
◆
⑩✒➲ ➫ ✷✹❍❇❑❇❑❇❑♦❍❖✷❇▲❷❶❘❑
➥✹➩❯➫ ✷ → ➥ ➭➦ → ➥ ➭ →➋➥✣➩➭➯❇➯❇➯✰→ ➥ ➭➦✗➧
➭
♣ ❀❃❂✿❩✂P❈❤✐❤⑧✷❇▲✰➍✟❄❢❭▼❣✝❤❙✾❖❊ ⑩ ➥ ➦ ❍ ➥ ➭ ❍❇❑❇❑❇❑♦❍ ➥ ➦✗➧ ❶❛❄❢✼✿P✹❄✻P✹❉❢✾q❊❅❚❱❤❃❭✿❄❢❀❙❚❱❂✿❊➙❚❈⑤✯❄❢✼✿❀❃❊✽❊❅❴s❊❅❄❅✾❖❜✂✸
✷
© Canadian Mathematical Society 2016
Draft Solutions - 2016 CMO
❽✌❾❈❿➁➀▼➂✎➃➄❾✹➅✌➆
♣ ❈❚ ❉❁P❨❣✌P✹❉❢❄❢❀❃❪❇❭✿❤❃P✹❉♠❊❅❚❱❤✐❭▼❄❢❀❙❚❱❂ ⑩ ➥ ➦ ❍ ➥ ➭ ❍❇❑❇❑❇❑☎❍ ➥ ➦✗➧ ❶❘✉✕❤❙✾❇❄❺➳ ➫➤➥✣➭➦ → ➥✣➭➭ →➵➯❇➯❇➯❷→ ➥✣➭➦✗➧ ✸
✺✽✼✿✾❖❂
◆
➥ ➩ ➫ ✷ → ➥➳ ➩➭ ➸ ◆ ➥ ➩➭ ➛✈➳ ➥ ➩ → ➳ ➫ ▲s❑
➺ ✾❇❄ ➔ P❈❂✝❩✙➣✽❳❝✾❧❄❢✼▼✾⑦❉❢❚✶❚❈❄❢❊➎❚❈⑤✜❄❢✼▼✾♠❻✣❭✝P❈❩▼❉❋P✹❄❢❀❃❪❞◆✰➻ ➭ ➛❯➳♦➻ → ➳ ➫ ▲s✉✿❊❢❚❁⑤✒❚❈❉➙✾❖P❈❪❋✼
➲ ✉ ➥✹➩✱➫ ➔ ❚❈❉ ➥✹➩✱➫ ➣☎✸✮➒❡✾✮P❈❤✐❊❅❚❯✼✿P♦♥❈✾ ➔ ➣ ➫ ➳✰➼✹◆ ⑩ ❳✣❴❫➽⑦❀❙✾❇❄❢Ps➾➚❊q⑤✒❚❈❉❋❜✧❭✿❤❃Ps✉✕⑤➡❚❈❉
✾❥r▼P❈❜t❣✝❤❙✾☎❶❘✸
➜ ⑤✯P❈❤✐❤❝❄❢✼▼✾ ➥ ➩ P✹❉❋✾❺✾❖❻✣❭✝P❈❤➄✉▼❄❢✼▼✾❖❂
◆ ➞
➥✰➩✜➫ ✷ → ✷❇▲ ➫ ①
⑤✒❚❈❉✽P❈❤❃❤ ➲ ✸❛➪❞❄❢✼✿✾❇❉❢◗❧❀❃❊❅✾❈✉▼❤❙✾❇❄❧① →✇➶ ❚❈⑤✯❄❢✼▼✾ ➥✹➩ ❳❝✾ ➔ ✉✿P❈❂✝❩③❤❙✾❇❄✽①⑦➛ ➶ ❚❈⑤✕❄❢✼▼✾ ➥✹➩ ❳❝✾
➣☎✉▼◗❧✼▼✾❇❉❋✾❺▲✖➹ ➶✙➘ ➈▼✸❹✺❧✼▼✾❖❂✂❳✶❴❨❄❢✼▼✾❺⑥⑦➴③➍➬➷♠➴➮❀❃❂▼✾❖❻✶❭✿P❈❤❃❀❙❄❵❴❈✉
◆ ➔ ➣ ➫ ➳ ➫ ⑩ ① →➱➶ ❶ ➔ ➭ →➋⑩ ①♠➛ ➶ ❶✃➣ ➭❺❐ ■ ➔ ➣✹❒ ■❈①♠➛ ➶ ➭ ❑
♣ ❉❋❚❱❜❮❄❢✼▼✾✂❆❱❀❙♥❈✾❖❂✇✾❖❻✶❭✿P✹❄❢❀❙❚❱❂✿❊❖✉ ➥ ➩ ❐ ✷✙⑤➡❚❈❉❨P❈❤❃❤ ➲ ✉➎❊❢❚ ➔ P❈❂✿❩✥➣✮P✹❉❢✾✂❣❝❚❱❊❢❀❙❄❢❀❃♥❈✾❈✸
✺✽✼✿✾❖❂➱❰ ■❈①♠➛ ➶ ➭ ➘ ❏ ➸ ■❈①❨➛ ➶ ➭ ➘ ➝ ➸ ➶ ➭ ❐ ✷❖◆ ➸ ➶ ➫ ➈▼✸ÐÏ✻✾❖❂✿❪❥✾❈✉
◆ ➔ ➣ ➫ ➝ ➔ ➭ → ➣ ➭ ➸ ⑩ ➣➙➛➠❏ ➔ ❶ ➭❧➫ ▲ ➸ ➣ ➫ ❏ ➔ ✸
⑥❞❩✿❩✿❀❃❂✿❆✧P❈❤❃❤✜❆❱❀❙♥❈✾❖❂✂❄❅✾❖❂③✾❖❻✶❭✿P✹❄❢❀❙❚❱❂✝❊❇✉▼◗❛✾q❆❈✾❇❄
➥ ➦ → ➥ ➭ →➉➯❇➯❇➯✹→ ➥ ➦✗➧ ➫ ✷❖◆✶❑
Ñ ❭▼❄ ➥ ➦ → ➥ ➭ →➤➯❇➯❇➯▼→ ➥ ➦✗➧ ➫ ➝ ➔✮→ ➣ ➫ ✷❖■ ➔ ✉✽❊❢❚ ➔ ➫ ✷❖◆❱➼✶✷❖■ ➫ ➈✣➼✹❏✈P❈❂✿❩
➣ ➫ ➈▼✸✩✺❧✼▼✾❇❉❢✾❇⑤✒❚❈❉❢✾❈✉❧❄❢✼▼✾❡❊❅❚❱❤✐❭▼❄❢❀❙❚❱❂✿❊✙P✹❉❢✾ ⑩ ➞❱➼✹①✶❍❋➞❱➼✹①✶❍❇❑❇❑❇❑❈❍❋➞❱➼✹①❱❶✙P❈❂✿❩➋P❈❤❃❤❞❄❅✾❖❂
❣❝✾❇❉❋❜✧❭▼❄❢P✹❄❢❀❙❚❱❂✿❊➙❚❈⑤ ⑩ ➈✣➼✹❏✶❍❅➈✣➼✹❏✶❍❇❑❇❑❇❑❱❍❅➈✣➼✹❏✶❍❅➈✣❶❘✸
❏✶✸ ♣ ❃❀ ❂✿❩✧P❈❤❃❤✣❣❝❚❱❤❙❴✶❂✿❚❱❜✖❀❃P❈❤❃❊✕Ò ⑩ ➻❝❶✕◗❧❀❃❄❢✼❁❀❃❂✣❄❅✾❇❆❈✾❇❉✵❪❥❚✶✾❇Ó➇❪❇❀❙✾❖❂✣❄❢❊⑧❊❢❭✿❪❋✼●❄❢✼✝P✹❄ÔÒ ⑩ Ò ⑩✒Õ ❶ →
Õ ❶➎❃❀ ❊✱P✖❣✿❉❋❀❃❜t✾♠❂✶❭✿❜❁❳✳✾❇❉✱⑤✒❚❈❉❧❀❃❂ ❼ ❂✿❀❙❄❅✾❖❤❙❴✖❜➇P❈❂❷❴✂❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊ Õ ✸
Ö ➅✿×❘Ø❛Ù❥Ú❇➆ Ò ⑩✒Õ ❶ ➫ÜÛ ◗❧✼▼✾❇❉❢✾ Û ❀✐❊❺P✂❣✿❉❋❀✐❜t✾t❂✣❭✿❜❁❳❝✾❇❉❺P❈❂✿❩❡Ò ⑩✒Õ ❶ ➫ ➛❧■ Õ③→ ➣
◗❧✼✿✾❇❉❢✾❺➣⑦❀❃❊➙❚✶❩✝❩☛✸
❽✌❾❈❿➁➀▼➂✎➃➄❾✹➅ ➨➉Ý ❚❈❄❅✾➠❄❢✼✿P✹❄❯❀❙⑤●Ò ⑩✒Õ ❶ ➫ ▲➱❄❢✼✿✾❖❂ÞÒ ⑩ Ò ⑩✒Õ ❶ →ßÕ ❶ ➫ Ò ⑩✒Õ ❶ ➫ ▲
◗❧✼✝❀❃❪❋✼✈❀✐❊❺❂▼❚❈à❄q❣✝❉❋❀❃❜t✾❈✸ ➺à☎á ✾❇❄❁Ò ⑩ ➻❝❶⑦❳❝✾➇P④❩▼✾❇❆❈❉❢✾❇✾ ➶ ❣❝❚❱❤❙❴s❂▼❚❱❜✖❀❃P❈❤✯❚❈⑤✱❄❢✼▼✾❨⑤➡❚❈❉❋❜
Ò ⑩ ➻☛❶ ➫ ➔✶à ➻ →✇➔✶à☎á ➦ ➻ ➦ →➉➯❇➯❇➯✰→✇➔ ➧ P❈❂✿❩✂❂▼❚❈❄❅✾❺❄❢✼✝P✹❄❧❀❙⑤ÔÒ ⑩✒Õ ❶❞➫ â ▲t❄❢✼▼✾❖❂
Ò ⑩ Ò ⑩✒Õ ❶ →ÐÕ ❶✯➛✈Ò ⑩✒Õ ❶ ➫
➔✣àsã❙⑩ Ò ⑩✒Õ ❶ →ÐÕ ❶ à ➛ Õ à♦ä →✇➔✣à☎á ➦ ã❙⑩ Ò ⑩✒Õ ❶ →✇Õ ❶ à♦á ➦ ➛ Õ à☎á ➦ ä →å➯❇➯❇➯✹→➱➔ ➦ Ò ⑩✒Õ ❶
■
© Canadian Mathematical Society 2016
Draft Solutions - 2016 CMO
◗❧✼✝❀❃❪❋✼❨❀❃❊❹❩✿❀❃♥✶❀❃❊❋❀❙❳✝❤❙✾➙❳✶❴ ⑩ Ò ⑩✒Õ ❶ →④Õ ❶✝➛ Õ ➫ Ò ⑩✒Õ ❶❘✸⑧✺✽✼▼✾❇❉❋✾❇⑤➡❚❈❉❢✾⑦❀❙⑤☛Ò ⑩ Ò ⑩✒Õ ❶ →④Õ ❶
❀❃❊✵❣✿❉❋❀❃❜t✾➙❄❢✼▼✾❖❂➇✾❖❀❙❄❢✼▼✾❇❉❹Ò ⑩✒Õ ❶ ➫Þæ ✷✱❚❈❉➎Ò ⑩ Ò ⑩✒Õ ❶ →✂Õ ❶ ➫Þæ Ò ⑩✒Õ ❶ ➫ÐÛ ⑤✒❚❈❉❛❊❅❚❱❜t✾
❣✿❉❘❀❃❜t✾✵❂✶❭✿❜q❳❝✾❇❉ Û ✸⑧↕▼❀❃❂✿❪❥✾❹Ò ⑩ ➻☛❶✜❀✐❊✏P✻❣✳❚❱❤❃❴✶❂▼❚❱❜➇❀❃P❈❤➄✉♦❀❙❄✕⑤➡❚❱❤✐❤❙❚♦◗❧❊✕❄❢✼✿P✹❄✯Ò ⑩✒Õ ❶ ➫Þæ ✷
⑤✒❚❈❉Ô❚❱❂✿❤❙❴ ❼ ❂✿❀❃❄❅✾❖❤❙❴❺❜✖P❈❂✣❴❁❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊ Õ ✸✯✺❧✼▼✾❇❉❢✾❇⑤✒❚❈❉❢✾➙✾❖❀❙❄❢✼▼✾❇❉⑧Ò ⑩✒Õ ❶ ➫ Ò ⑩ Ò ⑩✒Õ ❶ →●Õ ❶
⑤✒❚❈❉❺❀❃❂ ❼ ❂✝❀❙❄❅✾❖❤❙❴✂❜✖P❈❂✣❴④❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊ Õ ❚❈❉❺Ò ⑩✒Õ ❶ ➫ ➛✽Ò ⑩ Ò ⑩✒Õ ❶ →✥Õ ❶❧⑤➡❚❈❉❺❀❃❂ ❼ ❂✿❀❙❄❅✾❖❤❙❴
❜✖P❈❂✣❴q❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊ Õ ✸Ô↕▼❭▼❣✿❣❝❚❱❊❅✾✵❄❢✼✿P✹❄⑧Ò ⑩✒Õ ❶ ➫ Ò ⑩ Ò ⑩✒Õ ❶ →✧Õ ❶✏⑤✒❚❈❉✯❀❃❂ ❼ ❂✿❀❙❄❅✾❖❤❙❴⑦❜✖P❈❂✣❴
❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊ Õ ✸✵✺❧✼✿❀❃❊✱❀❃❜t❣✌❤❃❀❙✾❖❊➎❄❢✼✿P✹❄✽❄❢✼▼✾♠❣❝❚❱❤❙❴s❂▼❚❱❜✖❀✐P❈❤✳Ò ⑩ Ò ⑩ ➻❝❶ → ➻❝❶✯➛✈Ò ⑩ ➻☛❶➙✼✿P❈❊
❀❃❂ ❼ ❂✿❀❙❄❅✾❖❤❙❴❁❜✖P❈❂❷❴t❉❋❚✣❚❈❄❢❊➎P❈❂✿❩❨❄❢✼✣❭✿❊❛❀❃❊❹❀✐❩▼✾❖❂❷❄❢❀✐❪❇P❈❤❃❤❙❴✧ç❇✾❇❉❢❚▼✸❹✺✽✼▼✾❇❉❋✾❇⑤➡❚❈❉❢✾✻Ò ⑩ Ò ⑩ ➻❝❶ →
➻☛❶ ➫ Ò ⑩ ➻❝❶➎✼✿❚❱❤❃❩✿❊➎❀❃❩▼✾❖❂✣❄❢❀❃❪❇P❈❤❃❤❙❴❈✸ Ý ❚♦◗Þ❂▼❚❈❄❅✾♠❄❢✼✿P✹❄✱❀❙⑤ ➶ ❐ ■❁❄❢✼▼✾❖❂✂Ò ⑩ Ò ⑩ ➻☛❶ → ➻❝❶
✼✿P❈❊✯❩▼✾❇❆❈❉❢✾❇✾ ➶ ➭ ◗✻✼✿❀❃❤❙✾✵Ò ⑩ ➻☛❶✜✼✝P❈❊✯❩✿✾❇❆❈❉❢✾❇✾ ➶ ✉✹◗❧✼✝❀❃❪❋✼❁❀❃❊✕❂✿❚❈❄✕❣❝❚❱❊❢❊❢❀❃❳✝❤❙✾❈✸✜✺✽✼▼✾❇❉❋✾❇⑤➡❚❈❉❢✾
Ò ⑩ ➻☛❶✽❀✐❊❧P✹❄⑦❜t❚❱❊❅❄⑦❤❃❀❃❂▼✾❖P✹❉✻◗❧❀❙❄❢✼❯Ò ⑩ ➻❝❶ ➫ ➔ ➻ → ➣⑦⑤➡❚❈❉⑦❊❅❚❱❜t✾●❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊ ➔ P❈❂✿❩❫➣♦✸
Ý ❚♦◗è❂▼❚❈❄❅✾❺❄❢✼✿P✹❄
Ò ⑩ Ò ⑩ ➻☛❶ → ➻☛❶ ➫ ➔✳⑩✎➔❺→ ✷♦❶❵➻ →✇➔ ➣ → ➣
P❈❂✿❩➠❄❢✼✣❭✝❊ ➔ ➫ ➔❝⑩✎➔●→ ✷♦❶qP❈❂✿❩ ➔ ➣ → ➣ ➫ ➣☎✸ ➜ ❄q⑤✒❚❱❤❃❤❙❚☎◗❧❊❺❄❢✼✿P✹❄ ➔ ➫ ▲③◗✻✼✿❀❃❪❋✼
❤❙✾❖P❈❩✝❊q❄❅❚❲❄❢✼▼✾✙❊❅❚❱❤❃❭▼❄❢❀❙❚❱❂ÐÒ ⑩✒Õ ❶ ➫ßÛ ◗❧✼✿✾❇❉❢✾ Û ❀❃❊❁P④❣✝❉❋❀❃❜t✾➇❂✶❭✿❜❁❳❝✾❇❉❖✸ Ñ ❴✈❄❢✼▼✾
❊❢P❈❜✖✾●P✹❉❋❆❱❭✿❜t✾❖❂✣❄❺❀❙⑤❛Ò ⑩✒Õ ❶ ➫ ➛❧Ò ⑩ Ò ⑩✒Õ ❶ →➱Õ ❶✻⑤✒❚❈❉❺❀❃❂ ❼ ❂✝❀❙❄❅✾❖❤❙❴✂❜✖P❈❂✣❴❲❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊
Õ ❄❢✼▼✾❖❂➋Ò ⑩ ➻❝❶ ➫ ➛✽Ò ⑩ Ò ⑩ ➻❝❶ → ➻❝❶❨✼▼❚❱❤✐❩✿❊❨❀✐❩▼✾❖❂❷❄❢❀✐❪❇P❈❤❃❤❙❴➱P❈❂✿❩➉Ò ⑩ ➻❝❶❨❀❃❊✙❤❃❀❃❂✿✾❖P✹❉
◗❧❀❃❄❢✼❡Ò ⑩ ➻❝❶ ➫ ➔ ➻ → ➣☎✸ ➜ ❂❡❄❢✼✿❀✐❊❺❪❇P❈❊❅✾❨❀❃❄❺⑤➡❚❱❤✐❤❙❚♦◗❧❊❺❄❢✼✿P✹❄ ➔ ➫ ➛ ➔✳⑩✎➔●→ ✷♦❶♠P❈❂✿❩
➔ ➣ → ➣ ➫ ➛❧➣☎✸✵✺✽✼✿❀❃❊❛❀❃❜t❣✝❤❃❀❃✾❖❊Ô❄❢✼✿P✹❄➙✾❖❀❙❄❢✼▼✾❇❉ ➔ ➫ ▲❁❚❈❉ ➔ ➫ ➛❧■✶✸ ➜ ⑤ ➔ ➫ ➛✻■♠❄❢✼▼✾❖❂
Ò ⑩✒Õ ❶ ➫ ➛❧■ Õ③→ ➣❁◗❧✼✝❀❃❪❋✼❫❀❃❊⑦❣✿❉❘❀❃❜t✾●⑤➡❚❈❉❁❊❅❚❱❜t✾✖❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊ Õ ❚❱❂✿❤❃❴④❀❙⑤➙➣❁❀✐❊⑦❚✶❩✿❩✜✸
Ý ❚❈❄❅✾⑦❄❢✼✿P✹❄✱❀❃❂✖❄❢✼✿❀✐❊❛❪❇P❈❊❅✾♠Ò ⑩ Ò ⑩✒Õ ❶ →❫Õ ❶ ➫ ■ Õ ➛④➣✱◗❧✼✿❀✐❪❋✼✮❀❃❊❛❀✐❂✿❩▼✾❇✾❖❩❨❣✿❉❋❀❃❜t✾❧⑤➡❚❈❉
❀❃❂ ❼ ❂✿❀❙❄❅✾❖❤❙❴✖❜➇P❈❂❷❴✙❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊ Õ P❈❊✽❤❃❚❱❂▼❆✖P❈❊✻➣❞❀❃❊✱❚s❩✿❩☛✸
é
➈▼✸ ➺ ❖P ♥✹P❈❜✖P❈❂③♥❈✾❇❉❋❊❋❭✿❊✱❄❢✼▼✾ ♣ ❤❙✾❖P ✸ ➺ ✾❇❄❺ê❁✉✜ë❨✉✜P❈❂✝❩❲ìí❳❝✾✧❣❝❚❱❊❢❀❙❄❢❀❙♥❈✾●❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊❇✉✏P❈❂✿❩
P❈❊❢❊❋❭✿❜t✾●êî➹Þëï➹ð■✹ê❁✸q⑥➤ñ✝✾❖P✂❀❃❊⑦P✹❄❞❄❢✼✿✾●❂✶❭✿❜❁❳❝✾❇❉⑦▲✮❚❱❂❫❄❢✼▼✾t❂✶❭✿❜❁❳✳✾❇❉⑦❤❃❀❃❂✿✾❈✸
✺✽✼✿✾③ñ✿✾❖P➠❪❇P❈❂➉❜t❚☎♥❈✾④❳✶❴④↔❅❭✿❜✖❣✝❀❃❂▼❆❡❄❅❚➊❄❢✼▼✾❯❉❋❀❃❆❱✼❷❄✖❳✶❴✥ê❦❚❈❉➇❳✶❴✥ë✮✸ Ñ ✾❇⑤✒❚❈❉❢✾
❄❢✼▼✾❧ñ✿✾❖Pq❊❅❄❢P✹❉❋❄❢❊❝↔❢❭✿❜t❣✝❀❃❂✿❆▼✉ ➺ P❖♥✹P❈❜✖P❈❂✮❪❋✼▼❚✶❚❱❊❅✾❖❊ ❼ ❂✿❀❙❄❅✾❖❤❃❴✧❜✖P❈❂✣❴t❀❃❂✣❄❅✾❇❉❢♥✰P❈❤✐❊ ➏❖ò➋→
✷✹❍ ò➵→ ■✶❍❇❑❇❑❇❑♦❍ ò➵→ êq➐⑦❪❥❚❱❂✝❊❢❀❃❊❅❄❢❀❃❂✿❆q❚❈⑤✜êÞ❪❥❚❱❂✿❊❢✾❖❪❇❭▼❄❢❀❙♥❈✾❞❣✳❚❱❊❋❀❙❄❢❀❙♥❈✾❧❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊❇✉▼P❈❂✿❩
❣✝❤✐P❈❪❥✾❖❊♠❤❃P❖♥✹P③P✹❄❺P❈❤❃❤Ô❚❈⑤➎❄❢✼▼✾✖❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊q❀✐❂④❄❢✼✿✾✖❀❃❂❷❄❅✾❇❉❋♥✰P❈❤❃❊❖✸●✺❧✼▼✾t❀❃❂❷❄❅✾❇❉❋♥✰P❈❤❃❊♠❜✧❭✿❊❅❄
❳❝✾♠❪❋✼▼❚❱❊❢✾❖❂❯❊❅❚✖❄❢✼✝P✹❄ ➨
⑩ ➃ ❶➙P❈❂❷❴✮❄✗◗➎❚❨❩✝❀❃❊❅❄❢❀❃❂✿❪❥❄✱❀❃❂✣❄❅✾❇❉❢♥✹P❈❤❃❊✱P✹❉❢✾q❩✝❀❃❊✒↔❅❚❱❀❃❂❷❄✱P❈❂✿❩❯❂▼❚❈❄✽P❈❩✰↔❢P❈❪❥✾❖❂❷❄❇ó
⑩ ➃✒➃ ❶✕❄❢✼▼✾❇❉❋✾✽P✹❉❢✾❧P✹❄✵❤❙✾❖P❈❊❅❄✵ìÞ❣✳❚❱❊❋❀❙❄❢❀❙♥❈✾✱❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊⑧◗❧❀❙❄❢✼t❂✿❚❺❤❃P❖♥✹P⑦❳❝✾❇❄✗◗➎✾❇✾❖❂✖P❈❂❷❴●❄❵◗❛❚
❀❃❂✣❄❅✾❇❉❢♥✹P❈❤❃❊❇ó▼P❈❂✿❩
⑩ ➃✒➃➡➃ ❶➙❂▼❚✖❤❃P♦♥✰P➇❀❃❊➙❣✝❤❃P❈❪❥✾❖❩③P✹❄✻P❈❂❷❴✂❀❃❂✣❄❅✾❇❆❈✾❇❉❧❤❙✾❖❊❢❊✽❄❢✼✿P❈❂③ìt✸
❸❹❉❢❚♦♥❈✾③❄❢✼✝P✹❄✖❄❢✼▼✾❯❊❢❜➇P❈❤❃❤❙✾❖❊❅❄tìï⑤✒❚❈❉✖◗❧✼✿❀✐❪❋✼✇❄❢✼▼✾③ñ✝✾❖P✈❪❇P❈❂❲↔❢❭✿❜t❣➱❚♦♥❈✾❇❉✙P❈❤❃❤➙❄❢✼▼✾
❀❃❂✣❄❅✾❇❉❢♥✹P❈❤❃❊✙P❈❂✿❩ÜP❖♥❈❚❱❀✐❩ÞP❈❤❃❤❞❄❢✼▼✾✈❤❃P♦♥✰Ps✉❞❉❢✾❇❆❱P✹❉❋❩✿❤❃✾❖❊❢❊✙❚❈⑤❁◗✻✼✿P✹❄ ➺ P❖♥✹P❈❜✖P❈❂Þ❩✿❚✣✾❖❊❇✉
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© Canadian Mathematical Society 2016
Draft Solutions - 2016 CMO
❀❃❊❫ì ➫ ⑩✒Õ ➛í✷♦❶➬ê → ë✮✉●◗❧✼▼✾❇❉❢✾ Õ ❀✐❊④❄❢✼✿✾➱❣✳❚❱❊❋❀❙❄❢❀❙♥❈✾✇❀❃❂❷❄❅✾❇❆❈✾❇❉✈❊❋❭✿❪❋✼è❄❢✼✿P✹❄
ê
ê
Õ➇→ ✷ ➘ ëè➛❡êß➹ Õ ✸
❽✌❾❈❿➁➀▼➂✎➃➄❾✹➅✌➆ ➺ ✾❇❄❧ë ➫ ê →✥ô ◗❧✼▼✾❇❉❢✾❺ê♠➼ ⑩✒Õ✮→ ✷♦❶ ➘➉ô ➹✥ê♠➼ Õ ✸
♣ ❀❙❉❋❊❢❄❇✉✿✼▼✾❇❉❢✾❺❀❃❊✱P❈❂③❀✐❂▼⑤➡❚❈❉❘❜✖P❈❤☛❊❅⑨❈✾❇❄❢❪❋✼④❚❈⑤✯❄❢✼▼✾❺❣✿❉❢❚✶❚❈⑤➬✸
➺ P❖♥✹P❈❜✖P❈❂☛➾➚❊✱❊❅❄❅❉❋P✹❄❅✾❇❆❈❴ ➨ ➓❞❊❅✾♠❚❱❂✿❤❙❴✖❊❋P✹⑤➡✾♠❀❃ä ❂✣❄❅✾❇❉❢♥✹P❈❤❃❊➎◗❧❀❙❄❢✼ Õ ê →➊ô ➛➠✷♠❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊❇✸
✺✽✼✿✾❁ñ✿✾❖P❨◗❧❀❃❤✐❤✏❊❅❄❢P✹❉❢❄♠P✹❄♠❣❝❚❱❊❢❀❙❄❢❀❃❚❱❂ ã ✷✹❍ ô ⑤✒❉❢❚❱❜➑❄❢✼▼✾t❤❙✾❇⑤✒❄❇✉✳◗✻✼✿❀❃❪❋✼❲❣✝❭✿❄❢❊✻✼✿❀✐❜✫P✹❄
❣❝❚❱❊❢❀❙❄❢❀❃❚❱❂ ã Õ ê❁❍ Õ ê →åô ➛➉✷ ä ⑤✒❉❢❚❱❜✫❄❢✼▼✾●❉❋❀❙❆❱✼✣❄❇✸♠⑥❧⑤✒❄❅✾❇❉ Õ ➛➉✷✽↔❢❭✿❜t❣✌❊❇✉✳✼✿✾❁◗❧❀✐❤❃❤
❊❅❄❢❀✐❤❃❤☛✼✿P❖♥❈✾ Õ ê➉➛ ⑩✒Õ ➛✥✷♦❶ ⑩ ê →åô ❶ ➫ ê➋➛ ⑩✒Õ ➛å✷♦❶ ôöõ❦ô ❩✿❀❃❊❅❄❢P❈❂✝❪❥✾❺❄❅❚➇❆❈❚▼✉
◗❧✼✝❀❃❪❋✼t❀❃❊⑧❂✿❚❈❄✵✾❖❂▼❚❱❭▼❆❱✼t⑤✒❚❈❉❹P⑦❳✝❀❃❆➎↔❅❭✿❜✖❣t❄❅❚q❪❇❤❙✾❖P✹❉✵❄❢✼▼✾✻❤❃P❖♥✹Ps✸⑧✺✽✼✣❭✝❊❇✉❷✼▼✾✽❜✧❭✿❊❅❄✵❩▼❚
P✹❄❞❤❙✾❖P❈❊❅❄ Õ ↔❅❭✿❜✖❣✝❊✽❀❃❂✂❄❢✼✿✾❁❊❢P✹⑤✒✾❁❀❃❂✣❄❅✾❇❉❢♥✹P❈❤➄✉✝❳✌❭▼❄✽❄❢✼✿P✹❄❇➾➚❊✻❣❝❚❱❊❢❊❢❀❙❳✝❤❃✾❞❚❱❂✝❤❙❴✙◗❧❀❙❄❢✼③P❈❤❃❤
❊❢❜➇P❈❤❃❤s↔❢❭✿❜t❣✝❊❖✉✌P❈❂✿❩❲⑤✎❭▼❉❢❄❢✼▼✾❇❉❋❜✖❚❈❉❢✾❁❀❃❊❞❀❃❜t❣❝❚❱❊❢❊❋❀❙❳✝❤❙✾♠❀❙⑤⑧❄❢✼▼✾●❊❅❄❢P✹❉❢❄❢❀✐❂▼❆❨❣❝❚❱❊❢❀❃❄❢❀❙❚❱❂❯❀❃❊
ô ✸✵✺✽✼✿❀❃❊➙❆❱❀❙♥❈✾❖❊✻✼✿❀❃❜î❊❅❄❢P✹❉❋❄❢❀❃❂▼❆t❣❝❚❱❊❢❀❙❄❢❀❙❚❱❂❡✷q✼✿❀❙❆❱✼▼✾❇❉✽❀❃❂✙❄❢✼✿✾q❂▼✾❥rs❄❧❊❢P✹⑤✒✾q❀❃❂✣❄❅✾❇❉❢♥✰P❈❤✟✉
❊❅❚✖❊❢❚✣❚❱❂▼✾❇❉❧❚❈❉✻❤❃P✹❄❅✾❇❉✽❄❢✼✿✾❺ñ✿✾❖Pt❀❃❊✱❆❈❚❱❀✐❂▼❆t❄❅❚➇✼✿❀❙❄✱❄❢✼✿✾❺❤❃P❖♥✹Ps✸
♣ ❤❙✾❖Ps➾➚❊♠❊❅❄❅❉❘P✹❄❅✾❇❆❈❴ ➨ ✺✽✼▼✾●ñ✿✾❖Pq↔❢❭✿❊❅❄⑦❩▼❚✣✾❖❊❞❚❱❂▼✾t❀❃❂✣❄❅✾❇❉❢♥✹P❈❤ÔP✹❄⑦P✮❄❢❀❃❜✖✾❈✸ ➜ ⑤❹❄❢✼▼✾t❊❢P✹⑤➡✾
❀❃❂✣❄❅✾❇❉❢♥✹P❈❤▼✼✿P❈❊✵P✹❄➎❤❙✾❖P❈❊❅❄ Õ ê →③ô ❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊❛❀❃❂t❀❙❄❇✉❱❄❢✼▼✾❧ñ✿✾❖P❺✼✿P❈❊✵❩✝❀❃❊❅❄❢P❈❂✿❪❥✾✽÷ õ➱Õ ê
❄❅❚✙❆❈❚✙❄❅❚✙❄❢✼▼✾✧❂✿✾❥r✶❄♠❤❃P♦♥✰P✮◗❧✼▼✾❖❂❲❀❙❄⑦❊❅❄❢P✹❉❢❄❢❊❇✸❺ø❧✾❇❣❝✾❖P✹❄❅✾❖❩✿❤❙❴❯❩▼❚✮❳✝❀❙❆❺↔❅❭✝❜t❣✝❊❞❭✿❂❷❄❢❀✐❤
÷➊❀❃❊●❳❝✾❇❄✗◗➎✾❇✾❖❂Þ✷③P❈❂✿❩ ô ❜t❚✶❩✱ê❁✉➎❄❢✼▼✾❖❂✥❊❢❜✖P❈❤✐❤✜↔❢❭✿❜t❣✌❊✧❭✝❂❷❄❢❀❃❤➙❄❢✼✿✾✂❉❢✾❖❜✖P❈❀❃❂✿❀✐❂▼❆
❩✿❀✐❊❅❄❢P❈❂✿❪❥✾✙❀❃❊✧❳✳✾❇❄❵◗❛✾❇✾❖❂➵✷✮P❈❂✝❩ ô ✉✵❄❢✼▼✾❖❂✇P ❼ ❂✝P❈❤❹❳✝❀❃❆➇↔❢❭✿❜t❣✏✸✈✺✽✼✿❀❃❊❁◗❛❚❈❉❋⑨✶❊tP❈❊
❤❙❚❱❂✿❆❁P❈❊➙❄❢✼▼✾ ❼ ❉❋❊❅❄❛❣✝P✹❉❢❄➙❩▼❚✶✾❖❊❇✸✵Ï❧❚☎◗❛✾❇♥❈✾❇❉♦✉✿◗❛✾⑦❆❈✾❇❄✱P✹❄✱❤❙✾❖P❈❊❅❄ Õ ❳✌❀❙❆✱↔❢❭✿❜t❣✝❊➎❊❢❀❃❂✝❪❥✾
ñ✿❚✶❚❈❉ ⑩❅⑩ ÷●➛➌✷♦❶❅➼ùê♠❶✻❪❇P❈❂❫❂▼✾❇♥❈✾❇❉⑦❆❈❚✮❩▼❚☎◗❧❂④❄✗◗➎❚✮⑤✒❉❢❚❱❜➑P➇❳✌❀❙❆♠↔❢❭✿❜t❣ ⑩ ❚❈❉❞◗➎✾❈➾➚❩❲❳❝✾
❩▼❚❱❂✿✾❁❩▼❚❱❀❃❂✿❆✖❳✝❀❙❆❺↔❅❭✿❜✖❣✝❊❘❶❘✉✌❊❢❚❨◗➎ä ✾❁❆❈✾❇❄ Õ ❳✌❀❙❆♠↔❢ä ❭✿❜t❣✝❊❖✉✝P❈❂✿❩④❄❢✼✣❭✿❊✻◗➎✾✧P✹❉❢✾✧ä ❆❈❚✶❚✶❩
❀❙⑤❛÷❹❜t❚s❩✽êú❀❃❊♠❀❃❂❫P❈❂❷❴❲❚❈⑤ ã ✷✹❍ ô ✉ ã➚ôÜ→ ✷✹❍❋■ ô ✉❹✸❇✸❖✸ ã Õ✕ôÜ→ ✷✹❍ ⑩✒Õ❯→ ✷♦❶ ô ✉✏❳✝❭✿❄
❄❢✼✿P✹❄❇➾➚❊❧✾❇♥❈✾❇❉❢❴✶❄❢✼✿❀❃❂▼❆▼✸
é
➺ ✾❇❄ ô ä ➫ ëû➛➋êq✸➉➒❡✾④❊❢✼✝P❈❤❃❤✱◗✽❉❋❀❃❄❅✾✂❚❱❭▼❉❨❀❃❂✣❄❅✾❇❉❢♥✹P❈❤❃❊t❚❈⑤♠❤❃P♦♥✰P➊❀❃❂➱❄❢✼▼✾④⑤➡❚❈❉❋❜
⑩✎ü ➩ ❍❢ý ➩ ➫ ➏♦ü ➩ → ✷✹❍ ü ➩ → ■✶❍❇❑❇❑❇❑☎❍❢ý ➩ ➐❱✉▼◗✻✼▼✾❇❉❢✾♠ý ➩ ➫ ü ➩ → êèP❈❂✿❩✙ý ➩ á ➦ ➹ ü ➩
⑤✒❚❈❉♠✾❇♥❈✾❇❉❢❴ ➲ ❐ ✷✹✸➇➒❡✾➇P❈❤❃❊❅❚③❤❙✾❇❄❺ý ➧ ➫ ▲s✸t➒❡✾❨❊❢✼✿P❈❤❃❤⑧P❈❤❃❊❢❚✂❉❢✾❇❣✿❉❢✾❖❊❅✾❖❂✣❄❺P✙❣✝P✹❄❢✼
⑤✒❚❈❉♠❄❢✼▼✾tñ✿✾❖P③P❈❊qP✂❊❅✾❖❻✣❭✿✾❖❂✿❪❥✾t❚❈⑤✱❀❃❂❷❄❅✾❇❆❈✾❇❉❘❊♠➻ ➧ ❍❅➻ ➦ ❍❅➻ ➭ ❍❇❑❇❑❇❑✜◗❧✼▼✾❇❉❋✾t➻ ➧ ➫ ▲③P❈❂✿❩
➻sþ✱➛❡➻✶þ á ➦✽ÿ ➏ ê❁❍❢ë➇➐❁⑤➡❚❈❉✽✾❇♥❈✾❇❉❋❴✁ ❐ ▲s✸
Ý ❚♦◗è✼▼✾❇❉❢✾❺❀❃❊✱P✖❩✿✾❇❄❢P❈❀❃❤❙✾❖❩✙❣✿❉❢❚✶❚❈⑤➬✸
♣ ❀❙❉❋❊❢❄❇✉ÔP❈❊❋❊❢❭✿❜t✾✙ì❮➹ ⑩✒Õ ➛Þ✷♦❶➬ê → ë ⑩ ➫ Õ ê →Þô ❶ ➨ ◗❛✾✙❜❁❭✝❊❅❄❁❣✝❉❢❚♦♥❈✾✙❄❢✼✿P✹❄
➺ P❖♥✹P❈❜✖P❈❂➊✼✝P❈❊❁P❯◗❧❀❃❂✿❂✿❀✐❂▼❆✂❊❅❄❅❉❋P✹❄❅✾❇❆❈❴❈✸ ➺ ✾❇❄ ü ➩ ➫ ý ➩ á ➦ →åÕ ê →Þô ➛Ü✷➇⑤➡❚❈❉
✾❇♥❈✾❇❉❢❴ ➲ ❐ ✷✹✸ ⑩ ➪❞❳✝❊❅✾❇❉❋♥❈✾❺❄❢✼✿P✹❄ Õ ê →✥ô ➛✥✷ ❐ ìt✸➁❶
⑥❞❊❢❊❢❭✿❜✖✾❹❄❢✼✿P✹❄✯❄❢✼▼✾❹ñ✝✾❖P✻✼✿P❈❊✯P❈❂✧❀❃❂ ❼ ❂✿❀❃❄❅✾✵❣✝P✹❄❢✼❁❄❢✼✿P✹❄✯P♦♥❈❚❱❀❃❩✿❊ÔP❈❤❃❤❷❄❢✼▼✾➎❤✐P❖♥✹Ps✉✰◗✻✼✿❀❃❪❋✼
➈
© Canadian Mathematical Society 2016
Draft Solutions - 2016 CMO
ä
❜t✾❖P❈❂✝❊✱❄❢✼✿P✹❄✽➻sþ✖ÿ â ✎⑩ ü ➩ ❍❢ý ➩ ➡⑤ ❚❈❉✻P❈❤❃❤ ➲ ❍✂ ❐ ✷✹✸ ♣ ❚❈❉✻✾❖P❈❪❋✼ ➲ ❐ ✷✹✉✝❤❃✾❇❄
✄
➩ ➫
❜➇P✰r ➏ ➻ þ ➨ ➻ þ ➘✥ü ➩ ✵
➐ ❍ ò ➩ ➫ ✖
❜ ❀❃❂ ➏ ➻ þ ➨ ➻ þ õ ý ➩ ➐✵❍
➘ ü ➩ ✵
P❈❂✿❩ ☎ ⑩✒➲ ❶ ➫ ❜✖✰P r ➏ ➨ ➻ þ ➌
➐ ❑
⑥❞❤❃❊❅❚✖❤❙✾❇❄ ò ➧ ➫ ▲s✸➎✺✽✼▼✾❖❂✙⑤✒❚❈❉ ➲ ❐ ✷⑦◗❛✾q✼✝P❖♥❈✾
✄
➻✝✆✟✞ ➩✡✠
➩q➫
➻☛✆✟✞ ➩☞✠✍✌✕➦ ❑
ò ➩❺➫
P❈❂✝❩
⑥❞❤❃❊❅❚▼✉▼⑤✒❚❈❉✽✾❇♥❈✾❇❉❢❴ ➲ ❐ ✷✹✉✿◗❛✾❁✼✿P❖♥❈✾
⑩ P❷❶ ò ➩ ➫ ✄ ➩ → ë ⑩ ❳✳✾❖❪❇P❈❭✝❊❅✾ ✄ ➩ → ê ➘ûü ➩ → ê ➫ ý ➩ ❶❘ó
⑩ ❳✳❶ ü ➩ ❐ ✄ ➩ õûü ➩ ➛ ô➤⑩ ❊❢❀❃❂✿❪❥✾ ✄ ➩q➫ ò ➩ ➛➇ë õ ý ➩ ➛✖ë ➫ ü ➩ → ê❲➛✖ë✖❶❘ó
P❈❂✿❩
⑩ ❪♦❶➙ý ➩ ➹ ò ➩ ➘ ý ➩ →✥ôî⑩ ❊❢❀❃❂✝❪❥✾ ò ➩ ➫ ✄ ➩ → ë ➘ûü ➩ → ë ➫ ý ➩ →✥ô ❶❘✸
➟➙❤✐P❈❀❃❜❦✷ ➨ ☎ ⑩✒➲✵→ ✷♦❶ ➫ ☎ ⑩✒➲ ❶ →➋Õ❲→ ✷✙⑤➡❚❈❉t✾❇♥❈✾❇❉❢❴ ➲ ❐ ✷✹✸ ⑩ ✺❧✼✿P✹❄✖❀❃❊❇✉➎P✹⑤➡❄❅✾❇❉
↔❢❭✿❜t❣✝❀✐❂▼❆❺❚♦♥❈✾❇❉✱❚❱❂✿✾⑦❀✐❂❷❄❅✾❇❉❢♥✹P❈❤✌❚❈⑤✕❤❃P♦♥✰Ps✉s❄❢✼▼✾⑦ñ✿✾❖P✧❜❁❭✝❊❅❄✱❜✖P✹⑨❈✾⑦✾❥rsP❈❪❥❄❢❤❙❴ Õ ↔❢❭✿❜t❣✌❊
❳❝✾❇⑤✒❚❈❉❢✾➎↔❅❭✝❜t❣✝❀❃❂▼❆✧❚♦♥❈✾❇❉✻❄❢✼▼✾q❂▼✾❥rs❄✽❀❃❂✣❄❅✾❇❉❢♥✹P❈❤☛❚❈⑤⑧❤❃P❖♥✹Ps✸➁❶
✎❺✁✰✡☛✡✝✆ ➨
➻
✆✟✞ ➩☞✠✏✌✒✑✓✕✌ ➦
➘
→ ë Õ
ò ➩ → ë Õ
➫
➹öý ➩ →✥ôå→✕✔ ê →
➫
➻
✆✟✞ ➩☞✠✏✕✌ ➦
➻
ü ➩☞✌✕➦ → ê → ✷Ô❑
ê
Õ✗✖
Õ
Ñ ✾❖❪❇P❈❭✿❊❅✾q❚❈⑤✯❄❢✼✿✾q❊❅❄❅❉❋❀❃❪❥❄✽❀✐❂▼✾❖❻✣❭✝P❈❤❃❀❙❄✗❴❈✉▼◗➎✾q✼✿P♦♥❈✾❺➻ ✆✟✞ ➩☞✠✍✌✒✑✘✌✕➦ ➘ ý ☞➩ ✌✕➦ ✉✌P❈❂✿❩③✼▼✾❖❂✿❪❥✾
✆✟✞ ☞➩ ✠✍✌✒✑✘✌✕➦ ➘➱ü ☞➩ ✌✕➦ ✸✵✺✽✼▼✾❇❉❋✾❇⑤➡❚❈❉❢✙✾ ☎ ⑩✒➲ ❶ →ÐÕ❨→ ✷ ➘ ☎ ⑩✒➲☛→ ✷♦❘❶ ✸ Ý ❥✾ r✶❄❇✉✝◗❛✾q✼✝P❖♥❈✾
➻✝✆✟✞ ➩☞✠✍✌✒✑✘✌✕➦ ❐
➫
õ
➫
õ
➻✝✆✟✞ ➩✡✠✏✌✕➦ → ê Õ
ò ➩ → ê Õ
ý ➩ → ê Õ
ü ➩☞✕
✌ ➦ ➛ ô✥→ ✷
ü ➩☞✕
✌ ➦ ➛✈ê → ✷
→ ê
✺✽✼✿✾❇❉❢✾❇⑤➡❚❈❉❋✾q✝
➻ ✆✟✞ ➩☞✠✍✌✒✑✘✌ ❐ ➻✝✆✟✞ ➩☞✠✏✌✒✑✓✕
➭ ✷❺⑤➡❚❱❤❃❤❃❚♦◗❧❊❇✸ ✌ ➦
☎ ⑩✒➲ ❶ →✇Õ❨→ ■✶✸➎➟➙❤❃P❈❀✐❜➑
õ
⑩ ❊❢❀❃❂✿❪❥✾ ô ➹✥ê⑦❶❥❑
ü ➩☞✕
✌ ➦ ✉❝P❈❂✝❩④✼▼✾❖❂✝❪❥✾✁☎ ⑩✒➲✏→ ✷♦❶❲➹
➟➙❤✐P❈❀❃❜î■ ➨ ➻ þ ✌✕➦ ➛✈➻ þ ➫ ê ⑤➡❚❈❉⑦P❈❤❃❤✒ ➫ ☎ ⑩✒➲ ❶ → ✷✹❍❇❑❇❑❇❑♦✚❍ ☎ ⑩✒➲✏→ ✷♦❶✵➛➌✷✹✉❝⑤✒❚❈❉❞P❈❤❃❤
➲ ❐ ✷✹✸ ⑩ ✺✽✼✿P✹❄❺❀❃❊❇✉☛❄❢✼✿✾ Õ ❀✐❂❷❄❅✾❇❉❋❜✖✾❖❩✿❀❃P✹❄❅✾❞↔❢❭✿❜t❣✝❊❞❚❈⑤✽➟➙❤❃P❈❀❃❜➮✷t❜✧❭✿❊❅❄❺P❈❤❃❤Ô❳❝✾●❚❈⑤
①
© Canadian Mathematical Society 2016
Draft Solutions - 2016 CMO
❤❙✾❖❂✿❆❈❄❢✼✂êq✸➁❶
✎❺✁✰✡☛✡✝✆ ➨ ➜ ⑤⑧➟➙❤✐P❈❀❃❜û■●❀❃❊✱⑤✎P❈❤❃❊❅✾❈✉▼❄❢✼▼✾❖❂
✄ ☞➩ ✌✕➦ ➫ ➻ ✆✟✞ ☞➩ ✌✕✛➦ ✠ ➫ ➻ ✆✟✞ ✡➩ ✠✏✌✒✑✓✌✕➦ ❐ ➻ ✆✟✞ ☞➩ ✠✏✌✕➦ →Ü⑩✒Õ ➛✥✷♦❶➬ê → ë
õ ý ➩ →ÐÕ ê →✥ô
➫ ü ☞➩ ✌✕➦ → ✷
õ ✄ ☞➩ ✕✌ ➦
◗❧✼✝❀❃❪❋✼✂❀❃❊✱P➇❪❥❚❱❂❷❄❅❉❋P❈❩✝❀❃❪❥❄❢❀❙❚❱❂☛✸❹✺❧✼✿❀❃❊➎❣✿❉❢❚☎♥❈✾❖❊❞➟➙❤❃P❈❀✐❜û■✶✸
➒❡✾q❪❇P❈❂④❂▼❚♦◗ð❪❥❚❱❂✿❪❇❤❃❭✿❩▼✾♠❄❢✼✝P✹❄
➻ ✆✟✞ ☞➩ ✌✕✛➦ ✠✍✌✕➦ ➫ ➻ ✆✟✞ ☞➩ ✠✏✌✒✑✓✌ ➭ ➫ ➻ ✆✟✞ ☞➩ ✠✏✌✕➦ →ÐÕ ê → ë➌ó
❀➄✸ ✾❈✸❙✉ ò ☞➩ ✌✕➦ ➫ ò ➩ →ÐÕ ê → ë ⑤✒❚❈❉✽✾❖P❈❪❋✼ ➲ ❐ ✷✹✸
✺✽✼✿✾❇❉❢✾❇⑤➡❚❈❉❋✾
ò ✡➩ ✌✕➦ ➛✈ý ☞➩ ✌✕➦ ➫ ò ➩ →ÐÕ ê → ë➮➛ ⑩ ý ➩ →✇Õ ê →✥ô ➛➱✷ → ê♠❶
➫ ò ➩ ➛✈ý ➩ → ✷Ô❑
Ï✻✾❖❂✿❪❥✾
ô ❐ ✢
ò ✜ ✌✕➦ ➛✈ý ✜ ✌✕➦ ➫ ò ➦ ➛✈ý ➦ →✥ô õöô
◗❧✼✝❀❃❪❋✼✂❀❃❊❧P✖❪❥❚❱❂❷❄❅❉❘P❈❩✿❀❃❪❥❄❢❀❙❚❱❂☛✸❛✺✽✼▼✾❇❉❢✾❇⑤✒❚❈❉❢✾q❂▼❚t❣✌P✹❄❢✼③⑤✒❚❈❉✽❄❢✼▼✾❺ñ✿✾❖P➇P❖♥❈❚❱❀❃❩✿❊✻P❈❤❃❤☛❄❢✼▼✾
❤❃P♦♥✰Ps✸➠➒❡✾✂❚❈❳✝❊❅✾❇❉❢♥❈✾✂❄❢✼✿P✹❄ ➺ P❖♥✹P❈❜✖P❈❂➱❚❱❂✿❤❙❴➊❂▼✾❇✾❖❩✿❊●❄❅❚❫❣✝❭▼❄t❤❃P♦♥✰P❲❚❱❂✇❄❢✼▼✾ ❼ ❉❋❊❢❄
ô✥→ ✷❺❀❃❂✣❄❅✾❇❉❢♥✰P❈❤✐❊❇✸
Ý ❚♦◗èP❈❊❢❊❢❭✿❜t✾❺ì ❐ ⑩✒Õ ➛➱✷♦❶➬ê → ë✮✸⑧➒❡✾❺◗❧❀❃❤❃❤❝❊❢✼✿❚♦◗Þ❄❢✼✿P✹❄❧❄❢✼▼✾♠ñ✿✾❖P✖❪❇P❈❂④P❖♥❈❚❱❀❃❩
P❈❤❃❤❝❄❢✼✿✾❺❤❃P❖♥✹Ps✸❹➒❡✾q❊❋✼✿P❈❤❃❤☛❂▼✾❇✾❖❩✂❄❢✼✿✾❺⑤➡❚❱❤✐❤❙❚♦◗❧❀✐❂▼❆❁❉❋✾❖❊❢❭✿❤❙❄ ➨
➟➙❤✐P❈❀❃❜î❏ ➨ ➺ ✾❇❄⑦÷ ❐ Õ ê❁✸❺✺✽✼▼✾❖❂④❄❢✼▼✾❇❉❢✾●✾❥r▼❀❃❊❅❄♠❂▼❚❱❂✿❂✿✾❇❆❱P✹❄❢❀❙♥❈✾t❀❃❂❷❄❅✾❇❆❈✾❇❉❘❊❺➳tP❈❂✿✤
❩ ✣
❊❢❭✝❪❋✼✂❄❢✼✿P✹❄❞➳♦ê → ✣✃ë ÿ ⑩ ÷✧➛ ô ❍❢÷ ä ✸
➒❡✾q❊❋✼✿P❈❤❃❤❝❣✿❉❢❚☎♥❈✾❺❄❢✼✿❀❃❊➙❉❋✾❖❊❢❭✿❤❙❄✽P✹❄✽❄❢✼✿✾♠✾❖❂✿❩☛✸
♣ ❀❙❉❋❊❢❄❇✉✕❚❈❳✝❊❢✾❇❉❢♥❈✾➇❄❢✼✿P✹❄ ü ➦ ❐ Õ êq✸ Ñ ❴➊➟➙❤❃P❈❀✐❜ ❏✶✉✵❀❙❄q❀❃❊q❣❝❚❱❊❢❊❢❀❙❳✝❤❃✾✧⑤✒❚❈❉✧❄❢✼▼✾➇ñ✿✾❖P
❄❅❚✖❜✖P✹⑨❈✾❁Pt❊❅ä✾❖❻✣❭✿✾❖❂✿❪❥✾♠❚❈⑤▼↔❢❭✿❜t❣✝❊➙❊❅❄❢P✹❉❋❄❢❀❃❂▼❆t⑤✒❉❢❚❱❜î▲tP❈❂✿❩✙✾❖❂✿❩✿❀✐❂▼❆✧P✹❄✽P●❣❝❚❱❀❃❂✣❄➙❚❈⑤
⑩✎ü ➦ ➛ ô ❍ ü ➦ ✸ ♣ ❉❢❚❱❜➑P❈❂❷❴✂❣❝❚❱❀❃❂✣❄✱❚❈⑤⑧❄❢✼✿❀❃❊✻❀❃❂❷❄❅✾❇❉❋♥✰P❈❤➄✉✌P➇❊❢❀❃ä ❂✿❆❱❤❙✾➎↔❅❭✝❜t❣③❚❈⑤✵❊❢❀❙ç❇✾✧ë
❄❢P✹⑨❈✾❖❊Ô❄❢✼▼✾➙ñ✿✾❖P✻❚♦♥❈✾❇❉ ⑩✎ü ➦ ❍❢✦ý ✣✥ ❄❅❚♠P❞❣❝❚❱❀❃❂❷❄✯❀❃❂ ⑩ ý ➦ ❍❢ý ➦ →❨ô ✉✹◗❧✼✿❀✐❪❋✼✧❪❥❚❈❉❢❉❢✾❖❊❅❣❝❚❱❂✿❩✿❊
❄❅❚✖❄❢✼✿✾♠❣✳❚❱❀✐❂❷❄✱➻ ✆✟✞ ➦✛✠✍✌✕➦ ⑩ ➫ ò ➦ ❶❛❚❱❂③❄❢✼▼✾♠ñ✿✾❖Ps➾➚❊❧❣✝P✹❄❢✼☛✸
Ý ❚♦◗➤◗➎✾t❭✿❊❅✾t❀❃❂✿❩✝❭✿❪❥❄❢❀❙❚❱❂④❄❅❚✂❣✿❉❢❚♦♥❈✾➇❄❢✼✿P✹❄❇✉✜⑤✒❚❈❉♠✾❇♥❈✾❇❉❢❴ ➲ ❐ ✷✹✉✜❄❢✼▼✾❇❉❢✾t❀❃❊♠P✮❣✝P✹❄❢✼
❊❢❭✝❪❋✼✖❄❢✼✝P✹❄❛➻ þ P♦♥❈❚❱❀❃❩✿❊➎❤❃P❖♥✹P❺⑤✒❚❈❉➎P❈❤✐❤✧ ➘ ☎ ⑩✒➲ ❶ → ✷✹✸❹✺✽✼✿✾✻❪❇P❈❊❅✾ ➲ ➫ ✷✻❀❃❊❹❩▼❚❱❂✿✾❈✉✶❊❅❚
◆
© Canadian Mathematical Society 2016
Draft Solutions - 2016 CMO
P❈❊❢❊❋❭✿❜t✾✧❄❢✼✿P✹❄♠ä ❄❢✼▼✾tP❈❊❢❊❅✾❇❉❢❄❢❀❃❚❱❂❲✼✿❚❱❤❃❩✿❊✻⑤➡❚❈❉❺P✮❆❱❀❙♥❈✾❖❂ ➲ ✸❁✺✽✼▼✾❖❂❲➻ ✆✟✞ ➩✡✠✏✌✕➦ ➫ ò ➩③ÿ
⑩ ý ➩ ❍❢ý ➩ →✥ô ✸✵✺✽✼▼✾❇❉❢✾❇⑤✒❚❈❉❢✾
ü ☞➩ ✌✕➦ ➛ ò ➩ ❐ ý ➩ → ìÞ➛ ⑩ ý ➩ →➱ô ❶ ➫ ìÞ➛ ô ❐ Õ ê➠❑
⑥✻❣✿❣✝❤❙❴s❀❃❂▼❆✽➟➙❤❃P❈❀✐❜➋❏✱◗❧❀❙❄❢✼♠÷ ➫ ü ☞➩ ✌✕➦ ➛ ä ò ➩ ❊❢✼▼❚☎◗❧❊☛❄❢✼✿P✹❄✜❄❢✼✿✾✵ñ✿✾❖P✽❪❇P❈❂✱↔❅❭✿❜✖❣♠⑤➡❉❢❚❱❜
ò ➩ ❄❅❚❨P➇❣❝❚❱❀❃❂❷❄✱❚❈⑤ ⑩✎ü ✡➩ ✌✕➦ ➛ ô ❍ ü ☞➩ ✌✕➦ ✸✱⑥ß❊❢ä ❀❃❂▼❆❱❤❃✾➎↔❅❭✿❜✖❣③❚❈⑤✵❊❢❀❙ç❇✾qëí❄❢✼✿✾❖❂③❄❢P✹⑨❈ä ✾❖❊
❄❢✼▼✾♠ñ✝✾❖P●❄❅❚➇Pt❣❝❚❱❀❃❂✣❄➎❚❈⑤ ⑩ ý ☞➩ ✕✌ ➦ ❍❢ý ☞➩ ✌✕➦ →➱ô ⑩ ◗❧❀❙❄❢✼▼❚❱❭▼❄✱♥s❀❃❊❢❀❃❄❢❀❃❂▼❆ ⑩✎ü ☞➩ ✌✕➦ ❍❢ý ☞➩ ✌✕➦ ❶❘✉
P❈❂✿❩✙❄❢✼✝❀❃❊✱❣❝❚❱❀❃❂✣❄➙❊❢✾❇❉❢♥❈✾❖❊❧P❈❊❧➻ ✆✟✞ ☞➩ ✌✕✛➦ ✠✏✌✕➦ ✸✵✺✽✼✿❀❃❊✱❪❥❚❱❜t❣✌❤❙✾❇❄❅✾❖❊✽❄❢✼▼✾❺❀✐❂✿❩✿❭✿❪❥❄❢❀❙❚❱❂✜✸
✎❺✁✰✡☛✡✝✆❞✡✝✆q✤✧☞✒✄✝✑✩★✫✪ ➨ ➺ ✾❇✭
❄ ✬❲❳✳✾♠❄❢✼✿✾❺❆❈❉❢✾❖P✹❄❅✾❖❊❅❄⑦❀❃❂✣❄❅✾❇❆❈✾❇❉❧❄❢✼✿P✹❄❞❀❃❊✽❤❙✾❖❊❢❊✽❄❢✼✿P❈❂③❚❈❉
✾❖❻✶❭✿P❈❤✜❄❅❚✮÷✶➼ùê❁✸❧✺✽✼▼✾❖✮
❂ ✬ ❐ Õ P❈❂✝❩✮✬✌ê ➘ ÷✂➹ ⑩ ✬ → ✷♦❶➬êq✸ ♣ ❚❈❉ ➥❨➫ ▲s❍❇❑❇❑❇❑♦❍ Õ ✉
❤❙✾❇❄
✯✱✰
➫ ⑩ ✬✙➛ ➥ ❶➬ê → ➥ ë ➫ ✬✌ê → ➥ ô ❑
✺✽✼✿✾❖❂
✯
✯ ➧➉➫ ✬✌ê ➘ ÷✝❍
✑ ✯✱➫ ✰ ✬✌ê ✯✱→Ð
✰ Õ✕ô ➫ ✬✌ê →Ü⑩✒Õ❨→ ✷♦❶ ô ➛ ô ❐ ⑩ ✬ → ✷♦❶➬êÜ➛ ô õ ÷q➛ ô ❑
P❈❂✿❩ ✌✕➦ ➛ ➫ ô ⑤✒❚❈❉ ➥✖➫ ▲s❍❇❑❇❑❇❑♦❍ Õ ➛✥✷✹✸
✯ ✰
✺✽✼✿✾❇❉❢✾❇⑤➡❚❈❉❋✾♠◗❛✾q❜✧❭✿❊❅❄✽✼✝P❖♥❈✾ ÿ ⑩ ÷✧➛ ô ❍❢÷ ä ⑤➡❚❈❉✻❊❅❚❱❜t✾ ➥ ❀❃❂ ➏ ▲s❍❖✷✹❍❇❑❇❑❇❑✰❍ Õ ➐❱✸✂é
①✶✸ ➺ ✾❇❄✳✲✙ê❞ë ô ❳❝✾✮P❈❂✇P❈❪❇❭▼❄❅✾❥➍✗P❈❂▼❆❱❤❃✾❖❩✇❄❅❉❋❀❃P❈❂▼❆❱❤❙✾✮◗❧❀❙❄❢✼ÐP❈❤❙❄❢❀❙❄❢❭✝❩▼✾❖❊❁ê✵✴ P❈❂✿❩✇ë✙✶
❜t✾❇✾❇❄❢❀✐❂▼❆✂P✹✸❄ ✷➠✸ ➺ ✾❇❄ ✄ ❳✳✾●❄❢✼▼✾t❜✖❀✐❩▼❣❝❚❱❀❃❂❷❄✻❚❈⑤➎❊❢✾❇❆❱❜t✾❖❂❷❄❺ê❞ë✮✉✏P❈❂✿❩❫❊❢❭▼❣✝❣✳❚❱❊❢✾
❄❢✼✿P✹❄⑦❄❢✼▼✾t❪❇❀❙❉❘❪❇❭✿❜✖❪❇❀❙❉❋❪❇❤❃✾❖❊❧❚❈✹⑤ ✲✺✴✻✶ ✄ P❈❂✿❩✼✲✙ê❞✙ë ✷ ❜t✾❇✾❇❄❺P✹❄⑦❣❝❚❱❀❃❂❷❄❢❊⑦ÒîP❈❂✿❩
✽ ◗❧❀❙❄❢✼ÐÒ➑❚❱❂Ð❄❢✼▼✾✂❊❢P❈❜t✾✙❊❢❀❃❩▼✾❨❚❈⑤ ô ✷➢P❈❊●ê❁✸❡❸✵❉❢❚☎♥❈✾✮❄❢✼✝P✹❄✧❄❢✼✿✾✮❤❃❀✐❂▼✾❖❊✗✶✗✴❯✉
Ò ✷➊✉ÔP❈❂✿❩ ✄ ✽ P❈❤❃❤✵❣✝P❈❊❋❊❺❄❢✼▼❉❢❚❱❭✿❆❱✼➊P④❊❢❀❃❂▼❆❱❤❙✾t❣❝❚❱❀❃❂✣❄❺❚❱❂➊❄❢✼▼✾❨❪❇❀❃❉❋❪❇❭✿❜✖❪❇❀❙❉❘❪❇❤❙✾t❚❈⑤
✸
✲✙ê❞ë ô ✸
❽✌❾❈❿➁➀▼➂✎➃➄❾✹➅✌➆
✾
© Canadian Mathematical Society 2016
Draft Solutions - 2016 CMO
✜
❃
❑▼▲
❄❆❅
❈
❉❋❊
❇
●☞❍
■✂❏
❀ ✿
◗
❘
◆P❖
❁✟❂
❏☛❙ ❚
➺ ❇✾ ❄❛ý➵❩✿✾❖❂▼❚❈❄❅✾✻❄❢✼✿✾❞❀❃❂✣❄❅✾❇❉❋❊❅✾❖❪❥❄❢❀❃❚❱❂➇❚❈⑤✜❤❃❀✐❂▼✾❖❊ îP❈❂✿❩❨Ò ➠✸❹↕s❀❃❂✝❪❥✾❧❻✣❭✝P❈❩▼❉❋❀❃❤❃P✹❄❅✾❇❉❢➍
P❈❤❃❊ ô ➑P❈❂✿❩➇ê❞Ò ❲ëèP✹❉❋✾❧❪❥❴✶❪❇❤✐❀❃❪✹✉❷◗➎✾❧✼✿P♦♥❈✾ ➙ý ✮ê ➫ ✷❖➞✹▲ ✜➛ ✮ê ➫
✷❖➞✹▲ ❧➛
➫ ✷❖➞✹▲ ❧➛ ô ➫ ➝✹▲ → êq✉❝P❈❂✿❩ ➙ý♠Ò❺ê ➫ ❲Ò❺ê ➫
✷❖➞✹▲ ➛ ④ë●ê ➫ ➝✹▲ → ê❁✸✯✺❧✼▼✾❇❉❢✾❇⑤✒❚❈❉❢✾❈✉❷ê❞Ò ❨ý➉❀❃❊Ô❪❥❴s❪❇❤❃❀❃❪✹✸⑧✺✽✼✿❀❃❊✯❀❃❂✧❄❢❭▼❉❋❂●❀❃❜t➍
❣✝❤✐❀❙✾❖❊➙❄❢✼✿P✹❄ ✱Ò❺ë ➫ ➙Ò❺ë ➫ ➙Òqê ➫ ➙Ò❺ê ➫ ➙Ò❺ý ➫ ➙Ò❺ý ✖✉
P❈❂✿❩✂❊❢❚➇Ò❺ë●ý ✩❀❃❊✱P❈❤✐❊❅❚➇❪❥❴s❪❇❤❃❀❃❪✹✸
➺ ✾❇❄⑦ìî❩✿✾❖❂▼❚❈❄❅✾t❄❢✼▼✾t❳✝P❈❊❅✾●❚❈⑤❛❄❢✼▼✾✖P❈❤❙❄❢❀❙❄❢❭✿❩✿✾❁⑤✒❉❢❚❱❜ ô ❄❅❚③ê❞ë❨✸✧✺✽✼▼✾❖❂ ④❍ ✖❍❢ì✱❍
P❈❂✿❩ P❈❤❃❤✣❤❃❀❙✾❹❚❱❂❁❄❢✼▼✾✱➝☎➍✟❣❝❚❱❀❃❂❷❄✯❪❇❀❙❉❋❪❇❤❙✾❹❚❈⑤ ✙ê❞ë ô ✉✰P❈❂✝❩❁❊❅❚⑦P✹❉❢✾➙❪❥❴✶❪❇❤✐❀❃❪✹✸✏➒❡✾✱P❈❤❃❊❅❚
⑨s❂▼❚♦◗ ê❞Ò ✮ý●✉✜Ò❺ë●ý ✖✉✜ë ô ❁ìt✉☛P❈❂✿❩❫ê ô ❨ìíP✹❉❢✾✖❪❥❴s❪❇❤❃❀❃❪✹✉❝◗❧✼✿❀✐❪❋✼❫❀❃❜t❣✝❤✐❀❙✾❖❊
➙ê❞ý♠ë ➫ ✱Ò❺ý♠ëß➛ ➙Ò❺ý♠ê ➫ ➙Ò që ➛ ➙Ò ✮ê ➫ ➙Ò ❁ì → ➙ì qëß➛
➙Ò ❨ì → ➙ê ✮ì ➫ ✱ì që → ✱ê ✮ì ➫ ➙ì ô ë → ➙ê ô ì ➫ ô ✸⑧✺✽✼▼✾❇❉❢✾❥➍
⑤✒❚❈❉❢✾❈✉✿ý ❤❃❀❙✾❖❊➙❚❱❂❯❄❢✼▼✾❺❪❇❀❙❉❋❪❇❭✝❜✖❪❇❀❙❉❋❪❇❤❙✾⑦❚❈⑤ ✙ê❞ë ô ✸
◗❧❀❙❄❢✼❲❄❢✼▼✾✖❪❇❀❙❉❢➍
Ý ❚♦◗✩❤❃✾❇❄ P❈❂✝❩❡ý ❩▼✾❖❂▼❚❈❄❅✾t❄❢✼✿✾✖❀❃❂❷❄❅✾❇❉❘❊❅✾❖❪❥❄❢❀❙❚❱❂✿❊⑦❚❈⑤➎❤❃❀❃❂▼✾
❪❇❭✿❜➇❪❇❀❙❉❋❪❇❤❙✾✱❚❈⑤ ✙ê❞ë ô ✉❷❪❘✼▼❚❱❊❅✾❖❂✮❊❅❚q❄❢✼✿P✹❄ ❍ ❍ ❍❢ý ❤❃❀❙✾✱❚❱❂➇❄❢✼▼✾✻❤❃❀✐❂▼✾✱❀❃❂t❄❢✼✿P✹❄
❚❈❉❋❩✿✾❇❉❖✸✕➒❡✾❧◗❧❀❃❤❃❤✶❊❢✼▼❚☎◗➱❄❢✼✿P✹❄❹ý ➫ ý●✉❱◗❧✼✿❀✐❪❋✼✧◗❧❀❃❤❃❤✶❪❥❚❱❜t❣✝❤❃✾❇❄❅✾✱❄❢✼▼✾✱❣✿❉❢❚✶❚❈⑤❵✸ÔÏ✻❚♦◗✱➍
✾❇♥❈✾❇❉❖✉ ❼ ❉❋❊❅❄✻❂▼❚❈❄❅✾❺❄❢✼✿P✹❄✻❄❢✼▼✾❁❪❇❀❙❉❘❪❇❭✿❜✖❪❇❀❙❉❋❪❇❤❃✾❞❚❈⑤ ✙ê❞ë ô ✼✿P❈❊❧❉❋P❈❩✿❀❃❭✝❊
✉✌P❈❂✿❩
➭
❄❢✼▼✾⑦❪❇❀❙❉❋❪❇❭✝❜✖❪❇❀❙❉❋❪❇❤❙✾✻❚❈⑤ ✙ê❞ë ✼✿P❈❊➎❉❋P❈❩✿❀✐❭✿❊
á ✸✵✺❧✼✣❭✿❊
å➫
❄❢✼▼✾✮❄✗◗➎❚❡❪❇❀❙❉❋❪❇❤❙✾❖❊✧✼✿P♦♥❈✾✂✾❖❻✣❭✿P❈❤❛❉❋P❈❩✿❀✐❭✿❊❇✉⑧P❈❂✿❩Ð➭ ❊❅❚④❄❢✼✿✾❇❴➊❜✧❭✿❊❅➭ ❄✧❳✳✾✮➦ ❅➧❊❅❴s❜✖❜t✾❇❄❅❉❋❀❃❪❇P❈❤
P✹❳❝❚❱❭▼❄✱❄❢✼▼✾♠❣❝❚❱❀❃❂✣❄ ✸ ➜ ❂✂❣✌P✹❉❢❄❢❀❃❪❇❭✿❤❃P✹❉♦✉
✸
➫
↕s❀✐❂✿❪❥✾ ✱ê ❁ë ➫ ➙ê ✮ë ➫ ➝✹▲ ✰✉❱◗➎✾✽⑤✎❭▼❉❢❄❢✼▼✾❇❉❋❜✖❚❈❉❢✾❛⑨s❂▼❚☎◗✥❄❢✼✿P✹❄ ❀❃❊✯❄❢✼▼✾❧❪❇❀❙❉❢➍
❪❇❭✿❜➇❪❥✾❖❂❷❄❅✾❇❉⑦❚❈⑤⑧❳❝❚❈❄❢✼ ✙ê që P❈❂✿❩ ✂ê ✮ë✮✸✽✺✽✼✣❭✝❊❇✉ ê ➫
➫
➫
➯ ý ➫
➯ ý ➫
ë✮✸ Ñ ❴✂❸✯❚♦◗➎✾❇❉❞❚❈⑤⑧P❨❸✕❚❱❀✐❂❷❄❇✉✌◗❛✾❁❄❢✼▼✾❖❂❯✼✿P♦♥❈✾
ê ➯ ë ➫
➭ ✸ ➜ ❂✂❣✝P✹❉❢❄❢❀✐❪❇❭✿❤❃P✹❉❖✉▼❄❢✼✿❀✐❊✽❜t✾❖P❈❂✿❊❧❄❢✼✿P✹❄❧❄❢✼▼✾❁❪❇❀❙❉❋❪❇❭✿❜✖❪❇❀❃❉❋❪❇❤❙✾⑦❚❈⑤
➞
❯✶✗✴
❯✶
✴❲✷
❬❩
❱✷
✸✷
✷✹✸✻✺✽✼✿✾❁❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊●✷✹❍❋■✶❍❋❏✶❍❇❑❇❑❇❑☎❍❋■✹▲▼✷❖◆✙P✹❉❋✾❁◗✽❉❘❀❙❄❅❄❅✾❖❂❯❚❱❂❲P❨❳✳❚❱P✹❉❘❩☛✸✽❬✵❚❱❭❫❪❇P❈❂❫❪❋✼▼❚✶❚❱❊❅✾
P❈❂✣❴❯❄❵◗❛❚❯❂✶❭✿❜❁❳❝✾❇❉❋❊❞❚❱❂❡❄❢✼▼✾●❳❝❚❱P✹❉❋❩❡P❈❂✿❩❫❉❢✾❇❣✝❤✐P❈❪❥✾✧❄❢✼✿✾❖❜❦◗❧❀❙❄❢✼❫❄❢✼▼✾❖❀❙❉♠P♦♥❈✾❇❉❋P✹❆❈✾❈✸
♣ ❚❈❉q✾❥rsP❈❜t❣✌❤❙✾❈✉☛❴❈❚❱❭✈❪❇P❈❂❡❉❢✾❇❣✌❤❃P❈❪❥✾✂✷tP❈❂✿❩✈■✮◗❧❀❙❄❢✼✇✷✹❑②①✶✉✏❚❈❉q❴❈❚❱❭❡❪❇P❈❂❡❉❢✾❇❣✝❤❃P❈❪❥✾③✷
P❈❂✿❩❫❏t◗✻❀❙❄❢✼④P✮❊❅✾❖❪❥❚❱❂✿❩❲❪❥❚❈❣✶❴③❚❈⑤❛■✶✸⑦⑥❧⑤✒❄❅✾❇❉⑦■✹▲▼✷❖①✮❉❋✾❇❣✝❤❃P❈❪❥✾❖❜t✾❖❂✣❄❢❊❞❚❈⑤⑧❄❢✼✿❀❃❊✻⑨✶❀❃❂✝❩☛✉
❄❢✼▼✾♠❳❝❚❱P✹❉❋❩③◗❧❀❃❤❃❤✳✼✿P❖♥❈✾❁❚❱❂✿❤❙❴❨❚❱❂▼✾q❂✶❭✿❜q❳❝✾❇❉✽❤❙✾❇⑤✒❄✱❚❱❂③❀❙❄❇✸
⑩ P❷❶✂❸❹❉❢❚♦♥❈✾✻❄❢✼✿P✹❄✵❄❢✼▼✾❇❉❢✾❞❀❃❊✵P❺❊❢✾❖❻✣❭▼✾❖❂✝❪❥✾✽❚❈⑤☛❉❢✾❇❣✝❤❃P❈❪❥✾❖❜✖✾❖❂❷❄❢❊❹❄❢✼✿P✹❄✵◗❧❀❃❤✐❤▼❜✖P✹⑨❈✾❧❄❢✼▼✾
❼ ❂✝P❈❤☛❂✣❭✿❜❁❳❝✾❇❉✱✾❖❻✣❭✝P❈❤☛❄❅❚❨■✶✸
⑩ ❳✳❶✙❸❹❉❢❚♦♥❈✾✻❄❢✼✿P✹❄✵❄❢✼▼✾❇❉❢✾❞❀❃❊✵P❺❊❢✾❖❻✣❭▼✾❖❂✝❪❥✾✽❚❈⑤☛❉❢✾❇❣✝❤❃P❈❪❥✾❖❜✖✾❖❂❷❄❢❊❹❄❢✼✿P✹❄✵◗❧❀❃❤✐❤▼❜✖P✹⑨❈✾❧❄❢✼▼✾
❼ ❂✝P❈❤☛❂✣❭✿❜❁❳❝✾❇❉✱✾❖❻✣❭✝P❈❤☛❄❅❚❯✷❇▲❈▲❈▲s✸
✌❽ ❾❈❿➁➀▼➂✎➃➄❾✹➅✌➆ ⑩ P❷❶ ♣ ❀❃❉❋❊❅❄➇❉❋✾❇❣✝❤❃P❈❪❥✾❲■✹▲▼✷❥➈✥P❈❂✝❩➉■✹▲▼✷❖◆➊◗❧❀❙❄❢✼➋■✹▲▼✷❖①✶✉✻P❈❂✝❩➌❄❢✼▼✾❖❂➉❉❢✾❥➍
❣✝❤✐P❈❪❥✾t❄❢✼▼✾✖❄✗◗➎❚④❪❥❚❈❣✌❀❙✾❖❊❺❚❈⑤✱■✹▲▼✷❖①③◗✻❀❙❄❢✼❡P③❊❢❀❃❂✿❆❱❤❙✾✖❪❥❚❈❣✣❴❈✸✙✺✽✼✿❀✐❊❞❤❃✾❖P❖♥❈✾❖❊✧❭✿❊♠◗❧❀❙❄❢✼
➏ ✷✹❍❋■✶❍❇❑❇❑❇❑✰❍❋■✹▲▼✷❖❏✶❍❋■✹▲▼✷❖①✣➐❱✸ ♣ ❉❢❚❱❜➑✼▼✾❇❉❢✾❈✉✳◗➎✾✧❪❇P❈❂④❉❢✾❇❣✝❤❃P❈❪❥✾●■✹▲▼✷❖❏✮P❈❂✿❩④■✹▲▼✷❖①❨◗❧❀❙❄❢✼
■✹▲▼✷❥➈t❄❅❚●❆❈✾❇❄ ➏ ✷✹❍❋■✶❍❇❑❇❑❇❑✰❍❋■✹▲▼✷❖■✶❍❋■✹▲▼✷❥➈s➐❱✸✱➒❡✾❺❪❇P❈❂✙❄❢✼▼✾❖❂✮❉❢✾❇❣✝❤❃P❈❪❥✾❺■✹▲▼✷❖■●P❈❂✿❩✂■✹▲▼✷❥➈
◗❧❀❃❄❢✼❲■✹▲▼✷❖❏✶✉✏P❈❂✝❩❲❊❢❚✮❚❱❂✜✉❝❭✝❂❷❄❢❀❃❤✕◗➎✾✧✾❇♥❈✾❖❂✣❄❢❭✿P❈❤❃❤❃❴③❆❈✾❇❄♠❄❅❚ ➏ ✷✹❍❋❏✣➐❱✸q➒❡✾ ❼ ❂✿❀✐❊❢✼③❳✶❴
❉❢✾❇❣✌❤❃P❈❪❇❀❃❂▼❆✂✷♠P❈❂✿❩❯❏t◗❧❀❙❄❢✼③■✶✸
⑩ ❳✳❶✻➓❞❊❢❀❃❂▼❆✮❄❢✼▼✾✖❊❢P❈❜✖✾✖❪❥❚❱❂✿❊❅❄❅❉❋❭✿❪❥❄❢❀❃❚❱❂❲P❈❊q❀❃❂ ⑩ P❷❶❘✉✏◗➎✾●❪❇P❈❂ ❼ ❂✿❩❫P✙❊❅✾❖❻✣❭✿✾❖❂✿❪❥✾t❚❈⑤
❉❢✾❇❣✌❤❃P❈❪❥✾❖❜t✾❖❂✣❄❢❊❧❄❢✼✿P✹❄✻❉❢✾❖❩✿❭✿❪❥✾❖❊ ➏♦➔ ❍ ➔q→ ✷✹❍❇❑❇❑❇❑♦❍❋➣♦➐●❄❅❚♠↔❢❭✿❊❅❄ ➏♦➔q→ ✷✰➐❱✸✻↕s❀❃❜✖❀❃❤✐P✹❉❋❤❙❴❈✉
❪❇P❈❂✂P❈❤❃❊❢❚ ❼ ❂✿❩✙Pt❊❅✾❖❻✣❭✿✾❖❂✿❪❥✾⑦❚❈⑤✕❉❢✾❇❣✌❤❃P❈❪❥✾❖❜t✾❖❂✣❄❢❊➙❄❢✼✿P✹❄✱❉❢✾❖❩✿❭✿❪❥✾❖❊ ➏♦➔ ❍ ➔❞→ ✷✹❍❇❑❇❑❇❑♦❍❋➣♦➐
❄❅❚⑦↔❢❭✿❊❅❄ ➏ ➣➎➛✥✷✰➐❱✸
➜ ❂⑦❣✝P✹❉❢❄❢❀❃❪❇❭✝❤❃P✹❉❖✉❖◗➎✾❹❪❇P❈❂ ❼ ❂✝❩♠❊❅✾❖❻✣❭✿✾❖❂✿❪❥✾❖❊✜❚❈⑤✶❉❢✾❇❣✝❤❃P❈❪❥✾❖❜✖✾❖❂❷❄❢❊✏❄❢✼✿P✹❄☛❉❢✾❖❩✿❭✿❪❥✾ ➏ ✷✹❍❋■✶❍❇❑❇❑❇❑✹❍❋➝❈➝❈➝✣➐
❄❅❚❁↔❢❭✿❊❅❄ ➏ ➝❈➝❈➞✣➐❱✉✏P❈❂✿❩❲❄❢✼✿P✹❄⑦❉❢✾❖❩✿❭✿❪❥✾ ➏ ✷❇▲❈▲▼✷✹❍❖✷❇▲❈▲❱■✶❍❇❑❇❑❇❑❱❍❋■✹▲▼✷❖◆✣➐➇❄❅❚❁↔❢❭✿❊❅❄ ➏ ✷❇▲❈▲❱■✣➐❱✸
✺✽✼✝❀❃❊♠❤❙✾❖P❖♥❈✾❖❊●❭✿❊q◗❧❀❙❄❢✼ ➏ ➝❈➝❈➞✶❍❖✷❇▲❈▲❈▲s❍❖✷❇▲❈▲❱■✣➐❱✸✈➒❡✾✮❪❇P❈❂➊❉❢✾❇❣✝❤✐P❈❪❥✾❨➝❈➝❈➞❯P❈❂✝❩➌✷❇▲❈▲❱■
◗❧❀❃❄❢✼❲P✂❊❢✾❖❪❥❚❱❂✿❩✈❪❥❚❈❣✶❴❲❚❈⑤❞✷❇▲❈▲❈▲s✉✯P❈❂✝❩❡❄❢✼▼✾❖❂❫❉❢✾❇❣✝❤✐P❈❪❥✾t❄❢✼▼✾t❄✗◗➎❚❯❪❥❚❈❣✝❀❙✾❖❊⑦❚❈⑤⑦✷❇▲❈▲❈▲
◗❧❀❃❄❢✼✂Pt❊❢❀❃❂▼❆❱❤❙✾♠❪❥❚❈❣✶❴✂❄❅❚➇❪❥❚❱❜t❣✝❤❃✾❇❄❅✾❺❄❢✼▼✾❺❪❥❚❱❂✿❊❅❄❅❉❘❭✿❪❥❄❢❀❙❚❱❂☛✸
■✶✸⑦➟➎❚❱❂✝❊❢❀❃❩▼✾❇❉✂❄❢✼▼✾➠⑤➡❚❱❤❃❤❃❚♦◗❧❀❃❂✿❆✥❊❅❴✶❊❢❄❅✾❖❜➢❚❈⑤✂✷❇▲✥✾❖❻✶❭✿P✹❄❢❀❙❚❱❂✿❊④❀❃❂➤✷❇▲➌❉❢✾❖P❈❤♠♥✹P✹❉❋❀❃P✹❳✝❤❙✾❖❊
➥✶➦ ❍❇❑❇❑❇❑♦❍ ➥✶➦✗➧✹➨
◆
⑩✒➲ ➫ ✷✹❍❇❑❇❑❇❑♦❍❖✷❇▲❷❶❘❑
➥✹➩❯➫ ✷ → ➥ ➭➦ → ➥ ➭ →➋➥✣➩➭➯❇➯❇➯✰→ ➥ ➭➦✗➧
➭
♣ ❀❃❂✿❩✂P❈❤✐❤⑧✷❇▲✰➍✟❄❢❭▼❣✝❤❙✾❖❊ ⑩ ➥ ➦ ❍ ➥ ➭ ❍❇❑❇❑❇❑♦❍ ➥ ➦✗➧ ❶❛❄❢✼✿P✹❄✻P✹❉❢✾q❊❅❚❱❤❃❭✿❄❢❀❙❚❱❂✿❊➙❚❈⑤✯❄❢✼✿❀❃❊✽❊❅❴s❊❅❄❅✾❖❜✂✸
✷
© Canadian Mathematical Society 2016
Draft Solutions - 2016 CMO
❽✌❾❈❿➁➀▼➂✎➃➄❾✹➅✌➆
♣ ❈❚ ❉❁P❨❣✌P✹❉❢❄❢❀❃❪❇❭✿❤❃P✹❉♠❊❅❚❱❤✐❭▼❄❢❀❙❚❱❂ ⑩ ➥ ➦ ❍ ➥ ➭ ❍❇❑❇❑❇❑☎❍ ➥ ➦✗➧ ❶❘✉✕❤❙✾❇❄❺➳ ➫➤➥✣➭➦ → ➥✣➭➭ →➵➯❇➯❇➯❷→ ➥✣➭➦✗➧ ✸
✺✽✼✿✾❖❂
◆
➥ ➩ ➫ ✷ → ➥➳ ➩➭ ➸ ◆ ➥ ➩➭ ➛✈➳ ➥ ➩ → ➳ ➫ ▲s❑
➺ ✾❇❄ ➔ P❈❂✝❩✙➣✽❳❝✾❧❄❢✼▼✾⑦❉❢❚✶❚❈❄❢❊➎❚❈⑤✜❄❢✼▼✾♠❻✣❭✝P❈❩▼❉❋P✹❄❢❀❃❪❞◆✰➻ ➭ ➛❯➳♦➻ → ➳ ➫ ▲s✉✿❊❢❚❁⑤✒❚❈❉➙✾❖P❈❪❋✼
➲ ✉ ➥✹➩✱➫ ➔ ❚❈❉ ➥✹➩✱➫ ➣☎✸✮➒❡✾✮P❈❤✐❊❅❚❯✼✿P♦♥❈✾ ➔ ➣ ➫ ➳✰➼✹◆ ⑩ ❳✣❴❫➽⑦❀❙✾❇❄❢Ps➾➚❊q⑤✒❚❈❉❋❜✧❭✿❤❃Ps✉✕⑤➡❚❈❉
✾❥r▼P❈❜t❣✝❤❙✾☎❶❘✸
➜ ⑤✯P❈❤✐❤❝❄❢✼▼✾ ➥ ➩ P✹❉❋✾❺✾❖❻✣❭✝P❈❤➄✉▼❄❢✼▼✾❖❂
◆ ➞
➥✰➩✜➫ ✷ → ✷❇▲ ➫ ①
⑤✒❚❈❉✽P❈❤❃❤ ➲ ✸❛➪❞❄❢✼✿✾❇❉❢◗❧❀❃❊❅✾❈✉▼❤❙✾❇❄❧① →✇➶ ❚❈⑤✯❄❢✼▼✾ ➥✹➩ ❳❝✾ ➔ ✉✿P❈❂✝❩③❤❙✾❇❄✽①⑦➛ ➶ ❚❈⑤✕❄❢✼▼✾ ➥✹➩ ❳❝✾
➣☎✉▼◗❧✼▼✾❇❉❋✾❺▲✖➹ ➶✙➘ ➈▼✸❹✺❧✼▼✾❖❂✂❳✶❴❨❄❢✼▼✾❺⑥⑦➴③➍➬➷♠➴➮❀❃❂▼✾❖❻✶❭✿P❈❤❃❀❙❄❵❴❈✉
◆ ➔ ➣ ➫ ➳ ➫ ⑩ ① →➱➶ ❶ ➔ ➭ →➋⑩ ①♠➛ ➶ ❶✃➣ ➭❺❐ ■ ➔ ➣✹❒ ■❈①♠➛ ➶ ➭ ❑
♣ ❉❋❚❱❜❮❄❢✼▼✾✂❆❱❀❙♥❈✾❖❂✇✾❖❻✶❭✿P✹❄❢❀❙❚❱❂✿❊❖✉ ➥ ➩ ❐ ✷✙⑤➡❚❈❉❨P❈❤❃❤ ➲ ✉➎❊❢❚ ➔ P❈❂✿❩✥➣✮P✹❉❢✾✂❣❝❚❱❊❢❀❙❄❢❀❃♥❈✾❈✸
✺✽✼✿✾❖❂➱❰ ■❈①♠➛ ➶ ➭ ➘ ❏ ➸ ■❈①❨➛ ➶ ➭ ➘ ➝ ➸ ➶ ➭ ❐ ✷❖◆ ➸ ➶ ➫ ➈▼✸ÐÏ✻✾❖❂✿❪❥✾❈✉
◆ ➔ ➣ ➫ ➝ ➔ ➭ → ➣ ➭ ➸ ⑩ ➣➙➛➠❏ ➔ ❶ ➭❧➫ ▲ ➸ ➣ ➫ ❏ ➔ ✸
⑥❞❩✿❩✿❀❃❂✿❆✧P❈❤❃❤✜❆❱❀❙♥❈✾❖❂✂❄❅✾❖❂③✾❖❻✶❭✿P✹❄❢❀❙❚❱❂✝❊❇✉▼◗❛✾q❆❈✾❇❄
➥ ➦ → ➥ ➭ →➉➯❇➯❇➯✹→ ➥ ➦✗➧ ➫ ✷❖◆✶❑
Ñ ❭▼❄ ➥ ➦ → ➥ ➭ →➤➯❇➯❇➯▼→ ➥ ➦✗➧ ➫ ➝ ➔✮→ ➣ ➫ ✷❖■ ➔ ✉✽❊❢❚ ➔ ➫ ✷❖◆❱➼✶✷❖■ ➫ ➈✣➼✹❏✈P❈❂✿❩
➣ ➫ ➈▼✸✩✺❧✼▼✾❇❉❢✾❇⑤✒❚❈❉❢✾❈✉❧❄❢✼▼✾❡❊❅❚❱❤✐❭▼❄❢❀❙❚❱❂✿❊✙P✹❉❢✾ ⑩ ➞❱➼✹①✶❍❋➞❱➼✹①✶❍❇❑❇❑❇❑❈❍❋➞❱➼✹①❱❶✙P❈❂✿❩➋P❈❤❃❤❞❄❅✾❖❂
❣❝✾❇❉❋❜✧❭▼❄❢P✹❄❢❀❙❚❱❂✿❊➙❚❈⑤ ⑩ ➈✣➼✹❏✶❍❅➈✣➼✹❏✶❍❇❑❇❑❇❑❱❍❅➈✣➼✹❏✶❍❅➈✣❶❘✸
❏✶✸ ♣ ❃❀ ❂✿❩✧P❈❤❃❤✣❣❝❚❱❤❙❴✶❂✿❚❱❜✖❀❃P❈❤❃❊✕Ò ⑩ ➻❝❶✕◗❧❀❃❄❢✼❁❀❃❂✣❄❅✾❇❆❈✾❇❉✵❪❥❚✶✾❇Ó➇❪❇❀❙✾❖❂✣❄❢❊⑧❊❢❭✿❪❋✼●❄❢✼✝P✹❄ÔÒ ⑩ Ò ⑩✒Õ ❶ →
Õ ❶➎❃❀ ❊✱P✖❣✿❉❋❀❃❜t✾♠❂✶❭✿❜❁❳✳✾❇❉✱⑤✒❚❈❉❧❀❃❂ ❼ ❂✿❀❙❄❅✾❖❤❙❴✖❜➇P❈❂❷❴✂❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊ Õ ✸
Ö ➅✿×❘Ø❛Ù❥Ú❇➆ Ò ⑩✒Õ ❶ ➫ÜÛ ◗❧✼▼✾❇❉❢✾ Û ❀✐❊❺P✂❣✿❉❋❀✐❜t✾t❂✣❭✿❜❁❳❝✾❇❉❺P❈❂✿❩❡Ò ⑩✒Õ ❶ ➫ ➛❧■ Õ③→ ➣
◗❧✼✿✾❇❉❢✾❺➣⑦❀❃❊➙❚✶❩✝❩☛✸
❽✌❾❈❿➁➀▼➂✎➃➄❾✹➅ ➨➉Ý ❚❈❄❅✾➠❄❢✼✿P✹❄❯❀❙⑤●Ò ⑩✒Õ ❶ ➫ ▲➱❄❢✼✿✾❖❂ÞÒ ⑩ Ò ⑩✒Õ ❶ →ßÕ ❶ ➫ Ò ⑩✒Õ ❶ ➫ ▲
◗❧✼✝❀❃❪❋✼✈❀✐❊❺❂▼❚❈à❄q❣✝❉❋❀❃❜t✾❈✸ ➺à☎á ✾❇❄❁Ò ⑩ ➻❝❶⑦❳❝✾➇P④❩▼✾❇❆❈❉❢✾❇✾ ➶ ❣❝❚❱❤❙❴s❂▼❚❱❜✖❀❃P❈❤✯❚❈⑤✱❄❢✼▼✾❨⑤➡❚❈❉❋❜
Ò ⑩ ➻☛❶ ➫ ➔✶à ➻ →✇➔✶à☎á ➦ ➻ ➦ →➉➯❇➯❇➯✰→✇➔ ➧ P❈❂✿❩✂❂▼❚❈❄❅✾❺❄❢✼✝P✹❄❧❀❙⑤ÔÒ ⑩✒Õ ❶❞➫ â ▲t❄❢✼▼✾❖❂
Ò ⑩ Ò ⑩✒Õ ❶ →ÐÕ ❶✯➛✈Ò ⑩✒Õ ❶ ➫
➔✣àsã❙⑩ Ò ⑩✒Õ ❶ →ÐÕ ❶ à ➛ Õ à♦ä →✇➔✣à☎á ➦ ã❙⑩ Ò ⑩✒Õ ❶ →✇Õ ❶ à♦á ➦ ➛ Õ à☎á ➦ ä →å➯❇➯❇➯✹→➱➔ ➦ Ò ⑩✒Õ ❶
■
© Canadian Mathematical Society 2016
Draft Solutions - 2016 CMO
◗❧✼✝❀❃❪❋✼❨❀❃❊❹❩✿❀❃♥✶❀❃❊❋❀❙❳✝❤❙✾➙❳✶❴ ⑩ Ò ⑩✒Õ ❶ →④Õ ❶✝➛ Õ ➫ Ò ⑩✒Õ ❶❘✸⑧✺✽✼▼✾❇❉❋✾❇⑤➡❚❈❉❢✾⑦❀❙⑤☛Ò ⑩ Ò ⑩✒Õ ❶ →④Õ ❶
❀❃❊✵❣✿❉❋❀❃❜t✾➙❄❢✼▼✾❖❂➇✾❖❀❙❄❢✼▼✾❇❉❹Ò ⑩✒Õ ❶ ➫Þæ ✷✱❚❈❉➎Ò ⑩ Ò ⑩✒Õ ❶ →✂Õ ❶ ➫Þæ Ò ⑩✒Õ ❶ ➫ÐÛ ⑤✒❚❈❉❛❊❅❚❱❜t✾
❣✿❉❘❀❃❜t✾✵❂✶❭✿❜q❳❝✾❇❉ Û ✸⑧↕▼❀❃❂✿❪❥✾❹Ò ⑩ ➻☛❶✜❀✐❊✏P✻❣✳❚❱❤❃❴✶❂▼❚❱❜➇❀❃P❈❤➄✉♦❀❙❄✕⑤➡❚❱❤✐❤❙❚♦◗❧❊✕❄❢✼✿P✹❄✯Ò ⑩✒Õ ❶ ➫Þæ ✷
⑤✒❚❈❉Ô❚❱❂✿❤❙❴ ❼ ❂✿❀❃❄❅✾❖❤❙❴❺❜✖P❈❂✣❴❁❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊ Õ ✸✯✺❧✼▼✾❇❉❢✾❇⑤✒❚❈❉❢✾➙✾❖❀❙❄❢✼▼✾❇❉⑧Ò ⑩✒Õ ❶ ➫ Ò ⑩ Ò ⑩✒Õ ❶ →●Õ ❶
⑤✒❚❈❉❺❀❃❂ ❼ ❂✝❀❙❄❅✾❖❤❙❴✂❜✖P❈❂✣❴④❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊ Õ ❚❈❉❺Ò ⑩✒Õ ❶ ➫ ➛✽Ò ⑩ Ò ⑩✒Õ ❶ →✥Õ ❶❧⑤➡❚❈❉❺❀❃❂ ❼ ❂✿❀❙❄❅✾❖❤❙❴
❜✖P❈❂✣❴q❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊ Õ ✸Ô↕▼❭▼❣✿❣❝❚❱❊❅✾✵❄❢✼✿P✹❄⑧Ò ⑩✒Õ ❶ ➫ Ò ⑩ Ò ⑩✒Õ ❶ →✧Õ ❶✏⑤✒❚❈❉✯❀❃❂ ❼ ❂✿❀❙❄❅✾❖❤❙❴⑦❜✖P❈❂✣❴
❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊ Õ ✸✵✺❧✼✿❀❃❊✱❀❃❜t❣✌❤❃❀❙✾❖❊➎❄❢✼✿P✹❄✽❄❢✼▼✾♠❣❝❚❱❤❙❴s❂▼❚❱❜✖❀✐P❈❤✳Ò ⑩ Ò ⑩ ➻❝❶ → ➻❝❶✯➛✈Ò ⑩ ➻☛❶➙✼✿P❈❊
❀❃❂ ❼ ❂✿❀❙❄❅✾❖❤❙❴❁❜✖P❈❂❷❴t❉❋❚✣❚❈❄❢❊➎P❈❂✿❩❨❄❢✼✣❭✿❊❛❀❃❊❹❀✐❩▼✾❖❂❷❄❢❀✐❪❇P❈❤❃❤❙❴✧ç❇✾❇❉❢❚▼✸❹✺✽✼▼✾❇❉❋✾❇⑤➡❚❈❉❢✾✻Ò ⑩ Ò ⑩ ➻❝❶ →
➻☛❶ ➫ Ò ⑩ ➻❝❶➎✼✿❚❱❤❃❩✿❊➎❀❃❩▼✾❖❂✣❄❢❀❃❪❇P❈❤❃❤❙❴❈✸ Ý ❚♦◗Þ❂▼❚❈❄❅✾♠❄❢✼✿P✹❄✱❀❙⑤ ➶ ❐ ■❁❄❢✼▼✾❖❂✂Ò ⑩ Ò ⑩ ➻☛❶ → ➻❝❶
✼✿P❈❊✯❩▼✾❇❆❈❉❢✾❇✾ ➶ ➭ ◗✻✼✿❀❃❤❙✾✵Ò ⑩ ➻☛❶✜✼✝P❈❊✯❩✿✾❇❆❈❉❢✾❇✾ ➶ ✉✹◗❧✼✝❀❃❪❋✼❁❀❃❊✕❂✿❚❈❄✕❣❝❚❱❊❢❊❢❀❃❳✝❤❙✾❈✸✜✺✽✼▼✾❇❉❋✾❇⑤➡❚❈❉❢✾
Ò ⑩ ➻☛❶✽❀✐❊❧P✹❄⑦❜t❚❱❊❅❄⑦❤❃❀❃❂▼✾❖P✹❉✻◗❧❀❙❄❢✼❯Ò ⑩ ➻❝❶ ➫ ➔ ➻ → ➣⑦⑤➡❚❈❉⑦❊❅❚❱❜t✾●❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊ ➔ P❈❂✿❩❫➣♦✸
Ý ❚♦◗è❂▼❚❈❄❅✾❺❄❢✼✿P✹❄
Ò ⑩ Ò ⑩ ➻☛❶ → ➻☛❶ ➫ ➔✳⑩✎➔❺→ ✷♦❶❵➻ →✇➔ ➣ → ➣
P❈❂✿❩➠❄❢✼✣❭✝❊ ➔ ➫ ➔❝⑩✎➔●→ ✷♦❶qP❈❂✿❩ ➔ ➣ → ➣ ➫ ➣☎✸ ➜ ❄q⑤✒❚❱❤❃❤❙❚☎◗❧❊❺❄❢✼✿P✹❄ ➔ ➫ ▲③◗✻✼✿❀❃❪❋✼
❤❙✾❖P❈❩✝❊q❄❅❚❲❄❢✼▼✾✙❊❅❚❱❤❃❭▼❄❢❀❙❚❱❂ÐÒ ⑩✒Õ ❶ ➫ßÛ ◗❧✼✿✾❇❉❢✾ Û ❀❃❊❁P④❣✝❉❋❀❃❜t✾➇❂✶❭✿❜❁❳❝✾❇❉❖✸ Ñ ❴✈❄❢✼▼✾
❊❢P❈❜✖✾●P✹❉❋❆❱❭✿❜t✾❖❂✣❄❺❀❙⑤❛Ò ⑩✒Õ ❶ ➫ ➛❧Ò ⑩ Ò ⑩✒Õ ❶ →➱Õ ❶✻⑤✒❚❈❉❺❀❃❂ ❼ ❂✝❀❙❄❅✾❖❤❙❴✂❜✖P❈❂✣❴❲❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊
Õ ❄❢✼▼✾❖❂➋Ò ⑩ ➻❝❶ ➫ ➛✽Ò ⑩ Ò ⑩ ➻❝❶ → ➻❝❶❨✼▼❚❱❤✐❩✿❊❨❀✐❩▼✾❖❂❷❄❢❀✐❪❇P❈❤❃❤❙❴➱P❈❂✿❩➉Ò ⑩ ➻❝❶❨❀❃❊✙❤❃❀❃❂✿✾❖P✹❉
◗❧❀❃❄❢✼❡Ò ⑩ ➻❝❶ ➫ ➔ ➻ → ➣☎✸ ➜ ❂❡❄❢✼✿❀✐❊❺❪❇P❈❊❅✾❨❀❃❄❺⑤➡❚❱❤✐❤❙❚♦◗❧❊❺❄❢✼✿P✹❄ ➔ ➫ ➛ ➔✳⑩✎➔●→ ✷♦❶♠P❈❂✿❩
➔ ➣ → ➣ ➫ ➛❧➣☎✸✵✺✽✼✿❀❃❊❛❀❃❜t❣✝❤❃❀❃✾❖❊Ô❄❢✼✿P✹❄➙✾❖❀❙❄❢✼▼✾❇❉ ➔ ➫ ▲❁❚❈❉ ➔ ➫ ➛❧■✶✸ ➜ ⑤ ➔ ➫ ➛✻■♠❄❢✼▼✾❖❂
Ò ⑩✒Õ ❶ ➫ ➛❧■ Õ③→ ➣❁◗❧✼✝❀❃❪❋✼❫❀❃❊⑦❣✿❉❘❀❃❜t✾●⑤➡❚❈❉❁❊❅❚❱❜t✾✖❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊ Õ ❚❱❂✿❤❃❴④❀❙⑤➙➣❁❀✐❊⑦❚✶❩✿❩✜✸
Ý ❚❈❄❅✾⑦❄❢✼✿P✹❄✱❀❃❂✖❄❢✼✿❀✐❊❛❪❇P❈❊❅✾♠Ò ⑩ Ò ⑩✒Õ ❶ →❫Õ ❶ ➫ ■ Õ ➛④➣✱◗❧✼✿❀✐❪❋✼✮❀❃❊❛❀✐❂✿❩▼✾❇✾❖❩❨❣✿❉❋❀❃❜t✾❧⑤➡❚❈❉
❀❃❂ ❼ ❂✿❀❙❄❅✾❖❤❙❴✖❜➇P❈❂❷❴✙❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊ Õ P❈❊✽❤❃❚❱❂▼❆✖P❈❊✻➣❞❀❃❊✱❚s❩✿❩☛✸
é
➈▼✸ ➺ ❖P ♥✹P❈❜✖P❈❂③♥❈✾❇❉❋❊❋❭✿❊✱❄❢✼▼✾ ♣ ❤❙✾❖P ✸ ➺ ✾❇❄❺ê❁✉✜ë❨✉✜P❈❂✝❩❲ìí❳❝✾✧❣❝❚❱❊❢❀❙❄❢❀❙♥❈✾●❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊❇✉✏P❈❂✿❩
P❈❊❢❊❋❭✿❜t✾●êî➹Þëï➹ð■✹ê❁✸q⑥➤ñ✝✾❖P✂❀❃❊⑦P✹❄❞❄❢✼✿✾●❂✶❭✿❜❁❳❝✾❇❉⑦▲✮❚❱❂❫❄❢✼▼✾t❂✶❭✿❜❁❳✳✾❇❉⑦❤❃❀❃❂✿✾❈✸
✺✽✼✿✾③ñ✿✾❖P➠❪❇P❈❂➉❜t❚☎♥❈✾④❳✶❴④↔❅❭✿❜✖❣✝❀❃❂▼❆❡❄❅❚➊❄❢✼▼✾❯❉❋❀❃❆❱✼❷❄✖❳✶❴✥ê❦❚❈❉➇❳✶❴✥ë✮✸ Ñ ✾❇⑤✒❚❈❉❢✾
❄❢✼▼✾❧ñ✿✾❖Pq❊❅❄❢P✹❉❋❄❢❊❝↔❢❭✿❜t❣✝❀❃❂✿❆▼✉ ➺ P❖♥✹P❈❜✖P❈❂✮❪❋✼▼❚✶❚❱❊❅✾❖❊ ❼ ❂✿❀❙❄❅✾❖❤❃❴✧❜✖P❈❂✣❴t❀❃❂✣❄❅✾❇❉❢♥✰P❈❤✐❊ ➏❖ò➋→
✷✹❍ ò➵→ ■✶❍❇❑❇❑❇❑♦❍ ò➵→ êq➐⑦❪❥❚❱❂✝❊❢❀❃❊❅❄❢❀❃❂✿❆q❚❈⑤✜êÞ❪❥❚❱❂✿❊❢✾❖❪❇❭▼❄❢❀❙♥❈✾❞❣✳❚❱❊❋❀❙❄❢❀❙♥❈✾❧❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊❇✉▼P❈❂✿❩
❣✝❤✐P❈❪❥✾❖❊♠❤❃P❖♥✹P③P✹❄❺P❈❤❃❤Ô❚❈⑤➎❄❢✼▼✾✖❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊q❀✐❂④❄❢✼✿✾✖❀❃❂❷❄❅✾❇❉❋♥✰P❈❤❃❊❖✸●✺❧✼▼✾t❀❃❂❷❄❅✾❇❉❋♥✰P❈❤❃❊♠❜✧❭✿❊❅❄
❳❝✾♠❪❋✼▼❚❱❊❢✾❖❂❯❊❅❚✖❄❢✼✝P✹❄ ➨
⑩ ➃ ❶➙P❈❂❷❴✮❄✗◗➎❚❨❩✝❀❃❊❅❄❢❀❃❂✿❪❥❄✱❀❃❂✣❄❅✾❇❉❢♥✹P❈❤❃❊✱P✹❉❢✾q❩✝❀❃❊✒↔❅❚❱❀❃❂❷❄✱P❈❂✿❩❯❂▼❚❈❄✽P❈❩✰↔❢P❈❪❥✾❖❂❷❄❇ó
⑩ ➃✒➃ ❶✕❄❢✼▼✾❇❉❋✾✽P✹❉❢✾❧P✹❄✵❤❙✾❖P❈❊❅❄✵ìÞ❣✳❚❱❊❋❀❙❄❢❀❙♥❈✾✱❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊⑧◗❧❀❙❄❢✼t❂✿❚❺❤❃P❖♥✹P⑦❳❝✾❇❄✗◗➎✾❇✾❖❂✖P❈❂❷❴●❄❵◗❛❚
❀❃❂✣❄❅✾❇❉❢♥✹P❈❤❃❊❇ó▼P❈❂✿❩
⑩ ➃✒➃➡➃ ❶➙❂▼❚✖❤❃P♦♥✰P➇❀❃❊➙❣✝❤❃P❈❪❥✾❖❩③P✹❄✻P❈❂❷❴✂❀❃❂✣❄❅✾❇❆❈✾❇❉❧❤❙✾❖❊❢❊✽❄❢✼✿P❈❂③ìt✸
❸❹❉❢❚♦♥❈✾③❄❢✼✝P✹❄✖❄❢✼▼✾❯❊❢❜➇P❈❤❃❤❙✾❖❊❅❄tìï⑤✒❚❈❉✖◗❧✼✿❀✐❪❋✼✇❄❢✼▼✾③ñ✝✾❖P✈❪❇P❈❂❲↔❢❭✿❜t❣➱❚♦♥❈✾❇❉✙P❈❤❃❤➙❄❢✼▼✾
❀❃❂✣❄❅✾❇❉❢♥✹P❈❤❃❊✙P❈❂✿❩ÜP❖♥❈❚❱❀✐❩ÞP❈❤❃❤❞❄❢✼▼✾✈❤❃P♦♥✰Ps✉❞❉❢✾❇❆❱P✹❉❋❩✿❤❃✾❖❊❢❊✙❚❈⑤❁◗✻✼✿P✹❄ ➺ P❖♥✹P❈❜✖P❈❂Þ❩✿❚✣✾❖❊❇✉
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© Canadian Mathematical Society 2016
Draft Solutions - 2016 CMO
❀❃❊❫ì ➫ ⑩✒Õ ➛í✷♦❶➬ê → ë✮✉●◗❧✼▼✾❇❉❢✾ Õ ❀✐❊④❄❢✼✿✾➱❣✳❚❱❊❋❀❙❄❢❀❙♥❈✾✇❀❃❂❷❄❅✾❇❆❈✾❇❉✈❊❋❭✿❪❋✼è❄❢✼✿P✹❄
ê
ê
Õ➇→ ✷ ➘ ëè➛❡êß➹ Õ ✸
❽✌❾❈❿➁➀▼➂✎➃➄❾✹➅✌➆ ➺ ✾❇❄❧ë ➫ ê →✥ô ◗❧✼▼✾❇❉❢✾❺ê♠➼ ⑩✒Õ✮→ ✷♦❶ ➘➉ô ➹✥ê♠➼ Õ ✸
♣ ❀❙❉❋❊❢❄❇✉✿✼▼✾❇❉❢✾❺❀❃❊✱P❈❂③❀✐❂▼⑤➡❚❈❉❘❜✖P❈❤☛❊❅⑨❈✾❇❄❢❪❋✼④❚❈⑤✯❄❢✼▼✾❺❣✿❉❢❚✶❚❈⑤➬✸
➺ P❖♥✹P❈❜✖P❈❂☛➾➚❊✱❊❅❄❅❉❋P✹❄❅✾❇❆❈❴ ➨ ➓❞❊❅✾♠❚❱❂✿❤❙❴✖❊❋P✹⑤➡✾♠❀❃ä ❂✣❄❅✾❇❉❢♥✹P❈❤❃❊➎◗❧❀❙❄❢✼ Õ ê →➊ô ➛➠✷♠❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊❇✸
✺✽✼✿✾❁ñ✿✾❖P❨◗❧❀❃❤✐❤✏❊❅❄❢P✹❉❢❄♠P✹❄♠❣❝❚❱❊❢❀❙❄❢❀❃❚❱❂ ã ✷✹❍ ô ⑤✒❉❢❚❱❜➑❄❢✼▼✾t❤❙✾❇⑤✒❄❇✉✳◗✻✼✿❀❃❪❋✼❲❣✝❭✿❄❢❊✻✼✿❀✐❜✫P✹❄
❣❝❚❱❊❢❀❙❄❢❀❃❚❱❂ ã Õ ê❁❍ Õ ê →åô ➛➉✷ ä ⑤✒❉❢❚❱❜✫❄❢✼▼✾●❉❋❀❙❆❱✼✣❄❇✸♠⑥❧⑤✒❄❅✾❇❉ Õ ➛➉✷✽↔❢❭✿❜t❣✌❊❇✉✳✼✿✾❁◗❧❀✐❤❃❤
❊❅❄❢❀✐❤❃❤☛✼✿P❖♥❈✾ Õ ê➉➛ ⑩✒Õ ➛✥✷♦❶ ⑩ ê →åô ❶ ➫ ê➋➛ ⑩✒Õ ➛å✷♦❶ ôöõ❦ô ❩✿❀❃❊❅❄❢P❈❂✝❪❥✾❺❄❅❚➇❆❈❚▼✉
◗❧✼✝❀❃❪❋✼t❀❃❊⑧❂✿❚❈❄✵✾❖❂▼❚❱❭▼❆❱✼t⑤✒❚❈❉❹P⑦❳✝❀❃❆➎↔❅❭✿❜✖❣t❄❅❚q❪❇❤❙✾❖P✹❉✵❄❢✼▼✾✻❤❃P❖♥✹Ps✸⑧✺✽✼✣❭✝❊❇✉❷✼▼✾✽❜✧❭✿❊❅❄✵❩▼❚
P✹❄❞❤❙✾❖P❈❊❅❄ Õ ↔❅❭✿❜✖❣✝❊✽❀❃❂✂❄❢✼✿✾❁❊❢P✹⑤✒✾❁❀❃❂✣❄❅✾❇❉❢♥✹P❈❤➄✉✝❳✌❭▼❄✽❄❢✼✿P✹❄❇➾➚❊✻❣❝❚❱❊❢❊❢❀❙❳✝❤❃✾❞❚❱❂✝❤❙❴✙◗❧❀❙❄❢✼③P❈❤❃❤
❊❢❜➇P❈❤❃❤s↔❢❭✿❜t❣✝❊❖✉✌P❈❂✿❩❲⑤✎❭▼❉❢❄❢✼▼✾❇❉❋❜✖❚❈❉❢✾❁❀❃❊❞❀❃❜t❣❝❚❱❊❢❊❋❀❙❳✝❤❙✾♠❀❙⑤⑧❄❢✼▼✾●❊❅❄❢P✹❉❢❄❢❀✐❂▼❆❨❣❝❚❱❊❢❀❃❄❢❀❙❚❱❂❯❀❃❊
ô ✸✵✺✽✼✿❀❃❊➙❆❱❀❙♥❈✾❖❊✻✼✿❀❃❜î❊❅❄❢P✹❉❋❄❢❀❃❂▼❆t❣❝❚❱❊❢❀❙❄❢❀❙❚❱❂❡✷q✼✿❀❙❆❱✼▼✾❇❉✽❀❃❂✙❄❢✼✿✾q❂▼✾❥rs❄❧❊❢P✹⑤✒✾q❀❃❂✣❄❅✾❇❉❢♥✰P❈❤✟✉
❊❅❚✖❊❢❚✣❚❱❂▼✾❇❉❧❚❈❉✻❤❃P✹❄❅✾❇❉✽❄❢✼✿✾❺ñ✿✾❖Pt❀❃❊✱❆❈❚❱❀✐❂▼❆t❄❅❚➇✼✿❀❙❄✱❄❢✼✿✾❺❤❃P❖♥✹Ps✸
♣ ❤❙✾❖Ps➾➚❊♠❊❅❄❅❉❘P✹❄❅✾❇❆❈❴ ➨ ✺✽✼▼✾●ñ✿✾❖Pq↔❢❭✿❊❅❄⑦❩▼❚✣✾❖❊❞❚❱❂▼✾t❀❃❂✣❄❅✾❇❉❢♥✹P❈❤ÔP✹❄⑦P✮❄❢❀❃❜✖✾❈✸ ➜ ⑤❹❄❢✼▼✾t❊❢P✹⑤➡✾
❀❃❂✣❄❅✾❇❉❢♥✹P❈❤▼✼✿P❈❊✵P✹❄➎❤❙✾❖P❈❊❅❄ Õ ê →③ô ❀❃❂✣❄❅✾❇❆❈✾❇❉❋❊❛❀❃❂t❀❙❄❇✉❱❄❢✼▼✾❧ñ✿✾❖P❺✼✿P❈❊✵❩✝❀❃❊❅❄❢P❈❂✿❪❥✾✽÷ õ➱Õ ê
❄❅❚✙❆❈❚✙❄❅❚✙❄❢✼▼✾✧❂✿✾❥r✶❄♠❤❃P♦♥✰P✮◗❧✼▼✾❖❂❲❀❙❄⑦❊❅❄❢P✹❉❢❄❢❊❇✸❺ø❧✾❇❣❝✾❖P✹❄❅✾❖❩✿❤❙❴❯❩▼❚✮❳✝❀❙❆❺↔❅❭✝❜t❣✝❊❞❭✿❂❷❄❢❀✐❤
÷➊❀❃❊●❳❝✾❇❄✗◗➎✾❇✾❖❂Þ✷③P❈❂✿❩ ô ❜t❚✶❩✱ê❁✉➎❄❢✼▼✾❖❂✥❊❢❜✖P❈❤✐❤✜↔❢❭✿❜t❣✌❊✧❭✝❂❷❄❢❀❃❤➙❄❢✼✿✾✂❉❢✾❖❜✖P❈❀❃❂✿❀✐❂▼❆
❩✿❀✐❊❅❄❢P❈❂✿❪❥✾✙❀❃❊✧❳✳✾❇❄❵◗❛✾❇✾❖❂➵✷✮P❈❂✝❩ ô ✉✵❄❢✼▼✾❖❂✇P ❼ ❂✝P❈❤❹❳✝❀❃❆➇↔❢❭✿❜t❣✏✸✈✺✽✼✿❀❃❊❁◗❛❚❈❉❋⑨✶❊tP❈❊
❤❙❚❱❂✿❆❁P❈❊➙❄❢✼▼✾ ❼ ❉❋❊❅❄❛❣✝P✹❉❢❄➙❩▼❚✶✾❖❊❇✸✵Ï❧❚☎◗❛✾❇♥❈✾❇❉♦✉✿◗❛✾⑦❆❈✾❇❄✱P✹❄✱❤❙✾❖P❈❊❅❄ Õ ❳✌❀❙❆✱↔❢❭✿❜t❣✝❊➎❊❢❀❃❂✝❪❥✾
ñ✿❚✶❚❈❉ ⑩❅⑩ ÷●➛➌✷♦❶❅➼ùê♠❶✻❪❇P❈❂❫❂▼✾❇♥❈✾❇❉⑦❆❈❚✮❩▼❚☎◗❧❂④❄✗◗➎❚✮⑤✒❉❢❚❱❜➑P➇❳✌❀❙❆♠↔❢❭✿❜t❣ ⑩ ❚❈❉❞◗➎✾❈➾➚❩❲❳❝✾
❩▼❚❱❂✿✾❁❩▼❚❱❀❃❂✿❆✖❳✝❀❙❆❺↔❅❭✿❜✖❣✝❊❘❶❘✉✌❊❢❚❨◗➎ä ✾❁❆❈✾❇❄ Õ ❳✌❀❙❆♠↔❢ä ❭✿❜t❣✝❊❖✉✝P❈❂✿❩④❄❢✼✣❭✿❊✻◗➎✾✧P✹❉❢✾✧ä ❆❈❚✶❚✶❩
❀❙⑤❛÷❹❜t❚s❩✽êú❀❃❊♠❀❃❂❫P❈❂❷❴❲❚❈⑤ ã ✷✹❍ ô ✉ ã➚ôÜ→ ✷✹❍❋■ ô ✉❹✸❇✸❖✸ ã Õ✕ôÜ→ ✷✹❍ ⑩✒Õ❯→ ✷♦❶ ô ✉✏❳✝❭✿❄
❄❢✼✿P✹❄❇➾➚❊❧✾❇♥❈✾❇❉❢❴✶❄❢✼✿❀❃❂▼❆▼✸
é
➺ ✾❇❄ ô ä ➫ ëû➛➋êq✸➉➒❡✾④❊❢✼✝P❈❤❃❤✱◗✽❉❋❀❃❄❅✾✂❚❱❭▼❉❨❀❃❂✣❄❅✾❇❉❢♥✹P❈❤❃❊t❚❈⑤♠❤❃P♦♥✰P➊❀❃❂➱❄❢✼▼✾④⑤➡❚❈❉❋❜
⑩✎ü ➩ ❍❢ý ➩ ➫ ➏♦ü ➩ → ✷✹❍ ü ➩ → ■✶❍❇❑❇❑❇❑☎❍❢ý ➩ ➐❱✉▼◗✻✼▼✾❇❉❢✾♠ý ➩ ➫ ü ➩ → êèP❈❂✿❩✙ý ➩ á ➦ ➹ ü ➩
⑤✒❚❈❉♠✾❇♥❈✾❇❉❢❴ ➲ ❐ ✷✹✸➇➒❡✾➇P❈❤❃❊❅❚③❤❙✾❇❄❺ý ➧ ➫ ▲s✸t➒❡✾❨❊❢✼✿P❈❤❃❤⑧P❈❤❃❊❢❚✂❉❢✾❇❣✿❉❢✾❖❊❅✾❖❂✣❄❺P✙❣✝P✹❄❢✼
⑤✒❚❈❉♠❄❢✼▼✾tñ✿✾❖P③P❈❊qP✂❊❅✾❖❻✣❭✿✾❖❂✿❪❥✾t❚❈⑤✱❀❃❂❷❄❅✾❇❆❈✾❇❉❘❊♠➻ ➧ ❍❅➻ ➦ ❍❅➻ ➭ ❍❇❑❇❑❇❑✜◗❧✼▼✾❇❉❋✾t➻ ➧ ➫ ▲③P❈❂✿❩
➻sþ✱➛❡➻✶þ á ➦✽ÿ ➏ ê❁❍❢ë➇➐❁⑤➡❚❈❉✽✾❇♥❈✾❇❉❋❴✁ ❐ ▲s✸
Ý ❚♦◗è✼▼✾❇❉❢✾❺❀❃❊✱P✖❩✿✾❇❄❢P❈❀❃❤❙✾❖❩✙❣✿❉❢❚✶❚❈⑤➬✸
♣ ❀❙❉❋❊❢❄❇✉ÔP❈❊❋❊❢❭✿❜t✾✙ì❮➹ ⑩✒Õ ➛Þ✷♦❶➬ê → ë ⑩ ➫ Õ ê →Þô ❶ ➨ ◗❛✾✙❜❁❭✝❊❅❄❁❣✝❉❢❚♦♥❈✾✙❄❢✼✿P✹❄
➺ P❖♥✹P❈❜✖P❈❂➊✼✝P❈❊❁P❯◗❧❀❃❂✿❂✿❀✐❂▼❆✂❊❅❄❅❉❋P✹❄❅✾❇❆❈❴❈✸ ➺ ✾❇❄ ü ➩ ➫ ý ➩ á ➦ →åÕ ê →Þô ➛Ü✷➇⑤➡❚❈❉
✾❇♥❈✾❇❉❢❴ ➲ ❐ ✷✹✸ ⑩ ➪❞❳✝❊❅✾❇❉❋♥❈✾❺❄❢✼✿P✹❄ Õ ê →✥ô ➛✥✷ ❐ ìt✸➁❶
⑥❞❊❢❊❢❭✿❜✖✾❹❄❢✼✿P✹❄✯❄❢✼▼✾❹ñ✝✾❖P✻✼✿P❈❊✯P❈❂✧❀❃❂ ❼ ❂✿❀❃❄❅✾✵❣✝P✹❄❢✼❁❄❢✼✿P✹❄✯P♦♥❈❚❱❀❃❩✿❊ÔP❈❤❃❤❷❄❢✼▼✾➎❤✐P❖♥✹Ps✉✰◗✻✼✿❀❃❪❋✼
➈
© Canadian Mathematical Society 2016
Draft Solutions - 2016 CMO
ä
❜t✾❖P❈❂✝❊✱❄❢✼✿P✹❄✽➻sþ✖ÿ â ✎⑩ ü ➩ ❍❢ý ➩ ➡⑤ ❚❈❉✻P❈❤❃❤ ➲ ❍✂ ❐ ✷✹✸ ♣ ❚❈❉✻✾❖P❈❪❋✼ ➲ ❐ ✷✹✉✝❤❃✾❇❄
✄
➩ ➫
❜➇P✰r ➏ ➻ þ ➨ ➻ þ ➘✥ü ➩ ✵
➐ ❍ ò ➩ ➫ ✖
❜ ❀❃❂ ➏ ➻ þ ➨ ➻ þ õ ý ➩ ➐✵❍
➘ ü ➩ ✵
P❈❂✿❩ ☎ ⑩✒➲ ❶ ➫ ❜✖✰P r ➏ ➨ ➻ þ ➌
➐ ❑
⑥❞❤❃❊❅❚✖❤❙✾❇❄ ò ➧ ➫ ▲s✸➎✺✽✼▼✾❖❂✙⑤✒❚❈❉ ➲ ❐ ✷⑦◗❛✾q✼✝P❖♥❈✾
✄
➻✝✆✟✞ ➩✡✠
➩q➫
➻☛✆✟✞ ➩☞✠✍✌✕➦ ❑
ò ➩❺➫
P❈❂✝❩
⑥❞❤❃❊❅❚▼✉▼⑤✒❚❈❉✽✾❇♥❈✾❇❉❢❴ ➲ ❐ ✷✹✉✿◗❛✾❁✼✿P❖♥❈✾
⑩ P❷❶ ò ➩ ➫ ✄ ➩ → ë ⑩ ❳✳✾❖❪❇P❈❭✝❊❅✾ ✄ ➩ → ê ➘ûü ➩ → ê ➫ ý ➩ ❶❘ó
⑩ ❳✳❶ ü ➩ ❐ ✄ ➩ õûü ➩ ➛ ô➤⑩ ❊❢❀❃❂✿❪❥✾ ✄ ➩q➫ ò ➩ ➛➇ë õ ý ➩ ➛✖ë ➫ ü ➩ → ê❲➛✖ë✖❶❘ó
P❈❂✿❩
⑩ ❪♦❶➙ý ➩ ➹ ò ➩ ➘ ý ➩ →✥ôî⑩ ❊❢❀❃❂✝❪❥✾ ò ➩ ➫ ✄ ➩ → ë ➘ûü ➩ → ë ➫ ý ➩ →✥ô ❶❘✸
➟➙❤✐P❈❀❃❜❦✷ ➨ ☎ ⑩✒➲✵→ ✷♦❶ ➫ ☎ ⑩✒➲ ❶ →➋Õ❲→ ✷✙⑤➡❚❈❉t✾❇♥❈✾❇❉❢❴ ➲ ❐ ✷✹✸ ⑩ ✺❧✼✿P✹❄✖❀❃❊❇✉➎P✹⑤➡❄❅✾❇❉
↔❢❭✿❜t❣✝❀✐❂▼❆❺❚♦♥❈✾❇❉✱❚❱❂✿✾⑦❀✐❂❷❄❅✾❇❉❢♥✹P❈❤✌❚❈⑤✕❤❃P♦♥✰Ps✉s❄❢✼▼✾⑦ñ✿✾❖P✧❜❁❭✝❊❅❄✱❜✖P✹⑨❈✾⑦✾❥rsP❈❪❥❄❢❤❙❴ Õ ↔❢❭✿❜t❣✌❊
❳❝✾❇⑤✒❚❈❉❢✾➎↔❅❭✝❜t❣✝❀❃❂▼❆✧❚♦♥❈✾❇❉✻❄❢✼▼✾q❂▼✾❥rs❄✽❀❃❂✣❄❅✾❇❉❢♥✹P❈❤☛❚❈⑤⑧❤❃P❖♥✹Ps✸➁❶
✎❺✁✰✡☛✡✝✆ ➨
➻
✆✟✞ ➩☞✠✏✌✒✑✓✕✌ ➦
➘
→ ë Õ
ò ➩ → ë Õ
➫
➹öý ➩ →✥ôå→✕✔ ê →
➫
➻
✆✟✞ ➩☞✠✏✕✌ ➦
➻
ü ➩☞✌✕➦ → ê → ✷Ô❑
ê
Õ✗✖
Õ
Ñ ✾❖❪❇P❈❭✿❊❅✾q❚❈⑤✯❄❢✼✿✾q❊❅❄❅❉❋❀❃❪❥❄✽❀✐❂▼✾❖❻✣❭✝P❈❤❃❀❙❄✗❴❈✉▼◗➎✾q✼✿P♦♥❈✾❺➻ ✆✟✞ ➩☞✠✍✌✒✑✘✌✕➦ ➘ ý ☞➩ ✌✕➦ ✉✌P❈❂✿❩③✼▼✾❖❂✿❪❥✾
✆✟✞ ☞➩ ✠✍✌✒✑✘✌✕➦ ➘➱ü ☞➩ ✌✕➦ ✸✵✺✽✼▼✾❇❉❋✾❇⑤➡❚❈❉❢✙✾ ☎ ⑩✒➲ ❶ →ÐÕ❨→ ✷ ➘ ☎ ⑩✒➲☛→ ✷♦❘❶ ✸ Ý ❥✾ r✶❄❇✉✝◗❛✾q✼✝P❖♥❈✾
➻✝✆✟✞ ➩☞✠✍✌✒✑✘✌✕➦ ❐
➫
õ
➫
õ
➻✝✆✟✞ ➩✡✠✏✌✕➦ → ê Õ
ò ➩ → ê Õ
ý ➩ → ê Õ
ü ➩☞✕
✌ ➦ ➛ ô✥→ ✷
ü ➩☞✕
✌ ➦ ➛✈ê → ✷
→ ê
✺✽✼✿✾❇❉❢✾❇⑤➡❚❈❉❋✾q✝
➻ ✆✟✞ ➩☞✠✍✌✒✑✘✌ ❐ ➻✝✆✟✞ ➩☞✠✏✌✒✑✓✕
➭ ✷❺⑤➡❚❱❤❃❤❃❚♦◗❧❊❇✸ ✌ ➦
☎ ⑩✒➲ ❶ →✇Õ❨→ ■✶✸➎➟➙❤❃P❈❀✐❜➑
õ
⑩ ❊❢❀❃❂✿❪❥✾ ô ➹✥ê⑦❶❥❑
ü ➩☞✕
✌ ➦ ✉❝P❈❂✝❩④✼▼✾❖❂✝❪❥✾✁☎ ⑩✒➲✏→ ✷♦❶❲➹
➟➙❤✐P❈❀❃❜î■ ➨ ➻ þ ✌✕➦ ➛✈➻ þ ➫ ê ⑤➡❚❈❉⑦P❈❤❃❤✒ ➫ ☎ ⑩✒➲ ❶ → ✷✹❍❇❑❇❑❇❑♦✚❍ ☎ ⑩✒➲✏→ ✷♦❶✵➛➌✷✹✉❝⑤✒❚❈❉❞P❈❤❃❤
➲ ❐ ✷✹✸ ⑩ ✺✽✼✿P✹❄❺❀❃❊❇✉☛❄❢✼✿✾ Õ ❀✐❂❷❄❅✾❇❉❋❜✖✾❖❩✿❀❃P✹❄❅✾❞↔❢❭✿❜t❣✝❊❞❚❈⑤✽➟➙❤❃P❈❀❃❜➮✷t❜✧❭✿❊❅❄❺P❈❤❃❤Ô❳❝✾●❚❈⑤
①
© Canadian Mathematical Society 2016
Draft Solutions - 2016 CMO
❤❙✾❖❂✿❆❈❄❢✼✂êq✸➁❶
✎❺✁✰✡☛✡✝✆ ➨ ➜ ⑤⑧➟➙❤✐P❈❀❃❜û■●❀❃❊✱⑤✎P❈❤❃❊❅✾❈✉▼❄❢✼▼✾❖❂
✄ ☞➩ ✌✕➦ ➫ ➻ ✆✟✞ ☞➩ ✌✕✛➦ ✠ ➫ ➻ ✆✟✞ ✡➩ ✠✏✌✒✑✓✌✕➦ ❐ ➻ ✆✟✞ ☞➩ ✠✏✌✕➦ →Ü⑩✒Õ ➛✥✷♦❶➬ê → ë
õ ý ➩ →ÐÕ ê →✥ô
➫ ü ☞➩ ✌✕➦ → ✷
õ ✄ ☞➩ ✕✌ ➦
◗❧✼✝❀❃❪❋✼✂❀❃❊✱P➇❪❥❚❱❂❷❄❅❉❋P❈❩✝❀❃❪❥❄❢❀❙❚❱❂☛✸❹✺❧✼✿❀❃❊➎❣✿❉❢❚☎♥❈✾❖❊❞➟➙❤❃P❈❀✐❜û■✶✸
➒❡✾q❪❇P❈❂④❂▼❚♦◗ð❪❥❚❱❂✿❪❇❤❃❭✿❩▼✾♠❄❢✼✝P✹❄
➻ ✆✟✞ ☞➩ ✌✕✛➦ ✠✍✌✕➦ ➫ ➻ ✆✟✞ ☞➩ ✠✏✌✒✑✓✌ ➭ ➫ ➻ ✆✟✞ ☞➩ ✠✏✌✕➦ →ÐÕ ê → ë➌ó
❀➄✸ ✾❈✸❙✉ ò ☞➩ ✌✕➦ ➫ ò ➩ →ÐÕ ê → ë ⑤✒❚❈❉✽✾❖P❈❪❋✼ ➲ ❐ ✷✹✸
✺✽✼✿✾❇❉❢✾❇⑤➡❚❈❉❋✾
ò ✡➩ ✌✕➦ ➛✈ý ☞➩ ✌✕➦ ➫ ò ➩ →ÐÕ ê → ë➮➛ ⑩ ý ➩ →✇Õ ê →✥ô ➛➱✷ → ê♠❶
➫ ò ➩ ➛✈ý ➩ → ✷Ô❑
Ï✻✾❖❂✿❪❥✾
ô ❐ ✢
ò ✜ ✌✕➦ ➛✈ý ✜ ✌✕➦ ➫ ò ➦ ➛✈ý ➦ →✥ô õöô
◗❧✼✝❀❃❪❋✼✂❀❃❊❧P✖❪❥❚❱❂❷❄❅❉❘P❈❩✿❀❃❪❥❄❢❀❙❚❱❂☛✸❛✺✽✼▼✾❇❉❢✾❇⑤✒❚❈❉❢✾q❂▼❚t❣✌P✹❄❢✼③⑤✒❚❈❉✽❄❢✼▼✾❺ñ✿✾❖P➇P❖♥❈❚❱❀❃❩✿❊✻P❈❤❃❤☛❄❢✼▼✾
❤❃P♦♥✰Ps✸➠➒❡✾✂❚❈❳✝❊❅✾❇❉❢♥❈✾✂❄❢✼✿P✹❄ ➺ P❖♥✹P❈❜✖P❈❂➱❚❱❂✿❤❙❴➊❂▼✾❇✾❖❩✿❊●❄❅❚❫❣✝❭▼❄t❤❃P♦♥✰P❲❚❱❂✇❄❢✼▼✾ ❼ ❉❋❊❢❄
ô✥→ ✷❺❀❃❂✣❄❅✾❇❉❢♥✰P❈❤✐❊❇✸
Ý ❚♦◗èP❈❊❢❊❢❭✿❜t✾❺ì ❐ ⑩✒Õ ➛➱✷♦❶➬ê → ë✮✸⑧➒❡✾❺◗❧❀❃❤❃❤❝❊❢✼✿❚♦◗Þ❄❢✼✿P✹❄❧❄❢✼▼✾♠ñ✿✾❖P✖❪❇P❈❂④P❖♥❈❚❱❀❃❩
P❈❤❃❤❝❄❢✼✿✾❺❤❃P❖♥✹Ps✸❹➒❡✾q❊❋✼✿P❈❤❃❤☛❂▼✾❇✾❖❩✂❄❢✼✿✾❺⑤➡❚❱❤✐❤❙❚♦◗❧❀✐❂▼❆❁❉❋✾❖❊❢❭✿❤❙❄ ➨
➟➙❤✐P❈❀❃❜î❏ ➨ ➺ ✾❇❄⑦÷ ❐ Õ ê❁✸❺✺✽✼▼✾❖❂④❄❢✼▼✾❇❉❢✾●✾❥r▼❀❃❊❅❄♠❂▼❚❱❂✿❂✿✾❇❆❱P✹❄❢❀❙♥❈✾t❀❃❂❷❄❅✾❇❆❈✾❇❉❘❊❺➳tP❈❂✿✤
❩ ✣
❊❢❭✝❪❋✼✂❄❢✼✿P✹❄❞➳♦ê → ✣✃ë ÿ ⑩ ÷✧➛ ô ❍❢÷ ä ✸
➒❡✾q❊❋✼✿P❈❤❃❤❝❣✿❉❢❚☎♥❈✾❺❄❢✼✿❀❃❊➙❉❋✾❖❊❢❭✿❤❙❄✽P✹❄✽❄❢✼✿✾♠✾❖❂✿❩☛✸
♣ ❀❙❉❋❊❢❄❇✉✕❚❈❳✝❊❢✾❇❉❢♥❈✾➇❄❢✼✿P✹❄ ü ➦ ❐ Õ êq✸ Ñ ❴➊➟➙❤❃P❈❀✐❜ ❏✶✉✵❀❙❄q❀❃❊q❣❝❚❱❊❢❊❢❀❙❳✝❤❃✾✧⑤✒❚❈❉✧❄❢✼▼✾➇ñ✿✾❖P
❄❅❚✖❜✖P✹⑨❈✾❁Pt❊❅ä✾❖❻✣❭✿✾❖❂✿❪❥✾♠❚❈⑤▼↔❢❭✿❜t❣✝❊➙❊❅❄❢P✹❉❋❄❢❀❃❂▼❆t⑤✒❉❢❚❱❜î▲tP❈❂✿❩✙✾❖❂✿❩✿❀✐❂▼❆✧P✹❄✽P●❣❝❚❱❀❃❂✣❄➙❚❈⑤
⑩✎ü ➦ ➛ ô ❍ ü ➦ ✸ ♣ ❉❢❚❱❜➑P❈❂❷❴✂❣❝❚❱❀❃❂✣❄✱❚❈⑤⑧❄❢✼✿❀❃❊✻❀❃❂❷❄❅✾❇❉❋♥✰P❈❤➄✉✌P➇❊❢❀❃ä ❂✿❆❱❤❙✾➎↔❅❭✝❜t❣③❚❈⑤✵❊❢❀❙ç❇✾✧ë
❄❢P✹⑨❈✾❖❊Ô❄❢✼▼✾➙ñ✿✾❖P✻❚♦♥❈✾❇❉ ⑩✎ü ➦ ❍❢✦ý ✣✥ ❄❅❚♠P❞❣❝❚❱❀❃❂❷❄✯❀❃❂ ⑩ ý ➦ ❍❢ý ➦ →❨ô ✉✹◗❧✼✿❀✐❪❋✼✧❪❥❚❈❉❢❉❢✾❖❊❅❣❝❚❱❂✿❩✿❊
❄❅❚✖❄❢✼✿✾♠❣✳❚❱❀✐❂❷❄✱➻ ✆✟✞ ➦✛✠✍✌✕➦ ⑩ ➫ ò ➦ ❶❛❚❱❂③❄❢✼▼✾♠ñ✿✾❖Ps➾➚❊❧❣✝P✹❄❢✼☛✸
Ý ❚♦◗➤◗➎✾t❭✿❊❅✾t❀❃❂✿❩✝❭✿❪❥❄❢❀❙❚❱❂④❄❅❚✂❣✿❉❢❚♦♥❈✾➇❄❢✼✿P✹❄❇✉✜⑤✒❚❈❉♠✾❇♥❈✾❇❉❢❴ ➲ ❐ ✷✹✉✜❄❢✼▼✾❇❉❢✾t❀❃❊♠P✮❣✝P✹❄❢✼
❊❢❭✝❪❋✼✖❄❢✼✝P✹❄❛➻ þ P♦♥❈❚❱❀❃❩✿❊➎❤❃P❖♥✹P❺⑤✒❚❈❉➎P❈❤✐❤✧ ➘ ☎ ⑩✒➲ ❶ → ✷✹✸❹✺✽✼✿✾✻❪❇P❈❊❅✾ ➲ ➫ ✷✻❀❃❊❹❩▼❚❱❂✿✾❈✉✶❊❅❚
◆
© Canadian Mathematical Society 2016
Draft Solutions - 2016 CMO
P❈❊❢❊❋❭✿❜t✾✧❄❢✼✿P✹❄♠ä ❄❢✼▼✾tP❈❊❢❊❅✾❇❉❢❄❢❀❃❚❱❂❲✼✿❚❱❤❃❩✿❊✻⑤➡❚❈❉❺P✮❆❱❀❙♥❈✾❖❂ ➲ ✸❁✺✽✼▼✾❖❂❲➻ ✆✟✞ ➩✡✠✏✌✕➦ ➫ ò ➩③ÿ
⑩ ý ➩ ❍❢ý ➩ →✥ô ✸✵✺✽✼▼✾❇❉❢✾❇⑤✒❚❈❉❢✾
ü ☞➩ ✌✕➦ ➛ ò ➩ ❐ ý ➩ → ìÞ➛ ⑩ ý ➩ →➱ô ❶ ➫ ìÞ➛ ô ❐ Õ ê➠❑
⑥✻❣✿❣✝❤❙❴s❀❃❂▼❆✽➟➙❤❃P❈❀✐❜➋❏✱◗❧❀❙❄❢✼♠÷ ➫ ü ☞➩ ✌✕➦ ➛ ä ò ➩ ❊❢✼▼❚☎◗❧❊☛❄❢✼✿P✹❄✜❄❢✼✿✾✵ñ✿✾❖P✽❪❇P❈❂✱↔❅❭✿❜✖❣♠⑤➡❉❢❚❱❜
ò ➩ ❄❅❚❨P➇❣❝❚❱❀❃❂❷❄✱❚❈⑤ ⑩✎ü ✡➩ ✌✕➦ ➛ ô ❍ ü ☞➩ ✌✕➦ ✸✱⑥ß❊❢ä ❀❃❂▼❆❱❤❃✾➎↔❅❭✿❜✖❣③❚❈⑤✵❊❢❀❙ç❇✾qëí❄❢✼✿✾❖❂③❄❢P✹⑨❈ä ✾❖❊
❄❢✼▼✾♠ñ✝✾❖P●❄❅❚➇Pt❣❝❚❱❀❃❂✣❄➎❚❈⑤ ⑩ ý ☞➩ ✕✌ ➦ ❍❢ý ☞➩ ✌✕➦ →➱ô ⑩ ◗❧❀❙❄❢✼▼❚❱❭▼❄✱♥s❀❃❊❢❀❃❄❢❀❃❂▼❆ ⑩✎ü ☞➩ ✌✕➦ ❍❢ý ☞➩ ✌✕➦ ❶❘✉
P❈❂✿❩✙❄❢✼✝❀❃❊✱❣❝❚❱❀❃❂✣❄➙❊❢✾❇❉❢♥❈✾❖❊❧P❈❊❧➻ ✆✟✞ ☞➩ ✌✕✛➦ ✠✏✌✕➦ ✸✵✺✽✼✿❀❃❊✱❪❥❚❱❜t❣✌❤❙✾❇❄❅✾❖❊✽❄❢✼▼✾❺❀✐❂✿❩✿❭✿❪❥❄❢❀❙❚❱❂✜✸
✎❺✁✰✡☛✡✝✆❞✡✝✆q✤✧☞✒✄✝✑✩★✫✪ ➨ ➺ ✾❇✭
❄ ✬❲❳✳✾♠❄❢✼✿✾❺❆❈❉❢✾❖P✹❄❅✾❖❊❅❄⑦❀❃❂✣❄❅✾❇❆❈✾❇❉❧❄❢✼✿P✹❄❞❀❃❊✽❤❙✾❖❊❢❊✽❄❢✼✿P❈❂③❚❈❉
✾❖❻✶❭✿P❈❤✜❄❅❚✮÷✶➼ùê❁✸❧✺✽✼▼✾❖✮
❂ ✬ ❐ Õ P❈❂✝❩✮✬✌ê ➘ ÷✂➹ ⑩ ✬ → ✷♦❶➬êq✸ ♣ ❚❈❉ ➥❨➫ ▲s❍❇❑❇❑❇❑♦❍ Õ ✉
❤❙✾❇❄
✯✱✰
➫ ⑩ ✬✙➛ ➥ ❶➬ê → ➥ ë ➫ ✬✌ê → ➥ ô ❑
✺✽✼✿✾❖❂
✯
✯ ➧➉➫ ✬✌ê ➘ ÷✝❍
✑ ✯✱➫ ✰ ✬✌ê ✯✱→Ð
✰ Õ✕ô ➫ ✬✌ê →Ü⑩✒Õ❨→ ✷♦❶ ô ➛ ô ❐ ⑩ ✬ → ✷♦❶➬êÜ➛ ô õ ÷q➛ ô ❑
P❈❂✿❩ ✌✕➦ ➛ ➫ ô ⑤✒❚❈❉ ➥✖➫ ▲s❍❇❑❇❑❇❑♦❍ Õ ➛✥✷✹✸
✯ ✰
✺✽✼✿✾❇❉❢✾❇⑤➡❚❈❉❋✾♠◗❛✾q❜✧❭✿❊❅❄✽✼✝P❖♥❈✾ ÿ ⑩ ÷✧➛ ô ❍❢÷ ä ⑤➡❚❈❉✻❊❅❚❱❜t✾ ➥ ❀❃❂ ➏ ▲s❍❖✷✹❍❇❑❇❑❇❑✰❍ Õ ➐❱✸✂é
①✶✸ ➺ ✾❇❄✳✲✙ê❞ë ô ❳❝✾✮P❈❂✇P❈❪❇❭▼❄❅✾❥➍✗P❈❂▼❆❱❤❃✾❖❩✇❄❅❉❋❀❃P❈❂▼❆❱❤❙✾✮◗❧❀❙❄❢✼ÐP❈❤❙❄❢❀❙❄❢❭✝❩▼✾❖❊❁ê✵✴ P❈❂✿❩✇ë✙✶
❜t✾❇✾❇❄❢❀✐❂▼❆✂P✹✸❄ ✷➠✸ ➺ ✾❇❄ ✄ ❳✳✾●❄❢✼▼✾t❜✖❀✐❩▼❣❝❚❱❀❃❂❷❄✻❚❈⑤➎❊❢✾❇❆❱❜t✾❖❂❷❄❺ê❞ë✮✉✏P❈❂✿❩❫❊❢❭▼❣✝❣✳❚❱❊❢✾
❄❢✼✿P✹❄⑦❄❢✼▼✾t❪❇❀❙❉❘❪❇❭✿❜✖❪❇❀❙❉❋❪❇❤❃✾❖❊❧❚❈✹⑤ ✲✺✴✻✶ ✄ P❈❂✿❩✼✲✙ê❞✙ë ✷ ❜t✾❇✾❇❄❺P✹❄⑦❣❝❚❱❀❃❂❷❄❢❊⑦ÒîP❈❂✿❩
✽ ◗❧❀❙❄❢✼ÐÒ➑❚❱❂Ð❄❢✼▼✾✂❊❢P❈❜t✾✙❊❢❀❃❩▼✾❨❚❈⑤ ô ✷➢P❈❊●ê❁✸❡❸✵❉❢❚☎♥❈✾✮❄❢✼✝P✹❄✧❄❢✼✿✾✮❤❃❀✐❂▼✾❖❊✗✶✗✴❯✉
Ò ✷➊✉ÔP❈❂✿❩ ✄ ✽ P❈❤❃❤✵❣✝P❈❊❋❊❺❄❢✼▼❉❢❚❱❭✿❆❱✼➊P④❊❢❀❃❂▼❆❱❤❙✾t❣❝❚❱❀❃❂✣❄❺❚❱❂➊❄❢✼▼✾❨❪❇❀❃❉❋❪❇❭✿❜✖❪❇❀❙❉❘❪❇❤❙✾t❚❈⑤
✸
✲✙ê❞ë ô ✸
❽✌❾❈❿➁➀▼➂✎➃➄❾✹➅✌➆
✾
© Canadian Mathematical Society 2016
Draft Solutions - 2016 CMO
✜
❃
❑▼▲
❄❆❅
❈
❉❋❊
❇
●☞❍
■✂❏
❀ ✿
◗
❘
◆P❖
❁✟❂
❏☛❙ ❚
➺ ❇✾ ❄❛ý➵❩✿✾❖❂▼❚❈❄❅✾✻❄❢✼✿✾❞❀❃❂✣❄❅✾❇❉❋❊❅✾❖❪❥❄❢❀❃❚❱❂➇❚❈⑤✜❤❃❀✐❂▼✾❖❊ îP❈❂✿❩❨Ò ➠✸❹↕s❀❃❂✝❪❥✾❧❻✣❭✝P❈❩▼❉❋❀❃❤❃P✹❄❅✾❇❉❢➍
P❈❤❃❊ ô ➑P❈❂✿❩➇ê❞Ò ❲ëèP✹❉❋✾❧❪❥❴✶❪❇❤✐❀❃❪✹✉❷◗➎✾❧✼✿P♦♥❈✾ ➙ý ✮ê ➫ ✷❖➞✹▲ ✜➛ ✮ê ➫
✷❖➞✹▲ ❧➛
➫ ✷❖➞✹▲ ❧➛ ô ➫ ➝✹▲ → êq✉❝P❈❂✿❩ ➙ý♠Ò❺ê ➫ ❲Ò❺ê ➫
✷❖➞✹▲ ➛ ④ë●ê ➫ ➝✹▲ → ê❁✸✯✺❧✼▼✾❇❉❢✾❇⑤✒❚❈❉❢✾❈✉❷ê❞Ò ❨ý➉❀❃❊Ô❪❥❴s❪❇❤❃❀❃❪✹✸⑧✺✽✼✿❀❃❊✯❀❃❂✧❄❢❭▼❉❋❂●❀❃❜t➍
❣✝❤✐❀❙✾❖❊➙❄❢✼✿P✹❄ ✱Ò❺ë ➫ ➙Ò❺ë ➫ ➙Òqê ➫ ➙Ò❺ê ➫ ➙Ò❺ý ➫ ➙Ò❺ý ✖✉
P❈❂✿❩✂❊❢❚➇Ò❺ë●ý ✩❀❃❊✱P❈❤✐❊❅❚➇❪❥❴s❪❇❤❃❀❃❪✹✸
➺ ✾❇❄⑦ìî❩✿✾❖❂▼❚❈❄❅✾t❄❢✼▼✾t❳✝P❈❊❅✾●❚❈⑤❛❄❢✼▼✾✖P❈❤❙❄❢❀❙❄❢❭✿❩✿✾❁⑤✒❉❢❚❱❜ ô ❄❅❚③ê❞ë❨✸✧✺✽✼▼✾❖❂ ④❍ ✖❍❢ì✱❍
P❈❂✿❩ P❈❤❃❤✣❤❃❀❙✾❹❚❱❂❁❄❢✼▼✾✱➝☎➍✟❣❝❚❱❀❃❂❷❄✯❪❇❀❙❉❋❪❇❤❙✾❹❚❈⑤ ✙ê❞ë ô ✉✰P❈❂✝❩❁❊❅❚⑦P✹❉❢✾➙❪❥❴✶❪❇❤✐❀❃❪✹✸✏➒❡✾✱P❈❤❃❊❅❚
⑨s❂▼❚♦◗ ê❞Ò ✮ý●✉✜Ò❺ë●ý ✖✉✜ë ô ❁ìt✉☛P❈❂✿❩❫ê ô ❨ìíP✹❉❢✾✖❪❥❴s❪❇❤❃❀❃❪✹✉❝◗❧✼✿❀✐❪❋✼❫❀❃❜t❣✝❤✐❀❙✾❖❊
➙ê❞ý♠ë ➫ ✱Ò❺ý♠ëß➛ ➙Ò❺ý♠ê ➫ ➙Ò që ➛ ➙Ò ✮ê ➫ ➙Ò ❁ì → ➙ì qëß➛
➙Ò ❨ì → ➙ê ✮ì ➫ ✱ì që → ✱ê ✮ì ➫ ➙ì ô ë → ➙ê ô ì ➫ ô ✸⑧✺✽✼▼✾❇❉❢✾❥➍
⑤✒❚❈❉❢✾❈✉✿ý ❤❃❀❙✾❖❊➙❚❱❂❯❄❢✼▼✾❺❪❇❀❙❉❋❪❇❭✝❜✖❪❇❀❙❉❋❪❇❤❙✾⑦❚❈⑤ ✙ê❞ë ô ✸
◗❧❀❙❄❢✼❲❄❢✼▼✾✖❪❇❀❙❉❢➍
Ý ❚♦◗✩❤❃✾❇❄ P❈❂✝❩❡ý ❩▼✾❖❂▼❚❈❄❅✾t❄❢✼✿✾✖❀❃❂❷❄❅✾❇❉❘❊❅✾❖❪❥❄❢❀❙❚❱❂✿❊⑦❚❈⑤➎❤❃❀❃❂▼✾
❪❇❭✿❜➇❪❇❀❙❉❋❪❇❤❙✾✱❚❈⑤ ✙ê❞ë ô ✉❷❪❘✼▼❚❱❊❅✾❖❂✮❊❅❚q❄❢✼✿P✹❄ ❍ ❍ ❍❢ý ❤❃❀❙✾✱❚❱❂➇❄❢✼▼✾✻❤❃❀✐❂▼✾✱❀❃❂t❄❢✼✿P✹❄
❚❈❉❋❩✿✾❇❉❖✸✕➒❡✾❧◗❧❀❃❤❃❤✶❊❢✼▼❚☎◗➱❄❢✼✿P✹❄❹ý ➫ ý●✉❱◗❧✼✿❀✐❪❋✼✧◗❧❀❃❤❃❤✶❪❥❚❱❜t❣✝❤❃✾❇❄❅✾✱❄❢✼▼✾✱❣✿❉❢❚✶❚❈⑤❵✸ÔÏ✻❚♦◗✱➍
✾❇♥❈✾❇❉❖✉ ❼ ❉❋❊❅❄✻❂▼❚❈❄❅✾❺❄❢✼✿P✹❄✻❄❢✼▼✾❁❪❇❀❙❉❘❪❇❭✿❜✖❪❇❀❙❉❋❪❇❤❃✾❞❚❈⑤ ✙ê❞ë ô ✼✿P❈❊❧❉❋P❈❩✿❀❃❭✝❊
✉✌P❈❂✿❩
➭
❄❢✼▼✾⑦❪❇❀❙❉❋❪❇❭✝❜✖❪❇❀❙❉❋❪❇❤❙✾✻❚❈⑤ ✙ê❞ë ✼✿P❈❊➎❉❋P❈❩✿❀✐❭✿❊
á ✸✵✺❧✼✣❭✿❊
å➫
❄❢✼▼✾✮❄✗◗➎❚❡❪❇❀❙❉❋❪❇❤❙✾❖❊✧✼✿P♦♥❈✾✂✾❖❻✣❭✿P❈❤❛❉❋P❈❩✿❀✐❭✿❊❇✉⑧P❈❂✿❩Ð➭ ❊❅❚④❄❢✼✿✾❇❴➊❜✧❭✿❊❅➭ ❄✧❳✳✾✮➦ ❅➧❊❅❴s❜✖❜t✾❇❄❅❉❋❀❃❪❇P❈❤
P✹❳❝❚❱❭▼❄✱❄❢✼▼✾♠❣❝❚❱❀❃❂✣❄ ✸ ➜ ❂✂❣✌P✹❉❢❄❢❀❃❪❇❭✿❤❃P✹❉♦✉
✸
➫
↕s❀✐❂✿❪❥✾ ✱ê ❁ë ➫ ➙ê ✮ë ➫ ➝✹▲ ✰✉❱◗➎✾✽⑤✎❭▼❉❢❄❢✼▼✾❇❉❋❜✖❚❈❉❢✾❛⑨s❂▼❚☎◗✥❄❢✼✿P✹❄ ❀❃❊✯❄❢✼▼✾❧❪❇❀❙❉❢➍
❪❇❭✿❜➇❪❥✾❖❂❷❄❅✾❇❉⑦❚❈⑤⑧❳❝❚❈❄❢✼ ✙ê që P❈❂✿❩ ✂ê ✮ë✮✸✽✺✽✼✣❭✝❊❇✉ ê ➫
➫
➫
➯ ý ➫
➯ ý ➫
ë✮✸ Ñ ❴✂❸✯❚♦◗➎✾❇❉❞❚❈⑤⑧P❨❸✕❚❱❀✐❂❷❄❇✉✌◗❛✾❁❄❢✼▼✾❖❂❯✼✿P♦♥❈✾
ê ➯ ë ➫
➭ ✸ ➜ ❂✂❣✝P✹❉❢❄❢❀✐❪❇❭✿❤❃P✹❉❖✉▼❄❢✼✿❀✐❊✽❜t✾❖P❈❂✿❊❧❄❢✼✿P✹❄❧❄❢✼▼✾❁❪❇❀❙❉❋❪❇❭✿❜✖❪❇❀❃❉❋❪❇❤❙✾⑦❚❈⑤
➞
❯✶✗✴
❯✶
✴❲✷
❬❩
❱✷
✸✷