✁✄✂✆☎✞✝✠✟☛✡✌☞✎✍✏✝✠✂✑✟✓✒✔✟☛✂✖✕✔✗✙✘✛✚✞✜ ✢ ✡✙✣✥✤✑✘✦✗★✧✪✩✫✝✬✩✞✡✭✗✮✗✌✘ ✿❘ ✰✶✸✷❀❋❞✷✳❭❝✻❆❙✸❖❡✶❚✯✱❉✮●❊✰✳❬✮✶✸✲✵✶◗✴✮✻❆❅❢❋❱✶✸✼❣✷✺❯❲✰✿●❊✹✌✰✳✶◗✲❤✻✽❳✮❉✭✼✾✻✸❳✌❖❃✰✿✶❨✷❀✶✺✷❁✰✳✶❃●❃✐❁✴❥❂❄❩❚✰✳❅❛✻❆❉✭❵❁❅❈✻✸❋✾❖❃✼❤❇❊✴✮❉✭✻◗✶✸❋✾●❊❋❫●✥❬▲❂❄❭❫✰✳●■✻◗❪❲❅❦❴❚❍❆✻❧❏▲✰✳❅❛❂♠❑▲▼✬❵❁❉✭✻❆❋❤❖❈✰✳♥❆●✥❖❃✰✿✷ ✻◆❭❜✻◗●♠✴✭❅❖☛✻▲✰✳●❝❬❝✻◗✷❀✻✻ ✷ ❘❵❁✻❆✲✾✼❢❉✭❋❞❭❆♦❤♣✸❇❥q✭✼r♣s❂t✰✉✼❤✴❧✼❤✴✮✶❱✴✮✶✸✷✿❳t❉❥❵▲✻✈❅❤✼✇✻✸✼✾✶✈❅✲✵✴✮❉❥✷✳✻◗❋✇❅r✴❥✰✿❳❫♣✽✴✮✶ ❂❄❍◗❏▲✻◗②❆❅❫❏t✻▲✻◗❇❥●❊✷❁✶❛❬❄✼✾✼❤❉❨✴✮✻✽✶◗✼✇●❜✼❢✶✸❙✽●❊❋✇✻▲❬❚❬❥❅q❊✶❆✻✸✲❢❉✭✼✾✶✺●❊❬✮✻✽✼③✻◗❋❞✼❤❭❚✴✮✶✺❅❢✼✾④❧q✌❬✭●✭✰✳✰ ✶◗❅①✶◗❋✇✰✿●❝❅❤✰❀✼❞❯❲❭❜✰✳✶◗❉❥❳✮❵③❳✌⑤①✶❦❉❥q❊q✌●❥✶◗✼✾●③✰✿✷❪ ❴✴❥t✰✿✶✬❅⑩❅❢❅✵✼✾❳❊✻◗✻◗❋✇❋✾✼✾✶⑥✶❆❬◆✼✾✰✳✼✾❖❃✶◗✻❆✶⑨✲✵✴❥✴✮✰✳✶⑨●❊❙❡❳❊❅❋✾✶✸✶❆❳❊✲❢❉✭✻◗●❊❋✾✶❆❬✮❬✖✻◗❋❞❭✫❵❶❉✭❋❡❅✲✵✼❤✴✮❘ ✴✮❉✮✶✏❉❥✷❦❷r✰✿✻❆●⑥❙✽❋✽❍◗✶◗❸ ❏✭❙✽⑦✸✻✸②✌✼✾✰✳❪⑨❉✭●①⑧❞● ❹ ❅✐❀✼❤✲✵✴✮✴✮✶✺❉✮❉❥✴❥❿❲✷✸✰✳❙✸✼❢❳✮✴✮✶✸✻❆✻▲✶◗❋➀✲✵❅❢✼❦✼❛✴✮❵❁✶✸✲❤❋✾❋✇❉✭❉✭❅❣❖☛❖➁✰✿●❜❳✌✼➀✶❺❂♠❼▲✼✾❋✇✰❀❉❄✻❆✼✾✰●❊❭✭❘ ✲❢✶◗✶❚✶❺✻◗♦❢✶✵❋✇♣❺❻◗❅❱❂♠✻◗✰✿❖☛●✠✴❥✰✳✰✿✯✫✲✵●✮✴❄✻✸✻◗✼✾✴✮❋❢✰✳✶❱❋✇❉✭✻❆●♠❳❊♥❆✻◗❵❶✶❆❅❉✭✲✵❅❋❣✴✙✶❆❅♣✙❬❚✶❆✯✥✰✳✲❢●♠❉✭❉✭●❊❍◗❋✾❬✮❉✮❏▲✻◗❽▲✲❤✲❢❋❞❾❊❭❉ ❪ ✐✿❵❁❋✾❉✭❖➂❍❆❏❥⑦▲⑦➃✼✾❉■❍◗❏✭⑦✸❏✽♦❢♣➄✴✮✶◆✴✭✻❆❅❈✻✸✷✉❂❄✻❺❭✮❅❝✼✇✻◗q✌❙✸✴▲✼♠✰✳●✫✼❤✴✮✶ ❅✵✻❆q✌●❊❇❥❬☛q❊❍❆❋❤②✭❇✭➅❥❅❣❇❥❉❥q✭❵❊✼➆⑤①✼❤❉❥✴✮q✌✶✸✶✸●①●①♣▲♣❆❵❶❼❥❉✭❋✇❋❣✻◗✼✾●❊✶◗✲❤●❄✶✮❭✭❪❛✶◗❴❄✻◗✶❦❋✇❅✽✰✳♣❥●✭✴✮✰✉✼✾✶✺✰✳✻✸❳❊✷✿✷❋✾❭❧✶✸❳❊✼✇✻◗✻◗q✌❋✾✶❆❙✸❬❈✴▲✼✙❅❢❅❤✼✾✼✾q✌q✌❬❊❬✮✶✸✶◗●❥●❥✼✾✼✇❅❦❅❣❵❶❇✮❉✭✶✽❋❣✼➀✼❤❂♠✴✮✶◗✶❜✶◗❷s●➄❇✭✼❤✻▲✴✮✲❢✶❲✲❢✻✸✻▲✷✳❙✸✻✸✶✸q❊❅③❋✽✶✸❸❉❥✻✸❵❥✼r❍❆❹◗▼ ♣ ✼❤q❊✴✮●❊✶✠❬❊✶✸❼❥❋✾❋✾❙✽✶✸❋✇●❊✻▲✲✵❬❥✴⑩q❊✻✸❅r✼✾✲✵✶❡✴✮❉✮✷✳✶❉❥✷✳➇✶✸✶❤✷➀❻❆❪⑩✰✉✼t✹✌✲❢✻✽✶◗✼✾❋➀✻✸✼✾✰✿✰✿✷❀➈✌✷❁✶❥✲❢➋ ✻✽❅t✼❢✶✮❅❢✼✾❪◆q✌❬❊➉✸✰✳✶✸●❊●❥✲❢✼✾✶✬❅t❍❆❅✵✰✉❏❆✼✺❏✸➊✌✲❤❉✭♣❦❖☛✴✮✶➃❳✌✶❺✴✭✼✾✻◗✰❀✼✾❅❝✰ ❇❊✶❡✶◗✶✸✶◗●❥●✠✼❢✼❢❋✇✻❆✶✸✻▲●❊✲✵✲❢✴❥✶❡✰✿●❊✶✵❻◗❙❃✻◗✻✽❖☛✼❚✰✿●✮✼❤✴✮✻▲✶ ➇ ✼✾❙✽✰❁❋✾❉✭❉❥●✮q❊❅③●❊✼❢❬①❉ ♣◗❘ ✻❆✻❆●❊❋✇❬❚✰❁❉❥✼❤q❊✴✮❅❛✶r❭❄✶◗❘●❊❅✵❳✌❙❺✰✿✶◗●❊✲❢✶❆✰✳✻✸✶✸✷✿❋✾✰✉✰✿➍✵●❊✶✈❙❧✰✳●❚❅r✲✵✼❢✴✮✶◗❉✮✲✵❉❥✴✭✷✿●❊❅◗❉❥❪①✷❁➌①❉✮✴✮❙r❭❄✶r❭❄❬❥✻◗q❊❋✾❋✾✶❨✰✿●❊❙✸❙✈✶✸✼❤●❊✴✮✶◗✶✽❋✇✰✳✻✸❋❫✷✿✷❅❭t❘✶❆✲❢❵❁❋✾❉✭❉✭●❊❖✎❬✮✻◗✻➄❋❞❭t❖❡❅❤❉✮✼✾q✌❬✮❬❥✶◗❅❢✰❁✶✸✼❣❅◗❇✮❪❛✻❆❴✺✲✵♥❥✰✿❅➇ ✼✾✰✳❖☛✻◗❅r❳❊♥♠❋✾❉✰✳❅❲✼❢✰✿●❊❉☛❙❨❇✮✼❤❋✇✴✮✰✳●❊✶✽❙❚✰✳❋❦✼❤♥▲✴✮●❊✶◗❉❆❖➁❂t✼❢✷✳✶❆❉❝❬✮✼❤❙✸✴✮✶✺✶♠✰✿❋✾❅❱✶◗➎✭➎✭q✮q✮✰✉✰✿✼❢❋✾✶t✶❆❬☛✶◗●▲✷✳➏❞✶ ❉✸❘ ❭❥✶✸✻▲✷✌❇❥✰✳●☛✷❁✶❥❖❃♣✭❅❤✻✽✰✳●❊✼❤✴✮✲❢✶❈✶◗❖☛❷r❖❃✻✸✼✾❉✭✰✳✲❢❅❤❅✸✼❦❪❦❉❥➌✙❵❊✴✮✼❤✴✮✶❝✶✸✲✵❖➐✴✭✻✸✻◗✷✿✷✳❋✾✶◗✶❨●❊❂♠❙✸✶♠✶✽✷✿❉❥✷✳➇ ❵ ❖❡❉▲✼✾✰ ❘ ❴❄✻✸❘ ✼✾✶■✶❆❬✙✴✭❹◗✻❆♣✭❅◆❅✻❺❇❊❭✮✶◗❅✺✶◗●✖✹✌✻✽❖☛✼✾✻✸✻❆✰✿✷❀❋✾✷❁❋✾✶✮✰✳❪✶❆❬✖❅❤✰✳●❊✲❢✶⑨❍❆❏❥⑦❆▼✮❪➑❴✺✰✿❅❡❂t✰✿❵❶✶✥✰✿❅✠✻◗✷✿❅r❉✛✻■❖❃✻✽✼❤✴✦✼❢✶✸✻▲✲✵✴✮✶✸❋ ✐❶❙✽✴✮✻❆❉❆❋✾❬✮❂♠✶◗✶ ●✭❘ ✰✿✶✸●❊❋r❙t♣❣●❊✴✮❉❆✶❡❂❄✲❢✻▲✷✿✻◗❬✭✰✿✻✽❖❃❭✭❅❨❅❢♦✵✼❤❪❣✴✭➌①✻✸✴✮✼❚✶r✴✮❭❃✶◗✴✭❋➄✻ ✰✳●❥✶✺✼✾✶◗✼➀❋✾❂♠✶✸❅❤❉☛✼✇❅✺✲✵✴❥✷✿✰✳✰❀✶☛✷❁❬✭❖❃❋✾✶◗❉✭●③❋✾❪ ✶☛✰✿●✬✼❤✴✮✶☛❋✾✶✸✻◗✷✿❖➒❉❥❵❦❇✮❉▲✼✾✻◗●▲❭◆✻◗●❊❬ ❘ ✻✭♣❆❋✾✶✸✻▲❬❥✰✳●❊❙➄✐✿✰✳●❈❼▲❋✾✶◗●❊✲✵✴♠❉✭❋❛➓❥●❊❙❺✷✿✰✿❅r✴❆♦❢♣▲✲❤❋✾❉✭❅✵❅❢❂♠❉✭❋✾❬✭❅✽♣ ✺ ❴ ✿ ✰ ❦ ❅ ✮ ✴ ✮ ❉ ❊ ❇ ❥ ❇ ❁ ✰ ✸ ✶ ➆ ❅ ✳ ✰ ❊ ● ❢ ✲ ❀ ✷ ✌ q ❊ ❬ ❨ ✶ ❢ ✲ ✳ ✰ ❊ ● ✸ ✶ ❃ ❖ ✷✿✴❥✰✿✰✳❅❤❅❄✼✾✶◗✷✳●✭✶✸✰✿✰✿●❊❅✵q❊❙❧❋✾✼❢✶☛❉♠✼✾❖☛✰✿❖❡q❊✶❜❅❤✰❁❂❄✲❲✐✿✻❆✰✳❅♠●❊❬❥❅✵✰❁❳✌✶❚✶✸❋✾●❥❉✮✼✺✲✵♥❄✰✿❖❃✻◗❳✮●❊❋✾❬❈❉ ✶✽✷❁✰✳●❊✶◗✲✾❙❡✼❢❋✾✴❥❉✭✰✿●✭❅❈✰❁✲✾➓❥✻✽●❊♦❢❙❺♣▲✷✿✻❆✰✿❅r●❊✴✙❬❄♣❣❂❄✷✿✰✿●❊✻✸✷❁❙❺♥◗q✮✰✳✰✳●❊❅❢❙✭✼✾✰❁❪✈✲✾❅✽❯❲♣➆q❊✻◗❋✾●❊✰✿●❊❬✫❙❧❇❥✼❤✴✮✰✿✷❀✶❝✷✿✰✿✻❆➋ ❏▲❋✾❬✭➔◗❅◗❅✽❪♣ ✯✥❉✭❅❤✼✈❴✺❉❥❵❊✰✿❅③✼❤✴✮➈❊✶◗❋✇❅❤❅✼①✶❄❳✮✻❆q✌✲✾❇❥✼✾✰ ✷✿❘ ✰✿❅r✰❀✼✾✴✮✰✳✶◗✶◗❬✺❅❱❅r✴✭❉❥✻ ✷❀❘ q✭✶t✼✾✰❁●❊❉✭❉❆●✺❂✎✻◗❘ ❳❊❇❊❳❊✶◗✶◗✶◗✻◗●☛❋✾✶❆❋✾✶◗❬❚❳✭✰✳✷✳●❜✻❆✲❤→✌✶◗➣❱❬❃↔❥❇◗↕⑩❭☛➙✈❳✮➛❶❋✾➜➞❉✮➝❄❇✭➟✛✷✳✶◗➠❱❖➐➡①❅r➢❚❉❥➤❊✷ ❘ ➟➥✰✿●❊❙✭✰✳●❝❪ ✯✫✻❆❋✾✲✵✴ ✴✭❍❆✹✌❏❆✻❆✻✸❏▲❅♠✼✇✻◗❏❊✲❤✰❀❪✎❉✭✷✿✷✳●❥✶☛➉✸✼❢✰✳✴✭❋✇●❊✰❁✻❆❇❥✲❢❅t✶✠q✭✼❢❇❊✼❤✶◗✶◗✴✮❬✱✲❤✶✸❉✭●①❖❃❖❃♣❨❉✭✶☛✴✮❋✾✶❃✶⑩✻❃✻❆❋✾✴✭●❊✶◗✻❆❙❺❬⑩❅❃q✮❖❃✷✳❬❊✻◗✶ ❋❨❉✭❘ ❋✾❅r✶✽✶❡❉❥✷❁❉✭✷ ❅❳❊❉❥✶◗✶❆✷✿❋➄q✭❬⑥✼✾✻◗✰✳✻✱●❊❉✭❬✠●✮❋✾❅◗✶✸❳✌❪✱✻◗❉✭✷✈✹✙❅✼✾✶◗✶◗✻◗❋❧❅❤❅❤✰❁✼✾✰✳❬✮●✠✶✠✶◗❅❅❈❵❶❉✭✶ →✌❋❜➣❱✶◗❳✮❋✇↔❥❋✾✻✸↕✎❉✮✷➆❇✭❳✮➙✈✷✳❋✾✶◗❉✮➛❶❖★➜➞❇✭➝✱✷✳✶◗❅r➟✛❖➦❉❥✷ ➠❱❘ ❅✰✿➡①✶❆●❊➢❚✲✾❙✠✼✾➤❊✰✳✻◗❉✭➟✆●❊●✮❬ ❅ ♣ ✐❿❲✳✰✿●❝❅❅✼❤❉✮✴✮✲❢✶❝✰✳✻✽✯◆✼✾✰❁❉✭❿➆●☛❿✫❉❥❳✮❵③q✌❼❥❇❥❋✾✷✿✶✸✰✳●❊✲❢✻✽✲✵✼✾✴✬✰❁❉✭✯✫●✮❅❺✻✽♣❆✼❤✴❈✼❤✴✮➌✮✶❝✶◗✯✫✻❆✲✵✻✽✴✮❘✼❤✶◗✴✮❋✇✶◗❅✾♦❖☛❪ ✻✸✼✾✰✳✲❢✻✸✷✌➧➆✻✸➍✵✶❺✼✇✼❢✶❥♣❥❘ ✻◗●❊❬♠✼❤✴✮✶❜✹❊q✮✷✿✷✳✶✽✼✾✰✿●❈❉❥❵❊✼❤✴✮✶ ✹❩❚✌✻✸❉▲✼✇✼❛✻◗❉✭✰❀✷✿❼❥●✭✷✳✶☛❋✾✷✉❭❄❉✭✰✿❖➨❅❄✼❤✴✭✻❡✻❆✻✽●❜✼r❋✾♣❥✶◗➓✭❇✭✻✸❬✭✷③q✭✰✉✼❛➏❞✼❢❉◗❉✭✴✮❭❫❋✶❚➋❪⑩❅➆❅✵❳❊q✌❴❚❉❥❇✭✰✳✰✳❅t❖☛●❥❅r✼❣✰✉✼✾❉❥❉❥❅❲✷❀❵q✭❳✮❘ ✼✾❋✾✰❁✰✳❉✭❉✭✶❺❳✌❂❈●✮❉✭❅t♣◗❅✵✴✭✻❆✻◗❋✾✻✷✿❅✈✶❃❘ ✰✳❉✭✻◗●❊●❝✷ ❙❱❂❄✻✺✻➄✻✽❋✾❭✭❋✾✶◗✶❆❅♠❙❺❙❺q✮✲❢q✮❉✭✷✳✷✿✻◗❋❢✻❆❋❦❋✾❋③✶◗❇✭✲❤✲✾❉✭✻❆✼✺❅❤●❥✻◗✰✳✼❢❅❺●❊❋✇♣✸✰❁❬✬❇❥❂♠q✭❂♠✴❥✼❢✰✳✶✽❉✭✲✵✷✿❋❫✴❝✷✳➇✷❀✻◗✰❁❉✭♥❆❋✾❋✾✶✺✶❚❙✽✻◗✯✱✻◗●✭✷ ❂❄✰❀✰✳➍✵✲✵✻✽✶◗✴✮❭✭❬③✶✸❅ ✷❪ ❂♠✴✭✻❆✶✸❅❣✷❀✷✽❇✮✼❤✶❆✴✮✶✸❉❥●✺q✌✻✈❙✸✴▲❋✾✼①✶◗✻✸❉❥✷◗q✭❳✭✼✙✷❁✻❆✶✸●❊✻❆❬✺❅❤q❊✶❤❋✾❻▲✶❲❳✭❵❶✷✳❉✭✻✸❋❊✰✳●❊✼❤✴❥✶◗❬①✰✿❅❛♣➓✭❂t❬✭✰❀✰✉✼❤✼❢✴❱❉✭➏❁❋❢q❊➇❤✰✿❅❢●❊✼✌➇✼❤➩①✴✮✴❥✶❲✰❁✻❆✶✽❳✮❵✸❳❊✼❢❉❚❋✾❉✭✴✭❳✮✻ ❋✾✰✿✶❱✻✸✼✾❬❊✶✈✶ ✻❆❖❃✶✽✷❁❉❥❉✭q❊❳❊●❥✶❆✼❫❬❚❉❥✻◗❵▲●t❬✮❉✭✶✽●❊✼✇✻◗❙✸✰❀❉❥✷✇❪➆✰✿●❊⑧❁❙ ✼ ❵❁❋✾✰✳✶◗●❊❬✭❅r✴❥✰✿❳❜❂t✰❀✼❤✴❈❅❤q✌✲✵✴❈✻✺❂❄✻◗❋❢❖➫✴❥q❊❖❃✻◗●❡❇✮✶✸✰✿●❊❙t✻◗●❊❬☛❅❤q✌✲✵✴☛❘ ✻♠❙✸❉✮❉✮❘ ❬❈❖❃✻✽✼❤✴✮✶◗❖☛✻✸✼✾✰✳✲❢✰✳✻◗●③❪ ✁✄✂✆☎✞✝✠✟☛✡☞☎✍✌✏✎ ✑✓✒✕✔✗✖✙✘✛✚✜✚✣✢✥✤ ✷✹❄✮✩✺✫▲❛✬✩✬✭■✫✣✲■❊❜❆◆✦★❀✮✭✼✧✪✩✜✵✽❊◗✾✣✩✬❖✽✫✮❝▲✹✭✰❞❢✷❈✯✱✲■❡✬❏◗✭✳✲✰❣✐✴✶✵✸✵✸❤❇❆❈✵❅✷✹❥❧✩✺✻✶❦❭♠❴✯✣✭✼✻✽✵✽♥♣✫✣❆❋❊❅✾✣♦❇❘❚✿❁❙✓❀❂qs❀✮r✸✻✶✵❱✾❃t✮❯✓♥✐✻❅✻✶❄✣✾✮❑❫❄✣✻✶❀✜✵❅✯✜✻❅✻✽✲■❊✙✭✣✻✶✷✹✾❳✻✶✩❂❊✕❲❨✴✶✩✬✵✶✫✣✻●✯✉✷❍❆❇❀✮✩✬❆❈✵❅✵✽✫✣❏❂✻❉❆❱✯✮✻✽✵✶✵❅✷❈✯✬❆❋✲✼✿✹❊✥✲✳✻✽✷❈✻●✩✜✭▲✷✞❆❈❩✶✲■✷❍✩✮✻❅❀✬✭✈✿✹✲■✲✰❊❇❛✣✵❬✷❭✩✜✷❈▼✱✲✼✻❅❏❃❆❈❪✰✯✇❯❫✵✜❲❴❑✮✷✹✵❅✩①❩❬❊❁❵✹❄▲❄✮❑✶❆❈✵❅✧✪✩❅✿✹②✬✲✼❀✬✻✽✲✰✲✼✭■✯✮✿❁✭ ❀▼✵ ✻❃❝▲❞❢✭✳✲✼❡✬❊✹✷③❣❿✩✬❤❇④❫❥✰④✰❦➀✻●♠❂②✜♥♣✩✬✫✣♦❢❆❋✲✳q★✷❈✵❃r✸❄✣t✣❆❈♥➁✩✮❛✬❘✜✭✰❩✽✵✽✲✼❏❃✾✣✿✹❊③✵❱④✰❆❈❏❂✩✜✻✶❏⑤✾❉▼❂❄✮✻✶✩✜❊✹✾❂✷✛✩✬✲✼❊❁✫✣❊❁❆✈✫▲✵✶✵✽✯✬❊❱✲✳✷❈✩✬✩✜④⑥❆❈✲■❝❉✻❅⑦❬✭▲⑧❬❛✣⑨⑩✩✜♥❷✻❅❆❈✯❃❶❉❦❭❀✜♠▲✻❬❸❬②✜❹✱✵➂❶❉✾✣❦❧❥❻✩❉❺❽✷❇❼❅❛✮⑦❽⑧✽✵✶❹❾✵✽✾❂✻❅✻✶✾✣❊✕✯ ❊❜✩✮➄★✿✹✵❅✲■✻✶✻✽✯✣✷❈✵✶✵✽✯❳❆❋❊●✧➂➉✶❯★✲➃✷❍❀✮❀①✩✬✲✰❯❫✿✹➄✍✵✜➊❇➅❇❊❜➆➇✵●✭✰④❧✵❅✩✜✿❈❆✸✷❈✲✰✻✶✩✜❊➂✾✪✭✼✩✬✩✜④✮✾✣④✰❖❂✻❬②✜✻❅✩✬❊❱✫✣❊❜❆❈✩✜✲➃✷✹❏❃✵✛✵◗❄✮❆❈✩✬✩✮④✈❛✜✩✬✭✼✵❅✫✣❏❂❆③❊◆❆❈✵✽④✰❆❈✻❉✩✜✯✮❏➋✵❅❆❋❊❬✩❅❑s②✜✧✪✵✽❆s✵◗✷❍❀✮✯✮✵❢✵✶✿✹▼✜✲✼✵❅✯✣✻❅✵❅❆❋✯⑩❊❚✧✪✷❍❀✜✩✬✻●✫✮✷③✭✰✯✻ ✭■✵❅➈ ✲✼❊❁✴✶❄▲✵✽✵❅✭ ▼✺✿✹✲✼❛✮✻✽✭■✵✸✭ ▼✺❏❃④❧✩✜✩✜❆❈❆✞✵✸④✰✻❬❄✮②✜❆❈✩✬✩✮✫✣✯✜✫▲❆❈✲➃✿❈✷✹✷❈✵✽✲ ❊❇②✜✵✜④✰❆❈❑❉✩✜✻❅❏➁❊❫✧✪✷❍❀✮✵●✵✪✭■✭✜✵✽✻❅✻✶❊✈❆❋❛✣✭■✲✼✵●✵❅✲✼❆❚✾✣②✜❖⑥✩✬❏◗✭✳✫✣✩✜❏◗❆❈✵✛✵✽❊❅❏◗❘ ✵✽✻✶✾✜✲■✾✣❖❬④✼✫✮✭❋❘❚➌⑩✵✛✻❅❆❈✵✛✭✰✩✮✩✮✴❅✲✼✾✣❖ ✭✰✶✵✽➓✶❏❃➓❉➔ ❊❃✲✼✦✈④✰❊❁❆❈❊❁➍✣✩✜✫▲❆❋❏→✵✬❊❍✷✛❑✬✷❍✻✶❏❃❀✮✾✣✵✉✵❅✯◗✾✬❄✣✦✓✷❈✻✶✲✼✧❱✩✜❖✽✾✣✻❅✵❅✵✶✾✬❊❳✯❳✷◆✩✬✷❈✷❍④✙✩✺❀✬❝✬✲■❆❈❊➂⑦❽✵✽⑧❬✲➃❄✮✷✹⑨➣✵✽❆❈✩✶❆❋♥❷✻✽➎❻✵✶✷❈✵③✿❈❶❉✷⑥❦❭✲✳♠▲✷✈✲✼✾⑩❸❬❀✮❹✱✵✽❏✺❆❈❶❉▼❃✵✣❦❧❥❻❘✈❺❽➏❚✦❧❼❅✵❅④✣⑦❽✻❅▼✜⑧✽❆➂✩✬❹↔✫❃➐✬✾✣✩✜❀✜✯✉❆↕✻●②✜➑❉❝▲✵➂✲■❞❢✾✮❊✕✻❅❡✬✩✜✭✼❣➙❏◗✵✱✵✸✲■❤❇✾❳④✰❥❧✻●❦❭✷❍♠♣②✜❀✮✩✬✵⑩♥♣✫✣➒❢❆❋♦❢✲✳✵✶✷❈✵➂q★✿✹✵❅r✸❄✣❏❃❆❈t✣✩✮❛✣♥✐❛✮✵✽❆ ❑ ❄✮✿❍✩✬✭✼✵❅✫✮✻❅✭✼✯❂❊❜✵③❄✮④❧❆❈✩✜✩❅❆❭②✬✧❱✲✰✯✮✻✶✵③❆❈✯✙✷❍❀✮✷❍❀✮✵❱✩✜❆❈❊❜✵●✵③④❧✵❅❄✣❆❈❆❈✵✽✩✮✾✣❛✬✿❍✭✰✵③✵✽✷❈❏➛✩✙✾✜✷❍❀✮✫✣✵✸❏❃②✜❛✣✩✬✵✽✭✳❆❋✫✣❊❇❏◗✷✹✩✙✵✬❑✬✫✣✲✼❊✽❊❁❊❁❘✈✫▲✦✓✵✬✧✪❑✬✻✶✩✬✾✣✫✮✯❂✭✰✯✱❄✮✻✽✻❉✭✼❖✽❊✕✩✺✵➂✾✜✻✶❄✮✫✣❄✣❏◗❆❈❛✮✵❅✵❅✿✹✲✼❆❋✻●❊❬✷✹❑✜✵③✲■✾❃✲■④✮✿✹▼✜✻❅✩✬❊❜✫ ✵ ✭✷✹✰✻◗✩✮✩❳✩✮✷❭▼✜❆❈✴❅✵✶❄▲✲✼✿✹✾✣✩✉✵✽❖✙✲ ✿❍②✜④❧❆❈✵✙✩✜✵✶❆⑥✵✽▼✜❄✣✩✬❊❜❊❱✩✜✫✣❏❃❆✥✲✼✾✬✭■✵❬✷❈✲■✧✪❊❍✩✇✷⑥❀✮✷❍✩✬✵✽❀✮④❫❆❈✵◗✵❃④✰✻❬✭✳❛✮✲✼②✜❊✹✵●✩✬✷❱✷➀✫✣✧✪✩✬❆❈④❇✲➃✵❅✷✹✵❅❄✜✵✽✾ ❊➂✫✣❆❈❛❅ ✵↕▼✉➓✶➓❄✮➑✬❆❈✻✶✵✶✩✮✾✣❛✜❛✜✯❆❈✭✼✫✣✵❅ ❅❏➜✻❅➓❉❆❻▼ ➓ ✾✜✫✣ ✶❄✣➝❏❃❆❈✩✮❑ ❛✣❛✬ ✶✵✽✭✰➓✶✵✽❆❋➓✜❊❅❏❃➞ ❘✉❊❬❘✱❑s❙✓➟❢✻✶❀✜✾✣✾✣✻✶✿✹✯◗✾✣✵✙✴◗✧✪✧✪▼✜✵✱✩✬✵❂✻✶✫❚❀✜❆❈❘✉✻❬✵❃②✜➌⑩❀✮✵✪✩✜✵◗▼✜❄✜✩✬✻✶✲✼✾✣✫✣❆❈✵❖ ❆ ④✰❄▲✻●✵✽②✜❆❍✩✬❀✜✫✣✻❅❆❋❄✣✲✳❊✥✷❈✵✺✻➂❄✮➡▲❆❈✻❅✩✮❊❜❛✜❀✺✭✼✵❅❏◗❏❂✵✽❊●❏◗❑✣✷❍✩✜❀✮❆❻✵✺▼❂❯❫❊✹✷❈❲❴✲✰✿❁✴●❩✙❵✕✲■❘ ❊✛❄✮✭■✻✶✾✮✾✜✲■✾✣❖❱✷❈✩◗❏❂✻❉✴✶✵❱✷❍❀✮✵❅❏➠✻●②✬✻✽✲■✭■✻❉❛✬✭✰✵✺✩✜✾↕❯❫➒➇❪❧✩✜❆ ➐✻❅✮✭■✯✬❏◗✲✳✷❈✩✜✩✜❊✹❆❢➟❇✷❚④❧✫✣❊❍✩✜✲✼❆✞❆✈✾✣❊❁❖❬✷❍✲■✭✼❀✮✾✣✵✶✵✙✿❍❁✵✽❀✜❄✣❆❈✻✶✻❅✵✸✾✣❊❍✷❍✷❢✯✮❀✜✵✶❊❁✻❅✯✬✲ ✾✣➤✪✭➃▼➂✴❉▼✜❊✥✲✼✵❅✾✬❖✽✻❅✷❈✩✱❆❋✩❱❊❅✩✬❘✸✻❅✫✜✾✙❩✶✷❫❀✜✵❁✷✹➤✶✻●✩✱✧❱✿✹✵✽❩✶✾↕✭✳❀✜✭✰✵✽✻❬❀✜✾✬✧❱✻✶✷✍❊✛✾❃④❧❏❃✵✽➢◆✻✽✩✬✩✮✷❈✭✰✫✣✯✜✯✮❆❈✲■✵✶✾★✵✥✯❂❑✶④❧✧✪✩✜✷❍❆❫❀✜❀✮✻✽❀✬✩✱✷✥✲✼❖✽❀✜❊✕❀❱✻❅✵✶❊✥❊❜✿❈✷❈✿❁❛✮✲✼❀✮✩✜✵✶✩✮✾❳✵✽✩✬✾✺✭❉✩✬✷❍❊❍④❫❀✮✷❈✩✬✫▲✵✺✫✣✯✮❲❨❆✞✵❅✾✬➎❻✻❬✩✬✷❋▼✣✫✣❊✞❀✮❆✹✻❅✾✮✵✽✾✣❏✻✽✯ ✭ ✷✹✾✣✵✽✩✶✻❉✧➧✿❁❀✮✻✶✵✽✾✣❆❋❊❅✯✱❘↕④❧✵●➥➂✭✳✷◆✩✶✲✳✧✪✷❫✵❜✧❱②✜✻❅✵❅❊✈❆✕❑❚✷❈✲✼❀✮❏❃✵◗✵✸❀✜✷✹✻❅✩✱❊❱❏❃❛✣✵❅✩❅✵❅②✜✾❳✵❱✩✜➎✰✫▲✾✍❖✽❘ ❖❬✭■✲■✾✣❖❂❏❂✻✶✾❉▼✇✯✜✫✜✷❈✲✼✵❅❊③④❧✩✜❆✸➦✜✫✮✲➃✷✹✵❂❊❜✩✜❏❃✵✺✷❈✲■❏◗✵ ✧✪➵ ❊✕✵✽❊❍✭✼✲✼✿❍❊✹✩✜✷❈➥➂❏❃✻❅✾✬✻❬✵❂✷❚②✜✷❈❲❨✲■✾✣✩↕❖✙✻❬➭✕▼✣✵●❊❁❀✮✻❅➯♣✵✽✲✼✯❃❏✐➥➂✷❍✩✮➐✜❀✜✩✜✯✜✻●❄▲✷❜✲➃✷✹✵✽❑▲✩✜❆③✷❍❆▲❀✮✻✶④❧✵◗✩✜❊③❆✮➨✓✷❍✷❍❀✮✩✜❀✮✵◗✻✶✵❇❆❈✾✣❄✣✯↕✵❬✻❅✩✬✧➲❊❍④❇✷✮▼✜❲❨❝✣✵❅❞✥✻●✻❅❡✬▼✮❆❜❣➩❑●❀✮❊✕✵❅✩✥❤❇❏➳✷❍❥✰❦➀❀✮♠⑩➐✮✵✞✯✬♥♣✷✹✲✳❆❋✷❈✻✶✩✜♦❇✾✮❆❬qs❊❍❘◗✲✳r✸✷❈➭✕✲✼t✮✵✽✩✜♥➫ ➯➣✾✸❊❜❀✜✵❍❀✮✻❅➤❅✩✬❊✪✷❈✫✮✵❅✻✶✭✼✾✣✯③✿❈✯✜✷✹❛✣✵❅❊✸✵❇✯①✻✪✻✽✻✶✧❱✭✼❊➂❏❃✻❅✷❍✩✜❆✹❀✮❏❊❍✵ ✷ ❊❜✵✽✻✶❏✱✭✰✵✽❊✕❊✽❘ ♥✾✣♣✩❉♦❢✷⑥q★✻❅❄✣r✸➏❚❄✣✩✬t✣✵❅✫❃♥➸✻❅✧➂❆❢✷❍✲✼✲✳❀✬✾↕✭■✭s✲■❊✛✵❜❀✜②✜✻❬❄✣✵❅②✜✻❅❆❻✵✺❊❍▼◗✷✞✾✣✲■▼✜❊✕✩❉✵❅❊❍✷❈✻❅✫▲✲✼✿❍❆❜✵✜✵❅❑▲❑✓✯❃✾✮❛✜✻✶✷❍✫✜❀✜❏❃✷✛✻●✵✽✩✜✷✈✭ ✾✜▼❂✧✪✭ ▼❂✷❍✵✪❀✮✧✪✲■✵ ✾✬❀✮✷✹➈ ❆❈✵✽❯★✩✮✾◗✩✜✯✜✷❍✾✬✫▲❀✮✷✹✿✹❆❈✵✽✵✶✲✼❆❈✯❳❛✜✵❱✫✜✻✺✧❱✷❈✩✜✾✣✻✶❆➂✵❽❊✛✧✐➺✣❊✕❆❈❄✮✩✬④❧✻❉✵✽➍✮✿✹✻✽✭✰✵✺✷❈✵✽✫✣❊●✻❬❆❈➊③②✬✵✪✻❅❊✕✷❈✲✳✵✶✭✼✩✇✿❈✻✶✷❈❛✜❝▲✲✼✭✼✩✜❞❢✵✣✾✍❘❱❡✬❘➂❣➩✦✈✦✰✷✥❀✜❤❇✻❬✯✜❥✰②✜❦➀✲✼✯♠✵ ❆❈❄▲✵✶✩✜✿✹❊❍✵✽✲✳✲✷❈②✜✲ ②✜✵✶✵✣✯✺❘❱✭■✲■✦◆❏❂❄▲✲➃✵✽✷✹✵❅❆❋❊❜✯✺✩✜④❧✾✮✵❅✻✽✵✶✭■✯✮✭ ▼↕❛✮✻✶❀✜✿❁✻❬✴✙②✜✵✺✩✜✾✙✻❅❄✣✷❍❀✬❄✮✲✼❆❈❊❇✵✶✾✣✿✹✲■✵❬✻✽✧❷✷❈✵✶④❧✯↕✵❅✻●❖✽✷❈✵❬✫✣✷❋✷❈❆❈✲✼✵✜✾✣❑✜❖➂❛✜✷✹✫✜✩❳✷✞✻❅✴✶✭✳✾✣✭✮✩✶✷❍✧➋❀✮✵③✻✺④❧✵❅✭✳✲✳✵✶✷❋✯✮✷❈✭✼❛✮✵✙✻✶❏❃✿❁✴✙✩✜❀✜❆❈✵✺✻✶❊❢✻❉❛✣❛✮✵❅✩✬✵❅✫✜✾ ✷ ❏❃✻❅✾❉▼❃✦❧✩✬④✜④s▼✜✩✬✩✬✫✣✫❂❆✞✿❈❆❈✻✶✵✶✾❱❖❬✷❍✫✮❀✬✭■✻✶✲✼✾✣❆❇✴❱✿✹✩✜✩✬✾✬④✜✷✹✧❱❆❈✲✼✻❬❛✜▼✮✫✜❊◆✷❈④❧✩✜✩✜❆❋❆❚❊✥✫✣✻✶❊◆✾✣✯✪✷✹✩✪✧✪❄✮❀✜❆❈✻✽✩✮✷◆✯✜✫▲✷❍❀✮✿✹✵③✵❜▼✱✻➂✭✼❛✮✩✮✵●✩✮✷❋✴❱✷❈✵❅✭✳✲✰❆❫✴✶❏❃✵✮❘ ✻✶❖●✻✽➻✹✲■✾✣✵✜❑✬❛✮✵✸❊❁✫✣❆❈✵⑥✷❈✩ ❊❜✵✽✾✣✯❂❏◗✵✸✷❍❀✮✩✜❊❜✵➂❊❍✫▲❖✽❖✽✵❅❊✹✷❈✲✰✩✜✾✮❊✽❘❚➌⑩✵➂✻✶❆❈✵➂✻❅✭ ✧❱✻●▼✜❊❇✲✼✾✬✷❈✵❅❆❈✵✽❊❍✷❈✵✶✯✺✲✼✾✙✧❱✻❬▼✮❊✈✷❈✩✺✲✼❏❂❄✣❆❈✩✽②✜✵✣❘ ✁✄✂ ☛ ✟ ✎➋☎❚✌✏✂ ☎✝✆✟✞✡✠☛✠ ↕☞ ✘✍✌✺✢✏✎❉✚✒✑✺✒✔✓❍✒❜✤✍✕☛✖⑥✒ ➺✮✭✰✵✽✻✶❊✕✵✸❊✕✵❅✾✣✯➂▼✜✩✬✫✣❆◆❊❜✩✬✭✳✫✜✷❈✲✰✩✜✾✮❊❫✷✹✩➂✷❍❀✮✵③❄✮❆❈✩✮❛✜✭✼✵❅❏❂❊✈✲■✾✪✷❍❀✬✲✼❊❇✵❅✯✜✲➃✷❈✲✰✩✜✾✺❛❅▼✘❢✗ ♦✥⑧✚✜✙ ⑧✜❅✛ ❦✣✢ ✤✦✥✧✥✩★ ❘ ➵ ✿✹✩✜❄✶▼➣✩✬④✫✪✭✬☛✮✩✯✱✰✲✪✭✬☛✮✩✳✣✴✦✬✏✵✶✪✭✬✸✷✹✯✫✰✺✪✼✻✾✽❀✿❂❁✡❃⑩✧➂✲■✭✳✭⑥❛✮✵✇❄✣❆❈✵✽❊❜✵✽✾✬✷✹✵❅✯❴✷❈✩ ✩✜✯✣✾✣✵❅✿✹✵❂✲✼❊❍❄✮✲✰✩✜❆❈✵✶✾❃✹✫✣✩✬✾✜④✣✲➃✷❍②✜❀✮✵✽✵✪❆❋❊❁✵❅✲➃✯✜✷❭▼⑩✲➃✷✹✩✜❆❈✵✽❆◆✻❉✲✼✯✮❊❇✵❅➍✣❆❢✾✮✧✪✻❅✭➀❀✮❘ ✩❳❊❜✵✽✾✣✯✮❊✸✲✼✾⑩❊✕✩✬✭■✫✜✷❈✲✼✩✜✾✮❊➂❛✮✵✽④❧✩✜❆❈✵✺✷❍❀✮✵❂✯✣✵✽✻❉✯✬✭■✲■✾✣✵✣❘✙❙✓❀✮✵ ✶✻❅ ④■➓✶✷❈➓✜✵❅❆❈❆◆❇✬✷❍➟❇ ❅❀✬➓❉✫✣✲■❊✥➓❀❆✓❉ ✿❍❄✣❘✇✩✜❆❈✾✬✩✮➌⑩✷❈❛✬✵❅✭✰✵❂❊✹✵✽✷❇❏❃✷❍❀✜✯✣❊✓✻✶✵✽✷❍✾✣❊❁❀✬✲✼✴ ❖●✲✼❊s✾✣➵ ❏◗✵✶✾✣✯✺✩✜✯✮✾✬④❧❆✚✩✜✵●✷❍❊ ❆✈❀✸❋✽❊✕✿❍✻✶✵✶✩✜❛✣✿✹❏❃✩✜✵●✾✣✵✈✭■✭✼✯✮✵✜④✰❆❈✻❅❑◆✩✜❆❻✩✬❏♣▼❂④❇❊✹✷❍❯s✷❈❀✮✫▲✩✬✵❢✯✣✭✳✭❅❯s✵✽✵❅❄ ✾✬✩✬❖✽✷❈✭✳✵✇❅✭ ❊❢✵❅❄ ❲❴④✰❖✽❆❈✵✛✩✜✩✜❏➋❲❴✾✬✷✹✩✜❏❃✷❍❀✮✾✬✩✜✷✹✵❍❏❃❆❈✵❅❋✽✩✜✾✣✻❬❆❈②✬✿❭✵❅▼★✻❅✾✣❑✍✭▲✿❭❆❈✧✪▼③✵✶❀✮❖❬❯s✩⑩✲✼✩✜✩✜✾✬✾✍✭✰✩✮✷❈❘ ✵❅✩✮❊✹✴✶✷❜❊ ❑ ■❑❏▼▲❖◆☛◗❏❙❘✲❚✏▲❙❯❲❱❨❳❍❏❙▲✡◆☛❚✏❩❬◆❪❭✟❫✫❫❖❴✟❵✫❫❙❛ ❜❈❝❈❞ ✻✹❡✲❢❖✽❈❣ ❝✚❤✝✐✩❝✚❥✱❦❈❧❀❧✚♠ ♥ ❘➇❙✓❀✮✵①❖✽✩✬✭✰✯✮✵❅✾ ❆❋✻✽✷❈✲✼✩ ❄✣❆❈✩✜❄✣✵❅❆➀✷❭▼♣✷❍❀✜✻●✷◗✲➃✷❈❊↕❘✈➑✶❏❂✲✼✾✣✫✮✯✱✭➃✷❈✻✶✲✼❄✜✾✣✭■✩❉✲✼✿✹✷❍✻●❀✮✷❈✵✽✲➃②✜❆✈✵❴✾✜✫✣✲■✾❉❏◗②✜❛✮✵❅✵❅❆❋❊✕❆❫✵ ✧➂✲➃✷❍❀✙✷❍❀✬✲✼✲■❊❢❊✇❄✮✵❅❆❈➦✜✩✜✫✣❄▲✻✽✵✽✭✸❆❋✷❻❀✜✷✹▼✍✩❿✻❅❘ ❊✺✲✳✷❋✷❍❊✇❀✮✵✇✯✣✵❅❆❈✿✹✵✽✲✼❏❃❏❂✻❅✻❅❆❍✭③✴❅✻✶❄✣❛✜✻❅✭✼❆❋✵ ✷ ❭ ❘◆❯s✩✜✾✮❊❍✲✰✯✮✵❅❆✞✻✙➦✬✫✣✻✶❆➀✷✹✵✽❆❇✿✹✲✼❆❈✿✹✭✼✵❱✩✬④★❆❋✻❉✯✬✲■✫✣❊ ❘ ❪✼✻●❵◗❄▲➑❉✲■✵✽✾✣❆❈✯♣✲■❏◗✻ ✵❬✷✹❆❈✵✽✵❅❆✈✿❈✻❅✷❈✻❅❊✈✾✣✷❍❖❬❀✮✭✰✵✪✵ ➦✬❀✜✫✣✻❬✻✶②✜❆➀✲■✷✹✾✣✵✽❖①❆❢✿✹✷❍✲■❀✮❆❈✵❴✿✹✭✰✵✮❊❁❘✻✶❏❃✵❴✻❅❆❈✵❅✻❴✻✶✾✣✯➣✷❍❀✮✵❴❊❁✻✶❏❃✵ ❪❧❛✶❵❳❊❁➑✜✫▲✩✜✿❁❆③❀◗✻↕✻✱✿✹❆❈✩✜✵❅❏❃✿❈✷❈❄✜✻❅✭✰✾✣✵❬❖❬✷✹✵❳✭✰✵✬❑s✿✹✲✼❀✜❆❈✿✹✻❬✭✼②✜✵❳✲■✾✣✩✬❖✙④❇✻✶❆❋✻❉✾❳✯✬✻✶✲■✫✣❆❈✵✽❊ ✻✱✻❅❑❫✾✣✲■✯↕❊❱✲✳✻✺✷③❄▲❄▲✵✽✩✜❆❈❊❁✲■❊❁❏◗✲✼❛✜✵❬✭✼✷✹✵❂✵✽❆⑥✷✹✩⑩✵✶➦✬➍▲✫✣✾✣✻❅✯ ✭ ✷✹✩✪✷❍❀✜✻✽✷❇✩✬④▲✷❍❀✮✵❱✿✹✲✼❆❈✿✹✭✼✵✧♦➂➭❍✫✣❊✹✷❈✲■④✳▼✪▼✜✩✬✫✣❆✈✻❅✾✮❊✹✧✪✵✽❆❬❘ ♣ ❘ ➵ ❛✜✻✶❆❈❆❈✵✽✭✬✲✼❊◆➍✮✭■✭✼✵✶✯③✧➂✲➃✷❍❀③✧❱✻●✷✹✵✽❆❬❘✍➌⑩✵✸✵✽❏❃❄✬✷❭▼✙❀✜✻❅✭✳④▲✩✬④✮✲➃✷❈❊✈✿✹✩✜✾✬✷✹✵✽✾✬✷❈❊✞✻✶✾✣✯➂✷❍❀✮✵❅✾✙✻❉✯✮✯✙✻ ✧➂✭■✻✽✲➃✷◆✷✹✲✳❆❈✷❍✷❍✵✸❀✱❀✮✩✬✵❱✷❍④✶❀✜❛✣✧❱❆❈✵❅✵✶✻✽❖❬✵✪✷❈✲✼✾✮✵❅✭■✲➃✾✜❆❽✷✹❘ ✲■❆❈✾✣✵❅➵ ✜❖❊✛④■♦ ✷❈✩✬✵❅④★❆❚✧❱✯✣✩✬✻●✷✹✲■✾✣✵✽❆❇❖❇✲■✷❍✾❃❀✬✲✼✷❍❊✞❀✮✩✜✵✺❄✣❛✮✵❅✻❅❆❋✻●❆✹❆❈✷❈✵●✲✰✩✜✭❋❘➂✾✪➥➂❊✕✵❜✩✶②✜✧✐✵❅✾✙❏❃✿✹✩✜✻❅✾✮✾❉▼◗❊❜✵❅✭✳✿✹✲✳✫✜✷✹❆❈✷❈✵✽✲➃②✜❊❢✵❢✧✪✷❈✵✽✲✼❏❃❆❈✵✪✵❅❊❬✲✼✾❂❑❬✧✪✷❍❀✮✵⑥✵✺✻✶❛✮❆❈✵✥✻❅❆✹✭✰✵●❆❈✵●④■✷ ✭ ❴ ❘❚❙✓❀✮✵③✻❅❆❈✵❅✻❅❊❇✩✬④✮✷❍❀✜❆❈✵✶✵✛④✰✻❉✿✹✵❅❊❇✩✬④▲✻③❆❈✵❅✿❈✷❈✻❅✾✣❖❬✫✮✭✼✻❅❆◆❄✣✻❅❆❋✻❅✭✼ ✭✰❪✰✵●✻❬✭✰❵❚✵✽✲✳❄✮✷❋❊✞✲■❄▲②✜✵❅✩✬✯❂✭■✫✣✻❅❏❃❆❈✵ ✩✵ r✇❪❧❛✶✿❍❏❵✍✷❍❀✮❑ ✵➂✭✼✵❅✾✣✿❍❏❖❽✷❍❀❂✻✶✩✬✾✣④✓✯ ✲✳✷❋❊✥✯✜✲■✿❍✻❉❏ ❖✽✩✜✾✮❘✈✻✽➑❉✭ ✲■✾✣✯❲q ❘ ❛ ❘✈➐✶②✬✻✽✭■✫✣✻●✷✹✵ ❘ s t❘ ❋❅✵●✷ ❑ ❑ ❑✍✻❅✾✣✯ ❛✣✵❃✿✹✩✬✭■✭✳✲✼✾✣✵✽✻✶❆⑥❄▲✩✬✲■✾✬✷❈❊➂❊❍✫▲✿❁❀ ✷❍❊❜❀✜✵✽❏❂✻✽✷ ✲✼✕✿✹✲■❆❈✿✹✭✰✵✽❊✈✩✬④✣❆❈✵❅❊❁❄▲✵❅✿❈✷❈✲➃②✜✵③✯✬✲✼✻❅❏◗✵❬✷✹❘ ✵✽❆❋❯s❊ ✩✜✾✮❊❍✲✰✯✮❑ ✵❅❆❱✷❍❀✜✻✶❆❈✾✣✵✶✯✵ ✧➂✿✹✫✣✲✳❆❻✷❍②✜❀❂❖❘✲✳✭■✿❍✲■❋❅✵✽✾✣✵❬✾✬✵❅✷ ✷✹✻❅❆❈❆▲✵❅✷✹❊ ❛✣❆❈✲■✵✛✻✶✾✣✷❍✻❅❀✮❖❬✾✣✵③✭✼✵ ✯ ✲■✾✬✷✹✵✽❘✥❆❋❊❜➒❢✵❅✵❬❪✼✿❈✷✹❊❜✷❈✵✽❀✜✲✰✩✜❆✹✻✶❏✱✾✱✯✣✵❅✲✼✩✬✾✣✯✸④✣✵✸✷❍✲✼✾✛❀✮✷❍❀✮✵③✷❍✵❱❀✮❊❜✵❢✵✽✻✶❏❂❆❈✯✮✵✽❆❋✲✼✻✙✻❬✕✧➂✿✹✩✬✲■④▲✲✼❆❈✾✣✿✹✷❍✭✰❖❜❀✮✵✽❵❁✵ ❊ ❘ √ 1+ N = ≈ 1.618033989 1/N 0.618033989 1 Q 18 40 80 PQ P 6+ 6+ √ + ··· A B C D AB = BC = CD = AC BD AD E B C AED q E q q q q A B C D ❆ ❄▲✻✶❏❂✩✜❘⑩❆➀❄✮✷❈❙★✲✰✭✼✩✜✵✜❀✮✾✮❑✣✵❳✻✽✷✹✩❂✩✜✭★❊✕✷✹✷✹✿✹✩❂❆❋✲■✲✼✭✳✷❍❄✜✭✼✻●❀✮✭✰✷❈✵❱✵✺✲✰✩✜✷❍❊❜❀✮✾✇➦✬✵✱✫✣❄▲✻✶✩✜✵✽❆❈❊❜❆❈✵✺✿✹✲✼✩✮✲✳❆❈✭■✯✉✭■✩✮✻✽✩❉✷❈✩✬✷✛✲✼④❇✩✜✩✬✾❳✻❳④✍❄✣❄✣✲➃✷❈✵❅✵❅❊✛❆❋✾✣✲✰✭✼✯✬✩✮✵❅✫✮✾✣✯✓✭■❖❽❑✮✫✣✷❍✧✪❏✄❀↕✵✺❪■✲■④❧❏✱❊✪✩✜❆⑥✫✮❄✮✭✳❆❈✵❁✷❈✩✮➤❉✲✼ ❄✮✵❅✾✬✭ ▼✪✷⑥✷❍✭✰❀✮✵✽✵✸✾✣❖❽✭✼✷❍✵❅❀✜✾✣❊➂❖❽✷❍✻❅❀✺❆❈✵❂❛❅▼✺❆❈✵●✾✜✭✰✵✽✲✼✾✣✻✶✵❽❊✕❵✕✵✶❘❚✯❳❙✬④✰✧✪❆❈✩✜✩✙❏ ❄✣✵❅✷❍✾✣❀✮✯✬✵✺✫✮✲✼✭■✾✜✫✣✲➃❏❂✷❈✲✼✻✽❊✥✭❚✩✬❄▲④✓✩✜✯✬❊❍✲■✲✳➯s✷❈✲✼✵❅✩✜❆✹✾ ❊❜✩✜❀✮❊❜✩✶✿✹✲✳✧❱✭■✭■✻✽✾✍✷❈❘⑥✲✼✩✜❙★✾③❀✮❄✣✵✙✵❅❊✕❆❋❀✮✲✰✩✮✩✜✯✛❆❋✷❈✲✼✵❅❊ ❆✥✩✜❊❜✾✣✵❅✵✙✿❍✩✜❏◗✾✣✵✽✯s✻✶❘s❊❍❙★✫✣❀✮❆❈✵❅✵✈❊ ✷➀✧✪✩✸✿❍❄✣❏↕✵❅✾✣❑✓✯✬✻❅✫✮✾✣✭■✯◗✫✣❏❂✲➃✷❈❊❊ ✲✼✭✰✻✶✾✩✜❆❈✾✣✵③✷❍❖✽✻✽❀✮✵✽✭■✵●❆✈✲✼✲✼❖●❆✙❄✣✾✣✵❅✵✶✲✼✾✜✾✣✯✺✲➃✯✬✷❈✻✶✫✮✲✼❖●✻✽✭■✫✣✻❅✭⑥✲■❏✇❄✣✾✺✩✜❘✈④❧❊❁✩✜❪ ✲➃➵❆✍✷❈✲✰✲✼✷❍✩✜❆✞❀✮✾✍❆❈✵✸❘➁✵✽➍▲❊❁✲■❆❋➑✶❊❍❊✹✲✼✷❋✷❚✾✣✻✶✷❈✯❴✾✣✲✼❏❃✿✹✷❍✵➂❀✮✵③✲■✵✇❊❢✻✽④■✾✣✭✰✷✹✵✽✵✶✵✽✾✣❖❬❆ ❖❽✭✼✵✶✷❍❀➣✿❈❊❜✷❈✵❅✵✶✩✬✿❍✯s④⑥✩✜❘ ✾✣❵✷❍❀✮✯✜✵❊ ✧➂✧✪✲✰❀✬❘◗✯✮✲■✭✼✵❱✦❻✵✙✾✇✻✽❏❃✷❋✻❂✷❋✻✶✻❉❆❈✴✶✵✽✿❁❀✮✲■➍✣✾✣✵❅✾✣❖✙✯✱✵❅✵❁❆❻✷❈▼✓➤❅✩✱✻✶❑✍✲➃✿❈✻◗✷❈✷❈❊✥✭➃✿❭▼ ▼✜✵❁➤✽✭✳✲✼✷✹✾✣✵✽✿❍✯✜❆❈✩✜✲✼❆❈❏❂✩✜✲✼✿✹❆❬❄✮✻✽❘◆✭✼✭❫✵●❙✓❊✹✷❈❀✮✷✹✵✙✩✜✵❱❆❈❆❋❊❍✵❜✻✶✷❋②✜❖✽✻❅✩✬✵✺✲■✭■❆❈✫✜✷❈✿✹✻❅✻❅✷❈✲✼✾✣❊❜✩✜✵✪✴❳✾✮❖✽❊❅❀✜✩✮❘➂✻❅✵✽❊➂✦❧❊❢④s✻❃④✰✷❍❆❈❀✮❊❁✩✜❄✮✵❱❏➁✲■✷❈❆❋✻❅✻❅✷❍✾✣✭❚❀✮✴❃✵✪❊❍✷❋❀✜❛✣✻❅✲■✻❅✩❉❆❈❊✛✷❋✿✹✷❈✻❅✻✱✩✜❊❜❏➁✵❂❀✮✵✽✩✜✷✹✲✼✩✙✾✣❖✽❀❉✵❂✷❍✷⑥❀✮❏❃✵③✩✬✵●④ ✷✹✷❈✩✜✵❅❄ ❆ ❏✐✻✶✾✣✯✪✻③✯✜✲■✻✶❏❃✵●✷❈✵❅❆❫✩✬④ ❏⑩❑❅➍✣✾✣✯❱✷❍❀✮✵⑥✭✼✵❅✾✣❖❽✷❍❀✙✩✬④✬✷❍❀✮✵✸✵❁➤✽✷✹✵✽❆❈✲✼✩✜❆❫✵❅✯✣❖✽✵✸✩✬④✜✷❍❀✮✵✛❊✹✷❈✻✽✲✼❆❈✿❈✻✶❊✕✵✣❘ ✁ ✂☎✄ ✆ ✄✝ ✞ ✟✡✠ ✄✆ ☛ ✄ ✠ ❨ 25 ✯ ✂ ❞ ❨ ✯ ❞ ☞ 10 ❳❍❏▼▲ ❯ ❏ ✍❘✺❩ ❑ ■ ❏❙▲✡◆✾◗❏▼❘❬❚❖▲ ❯❲❱ ❭✟❫✫❫❖❴✟❵✫❫❙❛ ❜✜❝❈❞ ✻✹❡ ✦✽❈❣ ❝✚❤✝✐❀❥✔❝ ❦✧❧✧❧✜♠ ✍✌ ✏✎ ♥ ❘✘❋❅✵♣✾✣✩✜❏❃❛✮❆❈✵➇✯✓➉ ✩✜❆ N = ✻♣✭✼✻♣❆❈✵❅❏❂✻✶❆❈➦✬✫✣✻❉❛✬✭✰✵ 618033989 ❄✣❆❈✩✜❄✮❆❈✲❅✵●❊ ✷✚✵✉❊ ➦✬✫▲✵①❊✕✩✜❘✬✾ ❙✣❆❈✲✼✾❉✩✬②✜✫✬✵✽②✜❆❋✵❅❊❜❆✞✵✇✫✣❏❂✾❂✫✮✻❅✭➃✫✜✷❈✷✹✲✼❄✜❆❈✵❱✭■✲✼✿✹✾✣✻●≈✩✜✷❈❏❃✲■④ ❛✮1,1/N ✽ ✵ ❍ ❊ t✵✶❊ ❖●✻✽✭ ✻①❄ ❊❁✻①❄✣✻❅❆❋✷❈✲✼✵✉✯✾✵✶❊ ✿✹✲■❏❃✻✽✭✰✵ ✷ ❆❈✵❱✻❬▼✬✻✶✾✬✷❇✿✹✵●✷❋✷❈✵❱❄✮❆❈✩✜❄✣❆❋✲❅✵❬❊ ✷✜✵✣❊ ❘ 0, 618033989 ❭ ❘✣❯s✩✜✾✮❊❍✲✰✯✾✵❅❊ ❆❈✩✜✾✮❊❇✫✣✾❃➦✜✫✣✻❅❆❋✷❇✯✣✵❱✿❍✵✽❆❈✿✹✭✰✵❱✯✣✵➂❆❋✻●▼✜✩✜✾ ❘ ❪✼✻●❵✺❙✣❏ ❆❈✵✽✩✬❏◗✫✬②✜✵➂✵❅❄❆③✵✽❊ ✭✰❆❈✵❳✲■❏●❆❈✵❅✵❬❄ ✿❈✷✹✷❈❆❈✻❅✵✪✾✣➦✬❖❬✫▲✭✰✵↕✵✪➦✜✿❍✫✮✵❱✲✈➦✜✻ ✫✣✻❅✻❳❄ ❆❋✷❇✭■✻❳✯✮✵✪④❧✩✬✿❍✲✼✵✽❊❱❆❈✿✹✭✼✻↕✭✰✵✮❏❘ ✵❅❏❃✵↕✻✽✲✼❆❈✵↕✵❬✷➂✭✼✵ ❪❧❛✶❵❳❄✮➐✜✫✮❊✹✷❍✲■❍❊✕✲✳❊✕✭❅✵✙❄▲✷✹✩✜❆❈❊❁✩✬❊❁✫✬✲✼❛✜②✜✭✼✵✽✵✥❆⑥➦✬✫✣✫▲✾❳✵✥✷❈❄✣✵✽✩✬✭❚✫✣❆❈❆★✵❅✫✣✿❈✷❈✾❱✻❅✾✣✿✹✵❅❖❬❆❈✭✰✿✹✵✱✭✼✵✥✻●▼✬✿❍✩✜✻❅✾✬❏❂✷✝❄✮✻✱❄✭✼✵●✭■✷s✻✱✯✣④❧✵❢✩✬✲✼❆❋❊✛✻❬▼✜✭✼✻❂✩✜✾ ❏ 1✵❅❑✽❏❃✩✜✾✵ ✻❅❆✚✵✽❊✲■❆❈❄▲✵✪✩✜✵●✾✮✷✞❊✕✵✣✭✰✵❱❘ ❏ ✵❅❏❃✵❱❄ ✵✽❊ ❆❈✲■❏●✵❬❄ ✷✹❆❈✵✙➦✬✫▲✵✙✿✹✵✙✿✹✵❅❆❈✿✹✭✼✵ ❇♦ ➭❍✫✣❊❍✷❈✲✳➍✓✵✽❆❫②✜✩❉✷✹❆❈✵ ♣ ❘❉➅✥✾✺❛✜✻✶❆❋✲■✭✜✵❅❊✹✷❚❆❈✵❅❏❂❄✮✭✳✲✮✯✓➉ ✵✽✻❅✫❚❘❉➟❢✾✺✵✽✾➂②✬✲✰✯✮✵⑥✭■✻✸❏❃✩✬✲✳✷❈✲❅✵✛❊ ✵●✷❫✩✜✾✪✻❬➎❻✩✬✫✜✷✹✵⑥✫✣✾❱✭■✲➃✷✹❆❈✵✸✯✓➉ ✵✽✻❅✫❚❘ ➵▼❂❄✣✻❬❆✚②✬✵❅❄ ✻❅❊◆✲➃✷❍✻❬❍②✜✲✳✭✓✩✬✯✣✲✼❆✍✵③✵✽✭■➯s✲➃✷✹✵✶❆❈✿❈✵❅✷❈❊✥✫ ✯✓✵✛❊ ➉ ✿✹✵✽✵●✻❅✷❋✫◗✷❈✵✛✯✮✩✜✻❅❄✾✮✵✽❊❢❊ ❆❋✭✼✻✽✵✪✷❈✲✼❛✜✩✜✻✶✾✪❆❋✲■❊✕✭▲✵❅✻✽❄✬✫◗✷s④❧✯✾✩✬✵✽❊ ✲✼❄✣❊❬❑●✻❅✲■❆❋✭✬✷✺✵❅♦ ✾❱❆❈✵❅❊✹✷✹✵❢✷✹❆❈✩✬✲■❊❫✭✳✲✳✷✹❆❈✵✽❊❅❘❅❯s✩✜❏❃❛✜✲✼✵❅✾ ❴ ❘ ❅❋ ✵❅❊✍✻✽✲✼❆❈✵✽❊❚✯✣✵✞✷✹❆❈✩✬✲✼❊❚✯✮✵❅❊s④✰✻✶✿❍✵✽❊❫✯✓➉ ✫✣✾➂❄✮✻✶❆❋✻✽✭■✭❅✵✽❊ ✭❅✵❅❊ ❄✜✲✼❄ ✵✶❄ ✯✮✵ Q ❆❙✣❈✵✶❆❈✿❈✩✬✷❋✫✬✻✶②✜✾✣✵❅❖❬❆ ✭✼✵➩③q ❊❜❪✰✩✜✻❬✾✬❵◗✷❨✭✰✵ ✯✮✵ ②✜✩✬18✭■✫✣❏❃✿❍✵❖❏ ➂r ❪✰❑ ❛❉40❵◗✭✼✻❿✿❍❏ ✭✰✩✜✾✣❖❬✵❬✫▲✷ ✵●✫✣80❆✉✯✮✿❍✵❿❏ ✭■✻ ❘ ❖●❆❋✻✶✾✣✯✮✵✙✯✬✲✼✻✶❖✽✩✜✾✮✻❅✭✼✵ P Q ❘ P ❛ ❘ ➐✶❊ ②✬✻✽✭■✫▲✵✽❆ + + √6 + · · · ❘ s ❘s❩✶✩✬✲➃✷ A ❑ B ❑ C ✵●✷ D ✯✣✵✽❊✸❄▲✩✬✲■✾✬✷❈❊✛✻❅✭✳✲✰❖●✾✾✵✽❊ ❊⑥✷✹✵●✭✼❊✸➦✜✫▲✵ ✲✼✯✾✵✽❄ ❆❈❑ ✵➂✷✹❆❈✩✬✲■❊⑥✵❬✷ ✯✣✵✽❏❂✲✼ ❘ AB ✿❩✶❍✵✽✩✬❆❈✲✳✿✹✷ ✭✰=✵✽❊✙BC ✯✭❭✮➉ ✲✼✵❳✾✬✷❈✯✬✵❅=✲✼✻❅❆❋❊✕❏●CD ✵✵✶✿❈❬❄ ✷❈✷✹✲✼❆❈✩✜✵❅=✾✉❊❱✯✣❆❈1✵❅✵✽❘▲❊❁❊✙❄▲➟❢✯✮✵❅✾❳✿❈✵❅✷❈❏✱✿✹✲■✩✜④✰✲✰❊ ✾✮❁❊❁✿❍AC q BD AD ✽ ✵ ❈ ❆ ✹ ✿ ✰ ✭ ✽ ✵ ✙ ❊ ✣ ✯ ◗ ✵ ❍ ✿ ✽ ✵ ✬ ✾ ✹ ✷ ❈ ❆ ❅ ✵ ❊ E ✵●✷ EC❪✼❄✣❘❢✻❅➒ ❆❋✷❈✵●❊✲✼✵✪✷❈✵❅❀✜❆❈❏❂✻❉✿❁✲■❀✬✾✣✫✣✵❅❆✚❆➂✵✶❊ ✭❻✵❱➉ ✻✽❊❍✲✼✫✣❆❈✵↕❆✞✭✼✯✜✵❱✫✉✯✣✵✽✷✹❆❈❊✕✲■❊❍✻✶✲✼✾✣✾❉❵❁❖❬❘✭✼✵⑩✿✹✫✣❆❻②✬✲■✭✳✲✰❖●✾✣✵ q q q q B A B C D AED √ 1+ ✒✑ ✓✑ ✒✑ ✒✑ 2 ❉ ✜✭✰✯ ✩✜✫✮❘s✾✣✭✼✵✇❖❬➟❢✫▲✾↕✵✽✵●❊❍✫✣❊✕✷❱✻✽❆✛✲✳❄✮✷⑥✯✮❆❈✵✱✩✜➦✬✫▲❄▲✿❍✵●✵✺✩✜✭■❆➀✫✮✭✼✷❈✻✱✲✼✲✰✕✩✜❄ ✿✹✾✮✲✍✵❅❊ ✾✣❆❋❪✰✵●❄✮✲✰✩✮✭■✻✶✭✼✯✮✵ ❆⑥✵❂✵❍✻↕❄ ➤✶✯✓✵❅✭✼➉ ✻↕✩✜❏❂❊✕❆❋❄✮✿✹✻❉✲■✭✼✭✳✿✹✵✜✭✼✲■✻●❑▲✾✣✷❈❄▲✵✇✲✰✩✜✩✬✿❈✾↕✫✣✻✶❆❇❆❈✯★❆✚✷✹➉ ✵❅❊❆❋✫✣✲✼✵⑩✾⑩❄✜✭✰✯✣❄✣✵✽✵❳❆❢✵❅✾✣✭■✭■✻✻ ✁ ❄❄✮✭✳✵✽❊ ✲✰❆❈✵✽✲✼❆❇✩✮✯✣❄✮✵❃✻✶❆❇✯★✾✣➉ ✩✜✵●❊❜✫✮✿✹④✍✲✳✭■❊❁✭■✻✪✻✽✷❈✭✼✲✼✩✜✩✜✾✣✾✇❖❬✫▲✯★✵✽➉ ✫✣✫✣✾①❆❭❵✕❄▲❘s✵✽➐✬✾✣✾❂✯✜✭ ✫✮✻✶✭✼✿❁✵✜❀✜❑★✻✶✲■✾✬✭✍✷❢④✰✻❅✯✣✫✜✵●✷✸✫❉➤✺❏✱❄✣✫✮✵❅✭✳✷❈✾✣✲✼ ✲✼✭✰✯✜✾✣✵✙✫✮✯✬❄✜✭✼✲✰✵❅✭■➦✬❊❚✫✣✫ ❊✸✯✣✵✶❊ ✵✈✿✹✵✺✩✬✭✼✩✜✵❬✫✣✷✥✾✣❆❋✷✥❖❬✵❅✫▲✾◗❏❃✵✽✾✾✵❅✫✣❊❍✵❅❆❋❊ ✫✣❊❚❖❬❆❈✭■✯✜✲✼✵ ❖✽✲✳➯✡✵❅✵✽✻❅❊ ❆❈✾✬✵❅✿❍✷❢✾✬❏↕✭■✷❈✻✙✵❅❑✓❊✏✚❆➦✬✵✽❊✻❇✫▲❄ ❊❁✵✺✭✼✲■✻❇❊❍❊❁✷❋❄▲✻✺✻✶✩✜✾✣❄ ❊❍✿✹✲✳✵✽❊✵✺✷❈❆❈✲✼✲✼✯✮✩✜✩✮✵✪✾✛✯✣✲✼✭❻✵✺✾✜➉ ✻✽✲➃✯✓✲✼✷❈❆✕➉✲✼✩✜✻✽❑✣❊✕✭✰❊❁✵ ✲ ✂ ❆④❧❈✿✹✩✬✵●✲■✭✳✲✼✷✹✭✼❊✱❆❈✻●✩✬✷❈✻✽✲✰✫✬✫➣✩✜②✜✾✇✵✽❛✮✾✬✵❅✩✬✍✷ ❊✹✫✜✷③✻✙❄✷✺✯✣✾✣✯✣✵✱✩✬✵⑩✫✬✫✣❊❜②✜✾✣✵✽✵✽✵❃❄✜✻❅✷✪✫❳❊❜✵❅❊❜❄✮✿❍✵❅✩✜✻✶✿❍✾✣❆❋✩✜✻✽✯✮✾✣✭■✵◗❅✭✯✮✵✽❄ ✵❅✵●✭✼✵❅❊❬✷③❑◆❊✥➦✬✲✳❄✣✭✼✫▲❊✱✩✬✵❂✫✣✩✮✭✰❆✈✿❍✩✜✿✹✭✼❆❋✻✪✫✣❊✕❄✣➦✜❄✣✵❅✫s❆❈✾✬✵✽➉ ✩✜✷✪❏❂✾✇✭✰❅✲ ✵●✵❅❄ ✭✼✫✣✵❅❆❈✵ ❊❆ ❄▲✩✜❊❍✲✳✷❈✲✼✩✜✾❱✲■✾✜✲✳✷❈✲■✻❅✭✼✵✜❑✺✵❽❊ ②✬✻❅✭✳✫▲✵❅❆s✭✼✻⑥✭✼✩✜✾✣❖❬✫▲✵✽✫✣❆❚✯✜✫✙❄✮✭✳✫✣❊❫✭✰✩✜✾✣❖✜❘ ✿✫✣✹✻✽✾✣❘✍✭■✵③✲✼➒❇✵❅❊❁❆❢✻✶❄✮✾✮✯★✲■❆❋❊❢➉ ✫✣✻❅✫✣✭✼✾↕✵③✾✣❏●✵✪✵●✷◆❆❋✵●❄ ☎✻✵❅✷✹✄◗❆❈✾✪✵✙④✰✾✣✻❅✯✣✵✽✲➃✵✪❆❈✷✞✲✼✭■✵❍✵✜✻✶➤❅❑✜❆❈✻❉✫✣❖✽✿❈✵✺✾◗✷❈✵❅➦✬✚❆ ❏❃✵❅✫✮❊ ❊✕✵❅✲★✵❅✾✬✭✼❆❻✩✜✷◆②✜✾✣✯✣✩✬❖✽✲■✵●✵✺❆❢✫❉✯✓➤③✯✣➉ ✵✺④❧✵✽✩✬❊✕❛✜✲✼❊✕❊◆✻✶✵❅❊✛✭✰✾✣✵⑥✿✹✵❅✷❈✵✙✾❳✩✬✿➀✫✣❀✜▼✬❆❬✻✽✭■❘✬✲■✫✜✾✣❩❅✷❢✯✮✻❉✭✼❆❋✿❁✻✙✲✰❀✜➦✬❄✮✻❅✫▲✻✶✾✬✵✙❆❈✷✞✩✬✵❅➦✜✲✍❊✹✫▲✷❇✵❍✵✛➤❅❏✱✷✚✭✰✵✽✵✛❊✫✣❆❈✾✜✚❆ ✲✼✵✽✵✽✲s❊ ✫✣❊❜✯★✵✽❆❈➉ ❆❻✵✺✫✣②✜✾❳✯✜✩✬✻✶✲✼✵❅❆❫✾✮❊✕✻ ❊ ✵❁➤✽✷✜✵❅●❏❊ ❆❋✲✰✵●❄ ✵●✷✹✫✣❆❈✵✽❆❈✵✪❊✥✯✣✯✣✵❱✵③❀✜✭❭➉ ✻❅✵❅✫✜❊✕✿✹✷❇✻✽✵❬✭■✷✲✼✵❅❆❽❘ ●❏ ✵❬❄ ✷✹❆❈✵❅❊✥✯✣✵❱✯✜✲■✻✶●❏ ✵●❄ ✷✹❆❈✵✬❑❉✷✹❆❈✩✬✫✬②✜✵✽❆✞✭✼✻➂✭✼✩✜✾✣❖❬✫▲✵✽✫✣❆✈✯✣✵③✭✼✻❱❆❋✻❅❏❃❄✣✵ ✠ ✄✆ ☛ ✄ ✠ ✂ ✄ ✂☎✄ ✆ ✄✝ ✞ ✑ ❨ ✯ ❞ 25 ❨ ✯ ❞ ☞ 10 ❨❲ ✻✽✷❍❀✮✵✽✆ ❏❃✵❍➤❅✻●✷❚✷❈✲✰✧✪✿❈❊⑥✵❱❯★❖❬✩✜✲ ②✜✾✬✵✸✷✹✵✽✷❍❊❍❀✮✷✥✵➂➑✶❊❜✲✼✩✬✾✮✭✳✻✽✫✜✭❚✷❈➄★✲✰✩✜✩✬✾✮✫✣❊✞✾✣✷❈✯✇✩✙➺✬✷❍✻❅❀✮❆❋✵ ✷✛ ❅➨✞➓❉❀➓ ✝ ❉ ❅➓❉➨✓➓✶❯❳➔ ❯sq ✩✬✠ ✭✳✟ ✭✰ ✵❅❇ ❖✽✠ ✵❅✟❊❢❉☛❩✶✡ ✵❅❘ ✾✜✲✼✩✜❆❢➥③✲✼❖✽❀❂❩✶✿❁❀✮✩✮✩✬✭ ♥ ❘✺❋❅✵❅❊✞✿❁❀✬✲✳➯s❆❈✵✽❊ ❑ ❑ ❑ ✵❬✷ ❊❜✩✜✾✬✷★✷✹✩✬✫✣❊❫✫✜✷❈✲■✭✳✲✼❊✧✵❅❊ ❊❫✫✣✾✣✵⑥④❧✩✬✲■❊◆❄▲✩✬✫✣❆✏✵❅❊ ✿❍❆❈✲■❆❈✵⑥✫✣✾✪✾✣✩✜❏◗❛✜❆❈✵ ✻③❄ ✿✹✲✼✾✣✵✽➦✺❊❍✷✈✿❁❀✬✯✜✲✳✲ ➯s②✜❆❈✲■❊❁✵✽✲✼❊ ❛✜✭✼✵➂❄✣✻❅❆ ✷❈✵✽❑ ✭✮➦✜✫▲✵✛✵✽✭✰❊❍✵✛✷✈✾✣✯✜✩✜✲ ②✜❏◗✲■❊❁❛✜✲✼❆❈❛✜✵❍✭✼✵➂✻✛❄ ❄✣✷✹❆❈✻❅✩✬❆ ✲■❊❇❘✌✿❁❀✬☞❇✲✳✫▲➯s✵●❆❈✭✓✵✽❊ ✵❅❊✹✷✞✭✰✵❱✵❅✿❁❊✹❀✬✷◆✲✳➯s✯✜✲❆❈②✜✵ ✲■❊❁✲✼❛✜♦ ✭✼✵✸❄✮✻✶❆ ❑ abcde cde bcd abc a ✍✏✎✠✑✓✒✕✔✗✖✘✎✚✙✜✛✣✢✥✤✧✦☛★✩✢✥✙✣✪✬✫✭✢✏✮✯✖✱✰✳✲✴✎✵✒☎✑✘★✩✶✸✚✷★ ✔✹✒✺✰✥✖✻✢✠✙✼✔✻✶✾✽✿✯✷★ ❀✏★✬✛❁✰✌★❂✫✿✤✯✒✠❃❄❃❅✎✚✙✣✰✌✮✯✖❆✑❇✑✱★✯✶ ✫✿✤✯✒✠❃❄❃❅✎✚✙✣✰✌✮✯✖❆✑❇✑✱★✯✶❉❈✿✽✼❊ ❁❊❜✿ ❀✮✵➂✩✬✷❈✲ ✵❅➤✛❆❈❏❂➺✮❄✣✫✮✩✜✲■✲✼✾✣❊✕❊✕❊❍✵✪➦✜✲✰❛✬✫▲❄✣✭✰✵✻❅✵✽❆ ❊❫✯✣✯✬✵⑥✩✬✲➃②✬✿❁✫ ❀✬✲✼❊✕✲■➯★✵ ❑✮❆❈❊❜✵❅✵●❊❬✭✰❑❅✩✜✯✮❑❚✾◗✩✜✾✣➦✜✾✣✩✬✫▲✿✫✣✵ ❊✪✯✮✵❜✵✽②✜❊❍✩✜✷❇❘✜✾✮❄✮➺✮❊✪✻❅✫✮✲■✻❬✲✼❆❢❊✕②✜➦✜✩✬✩✬✫▲✫◗✲✼❆✵ ✲■❏❃❄✮✯✬✯✜✻❅✲➃✲ ②✬②✜✲■❆❬✲✼✲■❊✕❊❁❘✍✵ ✲✼➺✮❛✜✫✮✭✼✵❃✲✼❊✕❄✣➦✜❑✽✻❅✫▲✻✽❆ ✭✰✵ ✩✜❆❋❊❫✵●✯✬✿❍✷➂✲➃✵✥②✬✯✜✲✼✾✣❊✕✻✶✩✜✵ ✾✮❏◗❊➂❛✜✭✰✵✽❆❈✵ ❊ ❑ ✻❅➒❢✭✼✩✜✩✜✾✣❆❋❊ ✿❢✯✣✵●✯✬✫❉✲➃②✬➤✙✲✼❄▲❊✕✵❱✩✜❊❁✭✼✻✙❊❁✲✼❛✜❊❜✩✜✲✳✭■❏❂✲➃✷✜✵❅❊❏◗❊❇✵ ❊❬➉ ✩✬➯★❆❈✵❅✾✬✷▼✻❱❄ ✾✣❑▲✩✬✵❬✫✣✷✥❊❙✯✣q✩✜✾✣✿ ❪✰✿✹✻❅❆ ❵❁❘ ❩✽❩✽✲✲ ❑✽❑✜✻❅✻❅✭✼✭✼✩✜✩✜❆❋❆❋❊ ❊ ✻❬✻●②✜②✜✵✶✵❅✿ ✿ ❑ ❑ ❑ ❑ ❑❅❑✣✿❍✵⑥✯✮✩✜➦✜✾✣✫✮✿ ✲✬✵❅❊✹✷s✲✼❏❂❘❄▲✩✜❊❁❊❁✲✼❛✜✭✼✵✣❘ ❑✮✿❍✵❱➦✜➒❢✫✮✩✜✲✣✾✣✭✼✻✽✿ ✲✼❊❁❊❜✵ ❑✶✿❍✵✛➦✜❘s✫✮➒❢✲✬✩✜✲■❏❃✾✣✿❽❄✜❑ ✭■✲✼➦✜✫▲✵✛➦✜✫▲✵ ✵❅❊✹✷❚❄✮✻❅❑✣✲■✵●❆❚✷ ✻●②✜✵❅✿ ❘ ❑ ❘✣➒❢✩✜✾✣✿ ②✬✻❅✫✜✷ bcd d=5 b c+d+e d abc c + e ≡ (mod 3) cde d=5 c=2 e ≡ (mod 3) e ∈ {1 4} c=4 e ≡ (mod 3) e ∈ {1 3} e=3 c=4 b b ∈ {1 2} b a=1 abcde = 12453 a=1 ❋❍●❏■❇❑❍▲ ▼❇●❖◆◗❘✿ ❙ ❑❖❚✘❚❯▼✏❱ ❙❖❲❨❳ ■ ❘ ◆✬▼ ❘❩❲❭❬✩❪✵❫✥❘ ■ ■❴●✯❵ ❫ ❑ ❲❆❛ ❑ ❘ ▲ ❜❝❘ ▲✻❑✬❞❖▼ ❙ ◆☎▲✣❡❍▲❆●❢❞ ❘❤❣ ❋❍✐✚❜✼❥❴◆❍❚❩▲ ▼❴▲✻❑✬▲❆●❢❦ ❘❩❲✻❧♠❙ ❵ ◆☎●❖❚♦♥♦♣ ❣ ■ q☎❑sr ❙❖❲ ❜ ❬ ● ❛❖❲ ●✏☛t◆ ●❏❜ ❣ ❚✻❱ ❙✹❛ ◆ ❘☎✉✺✈✣❙ ▲ ❙ ■ ▼ ❙ ♥ ❘ ❚✘r✹❜❉❘ ▲✻❑✬❞❖▼ ❙ ◆☎▲ ❜✺❋s✇②①❨③ ❙✬④❖❘❩❲ ▼❇❑❖❚⑥⑤❩❜✕① ❙ ■ ❘❩❧✿⑦s❙ ◆ ❛ ❜ ❬ ♣ ❥ ◆✠❵ ❞✯●❖◆◗❘ ❚❯▼ ❘☛✉✩❘ ▲❖⑧✏■ ❘✗⑨♦⑩✚❘❩❧ ● ❲ ● ④ ❜✵❘ ▲✻❑✬❞❖▼ ❙ ◆☎▲ ❜✥❶✼▼ ■ ■ ▼ ❙❖❧❅❬ rs●❖◆❷✇ ❙❍❸❆❹✯❘ ◆s❺✻▼ ❘✣❻ ●❏■ ■ ❘✓❛ ▼ ❙ ▲ ❘ ❥ ◆✯❚❩▲ ▼❇▲✗❑❍▲ ❘ ❜✠❼❖● ❲ ●❖◆☛▲❆●❏❜☎✐ ✈❉❪ ❭ ❘ ➵ ✾✱✫✣❆✹✾❃✿❍✩✜✾✬✷❋✻❅✲■✾✮❊❇✷❍❀✜❆❈✵❅✵✸✧✪❀✬✲➃✷✹✵✬❑✜❊❁✲ ➤✪❆❈✵✶✯✓❑✜✻✶✾✣✯✱④❧✩✬✫✣❆✈❛✜✭■✻❉✿❁✴✙❛✮✻✽✭■✭■❊❅❘ ❪✼✻●❵◗❊❜✦❧④◆✵●✭✰✩✜✵❅✾✣✿❈✷✹✵✱✵❅❛✮✯✺✻✽✲✼✭■❊❢✭s❆❈✲■❊✸✵❅✯✺❊❜♦ ✵●✭✰✵❅✿❈✷✹✵❅✯❳✻✽✷✥❆❋✻✶✾✣✯✮✩✜❏⑩❑✣✧✪❀✜✻✽✷❢✲✼❊⑥✷❍❀✮✵✺❄✣❆❈✩✮❛✜✻❉❛✬✲■✭✳✲✳✷❻▼◗✷❍❀✜✻●✷❢✷❍❀✮✵✱❛✮✻✽✭■✭ ➔ ❪❧❛✶❵❳✻✶✦❧④❫❆❈✵❱✷➀✧✪❛✣✩✇✩❉✷❍❛✜❀❂✻❅❛✬✭✳✭✼✭✼❊➂✻✶✿❁✻✶✴ ❆❈♦✵❂❊❜✵●✭✰✵❅✿❈✷✹✵❅✯⑩✻✽✷✸❆❋✻❅✾✣✯✣✩✜❏↕❑★✧✪❀✜✻✽✷✛✲■❊✸✷❍❀✮✵❃❄✣❆❈✩✮❛✜✻❉❛✬✲■✭✳✲✳✷❻▼↕✷❍❀✜✻●✷⑥✷❍❀✮✵❜▼ ❪❧✿❋❵◗✻✶✦❧④❫❆❈✵❱✷➀✧✪❛✣✩✇✩❉✷❍❛✜❀❂✻❅❛✬✭✳✭✼✭✼❊➂✻✶✿❁✻✶✴✣❆❈❑✮✵❂❖❬❊❜✲ ②✜✵●✭✰✵❅✵❅✾✺✿❈✷✹✷❍✵❅❀✜✯⑩✻●✷◆✻✽✷❍✷✸❀✮❆❋✵❜✻❅▼❂✾✣✯✣✻✶❆❈✩✜✵✸❏↕✷❍❑★❀✮✧✪✵❱❀✜❊❁✻✶✻✽❏❃✷✛✲■✵❱❊✸✿❍✷❍✩✬❀✮✭✼✵❃✩✬✫✣❄✣❆ ❆❈♦ ✩✮❛✜✻❉❛✬✲■✭✳✲✳✷❻▼↕✷❍❀✜✻●✷⑥✷❍❀✮✵❜▼ ✁✄☛ ✄✗✂✆✖ ✎ ☎ ✙✘ ✚✂ ✞✛✝ ✝ ✠✟ ✢✜ ✡✟ ☞☛ ❪✰✻❬❵ ❆❈✵❅✯ ✮ ✤ ✮✰✯✭✱ ✣✥✣✥✤✧✤✧✦✲✦✩✤✧★✫✳✪✭✮✵✬ ✴ ✤✷✶✥✪✡✸ ✍✏✎✠✑✓✒✕✔✗✖✘✎✚✙ ✲✴★☛✤❏❃❄✢✠✙✣✎ ◗✑✱★✩✙✏✖✘★☛✤ ✕❊ ⑥★❍✑❆✑✘✎✌✪ ❷✒✠✤❩❀✕✒✏★✥✔✻✶ ✠✑ ✼✒ ☎✢✠✤✹✶ ✺✎✥✥ ❀ ✤❖✎ ✙◗✎✥✶✼✍✩✛✣✢☎✖✗✙⑥❊ ✫ P( ) = ❧❪ ❛✶❵ ❛✜✭■✻❉✿❁✴ ❪❧✿❋❵❚➌⑩✵❱❀✜✻❬②✜✵ ❛✬✭✼✻✶✿❁✴ ❊✕✻❅❏◗✵ P (2 |2 P (2 = ) = ) = = ❋ ❊❉ 13 = ✍✌ ☛✔✹✒✏✰✌★✩✙✼✔ ✌✔✱✎ ✏✎ ✒✑ ✔✓ ✕☛ ✰✌★ ✺✍ ❷✶ 6 = 3+6+4 13 = 78 13 ❘ ✘✙ ☎✔✗✖✱✔✹✒✕✔✘✎ ❘ ✣✥✤✧✦✲✹ ✱ ✯✧✴❀✺✽✼✾✯✭✿ ★✙✸❂❁❃✣✥✣✥✤✧✤✧✦✕✦✩✹ ★✫✱ ✪✭✯✧✬ ✴✻✺✽✼✾✼✾✯✭✯✭✿ ★✙✿ ✸❄★✙✸ ❁❅✣✥✤✧✦✄❆✽❇ ✿ ✮ ✪ ✼✷✯✭✿ ★❈✸ = + + ❘ ⑧✏■ ❚❩●❭❚❩●❏■ ④❖❘ ❞ ⑦ r ❘❩❙ ◆✠❵❯♥ ❙✬④ ▼❴❞✿❦♦●✯❑❏■ ❘ ❜✚❚❩▲✻❑✬❞ ❘ ◆☎▲ ❜ ❻ ❘✓ ❛❍❘ ❱ ❞ ❘ ♥ ❲ ❑ ❧♠❧ ❖● ◆☛❞ ④ ▼ ■ ■ ❘ ❜✕♥ ❲ ❑ ♠ ❧ ❧ ❖● ◆☎❞ ④ ▼ ■ ■ ❘ ❜ ❞ ⑧✏■ ❘✗⑨⑥⑩✚❘❩❧ ● ❲ ● ④ ❜❍❩❚ ▲✗❑❍❞ ❘ ◆☛▲ ❜ s❻ ✉✯✈✣❙ ▲ ❙ ■ ▼ ❙ ♥ ❘ ❚✘r✹❜❍❚❩▲✻❑✬❞ ❘ ◆☎▲ ❜✯❋s✇②①✼③ ❙❍④❏❘❩❲ ▼❆❑❏❚✏⑤✓❜☛① ❙ ■ ❘❩❧✿⑦s❙ ◆ ❛ ❜☛❥❴◆☛❞✯●❖◆ ❘ ❯❚ ▼ ❙✹✉✯❙ ◆☎✕ ❶⑥▼ ■ ■ ▼ ❙❏❧ ❬ sr ●❖◆✭✇ ❙✬❸✗❹❍❘ ◆✬❺✻▼ ❘❷❻ ●❏■ ■ ❘✓❛ ▼ ❙ ▲ ❘ ❥❴◆❍❚❩▲ ▼❴▲✻❑✬▲ ❘ ❜✏❼❏● ❲ ●❖◆☎▲❇●❏❜✺✐ ✈❉❪ ♣ ❘✙✦❻✾◗✷❍❀✮✵✱✯✬✲✼✻✶❖●❆❋✻✶❏↕❑ ABC ✲✼❊✛✻✱❆❋✲✰❖✽❀❉✷❍❈✷✹❆❈✲■✻✶✾✣❖❬✭✼✵ ✧➂✵❅✻✶✲✳✿❁✷❍❀❀❳✭■✲■✾✣✵❂❊✕✵✶❖●❏❃✵❅✻✶✾✬✾✣✷ ✯ ✲✼❊③❄✣❘⑩✵❅❆❈➑❉❄▲✫✣✵✽✾✣❆➀✷❍✯✜❀✮✲✼✵❅✿✹✫✮❆❈❏◗✭✼✻❅✩✜❆⑥❆❈✷❈✵✬✩ ❑ ➑✯✜❉✲■✲■✾✣✻❉❖✽✯❃❑ ✩✜✷❍✾✮❀✮✻✽✵✪✭✓❛✜✷✹❊✕✲■✩❉✵✶❊❜✷❋❖●✵❅✻❅❏❃✿❈✭★✷❈✵❅❊✭✰✵✽✾✬✾✣✷❋❊✜❖❽q✷❍❀↕❑✜✻✶✩✬✾✣④s✯ ✷❍❀✮✵✺❊❜✵❅➦✜❛✜✫▲✲■✵✽❊❜✾✣✵❅✿❍✿❈✷❈✵✱❊ ✩✬④s✷❍❀✮✵ ❘ AB = AC A1 AC = Ai Bi AC Ai+1 B Ai C B B B BA1 + B1 A2 + B2 A3 + · · · ✑✛●■❍ ❏☛ ✓✖✻✢☛✑ ☛✎✠✑✓✒✕✔✗✖✘✎✚✙⑥❊ ✽❩ ✲✼✾✣✿✹✵ ✻❅✾✣✯❂✷❍❛✜❀✮✲■✵✪❊❜✵❅✭✼✵❅✿❈✷❈✾✣❊③❖❽✷❍✭✳✲✼❀◗✾✣✵❂✩✬④★❊✕✷❍✵✶❀✮❖●✵✪❏❃❊❜✵❅✵❅✾✬❖●✷ ❏◗✵✽✾✬✷ ❑★✷❍❀✮✵✱✭✰✵✽✾✣✲■❊ ❖❽✷❍❀✇✩✬④❫✷❍❀✮✵❂❊❜✵❅❘❢❖●❙✓❏◗❀✮✵✽✵✪✾✬✭✰✷ ✵✽✾✣❖❽✷❍❀◗✩✬✲■❊ ④ ✷❍❀✮✵❱❊❜✵❅❖●❏◗✵✽✾✬✷ ✲■❊✥❖❬✲➃②✜✵✽✾◗❛❅▼✙✷❍❀✮✵➂❙★❀✮✵❅✩✜❆❈✵❅❏➛✩✬④◆➺✬▼✬✷❍❀✜✻✶❖✽✩✜❆❋✻✶❊❢✻❅❊ ❘ ✯✜➐✜✲■✻✶❏◗✿❁❀ ✵✽✾✮✩✬❊❁④❢✲✼✩✜✷❍✾✮❀✮❊③✵↕❆❈✷✹✵❅❆❋✯✜✲✼✻❅✫▲✾✣✿✹❖❬✵✶✭✰✯⑩✵✽❊ ❛❅▼↕✻✺④✰✻❉✿❈✷❈✩✜❆✛✩✬④✲✼❊✱❊❍✲✼✻●❏✱✷✛✲■✵❅✭■✻✶✻✶❆③✿❁❀❳✷✹✩①❊❍✷❈✷❍✵❅❀✮❄✍✵↕❘➂✷✹❙★❆❋❀✬✲✼✻❅✫✣✾✣❊❬❖❬❑✓✭✰✷❍✵ ❀✮✵✪✷✹✩❉✷❋✻❅✭s❑✞✭✼✵❅✧➂✾✣✲➃❖❽✷❍✷❍❀❴❀↕✻❅✩✬✭✳④✭ A A1 A1 A2 A3 AC AA1 = B A1 B1 A = BA1 BA1 = 2 BA + AA1 = Bi Ai Ai+1 + 22 = √ 13 BAA1 AA1 C ➞ ✷❍❀✮✵❱❊❜✵❅➦✜✫▲✵✽✾✣✿❍✵✪✩✬④★✯✜✲■✻❉❖✽✩✜✾✮✻✽✭✓❊✕✵✶❖●❏❃✵❅✾✬✷❋❊❢✲■❊ √ √ √ √ 13 13 13 13 + + + ··· + n + ··· ❋ = √ 2 ❊❉ 13 + 1 + ··· + i + ··· 2 √ 13 = ⑧✏■ ❚❩●❭❚❩●❏■ ④❖❘ ❞ ⑦ r ❘❩❙ ◆✠❵❯♥ ❙✬④ ▼❴❞✿❦♦●✯❑❏■ ❘ ❜✚❚❩▲✻❑✬❞ ❘ ◆☎▲ ❜ ❻ ❘✓ ❛❍❘ ❱ ❞ ❘ ♥ s❻ ✉✯✈✣❙ ▲ ❙ ■ ▼ ❙ ♥ ❘ ❚✘r✹❜❍❚❩▲✻❑✬❞ ❘ ◆☎▲ ❜✯❋s✇②①✼③ ❙❍④❏❘❩❲ ▼❆❑❏❚✏⑤✓❜☛① ❙ ■ ❘❩❧✿⑦s❙ ◆ ❛ ❜☛❥❴◆☛❞✯●❖◆ ❶⑥▼ ■ ■ ▼ ❙❏❧ ❬ rs●❖◆✭✇ ❙✬❸✗❹❍❘ ◆✬❺✻▼ ❘❷❻ ●❏■ ■ ❘✓❛ ▼ ❙ ▲ ❘ ❥❴◆❍❚❩▲ ▼❴▲✻❑✬▲ ❘ ❜✏❼❏● ❲ ●❖◆☎▲❇●❏❜✺✐ ✈❉❪ 1− √ = 13 ❘ ❲ ❑ ❧♠❧ ❖● ◆☛❞ ④ ▼ ■ ■ ❘ ❜✕♥ ❲ ❑ ♠ ❧ ❧ ❖● ◆☎❞ ④ ▼ ■ ■ ❘ ❜ ❞ ⑧✏■ ❘✗⑨⑥⑩✚❘❩❧ ● ❲ ● ④ ❜❍❩❚ ▲✗❑❍❞ ❘ ◆☛▲ ❜ ❘ ❯❚ ▼ ✹❙ ✉✯❙ ◆☎✕ ❴ ❘❚❙★❀✮✵✪✵❅➦✜✫✣✻●✷❈✲✰✩✜✾ ❀✜✻✶❊❫✷➀✧✪✲❤✩✪✖✻✙✣❆❈✵✽✔✄✂✻❅✭✜✦✰✷✞❆❈✩✮✲✼❊✥✩❉✷❋❛✮❊●✵❅❑ ❊✹✷✈✾✣✻❅✾✣✩❉✷◆✯ ✷✹✩✙❘✁✫✣ ✍❊❜✾✣✵✸✩✶✷❍✧➂❀✮✵✪✲■✾✣➦✬❖❢✫✣✷❍✻❉❀✜✯✜✻✽❆❋✷ ✻✽✷❈✲✼✿❇④❧✩✜❆❈❏❂✫✮✭■✻▲❘ ❑✶➍✣✾✣✯❱✷❍❀✮✵❢②✬✻❅✭✳✫▲✵③✩✬④ ❘ ■ ✄✂✆☎ ✍✏✎✠✑✓✒✕✔✗✖✘✎✚✙ ➌⑩✵✇❀✜✻●②✜✵ α x2 − 3x + q = α3 + β = 81 β ✭☛✡✂ ❏☛ ✷☛☞☛ ✌✟ ■ ✢✏✔✻✢☛✑❆✖✗✢ ✫ ★ ✠✶ ☎✔✹✒✺✰✌★☛✙✣✔✻✶✼✍✞✝✠✟✆✡❨✢✏✮✠★✩✤❩✖❩✒ ✻✶✾✣✯✩✜❆ ✓ ☛ q ✩✶ ✏✢☛✑✱★✩❃ ✚✢✥✙✏❀✠✶ ❯✙✣✰✌✎✚✙◗★ ❖✖✻✢✕❊ ❘➇❩✽✲✼✾✣✿✹❑✶✵ ✧✪❀✬✲✼✿❁❀❃❖❬✲ ②✜✵❅❊ ❑✞✧✪❘ ✵✇❀✜✻❬②✜✵ α+β = αβ = q α3 + β = 81 (α + β) − 3αβ(α + β) = 81 27 − 9q = 81 q = −6 ⑧✏■ ❚❩●♠❚❩●❏■ ④❏❘ ❞ ⑦ r ✈✣❙ ▲ ❙ ■ ▼ ❙ ♥ ❘ ❚✘r✹❜✏❚❩▲✗❑❍❞ ❘ ◆☛▲ ❜✏❋s✇②①✭③ ❙❍④❏❘❩❲ ▼❆❑❏❚❷⑤✓❜✵① ❙ ■ ❘❩❧✿⑦s❙ ◆ ❛ ❜✵❥ ◆☎❞✯●❖◆ ❘ ❚❯▼ ❙ ❚ ❘❖❸ ●❖◆☎❞ ❚❩●❏■❆❑✬▲ ▼❴●❖◆ ✉ ❘❩❙ ◆☎❵❯♥ ❙❍④ ▼❇❞♦❦♦●✯❑❏■ ❘ ❜✯❚❩▲✗❑❍❞ ❘ ◆☛▲ ❜ ❻ ❘✓ ❛✬❘ ❱❨❞ ❘ ♥ ❲ ❑ ❧♠❧ ●❖◆☛❞ ④ ▼ ■ ■ ❘ ❜☛♥ ❲ ❑ ❧♠❧ ●❖◆☛❞ ④ ▼ ■ ■ ❘ ❜ ❻s✉✬❙ (◆☛❞✵⑧✺■ ❘✗⑨ ) ⑩✚❘❩❧ ● ❲ ● ④ ❜☎❚❩▲✗❑❍❞ ❘ ◆☛▲ ❜✏❶✼▼ ■ ■ ▼ ❙❖❧ ❬ r✬●❖◆❨✇ ❙❍❸❆❹✯❘ ◆s❺✻▼ ❘❷❻ ●❏■ ■ ❘✓❛ ▼ ❙ ▲ ❘ ❥ ◆✯❚❩▲ ▼❇▲✗❑❍▲ ❘ ❜✏❼❖● ❲ ●❖◆☛▲❆●❏❜✺✐ ✈❉❪ ❉ ✷❋ ❛ ❘ ➵ ④❧✩✬✫✣❆✹✕✯✬✲✰❖❬✲➃✷❚✾✜✫✣❏❃❛✣✵✽❆s✧✪❀✬✲✼✿❁❀➂✲✼❊◆✻✛❄✣✵❅❆❋④❧✵✶✿❈✷❚❊❜➦✬✫✣✻✶❆❈✵✥✲✼❊✞✿❍❆❈✵✽✻✽✷❈✵✶✯✪❛❅▼③✧❱❆❈✲➃✷❈✲✼✾✣❖ ➵ ✾✮✾✣✵✜➉ ❊ ✼✲✻❉❄✣✾✪❖✽❆❈✵✽✵⑥✷❍❊❜❀✮✲■✵✽✾➂✵③✾✬▼✜❊✕✷❇✻❅✵✽✻✶❏◗✻✶❖✽❆❋✵❅✵③❊❫❊✥④❧✩✜✩✬✩✬❆❈✭■✯✣④ ✭✼✩✶✵✽➵ ✧✪❆✍✾✮✧➂✵✶✾✣✯✪✵➂✲■✭✳✭✮❛❅✻✶✻❉▼❱✾✣❖●✯✱❙✣✻✽✩✜✲✼❙✮✾✙❏↕✩✜④❧❏①➉ ✩✜❊✞❆✹❘ ✻✶❏➁❖✽✵⑥✻✸✲■④❧✾➂✩✬▼✜✫✣✵✽❆❍❁✻✶✯✜❆❋❊✽✲✼❖❬❘❫✲✳✷◆❩✽✲✼❄▲❏✱✵✽✲■❆❈✭■④❧✻✶✵❅❆❋✿❈✭➃✷✞▼✓❑❅❊✕➦✜✲■✾ ✫✣✻❅❆❈✵✣▼✜❘❢✵❅➒❢✻❅✵❬❆❋❊●✷✹❑✽✵✽✷❍❆✹❏✱❀✮✵●✲✼✲✼✾✣❆❚✵✛✻✶✷❍❖✽❀✮✵❅✵❊ 31 ✚✂ ✓ ✕☛ ✑ ❘ ✍☛ ✍✏✎✠✑✘✮✠★✥✰ ✎✍❭✑✘★✑✏✓✒❷★✩❃❅✎✚✤s✎✺✮☛✶ ✘✙ ☎✔✗✖✱✔✹✒✕✔✘★✩✶✛✚✠✎✚✤❖✎✚✙✼✔✘✎✥✶ ✜☎ ☛✔✹✒✏✰✌★✩✙✼✔✻✶✕✔ ✖❆✑❇✑❆✖✗✢✥❃ ✝✄✂ ✕❍ ✚✎✚✙✖✝✞✢ ✑✗✏★☛✙✙✘❏✖✱★ ✽◗✎✠✑❇✑✱★✯❀☎✖✗✢✕✔✘★ ✜▼✜❀ ✵❅✻●✻❅②✜❆❋✵ ❊●❑ ❋❅✵●✯✬✷ ❏✱✲✰➵ ❖❬✫✣✲➃✾✮✷❈❊❍❊✽✾✣✷❢❘❢✵✜❛✮➉➌⑩❊✛✵✪✵✙✻✶✻ ❖✽✴❉✵✙✾✣❇ ❛✣✩✶✯✜✵✧➧✲✼❖❬✲✳✷❍✷❇❀✜✻✶✾✜✻●✾✣✷✫✣✯❂❏◗❙✮❛✮✩✜✵❅❏⑩❆❽❘❢➉ ❊✛❙✓✻✶❀✬❖✽✫✣✵✙❊●❑✜❘✸❛✣✷❍✵ ✦❧❀✮④ ✵✪❘✫❊✕❋❅✵✶✵●✿✹✷✩✜✾✣✯ ❑✮❀✜✷❍✻❬❀✮❇ ②✜✵✽✯✜✵✾ ✲✼❖❬✲✳✷❇✯✬✾✜✲✰✫✣❖❬❏◗✲➃✷❈❘➂❊✥❛✮➨✣✵❅✻✶✫✜❆◆✾✣✷❇✧✪✯❂✲✼✩✬✾✭✰✵❬✫✮✷ ✭✰✯ ✷❍❀✜❀✜✻●✻✶②✜✾ ✵➂✻✽❋❅✷◆❑✶✵●✧✪✷✮✭✼✵❅✷❍❀✬✻❅❀✮✲✼❊❍✵❇✿❁✷ ❀✱❊✕➦✜❏◗✫✣✯✬✵✽✻❅✲✰❖❬✻✶❆❈✲➃✵❅✾✮✷❈❊✍❊❬❊ ❑✜❛✣✻❅✵ ✾✱✲■❏❃✻✶❄✣✾✣✩✜✯ ❊✕❊❍✲✰❛✬❘ ✲■✲■✾⑥✭✳✲✳✷❻✷❍▼✍❀✜❘✈✻✽✷✓➨✜✩✜▼✱❆❈✯✣❊❋▼✮✵✽❏❂❆❜❑✕❏◗✧✪✵❬❀✮✷✹✵❅❆❻❆❈▼★✵ ❑❅✧✪✵❱✿✹✻❅✾✮✾✣✩❉✷❇❘✍❀✜❩✽✻❬✲■②✜✾✣✵ ✿❍✵❢❛✮✭✼✩❉✵❅✷❍❊❁❀ ❊ ❊❜❩✽➦✬✲✼✾✣✫✣✿✹✻✶✵ ❆❈✵✽❊✞❀✜✻❬②✜✵ ✯✜✲✼❖❬✲✳✷❋❑❇❊●❑❽✧✪✧✪✵①✵⑥❖✽❀✜✵●✻●✷ ②✜✵ ❑❉✩✜❆ ❘ ✶ ✻ ✣ ✾ ✯ ✠ ❘ ✓ ❙ ✬ ❀ ✼ ✲ ✺ ❊ ✬ ▼ ✰ ✲ ● ✵ ✰ ✭ ✜ ✯ ❊ ✻✶✾✣✯ ❑s✧✪❀✬✲✰✿❁❀↕✲✼❏❂❄✮✭✳✲✰✵✽❊③✷❍❀✜✻●✷ ➵ ✾✮✾✣✵➂✲✼❊ ✻✶✾✣✯✺❙✣✩✜❘✇❏➁➥❱✲✼❊ ✵✽✾✣✿❍✵✬❘ ❑ a n a m+n = t a m0 ✦ ❘❱❯✟◗✿❢❣◗◆❛▲❪❜❢❣✓ ◗ S ❳❖❝❞❭✪❪❜❝❱❳⑤❘❣❳ ❧❜◗ ✦ ❴❅◗✚➈ ❡ ❘❣❳➒❧❜◗ ✦ ❪❜❢ ❈✱◗✳❢❣❪ ❡ ✐☎✐❱❪❜❢❣❨❫❳❖❴❅➈ ❡ ❝ ∠A ❳■❝✔❪ ❙ ❘❣➊❅❝❹◗ ✦ ❡ ✐☎➊❜❘☎◗ ✦ ❪❜❢ ❢✘❳❩➈✳❯✜❘ ☛ ✂☎✄ ✆✞✝✟✆✞✝✠✄☛✡✠☞✍✌✏✎✒✑✔✗✓✕ ✖✙✘✛✚✜✖✣✢✤✂✦✥✧✕✗★✞✂✪✩✪✂✬✫✮✭✗✰✛✯ ✱✳✲✣✴✛✵✶✡✍☞✞✷✸✆✹✵✶✺✶✑✻✝✠✼✽✕✠✾✍✷✸✾❀✿✹☞✠✡✍✝✠✱❁✕✠✾✍✷✸✾❀✿✹☞✠✡✍✝✠✱❃❂✧✝✍✆❀✴✛✵ ✾✟✾✠✴✣✌ ❄ ☞✞✷❆❅❇☞❇❈✗✝✗✡✞✵ ✴❉✾✞❊✟✥☎❋❍●■❂✛✂✦✥✜❂✛✂❏✥✙✂☎✱✁✲❑✴✗✵☛✡✠☞✞✷✸✆✹✵☛✺ ✑▲✝✠✼❃❂✧☞❇✵ ✷❇▼✛✺ ✱✤❂✧☞❇✵ ✷❇▼✛✺ ✱◆★✛☞✍✎✒✾✠✴✣✝✍✴✒❊❖✻✭✸✰ ❄ ✖❉★◗❂✛✂❏✢❀✂✦✭❆★✛★✗✖✒✱ ✥✮✝❉▼✛☞✞✴✗✱❘✘✍✷✸✾✍✴❚❙✹☞❯❊❘✥✜✭❱✰ ❄ ✂✦✥✜❲■✭✹❳ ❄ ✖✛✕✍✕✠✱❏✥✣✾✍✴❚❙✸❨❚✝✁❩✛✾❬✄☛✝✍✆❪❭❚☞✞✷❀✌❉☞✞✆✹✱✬✰✍✂☎✱❫✲✧✕❇✂✜❊❑❴✍❋❍✖ ❄ ❋✦❵✬✂❏✥✜❲❍✱✦❩❚✝✍✷❯✺ ✾❬✄☛☞✞✆✹✱ ✩❛❖❜✱✜✲✧✕❇✂✜❊✧❵✦✂✦★✗✢ ❄ ✖✙✥❫❴✹✂✪✩❝❋❛✲❑✕✠✱✮✲❑✷❱✆❬▼❬✄ ✵ ✴✣☞✞✴✗❈✒✑✠❞❪✴❉✾✠✆✹✵❆▼✠❞❡✱✣✭ ✴❉✴✛✆✞✎✒✷❇▼✛❙❆❢✒✱✒✂✦▼✍✆✞✺ ✷✸✵ ✾✞❊❑❭✧✖❉❲✽✲✜★❇✂❜✩✽❳✒❖❣✲✜✥✪✢✒●❉✱ ❲✬✝✗✡✍☞✞✷✸✱✧❩✠✂☎✱✪✲❑✕✞✂✮❊✮❲❛❳ ❴✍❳❉✕✒❖✻✖✙✖✛✩❫✚✣✱✙❤❬✾❬✄☛✺❆✎❉✝✍❞❪❞✐☞❇✄ ✱✧✺✶❨❚☞❘✩❫☞✍✺✶❨❚☞✞✷✸✄ ✾✠✴✣✌✒✆❱❊✮✫✮✖❚❋❍✥☎✫✮✖✐✢✍✕❇✂✦❩✠✂❏✚✜✭❆❲✬✭☛✕❬✱❉✂✜❈✍✷✸✵ ✴✛✵✶✝✠✱ ✫✣✷❀☞✍☞✍❙✹☞❯❊✽❩❑✖❑✢✒✖✙✥❥●✮❳✬❵❪❋✪❋✮✱❘❂✙✵☛✝✠✄ ✾▲✲✣✴✛✵✶✡✍☞✞✷✸✆✹✵✶✺✶✑❬✱❫★✠✾❜❖❜✵ ✷✸✾✛✌✒✾❬✱✽✰✍✂✜✱❪✲✧✕❇✂✜❊❘★✗✭✦❤ ❄ ❋❛✲❚✱❛❩❚✝✠✄ ❢✻✰❚✝✍❞❪❞◆▼✠✴✛✵✶✺✶✑ ✰❚✝✠✄ ✄✶☞❇❈✛☞❇✱☎❵❛✵ ✴✣✺☛☞✞✷ ❄ ✾✗✡✍☞✞✴✗✱✮✘✛★✠✱❏✲❑✕✞✂✮❊❍✢✒✭❀✢❚✲✐❤✗❭❚❋❍✩✪✂❏✥✦✲✣✱✜✰❚✝✍❞ ✾✠✯ ✴✣☞✛✆✞❦ ✺ ✵ ✱✮✥✮✝✍❞❪✾✠✴✛✵ ✾❬❊✣✾✠✴✣✌❪✺✶❨❚☞❏❧❚✷❀✝❬❧✧✝✍✆✞☞✞✷✞❳ ➔❉➓ ♣ ♥✫♥ ❫ ❘ ✝ ❅➓❉➓✶➔ q ❆✮➔ ❑ ❆ ✟ ✡ ✟✵✤s✎☎✛♦✎✄☛✩★✥✰ ✄✂ ✦✠✒✠✢✥✙✣✪✡✟⑥✎✄☛ ❍❖✎ ✒♦✎✚❃❅★☛✤❖✎ ✝ ✠✷✢ ✤ ✡◗✒✺★ ✘❖✶ ➆⑥✙✺✖✱✮✠★✩✤✻☛❖✖✱✰✺✢✺✰ ✰✌★ ✠✏✢☎✑❇✑✻✢✺✰✌✎✠✑❆✖✘✰✚✶ ✠✏✢☎✑❇✑✻✢✺✰✌✎✠✑❆✖✘✰✚✶⑥✍✩✛✣✢☎✖✗✙ ❘ ❅❋ ✵●✷ m ❛✮✵⑥✷❍❀✮✵✸✭✼✵❅✾✣❖❽✷❍❀✺✩✬④✜✷❍❀✮✵✸❏◗✵❅✯✜✲■✻✶✾✪✷❈✩✙❊❍✲✰✯✮✵ b ✲■✾ ABC ❑❉✻❅✾✣✯✺✯✣✵●➍▲✾✣✵ m ✻❉❁❊ ✲■✿✹❏❂✿❍✩✜✲✳✭✼❆❈✻❅✯✬❆❈✲✼✭✾✣▼✍❖③❘✱✻✶➺✣❊❢❆❈✻✶✩❅✾✣②✜❖❬✵✪✭✼✵✷❍❀✜✻●✷ ✲■❊❢✻❉✿✹✫✜✷❈✵✜❑✮✩✮❛❉✷❈✫✣❊❜✵✬❑✮✩✜❆✞❆❈✲✼❖✽❀❉✷❍❁✻❅✲✼✾✣❊✸❖❬❄✣✭✰✵❅✩✜✯✍❊❁✲➃❘ ✷❈✲➃②✜✵✬❑★✾✣✵❅❖●✻✽✷❈✲ ②✜✵✜❑s✩✜❆❇➻❁✵✽❆❈✩ ✍✏✎✠✑✓✒✕✔✗✖✘✎✚✙■ ✄✂❤✦☛✎✕★❍✑✼✍✕❍❀✘✺✑✱✎✄☛✧ ✕★☛✤✓❀✠✶✛✟◗✢✄✂✾☛❖✖✱✰✌★✯✶✌☎ ✓✕✶✝➆⑥✍✥✍ ❘ ✻❅✾✣✯ ➅✥❊❁✲■✾✣❖➠✷❍❀✮✵ ✧✪✵●✭■✭✼✕✴✶✾✣✩✶✧❱✾➳④❧✩✜❆✹❏✱✫✮✭✼✻❅❊ ❑✶✧✪✵❱❀✜✻●②✜✵ b A c 4a4 + 9b2 c2 − 16m2b m2c mb √ mc = 12 2a2 + 2b2 − c2 = √ 2a2 − b2 + 2c2 4a4 + 9b2 c2 − 16m2b m2c = 4a4 + 9b2 c2 − (2a2 − b2 + 2c2 )(2a2 + 2b2 − c2 ) = 4a4 + 9b2 c2 − 4a4 − 2a2 (2b2 − c2 ) + (2c2 − b2 ) + (b2 − 2c2 )(2b2 − c2 ) = −2a (b + c ) + 9b2 c2 + (2b4 + 2c4 − 5b2 c2 ) = −2a2 (b2 + c2 ) + 2(b2 + c2 )2 = 2(b2 + c2 )(b2 + c2 − a2 ) 2 ❘ ❊❁❯✓✩✮✲✼❛✬✭✼❖●✵❅✷❈✾❳✻❅✫✣❆❈❊✕✻❅✭ ✵✜▼★❊ ❑✮❑ ✩✜❆✞❆❈✲✼❖✽❀❉✷❜❑✜❆❈✵❅❊❁✲■❄▲❊❇❑✜✵❅❄▲✧✪✿❈✩✜✷❈❀✬❊❍✲➃②✜✲✰✲✳✿❁✷❈✵●❀❂✲ ✭➃②✜▼s✵✜✲✼❘❊⑥❑❉❄▲❊❜✩❱✩✜❊❍✷❍✲✳❀✜✷❈✻✽✲ ②✜✷ ✵✜❑✣✾✣✵✶❖●✻●✷❈✲➃②✜✵✬❑✓✩✜❆✈➻❁✵❅❆❈✩❂✲✳④ ✜❀ ✻✶❊✞✷❍■✲ ❀✮❊✛✵③✻✶❊✕✿✹✻❅✫✜❏◗✷✹✵✬✵ ❑ 2(b2 + c2 ) b + c − a2 4a4 + 9b2 c2 − 16m2b m2c ∠BAC ✂ ☎✄ ✝✆ ✝✆ ⑧✏■ ❚❩●✴❚❩●❏■ ④❏❘ ❞ ⑦ r ❋ ❣ ✂ ❣ ❼❤⑧ ⑩ ❋ ❬ ⑧ ✈ ⑧ ❳ ❥ ❻ ❜ ☎◆❍▼ ④❖❘❩❲ ❚❯▼❴▲❴r②● ❉❋ ❙❖❲✗❙ ❯❘❏④ ●❏❜❨❋ ❙❖❲✗❙ ❯❘❏④ ●❏❜ ❫ ●❖❚✘◆❍▼ ❙✴❙ ◆☎❞ ❦ ❩❘ ❲ ❺ ❘✓❛ ● ④ ▼ ◆ ❙❩✉✴⑩ ✐ ❫ ⑧ ⑩✺❫ ⑧ ⑩ ⑧✏❜✞ ✠◆✬▼ ④❏❘❩❲ ❚❯▼❇▲ r ● ❫✥❘ ▼ ❲ ❑❍▲ ❜ ❫✥❘ ▼ ❲ ❑❍▲ ❜ ❬❍❘❖⑦s❙ ◆☎●❖◆ ✉ ✇②❥ ❻ ❦ ❣✯❬❄❫ ⑧✵❼✘⑧◗❥ ❬❍❬✬❣ ❜ ⑩ ●✯❑ ❘ ◆✬❜ ❲✗❙ ◆ ❸❯❘✱✉ ♥✣❥✻✐ ✈⑥✈♦❣②❫ ⑧◗❥ ❬❍❣ ✯❜ ❣✯❬ ❋✬❥ ❣②❻ ⑧✌✇②① ❫✥❣✩❬✬❬ ❜ ❙ ◆☎❞ ❻ ❦✌⑧ ⑩✺❬✬❣ ❋ ⑩✕❪ ♥✣❥❇✇②❥ ✈⑥✈ ❥ ❣ ❜✌⑧✏◆ ❛✬❘ ■❇● ❋❍▲ ❙ ▲ ❘ ✠◆✬▼ ④❏❘❩❲ ❚❯▼❇▲ r❏❜♦❋ ❙ ◆ ⑧✏◆ ❛✬❘ ■❇●❏❜✭❼✘③✏❜ ✥❋✓⑧ ✉❷❻ ❦♦❥✗① ❻ ⑩ ❼✬❥❴❋❏❜⑥✇ ▼ ❚✘❚❩●✯❑ ❲ ▼✣❋❍●✯❑✬▲❴q ❘❩❲ ◆②❋✬▲ ❙ ▲ ❘ ☎◆❍▼ ④❖❘❩❲ ❚❯▼❴▲❴r✹❜ ❏●✹❱✩■ ▼ ◆✬❜❄✇ ✐✥❜ ✠❋❩⑧ ✉✴⑩ ❥ ❻ ❦✵⑧ ⑩ ♥ ❥ ❪ ❦ ❣ ❋s❋✹❜ ⑩☎❙ ◆ ❸ q☛●✞① ❙ ■❴●❖❚ ☎ ❘❩❲ ❞ ❘ ❚❯❜ ❻ ⑧✺❜ ✥❋✓⑧ ✉ ❏✐ ❣ ❦❷✐✣❶◗⑧ ⑩ ♥✕❜ ①☛● ❲ ▲ ❙ ■ ❘ ❚❯❜ ✈ ✇ ❜✄ ✥❋✓⑧ ✉ ❶✣⑧ ❬ ❼s❦ ❣✯⑩ ❯⑧ ✈ ✐ ✠❋❏❜ ❲ ❚✹❑✹■ ▼ ◆ ❘ ◆ ❛ r ❧ ◆ ❙ ❚❯▼❆❑ ❧ ❜✥❥ ◆✯◆❍❚ ⑦s❲ ❑ ❸❆❹ ❜✬⑧✣❑❏❚❩▲ ❲ ▼ ❙✹✉✥❳✚❣ ✐ ⑩✺❣ ⑧ ❪ ✂✬⑧ ✈ ♥✚⑧ ❬❍❬ ❜✕❦ ❙❖❧ ❞ ❘ ◆✬❜ ❻ ❼✘❜✆ ✠❋❩⑧ ✉ ❼✘⑧◗❥ ❻ ❦❷❥✕✇❤⑧ ❣ ✂s⑧⑥❶◗⑧✏❜◗❼ ❙❏❹s❙ ▲ ❚✹❑ ❹ ▼ ❻ ▼❴▲❴r✹❜✵✐✚❚ ❙❖❹❖❙ ❜ ❙ ❱ ❙ ◆ ✉ ☎ ❣ ♥ ❬ ⑧ ✈❉❪ ✇ ⑩ ❼ ✯❜◗♥✣● ④❏❘❩❲ ❜✣①❏⑧✏❜ ✠❋❩⑧ ✉ ♥ ❪ ❪ ❋✬✇ ❣✯❣✯✈ ✂☛❜ ❙ ■❇▲ ⑦ ● ❧♠❧✿❘ ■ ❜✣▲❴q ❘❨✈♦❘ ▲❴q ❘❩❲ ■ ❙ ◆☎❞s❚ ✉ ① ❣ ❼ ❣✩⑩ ❪ ❶❭✐✵✐✥❜ ❜ ✠❋❩⑧ ✉✏❬ ❥ s❦❷✐ ☛❜✏①☛●❏■ ❹❝❻ ● ❧♠❧ ❑❏◆❍▼❴▲❴r ❻ ●❏■ ■ ❘✓❛✬❘ ❜◗❶✼▼ ◆☛▲ ❘❩❲ ❦ ❙❍④❏❘ ◆❍❜ ❬ ❜ ❫ ▼❇●❏■ ❙ ✠◆✬▼ ④❏❘❩❲ ❚❯▼❇▲ r❏❜ ❬❖❙ ✇ ▼ ❲✗❙ ❞ ❙ ❜ ❻ ⑧✏✁ ✥❋✓⑧ ✉ ❼✬❥❯❼ ☎☛✐ ✈ ⑧ ⑩ ☎❜ ❻ ● ❧ ❙ ◆ ❘ ❚❩▲ ▼ ❜ ⑩ ● ❧♠❙ ◆✬▼ ❙❩✉✠❙ ◆☎❞♠▲ q ❘ ❱ ❲ ●✹❱✥●❖❚ ❘❩❲❩❪ ✟✞ ✂ ❉ ✞ ❉ ✱✞ ❥✠ ☎✄ ❉ ❹☛ ❉ ✷❉ ✚✠ ❜✠ ✂☞✂ ✞ ✞ ✂ ✌✎✍✥✍✥✍➇✓✥✕✘✗✚✙✚✙✜✛r✢✿✤✟✛❜✦✜✤✁ ✳✫✣✭✪✯❱✰✳✲✴✰✚✵◆✶✿✷➄✸✜✺ ①❩❑❵❉◆❿ ❊▲✺ ♦✶◆❃ ✍✌☞✶✬❆ ✧✯✘❉✿✵✬✰❜❾✬❆❵➂✴✰✜❁❍❉✚❃❫①❩❉✟✓ ✿◗◆❘ a ✦ ✦ b ✦ ❡ ❴❅❨ c ❙ ✦ ◗❞❘❱❯✟◗✔♠❩◗✳❴❅➈☞❘❱❯❜❝❥❪❫❛❅❘❱❯✟◗✑❝❱❳❩✦ ❨✟◗✿❝✎❪❜❭✟❭✪❪❜❝❱❳⑤✦ ❘❣◗✥✦❘❱❯✟◗❞❧❜✦ ◗✳❢✘❘❣✓ ❳❖✐❱◗✳❝ A ✦ ✦ ❴❅❨ ❢❣◗✳❝❦❭❅◗✚✐❣❘❣❳ ❧❜◗✳♠ ♥ ❳■❴r❘☎❢✘❳ ❴❅➈✬♠❩◗ ❛▲❪❜❢ ▲❛ ❳■❝ B ❡ ❘❱❯✟◗✑❳❖❴✟❢ ❡ ❨❜C❳⑤➊❅❝❚❪❫❛✪❘☎❢✘❳ ❡ ❴❅➈✬♠❩◗ A B C ❡ ❡ ❴❅❨★A❳⑤❛ RB ❳■C❝s❳⑤❘✘❝❚✐☎❳■❢❣✐☎k➊❅❲❍=❢ ❡1❨❜❳⑤2➊❅❝ ✦ ❭❅.❢❣❪✳❧❜n◗✔❘❱❯ ❡ r❘ ☎ ✁ k k k ☎ k k k n n rk k=1 k ✟ k √ ✟✂ k k n k ak + bk k=1 n n Rk k=1 n n k=1 √ ≤ 3 k k n ≤ k ✓ n n + ck k=1 ➔ ■ ✄✂✆✝ ✫❍❀✘ ✛✟ ✖ ◗★✯✑ ◗✢✏✔✻✢☛✖❆✑❇✑✱★✯✶✁✒♦✎✵✒✏★✩✙✕✶ ✍✏✎✠✑✓✒✕✔✗✖✘✎✚✙ ❏❍ ❘ ✤❩✢✥✙ ❖★ ❙★❀✮✵❱❆❋✲✰❖✽❀❉✷✞✲■✾✣✵✶➦✬✫✣✻❅✭✳✲✳✷❻▼❂❏❂✻●▼❂❛✣✵✛✧❱❆❈✲➃✷❋✷✹✵✽✾❃✵✶➦✬✫✮✲➃②✬✻✽✭✰✵✽✾✬✷❈✭➃▼❂✻❅❊ ❘ ❪ ❵ ➅❊❁✥✲✼✯✣❊❁✲■✵❱✾✣❖❇✩✬④s✷❍❀✮❪ ✵✍❵❚❋✽✲■✻●❊✥✧➇✵✶➦✬✩✬✫✣④✣✻❅❩✽✭✣✲✼✾✣✷❈✩ ✵✽❊✞✻❅✾✣✯➂✷❍❀✮✵❅✾➂✷❍❀✮✵ ➵ ❲ ❇ ➢❢❲✄✦❻✾✣✵❅➦✜✫✣✻✽✭■✲➃✷❭▼✓❑✽✧✪✵✥➍✣✾✣✯➂✷❍❀✜✻✽✷s✷❍❀✮✵✥✭✰✵●④■✷ n n ak k=1 n n sin Ak n n Rk √ ≤ 3 n n + sin Ck k=1 k=1 n ≤ n n sin Ak + k=1 n n sin Bk + k=1 n = ck k=1 sin Bk k=1 n + Rk n + n bk + Rk k=1 n (sin Ak + sin Bk + sin Ck ) ❘ n n sin Ck k=1 ✧✪✿❍✩✜❀✬✶✩ ✾❉✲✰✧➛②✜✿❁❀✸✵✽✧✪✾✜✲■✲✰❊★✵◗✵✽②✬✾✣✫✣✻✽✿❍❊❜✭■✵✺✲✼✵❃✯✸✩✬✷❍④❧④✍❀✮✩✜✻✶❆★✵❃✾❉✻✶✧✪▼◗✾➂✵●❆❈✻❅✭■✵✽❆❍✭✼✻❉✕❛✬✴✶✯✮✲✳✷✹✾✣✵❅❆❋✩✶❆❋✻❅❊❢✧❱❆❻▼✸✧✪✾✉✷✹❀✮❆❋✲■✩❃✲✼✾✣✻❅✵✶✻✶✾✣➦✬❆❈❖❬✫✣✵✪✭✰✵✞✻❅✾✣✭✳✧➂✲✳✩❉✷❻✷❢✲➃▼ ✷❍④✰❀③✻❅✻❅❏❂✾✣✲✳❖❬✭■✲■✭✰✻✶✵✽❆◆❊ ✧➂✲➃❑ ✷❍❀❂✷❍❑✽❀✬✻❅✲■✾✣❊✛✯ ❆❈✵✽❊❁❘✿✫✮✭➃✝ ✷❜✎✕❑✜✰◗✧✪❊ ✂✵✪➑✜❊❁✩✜✫✣❆▲❄✮✷❍❄✮❀✮✭ ▼✵ ❑ ✻✇❄✮❆❈✩✮✩✬④ q❂❊❍✲✼✾✣✿✹✵◗✷❍❀✮✵⑩❊❍✲✼✾✣✵❳④✼✫✣✾✣✿❈✷❈✲✼✩✜✾❨✲✼❊✺✿❍✩✜✾✣✿❈✻●②✜✵⑩✩✜✾✉✷❍❀✮✵❳✲✼✾✬✷❈✵❅❆❻②✬✻✽✭ ❑❫✧✪✵⑩❀✜✻❬②✜❘ ✵✡ ❙★❀✬✫✣❊❬❑ ❘ ✆ ✩✶✧➇✿❍✩✜✾✮❊❍✲✰✯✮✵❅❆★✷❍❀✮✵✥✭✰✵●④■✷✍✲✼✾✣✵❅➦✜✫✣✻✽✭■✲➃✷❭▼s❘❫➅✥❊❍✲✼✾✣❖❇✷❍❀✮✵ ➵ ❲ ❇ ➢❇❲✄✦❻✾✣✵❅➦✜✫✣✻✽✭■✲➃✷❭▼✓❑✽✧✪✵✛❖✽✵❬✷ ❘ ❪ ❵ ✷❍➌⑩❀✮✵✙✵↕✧➂❊✕✵❅✲■✭✳❏✱✭❚✲✼❏❃❄✣✵❅✻✶❆❋✴✶✲✼✵✱❏❃✫✣✵●❊❜✷❈✵❅✵❃❆❽✩✬❘❨④◆❙✓❊❜❀✮✩✜✵❅❏❃✾①✵❂❪❻➥❱✷❍❏❃❀✮✵✽✵↕❆❈✩✜✩✜❆❈✻❅✾★✵❂❆❈➉ ✵❅❊✸❊❍✻⑩✷❋➑✬✻✶✩✬✩✜✾✣④✞❆✹✯✜❏✱✻✶✷❍❀✮❆❈✫✮✯⑩✵◗✭✼✻❬❆❈✷✹❵✕✵❅❆❋❘❚✲✼❊❍✻❅❙✓✫✮✾✣❀✮✭✳✷❋❖❬✵❅❊➂✭✰❆❈✵◗✵●✻✶④❧❛✣✲✼✩✜❊✙✩✬❆❈✫✜❖❬✵✜✲➃✷✥❑ ②✜✷✹✵✽❆❈✾❴✲■✻✶❛❅✾✣▼❴❖❬✭✼✵❅❛✣❊✽✩❉t❘ ✷❍❀ ❋❅✵❬✷ ✻❅✾✣❛✣✯✵ k=1 ✆ √ sin A + sin B + sin C ≤ 3/2 A B C sin A + sin B + sin C ≤ sin n k=1 ak Rk n n + k=1 n n ak bk Rk n + [0, π] √ + B + C) = sin(π/3) = 3/2 n + k=1 n n k=1 (A bk ck Rk + ≤ n n n k=1 n ck k=1 √ 3 n· √ = 3 3n n ≥ ak b k c k k=1 s rs s(s − a)(s − b)(s − c) r = (s − a)(s − b)(s − c) s ❘ ✻❅❪❭➥➂✭■❊❜✵❅✩✺❆❈✵❂❀✜✻●✧✪②✜✵✵◗❀✜✻●②✜✵❃✫✣❊❜✵❅✯✇✷❍❀✮✻✶✵ ✾✣➵✯ ❲ ❇ ❇➢ ❲ ✦❻❪❻✾✣➐✬✵❅✫✮➦✜✭✼✫✣✵❅❆✕✻✽➉ ✭■❊✛✲➃✷❭✦❻▼①✾✣✵❅✻❉➦✜❖●✻✽✫✣✲✼✻✽✾✍✭■✲➃❘ ✷❭❵➣▼✶❵✕❙✓❘◆❀✬❯s✫✣✩✜❊ ❏❃❛✜✲■✾✜✲✼✾✣❖✸✻❅✭✳✭✜✷❍⑩❘ ❀✮✵❅➌⑩❊✕✵✵ ≤ s (s − a) + (s − b) + (s − c) abc = 4Rrs R ≥ 2r = s s 3 s √ 3 √ s ≥ 3r = ✬➔ ❆❈✵❅❊❍✫✮✭✳✷❋❊●❑✶✧✪✵❱✩✮❛✬✷❋✻❅✲■✾ √ √ abc ≥ 4(2r)r(3 3r) = (2 3r)3 n n ak n n + bk k=1 n n + n ≥ ck k=1 ❑✶✧✪❀✬✲✰✿❁❀✪✧✪✵③✫✣❊❜✵③✲■✾❃❪ ❵ q k=1 (2 3rk ) k=1 ❉ ✂ n n √ = ⑧✏■ ❚❩●✴❚❩●❏■ ④❏❘ ❞ ⑦ r ❦ ❘❩❲ ❺ ❘✓❛ ● ④ ▼ ◆ ❙❩✉✕❻ ❦♦❥✗① ❻ ❯⑧ ✈ ✐ ✠❋❏❜ ❲ ❚✹❑✹■ ▼ ◆ ❘ ◆ ❛ r ✇ ▼ ❲✗❙ ❞ ❙ ❜ ❻ ⑧✺❜ ✥❋✓⑧ ✉✿❬ ❥ ❱ ❲ ●✹❱✥●❖❚ ❘❩❲❩❪ 3n √ rk k=1 ☎✄ ✝❉ ✆ ✞ ❘ ✝✆ ✂ ❋ ❣ ✂ ❣ ❼❤⑧ ⑩ ❋ ❬ ⑧ ✈ ⑧ ❳ ❥ ❻ ❜ ☎◆❍▼ ④❖❘❩❲ ❚❯▼❴▲❴r②● ❉❋ ❙❖❲✗❙ ❯❘❏④ ●❏❜❨❋ ❙❖❲✗❙ ❯❘❏④ ●❏❜ ❫ ●❖❚✘◆❍▼ ❙✴❙ ◆☎❞ ⑩ s❼ ❥❇❋✹❜✚✇ ▼ ❚✘❚❩●✯❑ ❲ ▼✠❋✬●✯❑❍▲ q ❘❩❲ ◆❨❋✬▲ ❙ ▲ ❘ ✠◆✬▼ ④❏❘❩❲ ❚❯▼❇▲ r❏❜ ✹●✹❱✩■ ▼ ◆❍❜✚✇ ✐✥❜ ✠❋❩⑧ ✉ ❶◗⑧ ❬ ❼✬❦ ❣✩⑩ ❧ ◆ ❙ ❚❯▼❆❑ ❧ ❜❢❥ ◆✯◆❍❚ ⑦s❲ ❑ ❸❆❹ ❜✭⑧◗❑❖❚❩▲ ❲ ▼ ❙❩✉ ① ❣ ❼ ❣✩⑩ ❪ ❶❭✐✵✐✥❜ ❫ ▼❴●❏■ ❙ ☎◆❍▼ ④❖❘❩❲ ❚❯▼❴▲❴r✹❜ ❬❖❙ s❦❉✆ ✐ ☎❜✿①☛●❏■ ❹❅❻ ● ❧♠❧ ❑❖◆✬▼❇▲ r ❻ ●❏■ ■ ❘✓❛❍❘ ❜✴❶⑥▼ ◆☎▲ ❘❩❲ ❦ ❙✬④❖❘ ◆✬❜ ❬ ❜ ✥❋✓⑧ ✉♠❙ ◆☎❞ ▲❴q ❘ ✠ ✌✎✍✥✍➅✒ ✓ ✕✘✗✚✙✚✙✜✛ ✢⑨✤✟✛❜✦❼✤ ✿✫ ✭✪✯❱✰✳✲✴✰✚✵◆✶✿✷✁✸✚✺ ✰✜❑❅❉✿❁❍❁❂✶✚✷❂⑩s❉✳✵❦✵❱①■❊❖❉✿❆✥❀❜❋▲✯❣❉✳✵✳✸✟✰✪⑥✿✯ ☎✳❆ ✜✯✘❉✚❃ ❱✶✜✓ ✿◗◆❘ M✓ ABC ❙ ◗ ❡ ❘❣◗◆❘☎❢ ❡ ❯✟◗✚❨❜❢❣❪❜❴ ✦ ❡ ❴❅❨r♠❩◗✬❘ M ❙ ◗ ✓ ❡ ❴✜♥❍❭✪❪❫❳■❴❫❘s❳■❴r❘❱❯✟◗③❳❖❴❫❘❣◗✿❢✘❳❩❪❜❢ ❪❫❛ ABC ✎◗✳❴❅❪✜❘☎◗✥❘❱❯✟◗ ❡ ❢❣◗ ❡ ❪❫❛ XY Z ❙ ♥ [XY Z] ❢❣❪✳❧❜◗✔❘❱❯ ❡ ❘ ✝ ✁ ✟✂ ✡✠ ✁ (M M )2 = M A2 [BM C] [ABC] AB − + M B2 [CM A] [ABC] [BM C][CM A] [ABC]2 + CA2 + M C2 + BC [AM B] [ABC] [CM A][AM B] [ABC]2 ✓ [AM B][BM C] [ABC]2 ✉✧✰✜❁❍❁❂✶◆❃⑨❋ ❯❫❳■❝✡❢❣◗✿❝❱➊✟♠⑤❘✈❛▲❪❜❢ ❡ ❘❣◗◆❘☎❢ ❡ ❯✟◗✚❨❜❢❣❪❜❴③❳■❝ ❹❝✱❳■❲r❳⑤♠ ❡ ❢ ✡❘☎❪ ♣ ❘☎◗☞q ❡ ❢②❘ ❝ ✟❯ ◗✿❪❜❢❣◗✿❲ ❛▲❪❜❢ ❡ ❘☎❢✘❳ ❡ ❴❅➈✬♠❩◗ ✓ ▲❛ M = G ✦ ❘❱❯✟◗➅✐❱◗✳❴❫❘☎❢❣❪❫❳❩❨r❪❫❛ ABC ✦ ❘❱❯✟◗✳❴✇❘❱❯✟◗③❢❣◗◆♠ ❡ ❘❣❳❩❪❜❴ ❙ ✚◗ ✐☎❪❜❲➄◗✳❝ ✦ 1 M G = (M A + M B + M C ) − (AB + BC + CA ) ✓ q➅❯❫❳❩✐✱❯✇❳❖❝❼q➅◗◆♠■♠ ➉ ❴❅❪✚q③❴ ✵✱✵◆✶✬❃✈❋❖①❖❉✳❊⑤❊▲✺✔❋▲❑✧✶❼✵❦❉✚❁❂✶❵✵◆✰✿❊②⑥✟❋❖①✝✰✜❃✇✸✚✺ ① ❣❑✧✶☞❊ ✧❉✟❋❖❉✳①⑤❊■❊✝✶✬❆✧➂✴✰✪⑥❜✶✬❃✟❆ ✚✯❣❉✿❃ ☎✶☞❻❜❉✚❃✧✷ ◆①✿❇✳❑⑨✰✪⑥✳❆ ✭✪✰✿❊❩❄ ✉✧✰✜❁❍❁❽⑥✿❃❜①●❋▲✺❍✉✧✰✿❊⑤❊●✶ ☎❜✶☞❆ ✔①❩❃⑨❋●✶✬✯❚❷✔❉❜❾✿✶✬❃✟❆ ✳❆❥❺✴❀☞⑩✥✓ ♣ ❳❖❴❅✐☎◗ M ❳■❝✎❳■❴★❘❱❯✟◗✑❳■❴❫❘☎◗✳❢❣❳❖❪❜❢s❪❫❛ ABC ✦ q➅◗③❯ ❡ ❧❜◗ M = αA + βB + γC ❛▲❪❜❢❥❝❹❪❜❲❍◗✇❭❅❪❜❝✱❳➒❘❣❳➒❧❜◗➇❢❣◗ ❡ ♠✈❴❜➊❅❲ ❙ ◗✿❢✘❝ α ✦ β ✦ γ ❝❱➊✪✐✱❯❍❘❱❯ ❡ ❘ α + β + γ = ✓❞➌ ✐❣❘❣➊ ❡ ♠■♠ ♥ ❢❣◗✥❘❱❯✟◗ ❡ ❢❣◗ ❡ ♠✧✐☎❪✟❪❜❢❣❨❜❳■❴ ❡ ❘❣◗✿❝❥❪❫❛ M ❢❣◗◆♠ ❡ ❘❣❳➒❧❜◗✥❘❣❪ (A, B, C) ❡ ❴❅❨ ❡ ❢❣◗③➈✬❳ ❧❜◗✿❴ (α, β, γ) ❡ ♥❙ ➋✝→☞➏ [BM C] ✦ [CM A] ✦ [AM B] ✓ α = β = γ = ✄✂ ☛ ✆☎ ✞✝ ✞✟ ☛ ✟ 2 ✠ ☛ [ABC] ✁ ✄✂ ☎ ✟✝ ✂ ✝ ✌ [ABC] [ABC] ✌ ➔❀ ❙★❀✮✵✽✾ −−−→ −−→ −−→ −−→ M M = αM A + β M B + γ M C (M M )2 ❩✽✲✼✾✣✿✹✵ ❑✬✻✶✾✣✯✙✧✪✵❱❀✜✻●②✜✵ −−−→ −−−→ = MM · MM = α M A2 + β M B + γ M C −−→ −−→ −−→ −−→ −−→ −−→ + 2αβ M A · M B + 2βγ M B · M C + 2γαM C · M A = α M A2 + β M B + γ M C + αβ(M A2 + M B − AB ) + βγ(M B + M C − BC ) + γα(M C + M A2 − CA2 ) = (α + β + γ)(αM A2 + βM B + γM C ) − αβAB − βγBC − γαCA2 α+β+γ =1 ❘ ❑✶✧✪✵✪✿✹✩✜✾✣✿✹✭■✫▲✯✮✵③✷❍❀✜✻✽✷ (M M )2 = αM A2 + βM B + γM C − (αβAB + βγBC + γαCA2 ) ✻✶✾✣✯✺✷❍❀✮✵➂❄✣❆❈✩✮✩✬④✓✲■❊✥✿❍✩✜❏❂❄✮✭✼✵●✷❈✵✪❛❅▼❃❪ ❵❫✻✶❛✣✩✽②✜✵✣❘ ❉ ⑧✏■ ❚❩●♦❚❩●❏■ ④❏❘ ❞ ⑦ r✭❶◗⑧ ❬ ❼✬❦ ✩❣ ⑩ ✻⑧ ✈ ✐✆ ✥❋✹❜ ❋ ❻ ❦ ❬ ✐✺❋ ❫✥❣✯⑩✏❳ ❜ ❫☛❙ r❏❯❚ ▼❇❞ ❘ ❜ ✈ ✯❜ ✥✓❋ ⑧ ❪ ✞ ❑ ❉ ❲ ❚✹❑✹■ ▼ ◆ ❘ ◆ ❛ r ❧ ◆ ❙ ❚❯▼❆❑ ❧ ❜✚❥ ◆✯◆❍❚ ⑦s❲ ❑ ❸❆❹ ❜✯⑧◗❑❖❚❩▲ ❲ ▼ ❙❩✉☛❙ ◆☛❞ ❖✐ ❣✩❬ ✌✎✍✥✍➅✌✔✓❚✕✘✗✚✙✚✙✜✛➅✢❹✤✪✩✜✦◆✤✁ ➎ →❫✩✿→❜✦✜→❫✩☞✤❫✫ ✭✪✯❱✰✳✲✴✰✚✵◆✶✿✷✇✸✚✺ ✾✿⑥✚❉✿❃ ❅ ♦✰✚✵ ☎✰➇➂✴✰✜❁❂✶◆✯❱✰ ❉✿❿ ✯✯❇✧⑥❫✶ ✌❱❆ ❺♦❃❫①●❾✿✶✬✯✘✵❱①●✷❫❉❫✷➄✷❅✶ ☎❜❉✳❊⑤❊❩❉❫✷❅✰✿❊■①✝✷✜❆ ☎❜❉✳❊⑤❊❩❉❫✷❅✰✿❊■①✝✷✜❆♦❀✬✲⑨❉✳①❖❃♦✓ ◗ ❘☎❢✘❳ ❴❅➈✬♠❩◗ ❴❅❨✇♠❩◗✬❘ ◗✥❘❱❯✟◗✔♠❩◗✳❴❅➈☞❘❱❯r❪❫❛❅❘❱❯✟◗✑❝✱❳❖✦ ❨❅◗✑❪❜❭❅❭❅❪❜❝✱❳➒❘☎◗✥❘❱❯✟◗ ✿◗◆❘ ❧❜◗✿❢②❘☎◗✁ A ✓ ABC ▲❛ m ❙ ❳■❝❚❡ ❘❱❯✟◗➇❡ ♠❖◗✿✦ ❴❅➈☞❘❱❯❂❡ ❪❫❛✈❘❱❯✟◗✇a❲❍❙ ◗✚❨❫❳ ❡ ❴ ❛❩❢❣❪❜❲ A ❘❣❪ BC ❴❅❨ ❳⑤❛ ❳■❝❚❘❱❯✟◗ ✐☎❳❖❢❣✐☎➊❅❲r❢ ❡ ❨❫❳■➊❅❝➅❪❫❛ ABC ❭✟✦ ❢❣❪✿❧❜◗★❘❱✦ ❯ ❡ ❘ m − R ❳■❝✑❭✪✓ ❪❜❝❱❳⑤❘❣❳ ❧❜◗ ✦ ❴❅◗✿➈ ❡ ❡ ❘❣❳ ❧❜◗ ✦ R❪❜✝❢ ❈✱◗✳❢❣❪ ✦ ❡ ✐☎✐❱❪❜❢❣❨❫❳❖❴❅➈ ❡ ❝ ∠A ❳❖❝❥❪ ❙ ❘❣➊❅❝❹◗ ❡ ✐☎➊❜❘☎◗ ❪❜❢❼❢✘❳❩➈✳❯✜❘❱❬ ❡ ❴❅➈✬♠❖◗✚❨ ✉✧✰✜❁❈✸✿①❩❃✟❉✟❋❖①✝✰✜❃➅✰✚④✳✵✱①❖❁★①■❊❖❉✚✯❅✵◆✰✿❊②⑥✟❋❖①✝✰✜❃✟✵❼✸✚✺✥➂✴✰✜✺ ✧❉✚✯❹✸✜❉✿✯❣❉✳❆✡❺✈❃❜①✝❾✿✶◆✯②✵✱①●❋▲✺❞✰✚④ ♦✶☞①❖✯☞⑥✟❋❩❆ ♦✶☞①❖✯☞⑥✟❋❩❆ ❫✶✜✸✜❉✿❃⑨✰✜❃❫❻✈❉✿❃⑨✷➄➂⑨① ❣❑❅❉✿✯✱✷ ✝ ✽✈❷➅✶✬✵✱✵❦❆✡➂⑨❉✚❃ ❣❑⑨✰❍✭❜❉◆❊●✰✚✵ ☎✪✶✬✯✱✷❅✶☞✵❦❆❵✉✚⑩ ❆❥❺✴❀☞⑩✥✓ ➆ ◗❂♠❩◗✬❘ M ❙ ◗ ❘❱❯✟◗❈❲r❳❖❨❅❬❱❭✪❪❫❳■❴❫❘✇❪❫❛ ✓✎BC ❴❅❨✁✐☎❪❜❴✟❝✱❳❖❨❅◗✳❢✑❘❱❯✟◗❂❘☎❢✘❳ ❴❅➈✬♠❩◗ A q✑❳⑤❘❱❯❍❝✱❳❖❨❅◗✳❝ AM = m ❡ ❴❅❨ AO = R ➌ ✐❱✐☎❪❜❡❢❣❨❜❳■❴❅➈✑❘☎❪✄✂✜➊✪✐☎♠■❳❖❨ ✦ ❘❱❯✟✓◗➇❢❣❡ ◗✳♠ ❡ ❘❣❳ ❧❜OM ◗➇❝✱❳ ❈✱◗✿❝ ❪❫❛✪❘❱❯✟◗✿❝❦◗✔❘②q➅❪✇❝✱❳❖❨❅◗✳❝❥❨❅◗✳❭✪◗✳❴❅❨✟❝❚❪❜❴✇❘❱❯✟◗✑❝✱❳ ❈✱◗③❪❫❛✪❘❱❯✟◗③❪❜❭❅❭❅❪❜❝✱❳➒❘☎◗ ❡ ❴❅➈✬♠❩◗✳❝ ☎✝✆✟✞✡✠☞☛ ✓ ❳❖❝❥❪ ❙ ❘❣➊❅➋ ❝❹◗ ✓ A ✌❵◗✿❢②❘☎✁◗ ❪❜❴ ❘❱❯✟◗ ✐☎❳■❢❣✐☎➊❅❲➄✐☎❳■❢❣✐☎♠❩◗ ➏ ❳■❝ A A ❝❹◗✳❭ ❡ ❢ ❡ ❘❣◗✚❨✇❛❩✓ ❢❣❪❜♣ ❲✣ ❘❱❯✟◗③✐☎❳■❢❣✐☎➊❅❲➄✐☎◗✿❴❫❘☎❢❣◗ ✦ O ❙ ♥➇❘❱❯✟◗ ✐✱❯✟❪❜❢❣❨ ❳❖❴❅✐☎◗ m R C ❳❖❝✔❪ ❙ ❘❣➊❅BC ❝❹◗❫➎✟q➅❯✟◗✿❴❅✐☎◗ ✦ OM ❘❱❯✟◗★❪❜❭✟⊥❭✪❪❜BC ❝❱❳⑤❘❣◗✇❝✱❳❖∠OM ❨❅◗ R A❳■❝ ✦ B M ♠❩❪❜❴❅➈✳◗✳❢❼❘❱❯ ❡ ❴r❘❱❯✟◗ ❡ ❨✜❤ ❡ ✐❱◗✳❴❫❘✎❝✱❳❖❨❅◗ ✦ m ➎❫❘❱❯ ❡ ❘s❳■❝ ✓ q➅❯✟◗✿❴ A ❳■❝✡❪ ❙ ❘❣➊❅❝❹◗ ❡ ❝✡✐☎♠ ❡ ❳■❲➄◗✿❨ m −R < O ✓ ✓ ☎✝✆✟✞✡✠✎✍ A = 90 ✏③◗✳❢❣◗ ❳❖❝ ❨❜❳ ❡ ❲❍◗◆❘❣◗✿❢❦➎✚❘❱❯❫➊❅❝ ✦ m = R ✦ ❡ ❴❅❨ m − R = ✓ BC ❡ ☎ ✂ ✁ ✟ a a ☞☎ ✌ ✂ ☎ ☎ ✟✂ a a a a ◦ a a ➔✜❆ ✴❧✆ ✣✚❝✁ ❘ A ✲✼❊❢✻✶✿✹✫✜✷✹✵✮❘ ✹✿ ✻❅✾❷✲✼❊✺❛✣✻❉❙★✵ ✿✹❀✮✫✜❄▲✵⑩✷❈✩✜✵✜❄✮❊❍❑❇✲✳❆❈✷❈✻✶✩✜✲ ❊✙②✜❄▲✵✜✩✜④❧❑✸✩✬❊❁✻❅✭■➻❁✭✼✭✈✵✽✩✶❆❈✲■✧❱❊✺✩✓❊❅❑✪✲■❘✒✾✣✩✜✿❍❆◗❋❅✩✜✵❬❆❈✾✣❆❈✷ ✵✶✵❅✿❈❖●✷ ✻✽q ✷❈✲ ②✜✵❨❛✣✧✪✵❳❀✮✷❍✵✽❀✮✾✵ ✷❍❄▲❀✮✵✽✵✇❆✹❄✣✻❅✭✰✾✣✵❅✩✜✾✣✯ ✾✣✯✬❖⑩✲✰✿✹✻✶✫✮❆❈❑★✭■✿❂✻✶✧✪❆✸✩✬✵◗④✥❛✬❀✜✲✼✷❍❊✕✻●❀✮✵✶②✜✵✉✿❈✵ ✷❈✩✜✿✹✲✼❆✛❆❈✿✹✩✬✫✣④ ❏❃✿✹✲✼❆❈✿✹✭✼❘✱✵✉➑✜❛✣✩✜✵❬❆ ✷➀✧✪✵✶✩✜✵✽✾✾ r ✧✪❀✮❑✍✵❅✾✣✧✪✿✹✵❳✵ ❖✽✵❬✷ ❘◗➑✜✩✜❘❴❆ ➑❉✲■✾✮✻●✻❅✷ ✭✳✭➃▼✓❑✍✧✪✩✜❆✛❀✮✵✽✻✽✾ ✷ ✧✪❄✣❆❈✵✱✩✜✭■❄✣✲✼❊❜✵❅✵❅✩✜❊❳✵✙❊✕✻✽✩✜✷❍✭➣❀✜✾❿✻✽❛✣✵✽✷ ✵❅✲➃✷❍✿❍❀✮✩✜✵❅❏❃❆✱✵❅❊❜❊➁❀✮✩✜✿✹❆➀✩✜✷❂❆✹❆❈✻❅✵❅❆❈✿❈✿❘ ✷➧✆ ④❧✩✜✩❉❆❷✷✹✵✙✷✹✩✜✷❍❆❋❀✜❆✲✼✻❅✻●✾✣✷✥❖❬✷❍✭✰❀✮✵✽✵❊ ❑ ✧✪❀✬✲✰✿❁❀✓❑✬✲✼✾⑥✻❅❊◆✧✪✧✪❀✬✲✰✵➂✿❁❀❀✜✻●✢☛②✜✑❆✑ ✵⑥✻✶➎✰✾✣✫✣❖❬❊✹✭✼✷✞✵❅❊❜❊❚✵❅✻✶✵❅❆❈✾★✵❢❑✶✻❉④❧✿✹✩✜✫✜❆❈✷❈✿❍✩✵✵✽❊r❽✷❍❀✮✵❅✾ ✧➂✲■✭✳✭✶✾✣✵❅❑✬✿❍✻✶✵✽❊❇❊✕❊❁✿✹✻✶✭✼✻✽❆❋✲✼✲■❏❃✭ ▼✸✵✶✭■✲✼✯✙✵✥✲■✩✜✾✙✾✸✷❍✷❍❀✮❀✮✵③✵✥❄✣✻❅❆❈❆❈✩✜✿ ❄✣✩✜❊✕✻✽✭❋❘ ma − R A BC OM B A C ∠M OA > ∠OM A ma − R > A B ma − R = BB CC ma − R < C A A a R m O B C B C M A BC ma − R > ⑧✏■ ❚❩●✭❚❩●❏■ ④❏❘ ❞ ⑦ r❝✇②❥ ❻ ❦ ❣✯❬❤❫ ⑧✵❼✘⑧◗❥ ❬❍❬✬❣ ❜ ⑩ ●✯❑ ❘ ◆❍❜ ❲✗❙ ◆ ❸✻❘✱✉♦❻ ❦♦❥✗① ❻ ⑩ ❼s❥❇❋✹❜✵✇ ▼ ❚✘❚❩●✯❑ ❲ ▼✕❋✬●✯❑❍▲ q ❘❩❲ ◆ ❋❍▲ ❙ ▲ ❘ ☎◆❍▼ ④❖❘❩❲ ❚❯▼❴▲❴r✹❜ ❏●✹❱✩■ ▼ ◆✬❜ ✇ ✐✚❜ ✥❋✓⑧ ✉ ❶✣⑧ ❬ ❼s❦ ❣✯⑩ ❯⑧ ✈ ✐ ✠❋❏❜ ❲ ❚✹❑✹■ ▼ ◆ ❘ ◆ ❛ r ❧ ◆ ❙ ❚❯▼❆❑ ❧ ❜❢❥❴◆❍◆✯❚ ⑦s❲ ❑ ❸✗❹ ❜ ⑧✣❑❏❚❩▲ ❲ ▼ ❙✹✉✌❳✚❣ ✐ ⑩✺❣ ♠⑧ ❪ ✂s⑧ ✈ ♥✥⑧ ❬✬❬ ❜✏❦ ❙❖❧ ❞ ❘ ◆✬❜ ❻ ❼✱✁ ❜ ✠❋❩⑧ ✁ ✉ ☎ ❣ ♥ ❬ ⑧ ✈❉❪ ✇ ⑩ ❼ ✯❜✏♥✣● ④❖❘❩❲ ❜✏①❏⑧✺❜✁ ✥❋✓⑧ ✉ ① ❣ ❼ ❣✩⑩ ❪ ❶♠✐✌✐✚❜ ❫ ▼❴●❏■ ❙ ✠◆✬▼ ④❏❘❩❲ ❚❯▼❇▲ r❏❜ ❬❏❙ ✇ ▼ ❲✗❙ ❞ ❙ ❜ ❻ ⑧✺❜ ✥❋✓⑧ ✉✵❬ ❥ s❦❷✐ ☛❜✵①☛●❏■ ❹ ❻ ● ❧♠❧ ❑❖◆✬▼❇▲ r ❻ ●❏■ ■ ❘✓❛❍❘ ❜ ❶⑥▼ ◆☎▲ ❘❩❲ ❦ ❙✬④❖❘ ◆✬❜ ❬ ❜ ✠❋❩⑧ ✉ ❼s❥✘❼ ☎☎✐ ✈ ⑧ ⑩ ☛❜ ❻ ● ❧ ❙ ◆ ❘ ❚❩▲ ▼ ❜ ⑩ ● ❧♠❙ ◆✬▼ ❙ ☛✌▼❇▲ q❭▲ ☛✼●✵❱ ❲ ●❍● ❚ ❆● ❲ ▲❴q ❘ ● ⑦ ▲✻❑❏❚ ❘ ❵ ( ❙ ◆ ❛ ■ ❘♦❸❆❙ ❚ ❘ )✉☛❙ ◆☛❞♠▲❴q ❘✂ ❱ ❲ ●✹❱✥●❖❚ ❘❩❲❩❪ ❼✹q ❘◗❙ ❚❩▲ ❘❩❲ ▼ ❚ ❹ ❞ ❘ ❚❯▼ ❛ ◆ ❙ ▲ ❘ ❚✕❚❩●❏■❆❑✬▲ ▼❴●❖◆✯❚✣▲ q ❙ ▲ ☛ ❘❩❲✘❘⑥❸ ● ❲✻❲✘❘❖❸ ▲ ❜ ⑦ ❑❍▲ ☛✵q☛●❖❚ ❘ ◆❙ ❙ ■ r❖❚❯▼ ❚♦● ✏▲ q ❘⑥❙✬❸ ❑✬▲ ❘ ❵ ❙ ◆ ❛ ■ ❘❉❸❇❙ ❚ ❘ ☛ ❙ ❚✵▼ ◆ ❸ ● ❧ ❱✩■ ❘ ▲ ❘✬❪ ❥❴◆ ❙ ❞✯❞❖▼❇▲ ▼❇●❖◆ ☎ ⑧ ❻☛❬ ✄ ⑧ ☎ ✂✚✐ ✈♦❣ ❻✩✈ ✯ ❜ ❫ ▼ ❛♠⑩☎❙ ❱✩▼❴❞s❚❯❜ ✇②❥ ❜✄ ✥❋✓⑧❤❱ ❲ ● ④ ▼❇❞ ❘ ❞ ❙❷❸ ●✯❑❖◆☛▲ ❘❩❲✘❘✗⑨❖❙❏❧ ❱✩■ ❘ ❚✘q☛● ☛✌▼ ◆ ❛ ▲❴q ❙ ▲✥▲ q ❘✼❸ ●❖◆ ❸ ■❇❑❖❚❯▼❴●❖◆✭▲❇●❉▲ q ❘✼❸ ● ❲✻❲✘❘❖❸ ▲ ❘ ❞✣❱ ❲ ●✹❱✥●❖❚ ❙ ■✯❚❩▲ ▼ ■ ■ ☛ ❙ ❚☎✄ ❙ ☛ ❘ ❞ ❪ ❼✹q ❘❩❲✘❘ ☛ ❘❩❲✘❘ ▲❴q ❲✘❘❖❘ ▼ ◆ ❸ ● ❲✻❲✘❘❖❸ ▲☎❚✹❑ ⑦s❧ ▼ ❚✘❚❯▼❇●❖◆❍❚ ❪ ABC ✞ ☞✂ ✂ ✂ ☎✄ ❉✞ ❉✂ ❼✠ ✆✞✝✠✟☛✡ ☞☛ ☞ ✣✥✤✧✦✩★✫✪ ✱✞ ❜✠ ☎✄ ✚✄ ✞ ✌✎✍✑✏✓✒✕✔✖✌✎✍✘✗✚✙✑✛✓✝✜✟✢✔ ✬✭✦✮★✰✯✲✱✳✬✩✦✴✤✶✵✩✬✠✷✸✪ ✬✠✹✎★✰✯✲✱ ✺✼✻✾✽ ✿❁❀✾❂☎✺✼❃❅❄✧❂❆✽ ✿❈❇❊❉☎❋❍●☎❄✧■ ✻✼❏▲❑❆✿▼❄◆❇❊❂❁❖❄✧❃❅❄✧❂❆✽ ✿❈❇❊❉❘◗❚❙❯❂❈❇❊❑▼❄❲❱❊❳ ●☎❳✾❨❬❩✼❏◆❭✁❪✾❄✧❂ ❫❵❴❅❛✘❜ ✪ ✬✭✦✮★✰✯✲✱✳✬✩✦✴✤✶✵✭❝ ❴❅❛ ✱ ❞✼❀▲❇❯❡❊✻✾✽ ❡❊❢❣✺❯✻✾✽ ✿▼❀✾❂❆❉☎❋❍●☎❄✧■ ✻✼❏▲❑❆✿▼❄◆❇❊❂❆❉❤❖❁✐❥❀✾❡❯✻✼❏❘✿❁❄❘❇❯❂❈❉❘◗❦❱❊❄✧■ ❀✾❧❊❀▲♠ ✻❍❨✧❏❘❇✼♥♦❄q■ ♣r❞✾❂❈❄❬✻❯❄❘❂❈✽ ❑s✉t✈❳ ❙❦❳❊✇✮❏✧❉❤s▲❄◆♠ ♠ ✺❯✻✾✽ ✿▼❀✾❂❆❉①❄✧❃❅❄✧❂❆✽ ✿❈✽②❋❲●①❄❬■ ✻✼❏❬❑❈✿❁❄❘❇❯❂❈❉❤❖❄❘❃✂❄❘❂❈✽ ✿❁✽③◗④t✈❳ ⑤✩❳✼❨✧❏✧❡❊✻✼❉❘⑥❊●☎❳ ✺④❳▲⑤⑦❀✼❀✼✻✼❂❈❀❬❭❍⑥❊❙❊❂❆❇②❑❁❄❍❱❯❳ ●☎❳✼❨❬❩✼❏◆❭✁❪✾❄✧❂ ✪ ✬✭✦✮★✰✯✲✱✳✬✩✦✴✤✶✵✩✬✠✷⑧✪ ✬✜✹⑨★✰✯✲✱ ❞✼❀▲❇❯❡❊✻✾✽ ❡❊❢❣✺❯✻✾✽ ✿▼❀✾❂❆❉☎❋❍●①❄❬■ ✻✼❏❬❑❈✿❁❄❘❇❯❂❈❉❤❖❁✐ ❀✾❡❊✻✼❏◆✿▼❄◆❇❊❂❆❉❘◗❦⑩✾❏❘✿❁❂❈✽ ❑sq❨❘❇❊❂❈❂③❪❅♣✠●②❏◆♥✾✽✼❶♦❏▲s✧✽ ♠ ✺❯✻✾✽ ✿▼❀✾❂❆❉①❄✧❃❅❄✧❂❆✽ ✿❈✽②❋❲●①❄❬■ ✻✼❏❬❑❈✿❁❄❘❇❯❂❈❉❤❖❄❘❃✂❄❘❂❈✽ ✿❁✽③◗❚⑩❊❩▲✽ ♠ ✽ ❧✉❷❀✾❡❯❢✾⑥✧❷❄❘❸✴❹✁✽ ❢❘❩✼❏✧❃⑦⑥ ❷✧❳ ⑩✶❳✼t❦❂❈❀✾❉❉❃❅❏✧❡④⑥✧❺❚❡❊✻✼❂❈❄q❻✶❩✾❏❬❡❯❢✾⑥❯❼✈❏❬❀✼s❬✽②❨✧❏◆✿▼❀②⑥✾❻✼❪✼❂❆❇❊❉✚❹❽❉▼✽ ❏