✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆ ❚❘❺◆ ❚❘×❮◆● ❙■◆❍ ❇❻❚ P❍×❒◆● ❚❘➐◆❍ ❉■❖P❍❆◆❚❊ ❚❯❨➌◆ ❚➑◆❍ ❈❤✉②➯♥ ♥❣➔♥❤✿ P❍×❒◆● P❍⑩P ❚❖⑩◆ ❙❒ ❈❻P ▼➣ số ò ì ì ❉❼◆ ❑❍❖❆ ❍➴❈ ●❙✳❚❙❑❍ ◆●❯❨➍◆ ❱❿◆ ▼❾❯ ❍⑨ ◆❐■ ✲ ✷✵✶✺ ▼ö❝ ❧ö❝ ▼ð ✤➛✉ ✷ ✶ ▼ët sè ❦✐➳♥ tự ìợ số ợ t ❚❤✉➟t t♦→♥ ❊✉❝❧✐❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷ ▲✐➯♥ ♣❤➙♥ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✸ P❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✸✳✶ ❚➻♠ ♥❣❤✐➺♠ r✐➯♥❣ ❞ü❛ ✈➔♦ ❣✐↔♥ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✸✳✷ ❚➻♠ ♥❣❤✐➺♠ r✐➯♥❣ ❞ü❛ ✈➔♦ t❤✉➟t t♦→♥ ❊✉❝❧✐❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ◆❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✷✹ ✶✳✹ ✷ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✷✻ ✷✳✶ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✷ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✧❜à ❝❤➦♥✧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✸ ◆❣❤✐➺♠ ♥❣✉②➯♥ ❞÷ì♥❣ ❝õ❛ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✸✹ ✷✳✸✳✶ ▼ët sè ✈➼ ❞ö ❧✐➯♥ q✉❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✸✳✷ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ ❞↕♥❣ ❧✐➯♥ ♣❤➙♥ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸ ▼ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✸✳✶ ✹✸ ◆❣❤✐➺♠ ♥❣✉②➯♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❧÷đ♥❣ ❣✐→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷ ✸✳✸ ✹✸ P❤÷ì♥❣ tr➻♥❤✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤✱ ❤➺ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❧÷đ♥❣ ❣✐→❝ ❝â ✤✐➲✉ ❦✐➺♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ❳→❝ ✤à♥❤ ♣❤➙♥ t❤ù❝ ❝❤➼♥❤ q✉② tọ trữợ ✳ ✳ ✺✻ ❑➳t ❧✉➟♥ ✻✹ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✻✺ ✶ ▼ð ✤➛✉ P❤÷ì♥❣ tr➻♥❤ ♥❣❤✐➺♠ ♥❣✉②➯♥ ❤❛② ❝á♥ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❞↕♥❣ t♦→♥ ❧➙✉ ✤í✐ ♥❤➜t ❝õ❛ ❚♦→♥ ❤å❝✳ ❚❤ỉ♥❣ q✉❛ ✈✐➺❝ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡✱ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ✤➣ t➻♠ r❛ ✤÷đ❝ ♥❤ú♥❣ t➼♥❤ ❝❤➜t s➙✉ s➢❝ ❝õ❛ sè ♥❣✉②➯♥✱ sè ❤ú✉ t➾✱ sè ✤↕✐ sè✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ ✤➣ ✤÷❛ ✤➳♥ sü r❛ ✤í✐ ❝õ❛ ❧✐➯♥ ♣❤➙♥ sè✱ ỵ tt ữớ t ỵ tt t t❤➦♥❣ ❞÷ ❜➻♥❤ ♣❤÷ì♥❣✱ sè ❤å❝ ♠♦❞✉❧❛r✱✳ ✳ ✳ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ t❤ü❝ ❝❤➜t ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦✲ ♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ❝â ❝❤ù❛ t❤❛♠ sè✳ ❈â t❤➸ ♥â✐ ✤➙② ❧➔ ♠ët ❞↕♥❣ t♦→♥ ❦❤→ ♠ỵ✐ ♠➫ ✈➔ ❝❤÷❛ ♣❤ê ❜✐➳♥ tr♦♥❣ ❝→❝ ❦ý t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐ ❜➟❝ ♣❤ê t❤æ♥❣✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ t→❝ ❣✐↔ ❦❤æ♥❣ ❝â t❤❛♠ ✈å♥❣ ❜❛♦ q✉→t ❤➳t ❝→❝ ✈➜♥ ✤➲ ✈➲ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ♠➔ ❝❤õ ②➳✉ ✤✐ s ự t ữỡ tr ợ ❜✐➳♥✱ ❜❛ ❜✐➳♥ ❤♦➦❝ ❜è♥ ❜✐➳♥✳ ❍✐ ✈å♥❣ ✤➙② s➩ ❧➔ ♠ët t➔✐ ❧✐➺✉ ❜ê ➼❝❤ ❝❤♦ ❝→❝ t❤➛② ❝æ ❣✐→♦ ✈➔ ❝→❝ ❡♠ ❤å❝ s✐♥❤ tr♦♥❣ q✉→ tr➻♥❤ æ♥ ❧✉②➺♥ t❤✐ ❤å❝ s✐♥❤ ❣✐ä✐✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ✸ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❈❤÷ì♥❣ ✷✳ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t❡ t✉②➳♥ t➼♥❤ ❈❤÷ì♥❣ ✸✳ ▼ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥✳ ◆❤➙♥ ✤➙②✱ t→❝ ❣✐↔ ①✐♥ ❜➔② tä sü ❦➼♥❤ trå♥❣ ✈➔ ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ ●❙✳❚❙❑❍ ◆❣✉②➵♥ ❱➠♥ ▼➟✉✳ ❚❤➛② ✤➣ ❞➔♥❤ tớ ữợ ụ ữ t❤➢❝ ♠➢❝ ❝õ❛ ❤å❝ trá tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♥➔②✳ ❚→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ♥❤➜t tỵ✐ ❇❛♥ ❣✐→♠ ❤✐➺✉✱ P❤á♥❣ ✣➔♦ t↕♦ ❙❛✉ ✤↕✐ ❤å❝✱ ❑❤♦❛ ❚♦→♥ ✲ ❈ì ✲ ❚✐♥ ❤å❝✱ ❝→❝ t❤➛② ❝æ ❣✐→♦ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ✤➸ t→❝ ❣✐↔ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ♥❤✐➺♠ ✈ư ❝õ❛ ♠➻♥❤✳ ✷ ❚→❝ ❣✐↔ ①✐♥ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣ ❧✉ỉ♥ q✉❛♥ t➙♠✱ ✤ë♥❣ ✈✐➯♥✱ ❝ê ✈ơ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ♠➔ t→❝ ❣✐↔ ❤å❝ t➟♣ t↕✐ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ❚ü ♥❤✐➯♥ ✲ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✳ ▼➦❝ ❞ò ✤➣ ❝â ♥❤✐➲✉ ❝è ❣➢♥❣ ♥❤÷♥❣ ❞♦ t❤í✐ ❣✐❛♥ ✈➔ tr➻♥❤ ✤ë ❝á♥ ♥❤✐➲✉ ❤↕♥ ❝❤➳ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤â tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❱➻ ✈➟② t→❝ ❣✐↔ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü ❣â♣ þ ❝õ❛ ❝→❝ t❤➛② ❣✐→♦✱ ❝ỉ ❣✐→♦ ❝ơ♥❣ ♥❤÷ ❝→❝ ỗ ữủ t ❤ì♥✳ ❚→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❍➔ ◆ë✐✱ t❤→♥❣ ✵✾ ♥➠♠ ✷✵✶✺ ❍å❝ ✈✐➯♥ t❤ü❝ ❤✐➺♥ ❚r➛♥ ❚r÷í♥❣ ❙✐♥❤ ✸ ❈❤÷ì♥❣ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ìợ số ợ t t t ♥❣❤➽❛ ✶✳✶ ✭①❡♠ ❬✶❪✮✳ ❙è ♥❣✉②➯♥ ❝ ✤÷đ❝ ❣å✐ ❧➔ ởt ữợ số số ổ ỗ tớ ổ t ❛ ✈➔ ❝ ❝❤✐❛ ❤➳t ❜✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷ ✭①❡♠ ởt ữợ số số ổ ỗ tớ ổ ữủ ữợ số ợ t ữợ số ữợ ú ỵ ữợ số ợ t t d ụ ữợ số ợ t t q ữợ r ữợ số ❝❤✉♥❣ ❧ỵ♥ ♥❤➜t ❝õ❛ ❛ ✈➔ ❜ ❧➔ sè ♥❣✉②➯♥ ữỡ ìợ số ợ t số ữủ ỵ rtst ❝♦♠♠♦♥ ❞✐✈✐s♦r✮✳ ◆❤÷ ✈➟② ❞ ❂ ✭❛✱❜✮ ❤❛② ❞ ❂ ❣❝❞✭❛✱❜✮✳ ❱➼ ❞ö ✶✳✶✳ ✭✷✺✱✸✵✮ ❂ ✺✱ ✭✷✺✱✲✼✷✮ ❂ ✶✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✸ ✭①❡♠ ❬✶❪✮✳ ▼ët sè ♥❣✉②➯♥ ❝ ✤÷đ❝ ởt ữợ số số a1 , a2 , a3 , , an ổ ỗ tớ ổ ữợ ❝õ❛ ♠é✐ sè ✤â✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✹ ✭①❡♠ ❬✶❪✮✳ ▼ët ÷ỵ❝ sè ❝❤✉♥❣ ❞ ❝õ❛ ♥ sè ♥❣✉②➯♥ a1, a2, a3, , an ổ ỗ tớ ổ ữủ ữợ số ợ t a1 , a2 , a3 , , an ữợ số a1 , a2 , a3 , , an ✤➲✉ ữợ ữỡ tỹ t ụ q ữợ r ữợ số ợ t số a1 , a2 , a3 , , an số ữỡ ìợ số ợ t ❝õ❛ a1 , a2 , a3 , , an ỵ a1 , a2 , a3 , , an ✮ ❤❛② ❣❝❞✭a1 , a2 , a3 , , an ✮✳ ◆❤÷ ✈➟② ❞ ❂ (a1 , a2 , a3 , , an ) ❤❛② ❞ ❂ ❣❝❞✭a1 , a2 , a3 , , an sỹ tỗ t ữợ sè ❝❤✉♥❣ ❧ỵ♥ ♥❤➜t ❝õ❛ ♥❤✐➲✉ sè✱ ①❡♠ ❬✶❪✮ ❈❤♦ ❝→❝ sè ♥❣✉②➯♥ a1 , a2 , a3 , , an ổ ỗ tớ ổ õ tỗ t ữợ số ợ t a1 , a2 , a3 , , an ✳ ❚➼♥❤ ❝❤➜t ✶✳✶ ✭①❡♠ ❬✶❪✮✳ ❈❤♦ ❛✱ ❜✱ q✱ r ❧➔ ❝→❝ sè ♥❣✉②➯♥ ✭a2 + b2 = 0✮✳ ◆➳✉ a = bq + r ✈➔ ≤ r < |b| t❤➻ ✭❛✱❜✮ ❂ ✭❜✱r✮✳ ❚❤✉➟t t♦→♥ ❊✉❝❧✐❞ ✭t❤✉➟t t t ữợ số ợ t số ♥❣✉②➯♥ ❞÷ì♥❣ ✮✳ ●✐↔ sû r0 = a, r1 = b ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ❚❛ →♣ ❞ư♥❣ ❧✐➯♥ t✐➳♣ t❤✉➟t t♦→♥ ❝❤✐❛ ri = ri+1 qi+1 + ri+2 , tr♦♥❣ ✤â ≤ ri+2 < ri+1 , ∀i = 0, 1, 2, ✈➔ ♥❤➟♥ ✤÷đ❝ ❝→❝ ♣❤➛♥ ❞÷ r1 , r2 , ✈ỵ✐ r1 > r2 > ✤➳♥ ❦❤✐ ❧➛♥ ✤➛✉ t✐➯♥ ♥❤➟♥ ✤÷đ❝ ♣❤➛♥ ❞÷ rn = ✭n ≥ 2, < ri+2 < ri+1 , ∀i = 0, 1, , n − 3✮✳ ❑❤✐ ✤â (a, b) = (r0 , r1 ) = (r1 , r2 ) = = (rn−2 , rn−1 ) = (rn−1 qn−1 , rn−1 ) = rn−1 ❱➟② (a, b) = rn−1 ❱➼ ❞ö ũ tt t t ữợ số ợ ♥❤➜t ❝õ❛ ✸✹✽✹ ✈➔ ✸✷✼✻✳ ▲í✐ ❣✐↔✐✳ ❚❛ ❝â 3484 = 3276.1 + 208 3276 = 208.15 + 156 208 = 156.1 + 52 156 = 52.3 + ❱➟② gcd(3484, 3276) = 52 ❱➼ ❞ö ✶✳✸✳ ❚➻♠ ♠ët ❝➦♣ sè ♥❣✉②➯♥ ①✱ ② ✤➸ 3484x + 3276y = 52 ▲í✐ ❣✐↔✐✳ ❚❤❡♦ ✈➼ ✈ư tr➯♥ t❛ ❝â ✺ 52 = 208 − 156.1 ⇒ 52 = 208 − (3276 − 208.15) = 16.208 − 3276 156 = 3276 − 208.15 52 = −3276 + 16.208 ⇒ 52 = −3276 + 16 (3484 − 3276.1) = 16.3484 − 17.3276 208 = 3484 − 3276.1 ❉♦ ✤â 3484.16 + 3276.(−17) = 52 ❱➟② (x; y) = (16; −17) ✶✳✷ ▲✐➯♥ ♣❤➙♥ sè ✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ ✭▲✐➯♥ ♣❤➙♥ sè ❤ú✉ ❤↕♥✱ ①❡♠ ❬✸❪✮ ▲✐➯♥ ♣❤➙♥ sè ❤ú✉ ❤↕♥ ❝â ✤ë ❞➔✐ ♥ ✭n ∈ N✮ ❧➔ ❜✐➸✉ t❤ù❝ ❝â ❞↕♥❣ a0 + a1 + a2 + + an−1 + an tr♦♥❣ ✤â a0 ❧➔ sè ♥❣✉②➯♥✱ ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣ ✭∀i = 1, 2, , n✮✱ an > ợ n > số tr ữủ ỵ ❤✐➺✉ ❧➔ [a0 ; a1 , a2 , , an ]✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ✭▲✐➯♥ ♣❤➙♥ sè ✈æ ❤↕♥✱ ①❡♠ ❬✸❪✮ ❈❤♦ a0, a1, a2, ❧➔ ❞➣② ✈æ ❤↕♥ ❝→❝ sè ♥❣✉②➯♥✱ > ✈ỵ✐ ∀i ≥ 1✳ ❱ỵ✐ ♠é✐ ❦✱ ✤➦t Ck = [a0 ; a1 , a2 , , ak ] õ tỗ t ợ lim Ck = k+ ) ữợ ỹ tỗ t ♥➔② s➩ ✤÷đ❝ ♥â✐ rã tr♦♥❣ t➼♥❤ ❝❤➜t ( ▲ó❝ ♥➔② t❛ ❣å✐ α ❧➔ ❣✐→ trà ❝õ❛ ❧✐➯♥ ♣❤➙♥ sè ✈æ ❤↕♥ [a0 ; a1 , a2 , ] ỵ = [a0 ; a1 , a2 , ] ❚➼♥❤ ❝❤➜t ✶✳✷ ✭①❡♠ ❬✸❪✮✳ ▼é✐ sè ❤ú✉ t➾ ❧➔ ♠ët ❧✐➯♥ ♣❤➙♥ sè ❤ú✉ ❤↕♥✳ ❈❤ù♥❣ ♠✐♥❤✳ a , b > 0, a, b ∈ Z✳ ✣➦t r0 = a, r1 = b t❛ ❝â b r0 = r1 q1 + r2 (0 < r2 < r1 ) r1 = r2 q2 + r3 (0 < r3 < r2 ) rn−2 = rn−1 qn−1 + rn (0 < rn < rn−1 ) rn−1 = rn qn +0 ●✐↔ sû x = ✻ ❙✉② r❛ x= a r0 r2 1 = = q1 + = q + r1 = q + = = q1 + r b r1 r1 q2 + q2 + + r2 r2 qn−1 + qn ⇒ x = [q1 ; q2 , , qn ] 243 62 ❱➼ ❞ö ✶✳✹✳ ❍➣② ❜✐➸✉ ❞✐➵♥ ❝→❝ sè ❤ú✉ t✛ 327 ✱ 243 ✱− ✱ t❤➔♥❤ ❧✐➯♥ ♣❤➙♥ sè✳ 37 37 23 ▲í✐ ❣✐↔✐✳ ❚❛ ❝â 32 = 4.7 + = 1.4 + = 1.3 + = 3.1 ⇒ 32 = [4; 1, 1, 3] = + ❧➔ ❧✐➯♥ ♣❤➙♥ sè ❝â ✤ë ❞➔✐ ✸✳ ❚÷ì♥❣ tü t❛ ❝ô♥❣ ❝â 1+ 243 243 62 = [6; 1, 1, 3, 5]✱ − = [−7; 2, 3, 5]✱ = [2; 1, 2, 3, 2]✳ 37 37 23 1+ ❚➼♥❤ ❝❤➜t ✶✳✸✳ ✭❱➲ t➼♥❤ ❞✉② ♥❤➜t ❝õ❛ ❧✐➯♥ ♣❤➙♥ sè ❤ú✉ ❤↕♥✱ ①❡♠ ❬✸❪✮ ❙ü ❜✐➸✉ ❞✐➵♥ ởt số ỳ t q ữợ số [a0 ; a1 , a2 , , an ] ❧➔ ❞✉② ♥❤➜t✳ ❚➼♥❤ ❝❤➜t ✶✳✹✳ ✭❈æ♥❣ t❤ù❝ t➼♥❤ ❣✐↔♥ ♣❤➙♥✱ ①❡♠ ❬✸❪✮ ❈❤♦ ❧✐➯♥ ♣❤➙♥ sè ❤ú✉ ❤↕♥ [a0 ; a1 , a2 , , an ]✳ ❳➨t ❤❛✐ ❞➣② (pk )nk=0 ✈➔ (qk )nk=0 ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉ p = a0 p = a1 a0 + pk = ak pk−1 + pk−2 , q0 = q = a1 qk = ak qk−1 + qk−2 , ∀k = 2, 3, pk ❑❤✐ ✤â ❣✐↔♥ ♣❤➙♥ t❤ù ❦ ❝õ❛ ❧✐➯♥ ♣❤➙♥ sè [a0 ; a1 , a2 , , an ] ❧➔ Ck = [a0 ; a1 , , ak ] = ✳ qk pk ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ Ck = [a0 ; a1 , , ak ] = ❜➡♥❣ q✉② ♥↕♣ t❤❡♦ ❦✱ qk ợ ữ ỵ Ck ữủ s✉② r❛ tø Ck−1 ❜➡♥❣ ❝→❝❤ t❤❛② ak−1 ❜ð✐ ak−1 + ✳ ak ❚❤➟t ✈➟② a0 p0 C0 = [a0 ] = a0 = = , q0 a1 a0 + p1 C1 = [a0 ; a1 ] = a0 + = = , a1 a1 q1 ✼ a1 + C2 = [a0 ; a1 , a2 ] = a2 a1 + = a0 + 1 a2 a2 (a1 a0 + 1) + a0 a2 p + p p2 = = a2 a1 + a2 q + q q2 ●✐↔ sû Ck = [a0 ; a1 , , ak ] = ak pk−1 + pk−2 pk = , k ≥ ak qk−1 + qk−2 qk ❑❤✐ ✤â ak + Ck+1 = ak + = ak+1 ak+1 pk−1 + pk−2 qk−1 + qk−2 ak+1 pk + pk−1 ak+1 (ak pk−1 + pk−2 ) + pk−1 = ak+1 (ak qk−1 + qk−2 ) + qk−1 ak+1 qk + qk−1 ❉♦ ✤â Ck+1 = pk+1 qk+1 ❱➟② t❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ Ck = [a0 ; a1 , , ak ] = pk qk ❱➼ ❞ö ✶✳✺✳ ❚➻♠ ❝→❝ ❣✐↔♥ ♣❤➙♥ ❝õ❛ ❧✐➯♥ ♣❤➙♥ sè [6; 1, 1, 3, 5]✳ ▲í✐ ❣✐↔✐✳ ❚❛ ❝â ❜↔♥❣ s❛✉ ❦ ak pk qk ✵ ✻ ✻ ✶ ✶ ✶ ✼ ✶ ✷ ✶ ✶✸ ✷ ✸ ✸ ✹✻ ✼ ✹ ✺ ✷✹✸ ✸✼ ❱➟② C0 = 6, C1 = 7, C2 = ❚➼♥❤ ❝❤➜t ✶✳✺ 13 46 243 , C3 = , C4 = 37 ✳ ✭①❡♠ ❬✸❪✮ ❈❤♦ Ck ❧➔ ❣✐↔♥ ♣❤➙♥ t❤ù ❦ ❝õ❛ [a0 ; a1 , a2 , , an ]✱ ✈ỵ✐ ✶✳✹)✳ ❑❤✐ ✤â ≤ k ≤ n ✈➔ pk , qk ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ tr♦♥❣ t➼♥❤ ❝❤➜t ( pk qk−1 − pk−1 qk = (−1)k−1 ❚➼♥❤ ❝❤➜t ✶✳✻ ✭①❡♠ ❬✸❪✮✳ ●✐↔ sû {Ck } ❧➔ ❞➣② ❣✐↔♥ ♣❤➙♥ ❝õ❛ ❧✐➯♥ ♣❤➙♥ sè ❤ú✉ ❤↕♥ [a0 ; a1 , a2 , , an ]✳ ❑❤✐ ✤â t❛ ❝â ❝→❝ ♠è✐ ❧✐➯♥ ❤➺ s❛✉ ✽ ✐✮ Ck − Ck−1 (−1)k−1 = ✱ ✈ỵ✐ ≤ k ≤ n qk qk−1 ✐✐✮ Ck − Ck−2 = ak (−1)k ✱ ✈ỵ✐ ≤ k ≤ n qk qk−2 ❚➼♥❤ ❝❤➜t ✶✳✼ ✭①❡♠ ❬✸❪✮✳ ❱ỵ✐ ❝→❝ ❣✐↔♥ ♣❤➙♥ Ck ❝õ❛ ❧✐➯♥ ♣❤➙♥ sè ❤ú✉ ❤↕♥ [a0; a1, a2, , an] t❛ ❝â ❝→❝ ❞➣② ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ✐✮ C1 > C3 > C5 > ✐✐✮ C0 < C2 < C4 < ✐✐✐✮ ♠é✐ ❣✐↔♥ ♣❤➙♥ ❧➫ C2j−1 ✤➲✉ ❧ỵ♥ ❤ì♥ ♠é✐ ❣✐↔♥ ♣❤➙♥ ❝❤➤♥ C2i ✳ ❚➼♥❤ ❝❤➜t ✶✳✽ ✭①❡♠ ❬✸❪✮✳ ❱ỵ✐ ♠å✐ k = 0, 1, , n t❤➻ (pk , qk ) = ✭tù❝ ❧➔ pk , qk ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉✮✳ ❚➼♥❤ ❝❤➜t ✶✳✾ ✭①❡♠ ❬✸❪✮✳ ❈❤♦ a0, a1, a2, ❧➔ ❞➣② ✈ỉ ❤↕♥ ❝→❝ sè ♥❣✉②➯♥✱ > ✈ỵ✐ ∀i ≥ 1✳ ❱ỵ✐ ♠é✐ ❦✱ ✤➦t Ck = [a0 ; a1 , a2 , , ak ] õ tỗ t ợ lim Ck k→+∞ ❚❤❡♦ t➼♥❤ ❝❤➜t ( ❈❤ù♥❣ ♠✐♥❤✳ ✶✳✼) t❛ ❝â C1 > C3 > C5 > > C2n−1 > C2n+1 > C0 < C2 < C4 < < C2n−2 < C2n < C2j−1 > C2i , ✈ỵ✐ ♠å✐ ✐✱ ❥✳ ❚ø ✤â s✉② r❛ ❞➣② {C2k+1 }✱ k = 0, 1, ❧➔ ❞➣② ❣✐↔♠ ✈➔ ữợ C0 ỏ {C2k } k = 0, 1, ❧➔ ❞➣② t➠♥❣ tr C1 ỵ tt ợ số t tỗ t ❣✐ỵ✐ ❤↕♥ lim C2k+1 = α , k→+∞ lim C2k = β k→+∞ ✶✳✻) t❛ ❝â ❚❤❡♦ t➼♥❤ ❝❤➜t ( C2k+1 − C2k (−1)2k = > = q2k+1 q2k q2k+1 q2k ✾ ✭❛✮ ❱➟② π x = − + (a + t)2π y= π + t2π tr♦♥❣ ✤â a, t ∈ Z, a ≥ 6✳ ✲ ❚r÷í♥❣ ❤đ♣ ✷✿ ❳➨t ❤➺ ♣❤÷ì♥❣ tr➻♥❤ π 2x + y = + k2π x + y = − π + h2π ❝â ♥❣❤✐➺♠ ❧➔ π + (k − h)2π x= y = − 5π + (2h − k)2π ❑➳t ❤ñ♣ ✤✐➲✉ ❦✐➺♥ x − y ≥ 10π t❛ t❤✉ ✤÷đ❝ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ 2k − 3h ≥ 13 t ữỡ tr tữỡ ữỡ ợ 2k 3h = b tr♦♥❣ ✤â b ∈ Z, b ≥ 5✳ ❚ø ✤â t❛ ❝â k = 2b + 3t h = b + 2t ❱➟② x= (t ∈ Z) π + (b + t)2π y = − 5π + t2π tr♦♥❣ ✤â b, t ∈ Z, b ≥ 5✳ ✲ ❚r÷í♥❣ ❤đ♣ ✸✿ ❳➨t ❤➺ ♣❤÷ì♥❣ tr➻♥❤ 5π 2x + y = + m2π x + y = π + l2π ✺✶ ❝â ♥❣❤✐➺♠ ❧➔ π + (m − l)2π x= y = − π + (2l − m)2π ❑➳t ❤ñ♣ ✤✐➲✉ ❦✐➺♥ x − y ≥ 10π t❛ t❤✉ ✤÷đ❝ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ 2m − 3l ≥ 14 t ữỡ tr tữỡ ữỡ ợ 2m − 3l = c tr♦♥❣ ✤â c ∈ Z, c ≥ 5✳ ❚ø ✤â t❛ ❝â m = 2c + 3t l = c + 2t ❱➟② (t ∈ Z) π + (c + t)2π x= y=−π + t2π tr♦♥❣ ✤â c, t ∈ Z, c ≥ 5✳ ✲ ❚r÷í♥❣ ❤đ♣ ✹✿ ❳➨t ❤➺ ♣❤÷ì♥❣ tr➻♥❤ 5π 2x + y = + m2π x + y = − π + h2π ❝â ♥❣❤✐➺♠ ❧➔ x= 7π + (m − h)2π y = − 3π + (2h − m)2π ❑➳t ❤ñ♣ ✤✐➲✉ ❦✐➺♥ x − y ≥ 10π t❛ t❤✉ ✤÷đ❝ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ 2m − 3h ≥ 11 t ữỡ tr tữỡ ữỡ ợ 2m − 3h = d ✺✷ tr♦♥❣ ✤â d ∈ Z, d ≥ 4✳ ❚ø ✤â t❛ ❝â m = 2d + 3t h = d + 2t ❱➟② x= (t ∈ Z) 7π + (d + t)2π y = − 3π + t2π tr♦♥❣ ✤â d, t ∈ Z, d ≥ 4✳ ❚â♠ ❧↕✐ ❝â ✹ ❤å ♥❣❤✐➺♠ (x; y) tọ ỗ x= + (b + t)2π x = − + (a + t)2π y= x= ✱ π + t2π π + (c + t)2π y=−π + ✱ t2π y = − 5π + t2π 7π + (d + t)2π x= ✱ y = − 3π + t2π tr♦♥❣ ✤â a, b, c, d, t ∈ Z, a ≥ 6, b ≥ 5, c ≥ 5, d ≥ 4✳ ❱➼ ❞ö ✸✳✺✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ sin(2x + y) = cos(x + y) = tr➯♥ ✤♦↕♥ [−6π; 6π] ✈➔ t❤ä❛ ♠➣♥ x − y ≥ 10π ✳ ❍➺ ✤➣ ❝❤♦ ❝â ✹ ❤å ♥❣❤✐➺♠ (x; y) t❤ä❛ ♠➣♥ x − y ≥ 10π ỗ + (b + t)2 π x = − + (a + t)2π x= y= x= y = − 5π + t2π 7π + (d + t)2π x= ✱ π + t2π π + (c + t)2π y=−π + t2π ✱ y = − 3π + t2π tr♦♥❣ ✤â a, b, c, d, t ∈ Z, a ≥ 6, b ≥ 5, c ≥ 5, d ≥ 4✳ ✺✸ ✱ ✲ ❚r÷í♥❣ ❤đ♣ ✶✿ π x = − + (a + t)2π y= π + (a, t ∈ Z, a ≥ 6)✳ t2π ❱ỵ✐ ✤✐➲✉ ❦✐➺♥ x, y ∈ [−6π; 6π] t❛ ❝â π −6π ≤ − + (a + t)2π ≤ 6π π −6π ≤ + a, t ∈ Z, a ≥ t2π tữỡ ữỡ ợ 1 −3≤a+t≤ +3 12 12 1 − −3≤t≤− +3 4 a, t ∈ Z, a ≥ s✉② r❛ 6≤a≤6+ ⇒ a = a∈Z ❱ỵ✐ a = 6, t = −3 t❛ ✤÷đ❝ 35π 11π ;− (x; y) = ✲ ❚r÷í♥❣ ❤đ♣ ✷✿ x= π + (b + t)2π y = − 5π + (b, t ∈ Z, b ≥ 5)✳ t2π ❱ỵ✐ ✤✐➲✉ ❦✐➺♥ x, y ∈ [−6π; 6π] t❛ ❝â π −6π ≤ + (b + t)2π ≤ 6π 5π −6π ≤ − + t2π ≤ 6π b, t ∈ Z, b ≥ tữỡ ữỡ ợ 1 − − ≤ b + t ≤ − +3 4 5 −3≤t≤ +3 12 12 b, t ∈ Z, b ≥ s✉② r❛ 5≤b≤5+ ⇒ b = b∈Z ❱ỵ✐ b = t❛ ✤÷đ❝ −3≤ t≤− −2 12 ⇒ t ∈ ∅ t∈Z ❱➟② tr÷í♥❣ ❤đ♣ ♥➔② t❛ ❦❤ỉ♥❣ t❤✉ ✤÷đ❝ ♥❣❤✐➺♠ (x; y) t❤ä❛ ♠➣♥ ②➯✉ ❝➛✉ ❜➔✐ t♦→♥✳ ✲ ❚r÷í♥❣ ❤đ♣ ✸✿ x= π + (c + t)2π y=−π + (c, t ∈ Z, c ≥ 5)✳ t2π ❱ỵ✐ ✤✐➲✉ ❦✐➺♥ x, y ∈ [−6π; 6π] t❛ ❝â −6π ≤ π + (c + t)2π ≤ 6π π −6π ≤ − + c, t ∈ Z, c ≥ t2π ≤ 6π ❍➺ ♥➔② tữỡ ữỡ ợ 1 3c+t +3 4 1 −3≤t≤ +3 12 12 c, t ∈ Z, c ≥ s✉② r❛ 5≤c≤5+ c∈Z ✺✺ ⇒ c = ❱ỵ✐ c = t❛ ✤÷đ❝ 1 −3≤ t≤− −2 12 ⇒ t ∈ ∅ t∈Z ❱➟② tr÷í♥❣ ❤đ♣ ♥➔② t❛ ❦❤ỉ♥❣ t❤✉ ✤÷đ❝ ♥❣❤✐➺♠ (x; y) t❤ä❛ ♠➣♥ ②➯✉ ❝➛✉ ❜➔✐ t♦→♥✳ ✲ ❚r÷í♥❣ ❤đ♣ ✹✿ x= 7π + (d + t)2π y = − 3π + (d, t ∈ Z, d ≥ 4)✳ t2π ❱ỵ✐ ✤✐➲✉ ❦✐➺♥ x, y ∈ [−6π; 6π] t❛ ❝â 7π + (d + t)2π ≤ 6π −6π ≤ 3π −6π ≤ − + d, t ∈ Z, d ≥ t2π ≤ 6π ❍➺ ♥➔② t÷ì♥❣ ữỡ ợ 7 3d+t +3 12 12 3 −3≤t≤ +3 4 d, t ∈ Z, d ≥ s✉② r❛ 4≤d≤4+ ⇒ d = d∈Z 31π 11π ;− ❚â♠ ❧↕✐ ❜➔✐ t♦→♥ ✤➣ ❝❤♦ õ (x; y) ỗ ợ d = 4, t = −2 t❛ ✤÷đ❝ (x; y) = 35π 11π ;− , 31π 11π ;− ✸✳✸ ❳→❝ ✤à♥❤ ♣❤➙♥ t❤ù❝ ❝❤➼♥❤ q✉② t❤ä❛ ♠➣♥ trữợ P tự q✉② ♠ët ❜✐➳♥✱ ①❡♠❬✺❪✮ ❈❤♦ > 0✱ αi ∈ R ✈ỵ✐ n ∀i = 1, 2, , n✳ ❑❤✐ ✤â f (x) = ✭♠ët ❜✐➳♥ ①✮ n xi ợ x > ữủ ❧➔ ♣❤➙♥ t❤ù❝ ❝❤➼♥❤ q✉② i=1 αi = 0✳ i=1 ú ỵ P tự q f (x) ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t t↕✐ x = ✣à♥❤ ♥❣❤➽❛ ✸✳✷✳ ✭P❤➙♥ t❤ù❝ ❝❤➼♥❤nq✉② ❤❛✐ ❜✐➳♥✱ ①❡♠ ❬✺❪✮ ❈❤♦ > 0✱ αi, βi ∈ R ✈ỵ✐ xαi y βi ✈ỵ✐ x > 0, y > ✤÷đ❝ ❣å✐ ❧➔ ♣❤➙♥ t❤ù❝ ∀i = 1, 2, , n✳ ❑❤✐ ✤â f (x, y) = n ❝❤➼♥❤ q✉② ✭❤❛✐ ❜✐➳♥ ①✱ ②✮ ♥➳✉ i=1 n αi = i=1 βi = 0✳ i=1 ❈❤ó þ ✸✳✷✳ P❤➙♥ t❤ù❝ ❝❤➼♥❤ q✉② f (x, y) ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t t↕✐ x = y = ❚r♦♥❣ ❝→❝ ✈➼ ❞ö s❛✉ t❛ ①➨t αi , βi ∈ Z✳ ❱➼ ❞ö ✸✳✻✳ ❳➨t ♣❤➙♥ t❤ù❝ ❝❤➼♥❤ q✉② f (x) = xα1 + 2xα2 + 3xα3 + 5xα4 + 7xα5 ❚➻♠ (α1 ; α2 ; α3 ; α4 ; α5 ) s❛♦ ❝❤♦ α1 + α2 + 2α3 + α4 − α5 > ▲í✐ ❣✐↔✐✳ ✭✷✾✮ ❉♦ f (x) ❧➔ ❤➔♠ ♣❤➙♥ t❤ù❝ ❝❤➼♥❤ q✉② ♥➯♥ α1 + 2α2 + 3α3 + 5α4 + 7α5 = ✭✷✾❛✮ ❚ø (29a) ✈➔ (29) t❛ t❤✉ ✤÷đ❝ α2 + α3 + 4α4 + 8α5 < −4 ❇➜t ♣❤÷ì♥❣ tr tữỡ ữỡ ợ + + 44 + 8α5 = m tr♦♥❣ ✤â m ∈ Z, m < −4✳ ✣➦t α3 = a, α4 = b, α5 = c, ❦➳t ❤đ♣ (29a) ✈➔ (29b) t❛ t❤✉ ✤÷đ❝ α1 = −2m − a + 3b + 9c α2 = m − a − 4b − 8c α3 = a α4 = b α5 = c tr♦♥❣ ✤â m, a, b, c ∈ Z, m < −4✳ ❈❤➥♥❣ ❤↕♥ ✈ỵ✐ m = −5, a = 1, b = 2, c = −4 t❛ ❝â (α1 ; α2 ; α3 ; α4 ; α5 ) = (−21; 18; 1; 2; −4) ✺✼ ✭✷✾❜✮ ❑❤✐ ✤â f (x) = 2x18 + 5x2 + 3x + + 21 x x ❱➟② t❛ ①➙② ❞ü♥❣ ✤÷đ❝ ❜➔✐ t♦→♥ s❛✉ ❇➔✐ t♦→♥ số tỹ ữỡ tũ ỵ ự ♠✐♥❤ r➡♥❣ 2x18 + 5x2 + 3x + + 21 ≥ 18 x x ❱➼ ❞ö ✸✳✼✳ ❳➨t ♣❤➙♥ t❤ù❝ ❝❤➼♥❤ q✉② f (x, y) = xα1 y β1 + 2xα2 y β2 + 3xα3 y β3 ❚➻♠ (α1 ; α2 ; α3 ) ✈➔ (β1 ; ; ) s tọ ỗ t❤í✐ ❝→❝ ❤➺ t❤ù❝ s❛✉ ▲í✐ ❣✐↔✐✳ α1 + 4α2 − 3α3 > 0, ✭✸✵✮ β1 − 3β2 + β3 ≤ ✭✸✶✮ ❉♦ f (x, y) ❧➔ ❤➔♠ ♣❤➙♥ t❤ù❝ ❝❤➼♥❤ q✉② ♥➯♥ t❛ ❝â α1 + 2α2 + 3α3 = β1 + 2β2 + 3β3 = t õ ỗ tớ = (22 + 3α3 ) , ✭✸✵❛✮ β1 = − (2β2 + 3β3 ) ✭✸✶❛✮ α2 − 3α3 > ✭✸✵❜✮ ❚❤➳ (30a) (30) t ữủ (30b) tữỡ ữỡ ợ − 3α3 = m tr♦♥❣ ✤â m ∈ Z+ ✳ ❚ø ✤â t❛ t❤✉ ✤÷đ❝ α2 = 3m + 2a α3 = m + a ✺✽ tr♦♥❣ ✤â m, a ∈ Z, m ≥ 1✳ ❚❤➳ (31a) ✈➔♦ (31) t❛ ✤÷đ❝ ✭✸✶❜✮ 5β2 + 2β3 ≥ −3 (31b) t÷ì♥❣ ✤÷ì♥❣ ✈ỵ✐ 5β2 + 2β3 = n tr♦♥❣ ✤â n ∈ Z, n ≥ −3✳ ❚ø ✤â t❛ t❤✉ ✤÷đ❝ β2 = n + 2b β3 = −2n − 5b tr♦♥❣ ✤â n, b ∈ Z, n ≥ −3✳ ❱➟② t❛ õ ỗ tớ (1 ; ; ) = (−9m − 7a; 3m + 2a; m + a) , (β1 ; β2 ; β3 ) = (4n + 11b; n + 2b; −2n − 5b) tr♦♥❣ ✤â m, n, a, b ∈ Z, m ≥ 1, n ≥ −3✳ ❈❤➥♥❣ ❤↕♥ ✈ỵ✐ m = 1, n = −3, a = 1, b = t❛ ❝â ♣❤➙♥ t❤ù❝ ❝❤➼♥❤ q✉② f (x, y) = y 10 3x2 + 2x y + x16 y4 ❝â ❣✐→ trà ♥❤ä ♥❤➜t f (x, y) = f (1, 1) = ◆➳✉ t❤❛② x = a b , y = t❤➻ t❛ t❤✉ ✤÷đ❝ ❜➔✐ t♦→♥ ❝â ❦➳t q✉↔ t÷ì♥❣ tü s❛✉ 2 ❇➔✐ t♦→♥ ✸✳✷✳ ❈❤♦ a, b ❧➔ số tỹ ữỡ tũ ỵ tr ọ ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ M= 64b10 a5 b 12a2 + + a16 32 b ❱➼ ❞ö ✸✳✽✳ ❈❤♦ ❤➔♠ ♣❤➙♥ t❤ù❝ ❝❤➼♥❤ q✉② f (x) = ax2 + bx4 + c x2 ✈ỵ✐ a, b, c ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ❚➻♠ ❜ë sè (a, b, c) s❛♦ ❝❤♦ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ f (x) ❦❤ỉ♥❣ ✈÷đt q✉→ ✶✶✳ ▲í✐ ❣✐↔✐✳ ❉♦ f (x) ❧➔ ❤➔♠ ♣❤➙♥ t❤ù❝ ❝❤➼♥❤ q✉② ♥➯♥ t❛ ❝â 2a + 4b − 2c = ✺✾ ❤❛② c = a + 2b ✭✸✷❛✮ ▼➦t ❦❤→❝✱ →♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❆▼ ✕ ●▼ s✉② rë♥❣ t❛ ❝â f (x) a b c = x2 + x4 + x−2 f (1) a+b+c a+b+c a+b+c 2a 4b −2c ≥ x a + b + c x a + b + c x a + b + c = ❉♦ ✤â f (x) = f (1) = a + b + c ❚❤❡♦ ❜➔✐ r❛ t❛ ❝â a + b + c ≤ 11 ✭✸✷❜✮ 2a + 3b ≤ 11 ✭✸✷❝✮ 2a + 3b = m ✭✸✷❞✮ ❚❤➳ ✭✸✷❛✮ ✈➔♦ ✭✸✷❜✮ t❛ ✤÷đ❝ tữỡ ữỡ ợ ợ m Z, m ≤ 11✳ P❤÷ì♥❣ tr➻♥❤ ✭✸✷❞✮ ❝â ♥❣❤✐➺♠ ✭❛✱❜✮ ❧➔ a = −m + 3t b = m − 2t ❱➟② t❛ ❝â (t ∈ Z) a = −m + 3t b = m − 2t c = m − t ❉♦ ❛✱ ❜✱ ❝ ❞÷ì♥❣ ♥➯♥ m m