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✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❉×❒◆● ❈➷◆● ❈Ø ❇❻❚ ✣➃◆● ❚❍Ù❈ ❱⑨ ❈Ü❈ ❚❘➚ ❙■◆❍ ❇Ð■ ❈⑩❈ ✣❆ ❚❍Ù❈ ✣❸■ ❙➮ ❇❆ ❇■➌◆ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✾ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❉×❒◆● ❈➷◆● ❈Ø ❇❻❚ ✣➃◆● ❚❍Ù❈ ❱⑨ ❈Ü❈ ❚❘➚ ❙■◆❍ ❇Ð■ ❈⑩❈ ✣❆ ❚❍Ù❈ ✣❸■ ❙➮ ❇❆ ❇■➌◆ ❈❤✉②➯♥ ♥❣➔♥❤✿ P❍×❒◆● P❍⑩P ❚❖⑩◆ ❙❒ ❈❻P ▼➣ sè✿ ✻✵ ✹✻ ✵✶ ✶✸ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ữớ ữợ ❱➠♥ ▼➟✉ ❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✾ ✐ ▼ö❝ ❧ö❝ ▼Ð ✣❺❯ ❈❤÷ì♥❣ ✶✳ ✣❛ t❤ù❝ ✈➔ ❝→❝ ❤➺ t❤ù❝ ❧✐➯♥ q✉❛♥ ✶ ✸ ✶✳✶ ▼ët sè ❜➜t ✤➥♥❣ t❤ù❝ ❝ê ✤✐➸♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ✤❛ t❤ù❝ ✶✳✷ ✣❛ t❤ù❝ ❜➟❝ ❜❛ ✈➔ ♠ët sè ❤➺ t❤ù❝ ❝ì ❜↔♥ ✶✳✸ ✳ ✳ ✳ ✳ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷✳✶ ❈æ♥❣ t❤ù❝ ❱✐➧t❡ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✷✳✷ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ✤è✐ ①ù♥❣ ❜❛ ➞♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✷✳✸ P❤➙♥ t➼❝❤ ✤❛ t❤ù❝ t❤➔♥❤ ♥❤➙♥ tû ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✷✳✹ ❚➼♥❤ ❝❤✐❛ ❤➳t ❝õ❛ ❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✣❛ t❤ù❝ ❜➟❝ ❜❛ ✈➔ ❝→❝ ❤➺ t❤ù❝ tr♦♥❣ t❛♠ ❣✐→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ❈❤÷ì♥❣ ✷✳ ❈→❝ ❜➜t ✤➥♥❣ t❤ù❝ s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥ ✷✷ ✷✳✶ ✷✳✷ ✷✳✸ ❇➜t ✤➥♥❣ t❤ù❝ s✐♥❤ ❜ð✐ ✤❛ t❤ù❝ ❜➟❝ ❜❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✶✳✶ ❈→❝ ❦❤→✐ ♥✐➺♠ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ ỡ ❜↔♥ ❝õ❛ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥ ✳ ✳ ✳ ✳ ✷✹ ❈→❝ ❜➜t ✤➥♥❣ t❤ù❝ s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥ ✳ ✳ ✳ ✷✽ ✷✳✷✳✶ ▼ët sè ♠➺♥❤ ✤➲ ❜➜t ✤➥♥❣ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✷✳✷ ⑩♣ ❞ö♥❣ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ ✸✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼ët sè ❞↕♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❜❛ ❜✐➳♥ tr♦♥❣ ♣❤➙♥ t❤ù❝ ✳ ✳ ✳ ✳ ✸✺ ❈❤÷ì♥❣ ✸✳ ❈→❝ ❞↕♥❣ t♦→♥ ❝ü❝ trà s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥ ✸✽ ✸✳✶ ❈ü❝ trà t❤❡♦ r➔♥❣ ❜✉ë❝ tê♥❣ ✈➔ t➼❝❤ ❜❛ sè ✸✳✷ ❈→❝ ❞↕♥❣ t♦→♥ ❝ü❝ trà s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥ ✳ ✹✶ ✸✳✸ ▼ët sè ❞↕♥❣ t♦→♥ ❧✐➯♥ q✉❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ❑➌❚ ▲❯❾◆ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✹✼ ✹✽ ✶ ▼ð ✤➛✉ ❈❤✉②➯♥ ✤➲ ❜➜t ✤➥♥❣ t❤ù❝ ❝â ✈❛✐ trá r➜t q✉❛♥ trå♥❣ ð ❜➟❝ tr✉♥❣ ❤å❝ ♣❤ê t❤æ♥❣✳ ❇➜t ✤➥♥❣ t❤ù❝ ❦❤æ♥❣ ❝❤➾ ❧➔ ✤è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉ trå♥❣ t➙♠ ❝õ❛ ✣↕✐ sè ✈➔ ●✐↔✐ t➼❝❤ ♠➔ ❝á♥ ❧➔ ❝ỉ♥❣ ❝ư ✤➢❝ ❧ü❝ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❦❤→❝ ❝õ❛ t♦→♥ ❤å❝✳ ❚❛ ✤➣ ❜✐➳t r➡♥❣ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ✤❛ t❤ù❝ ✤➣ ✤÷đ❝ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ ❦❤↔♦ s→t ♥❤÷ ◆❡✇t♦♥✱ ▲❛❣r❛♥❣❡✱ ❇❡rst❡✐♥✱ ▼❛r❦♦✈✱ ❑♦❧♠♦❣♦r♦✈✱ ▲❛♥❞❛✉✱ ✳ ✳ ✳ ❈→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❞↕♥❣ ♥➔② ❝ơ♥❣ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ❜➡♥❣ ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ❦❤→❝ ♥❤❛✉ ❝õ❛ ❤➻♥❤ ❤å❝ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣ ✈➨❝tì ✈➔ ♣❤÷ì♥❣ ♣❤→♣ tå❛ ✤ë✱ ♣❤÷ì♥❣ ♣❤→♣ sè ♣❤ù❝✱✳ ✳ ✳ ❚✉② ♥❤✐➯♥✱ ❝→❝ ❞↕♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ự ợ ợ tự tờ qt t ữớ t ổ t t ỗ ❧ã♠✮ ✤➸ ❦❤↔♦ s→t ❝❤ó♥❣✳ ✣➸ ✤→♣ ù♥❣ ♥❤✉ ❝➛✉ ỗ ữù ỗ ữù s ọ ✈➔ ♥➙♥❣ ❝❛♦ ♥❣❤✐➺♣ ✈ö ❝õ❛ ❜↔♥ t❤➙♥ ✈➲ ❝❤✉②➯♥ ✤➲ ❜➜t ✤➥♥❣ t❤ù❝ ✈➔ ❝ü❝ trà s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥✱ tæ✐ ❝❤å♥ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ✧❇➜t ✤➥♥❣ t❤ù❝ ✈➔ ❝ü❝ trà s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥✧✳ ▲✉➟♥ ✈➠♥ ♥➔② ♥❤➡♠ ❝✉♥❣ ❝➜♣ ♠ët sè ❞↕♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✈➔ ❝ü❝ trà s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❝ò♥❣ ởt số q ỗ ✤➛✉✱ ❦➳t ❧✉➟♥ ✈➔ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶✳ ✣❛ t❤ù❝ ✈➔ ❝→❝ ❤➺ t❤ù❝ ❧✐➯♥ q✉❛♥✳ ❈❤÷ì♥❣ ✷✳ ❈→❝ ❜➜t ✤➥♥❣ t❤ù❝ s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥✳ ❈❤÷ì♥❣ ✸✳ ❈→❝ ❞↕♥❣ t♦→♥ ❝ü❝ trà s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ❧➔ ❦❤↔♦ s→t ♠ët sè ❧ỵ♣ ❜➜t ✤➥♥❣ t❤ù❝ ✈➔ ❝ü❝ trà s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥ ✈➔ ①➨t ❝→❝ ♠ð rë♥❣ ❝õ❛ ❝❤ó♥❣ ✤➸ →♣ ❞ư♥❣ tr♦♥❣ ❦❤↔♦ s→t ❝→❝ ❜➔✐ t♦→♥ ❝ü❝ trà ❧✐➯♥ q✉❛♥✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ ●❙✳❚❙❑❍✳ ◆❣✉②➵♥ ❱➠♥ ▼➟✉ t t ữợ ú ù t tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❧✉➟♥ ✈➠♥✳ ❚→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ t tợ ổ tr trữớ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt t❤í✐ t t rữớ ỗ tớ t ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ ❣✐❛ ✤➻♥❤ ✈➔ ỗ ổ ổ ú ù ✈✐➯♥ tỉ✐ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ ✈➔ tr♦♥❣ q✉→ tr➻♥❤ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❚❤→✐ ◆❣✉②➯♥✱ ✶✷ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✾✳ ❚→❝ ❣✐↔ ❉÷ì♥❣ ❈ỉ♥❣ ❈ø ✸ ❈❤÷ì♥❣ ✶✳ ✣❛ t❤ù❝ ✈➔ ❝→❝ ❤➺ t❤ù❝ ❧✐➯♥ q✉❛♥ ▼ö❝ ✤➼❝❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ tr➻♥❤ ❜➔② ♠ët sè ❜➜t ✤➥♥❣ t❤ù❝ ❝ê ✤✐➸♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ✤❛ t❤ù❝ ♥â✐ ❝❤✉♥❣✱ ✤❛ t❤ù❝ ❜➟❝ ❜❛ ♥â✐ r✐➯♥❣ ✈➔ ①➨t ♠ët sè ❤➺ t❤ù❝ ❝ì ❜↔♥✳ ▼ët ♣❤➛♥ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ ❞➔♥❤ ✤➸ ♥➯✉ ✈➲ ✤❛ t❤ù❝ ❜➟❝ ❜❛ ✈➔ ❝→❝ ❤➺ t❤ù❝ tr♦♥❣ t❛♠ ❣✐→❝✳ ❈→❝ ❦➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ t❤❛♠ ❦❤↔♦ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✷❪✱ ❬✸❪✳ ✶✳✶ ▼ët sè ❜➜t ✤➥♥❣ t❤ù❝ ❝ê ✤✐➸♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ✤❛ t❤ù❝ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ A ❈❤♦ ❜➟❝ n ❜✐➳♥ x ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à✳ ❚❛ ❣å✐ ✤❛ t❤ù❝ ❧➔ ♠ët ❜✐➸✉ t❤ù❝ ❝â ❞↕♥❣ fn (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 (an = 0), tr♦♥❣ ✤â ❝→❝ ∈ A ✤÷đ❝ ❣å✐ ❧➔ ❤➺ sè✱ an ❧➔ ❤➺ sè ❝❛♦ ♥❤➜t ✈➔ a0 ✭✶✳✶✮ ❧➔ ❤➺ sè tü ❞♦ ❝õ❛ ✤❛ t❤ù❝✳ fn (x) ❧➔ sè ♠ô ❝❛♦ ♥❤➜t ❝õ❛ ❧ô② t❤ø❛ ❝â ♠➦t tr ữủ ỵ deg(f ) ✤â ♥➳✉ tr♦♥❣ ✭✶✳✶✮ an = t❤➻ deg(f ) = n ◆➳✉ = 0, i = 1, , n ✈➔ a0 = t❤➻ t❛ ❝â ❜➟❝ ❝õ❛ ✤❛ t❤ù❝ ❧➔ 0✳ ◆➳✉ = 0, i = 0, , n t❤➻ t❛ ❝♦✐ ❜➟❝ ❝õ❛ ✤❛ t❤ù❝ ❧➔ −∞ ✈➔ ❣å✐ ✤❛ ❇➟❝ ❝õ❛ ✤❛ t❤ù❝ t❤ù❝ ❦❤æ♥❣ ✭♥â✐ ❝❤✉♥❣ t❤➻ ♥❣÷í✐ t❛ ❦❤ỉ♥❣ ✤à♥❤ ♥❣❤➽❛ ❜➟❝ ❝õ❛ ✤❛ t❤ù❝ ❦❤ỉ♥❣✮✳ ❚➟♣ ❤đ♣ t➜t ❝↔ ❝→❝ ✤❛ t❤ù❝ ✈ỵ✐ ❤➺ sè tr A[x] A=K A ữủ ỵ K[x] ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à✳ ❚❛ t❤÷í♥❣ ①➨t A = Z✱ ❤♦➦❝ A = Q ❤♦➦❝ A = R ❤♦➦❝ A = C✳ ❑❤✐ ✤â✱ t❛ ❝â ❝→❝ ✈➔♥❤ ✤❛ t❤ù❝ t÷ì♥❣ ù♥❣ ❧➔ Z[x], Q[x], R[x], C[x]✳ ❑❤✐ ❧➔ ♠ët tr÷í♥❣ t❤➻ ✈➔♥❤ ✹ ❈→❝ ♣❤➨♣ t➼♥❤ tr➯♥ ✤❛ t❤ù❝ ❈❤♦ ❤❛✐ ✤❛ t❤ù❝ f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 , g(x) = bn xn + bn−1 xn−1 + · · · + b1 x + b0 ❚❛ ✤à♥❤ ♥❣❤➽❛ ❝→❝ ♣❤➨♣ t➼♥❤ sè ❤å❝ f (x) + g(x) = (an + bn )xn + · · · + (a1 + b1 )x + a0 + b0 , f (x) − g(x) = (an − bn )xn + · · · + (a1 − b1 )x + a0 − b0 , f (x)g(x) = c2n x2n + c2n−1 x2n−1 + · · · + c1 x + c0 , tr♦♥❣ ✤â ck = a0 bk + a1 bk−1 + · · · + ak b0 , k = 0, , n t t ỡ ỵ sû A ❧➔ ♠ët tr÷í♥❣✱ f (x) ✈➔ g(x) = ❧➔ ❤❛✐ ✤❛ t❤ù❝ A[x]✱ t❤➳ A[x] s❛♦ ❝❤♦ ❝õ❛ ✈➔♥❤ t❤✉ë❝ t❤➻ ❜❛♦ ❣✐í ❝ơ♥❣ ❝â ❝➦♣ ✤❛ t❤ù❝ ❞✉② ♥❤➜t f (x) = g(x)q(x) + r(x) ◆➳✉ r(x) = t❛ ♥â✐ f (x) ✈ỵ✐ ❝❤✐❛ ❤➳t a tỷ tũ ỵ n þ ❝õ❛ ✈➔♥❤ A[x]✱ ♣❤➛♥ tû f (a) = ✈➔ r(x) deg r(x) < deg g(x) g(x)✳ n ●✐↔ sû q(x) A✱ f (x) = x i ❧➔ ✤❛ t❤ù❝ tị② i=0 ai ❝â ✤÷đ❝ ❜➡♥❣ ❝→❝❤ t❤❛② x ❜ð✐ a i=0 f (x) t↕✐ a✳ ◆➳✉ f (a) = t❤➻ t❛ ❣å✐ a ❧➔ ♥❣❤✐➺♠ ❝õ❛ f (x)✳ ❇➔✐ t♦→♥ t➻♠ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ f (x) tr♦♥❣ A ❣å✐ ❧➔ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè ❜➟❝ n tr♦♥❣ A✳ ✤÷đ❝ ❣å✐ ❧➔ ❣✐→ trà ❝õ❛ an xn + an−1 xn−1 + · · · + a1 x + a0 = (an = 0) ỵ ✶✳✷✳ ●✐↔ sû A ❧➔ ♠ët tr÷í♥❣✱ a ∈ A ✈➔ f (x) ∈ A[x]✳ ❉÷ sè ❝õ❛ ♣❤➨♣ ❝❤✐❛ f (x) xa f (a) ỵ ✶✳✸✳ a ❧➔ ♥❣❤✐➺♠ ❝õ❛ f (x) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ f (x) ❝❤✐❛ ❤➳t ❝❤♦ (x−a)✳ a ∈ A✱ f (x) ∈ A[x] ✈➔ m ❧➔ ♠ët sè tü ♥❤✐➯♥ ❤ì♥ ❤♦➦❝ ❜➡♥❣ 1✳ ❑❤✐ ✤â a ❧➔ ♥❣❤✐➺♠ ❜ë✐ ❝➜♣ m ❝õ❛ f (x) ❦❤✐ ✈➔ ❝❤➾ f (x) ❝❤✐❛ ❤➳t ❝❤♦ (x − a)m ✈➔ f (x) ❦❤æ♥❣ ❝❤✐❛ ❤➳t ❝❤♦ (x − a)m+1 ✳ ●✐↔ sû A ởt trữớ ợ r trữớ ủ m = t❤➻ t❛ ❣å✐ a ❧➔ ♥❣❤✐➺♠ ✤ì♥ ❝á♥ ❦❤✐ m = t❤➻ a ✤÷đ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ❦➨♣✳ ❙è ♥❣❤✐➺♠ ❝õ❛ ♠ët ✤❛ t❤ù❝ ❧➔ tê♥❣ sè ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ✤❛ t❤ù❝ ✤â ❦➸ ❝↔ ❜ë✐ ❝õ❛ ❝→❝ ♥❣❤✐➺♠ ✭♥➳✉ ❝â✮✳ ❱➻ ✈➟②✱ ♥❣÷í✐ t❛ ❝♦✐ ♠ët ✤❛ t❤ù❝ ❝â ♠ët ♥❣❤✐➺♠ ❜ë✐ ❝➜♣ m ♥❤÷ ởt tự õ m trũ ữủ ỗ ❍♦r♥❡r ●✐↔ sû f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 A[x] ợ A ởt trữớ õ tữỡ ❣➛♥ ✤ó♥❣ ❝õ❛ ♠ët ✤❛ t❤ù❝ ❝â ❜➟❝ ❜➡♥❣ n − 1✱ f (x) ❝❤♦ (x − a) ❧➔ ❝â ❞↕♥❣ q(x) = bn−1 xn−1 + · · · + b1 x + b0 , tr♦♥❣ ✤â bn−1 = an , bk = abk+1 + ak+1 , k = 0, , n − 2, ✈➔ ❞÷ sè r = ab0 + a0 ỵ ✭✣à♥❤ ❧➼ ❱✐➧t❡✮ ❛✳ ●✐↔ sû ♣❤÷ì♥❣ tr➻♥❤ an xn + an−1 xn−1 + · · · + a1 x + a0 = (an = 0) ❝â n ♥❣❤✐➺♠ ✭t❤ü❝ ❤♦➦❝ ♣❤ù❝✮ x1 , x2 , , xn t❤➻ E1 (x) := x1 + x2 + · · · + xn E2 (x) := x1 x2 + x1 x3 + · · · + xn−1 xn En (x) := x1 x2 xn ❜✳ ◆❣÷đ❝ ❧↕✐ ♥➳✉ ❝→❝ sè x1 , x2 , , xn ✭✶✳✷✮ an−1 =− an an−2 = an a0 = (−1)n an ✭✶✳✸✮ t❤ä❛ ♠➣♥ ❤➺ tr➯♥ t❤➻ ❝❤ó♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮✳ ❍➺ ✭✶✳✸✮ ❝â k t❤➔♥❤ ♣❤➛♥ t❤ù k ❝â Cn sè ❤↕♥❣✳ n t❤➔♥❤ ♣❤➛♥ ✈➔ ð ✈➳ tr→✐ ❝õ❛ E1 (x), E2 (x), , En (x) ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ✭✤❛ t❤ù❝✮ ✤è✐ ①ù♥❣ ❜➟❝ 1, 2, , n✱ t÷ì♥❣ ù♥❣✳ ❝✳ ❈→❝ ❤➔♠ sỡ t ỵ ộ tự tỹ ❜➟❝ n ✤➲✉ ❝â ❦❤æ♥❣ q✉→ n ♥❣❤✐➺♠ t❤ü❝✳ ✻ ❍➺ q✉↔ ✶✳✶✳ ✣❛ t❤ù❝ ❝â ✈æ sè ♥❣❤✐➺♠ ❧➔ ✤❛ t❤ù❝ ❦❤æ♥❣✳ ❍➺ q✉↔ ✶✳✷✳ ◆➳✉ ✤❛ t❤ù❝ ❝â ❜➟❝ ≤ n ♠➔ ♥❤➟♥ ❝ị♥❣ ♠ët ❣✐→ trà ♥❤÷ ♥❤❛✉ t↕✐ n+1 ✤✐➸♠ ♣❤➙♥ ❜✐➺t ❝õ❛ ✤è✐ sè t❤➻ ✤â ❧➔ ✤❛ t❤ù❝ ❤➡♥❣✳ ❍➺ q✉↔ ✶✳✸✳ ❍❛✐ ✤❛ t❤ù❝ ❜➟❝ ≤ n ♠➔ ♥❤➟♥ n + trò♥❣ ♥❤❛✉ t↕✐ n + ✤✐➸♠ ♣❤➙♥ ❜✐➺t ❝õ❛ ✤è✐ số t ú ỗ t ỵ ▼å✐ ✤❛ t❤ù❝ f (x) ∈ R[x] ❝â ❜➟❝ n ✈➔ ❝â ❤➺ sè ❝❤➼♥❤ ✭❤➺ sè an = ❝❛♦ ♥❤➜t✮ ✤➲✉ ❝â t❤➸ ♣❤➙♥ t➼❝❤ ✭❞✉② ♥❤➜t✮ t❤➔♥❤ ♥❤➙♥ tû ❞↕♥❣ m s i=1 ✈ỵ✐ (x2 + bk x + ck ) (x − di ) f (x) = an k=1 di , bk , ck ∈ R✱ 2s + m = n, b2k − 4ck < 0, s, m, n ∈ N∗ ✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ✶✮ ▼å✐ ♥❣❤✐➺♠ x0 ❝õ❛ ✤❛ t❤ù❝ ✭✶✳✶✮ ✤➲✉ t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝ |x0 | ≤ + ✷✮ ◆➳✉ am A , |a0 | A = max |ak | 1≤k≤n ❧➔ ❤➺ sè ➙♠ ✤➛✉ t✐➯♥ ❝õ❛ ✤❛ t❤ù❝ ✭✶✳✶✮ t❤➻ sè n 1+ ❝➟♥ tr➯♥ ❝õ❛ ❝→❝ ♥❣❤✐➺♠ ❞÷ì♥❣ ❝õ❛ ✤❛ t❤ù❝ ✤➣ ❝❤♦✱ tr♦♥❣ ✤â B B am ❧➔ ❧➔ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ♠ỉ✤✉♥ ❝→❝ số fn (x) t ữợ fn (x) = g(x)q(x) ✈ỵ✐ deg(g) > ✈➔ deg(q) > t t õ g ữợ fn (x) ✈➔ t❛ ✈✐➳t g(x)|fn (x) ✳ ❤❛② fn (x)✳✳g(x)✳ ◆➳✉ g(x)|f (x) ✈➔ g(x)|h(x) t❤➻ t❛ ♥â✐ g(x) ❧➔ ÷ỵ❝ ❝❤✉♥❣ ❝õ❛ f (x) ✈➔ h(x)✳ ◆➳✉ ❤❛✐ ✤❛ tự f (x) h(x) õ ữợ ❝→❝ ✤❛ t❤ù❝ ❜➟❝ t❤➻ t❛ ♥â✐ r➡♥❣ ❝❤ó♥❣ ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉ ✈➔ ✈✐➳t (f (x), h(x)) = tự ỵ ❝➛♥ ✈➔ ✤õ ✤➸ ❤❛✐ ✤❛ t❤ù❝ f (x) ✈➔ h(x) tố ũ tỗ t t❤ù❝ u(x) ✈➔ v(x) s❛♦ ❝❤♦ f (x)u(x) + h(x)v(x) ≡ ❚➼♥❤ ❝❤➜t ✶✳✶✳ ❝→❝ ✤❛ t❤ù❝ g(x)h(x) ◆➳✉ ❝→❝ ✤❛ t❤ù❝ f (x) h(x) ✈➔ g(x) ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉ ✈➔ ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉ t❤➻ ❝→❝ ✤❛ t❤ù❝ f (x) ✈➔ ❝ơ♥❣ ♥❣✉②➯♥ tè ❝ị♥❣ ♥❤❛✉✳ ❚➼♥❤ ❝❤➜t ✶✳✷✳ f (x)h(x) ✈➔ f (x) ❝❤✐❛ ❤➳t ❝❤♦ ❝❤✐❛ ❤➳t ❝❤♦ g(x)✳ f (x), g(x), h(x) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ g(x)✱ g(x) ✈➔ h(x) ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉ t❤➻ f (x) ◆➳✉ ❝→❝ ✤❛ t❤ù❝ ✼ ❚➼♥❤ ❝❤➜t ✶✳✸✳ ◆➳✉ ✤❛ t❤ù❝ ✈ỵ✐ ♥❣✉②➯♥ tè ❝ị♥❣ ♥❤❛✉ t❤➻ g(x) h(x) ✈➔ ❚➼♥❤ ❝❤➜t ✶✳✹✳ m [f (x)] ✈➔ f (x) ◆➳✉ ❝→❝ ✤❛ t❤ù❝ n [g(x)] ❝❤✐❛ ❤➳t ❝❤♦ ❝→❝ ✤❛ t❤ù❝ f (x) f (x) g(x) ✈➔ ❝❤✐❛ ❤➳t ❝❤♦ g(x) ✈➔ h(x) g(x)h(x) ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉ t❤➻ s➩ ♥❣✉②➯♥ tè ❝ị♥❣ ♥❤❛✉ ✈ỵ✐ ♠å✐ m, n ♥❣✉②➯♥ ❞÷ì♥❣✳ ▼ët sè ❜➜t ✤➥♥❣ t❤ù❝ ✤↕✐ sè ❝ì ❜↔♥ ❚r♦♥❣ ♣❤➛♥ ♥➔② tr➻♥❤ ❜➔② ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❝ì ỵ sỷ t tự ❣✐ú❛ tr✉♥❣ ❜➻♥❤ ❝ë♥❣ ✈➔ tr✉♥❣ ❜➻♥❤ ♥❤➙♥✮ x1 , x2 , , xn ❧➔ ❝→❝ sè ❦❤æ♥❣ ➙♠✳ ❑❤✐ ✤â √ x1 + x2 + · · · + xn ≥ n x1 x2 xn n ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✭✶✳✹✮ x1 = x2 = = xn ❇➜t ✤➥♥❣ t❤ù❝ ✭✶✳✹✮ ❝â tr♦♥❣ ♥❤✐➲✉ t➔✐ ❧✐➺✉ ❜➡♥❣ t✐➳♥❣ ❱✐➺t ✈➔ ✤÷đ❝ ❣å✐ ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ❈æs✐ ✭❈❛✉❝❤②✮✳ ❚✉② ♥❤✐➯♥✱ tr♦♥❣ ❝→❝ t➔✐ ữợ t tự tr õ t t ❆♥❤ ❧➔ ✏❆▼✲●▼ ■♥❡q✉❛❧✐t②✑✱ ❝❤♦ ♥➯♥ ✈➲ s❛✉✱ t❛ ❣å✐ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✹✮ ❧➔ ✑❇➜t ✤➥♥❣ t❤ù❝ ❣✐ú❛ tr✉❣ ❜➻♥❤ ❝ë♥❣ ✈➔ tr✉♥❣ ❜➻♥❤ ♥❤➙♥✑✳ ❇➜t ✤➥♥❣ t❤ù❝ ✭✶✳✹✮ ❦❤→ q✉❡♥ t❤✉ë❝ ✈ỵ✐ ✤❛ sè ❜↕♥ ✤å❝ ✈➔ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ ♥❤✐➲✉ t➔✐ ❧✐➺✉ ❜➡♥❣ t✐➳♥❣ ❱✐➺t✱ ♥➯♥ ❝❤ó♥❣ tỉ✐ s➩ ❦❤ỉ♥❣ tr➻♥❤ ❜➔② ❝❤ù♥❣ ♠✐♥❤ ♠➔ ❝❤➾ ①➨t ✈➼ ❞ö →♣ ❞ö♥❣✳ ❱➼ ❞ö ✶✳✶✳ ❈❤♦ ❝→❝ sè ❦❤æ♥❣ ➙♠ x, y, z ✳ ❈❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ x y z + + ≥ x1/2 y 1/3 z 1/6 ▲í✐ ❣✐↔✐✳ t tự tữỡ ữỡ ợ 3x + 2y + z ≥ 6 x3 y z ❚❛ ✈✐➳t ✈➳ tr→✐ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ð ❞↕♥❣ 3x + 2y + z x+x+x+y+y+z = 6 ❚❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ❣✐ú❛ tr✉♥❣ ❜➻♥❤ ❝ë♥❣ ✈➔ tr✉♥❣ ❜➻♥❤ ♥❤➙♥ t❛ ❝â 3x + 2y + z x+x+x+y+y+z = ≥ 6 ❇➜t ✤➥♥❣ t❤ù❝ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ x3 y z ✸✹ ❑❤✐ ✤â deg Q = ∃i ∈ {1, 2, 3} s❛♦ ❝❤♦ P (i + 1) − P (i) a−1 − ≥ a−1 ♥➯♥ ❙✉② r❛ a−1 |(P (i + 1) − ai+1 ) − (P (i) − )| ≥ (a − 1) ❉♦ ✤â max {|ai+1 − P (i + 1)|} ≥ 0≤i≤3 a−1 ❱➟② max |P (i) − | ≥ 0≤i≤3 ( ❞♦ a−1 (a − 1) ≥ a−1 ) , ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❇➔✐ t♦→♥ ✷✳✹✳ ❈❤♦ ❝→❝ sè ❞÷ì♥❣ x, y, z ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ xyz ≥ (x + y − z)(z + x − y)(x + y − z) ▲í✐ ❣✐↔✐✳ ✣➦t σ1 = x + y + z, σ2 = xy + yz + zx, σ3 = xyz ❑❤✐ ✤â ❜➜t ✤➥♥❣ t❤ù❝ ❝â ❞↕♥❣ σ3 ≥ (σ1 − 2x)(σ1 − 2y)(σ1 − 2z) = σ13 − 2(x + y + z)σ12 + 2(xy + yz + xz)σ1 − 8xyz ⇔σ3 ≥ σ13 − 2σ13 + 4σ1 σ2 − 8σ3 ⇔σ13 − 4σ1 σ2 + 9σ3 ≥ ❇➜t ✤➥♥❣ t❤ù❝ ❝✉è✐ ❝ị♥❣ ✤ó♥❣ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✽✳ ❱➟② ❜➜t ✤➥♥❣ t❤ù❝ ✤➣ ❝❤♦ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ✭❞➜✉ ❜➡♥❣ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❇➔✐ t♦→♥ ✷✳✺✳ 1✳ ❈❤♦ x = y = z ✮✳ x, y, z ❧➔ ❝→❝ sè ❦❤æ♥❣ ➙♠ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ x+y+z = ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ 7(xy + yz + zx) ≤ + 9xyz ▲í✐ ❣✐↔✐✳ ❱➻ x + y + z = 1✱ ♥➯♥ t❛ ❝â t❤➸ ✈✐➳t ❧↕✐ ❜➜t ✤➥♥❣ t❤ù❝ ✤➣ ❝❤♦ ð ❞↕♥❣ 7(x + y + z)(xy + yz + zx) ≤ 2(x + y + z)3 + 9xyz ✣➦t σ1 = x + y + z, σ2 = xy + yz + zx, σ3 = xyz ự tữỡ ữỡ ợ 71 213 + 9σ3 ❚❤❡♦ ✭✷✳✽✮ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❑❤✐ ✤â ❜➜t ✤➥♥❣ t❤ù❝ ✸✺ ❇➔✐ t♦→♥ ✷✳✻✳ ●✐↔ sû a, b, c ❧➔ ❝→❝ ❝↕♥❤ ❝õ❛ ♠ët t❛♠ ❣✐→❝ ✈ỵ✐ ❞✐➺♥ t➼❝❤ S ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ▲í✐ ❣✐↔✐✳ ✣➦t √ a2 + b2 + c2 ≥ 3S a = y + z, b = z + x, c = x + y, x, y, z > 0✳ ❑❤✐ ✤â t❛ ❝â a + b + c = 2(x + y + z), a = y + z, b = z + x, c = x + y, S = (x + y + z)xyz ❉♦ ✤â ❜➜t ✤➥♥❣ t❤ù❝ ✤➣ ❝❤♦ ❝â ❞↕♥❣ (y + z)2 + (x + z)2 + (x + y)2 ✣➦t ≥ 48(x + y + z)xyz x + y + z = σ1 , xy + yz + zx = σ2 , xyz = σ3 ✳ ❑❤✐ ✤â t tự tr tữỡ ữỡ ợ (1 x)2 + (σ1 − y)2 + (σ1 − z)2 ≥ 48σ1 σ3 ⇔ 3σ12 − 2(x + y + z)σ12 + x2 + y + z 2 ≥ 48σ1 σ3 ⇔ (σ12 + s2 )2 ≥ 48σ1 σ3 ⇔ (σ12 + σ12 − 2σ2 )2 ≥ 48σ1 σ3 ⇔ (σ12 − σ2 )2 ≥ 12σ1 σ3 ⇔ σ14 − 2σ12 σ2 + σ22 − 12σ1 σ3 ≥ ❚❛ ✈✐➳t ❧↕✐ ❜➜t ✤➥♥❣ t❤ù❝ ❝✉è✐ ❝ò♥❣ ð ❞↕♥❣ (σ14 − 5σ12 σ2 + 4σ22 + 6σ1 σ3 ) + σ2 (σ12 − 3σ2 ) + 6(σ22 − 3σ1 σ3 ) ≥ ❚❤❡♦ ❝→❝ ❝æ♥❣ t❤ù❝ ✭✷✳✹✮ ✈➔ ✭✷✳✼✮✱ tø♥❣ sè ❤↕♥❣ ð ✈➳ tr→✐ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✤➲✉ ❦❤æ♥❣ ➙♠ ♥➯♥ s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ σ14 − 5σ12 σ2 + 4σ22 + 6σ1 σ3 = 0, tù❝ ❧➔ x = y = z✳ ❙✉② r❛ a = b = c✱ σ12 − 3σ2 = 0, ❤❛② ABC σ22 − 3σ1 σ3 = 0, ❧➔ t❛♠ ❣✐→❝ ✤➲✉✳ ✷✳✸ ▼ët sè ❞↕♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❜❛ ❜✐➳♥ tr♦♥❣ ♣❤➙♥ t❤ù❝ ❇➔✐ t♦→♥ ✷✳✼✳ x, y, z ❈❤♦ ❝→❝ sè ❞÷ì♥❣ (xy + yz + zx) ▲í✐ ❣✐↔✐✳ ✣➦t ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ 1 + + ≥ (x + y)2 (y + z)2 (z + x)2 σ1 = x + y + z, σ2 = xy + yz + zx, σ3 = xyz ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ trð t❤➔♥❤ σ2 σ14 − 2σ12 σ2 + σ22 + 4σ1 σ3 ≥ (σ1 σ2 − σ3 )2 ❑❤✐ ✤â✱ t tự tữỡ ữỡ ợ 414 − 17σ12 σ22 + 4σ23 + 34σ1 σ2 σ3 − 9σ32 ≥ ❚❛ ✈✐➳t ❧↕✐ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✤➙② ð ❞↕♥❣ σ1 σ2 (σ13 − 4σ1 σ2 + 9σ3 ) + σ2 (σ14 − 5σ12 σ2 + 4σ22 + 6σ1 σ3 ) + σ3 (σ1 σ2 − 9σ3 ) ≥ ❚❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ✭✷✳✺✮ ✈➔ ✭✷✳✼✮ t❤➻ ♠é✐ sè ❤↕♥❣ ð ✈➳ tr→✐ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ❧➔ ❦❤æ♥❣ ➙♠✱ ✈➻ ✈➟② ❜➜t ✤➥♥❣ t❤ù❝ ✤➣ ❝❤♦ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❇➔✐ t♦→♥ ✷✳✽✳ ❈❤♦ ❝→❝ sè ❞÷ì♥❣ x, y, z ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ x2 y2 z2 x+y+z + + ≥ y+z z+x x+y ▲í✐ ❣✐↔✐✳ ✣➦t σ1 = x + y + z, σ2 = xy + yz + zx, σ3 = xyz ❑❤✐ ✤â ❜➜t ✤➥♥❣ t❤ù❝ ✤➣ ❝❤♦ t÷ì♥❣ ữỡ ợ 2x2 (1 y)(1 z) + 2y (σ1 − x)(σ1 − z) + 2z (σ1 − x)(σ1 − y) ≥σ1 (σ1 − x)(σ1 − y)(σ1 − z) ⇔2σ12 s2 − 2σ1 O(x2 y) + 2σ1 σ3 ≥ σ1 (σ1 − σ1 σ3 ), tr♦♥❣ ✤â s2 = x2 + y + z , O(x2 y) = x2 y + x2 z + y x + y z + z x + z y t÷ì♥❣ ù♥❣ ❧➔ tê♥❣ ❧ơ② t❤ø❛ ✈➔ q✉ÿ ✤↕♦✳ ❙û ❞ư♥❣ ❝→❝ ❝ỉ♥❣ t❤ù❝ ❝õ❛ tê♥❣ ❧ô② t❤ø❛ ✈➔ q✉ÿ ✤↕♦ t❛ ❝â ❜➜t ✤➥♥❣ t❤ù❝ 2σ12 (σ12 − 2σ2 ) − 2σ1 (σ1 σ2 − 3σ3 ) + 2σ1 σ3 ≥ σ1 (σ1 σ2 − σ3 ) ⇔ σ1 (2σ13 − 7σ1 σ2 + 9σ3 ) ≥ ❇➜t ✤➥♥❣ t❤ù❝ ❝✉è✐ ❝ị♥❣ ✤ó♥❣ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✽✳ ❱➟② ❜➜t ✤➥♥❣ t❤ù❝ ✤➣ ❝❤♦ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❇➔✐ t♦→♥ ✷✳✾✳ ❈❤♦ ❝→❝ sè ❞÷ì♥❣ x = y = z a, b, c t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ abc = 1✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ 1 + + ≥ a3 (b + c) b3 (c + a) c3 (a + b) ✸✼ ▲í✐ ❣✐↔✐✳ ✣➦t a= 1 , b= , c= ✳ x y z ❚❛ ❝â xyz = = abc ❑❤✐ ✤â✱ ❜➜t ✤➥♥❣ t❤ù❝ ✤➣ ❝❤♦ t÷ì♥❣ ✤÷ì♥❣ ✈ỵ✐ y2 z2 x2 + + ≥ y+z z+x x+y ❙û ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❣✐ú❛ tr✉♥❣ ❜➻♥❤ ❝ë♥❣ ✈➔ tr✉♥❣ ❜➻♥❤ ♥❤➙♥✱ t❛ ❝â √ 3 xyz y2 z2 x+y+z x2 + + ≥ ≥ = , y+z z+x x+y 2 ♥❣❤➽❛ ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❇➔✐ t♦→♥ ✷✳✶✵✳ x + y + z = 3a✳ ❈❤♦ ❧➔ ❝→❝ số ữỡ tọ ự r ợ ♠å✐ sè tü ♥❤✐➯♥ x+ y ▲í✐ ❣✐↔✐✳ x, y, z n + y+ z n n + z+ x n t❤➻ ≥3 a+ a n ✣➦t 1 x1 = x + , x2 = y + , x3 = z + , y z x sn = xn1 + xn2 + xn3 , σ1 = x1 + x2 + x3 ❚❛ ❝â 1 + + x y z 9 ≥ (x + y + z) + = 3a + =3 a+ x+y+z 3a a σ1 = (x + y + z) + ❚❤❡♦ ✭✷✳✶✶✮✱ t❛ ❝â σ1n 1 sn ≥ n−1 ≥ n−1 a + 3 a n =3 a+ a ❚ø ✤â s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = y = z ✸✽ ❈❤÷ì♥❣ ✸✳ ❈→❝ ❞↕♥❣ t♦→♥ ❝ü❝ trà s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ♥➔② ❧➔ tr➻♥❤ ❜➔② ❝ü❝ trà t❤❡♦ r➔♥❣ ❜✉ë❝ tê♥❣ ✈➔ t➼❝❤ ❜❛ sè❀ ❝→❝ ❞↕♥❣ t♦→♥ ❝ü❝ trà s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥ ✈➔ ♠ët sè ❞↕♥❣ t♦→♥ ❧✐➯♥ q✉❛♥✳ ❈→❝ ❦➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ t❤❛♠ ❦❤↔♦ tø t➔✐ ❧✐➺✉ ❬✸❪✳ ✸✳✶ ❈ü❝ trà t❤❡♦ r➔♥❣ ❜✉ë❝ tê♥❣ ✈➔ t➼❝❤ ❜❛ sè ▼ö❝ ♥➔② ✈➟♥ ❞ö♥❣ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ✤↕✐ sè ❝ì ❜↔♥ ✈➔ ♠ët sè ♠➺♥❤ ✤➲ ❜➜t ✤➥♥❣ t❤ù❝ ✤➸ tr➻♥❤ ❜➔② ♠ët sè ✈➼ ❞ö ✈➲ ❝ü❝ trà t❤❡♦ r➔♥❣ ❜✉ë❝ tê♥❣ ✈➔ t➼❝❤ ❜❛ sè✳ ❱➼ ❞ư ✸✳✶✳ ❈❤♦ ❝→❝ sè ❞÷ì♥❣ t❤❛② ✤ê✐ x, y, z ✳ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ F (x, y, z) = + x + 1+ y + 1+ z tr♦♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉ x + y + z = ❜✮ xy + xz + zx = ❝✮ xyz = ❛✮ ▲í✐ ❣✐↔✐✳ ❚❛ ✈✐➳t ❧↕✐ ❜✐➸✉ t❤ù❝ ✤➣ ❝❤♦ ð ❞↕♥❣ (x + 1)(y + 1)(z + 1) xyz + (x + y + z) + (xy + yz + zx) + xyz = xyz F (x, y, z) = ✸✾ ✣➦t σ1 = x + y + z, σ2 = xy + yz + zx, σ3 = xyz ✱ + σ1 + σ F =1+ σ3 t❛ ❝â σ1 = x + y + z = ⑩♣ ❞ö♥❣ ▼➺♥❤ ✤➲ ✷✳✸✱ t❛ ❝â σ2 54 + ≥1+ +√ ≥ + + = 64 F =1+ 3 σ3 σ σ3 σ3 σ1 σ1 ❛✮ ❚r÷í♥❣ ❤đ♣ ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x + y + z = 1, x = y = z ⇔x=y=z= ❱➟②✱ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② t❛ ❝â 1 , , 3 F (x, y, z) = F = 64 σ2 = xy + yz + zx = 1✳ ❱➟♥ ❞ö♥❣ ▼➺♥❤ ✤➲ √ √ σ1 + ≥1+ = 10 + + F =1+ σ3 σ3 σ23 σ2 ❜✮ ❚r÷í♥❣ ❤ñ♣ ✷✳✸✱ t❛ ❝â ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ xy + yz + zx = 1, x = y = z > ⇔x=y=z= √ ❱➟②✱ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② t❛ ❝â F (x, y, z) = F ❝✮ ❚r÷í♥❣ ❤đ♣ σ3 = xyz = 1✳ 1 √ ,√ ,√ 3 √ = 10 + ❱➟♥ ❞ö♥❣ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ▼➺♥❤ ✤➲ ✷✳✸✱ t❛ ❝â F =1+ √ + σ + σ2 = + σ1 + σ2 ≥ + 3 σ3 + 3 σ32 = σ3 ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ xyz = 1, x = y = z > ⇔ x = y = z = ❱➟②✱ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② t❛ ❝â F (x, y, z) = F (1, 1, 1) = ✹✵ ❱➼ ❞ư ✸✳✷✳ ❈❤♦ ❝→❝ sè ❞÷ì♥❣ t❤❛② ✤ê✐ ❝õ❛ ❜✐➸✉ t❤ù❝ x, y, z ✳ ❍➣② t➻♠ ❣✐→ trà ♥❤ä ♥❤➜t x2 y2 z2 F (x, y, z) = + + y+z z+x x+y tr♦♥❣ ♠é✐ tr÷í♥❣ ❤đ♣ s❛✉ x + y + z = ❜✮ xy + xz + zx = ❝✮ xyz = ❛✮ ▲í✐ ợ số ữỡ x, y, z ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲ ❙❝❤✇❛r③✱ t❛ ❝â y √ z √ x √ y+z+ √ z+x+ √ x+y y+z x+y z+x x2 y2 z2 + + (y + z + z + x + x + y) y+z z+x x+y (x + y + z)2 = √ ≤ ◆❤÷ ✈➟②✱ t❛ ❝â x+y+z x = y = z✳ F (x, y, z) ≥ ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❛✮ ❚r÷í♥❣ ❤đ♣ x + y + z = ❑❤✐ ✤â✱ t❛ ❝â F (x, y, z) = F ❜✮ ❚r÷í♥❣ ❤đ♣ xy + yz + zx = 1✳ 1 , , 3 = ❚❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ t❤ù ♥❤➜t tr♦♥❣ ▼➺♥❤ ✤➲ ✷✳✸✱ t❛ ❝â (x + y + z)2 ≥ 3(xy + yz + zx) = ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ F (x, y, z) = F ❝✮ ❚r÷í♥❣ ❤đ♣ t❛ ❝â x = y = z✳ 1 √ ,√ ,√ 3 ❉♦ ✤â √ = xyz = 1✳ ❚❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ t❤ù ❤❛✐ tr♦♥❣ ▼➺♥❤ ✤➲ ✷✳✸✱ √ x + y + z ≥ xyz = ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = y = z✳ ❱➟②✱ t❛ ❝â F (x, y, z) = F (1, 1, 1) = ✹✶ ❱➼ ❞ư ✸✳✸✳ ❳➨t ❝→❝ sè t❤ü❝ ❞÷ì♥❣ x, y, z t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ (x + y + z)3 = 32xyz ❍➣② t➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ✈➔ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ x4 + y + z P (x, y, z) = (x + y + z)4 ▲í✐ ❣✐↔✐✳ ◆❤➟♥ ①➨t r➡♥❣ ợ ởt số tỹ ổ tũ ỵ t❛ ❧✉æ♥ ❝â P (αx, αy, αz) = P (x, y, z), ✈➔ ♥➳✉ x, y, z t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ✤➲ ❜➔✐✱ t❤➻ αx, αy, αz ❝ô♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✤â✳ ❱➻ t❤➳ ❦❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ ❝â t❤➸ ❣✐↔ t❤✐➳t r➡♥❣ x + y + z = 4✳ ❑❤✐ ✤â✱ ❦➳t ❤đ♣ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❝õ❛ ✤➲ ❜➔✐✱ t❛ ❝â xyz = 2✳ ❇➔✐ t♦→♥ trð t❤➔♥❤✿ ✏❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ✈➔ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ P = (x4 + y + z ) 256 ❦❤✐ ❝→❝ ❜✐➳♥ sè ❞÷ì♥❣ x, y, z t❤❛② ✤ê✐✱ s❛♦ ❝❤♦ 4 ✣➦t Q = x + y + z ✈➔ σ1 = x + y + z, σ2 x + y + z = 4, xyz = 2.✑ = xy + yz + zx, σ3 = xyz ❚❤❡♦ ❝æ♥❣ t❤ù❝ ❲❛r✐♥❣✱ t❛ ❝â Q = σ14 − 4σ12 σ2 + 2σ22 + 4σ1 σ3 = 256 − 64σ22 + 32 = 2(σ22 − 32σ2 + 144) ❚ø ❝→❝ ✤✐➲✉ ❦✐➺♥ ❝õ❛ x, y, z ✱ t❛ ❝â σ2 = x(y + z) + yz = x(4 − x) + x ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ (y + z)2 ≥ 4yz ✱ t❛ ❝â ⇔ x3 − 8x2 + 16x − ≥ ⇔ (x − 2)(x2 − 6x + 4) ≥ x √ x ∈ (0.4)✱ ♥➯♥ tø ✤➙② s✉② r❛ − ≤ x ≤ (4 − x)2 ≥ ❱➻ ✸✳✷ ❈→❝ ❞↕♥❣ t♦→♥ ❝ü❝ trà s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥ ▼ö❝ ♥➔② ♥➯✉ ♠ët sè ❜➔✐ t♦→♥ ❝ü❝ trà s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥✳ ❱➼ ❞ö ✸✳✹✳ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ F (x, y, z) = x y z y+z z+x x+y + + + + + y+z z+x x+y x y z ✹✷ tr➯♥ ♠✐➲♥ D = {(x, y, z) : x > 0, y > 0, z > 0} rữợ t t õ y+z z+x x+y + + x y z y z z x x y + + + + + ≥ + + = = y x z y x z P (x, y, z) = ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = y = z✳ ❚✐➳♣ t❤❡♦✱ ①➨t ❜✐➸✉ t❤ù❝ Q(x, y, z) = x y z + + y+z z+x x+y ❚❛ ❜✐➳♥ ✤ê✐ ❜✐➸✉ t❤ù❝ ♥➔② ♥❤÷ s❛✉ y z +1 + +1 z+x x+y 1 = (x + y + z) + + y+z z+x x+y 1 1 = [(y + z)(z + x)(x + y)] + + y+z z+x x+y Q+3= x +1 + y+z ❚❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ❣✐ú❛ tr✉♥❣ ❜➻♥❤ ❝ë♥❣ ✈➔ tr✉♥❣ ❜➻♥❤ ♥❤➙♥✱ t❛ ❝â [(y + z)(z + x)(x + y)] 1 + + y+z z+x x+y ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❙✉② r❛ Q≥ ≥ y +z = z +x = x+y ⇔ x = y = z ✈➔ ❞♦ ✤â 15 = 2 ❞ö x = y = z = 1✳ F (x, y, z) = P (x, y, z) + Q(x, y, z) ≥ + ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = y = z✱ ❱➟② t❛ ❝â F (x, y, z) = ❱➼ ❞ö ✸✳✺✳ ❦✐➺♥ D ❈→❝ sè ❞÷ì♥❣ xy + yz + zx = 1✳ x, y, z ✈➼ 15 t❤❛② ✤ê✐✱ ♥❤÷♥❣ ❧✉ỉ♥ ❧✉ỉ♥ t❤ä❛ ♠➣♥ ✤✐➲✉ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ x3 y3 z3 F (x, y, z) = + + y + z z + x2 x2 + y ✹✸ ▲í✐ ❣✐↔✐✳ ❚❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ❈❛✉❝❤②✲❙❝❤✇❛r③✱ t❛ ❝â 2 2 (x + y + z ) = x3/2 y2 + z2 x3 y3 z3 + + y + z z + x2 x2 + y ≤ x(y + z2) y 3/2 +√ z + x2 + z 3/2 x2 + y y(z + x2 ) z(x2 + y ) x(y + z ) + y(z + x2 ) + z(x2 + y ) ❙✉② r❛ y3 z3 (x2 + y + z )2 x3 + + ≥ y + z z + x2 x2 + y x(y + z ) + y(z + x2 ) + z(x2 + y ) ❈❤ó♥❣ t❛ s➩ ❝❤ù♥❣ tä (x2 + y + z )2 x+y+z ≥ x(y + z ) + y(z + x2 ) + z(x2 + y ) ỵ , , σ3 ❧➔ ❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ì sð ❝õ❛ ❝→❝ ❜✐➳♥ x, y, z ✱ t❤➻ ❜➜t ✤➥♥❣ t❤ù❝ tr tữỡ ữỡ ợ (12 22 )2 ⇔ 2σ14 − 8σ12 σ2 + 8σ22 ≥ σ12 σ2 − 3σ1 σ3 σ1 σ2 − 3σ3 ⇔ 2(σ1 + 4σ2 + 6σ1 σ3 − 5σ12 σ2 ) + σ1 (σ1 σ2 − 9σ3 ) ≥ ❇➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✤➙② ✤ó♥❣ tr➯♥ ❝ì sð ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ tr♦♥❣ ❝→❝ ▼➺♥❤ ✤➲ ✷✳✸ ✈➔ ▼➺♥❤ ✤➲ ✷✳✼✳ ◆❤÷ ✈➟②✱ t❛ ❝â y3 z3 x+y+z x3 + + ≥ F (x, y, z) = y + z z + x2 x2 + y 2 ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = y = z✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✸✱ t❛ ❝â (x + y + z)2 ≥ 3(xy + yz + zx) = ❉♦ ✤â t❛ ❝â ❦➳t q✉↔ F = F ❱➼ ❞ö ✸✳✻✳ ❈❤♦ x, y, z ∈ R✳ 1 √ ,√ ,√ 3 √ = ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝✿ S = 19x2 + 54y + 16z − 16xz − 24y + 36xy ✹✹ ▲í✐ ❣✐↔✐✳ ❇✐➳♥ ✤ê✐ S ⇔ f (x) = 19x2 − 2(8z − 18y)x + 54y + 16z − 24y ❚❛ ❝â x = g(y) = (8z −18y)2 −(54y +16z −24y) = −702y +168zy −240z ⇒ y = (84z)2 − 702.240z = −161424z ≤ ∀z ∈ R g(y) ≤ ∀y, z ∈ R✳ ❱➟② x ≤ ∀y, z ∈ R ⇒ f (x) ≥ 0✳ ❱ỵ✐ x = y = z = t❤➻ S = ❙✉② r❛ ❱➼ ❞ö ✸✳✼✳ ❚➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ F (a, b, c) = (a − b)4 + (b − c)4 + (c − a)4 , tr♦♥❣ ✤â a, b, c ▲í✐ ❣✐↔✐✳ ✣➦t ❧➔ ❝→❝ sè t❤ü❝ ❦❤ỉ♥❣ ❜➨ ❤ì♥ ✈➔ ❦❤ỉ♥❣ ❧ỵ♥ ❤ì♥ x = a − b, y = b − c, z = c − a −1 ≤ x, y, z ≤ 1, ❑➼ ❤✐➺✉ σ1 , σ2 , σ3 2✳ ❑❤✐ ✤â x + y + z = ❧➔ ❝→❝ ✤❛ t❤ù❝ ✤è✐ ①ù♥❣ ❝ì sð ❝õ❛ ❝→❝ ❜✐➳♥ x, y, z ✱ ♥❣❤➽❛ ❧➔ σ + = x + y + z, σ2 = xy + yz + zx, σ3 = xyz ❚❤❡♦ ❝æ♥❣ t❤ù❝ ❲❛r✐♥❣✱ t❛ ❝â F = x4 + y + z = σ14 − 4σ12 σ2 + 2σ22 + 4σ1 σ3 = 2σ22 = 2(xy + yz + zx)2 ❚❛ ❝â (1 − x)(1 − y)(1 − z) ≥ ⇔ − (x + y + z) + xy + yz + zx − xyz ≥ ⇔ + xy + yz + zx − xyz ≥ ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛✱ ✈➼ ❞ư ✈ỵ✐ x = 1, y = −1, z = ✭✸✳✶✮ ❚❛ ❝ô♥❣ ❝â ❚❛ ❝â (1 + x)(1 + y)(1 + z) ≥ ⇔ + x + y + z + xy + yz + zx + xyz ≥ ⇔ + xy + yz + zx + xyz ≥ ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛✱ ✈➼ ❞ư ✈ỵ✐ x = 1, y = −1, z = ❈ë♥❣ t❤❡♦ tø♥❣ ✈➳ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ✭✸✳✶✮ ✈➔ ✭✸✳✷✮✱ t❛ ✤÷đ❝ + 2(xy + yz + zx) ≥ ⇔ xy + yz + zx ≥ −1 ▼➦t ❦❤→❝✱ ❧↕✐ ❝â 3(xy + yz + zx) ≤ (x + y + z)2 = ✭✸✳✷✮ ✹✺ ❙✉② r❛ ≥ xy + yz + zx ≥ −1 ⇒ (xy + yz + zx)2 ≤ ❱➟②✱ t❛ ❝â F = 2(xy + yz + zx)2 ≤ ⇒ max F = 2, ❦❤✐ x = 1, y = −1, z = ❚❛ ❝â ❦➳t q✉↔ max F = ✤↕t t↕✐ a = 1, b = 0, c = ❱➼ ❞ö ✸✳✽✳ ❈❤♦ x, y, z > 0✳ S= ▲í✐ ❣✐↔✐✳ ❚➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ xyz(x + y + z + x2 + y + z ) (x2 + y + z )(xy + yz + zx) ❙û ❞ư♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❈ỉs✐ ✈➔ ❇✉♥❤✐❛❈ỉ♣s❦✐ t❛ ❝â ❝→❝ ✤→♥❤ ❣✐→ s❛✉✿ x2 + y + z ≥ 3 x2 y z ; √ xy + yz + zx ≥ 3 xy.yz.zx = 3 x2 y z ; √ x + y + z ≤ (12 + 12 + 12 )(x2 + y + z ) = x2 + y + z ❚ø ✤â s✉② r❛ ❱➟② √ xyz(1 + 3) x2 + y + z S≤ (x2 + y + z )3 x2 y z √ √ xyz 1+ = x2 + y + z √ √ √ xyz 1+ 3+ ≤ √ √ = 3 xyz √ 3+ max S = ✸✳✸ ▼ët sè ❞↕♥❣ t♦→♥ ❧✐➯♥ q✉❛♥ ❇➔✐ ✸✳✶✳ x, y, z ❈❤♦ ❝→❝ sè ❞÷ì♥❣ t❤❛② ✤ê✐ t❤ù❝ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ x4 y4 z4 F (x, y, z) = + + x + y y + z z + x4 ✹✻ tr♦♥❣ ♠é✐ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤➙② ❛✮ x + y + z = 1✳ ❇➔✐ ✸✳✷✳ ❜✮ ❈→❝ sè ❞÷ì♥❣ xy + yz + zx = 1✳ a, b, c t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝✮ xyz = 1✳ ab + bc + ca = ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ ❇➔✐ ✸✳✸✳ bc ca ab + + a b c ❈❤♦ ❝→❝ sè t❤ü❝ x, y, z t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ x(x − 1) + y(y − 1) + z(z − 1) ≤ ❚➻♠ ❣✐→ trà ❜➨ ♥❤➜t ❝õ❛ ❝→ ❤➔♠ sè ❛✮ f (x, y, z) = x + y + z ✳ ❇➔✐ ✸✳✹✳ ❈→❝ sè ❦❤æ♥❣ ➙♠ ❜✮ x, y, z g(x, y, z) = x2 + y + z ✳ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ x2 + y + z ≤ ❚➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❤➔♠ sè f (x, y, z) = x + y + z + xy + yz + zx + xyz ❇➔✐ ✸✳✺✳ ❈→❝ sè ❞÷ì♥❣ x, y, z t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ x + y + z ≤ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ F (x, y, z) = ❇➔✐ ✸✳✻✳ 18 11 + + 18(xy + yz + zx) x2 + y + z xy + yz + zx ❚➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ P = (x − y)(y − z)(z − x)(x + y + z), tr♦♥❣ ✤â x, y, z ❇➔✐ ✸✳✼✳ ❚➻♠ ❣✐→ trà ♥❤ä ♥❤➜t ✈➔ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ ❜✐➸✉ t❤ù❝ ❧➔ ❝→❝ sè t❤ü❝ t❤✉ë❝ ✤♦↕♥ P = tr♦♥❣ ✤â x, y, z [0; 1] x+y y+z z+x + + , 1+z 1+x 1+y ❧➔ ❝→❝ sè t❤ü❝ t❤✉ë❝ ✤♦↕♥ ;1 ✹✼ ❑➳t ❧✉➟♥ ✣➲ t➔✐ ❧✉➟♥ ✈➠♥ ✤➣ ✤➲ ❝➟♣ ✤➳♥ ✈✐➺❝ ❦❤↔♦ s→t ♠ët sè ❧ỵ♣ ❜➜t ✤➥♥❣ t❤ù❝ ✈➔ ❝ü❝ trà s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥✳ ỗ r tự ❝→❝ ❤➺ t❤ù❝ ❧✐➯♥ q✉❛♥✳ ✭✷✮ ❚r➻♥❤ ❜➔② ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥✳ ✭✸✮ ❚r➻♥❤ ❜➔② ❝→❝ ❞↕♥❣ t♦→♥ ❝ü❝ trà s✐♥❤ ❜ð✐ ❝→❝ ✤❛ t❤ù❝ ✤↕✐ sè ❜❛ ❜✐➳♥✳ ✹✽ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬❆❪ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ◆❣✉②➵♥ ❱➠♥ ▼➟✉ t tự ỵ ◆❳❇ ●✐→♦ ❞ö❝✳ ❬✷❪ ◆❣✉②➵♥ ❱➠♥ ▼➟✉ ✭✷✵✵✺✮✱ ✣❛ t❤ù❝ ✤↕✐ sè ✈➔ ♣❤➙♥ t❤ù❝ ❤ú✉ t✛✱ ◆❳❇ ●✐→♦ ❞ö❝✳ ❬✸❪ ◆❣✉②➵♥ ❱➠♥ ▼➟✉✱ ◆❣✉②➵♥ ❱➠♥ ◆❣å❝ ✭✷✵✵✾✮✱ ✣❛ t❤ù❝ ✤è✐ ①ù♥❣ ✈➔ →♣ ❞ö♥❣✱ ◆❳❇ ●✐→♦ ❞ö❝✳ ... ❝ò♥❣ ♥❤❛✉ t❤➻ s➩ ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉ ợ m, n ữỡ ởt số t t❤ù❝ ✤↕✐ sè ❝ì ❜↔♥ ❚r♦♥❣ ♣❤➛♥ ♥➔② tr➻♥❤ ❜➔② ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥ ❝→❝ ✤❛ t❤ù❝ số ỡ ỵ sỷ ✭❇➜t ✤➥♥❣ t❤ù❝ ❣✐ú❛ tr✉♥❣ ❜➻♥❤ ❝ë♥❣... t= 1 d+ d = m+ √ m2 + + m õ ỵ ỡ tr t s ỵ ỵ số ❚r♦♥❣ t❛♠ ❣✐→❝ ABC ✳ sin✮ t❛ ❧✉æ♥ ❝â a b c = = = 2R sin A sin B sin C m2 + ỵ ỵ số s r t ❆❇❈ t❛ ❧✉æ♥ ❝â a2 = b2 + c2 − 2bc cos A;... ❦❤→❝ ♥❤❛✉ ❝õ❛ ❜ë sè a✮✳ ✣➦t ỵ r!(n r)! Er (a) n! sỷ x1 , x2 , , xn ❧➔ n số tỹ ổ ỵ y1 , y2 , , yn ❧➔ ❜ë ❝→❝ số tỹ ổ ữủ ỵ Pr (a) = ✣à♥❤ ♥❣❤➽❛ ✷✳✽✳ ❜ð✐ ❜ð✐ (x)✮ ✈➔ (y)✮✳ (x) (y) ữủ