ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC - NGUYỄN THÀNH CÔNG KHAI THÁC MỐI QUAN HỆ HÌNH HỌC - ĐẠI SỐ VÀO GIẢI MỘT SỐ BÀI TOÁN DÀNH CHO HỌC SINH GIỎI LUẬN VĂN THẠC SĨ TOÁN HỌC THÁI NGUYÊN - 2019 ĐẠI HỌC THÁI NGUYÊN TRƯỜNG ĐẠI HỌC KHOA HỌC - NGUYỄN THÀNH CƠNG KHAI THÁC MỐI QUAN HỆ HÌNH HỌC - ĐẠI SỐ VÀO GIẢI MỘT SỐ BÀI TOÁN DÀNH CHO HỌC SINH GIỎI Chuyên ngành: Phương pháp Toán sơ cấp Mã số: 46 01 13 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC PGS.TS Trịnh Thanh Hải THÁI NGUN - 2019 ✐ ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ✶ ▲í✐ ♥â✐ ✤➛✉ ✶ ✶ ❑❤❛✐ t❤→❝ ❦✐➳♥ t❤ù❝ ✣↕✐ sè ✤➸ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥ ị tữ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳ ▼ët sè ✈➼ ❞ö ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✶✳ ❇➔✐ t♦→♥ ❝ü❝ trà ❍➻♥❤ ❤å❝ ✶✳✷✳✷✳ ❇➔✐ t♦→♥ q✉ÿ t➼❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✳ ✸ ✳ ✸ ✳ ✶✹ ✷ ❑❤❛✐ t❤→❝ ❦✐➳♥ t❤ù❝ ❍➻♥❤ ❤å❝ ✤➸ ❣✐↔✐ ởt số t số ị tữ ❝❤✉♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳ ▼ët sè ✈➼ ❞ö ♠✐♥❤ ❤å❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✶✳ ❇➔✐ t♦→♥ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✷✳ ❇➔✐ t♦→♥ ❜✐➺♥ ❧✉➟♥ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝â t❤❛♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✸✳ ❇➔✐ t♦→♥ ❝❤ù♥❣ ♠✐♥❤ ❜➜t ✤➥♥❣ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✹✳ ❇➔✐ t♦→♥ ❝ü❝ trà ✤↕✐ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑➳t ❧✉➟♥ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✶✾ ✳ ✶✾ ✳ ✶✾ ✳ ✸✼ ✳ ✹✸ ✳ ✺✵ ✺✽ ✺✾ ✶ ▲í✐ ❝↔♠ ì♥ ❚r♦♥❣ s✉èt q✉→ tr➻♥❤ ❧➔♠ ❧✉➟♥ ✈➠♥✱ tæ✐ ❧✉æ♥ ♥❤➟♥ ữủ sỹ ữợ ú ù P●❙✳ ❚❙✳ ❚rà♥❤ ❚❤❛♥❤ ❍↔✐✳ ❚❤➛② ❧✉æ♥ q✉❛♥ t➙♠✱ t❤❡♦ ❞ã✐ s→t s❛♦✱ ❞➔♥❤ ♥❤✐➲✉ t❤í✐ ❣✐❛♥ ❝❤➾ ❜↔♦ t➟♥ t ữợ t tỉ✐✳ ❚ỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ s➙✉ s➢❝ ♥❤➜t ✤➳♥ ❚❤➛②✳ ❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❝→❝ ❚❤➛②✱ ❈ỉ ❦❤♦❛ ❚♦→♥ ✕ ❚✐♥ ✈➔ ♣❤á♥❣ ✣➔♦ ❚↕♦ ❝õ❛ tr÷í♥❣ ✣↕✐ ❍å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ❝ơ♥❣ ♥❤÷ ❝→❝ ❚❤➛② ❈ỉ t❤❛♠ ❣✐❛ ❣✐↔♥❣ ❞↕② ❦❤â❛ ❤å❝ ❝❛♦ ❤å❝ ✷✵✶✼ ✕ ✷✵✶✾ ✤➣ t➟♥ t➻♥❤ ❝❤➾ ❜↔♦ tr✉②➲♥ ✤↕t ❦✐➳♥ t❤ù❝ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ t❤❡♦ ❤å❝✱ t❤ü❝ ❤✐➺♥ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❈✉è✐ ❝ị♥❣✱ tỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝→♠ ỡ tợ ỗ ổ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï✱ ❧➔ ❝❤é ❞ü❛ ✈ú♥❣ ❝❤➢❝ ✈➲ ✈➟t ❝❤➜t ✈➔ t✐♥❤ t❤➛♥ ❝❤♦ tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥ t❤↕❝ sÿ✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✶✵ ♥➠♠ ✷✵✶✾ ❚→❝ ❣✐↔ ◆❣✉②➵♥ ❚❤➔♥❤ ổ õ ỵ t➔✐ ❍➻♥❤ ❤å❝ ✈➔ ✣↕✐ sè ❧➔ ❤❛✐ ♥ë✐ ❞✉♥❣ q✉❛♥ trå♥❣ ①✉②➯♥ s✉èt ❝❤÷ì♥❣ tr➻♥❤ t♦→♥ ❚❍❈❙ ✲ ❚❍P❚ ❣â♣ ♣❤➛♥ ❝➜✉ t❤➔♥❤ ♥➯♥ ❜ë ♠æ♥ ❚♦→♥ ❤å❝✳ ❉♦ ✤â✱ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❦❤❛✐ t❤→❝ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❍➻♥❤ ❤å❝ ✈➔ ✣↕✐ sè ❧➔ ♠ët ✈➜♥ ✤➲ r➜t q t ỗ tớ tổ q õ t❛ ❝→✐ ♥❤➻♥ tê♥❣ t❤➸ ❤ì♥✱ ❣â♣ ♣❤➛♥ ❣✐ó♣ ❝❤ó♥❣ t❛ ❤✐➸✉ rã ❤ì♥ ✈➲ ❚♦→♥ ❤å❝ ❝ơ♥❣ ♥❤÷ ❣✐ó♣ ➼❝❤ ❝❤♦ ✈✐➺❝ ❞↕② ✈➔ ❤å❝ ❜ë ♠æ♥ ❚♦→♥ ❤å❝✳ t t ữợ ♣❤→t tr✐➸♥ tr➯♥ t❤➳ ❣✐ỵ✐ r➜t q✉❛♥ t➙♠ ❝❤ó trå♥❣ ✈✐➺❝ ❞↕② ✈➔ ❤å❝ ❧✐➯♥ ♠ỉ♥✿ ❣✐ú❛ ❝→❝ ♠ỉ♥ ✈ỵ✐ ♥❤❛✉ ✈➔ ❣✐ú❛ ❝→❝ ♣❤➙♥ ♠ỉ♥ tr♦♥❣ ❝ị♥❣ ♠ët ♠ỉ♥ ❤å❝✳ ◆➲♥ ❣✐→♦ ❞ư❝ ❝õ❛ ❱✐➺t ◆❛♠ ❦❤ỉ♥❣ ♥➡♠ ♥❣♦➔✐ ữợ tớ ♠➻♥❤ t✐➳♣ ❝➟♥ ❤å❝ ❤ä✐✱ s→♥❣ t↕♦ ✈➔ ù♥❣ ❞ö♥❣ ữợ ữỡ tr tr trữớ ❚❍❈❙ ✲ ❚❍P❚ ❤✐➺♥ ♥❛②✱ ❜ë ♠æ♥ ❚♦→♥ ❝❤ù❛ ❤❛✐ ♠↔♥❣ rã r➺t✿ ♣❤➛♥ ✶ ❧➔ ✣↕✐ sè✱ ♣❤➛♥ ✷ ❧➔ ❍➻♥❤ ❤å❝✳ ✣✐➲✉ ♥➔② ❝â ♠➦t t➼❝❤ ❝ü❝ ❧➔ ❣✐ó♣ ❤å❝ s✐♥❤ ♥❤➟♥ ❜✐➳t ♥❣❛② ✤÷đ❝ ❝➜✉ tró❝ ❝❤÷ì♥❣ tr➻♥❤ ✈➔ t✐➳♣ t❤✉ ❦✐➳♥ t❤ù❝ ♠ët ❝→❝❤ ❝â ❤➺ t❤è♥❣✳ ◆❤÷♥❣ ♥❣÷đ❝ ❧↕✐✱ ♥â ❧➔♠ ❝❤♦ ❤å❝ s✐♥❤ ❤✐➸✉ r➡♥❣ ✤➙② ❧➔ ❤❛✐ ♣❤➙♥ ♠ỉ♥ ✤ë❝ ❧➟♣ ✈ỵ✐ ♥❤❛✉✱ ❦❤ỉ♥❣ ❝â ♠è✐ q✉❛♥ ❤➺ t÷ì♥❣ trđ q✉❛ ❧↕✐✱ ❝ơ♥❣ ♥❤÷ ✈✐➺❝ ❣➢♥ ❦➳t ❤❛✐ ♣❤➙♥ ♠ỉ♥ ♥➔② tr♦♥❣ s→❝❤ ❣✐→♦ ❦❤♦❛ ❚❍❈❙✲ ❚❍P❚ ❧➔ ❝❤÷❛ ✤÷đ❝ ✤➲ ❝➟♣ rã r➔♥❣ ✤➛② ✤õ✳ ❚❤ü❝ t➳ q✉→ tr➻♥❤ ❞↕② ✈➔ ❤å❝ ✤➣ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣✱ ❤å❝ s✐♥❤ ❤✐➸✉ ❜✐➳t ✈➲ ♠è✐ q số ỡ ỗ ✈➔ ❣➛♥ ♥❤÷ ❤✐➸✉ ✤➙② ❧➔ ✷ ♣❤➙♥ ♠ỉ♥ r✐➯♥❣ ❜✐➺t✱ ❣â♣ ♣❤➛♥ t↕♦ ♥➯♥ ♠æ♥ ❚♦→♥ ❤å❝✳ ❈→❝ ❡♠ ❤å❝ ♣❤➙♥ ♠æ♥ ♥➔♦ t❤➻ ❤å❝ ✈➔ ❧➔♠ ❜➔✐ t➟♣ ♣❤➙♥ ♠ỉ♥ ✤â✱ ❝ơ♥❣ ♥❤÷ ❣✐→♦ ✈✐➯♥ ❞↕② ❤å❝ t❤❡♦ t✐➳t ♠æ♥ ❍➻♥❤ ❤å❝ t❤➻ ❝❤✉②➯♥ ❧➔♠ ❜➔✐ ✈➲ ❍➻♥❤ ❤å❝✱ ✣↕✐ sè t❤➻ ❝❤✉②➯♥ ❧➔♠ ❜➔✐ ✈➲ ✣↕✐ sè✱ ➼t ❤♦➦❝ ❦❤ỉ♥❣ ❤♦➦❝ ❝❤÷❛ ❝❤ó trå♥❣ ✤➲ ❝➟♣ ✤➳♥ sü ❧✐➯♥ ❦➳t ❣✐ú❛ ❍➻♥❤ ❤å❝ ✈➔ ✣↕✐ sè tr♦♥❣ ❣✐↔♥❣ ❞↕② ❝ơ♥❣ ♥❤÷ ❣✐↔✐ ❜➔✐ t➟♣✳ ✷ ❚❤ỉ♥❣ q✉❛ t➻♠ ❤✐➸✉ t❤ü❝ t➳✱ tæ✐ t❤➜② r➡♥❣ ✈✐➺❝ ❦❤❛✐ t❤→❝ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❤❛✐ ♣❤➙♥ ♠æ♥ ❍➻♥❤ ❤å❝ ✈➔ ✣↕✐ sè s➩ ❣â♣ ♣❤➛♥ q✉❛♥ trå♥❣ ❣✐ó♣ ❝→❝ ❡♠ ❤✐➸✉ ❜✐➳t ❤ì♥ ✈➲ ❜ë ♠ỉ♥ ❚♦→♥ ❤å❝✱ ❝ơ♥❣ ♥❤÷ trđ ❣✐ó♣ ❝→❝ ❡♠ ỉ♥ t❤✐ ✈➔ t❤✐ ❤å❝ s✐♥❤ ọ P õ ợ ữợ ✤✐ ♠ỵ✐✱ ❝→❝❤ t✐➳♣ ❝➟♥ ❧í✐ ❣✐↔✐ ♠ỵ✐✱ ♣❤♦♥❣ ♣❤ó ❤ì♥ tr♦♥❣ q✉→ tr➻♥❤ ỉ♥ ❧✉②➺♥ ✈➔ t❤✐ ♠ỉ♥ ❚♦→♥✳ ỳ ỵ tr tổ qt t➔✐✿ ✧❑❤❛✐ t❤→❝ ♠è✐ q✉❛♥ ❤➺ ❍➻♥❤ ❤å❝ ✲ ✣↕✐ sè ✈➔♦ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥ ❞➔♥❤ ❝❤♦ ❤å❝ s✐♥❤ ❣✐ä✐✧✳ ❚❤æ♥❣ q✉❛ ♥❣❤✐➯♥ ❝ù✉ ♥❤ä ♥➔②✱ tæ✐ ♠♦♥❣ r➡♥❣ ♠➻♥❤ s➩ ❣â♣ ♣❤➛♥ ❧➔♠ rã ❤ì♥ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❤❛✐ ♣❤➙♥ ♠æ♥ ❍➻♥❤ ❤å❝ ✈➔ ✣↕✐ sè✱ ♠è✐ q✉❛♥ ❤➺ t÷ì♥❣ trđ ❧➝♥ ♥❤❛✉ tr♦♥❣ q✉→ tr➻♥❤ ❣✐↔♥❣ ❞↕② ✈➔ ❤å❝ ❚♦→♥ ❝õ❛ ❜↔♥ t❤➙♥ ð ❚❍❈❙✳ ✷✳ ▼ö❝ ✤➼❝❤✱ ♥❤✐➺♠ ✈ö ❝õ❛ ❧✉➟♥ ✈➠♥ ▼ö❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ ❦❤❛✐ t❤→❝ ♠è✐ q✉❛♥ ❤➺ ❣✐ú❛ ❍➻♥❤ ❤å❝ ✈➔ ✣↕✐ sè ❣â♣ ♣❤➛♥ t✐➳♣ ❝➟♥ ữợ t ợ t ữớ ✈➟♥ ❞ö♥❣ t➼♥❤ ❝❤➜t ✣↕✐ sè ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ ❍➻♥❤ ❤å❝ ✈➔ ♥❣÷đ❝ ❧↕✐ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ❍➻♥❤ ❤å❝ ✈ỵ✐ ❝ỉ♥❣ ❝ư ✣↕✐ sè t❤ỉ♥❣ q✉❛ ✈✐➺❝ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥ ❞➔♥❤ ❝❤♦ ❤å❝ s✐♥❤ ❣✐ä✐✱ ❧➔ ✤➲ t❤✐ ❝❤å♥ ❤å❝ s✐♥❤ ❣✐ä✐ ❝→❝ t➾♥❤✱ t♦➔♥ q✉è❝ ✈➔ ❦❤✉ ✈ü❝✳ ▲✉➟♥ ✈➠♥ t➟♣ tr✉♥❣ ✈➔♦ ❤♦➔♥ t❤➔♥❤ s ã ị tữ t ❝→❝ t➼♥❤ ❝❤➜t✱ ❝ỉ♥❣ ❝ư ❝õ❛ ✣↕✐ sè ✤➸ ❣✐↔✐ t ữủ ã ữ t ♠ët ❜➔✐ t♦→♥✱ ✤➲ t❤✐ ✈➲ ✣↕✐ sè✱ ❍➻♥❤ ❤å❝ s ọ ã ữ r ❜➡♥❣ ❝→❝❤ ✈➟♥ ❞ư♥❣ t➼♥❤ ❝❤➜t✱ ❝ỉ♥❣ ❝ư ❝õ❛ ✣↕✐ sè ✤➸ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥ ❍➻♥❤ ❤å❝ ✈➔ ♥❣÷đ❝ ❧↕✐ ❦❤❛✐ t❤→❝ ❝→❝ t➼♥❤ ❝❤➜t✱ ♣❤÷ì♥❣ ♣❤→♣ ❍➻♥❤ ❤å❝ ✤➸ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ✣↕✐ sè ❞➔♥❤ ❝❤♦ ❤å❝ s✐♥❤ ❣✐ä✐✳ ✸✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ◆ë✐ ❞✉♥❣ ❧✉➟♥ ✈➠♥ ♥❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ ❦➳t t t s ỗ ữỡ ữỡ ✶✿ ❚r➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❦❤❛✐ t❤→❝ ❦✐➳♥ t❤ù❝ ✣↕✐ sè ✤➸ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥ ❍➻♥❤ ❤å❝ ❈❤÷ì♥❣ ✷✿ ❚r➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❦❤❛✐ t❤→❝ ❦✐➳♥ t❤ù❝ ❍➻♥❤ ❤å❝ ✤➸ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥ ✣↕✐ sè ✸ ❈❤÷ì♥❣ ✶ ❑❤❛✐ t❤→❝ ❦✐➳♥ t❤ù❝ ✣↕✐ sè ✤➸ ❣✐↔✐ ởt số t ị tữ ữỡ ỵ tữ t t ỵ ổ tr số tr♦♥❣ q✉→ tr➻♥❤ t➻♠ r❛ ❧í✐ ❣✐↔✐ ❝❤♦ ♠ët sè ❜➔✐ t♦→♥ ❍➻♥❤ ❤å❝ ❜➡♥❣ ❝→❝❤ ✤÷❛ r❛ ♠ët sè ✈➼ ❞ư sû ❞ư♥❣ ❦✐➳♥ t❤ù❝ ✣↕✐ sè ✤➸ ✤÷❛ r❛ ❧í✐ ❣✐↔✐ ❝❤♦ ♠ët sè ❜➔✐ t♦→♥ ❝❤å♥ ❧å❝ ❞➔♥❤ ❝❤♦ ❤å❝ s✐♥❤ ❣✐ä✐✱ ❧➔ ✤➲ t❤✐ ❝❤å♥ ❤å❝ s✐♥❤ ❣✐ä✐ ❝→❝ ✤à❛ ♣❤÷ì♥❣✱ t♦➔♥ q✉è❝ ❝ơ♥❣ ♥❤÷ ✤➲ t❤✐ ❝❤å♥ ❤å❝ s✐♥❤ ❣✐ä✐ ❦❤✉ ✈ü❝ ❈❤➙✉ ⑩ ✲ ữỡ ởt số ữợ ỹ ổ ❹✉✳ ▼ët tr♦♥❣ ♥❤ú♥❣ ❦❤➙✉ q✉❛♥ trå♥❣ ➞♥ tr♦♥❣ ♥❤ú♥❣ ✈➼ ❞ö ❧➔ ✈✐➺❝ ❜✐➳♥ ✤è✐ ❜➔✐ t♦→♥ ❜❛♥ ✤➛✉ ✤➸ ❝❤ó♥❣ ❜ë❝ ❧ë ♥❤ú♥❣ ✤✐➸♠ ❝â t❤➸ ✈➟♥ ❞ư♥❣ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ✣↕✐ sè ✤➸ ❣✐↔✐ q✉②➳t ✈➜♥ ✤➲✱ ❝â t❤➸ t↕♠ ❣å✐ ✤➙② ❧➔ q✉→ tr➻♥❤ ✧✣↕✐ sè ❤â❛ ❜➔✐ t♦→♥ ❤➻♥❤ ❤å❝✧ ✱ s❛✉ ✤â ❧➔ q✉→ tr➻♥❤ sû ❞ư♥❣ ❝ỉ♥❣ ❝ư ✣↕✐ sè ✤➸ ♣❤→t ❜✐➸✉ ❜➔✐ t♦→♥ ❍➻♥❤ ❤å❝ ❜❛♥ ✤➛✉✳ ✶✳✷✳ ▼ët sè ✈➼ ❞ö ♠✐♥❤ ❤å❛ ✶✳✷✳✶✳ ❇➔✐ t♦→♥ ❝ü❝ trà ❍➻♥❤ ❤å❝ ❳✉➜t ♣❤→t tø ❜➜t ✤➥♥❣ t❤ù❝ ✭❇✣❚✮ ✣↕✐ sè r➜t q✉❡♥ t❤✉ë❝ s❛✉ ✤➙②✳ ❇✣❚ ✶✳ ❱ỵ✐ ❝→❝ sè ❞÷ì♥❣ a✱ b✱ c ❝â 1 + + (a + b + c) a b c ≥ ✹ ✣➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ a = b = c✳ ▲í✐ ❣✐↔✐ 1 + + (a + b + c) a b c a a b b c c −9=1+ + + +1+ + + +1−9 b c a c a b a b b c c a = + −2 + + −2 + + −2 b a c b a c 2 (a − b) (b − c) (c − a) = + + ≥ ab bc ac ✣➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ a − b = b − c = c − a = ⇔ a = b = c✳ ❇✣❚ ✷✳ ợ số ữỡ a, b, c õ b c a + + ≥ b+c c+a a+b ✣➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ a = b = c✳ ▲í✐ ❣✐↔✐ ⑩♣ ❞ư♥❣ ❇✣❚ ✶ t❛ ❝â a b c +1 + +1 + +1 −3 b+c c+a a+b 1 −3 + + = (a + b + c) b+c c+a a+b 1 1 −3 + + = [(b + c) + (c + a) + (a + b)] b+c c+a a+b ≥ −3= 2 a b c + + = b+c c+a a+b ✣➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ b + c = c + a = a + b ⇔ a = b = c✳ ❈â t❤➸ ✈➟♥ ❞ö♥❣ ❤❛✐ ❜➜t ✤➥♥❣ t❤ù❝ tr➯♥ ✈➔♦ ❣✐↔✐ ✈➔ s→♥❣ t↕♦ ❝→❝ ❜➔✐ t♦→♥ ❝❤ù❛ ❜➜t ✤➥♥❣ t❤ù❝ ❍➻♥❤ ❤å❝ ❤♦➦❝ t➻♠ ❝ü❝ trà ❍➻♥❤ ❤å❝✳ ❙❛✉ ✤➙② ❧➔ ♠ët sè ✈➼ ❞ư ♠✐♥❤ ❤å❛ ✤÷đ❝ tr➼❝❤ ❞➝♥ tø ❚↕♣ ❝❤➼ ❚♦→♥ ❍å❝ ✈➔ ❚✉ê✐ ❚r➫ ✈➔ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❇➔✐ t♦→♥ ✶✳✷✳✶✳✶✳ ❈❤♦ t❛♠ ❣✐→❝ ✤➲✉ ABC ❝â ❝↕♥❤ ❜➡♥❣ a✳ ●å✐ ✤÷í♥❣ ✈✉ỉ♥❣ ❣â❝ tø ✤✐➸♠ M ♥➡♠ tr♦♥❣ t❛♠ ❣✐→❝ ✤➳♥ ❝→❝ ❝↕♥❤ BC, CA, AB ❧➛♥ ❧÷đt ❧➔ M D, M E, M F ✳ ❳→❝ ✤à♥❤ ✈à tr➼ ❝õ❛ M ✤➸✿ ❛✮ M1D + M1E + M1F ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t✳ ❚➼♥❤ ❣✐→ trà ✤â✳ ❜✮ M D +1 M E + M E +1 M F + M F +1 M D ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t✳ ❚➼♥❤ ❣✐→ trà ✤â✳ ✺ ▲í✐ ❣✐↔✐ ❍➻♥❤ ✶ √ a ●å✐ h = ❧➔ ✤ë ❞➔✐ ✤÷í♥❣ ❝❛♦ ❝õ❛ t❛♠ ❣✐→❝ ✤➲✉ ABC M E = y, M F = z ✳ ❚❛ ❝â ✈➔ ✤➦t M D = x, SABC = SM BC + SM AC + SM AB ⇔ ah = ax + ay + az ⇔ x + y + z = h ❛✮ ⑩♣ ❞ö♥❣ ❇✣❚ ✶ t❛ ❝â 1 + + (x + y + z) x y z ❜✮ ⑩♣ ❞ư♥❣ ❇✣❚ ✷ t❛ ❝â ❦❤ỉ♥❣ ✤ê✐✳ √ 1 ≥9⇒ + + ≥ = x y z h a 1 (x + y + y + z + z + x) + + x+y y+z z+x √ 1 3 + + ≥ = ⇔ x+y y+z z+x 2h a ≥9 ❚r♦♥❣ ❝↔ ❤❛✐ tr÷í♥❣ ❤ñ♣ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ x = y = z ✱ ❧ó❝ ✤â M ❧➔ t➙♠ ✤÷í♥❣ trá♥ ♥ë✐ t✐➳♣ ∆ABC ✳ ●å✐ H ❧➔ trü❝ t➙♠ t❛♠ ❣✐→❝ ABC ❝â ❜❛ ❣â❝ ♥❤å♥ ✈ỵ✐ ❜❛ ✤÷í♥❣ ❝❛♦ AA1; BB1; CC1✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❇➔✐ t♦→♥ ✶✳✷✳✶✳✷✳ BB1 CC1 AA1 + + ❛✮ HA HB HC 1 ≥9 ✻ HB1 HC1 + + ≥ ❜✮ HA HA HB HC ✣➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ♥➔♦❄ ▲í✐ ❣✐↔✐ ❍➻♥❤ ✷ ●å✐ ❞✐➺♥ t➼❝❤ t❛♠ ❣✐→❝ ABC, HBC, HAC, HAB ❧➛♥ ❧÷đt ❧➔ S, S1, S2, S3 t❤➻ S = S1 + S2 + S ❛✮ ❉➵ t❤➜② ❉♦ ✤â HA1 S1 HB1 S2 HC1 S3 = ; = ; = AA1 S BB1 S CC1 S HA1 HB1 HC1 + + = AA1 BB1 CC1 ⑩♣ ❞ö♥❣ ❇✣❚ ✶ ✤÷đ❝ AA1 BB1 CC1 + + ≥ HA1 HB1 HC1 ✣➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ HA1 HB1 HC1 = = = AA1 BB1 CC1 ⇔ S1 = S2 = S3 = S , ❧ó❝ ✤â H ✈ø❛ ❧➔ trü❝ t➙♠✱ ✈ø❛ ❧➔ trå♥❣ t➙♠ ❝õ❛ t❛♠ ❣✐→❝ ABC ♥➯♥ ABC ❧➔ t❛♠ ❣✐→❝ ✤➲✉✳ ✷✼ ❍➻♥❤ ✾ √ 2− ♥➯♥ t❤❡♦ ỵ s t õ cos 75 = cos(45 + 35 ) = r q √ AC = a2 − − 3ac + c2 ◦ ◦ ◦ p ❘ã r➔♥❣ AB + BC ≥ AC ♥➯♥ s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❉➜✉ ✤➥♥❣ t❤ù❝ ①↔② r❛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ A; B; C t❤➥♥❣ ❤➔♥❣✱ tù❝ ❧➔ ❦❤✐ ⇔ SOAB + SOBC = SOAC p p √ √ √ ! + + bc ac ac ab √ + = arcsin 750 = ⇔b= 4 a 2+c (1) ❱➟② (1) ❧➔ ✤✐➲✉ ❦✐➺♥ ✤➸ ❝â ❞➜✉ ❜➡♥❣ tr♦♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✤➣ ❝❤♦✳ ❇➔✐ t♦→♥ ✷✳✷✳✶✳✼✳ ❈❤♦ a1 ; a2 ; an ❧➔ n sè t❤ü❝ ❜➜t ❦ý✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ √ q q q p n a21 + (1 − a2 )2 + a22 + (1 − a3 )2 + + a2n−1 + (1 − an )2 + a2n + (1 − a1 )2 ≥ ▲í✐ ❣✐↔✐ ✷✽ ❍➻♥❤ ✶✵ ❚r÷í♥❣ ❤đ♣ ✶✳ ◆➳✉ ♥ ❝❤➤♥ ❳➙② ❞ü♥❣ ❝→❝ ✤♦↕♥ A1 A2 = A2 A3 = A3 A4 = = An An+1 = An+1 An+2 = ✭①❡♠ ❤➻♥❤ ✈➩✱ ð ✤➙② ✤➸ ✤ì♥ ❣✐↔♥ t❛ ✈➩ ✈ỵ✐ n = 4✮ ❚r➯♥ ✤♦↕♥ Ai Ai+1 ✭ i = 1, 2, 3, , n + 1✮ ❧➜② ✤✐➸♠ Bi s❛♦ ❝❤♦ Bi Ai + = ✭r✐➯♥❣ tr➯♥ ✤♦↕♥ An+1 An+2 ❧➜② Bn+1 s❛♦ ❝❤♦ Bn+1 An+2 = a1 ✱ tù❝ ❧➔ an+1 = a1 ) ❑❤✐ ✤â ✤✐➸♠ Bi s➩ ♥➡♠ ❜➯♥ tr ữợ Ai+1 > ❞ö tr♦♥❣ ❤➻♥❤ ♠✐♥❤ ❤å❛ tr➯♥ t❤➻ < a1 < 1, < a2 < 1, a3 > 1, a4 < 0✮✳ ❚❤❡♦ ❝→❝❤ ①➙② ❞ü♥❣ tr➯♥ t❤➻ Bi+1 Ai+1 = |1 − ai+1 | ♥➯♥ Bi Bi+1 = q Bi A2i+1 + Bi+1 A2i+1 q = a21 + (1 − a2i+1 ) ❚ø ✤â ✈➳ tr→✐ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ✤➣ ❝❤♦ ❧➔ ✤ë ❞➔✐ ✤÷í♥❣ ❣➜♣ ❦❤ó❝ B1 B2 Bn Bn+1 ✳ ❉♦ n ❝❤➤♥ ♥➯♥ n ✱ n CBn+1 = A1 A2 + A3 A4 + + An−1 An ❂ √ n ❱➟② tø t❛♠ ❣✐→❝ ✈✉æ♥❣ B1 CBn+1 ✱ t❛ ❝â B1 Bn+1 ❂ ❚ø ✤ë ❞➔✐ ✤÷í♥❣ ❣➜♣ ❦❤ó❝ ♥è✐ B1 , Bn+1 ✱ ❦❤ỉ♥❣ ♥❤ä ❤ì♥ ✤ë ❞➔✐ B1 Bn+1 ✱t❛ B1 C = A2 A3 + A4 A5 + + An An+1 = s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❚r÷í♥❣ ❤đ♣ ✷✳ ◆➳✉ ♥ ❧➫ ✷✾ ✣➦t a + n + = a1, an+2 = a2, , a2n = an ❑❤✐ ✤â →♣ ❞ư♥❣ tr÷í♥❣ ❤đ♣ 1✱ t❛ ❝â p ❤❛② 2( q a21 + (1 − a2 )2 a21 ✰ p a22 + (1 − a3 )2 ✰✳✳✳ ✰ √ p 2n a22n + (1 − a1 )2 ≥ √ p 2n + (1 − a2 )2 + + a2n + (1 − a1 )2 ) ≥ ✭✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✮ ❚❛ t❤➜② ❞➜✉ ❜➡♥❣ ❝â ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ B1; B2; ; Bn+1 t❤➥♥❣ ❤➔♥❣✳ ✣✐➲✉ ✤â ①↔② r❛ ❦❤✐ a1 = a3 = = an+1 = a ❀ a2 = a4 = = an = − a ♥➳✉ n ❝❤➤♥ a1 = a2 = = an ♥➳✉ ♥ ❧➫✳ ❇➔✐ t♦→♥ ✷✳✷✳✶✳✽✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ a > c; b > c ✈➔ c > t❤➻ ▲í✐ ❣✐↔✐ p c(a − c) + p c(b − c) ≤ √ ab ❈→❝❤ ✶✳ ✭❙û ❞ö♥❣ t➼♥❤ ❝❤➜t ✣↕✐ sè✮ ❇➜t ✤➥♥❣ t❤ù❝ ❇✉♥②❛❦♦✈s❦② ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❇✉♥②❛❦♦✈s❦② t❛ ❝â q q√ √ √ √ √ √ √ √ √ V T = c a − c+ b − c c ≤ ( c) + ( b − c)2 ( a − c).( c)2 = b.a = V P √ √ √ √ c b−c ⇔ c = a − c b − c ❉➜✉ ✧❂✧ ①↔② r❛ ⇔ √a − c = √ c 2 ⇒ c = (a − c).(b − c) ⇒ c = ab − ac − bc + c2 ⇒ ab − ac − bc = ab ⇒ ab = (a + b).c ⇒ c = a+b ❈→❝❤ ✷✳ ✭❙û ❞ư♥❣ t➼♥❤ ❝❤➜t ❍➻♥❤ ❤å❝✮√ ❚➼♥❤ ❝❤➜t ❝õ❛ ✈❡❝tì √ √ √ √ √ ❳➨t ❝→❝ ✈❡❝tì ~u❂✭ c❀ b − c✮ ✈➔ ~v❂✭ a − c❀ c✮✱t❛ ❝â ⑤~u⑤ ❂ b✱ ⑤~v⑤ ❂ a✳ ❚❛ ❜✐➳t r➡♥❣ tr♦♥❣ ❤➺ tå❛ ✤ë ❉❡s❝❛rt❡s ✈ỵ✐ ~u❂✭x1; y1; z1✮❀~v❂✭x2; y2; z2✮ t❤➻ ~u✳~v ❂ ⑤~u⑤✳⑤~v ⑤✳cos✭~u✱~v ✮ ≤ ⑤~u⑤✳⑤~v ⑤ ✭❞♦ cos✭~u✱~v ✮ ≤ 1✮✳ ⇔ x1 x2 + y1 y2 + z z ≤ q x21 ❉➜✉ ✧❂✧ ①↔② r❛ ⇔ ~u ❂ λ~v ✭ λ 6= ✵✮ z12 y12 q + x22 + y22 + z22 + x = λx2 ⇔ y1 = λy2 z = λz + ✸✵ ⑩♣ ❞ö♥❣ ✈➔♦ ❜➔✐ t♦→♥ t❛ ❝â p p √ √ √ √ √ √ √ c a − c + b − c c ≤ b a ⇔ c(a − c) + c(b − c) ≤ ab √c = λ.√a − c ab ✳ ⇒ ❝ ❂ ❉➜✉ ✧❂✧ ①↔② r❛ ⇔ √ b − c = λ.√c a+b ●å✐ α✱ β ❀ γ ❧➔ ❜❛ ❣â❝ t↕♦ ❜ð✐ ✤÷í♥❣ ❝❤➨♦ ❝õ❛ ♠ët ❤➻♥❤ ❝❤ú ♥❤➟t ✈ỵ✐ ❜❛ ❝↕♥❤ ①✉➜t ♣❤→t tø ❝ị♥❣ ♠ët ✤➾♥❤✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✶✮ cos2 α ✰ cos2 β ✰ cos2 γ ❂ ✶ ❇➔✐ t♦→♥ ✷✳✷✳✶✳✾✳ ✷✮ √ cos2 α + 1✰ p cos2 β + 1✰ p √ cos2 γ + 1≤ 21 ▲í✐ ❣✐↔✐ ❍➻♥❤ ✶✶ ❈→❝❤ ✶✳ ✭❙û ❞ö♥❣ t➼♥❤ ❝❤➜t ✣↕✐ sè✮ AA′ AC ′ AB ❳➨t ∆C ′ BA ❝â ✿ cos β = AC ′ AD ❳➨t ∆C ′ DA ❝â ✿ cos γ = AC ′ ❳➨t ∆A′ C ′ A ❝â ✿ cos α = AA′2 + AB + AD2 ❉♦ ✤â✱ cos α + cos β + cos γ = = AC ′2 A′ A2 + AC A′ C AC ′2 = = = = C ′ A2 C ′ A2 AC ′2 2 ✸✶ ❜✮ ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❇✉♥②❛❦♦✈s❦② t❛ ❝â✿ p p p V T = cos2 α + + cos2 β + + cos2 γ + p p ≤ 12 + 12 + 12 cos2 α + + cos2 β + + cos2 γ + p √ √ = 4(cos2 α + cos2 β + cos2 γ) + = 4.1 + = 21 ❉➜✉ ” = ” ①↔② r❛ ❦❤✐ √ 1 =p =p cos2 α + cos2 β + cos2 γ + ⇒ cos α = cos β = cos γ ⇒ α = β = γ ⇒ ❤➻♥❤ ❤ë♣ ❝❤ú ♥❤➟t trð t❤➔♥❤ ❤➻♥❤ ❧➟♣ ♣❤÷ì♥❣✳ ❈→❝❤ ✷✳ ✭❙û ❞ư♥❣ t➼♥❤ ❝❤➜t ❍➻♥❤ ❤å❝✮ ✶✮ ▲➜② ✤÷í♥❣ ❝❤➨♦ ❝õ❛ ❤➻♥❤ ❤ë♣ ❝❤ú ♥❤➟t ❧➔♠ ✈❡❝tì ✤ì♥ ✈à ~e✱ ❜❛ ❝↕♥❤ ①✉➜t ♣❤→t tø ❝ị♥❣ ♠ët ✤➾♥❤ ❝õ❛ ❤ë♣ ❧➔♠ ❜❛ trö❝ tå❛ ✤ë t❤➻ cos α, cos β, cos γ ❧➔ ❝→❝ tå❛ ✤ë ❝õ❛ ~e ❞♦ ✤â cos2 α + cos2 β + cos2 γ = ⑤~e⑤2 = 1✳ ✷✮ ❳➨t ❝→❝ ✈❡❝t♦ p p √ ~u❂✭ cos2 α + 1❀ cos2 β + 1❀ cos2 γ + 1✮ ✈➔ ~u= (1; 1; 1) t❛ ❝â p √ √ ⑤~u⑤ ❂ cos2 α + + cos2 β + + cos2 γ + ❂ ✈➔ ⑤~v ⑤ ❂ 3✳ ❚❛ ❜✐➳t r➡♥❣ tr♦♥❣ ❤➺ tå❛ ✤ë ❉❡s❝❛rt❡s ✈ỵ✐ ~u❂✭x1 ; y1 ; z1 ✮❀~v ❂✭x2 ; y2 ; z2 ✮ t❤➻ ~u✳~v ❂ ⑤~u⑤✳⑤~v ⑤ cos✭~u✱~v ✮ ≤ ⑤~u⑤✳⑤~v ⑤ ✭❞♦ cos✭~u✱~v ✮ ≤ 1✮✳ ✭✯✮ ⇔ x1 x2 + y1 y2 + z z ≤ q x21 + ❉➜✉ ✧❂✧ ①↔② r❛ ⇔ ~u ❂ λ~v ✭ λ 6= ✵✮ ⇔ y12 z12 + x = λx2 y1 = λy2 z = λz ⑩♣ ❞ö♥❣ ✈➔♦ ❜➔✐ t♦→♥ t❛ ❝â ~u.~v ≤ |~u|.|~v | ⇐⇒ p cos2 α + 1+ ❉➜✉ ” = ” ①↔② r❛ p p q + x22 + y22 + z22 cos2 β + 1+ p √ √ √ cos2 γ + ≤ = 21 p p cos2 β + = cos2 γ + 1 ⇔ cos α = cos β = cos γ = √ ⇔ cos2 α + = ✸✷ ❇➔✐ t♦→♥ ✷✳✷✳✶✳✶✵✳ √ ●✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ x−1+x−3≥ ▲í✐ ❣✐↔✐ p 2(x − 3)2 + 2x − √ ❱ỵ✐ x ≥ ①➨t ❝→❝ ✈❡❝tì ~u ❂ ( x − 1; x − 3) ✈➔ ~e = (1; 1) t❛ ❝â |~u| = p √ x − + (x − 3)2 ✈➔ |~e| = ❚r♦♥❣ ❤➺ tå❛ ✤ë ❉❡s❝❛rt❡s t❛ ❝â ~u.~v = |~u|.|~v | cos(~u, ~v ) ≤ |~u|.|~v | ✭❞♦ cos(~u, ~v ) ≤ 1) ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ (∗) ✈➔♦ ❜➔✐ t♦→♥ t❛ ❝â √ x−1+x−3≤ √ p √ x − + (x − 3)2 p x − + x − ≤ 2(x − 3)2 + 2x − (1) p √ ❉♦ ✤â✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ x − + x − ≥ 2(x − 3)2 + 2x − ①↔② r❛ ⇔ ❉➜✉ ” = ” ð ❜➜t ♣❤÷ì♥❣ tr➻♥❤ (1) ①↔② r❛ √ ⇔ x − = x − ⇔ x = ❱➟② ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❝â ♥❣❤✐➺♠ x = ⇔ ❇➔✐ t♦→♥ ✷✳✷✳✶✳✶✶✳ √ ●✐↔✐ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ x+1+ √ 2x − + √ 50 − 3x ≤ 12 ▲í✐ ❣✐↔✐ ❈→❝❤√ ✶✳ ✭❙û ❞ư♥❣ t➼♥❤ ❝❤➜t ✣↕✐ sè✮ a= x − ≥ 0; b = x − 3✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ p ⇔ a + b ≥ 2b2 + 2a2 ( a+b≥0 (1) ⇔ (a + b)2 ≥ 2b2 + 2a2 (2) ⇒ 2ab ≥ a2 + b2 ⇒ a2 + b2 − 2ab ≤ ⇒ (a − b)2 ≤ √ ⇒a=b⇒ x−1=x−3 ( ( x−3≥0 x≥3 ⇒ ⇒ x − = (x − 3)2 x = 2; x = ⇒ x = t❤ä❛ ♠➣♥ ✭✶✮ ✭✯✮ ✸✸ ❱➟② ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t x = 5✳ ❈→❝❤ ✷✳ ✭❙û ❞ö♥❣ t➼♥❤ ❝❤➜t ❍➻♥❤ ❤å❝✮ 50 ≤①≤ √ √ √ ❳➨t ❝→❝ ✈❡❝tì ~u ❂ ✭ x + 1❀ 2x − 3❀ 50 − 3x✮ ✈➔ ~v = (1; 1; 1) √ √ √ ❚❛ ❝â ⑤~u⑤ ❂ 48 = ✈➔ |~v | = 3✳ ❚➟♣ ①→❝ ✤à♥❤ ✈➳ tr→✐ ❧➔ ❚r♦♥❣ ❤➺ tå❛ ✤ë ❉❡s❝❛rt❡s t❛ ❝â ~u.~v = |~u|.|~v | cos(~u, ~v ) ≤ |~u|.|~v | ✭❞♦ cos(~u, ~v ) ≤ 1) ✭✯✮ ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✭✯✮ ✈➔♦ ❜➔✐ t♦→♥ t❛ ❝â √ x+1+ √ 2x − + √ √ √ 50 − 3x ≤ 3 = 12 ❉➜✉ ” = ” ①↔② r❛ √ √ √ 50 ≤x≤ 3 50 ❱➟② ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❝â ♥❣❤✐➺♠ ≤ x ≤ ✳ ⇔ x+1= ❇➔✐ t♦→♥ ✷✳✷✳✶✳✶✷✳ 2x − = 50 − 3x ⇔ π ❈❤♦ α✱ β ✱ γ ❧➔ ❜❛ ❣â❝ ❞÷ì♥❣ ❝â α + β + γ = ✳ ❚➻♠ ❣✐→ trà ❧ỵ♥ ♥❤➜t ❝õ❛ g= p p p + tan α tan β + + tan β tan γ + + tan γ tan α ▲í✐ ❣✐↔✐✿ π π ⇔ α + β = − γ✳ 2 π ❚❛ ❝â tan(α + β) = tan( − γ) ❤❛② ❚ø α + β + γ = tan α + tan β = ⇒ tan α tan β + tan β tan γ + tan γ tan α = 1 − tan α tan β tan γ ❳➨t ❝→❝ ✈❡❝tì ~u = ( ✈➔ p p p + tan α tan β; + tan β tan γ; + tan γ tan α) ~v = (1; 1; 1), t❛ ❝â |~u| = 2; |~v | = √ ❚r♦♥❣ ❤➺ tå❛ ✤ë ❉❡s❝❛rt❡s t❛ ❝â ~u.~v = |~u|.|~v | cos(~u, ~v ) ≤ |~u|.|~v | ✭❞♦ cos(~u, ~v ) ≤ 1) ✭✯✮ ✸✹ ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✭✯✮ ✈➔♦ ❜➔✐ t♦→♥ t❛ ❝â p p √ + tan β tan γ + + tan γ tan α ≤ ❉➜✉ ” = ” ①↔② r❛ ⇔ tan α = tan β = tan γ ⇔ α = β = γ = π6 √ ❱➟② ❜✐➸✉ t❤ù❝ g ✤↕t ❣✐→ trà ❧ỵ♥ ♥❤➜t ❜➡♥❣ 3✱ ✤↕t ✤÷đ❝ ❦❤✐ α ❂ β ❂ γ ❂ π6 g= p + tan α tan β + ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ p p sin x+ − sin2 x ✰ sin x − sin2 x = ❇➔✐ t♦→♥ ✷✳✷✳✶✳✶✸✳ (3) ▲í✐ ❣✐↔✐ ❈→❝❤ ✶✳ ✭❙û ❞ö♥❣ t➼♥❤ ❝❤➜t ✣↕✐ sè✮ p ✣➦t t = sin x + − sin2 x ⑩♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❇✉♥②❛❦♦✈s❦② t❛ ❝â |t| = |1 sin x + ▲↕✐ ❝â p 2 − sin x| ≤ p (12 + 12 ) ⇒ −2 ≤ t ≤ q (sin2 x + − sin2 x) = p t2 = sin2 x + sin x − sin2 x + − sin2 x p = + sin x − sin2 x p t2 − ⇒ sin x − sin2 x = P❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ trð t❤➔♥❤ t+ t2 − =3 ⇒ t2 + 2t − = ⇒ (t − 2)(t + 4) = ❙✉② r❛ t = 2✭♥❤➟♥✮ ❤♦➦❝ t = −4 ✭❧♦↕✐✮✳ p ❱ỵ✐pt = ⇒ sin x + − sin2 x = ⇒ − sin2 x = − sin x ⇒ − sin2 x = − sin x + sin2 x ⇒ sin2 x − sin x + = ⇒ (sin x − 1)2 = π ⇒ sin x = ⇒ x = + k2π(k ∈ Z) ❈→❝❤ ✷✳ ✭❙û ❞ö♥❣ t➼♥❤ ❝❤➜t ❍➻♥❤ ❤å❝✮ ✸✺ p p ❳➨t ~u = (sin x; 1; − sin2 x) ✈➔ ~v = (1; − sin2 x; sin x) √ ❚❛ ❝â ⑤~u⑤ ❂ ⑤~v ⑤ ❂ ❚❛ ❜✐➳t r➡♥❣ tr♦♥❣ ❤➺ tå❛ ✤ë ❉❡s❝❛rt❡s ✈ỵ✐ ~u❂✭x1 ; y1 ; z1 ✮❀~v ❂✭x2 ; y2 ; z2 ✮ t❤➻ |~u.~v | = ||~u|.|~v | cos(~u, ~v )| ≤ |~u|.|~v |(do cos(~u, ~v ) ≤ 1).(∗) q q 2 ⇔ |x1 x2 + y1 y2 + z1 z2 | ≤ x1 + y1 + z1 + x22 + y22 + z22 x = λx2 ❉➜✉ ✧❂✧ ①↔② r❛ ⇔ ~u ❂ λ~v ✭ λ 6= ✵✮ ⇔ y1 = λy2 z = λz ⑩♣ ❞ö♥❣ ✈➔♦ ❜➔✐ t♦→♥ t❛ ❝â p √ √ + sin2 x + sin x − sin2 x| ≤ 3 = sin x = λ p ❉➜✉ ✧❂✧ ①↔② r❛ ⇔ = λ − sin2 x p − sin2 x = λ sin x | sin x + p ⇒ λ = ✈➔ sin x = ⇒ x = ❇➔✐ t♦→♥ ✷✳✷✳✶✳✶✹✳ π + 2kπ(k ∈ Z) ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❤➺ s❛✉ ✤➙② ✈æ ♥❣❤✐➺♠ x4 + y + z = x2 + y + 2z = √7 ▲í✐ ❣✐↔✐ ❳➨t ❝→❝ ✈❡❝tì ~u = (x2 ; y ; z ) ✈➔ ~v = (1; 1; 2) √ ❚❛ ❝â ⑤~u⑤❂✶✱ ⑤~v ⑤❂ 6✳ √ √ ❚❤❡♦ ❤➺ tr➯♥✱ t❛ ❝â ~u.~v = x2 + y + 2z = ✈➔ |~u|.|~v | = ❉♦ ✤â✱ ~u.~v > |~u|.|~v | ▼➔ tr♦♥❣ ❤➺ tå❛ ✤ë ❉❡s❝❛rt❡s t❛ ❝â ~u.~v = |~u|.|~v | cos(~u, ~v ) ≤ |~u|.|~v | ✭❞♦ cos(~u, ~v ) ≤ 1) ❱➟② ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤➲ ❝❤♦ ❧➔ ✈æ ♥❣❤✐➺♠✳ ❇➔✐ t♦→♥ ✷✳✷✳✶✳✶✺✳ sin ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✈ỵ✐ ♠å✐ t❛♠ ❣✐→❝ ABC ❧✉ỉ♥ ❝â B C A + sin + sin ≤ 2 2 (1) ✸✻ ▲í✐ ❣✐↔✐ ❚❛ ❝â B C A + sin + sin ≤ 2 2 π B+C π A+C π A+B ⇔ sin + sin + sin ≤ − − − 2 2 2 A+C A+B B+C + cos + cos ≤ ⇔ cos 2 2 ✣➦t B +2 C = α; A +2 C = β; A +2 B = γ ✱ t❤➻ α; β; γ ❧↕✐ ❧➔ ❜❛ ❣â❝ ❝õ❛ sin ♠ët t❛♠ ❣✐→❝✳ ❑❤✐ ✤â✱ t❛ ✤✐ ❝❤ù♥❣ ♠✐♥❤ cos α + cos β + cos γ ≤ 23 ✈ỵ✐ α; β; γ ❧↕✐ ❧➔ ❜❛ ❣â❝ ❝õ❛ ♠ët t❛♠ ❣✐→❝✳ ◆❤÷ ✈➟②✱ ❜➔✐ t♦→♥ ❧↕✐ q✉❛② trð ✈➲ ❜➔✐ t♦→♥ ✷✳✷✳✶✳✶ ✈➔ ❞➵ ❞➔♥❣ ❝❤ù♥❣ ♠✐♥❤ ♥❤÷ tr➯♥✳ ❇➔✐ t♦→♥ ✷✳✷✳✶✳✶✻✳ x; y; z ❜➜t ❦ý✱ ❧✉ỉ♥ ❝â ❈❤ù♥❣ ♠✐♥❤ ✈ỵ✐ ♠å✐ t❛♠ ❣✐→❝ ABC ✈➔ ❜❛ sè t❤ü❝ x2 + y + z ≥ 2xy cos C + 2xz cos B + 2yz cos A (1) ▲í✐ ❣✐↔✐ ❈❤å♥ ✤✐➸♠ I ❜➜t ❦ý tr♦♥❣ ♠➦t ♣❤➥♥❣ (ABC) ✈➔ ❞ü♥❣ ❜❛ ✈❡❝tì v~1; v~2; v~3 ❝â ✤ë ❞➔✐ ✤ì♥ ữủt ổ õ ợ BC; AC; AB ❚r♦♥❣ ✤â✱ |v~1| = |v~2| = |v~3| = ⇒ v~12 = v~22 = v~32 = ⑩♣ ❞ö♥❣ t ổ ữợ tỡ xv~1; yv~2; z v~3 t❛ ✤÷đ❝ ≤ (xv~1 +y v~2 +z v~3 )2 = x2 v~1 +y v~2 +z v~32 +2(xy v~1 v~2 +yz v~2 v~3 +xz v~1 v~3 ) (2) ▲↕✐ ❝â |v~1| = |v~2| = |v~3| = ⇒ v~12 = v~22 = v~32 = ❉♦ ✤â✱ v~1 v~2 = cos(v~1 ; v~2 ) = − cos C; v~2 v~3 = cos(v~2 ; v~3 ) = − cos A; v~1 v~3 = cos(v~1 v~3 ) = − cos B ❑❤✐ ✤â (2) t÷ì♥❣ ✤÷ì♥❣ ≤ (xv~1 + y v~2 + z v~3 )2 = x2 + y + z − 2(xy cos C + xz cos B + yz cos A) ⇔ x2 + y + z ≥ 2xy cos C + 2xz cos B + 2yz cos A ✸✼ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✈ỵ✐ ♠å✐ t❛♠ ❣✐→❝ ABC ✈➔ ❜❛ sè ữỡ m, n, p tũ ỵ ổ õ t ✷✳✷✳✶✳✶✼✳ A B C mnp m sin + n sin + p sin ≤ 2 2 1 + 2+ 2 m n p (1) ▲í✐ ❣✐↔✐ ❇✐➳♥ ✤ê✐ ✈➳ ♣❤↔✐ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ t❛ ❝â mnp 1 + 2+ 2 m n p mnp mnp mnp = + + 2 m2 n p np mp mn = = + + (n2 p2 + m2 p2 + m2 n2 ) 2 m n p 2mnp ❉♦ ✤â✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ (1) ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ ✤÷ì♥❣ A B C m2 n2 + m2 p2 + n2 p2 ≥ 2mnp m sin + n sin + p sin 2 ✣➦t mn = x; mp = y; np = z ✱ ❜➜t ✤➥♥❣ t❤ù❝ (1′) trð t❤➔♥❤ A+C A+B B+C + xz cos + yz cos x + y + z ≥ xy cos 2 2 2 (1′ ) = 2xy cos α + 2xz cos β + 2yz cos γ ❱ỵ✐ α = B +2 C ; β = A +2 C ; γ = A +2 B t↕♦ t❤➔♥❤ ❜❛ ❣â❝ tr♦♥❣ ♠ët t❛♠ ❣✐→❝✳ ◆❤÷ ✈➟② ❜➔✐ t♦→♥ ✤➣ ✤÷đ❝ ❜✐➳♥ ✤ê✐ ✤÷❛ ✈➲ ❜➔✐ t♦→♥ ✷✳✷✳✶✳✶✻✱ ❞♦ ✤â t❛ ❤♦➔♥ t♦➔♥ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü ❜➔✐ t♦→♥ ✷✳✷✳✶✳✶✻✳ ✷✳✷✳✷✳ ❇➔✐ t♦→♥ ❜✐➺♥ ❧✉➟♥ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝â t❤❛♠ sè ❚r♦♥❣ t❤ü❝ t➳ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ✤à♥❤ t➼♥❤ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝â t❤❛♠ sè✱ ú t ữủ q ợ rt ữỡ ♣❤→♣ ♥❤÷ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✈➔ ✤õ✱ t❛♠ t❤ù❝ ❜➟❝ ❤❛✐✱ ❣✐→ trà ❧ỵ♥ ♥❤➜t ✈➔ ♥❤ä ♥❤➜t✳✳✳❚r♦♥❣ s ợ t ởt ữỡ ❜✐➺♥ ❧✉➟♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ❝â t số Pữỡ ỗ t ị t÷ð♥❣ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❧➔ ❞ü❛ ✈➔♦ ❝→❝ ✤➦❝ tr÷♥❣ ♠❛♥❣ t➼♥❤ ❍➻♥❤ ❤å❝ ❝õ❛ ❜➔✐ t♦→♥ ♥❤÷ ♣❤÷ì♥❣ tr➻♥❤ ✤÷í♥❣ t❤➥♥❣✱ ✤÷í♥❣ trá♥✱ ❝→❝ ❜✐➸✉ ❞✐➵♥ ❍➻♥❤ ❤å❝ ❝õ❛ t➟♣ ❤đ♣ ♥❣❤✐➺♠ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤✱ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ ♠➔ t➻♠ r❛ ❧í✐ ❣✐↔✐ t❤➼❝❤ ❤đ♣ ❝❤♦ ❜➔✐ t♦→♥✳ ❙❛✉ ✤➙② ❧➔ ♠ët sè ✈➼ ❞ö ♠✐♥❤ ❤å❛ tr➼❝❤ ❞➝♥ tø ❚↕♣ ❝❤➼ ❚♦→♥ ❍å❝ ✈➔ ❚✉ê✐ ✸✽ ❚r➫ ✈➔ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❇➔✐ t♦→♥ ✷✳✷✳✷✳✶✳ ❇✐➺♥ ❧✉➟♥ t❤❡♦ ♠ sè ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ √ − x2 ❂ mx + − m (1) ❚❛ ❜✐➳t r➡♥❣ sè ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (1) ❝❤➼♥❤ ❧➔ sè ❣✐❛♦ √ ✤✐➸♠ ❝õ❛ ✷ ✤÷í♥❣ y = − x2 ✈➔ y = mx + − m✳ √ √ ❱➻ ② ❂ − x2 ⇔ x2 + y = y ỗ t❤à ❝õ❛ ② ❂ − x2 ❧➔ ♥û❛ ✤÷í♥❣ trá♥ ✭♣❤➛♥ ♥➡♠ tr➯♥ trö❝ ❤♦➔♥❤✮ t➙♠ t↕✐ ❣è❝ tå❛ ✤ë✱ ❜→♥ ❦➼♥❤ ❜➡♥❣ 2✳ ❈á♥ y = mx + − m ❧➔ ♠ët ❤å ✤÷í♥❣ t❤➥♥❣ ❧✉ỉ♥ ✤✐ q✉❛ ✤✐➸♠ ❝è ✤à♥❤ A(1; 2) ✈ỵ✐ ♠å✐ m✳ ❚❛ t õ t t ợ ữớ trỏ tứ A ữớ t y = s s ợ trư❝ ❤♦➔♥❤ ✈➔ t✐➳♣ t✉②➳♥ AD ✭①❡♠ ❤➻♥❤ ✈➩✮✳ ▲í✐ ❣✐↔✐ ❍➻♥❤ ✶✷ ●å✐ B(−2; 0) ✈➔ C(2; 0) ❧➔ ❤❛✐ ✤➛✉ ♠ót ❝õ❛ ✤÷í♥❣ ❦➼♥❤ BOC ✳ ●✐↔ sû m1 ; m2 ; m3 ; m4 t÷ì♥❣ ù♥❣ ❧➔ ❝→❝ ❤➺ sè ❣â❝ ❝õ❛ ❝→❝ ✤÷í♥❣ t❤➥♥❣ [ = −2 ❀m2 = − tan DCO \= AC; AD; AB; AE t❤➻ ❞➵ ❞➔♥❣ t❤➜② m1 = − tan ACO [ = −2✮❀ m3 = tan ABO [ = 2❀ \ = − tan(2OAE) [ = −4 ✭✈➻ tan OAE − tan EAD ❝á♥ m4 = ❚ø ✤â s✉② r❛ −4 ✶✮ P❤÷ì♥❣ tr➻♥❤ (1) ❝â ♥❣❤✐➺♠ ❦❤✐ < m ≤ ❤♦➦❝ −2 ≤ m < 3 ✷✮ P❤÷ì♥❣ tr➻♥❤ (1) ❝â ♥❣❤✐➺♠ ❦❤✐ m > ❤♦➦❝ m < −2 ❤♦➦❝ m = −4 ❤♦➦❝ m = −4 < m < 0✳ ✸✮ P❤÷ì♥❣ tr➻♥❤ (1) ✈æ ♥❣❤✐➺♠ ❦❤✐ ❇➔✐ t♦→♥ ✷✳✷✳✷✳✷✳ p ❚➻♠ m ✤➸ ❜➜t ♣❤÷ì♥❣ tr➻♥❤ (4 + x)(6 − x) ≤ x2 − 2x + m (1) ✸✾ ✤ó♥❣ ✈ỵ✐ ♠å✐ −4 ≤ x ≤ 6✳ ▲í✐ ❣✐↔✐ t❛ ❝â y ≥ ✈➔ (x − 1)2 + y = 25 ✣➦t y = (4 + x)(6 − x)✱ t❤➻ p ỗ t số y = (4 + x)(6 − x) ❧➔ ♥û❛ ✤÷í♥❣ trá♥ ✭♣❤➛♥ ♥➡♠ tr➯♥ trö❝ ❤♦➔♥❤✮ t➙♠ t↕✐ ✤✐➸♠ O1(1; 0) ✈➔ ❜→♥ ❦➼♥❤ 5✳ ❈á♥ y = x2 − 2x + m ❧➔ ♣❛r❛❜♦❧ ❧✉ỉ♥ ❝â ❝ü❝ t✐➸✉ ♥➡♠ tr➯♥ ✤÷í♥❣ x = ✭①❡♠ ❤➻♥❤ ♠✐♥❤ ❤å❛✮✳ p ❍➻♥❤ ✶✸ ✣➸ (1) ✤ó♥❣ ✈ỵ✐ ♠å✐ −4 ≤ x ≤ 6✱ t❤➻ ♣❛r❛❜♦❧ y = x2 − 2x + m ❧✉æ♥ ❧✉æ♥ ♣❤↔✐ ♥➡♠ tr➯♥ ♥û❛ ✤÷í♥❣ trá♥✱ tù❝ ❧➔ ❦❤✐ ✤➾♥❤ ♣❛r❛❜♦❧ ð tr➯♥ ✤÷í♥❣ t❤➥♥❣ x = 1✱ ♣❤↔✐ ♥➡♠ tr➯♥ ✤✐➸♠ M (1; 5)✳ ∆ ❚ø ✤â t❛ ❝â − 4a ≥ ⇔ m − ≥ ⇔ m ≥ 6✳ ❱➟② ❣✐→ trà ❝õ❛ ♠ ❝➛♥ t➻♠ ❧➔ m ≤ 6✳ ❇➔✐ t♦→♥ ✷✳✷✳✷✳✸✳ t❤❡♦ a ▲í✐ ❣✐↔✐ ❇✐➺♥ ❧✉➟♥ t❤❡♦ m sè ♥❣❤✐➺♠ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ x2 + y = (1) √ (ay + x)(x − a 3) = (2) ❉➵ t❤➜② ♣❤÷ì♥❣ tr➻♥❤ (1) ❜✐➸✉ ❞✐➵♥ ✤÷í♥❣ trá♥ t➙♠ ❧➔ ❣è❝ tå❛ ✤ë O(0;√0)✱ ❜→♥ ❦➼♥❤ R = 3✱ ❝á♥ ♣❤÷ì♥❣ tr➻♥❤ (2) ❜✐➸✉ ❞✐➵♥ ❜ð✐ ❤❛✐ ❞÷í♥❣ t❤➥♥❣ x = a ✈➔ y = − a1 x ✭ ♥➳✉ a 6= ✮✳ ❙è ♥❣❤✐➺♠ ❝õ❛ ❤➺ ❝❤➼♥❤ ❧➔ sè ❣✐❛♦ ✤✐➸♠ ❝õ❛ ❤❛✐ ✤÷í♥❣ t ợ ữớ trỏ t a > ✭✈➻ ❦❤✐ a < t❛ ❝â ❦➳t ✹✵ q✉↔ t÷ì♥❣ tü✱ ❝á♥ ❦❤✐ a = t❤➻ (2) ⇔ x = ✈➔ ❧ó❝ ✤â ❤➺ ❝â ♥❣❤✐➺♠✮✳ √ a √ (x0 ; y0 ) ỗ q t t õ x20 + y02 = 9; x0 = a ✈➔ y0 ❂ − x0 ✱ ✈➔ a √ ❞♦ a > ♥➯♥ s✉② r❛ a = 2✳ ❚ø ✤â sè ❣✐❛♦ ✤✐➸♠ ❝õ❛ ❤❛✐ ✤÷í♥❣ t❤➥♥❣ tr➯♥ ❚❛ ①➨t ①❡♠ ❦❤✐ ♥➔♦ x = a 3✱ y = − x ✈➔ ữớ trỏ ỗ q ợ ữớ trỏ ữủ ổ t↔ tr♦♥❣ ❤➻♥❤ ✈➩ ♠✐♥❤ ❤å❛✳ ❱➟② t❛ ✤✐ ✤➳♥ ❦➳t ❧✉➟♥ √ √ ❛✮ ❍➺ ❝â ♥❣❤✐➺♠ ❦❤✐ < |a| < ✈➔ |a| = ❍➻♥❤ ✶✹❛ ❜✮ ❍➺ ❝â ♥❣❤✐➺♠ ❦❤✐ |a| = √ ❤♦➦❝ |a| = √ ❍➻♥❤ ✶✹❜ ❝✮ ❍➺ ❝â ♥❣❤✐➺♠ ❦❤✐ a = ❤♦➦❝ |a| > √ ✹✶ ❍➻♥❤ ✶✹❝ ❇➔✐ t♦→♥ ✷✳✷✳✷✳✹✳ ❈❤♦ ❝→❝ sè ❞÷ì♥❣ a, b, c t❤ä❛ ♠➣♥ √ √ a+ b+ c = ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t ▲í✐ ❣✐↔✐ √ √ √ √ y−a+ z−a=1 √ √ y−b+ x−b=1 √x − c + √z − c = ❳➨t t❛♠ ❣✐→❝ ✤➲✉ ABC ❝â ❝↕♥❤ ❜➡♥❣ 1✳ ❚r♦♥❣ ❤➻♥❤ ❤å❝ ♣❤➥♥❣✿ ✧◆➳✉ M ❧➔ ♠ët ✤✐➸♠ ❜➜t ❦ý tr♦♥❣ t❛♠ ❣✐→❝ ✤➲✉✱ t❤➻ tê♥❣ ❦❤♦↔♥❣ ❝→❝❤ tø M ①✉è♥❣ ❜❛ ❝↕♥❤ t❛♠ ❣✐→❝ ✤➲✉ ❜➡♥❣ ❝❤✐➲✉ ❝❛♦ ❝õ❛ ♥â✧✳ ❈❤ù♥❣ ♠✐♥❤✳ ●å✐ ❝↕♥❤ t❛♠ ❣✐→❝ ✤➲✉ ❧➔ a(a > 0)✳ √ a ❑➫ ✤÷í♥❣ ❝❛♦ AH ✳ ❙û ❞ư♥❣ ỵ Ptr t t ữủ AH = ●å✐ h1 , h2 , h3 ❧➛♥ ❧÷đt ❧➔ ❦❤♦↔♥❣ ❝→❝❤ tø M ✤➳♥ ❜❛ ❝↕♥❤ BC; AB; AC ❝õ❛ t❛♠ ❣✐→❝ ABC ✳ ❚❛ ❝â SABC = SOBC + SOAB + SOAC 1 ❂ h1 a + h2 a + h3 a 2 ❂ a.(h1 + h2 + h3 ) √ a a ▼➦t ❦❤→❝✱ SABC = AH.BC = √2 √ 1 a a ❉♦ ✤â a.(h1 + h2 + h3 ) ❂ a ⇒ h1 + h2 + h3 ❂ = AH ✭✤✐➲✉ 2 2 ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✮