Một số bài toán số học trong hình học phẳng

ĐẠI HỌC THÁI NGUYÊN TRƢỜNG ĐẠI HỌC KHOA HỌC  - LÊ PHƢƠNG THẢO MỘT SỐ BÀI TOÁN SỐ HỌC TRONG HÌNH HỌC PHẲNG LUẬN VĂN THẠC SĨ TOÁN HỌC THÁI NGUYÊN - 2019 ĐẠI HỌC THÁI NGUYÊN TRƢỜNG ĐẠI HỌC KHOA HỌC  - LÊ PHƢƠNG THẢO MỘT SỐ BÀI TỐN SỐ HỌC TRONG HÌNH HỌC PHẲNG Chuyên ngành: Phƣơng pháp Toán sơ cấp Mã số: 46 01 13 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC PGS.TS NGUYỄN VIỆT HẢI THÁI NGUYÊN - 2019 ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ▲➯ P❤÷ì♥❣ ❚❤↔♦ ▼❐❚ ❙➮ ❇⑨■ ❚❖⑩◆ ❙➮ ❍➴❈ ❚❘❖◆● ❍➐◆❍ ❍➴❈ P❍➃◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✾ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ì Pữỡ ❇⑨■ ❚❖⑩◆ ❙➮ ❍➴❈ ❚❘❖◆● ❍➐◆❍ ❍➴❈ P❍➃◆● ❈❤✉②➯♥ ♥❣➔♥❤✿ P❤÷ì♥❣ ♣❤→♣ t♦→♥ ❝➜♣ ▼➣ sè✿ ✽✹✻✵✶✶✸ ▲❯❾◆ ❱❿◆ ữớ ữợ P ◆●❯❨➍◆ ❱■➏❚ ❍❷■ ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✾ ✐ ▲í✐ ❝↔♠ ì♥ ✣➸ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ ❧✉➟♥ ✈➠♥ ♠ët ❝→❝❤ ❤♦➔♥ tổ ổ ữủ sỹ ữợ ú ✤ï ♥❤✐➺t t➻♥❤ ❝õ❛ P●❙✳❚❙✳ ◆❣✉②➵♥ ❱✐➺t ❍↔✐✱ ●✐↔♥❣ ✈✐➯♥ ❝❛♦ ❝➜♣ ❚r÷í♥❣ ✤↕✐ ❤å❝ ❍↔✐ P❤á♥❣✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ t❤➛② ✈➔ ①✐♥ ❣û✐ ❧í✐ tr✐ ➙♥ ♥❤➜t ❝õ❛ tỉ✐ ✤è✐ ✈ỵ✐ ♥❤ú♥❣ ✤✐➲✉ t❤➛② ✤➣ ❞➔♥❤ ❝❤♦ tỉ✐✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ♣❤á♥❣ ✣➔♦ t↕♦✱ ❑❤♦❛ ❚♦→♥ ❚✐♥✱ qỵ t ổ ợ ✲ ✷✵✷✵✮ ❚r÷í♥❣ ✤↕✐ ❤å❝ ❦❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t➟♥ t➻♥❤ tr✉②➲♥ ✤↕t ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ qỵ ụ ữ t tổ t❤➔♥❤ ❦❤â❛ ❤å❝✳ ❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t t tợ ỳ ữớ ❧✉ỉ♥ ✤ë♥❣ ✈✐➯♥✱ ❤é trđ ✈➔ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❳✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥✦ ❍↔✐ P❤á♥❣✱ t❤→♥❣ ✳✳✳ ♥➠♠ ✷✵✳✳✳ ◆❣÷í✐ ✈✐➳t ▲✉➟♥ ✈➠♥ ▲➯ P❤÷ì♥❣ ❚❤↔♦ ✐✐ ❉❛♥❤ ♠ö❝ ❤➻♥❤ ✶✳✶ ✶✳✷ ✶✳✸ ✶✳✹ ✶✳✺ ❚❛♠ ❣✐→❝ P②t❤❛❣♦r❡✿ BC = AB + AC ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚❛♠ ❣✐→❝ ❍❡r♦♥ [c, e, b + d]✱ ✤÷í♥❣ ❝❛♦ a ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚❛♠ ❣✐→❝ ❍❡r♦♥ t❤❡♦ sü t➠♥❣ ❞➛♥ ❝õ❛ ❝↕♥❤ ❧ỵ♥ ♥❤➜t ✳ ❚❛♠ ❣✐→❝ P②t❤❛❣♦r❡ ❝ì ❜↔♥ ✈➔ ❝→❝ ❜→♥ ❦➼♥❤ r, , rb , rc ❚➼♥❤ ❝❤➜t ❝→❝ ❝❡✈✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✳ ✽ ✳ ✶✷ ✳ ✶✸ ✳ ✶✾ ✷✳✶ ❍❛✐ ♥❣❤✐➺♠ ❧➔ t❛♠ ❣✐→❝ ✈✉ỉ♥❣ ✈ỵ✐ m = ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✷ ❍❛✐ ♥❣❤✐➺♠ ❧➔ t❛♠ ❣✐→❝ tò ✈ỵ✐ m = ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✸ ❚❛♠ ❣✐→❝ ❝↕♥❤ tü ♥❤✐➯♥ ♥❣♦↕✐ t✐➳♣ ✤÷í♥❣ trá♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✶ ✸✳✷ ✸✳✸ ✸✳✹ ✸✳✺ ✸✳✻ ❚ù ❣✐→❝ ❤ú✉ t✛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚ù ❣✐→❝ ❤ú✉ t✛ ❝õ❛ ❇r❛❤♠❛❣✉♣t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✣ë ❞➔✐ ✷ ✤÷í♥❣ ❝❤➨♦✱ ❝❤✉ ✈✐✱ ❞✐➺♥ t➼❝❤ tù ❣✐→❝ ✳ ❉ü♥❣ tù ❣✐→❝ ❇r❛❤♠❛❣✉♣t❛ tø t❛♠ ❣✐→❝ ❍❡r♦♥ ❍❛✐ ✤÷í♥❣ ❝❤➨♦ AB, BC ∈ P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✭■▼❖ ✶✾✻✽✱ #1✮✱ ❈→❝❤ ❣✐↔✐ t❤ù ❜❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✹ ✹✺ ✹✼ ✺✷ ✺✹ ✐✐✐ ❉❛♥❤ ♠ö❝ ❜↔♥❣ ✶✳✶ ❚r➼❝❤ ❞❛♥❤ s→❝❤ ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷ ❍å t❛♠ ❣✐→❝ ❍❡r♦♥ ♣❤ư t❤✉ë❝ λ✱ ✈ỵ✐ ✶✵ ❣✐→ trà λ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✶ ❇❛ ❝↕♥❤ ❧➔ ❝➜♣ sè ❝ë♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✷ ❇➔✐ t♦→♥ P = nS ✈ỵ✐ n = 31 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✷✳✸ ❇➔✐ t♦→♥ P = nS ✈ỵ✐ n = 42 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✐✈ ▼ư❝ ❧ư❝ ●✐ỵ✐ t❤✐➺✉ ❧✉➟♥ ✈➠♥ ✶ ❚❛♠ ❣✐→❝ P②t❤❛❣♦r❡ ✈➔ t❛♠ ❣✐→❝ ❍❡r♦♥ ✶✳✶ ❇➔✐ t♦→♥ t➻♠ t❛♠ ❣✐→❝ P②t❤❛❣♦r❡ ✈➔ t❛♠ ❣✐→❝ ❍❡r♦♥ ✶✳✶✳✶ ❈→❝ ❜ë ❜❛ P②t❤❛❣♦r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✷ ❈→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ❇➔✐ t♦→♥ HG✿ ❚❛♠ ❣✐→❝ ❍❡r♦♥ ✈ỵ✐ r, , rb , rc ∈ N ✳ ✳ ✶✳✸ ❍å ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ♣❤ö t❤✉ë❝ λ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❚❛♠ ❣✐→❝ ❝↕♥❤ ♥❣✉②➯♥ ✈ỵ✐ ❤➺ t❤ù❝ ❣✐ú❛ S ✈➔ P ✷✳✶ ❚❛♠ ❣✐→❝ ❝↕♥❤ ♥❣✉②➯♥ ✈ỵ✐ S = m.P, m ∈ N ✳ ✳ ✷✳✶✳✶ ❚❤✉➟t t♦→♥ ●♦❡❤❧ ✈➔ t❤✉➟t t♦→♥ ▼❛r❦♦✈ ✷✳✶✳✷ ❍❛✐ tr÷í♥❣ ❤đ♣ t❤❛♠ sè ♥❣✉②➯♥ ✳ ✳ ✳ ✳ ✷✳✷ ❚❛♠ ❣✐→❝ ❝↕♥❤ ♥❣✉②➯♥ ✈ỵ✐ P = nS, n ∈ N ✳ ✳ ✷✳✷✳✶ ❚r÷í♥❣ ❤đ♣ n ❧➔ sè ♥❣✉②➯♥ tè ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✷ ❚r÷í♥❣ ❤đ♣ n ❧➔ ❤đ♣ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✸ ❚r÷í♥❣ ❤đ♣ r✐➯♥❣✿ ❚❛♠ ❣✐→❝ P②t❤❛❣♦r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✹ ✳ ✹ ✳ ✹ ✳ ✼ ✳ ✶✵ ✳ ✶✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✹ ✷✺ ✸✷ ✸✺ ✸✻ ✸✽ ✸✾ ✸ ▼ët sè ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ✹✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✺✽ ✸✳✶ ❚ù ❣✐→❝ ❝â ❝↕♥❤ ✈➔ ✤÷í♥❣ ❝❤➨♦ ❤ú✉ t✛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✸✳✷ ❳→❝ ✤à♥❤ ❝→❝ ②➳✉ tè ❝õ❛ tù ❣✐→❝ ❇r❛❤♠❛❣✉♣t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✸✳✸ ●✐ỵ✐ t❤✐➺✉ ♠ët sè ❜➔✐ t♦→♥ t❤✐ ❖❧②♠♣✐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✶ ●✐ỵ✐ t❤✐➺✉ ❧✉➟♥ ✈➠♥ ✶✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ◆❤✐➲✉ ❜➔✐ t♦→♥✱ ❦❤→✐ ♥✐➺♠ tr♦♥❣ ❤➻♥❤ ❤å❝ ❧✐➯♥ q✉❛♥ ✤➳♥ sè ❤å❝✳ ✣➦❝ ❜✐➺t ❝â ♥❤ú♥❣ ❜➔✐ t♦→♥ ❤♦➔♥ t♦➔♥ t❤✉ë❝ ❧➽♥❤ ✈ü❝ sè ❤å❝ ♥❤÷ ❜ë ❜❛ P②t❤❛❣♦r❡✱ t❛♠ ❣✐→❝ ❍❡r♦♥✱✳✳✳✣➸ ❣✐↔✐ q✉②➳t ♥❤ú♥❣ ❜➔✐ t♦→♥ ♥➔② t❤÷í♥❣ ♣❤↔✐ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡✱ ♣❤÷ì♥❣ tr➻♥❤ P②t❤❛❣♦r❡✱ ♣❤÷ì♥❣ tr➻♥❤ P❡❧❧✱✳✳✳✈➔ ♥❤✐➲✉ ❦✐➳♥ t❤ù❝ s➙✉ ✈➲ sè ♥❣✉②➯♥ tè ♥â✐ r✐➯♥❣ ✈➔ sè ❤å❝ ♥â✐ ❝❤✉♥❣✳ ✣➲ t➔✐ ♥➔② tr➻♥❤ ❜➔② ♥❤✐➲✉ ✈➜♥ ✤➲ ❝õ❛ sè ❤å❝ →♣ ❞ö♥❣ ✈➔♦ ❤➻♥❤ ❤å❝✱ ♠❛♥❣ ❧↕✐ ♥❤ú♥❣ ❦➳t q✉↔ s➙✉ s➢❝ ✈➲ ❜➔✐ t♦→♥ ❤➻♥❤ ❤å❝ ❣✐↔✐ ❜➡♥❣ ❦✐➳♥ t❤ù❝ sè ❤å❝✳ ▼ö❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧➔✿ ✲ ❚r➻♥❤ ❜➔② ❤❛✐ ❜➔✐ t♦→♥✿ t➻♠ ❝→❝ t❛♠ ❣✐→❝ P②t❤❛❣♦r❡✱ t➻♠ ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t✳ ◆➯✉ r❛ ❝→❝ t❤✉➟t t♦→♥ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ❝→❝ ❜➔✐ t♦→♥ ✤➦t r❛✳ ❈→❝ tr÷í♥❣ ❤đ♣ r✐➯♥❣ ①→❝ ✤à♥❤ t❛♠ ❣✐→❝ ❍❡r♦♥✿ ❇➔✐ t♦→♥ HG t➻♠ t❛♠ ❣✐→❝ ❍❡r♦♥ ✈ỵ✐ r, , rb , rc ∈ N✱ t❛♠ ❣✐→❝ ❝â ❝→❝ t số ữợ t ❣✐→❝ ❍❡r♦♥✱✳✳✳ ✲ ❙û ❞ö♥❣ ❝→❝ ❦✐➳♥ t❤ù❝ ❝õ❛ sè ữ ỵ tt t sỹ t ởt sè tü ♥❤✐➯♥ t❤➔♥❤ ❝→❝ sè ♥❣✉②➯♥ tè✱ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❉✐♦♣❤❛♥t✐♥❡✱ ❝→❝ ❧➟♣ ❧✉➟♥ sè ❤å❝ ♥â✐ ❝❤✉♥❣✱✳✳✳✤➸ ♥❣❤✐➯♥ ❝ù✉ ♠ët sè tr÷í♥❣ ❤đ♣ r✐➯♥❣ q✉❛♥ trå♥❣ ❝õ❛ ❜➔✐ t♦→♥ t➻♠ t❛♠ ❣✐→❝ ❝↕♥❤ ♥❣✉②➯♥ t❤ä❛ ♠➣♥ ♠ët tr♦♥❣ ❜❛ ✤✐➲✉ ❦✐➺♥ s❛✉ S = mP ; P = nS ❤❛② R/r = N ∈ N ✲ ◆➯✉ r❛ ❝→❝ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✈➔ ❝→❝❤ ❣✐↔✐ q✉②➳t ú ự ỳ t tự rt ỗ ữù ♥➠♥❣ ❧ü❝ ❞↕② ❝→❝ ❝❤✉②➯♥ ✤➲ ❦❤â ð tr÷í♥❣ ❚❍❈❙ ✈➔ ❚❍P❚ ❣â♣ ♣❤➛♥ ✤➔♦ t↕♦ ❤å❝ s✐♥❤ ❣✐→✐ ♠æ♥ ❍➻♥❤ ❤å❝✳ ✷ ✷✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ✤➲ t➔✐✱ ♥❤ú♥❣ ✈➜♥ ✤➲ ❝➛♥ ❣✐↔✐ q✉②➳t ❉ü❛ ✈➔♦ ❝→❝ t➔✐ ❧✐➺✉ ❬✷❪✱ ❬✸❪✱ ❬✹❪✱ ❬✻❪ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ♠ët sè ❜➔✐ t♦→♥ ❤❛② ✈➲ t❛♠ ❣✐→❝ ♥❣✉②➯♥ ✈➔ ❝ô♥❣ ❧➔ ♥❤ú♥❣ ❜➔✐ t♦→♥ ❦❤â ❤❛② ❣➦♣ tr♦♥❣ ❝→❝ ❦ý t❤✐ s tr ữợ qố t ❞✉♥❣ ❧✉➟♥ ✈➠♥ ❝❤✐❛ ❧➔♠ ✸ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✳ ❚❛♠ ❣✐→❝ P②t❤❛❣♦r❡ ✈➔ t❛♠ ❣✐→❝ ❍❡r♦♥ ❇➔✐ t♦→♥ t➻♠ ❜ë ❜❛ P②t❤❛❣♦r❡ ❧➔ ❜➔✐ t♦→♥ sè ❤å❝ q✉❡♥ t❤✉ë❝✱ t✉② ♥❤✐➯♥ ❦❤æ♥❣ t❤➸ ❦❤æ♥❣ ♥❤➢❝ ❧↕✐ ❝→❝ ❦➳t q✉↔ ✤➣ ❝â tr♦♥❣ ♥❤✐➲✉ ❝ỉ♥❣ tr➻♥❤✳ ❱✐➺❝ ❧➔♠ ♥➔② ❝ơ♥❣ ❝♦✐ ❧➔ ❜ê s✉♥❣ ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✤➛✉ t✐➯♥ ❝õ❛ ❜➔✐ t♦→♥ ✤➦t r❛✳ ❇➔✐ t♦→♥ t➻♠ t❛♠ ❣✐→❝ r tợ trữớ ủ r tú ❦➳t t❤ó❝ ð ♠ët ❦➳t q✉↔ tê♥❣ q✉→t✿ ❍å ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ♣❤ư t❤✉ë❝ t❤❛♠ sè✳ ❈❤÷ì♥❣ ♥➔② ❜❛♦ ỗ t t t Ptr t ❣✐→❝ ❍❡r♦♥ ✶✳✷✳ ❇➔✐ t♦→♥ HG✿ ❚❛♠ ❣✐→❝ ❍❡r♦♥ ✈ỵ✐ r, , rb , rc ∈ N ✶✳✸✳ ❍å ❝→❝ t❛♠ ❣✐→❝ ❍❡r♦♥ ♣❤ư t❤✉ë❝ λ✳ ❈❤÷ì♥❣ ✷✳ ❚❛♠ ❣✐→❝ ❝↕♥❤ ♥❣✉②➯♥ ✈ỵ✐ ❤➺ t❤ù❝ ❣✐ú❛ S ✈➔ P ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ♥➔② ✤➲ ❝➟♣ ✤➳♥ ❤❛✐ ❜➔✐ t♦→♥ ✈➲ t➻♠ t❛♠ ❣✐→❝ ❝↕♥❤ ♥❣✉②➯♥ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ♣❤ư✿ ❚➻♠ t❛♠ ❣✐→❝ ❝↕♥❤ ♥❣✉②➯♥ ✈ỵ✐ S = mP ✈➔ t➻♠ t❛♠ ❣✐→❝ ❝↕♥❤ ♥❣✉②➯♥ ✈ỵ✐ P = nS ✳ ❈→❝ ❦ÿ t❤✉➟t sè ❤å❝ ✤÷đ❝ ✈➟♥ ❞ư♥❣ ữỡ tr t t tợ ❝→❝ t❤✉➟t t♦→♥ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➡♥❣ ❝→❝ ♣❤➛♥ ♠➲♠ t ữỡ ỗ s ❚❛♠ ❣✐→❝ ❝↕♥❤ ♥❣✉②➯♥ ✈ỵ✐ S = mP, m ∈ N ✷✳✷✳ ❚❛♠ ❣✐→❝ ❝↕♥❤ ♥❣✉②➯♥ ✈ỵ✐ P = nS, n ∈ N✳ ❈❤÷ì♥❣ ✸✳ ▼ët sè ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ❈❤÷ì♥❣ ✸ ①➨t ❜➔✐ t♦→♥ t❛♠ ❣✐→❝ ♥❣✉②➯♥ rở tự ỳ t ợ ữỡ t✐➳♣ ❝➟♥ t÷ì♥❣ tü ✷ ❝❤÷ì♥❣ ✶ ✈➔ ✷✳ P❤➨♣ ❞ü♥❣ tù ❣✐→❝ ❤ú✉ ✹✹ ❍➻♥❤ ✸✳✷✿ ❚ù ❣✐→❝ ❤ú✉ t✛ ❝õ❛ ❇r❛❤♠❛❣✉♣t❛ ◆❤÷ ✈➟②✱ (ξ, η) ❧➔ ♠ët ✤✐➸♠ ❤ú✉ t✛ tr➯♥ ✤÷í♥❣ ❝♦♥❣ ❜➟❝ ✸ αX (Y − c)2 − − γY (X + c)2 − = ✭✸✳✶✮ ✲ ●✐↔ sû r➡♥❣ (ξ, η) ❧➔ ♥❣❤✐➺♠ ❤ú✉ t✛ ❝õ❛ ✭✸✳✶✮ t❤➻ t❛ ❝â ✷ t❛♠ ❣✐→❝ ❤ú✉ t✛ ❆❇❊ ✈➔ ❇❊❈ ✳ ✣➸ ❝â tù ❣✐→❝ ❤ú✉ t✛ t❛ ❝➛♥ ❝→❝ t❛♠ ❣✐→❝ ❆❊❉ ✈➔ ❈❊❉ ❧➔ ❤ú✉ t✛✱ ❤➻♥❤ ✲ ●✐↔ sû AD = l, DC = m, ED = δ ✳ ❚❛♠ ❣✐→❝ AED ✤÷đ❝ ①→❝ ✤à♥❤ l + δ + cα ❜ð✐ t❤❛♠ sè ❤ú✉ t✛ x = ✈ỵ✐ α δ (x − c)2 − l x − c2 + = ✈➔ = α 2x α 2x m + c t õ ữỡ tỹ ợ t CED ❧➜② y = 2x δ (y + c)2 − m y − c2 + = ✈➔ = γ 2y γ 2y ❄❄ ❑❤✐ ✤â✱ (x, y) s➩ ❧➔ ✤✐➸♠ ❤ú✉ t✛ tr➯♥ ✤÷í♥❣ ❜➟❝ ❜❛ αY (X − c)2 − − γX (Y + c)2 − = ✭✸✳✷✮ ✹✺ ◆❣÷đ❝ ❧↕✐ ♥➳✉ (x, y) ❧➔ ✤✐➸♠ ❤ú✉ t✛ tr➯♥ ✤÷í♥❣ ❝♦♥❣ ❜➟❝ ❜❛ ✭✸✳✷✮ s❛♦ ❝❤♦ δ δ ❝→❝ sè ❤ú✉ t✛ ✈➔ ❧➔ sè ❞÷ì♥❣ t❤➻ t❛ ❝â ❝→❝ t❛♠ ❣✐→❝ ❤ú✉ t✛ AED ✈➔ α γ ❈❊❉ ✳ ◆❤÷ ✈➟②✱ ♥➳✉ (ξ, η) ❧➔ ✤✐➸♠ ❤ú✉ t✛ tr➯♥ ✤÷í♥❣ ❜➟❝ ❜❛ ✭✸✳✶✮ ❦➨♦ t❤❡♦ ABC ❧➔ t❛♠ ❣✐→❝ ❤ú✉ t✛✱ t❤➻ ♠å✐ ✤✐➸♠ ❤ú✉ t✛ tr➯♥ ✭✸✳✷✮ ❝❤♦ t❛♠ ❣✐→❝ ❤ú✉ t✛ ACD ✱ s➩ ①→❝ ✤à♥❤ ♠ët tù ❣✐→❝ ❤ú✉ t✛ ABCD✳ ✸✳✷ ❳→❝ ✤à♥❤ ❝→❝ ②➳✉ tè ❝õ❛ tù ❣✐→❝ ❇r❛❤♠❛❣✉♣t❛ ◆ë✐ ❞✉♥❣ ♣❤➛♥ ♥➔② tr➻♥❤ ❜➔② ❝→❝❤ ❞ü♥❣ tù ❣✐→❝ ❇r❛❤♠❛❣✉♣t❛ ❞ü❛ ✈➔♦ ❣â❝ ❍❡r♦♥✱ t❛♠ ❣✐→❝ ❍❡r♦♥✳ ❚❛ ❤➣② ♥❤➢❝ ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥✳ ❙❛✉ ✤➙② t❛ s➩ ❜ê s✉♥❣ t❤➯♠ ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲ ❣â❝ ❍❡r♦♥✳ ❍➻♥❤ ✸✳✸✿ ✣ë ❞➔✐ ✷ ✤÷í♥❣ ❝❤➨♦✱ ❝❤✉ ✈✐✱ ❞✐➺♥ t➼❝❤ tù ❣✐→❝ ❍➻♥❤ ✸✳✸ ❝❤➾ r❛ ♠ët ❝✉♥❣ AB ❝õ❛ ✤÷í♥❣ trá♥ ❜→♥ ❦➼♥❤ R✳ ●✐↔ sû C ✈➔ C ❧➔ ✷ ✤✐➸♠ tr➯♥ ✤÷í♥❣ trá♥ ♥➡♠ ✈➲ ✷ ♣❤➼❛ ❝õ❛ AB ✳ ❚❛ ❝â✿ ACB + AC B = π ✈➔ AB = 2R sin ỵ S, s t ♥û❛ ❝❤✉ ✈✐ tù ❣✐→❝ ABCD✳ t❛ ❝â (ac + bd)(ad + bc) e= ✭✸✳✸✮ ab + cd ✹✻ f= S= (ac + bd)(ab + cd) ad + bc ✭✸✳✹✮ s = (a + b + c + d) ✭✸✳✺✮ (s − a)(s − b)(s − c)(s − d) ✭✸✳✻✮ ◆➳✉ d = tù ❣✐→❝ trð t❤➔♥❤ t❛♠ ❣✐→❝ ✈➔ ❝æ♥❣ t❤ù❝ ✭✸✳✻✮ t❤➔♥❤ ❝æ♥❣ t❤ù❝ ❍❡r♦♥ ♥ê✐ t✐➳♥❣✳ ❈→❝ ❝æ♥❣ t❤ù❝ ✭✸✳✸✮✱ ✭✸✳✹✮✱ ✭✸✳✻✮ ❧➔ ♥❤ú♥❣ ❝ỉ♥❣ t❤ù❝ ✤➦❝ tr÷♥❣ ❝õ❛ tù ❣✐→❝ ♥ë✐ t✐➳♣ ❝ơ♥❣ ữ ổ tự Pt ố ợ tự t ✣à♥❤ ♥❣❤➽❛ ✸✳✷✳ ▼ët ❣â❝ ✤÷đ❝ ❣å✐ ❧➔ ❣â❝ ❍❡r♦♥ ♥➳✉ s✐♥ ✈➔ ❝æ s✐♥ ❝õ❛ ♥â ❧➔ ❝→❝ sè ❤ú✉ t✛✳ 2t − t2 θ ❱➻ sin θ = , cos θ = ✈ỵ✐ t = tan ♥➯♥ ❣â❝ θ ❧➔ ❤ú✉ t✛ 2 1+t 1+t θ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ tan ❧➔ sè ❤ú✉ t✛✳ ❘ã r➔♥❣ tê♥❣ ✈➔ ❤✐➺✉ ❝→❝ ❣â❝ ❍❡r♦♥ ❧➔ ❣â❝ ❍❡r♦♥✳ ◆➳✉ tr♦♥❣ t❛♠ ❣✐→❝ ❆❇❈ t❛ ✤➦t A B C t1 = tan , t2 = tan , t3 = tan 2 t❤➻ s➩ ❝â t✛ ❧➺ t❤ù❝ a : b : c = t1 (t2 + t3 ) : t2 (t3 + t1 ) : t3 (t1 + t2 ) ❚ø ✤â s✉② r❛ t❛♠ ❣✐→❝ ❧➔ ❤ú✉ t✛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❝→❝ ❣â❝ ❝õ❛ ♥â ❧➔ ❣â❝ ❍❡r♦♥✳ P❤➨♣ ❞ü♥❣ tù ❣✐→❝ ❇r❛❤♠❛❣✉♣t❛✳ ❱➻ ❝→❝ ❣â❝ ✤è✐ ❞✐➺♥ tr♦♥❣ tù ❣✐→❝ ♥ë✐ π π t✐➳♣ ❜ò ♥❤❛✉ ♥➯♥ t❛ ❝â t❤➸ ❣✐↔ sû A, B ≤ ✈➔ C, D ≥ ✳ ❚ù ❣✐→❝ ♥ë✐ t✐➳♣ 2 π ABCD ❧➔ ❤➻♥❤ ❝❤ú ♥❤➟t ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ A = B = ♥â ❧➔ ❤➻♥❤ t❤❛♥❣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ A = B ✳ ●✐↔ sû CAD = CBD = θ✳ ❚ù ❣✐→❝ ♥ë✐ t✐➳♣ ABCD ❧➔ tù ❣✐→❝ ❤ú✉ t✛ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❝→❝ ❣â❝ A, B, θ ❧➔ ❝→❝ ❣â❝ ❍❡r♦♥✳ ◆➳✉ ABCD ❧➔ ♠ët tù ❣✐→❝ ❇r❛❤♠❛❣✉♣t❛ ♠➔ ❝→❝ AD BC ổ s s ỵ E = AD ∩BC ✳ ❚r➯♥ ❤➻♥❤ ✱ ❣✐↔ sû EC = α, ED = β ✳ AB EB EA ❱➻ ∆EAB ∼ ∆ECD ♥➯♥ = = = λ ✭❝❤➥♥❣ ❤↕♥✮✳ ◆❣❤➽❛ ❧➔ CD ED EC a α+b β+d = = =λ c β α ❄❄ ✹✼ ❤❛② a = λc, b = λβ − α, d = λα − β, λ > max α β , β α ✭✸✳✼✮ ❍ì♥ ỳ t ỵ s e = 2R sin B = 2R sin D = R ·α ρ f = 2R sin A = 2R sin C = R ·β ρ ✭✸✳✽✮ tr♦♥❣ ✤â ρ ❧➔ ❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ t❛♠ ❣✐→❝ ECD ❚❤❡♦ ✤à♥❤ ❍➻♥❤ ✸✳✹✿ ❉ü♥❣ tự rt tứ t r ỵ Pt ac + bd = ef ✈➔ R2 · α · β = c2 λ + (βλ − α)(αλ − β) ρ P❤÷ì♥❣ tr➻♥❤ ♥➔② ✤÷đ❝ ✈✐➳t ❧↕✐ ♥❤÷ s❛✉ R ρ α2 + β − c2 =λ − λ+1 αβ = λ2 − 2λ cos E + = (λ − cos E)2 + sin2 E ✹✽ ❤❛② R R − λ + cos E + λ cos E = sin2 E ữ ỵ r➡♥❣ sin E, cos E ❧➔ ❤ú✉ t✛ ✈➻ E ❧➔ ❣â❝ ❍❡r♦♥✳ ❚❤❡♦ t❤ù tü t❛ t❤✉ ✤÷đ❝ ❝→❝ ❣✐→ trà ❤ú✉ t✛ ❝õ❛ R ✈➔ λ✱ t❛ ✤➦t R − λ − cos E = t sin E ρ R sin E + λ + cos E = ρ t ✈ỵ✐ sè ❤ú✉ t✛ t ♥➔♦ ✤â✳ ❚ø ✤â t❛ ❝â ρ c sin E t + = t+ t t 1 − − cos E λ = sin E t R= θ ❚ø ❝→❝ ❜✐➸✉ t❤ù❝ ❜✐➸✉ ❞✐➵♥ R ✈➔ λ rã r➔♥❣ t = tan D C ◆➳✉ ✤➦t t1 = tan , t2 = tan ✤è✐ ✈ỵ✐ ❝→❝ ❣â❝ ❍❡r♦♥ C ✈➔ D t❤➻ 2 (t1 + t2 ) (1 − t1 t2 ) (t1 + t2 )2 − (1 − t1 t2 )2 ✈➔ sin E = cos E = 2 (1 + t1 ) (1 + t2 ) (1 + t21 ) (1 + t22 ) ❇➡♥❣ ❝→❝❤ ❝❤å♥ c = t + t21 + t22 ✱ t❤➻ tø ✭✸✳✽✮ t❛ ❝â tt1 + t21 + t22 α= , (t1 + t2 ) (1 − t1 t2 ) tt2 + t21 + t22 β= (t1 + t2 ) (1 − t1 t2 ) ✈➔ tø ✭✸✳✼✮ t❛ ❝â ❝↕♥❤ ✈➔ ✤÷í♥❣ ❝❤➨♦✱ ❞✐➺♥ t➼❝❤ ❝õ❛ tù ❣✐→❝ ♥ë✐ t✐➳♣✿ a = (t (t1 + t2 ) + (1 − t1 t2 )) (t1 + t2 − t (1 − t1 t2 )) b = + t21 (t2 − t) (1 + tt2 ) c = t + t21 + t22 d = + t22 (t1 − t) (1 + tt1 ) e = t1 + t2 + t22 f = t2 + t2 + t21 S = t1 t2 2t (1 − t1 t2 ) − (t1 + t2 ) − t2 (t1 + t2 ) t + (1 − t1 t2 ) − t2 ✹✾ ✈➔ ✤÷í♥❣ ❦➼♥❤ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ tù ❣✐→❝ + t21 2R = + t22 + t2 n q v , t2 = ✈➔ t = ✈ỵ✐ m, n, p, q, u, v ∈ N ➔♦ m p u ❝→❝ ✤➥♥❣ t❤ù❝ tr➯♥ t❛ t❤✉ ✤÷đ❝ tù ❣✐→❝ ❇r❛❤♠❛❣✉♣t❛✳ ▼å✐ tù ❣✐→❝ ❇r❛❤✲ ♠❛❣✉♣t❛ ✤➲✉ ✤÷đ❝ ❞ü♥❣ t❤❡♦ ❝→❝❤ ♥➔②✳ ❇➡♥❣ ❝→❝❤ t❤❛② t1 = ❱➼ ❞ö ✸✳✷✳✶✳ ✭❉ü♥❣ tù ❣✐→❝ ❇r❛❤♠❛❣✉♣t❛ tø t❛♠ ❣✐→❝ ❍❡r♦♥✮ ❚ø t❛♠ ❣✐→❝ ❍❡r♦♥ ❊❈❉ ✈ỵ✐ c : α : β = 14 : 15 : 13✳ Ð ✤➙② t1 = v , t2 = ✭✈➔ t3 = ✮✳ ❇➡♥❣ ❝→❝❤ ✤➦t t = t➼♥❤ ✤÷đ❝ ❝→❝ ❝↕♥❤ tù ❣✐→❝ u ❇r❛❤♠❛❣✉♣t❛ a = (7u − 4v)(7u + 4v), b = 13(u − 2v)(2u + v), c = 65uv, d = 5(2u − 3v)(3u + 2v) e = 30 u2 + v , f = 26 u2 + v S = 24 2u2 + 7uv − 2v 7u2 − 8uv − 7v ◆➳✉ ❧➜② u = 3, v = t❤➻ ✤÷đ❝ (a, b, c, d, e, f, S) = (323, 91, 195, 165, 300, 260, 28416) ợ u = 11, v = t ữủ tù ❣✐→❝ ♠➔ ❝↕♥❤ ✈➔ ✤÷í♥❣ ❝❤➨♦ ✤➲✉ ❧➔ ❜ë✐ ❝õ❛ ✻✺✳ ❘ót ❣å♥ ❧↕✐ t❛ ✤÷đ❝ (a, b, c, d, e, f, S) = (65, 39, 33, 25, 52, 60, 1344) ❚ù ❣✐→❝ ♥ë✐ t✐➳♣ tr♦♥❣ ✤÷í♥❣ trá♥ ✤÷í♥❣ ❦➼♥❤ ✻✺✳ ❱➼ ❞ö ✸✳✷✳✷✳ ◆➳✉ ❧➜② ECD ❧➔ t❛♠ ❣✐→❝ ✈✉ỉ♥❣ ✈ỵ✐ ❝↕♥❤ CD : EC : ED = m2 + n2 : 2mn : m2 − n2 t❛ ✤÷đ❝ a = m2 + n2 u2 − v b = ((m − n)u − (m + n)v)((m + n)u + (m − n)v) c = m2 + n2 uv d = 2(nu − mv)(mu + nv) e = 2mn u2 + v f = m2 − n2 u2 + v S = mn m2 − n2 u2 + 2uv − v u2 − 2uv − v ✺✵ √ Ð ✤➙② u m m+n > , ✳ ❚❛ ❝â ✷ tù ❣✐→❝ ❇r❛❤♠❛❣✉♣t❛ tø ❝→❝❤ ❞ü♥❣ ♥➔②✿ v n m−n n/m v/u a b c d e f S 2R ✶✴✷ ✶✴✷ ✶✴✹ ✶✴✺ ✼✺ ✶✸ ✹✵ ✸✻ ✻✽ ✺✶ ✾✻✻ ✽✺ ✻✵ ✶✻ ✷✺ ✸✸ ✺✷ ✸✾ ✼✶✹ ✻✺ ❱➼ ❞ö ✸✳✷✳✸✳ ◆➳✉ ❣â❝ θ ✤÷đ❝ ❝❤å♥ s❛♦ ❝❤♦ A + B − θ = π2 t❤➻ ❝↕♥❤ BC ❧➔ ✤÷í♥❣ ❦➼♥❤ ❝õ❛ tù ❣✐→❝ ABCD✳ ❑❤✐ ✤â✱ θ − t3 t1 + t2 − + t1 t2 = = + t3 t1 + t2 + − t1 t2 n q (m + n)q − (m − n)p ✣➦t t1 = , t2 = ✈➔ t = t❛ t❤✉ ✤÷đ❝ tù ❣✐→❝ m p (m + n)p − (m − n)q ❇r❛❤♠❛❣✉♣t❛✿ t = tan a = m2 + n2 p2 + q b = m2 − n2 p2 + q c = ((m + n)p − (m − n)q)((m + n)q − (m − n)p) d = m2 − n2 p2 − q e = 2mn p2 + q f = 2pq m2 + n2 ❙❛✉ ✤➙② ❧➔ ❤❛✐ tù ❣✐→❝ ❝ö t❤➸ t1 t2 t a b c d e f s ✷✴✸ ✶✴✸ ✸✴✶✶ ✻✺ ✷✺ ✸✸ ✸✾ ✻✵ ✺✷ ✶✸✹✹ ✸✴✹ ✶✴✷ ✶✴✸ ✷✺ ✼ ✶✺ ✶✺ ✷✹ ✷✵ ✶✾✷ ✸✳✸ ●✐ỵ✐ t❤✐➺✉ ♠ët sè ❜➔✐ t♦→♥ t❤✐ ❖❧②♠♣✐❝ ▼ët số t s ữủ t ữợ ❞↕♥❣ ❜➔✐ t♦→♥ sè ❤å❝ ✈➔ ✤÷đ❝ ❣✐↔✐ ♥❤í ❝→❝ ❦ÿ t❤✉➟t ❝õ❛ sè ❤å❝✳ ❱➼ ❞ö ✸✳✸✳✶✳ ✭■▼❖ ✶✾✺✺✲✶✾✺✻✱ ❜➔✐ ✸✮ ❈❤♦ a, b, c ❧➔ ❝→❝ sè tü ♥❤✐➯♥ t❤ä❛ ♠➣♥ a2 + b2 = c2 ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣✿ ❛✳ ❈â ➼t ♥❤➜t ✶ tr♦♥❣ ✷ sè ❛✱❜ ❝❤✐❛ ❤➳t ❝❤♦ ✸❀ ❜✳ ❈â ➼t ♥❤➜t ✶ tr♦♥❣ ✷ sè ❛✱❜ ❝❤✐❛ ❤➳t ❝❤♦ ✹❀ ❝✳ ❈â ➼t ♥❤➜t ✶ tr♦♥❣ ✸ sè ❛✱❜✱❝ ❝❤✐❛ ❤➳t ❝❤♦ ✺✳ ✺✶ ❈❤ù♥❣ ♠✐♥❤✳ ❛✳ ◆➳✉ ❝↔ a ✈➔ b ❦❤ỉ♥❣ ❝❤✐❛ ❤➳t ❝❤♦ ✸ t❤➻ ❞÷ ❝õ❛ a2 + b2 ❦❤✐ ❝❤✐❛ ❝❤♦ ✸ s➩ ❧➔ ✷✳ ✣✐➲✉ ✤â ❧➔ ❦❤ỉ♥❣ t❤➸ ✈➻ a2 + b2 ❜➡♥❣ c2 ✈ỵ✐ c ∈ N ❜✳ ◆➳✉ (a, b, c) = 1✳ ❑❤✐ ✤â a, b ❝ơ♥❣ ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉✳ ❚❤➟t p ữợ tố õ a, b t❤➻ ✈➻ a2 + b2 = c2 s✉② r p ụ ữợ c2 c ❚r→✐ ✈ỵ✐ ❣✐↔ t❤✐➳t (a, b, c) = ❈→❝ sè a, b ❦❤ỉ♥❣ t❤➸ ❝ò♥❣ ❝❤➤♥✱ ❝❤ó♥❣ ❝ơ♥❣ ❦❤ỉ♥❣ ỗ tớ t a = 2k + 1, b = 2l + sè a2 + b2 = 4(k + l2 ) + 4(k + l) + 2✱ rã r➔♥❣ ❝â ❞÷ ✷ ❦❤✐ ❝❤✐❛ ❝❤♦ ✹✱ tr♦♥❣ ❦❤✐ ✤â✱ c2 ❦❤✐ ❝❤✐❛ ❝❤♦ ✹ ❝❤➾ ❝â ❞÷ ❧➔ ✵ ❤♦➦❝ ✶✳ ❉♦ ✤â✱ ♠ët tr♦♥❣ ✷ sè a, b ❧➔ ❝❤➤♥✱ sè ❦✐❛ ❧➔ ❧➫ ✭ c ❧➔ sè ❧➫✮✳ ❑❤æ♥❣ ♠➜t t➼♥❤ ❝❤➜t tê♥❣ q✉→t✱ ❝♦✐ a ✲❝❤➤♥✱ b ✲❧➫✳ ❚ø ❝æ♥❣ t❤ù❝ t❤ù❝ P②t❤❛❣♦r❡ t❛ a c+b c−b = · 2 c+b c−b ❱➻ b ✈➔ c ❧➫ ♥➯♥ , ∈ N✱ tê♥❣ ❤❛✐ sè ♥➔② ❜➡♥❣ sè ❧➫ ❝ ♥➯♥ ♠ët 2 tr♦♥❣ ❝❤ó♥❣ ♣❤↔✐ ❧➫✱ sè ❦✐❛ ♣❤↔✐ ❝❤➤♥✳ ❙✉② r❛ t➼❝❤ ❝õ❛ ❝❤ó♥❣ ❜➡♥❣ (a/2)2 ❧➔ sè ❝❤➤♥✱ tù❝ a/2 ❝❤➤♥ ❤❛② a ❝❤✐❛ ❤➳t ❝❤♦ ✹✳ ◆➳✉ (a, b, c) = d > 1✱ tự tỗ t a1 , b1 , c1 N, (a1 , b1 , c1 ) = s❛♦ ❝❤♦ a = da1 , b = db1 , c = dc1 ❚❛ rót r❛ a21 + b21 = c21 ♠➔ t❤❡♦ tr➯♥ ♠ët tr♦♥❣ ✷ sè✱ ❝❤➥♥❣ ❤↕♥ a1 ❝❤✐❛ ❤➳t ❝❤♦ ✹✱ tù❝ ❧➔ a ❝❤✐❛ ❤➳t ❝❤♦ ✹✳ ❝✳ ❙è ❦❤æ♥❣ ❝❤✐❛ ❤➳t ❝❤♦ ✺ ❝â ❞↕♥❣ 5k ± ❤♦➦❝ 5k ± ❚ø ✤â t❤➜② ♥❣❛② ❜➻♥❤ ♣❤÷ì♥❣ ✶ sè ❦❤✐ ❝❤✐❛ ❝❤♦ ✺ ❝❤♦ ❞÷ ❧➔ ✶ ❤♦➦❝ ✹✳ ◆➳✉ a ✈➔ b ✤➲✉ ❦❤ỉ♥❣ ❝❤✐❛ ❤➳t ❝❤♦ ✺ t❤➻ ❞÷ ❝õ❛ a2 + b2 ❦❤✐ ❝❤✐❛ ❝❤♦ ✺ ❧➔ +1 = 2, (1 + 4) − = 0, (4 + 4) − = ▼➦t ❦❤→❝✱ ✈➻ a2 + b2 = c2 ❦❤✐ ❝❤✐❛ ❝❤♦ ✺ ❝❤➾ ❝â ❞÷ ❧➔ ✵✱✶✱✹ ✳ ◆❤÷ ✈➟②✱ a2 + b2 = c2 ❦❤✐ ❝❤✐❛ ❝❤♦ ✺ ❝❤➾ ❝â ❞÷ ❜➡♥❣ ✵✱ tù❝ c ❧➔ ❜ë✐ ❝õ❛ ✺✳ ❚â♠ ❧↕✐ ❝â ➼t ♥❤➜t ✶ sè tr♦♥❣ ✸ sè a, b, c ❧➔ ❜ë✐ ❝õ❛ ✺✳ ❱➼ ❞ö ✸✳✸✳✷✳ ✭■▼❖ ✶✾✼✵✲✶✾✼✶✱ ❜➔✐ ✹✮ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ ❝→❝ sè tü ♥❤✐➯♥ x, y, z t❤ä❛ ♠➣♥ xn + y n = z n t❤➻ min(x, y) ≥ n✳ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ❝→❝ sè tü ♥❤✐➯♥ x, y, z, n t❤ä❛ ♠➣♥ xn + y n = z n ✳ ❑❤æ♥❣ ♠➜t t➼♥❤ ❝❤➜t tê♥❣ q✉→t✱ t❛ ❝♦✐ x ≤ y ✳ ❱➻ z n = xn + y n > y n ♥➯♥ z > y ✈➔ ❞♦ ✤â✱ z ≥ y + 1✳ ❚❤❡♦ ♥❤à t❤ù❝ ◆❡✇t♦♥ z n ≥ (y + 1)n = y n + C1n y n−1 + + ≥ y n + ny n−1 ✭✸✳✾✮ s ợ ữỡ tr xn ≥ ny n−1 ✱ ♥❤÷♥❣ ✈➻ x ≤ y ♥➯♥ xn ≥ nxn−1 ❤❛② x ≥ n✳ ◆❤÷ ✈➟②✱ min(x, y) = x ≥ n ✺✷ ❱➼ ❞ö ✸✳✸✳✸✳ ✭■▼❖ ởt n ỗ ữủ t ❝→❝ t❛♠ ❣✐→❝ ❜ð✐ ❝→❝ ✤÷í♥❣ ❝❤➨♦ ❝õ❛ ♥â t❤ä❛ ♠➣♥✿ ✭❛✮ ▼é✐ ✤➾♥❤ t❛♠ ❣✐→❝ ❝â ♠ët sè ❝❤➤♥ ❝→❝ ✤÷í♥❣ ❝❤➨♦ ✈➔ ✭❜✮ ❦❤ỉ♥❣ ❝â ✷ ✤÷í♥❣ ❝❤➨♦ ♥➔♦ ❣➦♣ ♥❤❛✉ ð ✤✐➸♠ tr♦♥❣✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ n t ự rữợ t t õ t s tr n ỗ A1 A2 An ❦➫ ✤÷đ❝ ♠ët sè ♥➔♦ ✤â ❝→❝ ✤÷í♥❣ ❝❤➨♦ ✈➔ tø ♠é✐ ✤➾♥❤ A1 A2 An−1 ✤➲✉ ❝â sè ❝❤➤♥ ❝→❝ ✤÷í♥❣ ❝❤➨♦ t❤➻ tø ✤➾♥❤ An ❝ơ♥❣ ❝â sè ❝❤➤♥ ❝→❝ ✤÷í♥❣ ❝❤➨♦✳ ✣➸ ❣✐↔✐ ❜➔✐ t♦→♥ t❛ q✉② ♥↕♣ t❤❡♦ n ≥ 3✳ ❑❤✐ n = 3✱ ❦➳t ❧✉➟♥ ❤✐➸♥ ♥❤✐➯♥✳ ●✐↔ sû n > ✈➔ ❜➔✐ t♦→♥ ✤ó♥❣ ✈ỵ✐ ♠å✐ sè tü ♥❤✐➯♥ r t❤ä❛ ♠➣♥ ≤ r ≤ n✳ ❑❤✐ ✤â ♥➳✉ ♠ët t➟♣ ❤đ♣ ❝→❝ ✤÷í♥❣ ❝❤➨♦ ❝õ❛ r− ❣✐→❝ ❝❤✐❛ ♥â t❤➔♥❤ ❝→❝ t❛♠ ❣✐→❝ ✈➔ t❤ä❛ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭❛✮ ✈➔ ✭❜✮ t❤➻ r ❝❤✐❛ ❤➳t ❝❤♦ ✸✳ ●å✐ P ❧➔ t➟♣ ❤đ♣ ❝→❝ ✤÷í♥❣ ❝❤➨♦ ❝õ❛ n✲ ❣✐→❝ ♠➔ ❝❤✐❛ ♥â t❤➔♥❤ ❝→❝ t❛♠ ❣✐→❝ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭❛✮ ✈➔ ✭❜✮✳ ●✐↔ sû tø ✤➾♥❤ A ♥➔♦ ✤â ❝õ❛ n✲ ❣✐→❝ ❝â ➼t ♥❤➜t ✷ ✤÷í♥❣ ❝❤➨♦ t❤✉ë❝ P ✳ ❚❛ ❝❤å♥ ✷ ✤÷í♥❣ ❝❤➨♦ AB, AC ①✉➜t ♣❤→t tø A✱ t❤✉ë❝ P s❛♦ ❝❤♦✿ ♣❤➛♥ tr♦♥❣ ❣â❝ A1 AB ❝❤ù❛ ♠ët sè ❝❤➤♥ ❝→❝ ✤÷í♥❣ ❝❤➨♦ ✤✐ tø A✱ t❤✉ë❝ P ✱ ❝á♥ ♣❤➛♥ tr♦♥❣ ❝õ❛ ❣â❝ BAC ❦❤ỉ♥❣ ❝â ♠ët ✤÷í♥❣ ❝❤➨♦ ♥➔♦✱ ✤✐ tø A t❤✉ë❝ t➟♣ ❤đ♣ P ✭❤➻♥❤ ✮✳ ❑❤✐ ✤â ✤÷í♥❣ ❝❤➨♦ BC t❤✉ë❝ t➟♣ ❤ñ♣ P ✳ ❳➨t ✤❛ ❣✐→❝ AA1 B ✱ tø ♠é✐ ✤➾♥❤ ❝õ❛ ♥â s➩ ❝â ♠ët sè ❝❤➤♥ ❝→❝ ✤÷í♥❣ ❝❤➨♦ t❤✉ë❝ P ✳ ❚❤❡♦ ♥❤➟♥ ①➨t tr➯♥ tø ✤➾♥❤ B ❝ô♥❣ s➩ ❝â ♠ët sè ❝❤➤♥ ❝→❝ ✤÷í♥❣ ❝❤➨♦ t❤✉ë❝ P ✳ ❄❄ ❍➻♥❤ ✸✳✺✿ ❍❛✐ ✤÷í♥❣ ❝❤➨♦ AB, BC ∈ P ✺✸ ✣✐➲✉ t÷ì♥❣ tü ①↔② r❛ ✤è✐ ✈ỵ✐ ❝→❝ ✤❛ ❣✐→❝ BB1 C ✈➔ CC1 A✳ ❱➔ ♠é✐ ✤❛ ❣✐→❝ AA1 B, BB1 C, CC1 A sè ❝↕♥❤ ✤➲✉ ♥❤ä ❤ì♥ n ♥➯♥ t❤❡♦ ❣✐↔ t❤✐➳t q✉② ♥↕♣ sè ❝↕♥❤ ♠é✐ ✤❛ ❣✐→❝ ❝❤✐❛ ❤➳t ❝❤♦ ✸✳ ❚ê♥❣ sè ❝↕♥❤ ❝õ❛ ✸ ✤❛ ❣✐→❝ ✤â ❧➔ n + ✈➔ ❝â t❤➯♠ ✸ ❝↕♥❤ AB, BC, CA✳ ❙è n + ❝❤✐❛ ❤➳t ❝❤♦ ✸✱ s✉② r❛ n ❝❤✐❛ ❤➳t ❝❤♦ ✸✳ ❙✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ ❱➼ ❞ö ✸✳✸✳✹✳ ✭■▼❖ ✶✾✻✽✱ ❜➔✐ ✶✮ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❝â ♠ët ✈➔ ❝❤➾ ♠ët t❛♠ ❣✐→❝ ❝â ✤ë ❞➔✐ ❝→❝ ❝↕♥❤ ❧➔ ❝→❝ sè tü ♥❤✐➯♥ ❧✐➯♥ t✐➳♣ ✈➔ ❝â ♠ët ❣â❝ ❣➜♣ ✤æ✐ ❣â❝ ❦❤→❝✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ tr➻♥❤ ❜➔② t❤❡♦ ✸ ❝→❝❤✿ ❈→❝❤ ✶✳ ●✐↔ sû ∆ABC ❝â BC = a, AC = b, AB = c, ABC = α ✈➔ BAC = ỵ s b a sin a a = =⇒ = cos α = =⇒ cos α = sin α sin 2α sin α b 2b ỵ ổ s cos = a a2 + c − b = =⇒ a2 c = b a2 + c2 − b2 2ac 2b ❱➻ a, b, c ∈ N ♥➯♥ b|a2 c✳ ✲ ◆➳✉ b ð ❣✐ú❛ a ✈➔ c t❤➻ b ♥❣✉②➯♥ tè ✈ỵ✐ a✱ ✈ỵ✐ c ✈➔ b ❦❤ỉ♥❣ t❤➸ ❧➔ ÷ỵ❝ ❝õ❛ a2 c✳ ❉♦ ✤â b ❤♦➦❝ ❧➔ sè ♥❤ä ♥❤➜t ❤♦➦❝ ❧➔ sè ❧ỵ♥ ♥❤➜t tr♦♥❣ ✸ sè tü ♥❤✐➯♥ ❧✐➯♥ t✐➳♣✳ ✲ ◆➳✉ b ♥❤ä ♥❤➜t t❤➻ b|b + ❤♦➦❝ b|(b + 2)2 tü② t❤❡♦ a = b + ❤❛② c = b+2✳ ◆➳✉ ①↔② r❛ a = b+2, b|b+2 t❤➻ b = ✭❞➝♥ ✤➳♥ t❛♠ ❣✐→❝ s✉② ❜✐➳♥✮ ❤♦➦❝ b = ❞➝♥ ✤➳♥ b · a2 + c2 − b2 = 42 + +32 − 22 = 42 = a2 c✱ ♠➙✉ t❤✉➝♥ ✈ỵ✐ a = 3, c = 4, b = 2✳ ❱➟② b ❦❤æ♥❣ ❝❤✐❛ ❤➳t b + 2✳ ◆❤÷ t❤➳ b|(b + 2)2 ✭❦❤✐ c = b + 2✮✳ ❚ø ✤➙②✱ b|(b + 2)2 − b2 − 4b = 4✱ tù❝ b = 1, 2, • b = 1, c = 3, a = t❛♠ ❣✐→❝ s✉② ❜✐➳♥❀ • b = 2, c = 4, a = ⇒ 2.(9 + 16 − 4) = 42 = a2.c = 36, ổ ỵ b = 4, c = 6, a = t❤ä❛ ♠➣♥ ❣✐↔ t❤✐➳t✳ ❱➟② t❛♠ ❣✐→❝ ❞✉② ♥❤➜t t❤ä❛ ♠➣♥ ✤➲ ❜➔✐ ❧➔ t❛♠ ❣✐→❝ (4, 5, 6) ❈→❝❤ ✷✳ ❚❛♠ ❣✐→❝ ❆❇❈ ❝â A = 2B ⇒ C = 180◦ − 3B ✱ ✈➟② sin C = sin 3B ❚❛ ❝â sin2 A = sin2 2B = sin B cos B sin 2B = sin B(sin B + sin 3B) = sin B(sin B + sin C) ỵ s✐♥✱ t❛ ❝â a2 = b(b + c) ✺✹ ◆➳✉ b ❧➔ ❝↕♥❤ ♥❤ä ♥❤➜t t❤➻ (b+2)2 = b2 +b(b+1), s✉② r❛ (b−4)(b−1) = 0✳ ❈❤➾ ❝â b = 4, c = 5, a = t❤ä❛ ♠➣♥✳ ❈→❝ tr÷í♥❣ ❤đ♣ ❦❤→❝ ✤➲✉ ❞➝♥ ✤➳♥ t❛♠ ❣✐→❝ s✉② ❜✐➳♥ ❤♦➦❝ ♠➙✉ t❤✉➝♥✳ ❈→❝❤ ✸✳ ✭❦❤ỉ♥❣ ❞ò♥❣ ❧÷đ♥❣ ❣✐→❝✮✳ ●✐↔ ❍➻♥❤ ✸✳✻✿ ✭■▼❖ ✶✾✻✽✱ #1✮✱ ❈→❝❤ ❣✐↔✐ t❤ù ❜❛ sû ✸ ❝↕♥❤ ❧➔ a, b, c ✈➔ C = 2B ✳ ❑➨♦ ❞➔✐ AC ✤➳♥ D s❛♦ ❝❤♦ CD = BC = a✳ ❑❤✐ ✤â✱ CDB = ACB = ABC ✳ ❚ø ✤â✱ ✷ t❛♠ ❣✐→❝ ABC ✈➔ ADB ỗ út r tự c2 = b(a + b)✱ ❧➟♣ ❧✉➟♥ ♥❤÷ ❝→❝❤ ✷✳ ❱➼ ❞ư ✸✳✸✳✺✳ ✭■▼❖ ✷✵✵✾✱ ❜➔✐ ✺✮ ❚➻♠ ❤➔♠ sè f tø t➟♣ sè ♥❣✉②➯♥ ❞÷ì♥❣ ✤➳♥ t➟♣ sè ♥❣✉②➯♥ ❞÷ì♥❣ s❛♦ ❝❤♦ ợ a, b ữỡ tỗ t t ❦❤æ♥❣ s✉② ❜✐➳♥ ❝â ✤ë ❞➔✐ ❝→❝ ❝↕♥❤ ❧➔✿ a, f (a) ✈➔ f (b + f (a) − 1) ❈❤ù♥❣ ♠✐♥❤✳ ✣➙② ❧➔ ❜➔✐ t♦→♥ ❦❤â ✈ø❛ t❤✉ë❝ ❝❤✉②➯♥ ✤➲ sè ❤å❝ ✈ø❛ t❤✉ë❝ ❝❤✉②➯♥ ✤➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✳ ❝â ♥❤✐➲✉ ❝→❝❤ ❧➟♣ ❧✉➟♥ tr♦♥❣ ❧í✐ ❣✐↔✐✱ tr♦♥❣ ❝→❝❤ ✶ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ❧í✐ ❣✐↔✐ ❝❤✐ t✐➳t✳ ❈→❝❤ ✶✳ ●✐↔ sû f t❤ä❛ ♠➣♥ ✤➲ ❜➔✐✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ f (1) = 1✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû K = f (1) − > 0✱ ❣å✐ m ❧➔ ❣✐→ trà ♥❤ä ♥❤➜t ❝õ❛ f ✈➔ b ❧➔ sè ❜➜t ❦ý s❛♦ ❝❤♦ f (b) = m ❱➻ [1, m = f (b) ✈➔ f (b + f (1) − 1) = f (b + k)] t↕♦ t❤➔♥❤ ♠ët t❛♠ ❣✐→❝ ♥➯♥ ♣❤↔✐ ❝â f (b + k) < + f (b)✳ ❚➼♥❤ ♥❤ä ♥❤➜t ❝õ❛ m ❦➨♦ t❤❡♦ f (b + k) = m ✈➔ ❜➡♥❣ q✉② ♥↕♣ t❛ ❝â f (n + nk) = m, ∀n N tỗ t t ợ [b + nk, f (1), f (m)] ♥➯♥ ♣❤↔✐ ❝â b + nK < f (1) + f (m) ✈ỵ✐ ♠é✐ n✳ ♠➙✉ t❤✉➝♥ ♥➔② ❦➨♦ t❤❡♦ f (1) = 1✳ ✺✺ ❇❛ sè [a, = f (1), f (f (a))] ❧➔ ❝↕♥❤ t❛♠ ❣✐→❝ ✈ỵ✐ ♠å✐ a✳ ❉♦ ✤â✱ a − < f (f (a)) < a + ♥➯♥ f (f (a)) = a ✈➔ f ❧➔ ♠ët s♦♥❣ →♥❤✳ ❚✐➳♣ t❤❡♦ t❛ ❝â f (a), f (b), f (b + a − 1) ❧➔ ✸ ❝↕♥❤ t❛♠ ❣✐→❝ ✈ỵ✐ ♠å✐ a, b ∈ N+ ✳ ✣➦t z = f (2)✱ rã r➔♥❣ z > 1✳ ❱➻ [f (z), f (z), f (2z − 1)] t↕♦ t❤➔♥❤ t❛♠ ❣✐→❝ ♥➯♥ ♣❤↔✐ ❝â f (2z − 1) < f (z) + f (z) = 2f (f (z)) = ◆❤÷ ✈➟②✱ f (2z − 1) ∈ 1, 2, 3✳ ❱➻ f ❧➔ s♦♥❣ →♥❤ ✈➔ f (1) = 1, f (2) = ♥➯♥ f (2z − 1) = ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ f (k) = (k − 1)z − k + ✈ỵ✐ ♠å✐ k ∈ N ▼➺♥❤ ✤➲ ✤â ✤â ✤ó♥❣ ✈ỵ✐ k = 1, 2✱ ❣✐↔ sû ♥â ✤ó♥❣ ✈ỵ✐ ♠å✐ sè 1, 2, , k ❱➻ [f ((k − 1)z − k + 1), f (z), f (kz − k + 1)] t↕♦ t❤➔♥❤ t❛♠ ❣✐→❝ ♥➯♥ ♣❤↔✐ ❝â f (kz − k + 1) ≥ k + 1✳ ❍➔♠ f ✤ì♥ →♥❤ ♥➯♥ f (kz − k + 1) = i trø ❦❤✐ kz − k + = (i − 1)z − i + 2, tù❝ k + = i ❱➟② f (kz − k + 1) = k + ❤♦➦❝ f (k + 1) = kz − k + ✈➔ ♣❤➨♣ q✉② ♥↕♣ ❤♦➔♥ t❤➔♥❤✳ ❚❤➯♠ ✈➔♦ ✤â✱ f ❧➔ ❤➔♠ t➠♥❣✿ ♥➳✉ z > t❤➻ = f (z) > f (2) = z, ✈æ ỵ z = f (k) = 2(k − 1) − k + = k ❈✉è✐ ❝ü♥❣ f (x) = x, ∀x ∈ N+ ✱ ❝â ✸ ✤ë ❞➔✐ x, y = f (y) ✈➔ z = f (y) + f (x) − = x + y − t↕♦ t❤➔♥❤ ♠ët t❛♠ ❣✐→❝ t❤ä❛ ♠➣♥ ✤➲ ❜➔✐✳ ❚❤➟t ✈➟②✱ ❱➻ x ≥ 1, y ≥ ♥➯♥ ♣❤↔✐ ❝â z ≥ max x, y = |x − y| ✈➔ z < x + y ✳ ❱➟② t❛♠ ❣✐→❝ ✈ỵ✐ ✸ ✤ë ❞➔✐ ♥â✐ tr➯♥ t❤ü❝ sỹ tỗ t ổ s õ t ỗ t tr N+ ỗ t f (x) = x ợ x ∈ N+ ✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ ❦❤↔ ♥➠♥❣ ổ t ữợ s B1 ❚❛ ❝â f (1) = B2 ❱ỵ✐ ♠å✐ z ∈ N+ ✱ t❛ ❝â f (f (z)) = z ✳ ❈❤➾ ✈✐➺❝ ✤➦t x = z, y = 1✳ B3 ❱ỵ✐ ♠å✐ z ∈ N, z ≥ 1✱ t❛ ❝â f (z) ≤ z B4 ❚❤❡♦ B2 , B3 ❝â z = f (f (z)) ≥ f (z) ≥ z ✈➔ f (z) = z ✈ỵ✐ ∀z ∈ N+ ✳ ❚❛ ①➨t ❜➔✐ t♦→♥ t❤✐ ❖❧②♠♣✐❝ q✉è❝ t➳ ✷✵✶✻ ✭■▼❖✳ ✷✵✶✻✱ ❜➔✐ ✸✮✳ ❚❤❡♦ ♥❤÷ t➟♣ t❤➸ ✤ë✐ t✉②➸♥ ❱✐➺t ♥❛♠ ♥❤➟♥ ①➨t✿ ✤➙② ❧➔ ♠ët ❜➔✐ t♦→♥ ❦❤â ♥❤➜t tr♦♥❣ ✤➲ t❤✐ ✷✵✶✻✱ ❞♦ ◆❣❛ ✤➲ ♥❣❤à✳ ❇➔✐ t♦→♥ ✤➲ ❝➟♣ ✤➳♥ ♠ët ✤❛ ❣✐→❝ ♥❣✉②➯♥✱ ❝→❝ ✤➾♥❤ ♥➡♠ tr➯♥ ✤÷í♥❣ trá♥ ✈➔ ❝â t➼♥❤ ❝❤➜t ✤➦❝ ❜✐➺t✳ ✣➸ ❣✐↔✐ ♥â✱ ❦✐➳♥ t❤ù❝ ❞ò♥❣ ✤➳♥ ❦❤ỉ♥❣ ❝➛♥ ❝❛♦ ①❛ ♥❤÷♥❣ t÷ ❞✉② ♣❤↔✐ r➜t s→♥❣ t↕♦✳ ✺✻ ❱➼ ❞ö ✸✳✸✳✻✳ ✭■▼❖✲✷✵✶✻✱ ❜➔✐ ✸✮ ❈❤♦ P = A1 A2 Ak ởt ỗ tr t ✤➾♥❤ ❝â tå❛ ✤ë ♥❣✉②➯♥ ✈➔ ♥➡♠ tr➯♥ ♠ët ✤÷í♥❣ trá♥✳ ▼ët sè tü ♥❤✐➯♥ n ❧➫ t❤ä❛ ♠➣♥ ❜➻♥❤ ♣❤÷ì♥❣ ✤ë ❞➔✐ ♠é✐ ❝↕♥❤ ❝õ❛ P ✤➲✉ ❝❤✐❛ ❤➳t ❝❤♦ n✳ ●å✐ S ❧➔ ❞✐➺♥ t➼❝❤ ❝õ❛ P ✱ ❝❤ù♥❣ ♠✐♥❤ 2S ❧➔ sè tü ♥❤✐➯♥ ❝❤✐❛ ❤➳t ❝❤♦ n ự ỵ tữ qt ♥❤÷ s❛✉✿ ❈♦✐ P ❧➔ ✤❛ ❣✐→❝ ❦❤ỉ♥❣ ❝â ✤÷í♥❣ ❝❤➨♦ ♥➔♦ ♠➔ ❜➻♥❤ ♣❤÷ì♥❣ ✤ë ❞➔✐ ❝❤✐❛ ❤➳t ❝❤♦ n ✰ ❚❛ ①→❝ ✤à♥❤ vp (x) = max k ∈ Z pk√x ✈ỵ✐ ♠å✐ x ∈ Q✳ ◆❤➢❝ ❧↕✐ √ √ r➡♥❣ ♠å✐ t❛♠ ❣✐→❝ ABC ✈ỵ✐ ❝→❝ ❝↕♥❤ a, b, c ❞✐➺♥ t➼❝❤ S = S(ABC) ✈➔ ❜→♥ ❦➼♥❤ ✤÷í♥❣ trá♥ ♥❣♦↕✐ t✐➳♣ R✱ t❛ ❝â✿ abc = 4(2S)2 R2 (i); 4(2S)2 = 2ab + 2bc + 2ca − a2 − b2 − c2 (ii) ❱ỵ✐ ♠å✐ ✤❛ F ỵ P t m S(F ) = n + − ✈ỵ✐ n ✈➔ m ❧➔ sè ❝→❝ ✤✐➸♠ ♥❣✉②➯♥ ð tr♦♥❣ ✈➔ tr➯♥ ❝↕♥❤ ✤❛ ❣✐→❝ F ✳ ❉♦ ✤â✱ 2S(F ) = 2n + m − ∈ Z+ ✱ ✤➦❝ ❜✐➺t 2S(P ) ∈ Z+ ✳ ❚❛ ❝â Ai ❧➔ ❝→❝ ✤✐➸♠ ♥❣✉②➯♥ ♥➯♥ Ai Aj ❧➔ sè ♥❣✉②➯♥✳ ự t ố ợ trữớ ❤đ♣ n ❧➔ ❧ơ② t❤ø❛ ❝õ❛ ♠ët sè ♥❣✉②➯♥ tè ❧➫ p✳ ✣➦t n = pm ✈➔ ❞♦ ✤â ❝â pm |Ai Aj ✈➔ ♠✉è♥ ❝❤ù♥❣ ♠✐♥❤ pm |2S(P )✳ √ √ √ ❛✳ ❚r÷í♥❣ ❤đ♣ k = ●✐↔ sû a = A1 A2 , b = A2 A3 , c = A3 A1 s❛♦ ❝❤♦ pm |a, b, c ❚❤❡♦ ❝æ♥❣ t❤ù❝ ✭✐✐✮✱ 4(2S)2 = 2ab + 2bc + 2ca − a2 − b2 − c2 ▼å✐ sè ❤↕♥❣ ❜➯♥ ♣❤↔✐ ❝❤✐❛ ❤➳t ❝❤♦ p2m ✱ ❞♦ ✤â✱ p2m |4(2S)✱ ❦➨♦ t❤❡♦ p2m |2S ❜✳ ❚r÷í♥❣ ❤ñ♣ k ≥ ❚❛ ♣❤↔✐ q✉② ♥↕♣ t❤❡♦ k ữỡ ợ t t tữỡ tỹ t ✈➲ t❛♠ ❣✐→❝ ❝↕♥❤ ♥❣✉②➯♥✱ ✤â ❧➔ ❜➔✐ t♦→♥ ✈➲ tù ❣✐→❝ ❤ú✉ t✛ ✈➔ tù ❣✐→❝ ❇r❛❤♠❛❣✉♣t❛✳ ❑➳t q✉↔ ❝ơ♥❣ t❤✉ ✤÷đ❝ ❧í✐ ❣✐↔✐ ❤♦➔♥ ❝❤➾♥❤ ✤➸ ①→❝ ✤à♥❤ ❝→❝ ②➳✉ tè ❝õ❛ tù ❣✐→❝ ❇r❛❤♠❛❣✉♣t❛✳ P❤÷ì♥❣ ♣❤→♣ sè ❤å❝ ♠ët ❧➛♥ ♥ú❛ ♠❛♥❣ ❧↕✐ ❦➳t q✉↔ tèt✳ ✺✼ ❑➳t ❧✉➟♥ ❝õ❛ ❧✉➟♥ ✈➠♥ ▲✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ✤÷đ❝ ❝→❝ ❦➳t q✉↔ s❛✉ ✶✳ ●✐ỵ✐ t❤✐➺✉ ❜➔✐ t♦→♥ t❛♠ ❣✐→❝ P②t❤❛❣♦r❡ ✈➔ t❛♠ ❣✐→❝ ❍❡r♦♥ ✈➔ ❝→❝ ❦➳t q✉↔ ❝õ❛ ❜➔✐ t♦→♥✳ ❘✐➯♥❣ ❜➔✐ t♦→♥ t❛♠ ❣✐→❝ ❍❡r♦♥ ①➨t ❝❤✐ t✐➳t ❤❛✐ tr÷í♥❣ ❤đ♣ ❝â ✤✐➲✉ ❦✐➺♥ ❜ê s✉♥❣✿ ❚❛♠ ❣✐→❝ ❍❡r♦♥ ✈ỵ✐ ❝→❝ ❜→♥ ❦➼♥❤ ♥ë✐ t✐➳♣ ✈➔ ❜→♥ ❦➼♥❤ ❜➔♥❣ t✐➳♣ ❧➔ ❝→❝ sè tü ♥❤✐➯♥ ✈➔ ❜➔✐ t♦→♥ ❦❤↔♦ s→t ❤å t❛♠ ❣✐→❝ ❍❡r♦♥ ♣❤ö t❤✉ë❝ t❤❛♠ sè✳ ✷✳ ❍❛✐ ❜➔✐ t♦→♥ t✐➳♣ t❤❡♦ ❧➔ t❛♠ ❣✐→❝ ❝↕♥❤ ♥❣✉②➯♥ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ S = mP ✈➔ P = nS ✿ ♥➯✉ r❛ ❝→❝ t❤✉➟t t♦→♥ ✈➔ ❝→❝ ✈➼ ❞ö ❣✐↔✐ ✷ ❜➔✐ t♦→♥ ♥➔②✳ ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ♥➔② t❤❛♠ ❦❤↔♦ ✈➔ ❤➺ t❤è♥❣ ❝→❝ ❦➳t q✉↔ ❣➛♥ ✤➙② tr♦♥❣ ❝→❝ ❜➔✐ ❜→♦ ❬✷❪✱ ❬✹❪ ✈➔ ❬✻❪✳ ❈→❝ ❜➔✐ t♦→♥ ♥➔② ✤➲✉ ✤÷đ❝ ❣✐↔✐ ❜➡♥❣ ❦✐➳♥ t❤ù❝ sè ❤å❝ ❝➜♣✳ ✸✳ ❚r➻♥❤ ❜➔② ❧í✐ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ t÷ì♥❣ tü ❧✐➯♥ q✉❛♥ ✤➳♥ tù ❣✐→❝✿ ❚ù ❣✐→❝ ❤ú✉ t✛ ✈➔ tù ❣✐→❝ ❤ú✉ t✛ ♥ë✐ t✐➳♣ ✤÷í♥❣ trá♥✳ ✣➙② ❝ơ♥❣ ❧➔ ❞à♣ tr➻♥❤ ❜➔② ❝→❝ ❜➔✐ t♦→♥ t❤✐ ❖❧②♠♣✐❝ ❚♦→♥ q✉è❝ t➳ ❝â ✈➟♥ ❞ö♥❣ ❦✐➳♥ t❤ù❝ ✈➲ sè ❤å❝✳ ◆❤÷ ✈➟② ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② trå♥ ✈➭♥ ✺ ❜➔✐ t♦→♥ ❝â ②➳✉ tè sè ❤å❝ tr♦♥❣ ❤➻♥❤ ❤å❝✳ ❈❤ó♥❣ tỉ✐ ♥❤➟♥ t❤➜② ❝â ❝→❝ ữợ ự t t t t tữỡ tỹ ố ợ ữớ trỏ rë♥❣ ❝→❝ ❦➳t q✉↔ tø t❛♠ ❣✐→❝ s❛♥❣ tù ❞✐➺♥✱ tø ❤➻♥❤ ❤å❝ ♣❤➥♥❣ s❛♥❣ ❤➻♥❤ ❤å❝ ❦❤æ♥❣ ❣✐❛♥✳ ▼➦❝ ❞ò ✤➣ r➜t ❝è ❣➢♥❣ ♥❤÷♥❣ ❧✉➟♥ ✈➠♥ ❦❤ỉ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❤↕♥ ❝❤➳✱ ❦❤✐➳♠ ❦❤✉②➳t✳ ❚→❝ ❣✐↔ r➜t ♠♦♥❣ sỹ õ ỵ s t ổ ỗ t q ♥❣❤✐➯♥ ❝ù✉ ❤♦➔♥ ❝❤➾♥❤ ✈➔ ❝â ➼❝❤ ❤ì♥✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✳ ✺✽ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❈r✉③✱ ❏✳❆✳❉✱ ●♦❡❤❧✱ ❏✳ ❋✳ ✭✷✵✶✼✮✱ ❚✇♦ ■♥t❡r❡st✐♥❣ ■♥t❡❣❡r P❛r❛♠❡t❡rs ♦❢ ■♥t❡❣❡r ✲s✐❞❡❞ ❚r✐❛♥❣❧❡s✱ ❋♦r✉♠ ●❡♦♠✳ ✶✼✱ ✹✶✶ ✲✹✶✼✳ ❬✷❪ ●♦❡❤❧✱ ❏✳ ❋✳ ✭✷✵✶✷✮✱ ❋✐♥❞✐♥❣ ■♥t❡❣❡r ✲❙✐❞❡❞ ❚r✐❛♥❣❧❡s ❲✐t❤ P = mA✱ ❋♦r✉♠ ●❡♦♠✳ ✶✷✱ ✷✶✶ ✲✷✶✸✳ ❬✸❪ ▼❛rr♦✇s✱ ❇✳ ❏✳✭✷✵✵✶✮✱ P②t❤❛❣♦r❡❛♥ ❛♥❞ ❍❡r♦♥✐❛♥ tr✐❛♥❣❧❡s✱ ❆✉tr❛❧✐❛♥ ❇❡♥✐♦r ▼❛t❤❡♠❛t✐❝s ❏♦✉r♥❛❧ ✷✶✳ ❬✹❪ ▼❛❝❧❡♦❞✱ ❆✳ ❏✳✭✷✵✵✾✮✱ ❖♥ ■♥t❡❣❡r ❘❡❧❛t✐♦♥s ❇❡t✇❡❡♥ t❤❡ ❆r❡❛ ❛♥❞ P❡r✐♠❡t❡r ♦❢ ❍❡r♦♥ ❚r✐❛♥❣❧❡s ❋♦r✉♠ ●❡♦♠✳ ✾✱ ✹✶ ✲✹✻✳ ❬✺❪ ◆❛❣❛r❛❥❛♥✱ ❑✳ ❘✳✱ ❙r✐❞❤❛r❛♥✱ ❘✳ ✭✷✵✵✻✮✱ ❖♥ ❇r❛❤♠❛❣✉♣t❛✬s ❛♥❞ ❑✉♠✲ ♠❡r✬s q✉❛❞r✐❧❛t❡r❛❧s✱ ❊❧❡♠✳ ▼❛t❤✳ ✹✺ ✲✺✼✳ ❬✻❪ ❩❤♦✉✱ ▲✳ ✭✷✵✶✽✮✱ Pr✐♠✐t✐✈❡ ❍❡r♦♥✐❛♥ ❚r✐❛♥❣❧❡s ❲✐t❤ ■♥t❡❣❡r ■♥r❛❞✐✉s ❛♥❞ ❊①r❛❞✐✉s✱ ❋♦r✉♠ ●❡♦♠✳ ✶✽✱ ✼✶ ✲✼✼✳ ❬✼❪ ❨✐✉✱ P✳ ✭✷✵✵✶✮✱ ❍❡r♦♥✐❛♥ tr✐❛♥❣❧❡s ❛r❡ ❧❛tt✐❝❡ tr✐❛♥❣❧❡s✱ ❆♠❡r✳ ▼❛t❤✳ ▼♦♥t❤❧②✱✶✵✽✱ ✷✻✶ ✲✷✻✸✳ ...ĐẠI HỌC THÁI NGUYÊN TRƢỜNG ĐẠI HỌC KHOA HỌC  - LÊ PHƢƠNG THẢO MỘT SỐ BÀI TOÁN SỐ HỌC TRONG HÌNH HỌC PHẲNG Chuyên ngành: Phƣơng pháp Toán sơ cấp Mã số: 46 01 13 LUẬN VĂN THẠC SĨ TOÁN... HỌC PHẲNG Chuyên ngành: Phƣơng pháp Toán sơ cấp Mã số: 46 01 13 LUẬN VĂN THẠC SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC PGS.TS NGUYỄN VIỆT HẢI THÁI NGUYÊN - 2019 ✣❸■ ❍➴❈ ì Pữỡ ▼❐❚ ❙➮ ❇⑨■ ❚❖⑩◆... ▲➯ P❤÷ì♥❣ ❚❤↔♦ ▼❐❚ ❙➮ ❇⑨■ ❚❖⑩◆ ❙➮ ❍➴❈ ❚❘❖◆● ❍➐◆❍ ❍➴❈ P❍➃◆● ❈❤✉②➯♥ ♥❣➔♥❤✿ P❤÷ì♥❣ ♣❤→♣ t♦→♥ ❝➜♣ ▼➣ số ữớ ữợ ❞➝♥ ❦❤♦❛ ❤å❝✿ P●❙✳❚❙ ◆●❯❨➍◆ ❱■➏❚ ❍❷■ ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✶✾ ✐ ▲í✐ ❝↔♠ ì♥ ✣➸ ❤♦➔♥ t❤➔♥❤