❚❙✳ ◆❣✉②➵♥ ❱➠♥ ▲đ✐ ✭❝❤õ ❜✐➯♥✮ − ◆❣ỉ ❚❤à ◆❤➣ ❉ r❛ ❢t ✶✵✽ ❇⑨■ ❚❖⑩◆ ❈❍➴◆ ▲➴❈ ▲❰P ✻ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✶ ▼Ư❈ ▲Ư❈ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ▼ư❝ ❧ö❝ ✶ ❙è tü ♥❤✐➯♥ ✸ ✷ ❙è ♥❣✉②➯♥ ✽ ✸ P❤➙♥ sè ✶✵ ✹ ❍➻♥❤ ❤å❝ ✶✹ ✺ ▲í✐ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥ ❝❤å♥ ❧å❝ ✶✾ ✷ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✶ ❙è tü ♥❤✐➯♥ ✶✳ ❈❤♦ t➟♣ ❤ñ♣ D = {0; 1; 2; 3; · · · ; 20} ❛✮ ❱✐➳t t➟♣ ❤ñ♣ ❉ ❜➡♥❣ ❝→❝❤ ❝❤➾ r❛ t➼♥❤ ❝❤➜t ✤➦❝ tr÷♥❣ ❝❤♦ ❝→❝ ♣❤➛♥ tû ❝õ❛ ♥â✳ ❜✮ ❚➟♣ ❤ñ♣ ❉ ❝â ❜❛♦ ♥❤✐➯✉ ♣❤➛♥ tû❄ ❝✮ ❱✐➳t t➟♣ ❤ñ♣ ❊ ❝→❝ ♣❤➛♥ tû ❧➔ sè ❝❤➤♥ ❝õ❛ ❉ ✭sè ❝❤➤♥ ❧➔ sè ❝❤✐❛ ❤➳t ❝❤♦ ✷✮✳ ❚➟♣ ❤ñ♣ ❊ ❝â ❜❛♦ ♥❤✐➯✉ ♣❤➛♥ tû❄ ❞✮ ❱✐➳t t➟♣ ❤ñ♣ ❋ ❝→❝ ♣❤➛♥ tû ❧➔ sè ❧➫ ❝õ❛ ❉ ✭sè ❧➫ ❧➔ sè ❦❤æ♥❣ ❝❤✐❛ ❤➳t ❝❤♦ ✷✮✳ ❚➟♣ ❤ñ♣ ❋ ❝â ❜❛♦ ♥❤✐➯✉ ♣❤➛♥ tû❄ ✷✳ ❚r♦♥❣ ♠ët ❧ỵ♣ ❤å❝✱ ♠é✐ ❤å❝ s✐♥❤ ✤➲✉ ❤å❝ t✐➳♥❣ ❆♥❤ ❤♦➦❝ t✐➳♥❣ P❤→♣✳ ❈â ✷✺ ♥❣÷í✐ ❤å❝ t✐➳♥❣ ❆♥❤✱ ✷✼ ♥❣÷í✐ ❤å❝ t✐➳♥❣ P❤→♣✱ ❝á♥ ✶✽ ♥❣÷í✐ ❤å❝ ❝↔ ❤❛✐ t❤ù t✐➳♥❣✳ ❍ä✐ ❧ỵ♣ ❤å❝ ✤â ❝â ❜❛♦ ♥❤✐➯✉ ❤å❝ s✐♥❤❄ ✸✳ ❈❤♦ ♠ët sè ❝â ✸ ❝❤ú sè ❧➔ abc ✭❛✱ ❜✱ ❝ ❦❤→❝ ♥❤❛✉ ✈➔ ❦❤→❝ ✵✮✳ ◆➳✉ ✤ê✐ ❝❤é ❝→❝ ❝❤ú sè ❝❤♦ ♥❤❛✉ t❛ ✤÷đ❝ ♠ët sè ♠ỵ✐✳ ❍ä✐ ❝â t➜t ❝↔ ❜❛♦ ♥❤✐➯✉ sè ❝â ✸ ❝❤ú sè ♥❤÷ ✈➟②❄ ✭❑➸ ❝↔ sè ❜❛♥ ✤➛✉✮✳ ✹✳ ◗✉②➸♥ s→❝❤ ❣✐→♦ ❦❤♦❛ ❚♦→♥ ✻ t➟♣ ♠ët ❝â ✶✸✷ tr❛♥❣✳ ❍❛✐ tr❛♥❣ ✤➛✉ ❦❤æ♥❣ ✤→♥❤ sè✳ ❍ä✐ ♣❤↔✐ ❞ò♥❣ t➜t ❝↔ ❜❛♦ ♥❤✐➯✉ ❝❤ú sè ✤➸ ✤→♥❤ sè ❝→❝ tr❛♥❣ ❝õ❛ q✉②➸♥ s→❝❤ ♥➔②❄ ✺✳ ❱ỵ✐ ✾ q✉❡ ❞✐➯♠ ❤➣② s➢♣ ①➳♣ t❤➔♥❤ ♠ët sè ▲❛ ▼➣✿ ❛✮ ❈â ❣✐→ trà ❧ỵ♥ ♥❤➜t✳ ❜✮ ❈â ❣✐→ trà ♥❤ä ♥❤➜t✳ ✻✳ ❱✐➳t ❝→❝ t➟♣ ❤ñ♣ s❛✉ ❜➡♥❣ ❝→❝❤ ❧✐➺t ❦➯ ❝→❝ ♣❤➛♥ tû ❝õ❛ ❝❤ó♥❣✿ ❛✮ ❚➟♣ ❤đ♣ ❆ ❝→❝ sè tü ♥❤✐➯♥ ① ♠➔ x − = 14✳ ❜✮ ❚➟♣ ❤ñ♣ ❇ ❝→❝ sè tü ♥❤✐➯♥ ① ♠➔ x + = 5✳ ❝✮ ❚➟♣ ❤ñ♣ ❈ ❝→❝ sè tü ♥❤✐➯♥ ❦❤ỉ♥❣ ✈÷đt q✉→ ✶✵✵✳ ✼✳ ❈❤♦ ❆ ❧➔ t➟♣ ❤ñ♣ ❝→❝ sè tü ♥❤✐➯♥ ❝❤✐❛ ❤➳t ❝❤♦ ✸ ✈➔ ♥❤ä ❤ì♥ ✸✵❀ ❇ ❧➔ t➟♣ ❤đ♣ ❝→❝ sè tü ♥❤✐➯♥ ❝❤✐❛ ❤➳t ❝❤♦ ✻ ✈➔ ♥❤ä ❤ì♥ ✸✵❀ ❈ ❧➔ t➟♣ ❤đ♣ ❝→❝ sè tü ♥❤✐➯♥ ❝❤✐❛ ❤➳t ❝❤♦ ✾ ✈➔ ♥❤ä ❤ì♥ ✸✵✳ ❛✮ ❱✐➳t ❝→❝ t➟♣ ❤ñ♣ ❆✱ ❇✱ ❈ ❜➡♥❣ ❝→❝❤ ❧✐➺t ❦➯ ❝→❝ ♣❤➛♥ tû ❝õ❛ ❝→❝ t➟♣ ❤ñ♣ ✤â✳ ❜✮ ❳→❝ ✤à♥❤ sè ♣❤➛♥ tû ❝õ❛ ♠é✐ t➟♣ ❤đ♣✳ ❝✮ ❉ò♥❣ ❦➼ ❤✐➺✉ ⊂ ✤➸ t❤➸ ❤✐➺♥ q✉❛♥ ❤➺ ❣✐ú❛ ❝→❝ t➟♣ ❤ñ♣ ✤â✳ ✸ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✽✳ ❚➻♠ ❤❛✐ sè ❜✐➳t tê♥❣ ❝õ❛ ❝❤ó♥❣ ❧➔ ✶✼✻❀ ♠é✐ sè ✤➲✉ ❝â ❤❛✐ ❝❤ú sè ❦❤→❝ ♥❤❛✉ ✈➔ sè ♥➔② ❧➔ sè ❦✐❛ ✈✐➳t t❤❡♦ t❤ù tü ♥❣÷đ❝ ❧↕✐✳ ✾✳ ❑❤ỉ♥❣ t➼♥❤ ❣✐→ trà ❝ö t❤➸✱ ❤➣② s♦ s→♥❤ ❤❛✐ ❜✐➸✉ t❤ù❝✿ ❛✮ A = 199 · 201 ✈➔ B = 200 · 200✳ ❜✮ C = 35 · 53 − 18 ✈➔ D = 35 + 53 · 34✳ ✶✵✳ ❚➻♠ ① ❜✐➳t✿ ❛✮ (x + 74) − 318 = 200 ❜✮ 3636 : (12x − 91) = 36 ❝✮ (x : 23 + 45) · 67 = 8911✳ ✶✶✳ ❈❤♦ S = + 10 + 13 + · · · + 97 + 100✳ ❛✮ ❚ê♥❣ tr➯♥ ❝â ❜❛♦ ♥❤✐➯✉ sè ❤↕♥❣❄ ❜✮ ❚➻♠ sè ❤↕♥❣ t❤ù ✷✷✳ ❝✮ ❚➼♥❤ ❙✳ ✶✷✳ ❱✐➳t ❝→❝ t➼❝❤ ❤♦➦❝ t❤÷ì♥❣ s❛✉ ữợ ụ tứ ởt số 25 ã 84 ❀ ❜✮ 256 · 1253 ❀ ❝✮ 6255 : 257 ❞✮ 123 · 33 ✳ ✶✸✳ ❚➻♠ x ∈ N ❜✐➳t✿ 10 x ❛✮ x = ❀ ❜✮ x10 = x ❀ ❝✮ (2x − 15)5 = (2x − 15)3 ❞✮ 2x − 15 = 17 ❀ ❡✮ (7x − 11)3 = 25 · 52 + 200✳ ✶✹✳ ❚➼❝❤ ❝→❝ sè ❧➫ ❧✐➯♥ t✐➳♣ ❝â t➟♥ ❝ò♥❣ ❧➔ ✼✳ ❍ä✐ t➼❝❤ ✤â ❝â ❜❛♦ ♥❤✐➯✉ t❤ø❛ sè❄ ✶✺✳ ❈❤♦ S = + 31 + 32 + 33 + · · · + 330 ❚➻♠ ❝❤ú sè t➟♥ ❝ò♥❣ ❝õ❛ ❙✱ tø ✤â s✉② r❛ ❙ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ sè ❝❤➼♥❤ ♣❤÷ì♥❣✳ ✶✻✳ ❚➼♥❤ ❣✐→ trà ❜✐➸✉ t❤ù❝✿ ❛✮ (102 + 112 + 122) : (132 + 142) ❜✮ 9! − 8! − 7!16· 822 ❝✮ 11 ·(3213· 4· 4· 112 −) 169 ✳ ✶✼✳ ❚➻♠ ① ❜✐➳t✿ ❛✮ (19x + · 52) : 14 = (13 − 8)2 − 42 ✹ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ❜✮ · 3x = 10 · 312 + · 274 ✶✽✳ ▼ët t❤ò♥❣ ❝â ✶✻ ❧➼t✳ ❍➣② ❞ò♥❣ ♠ët ❜➻♥❤ ✼ ❧➼t ✈➔ ♠ët ❜➻♥❤ ✸ ❧➼t ✤➸ ❝❤✐❛ ✶✻ ❧➼t t❤➔♥❤ ❤❛✐ ♣❤➛♥ ❜➡♥❣ ♥❤❛✉✳ ✶✾✳ ❚r♦♥❣ ❝→❝ sè s❛✉✱ sè ♥➔♦ ❝❤✐❛ ❤➳t ❝❤♦ ✷❀ ❝❤♦ ✹❀ ❝❤♦ ✽❀ ❝❤♦ ✺❀ ❝❤♦ ✷✺❀ ❝❤♦ ✶✷✺❄ 1010; 1076; 1984; 2782; 3452; 5341; 6375; 7800 ✷✵✳ ❱ỵ✐ ❝ò♥❣ ❝↔ ✹ ❝❤ú sè ✷❀ ✺❀ ✻❀ ✼✱ ✈✐➳t t➜t ❝↔ ❝→❝ sè✿ ❛✮ ❈❤✐❛ ❤➳t ❝❤♦ ✹❀ ❝✮ ❈❤✐❛ ❤➳t ❝❤♦ ✷✺❀ ❜✮ ❈❤✐❛ ❤➳t ❝❤♦ ✽❀ ❞✮ ❈❤✐❛ ❤➳t ❝❤♦ ✶✷✺✳ ✷✶✳ ❈â ❜❛♦ ♥❤✐➯✉ sè tü ♥❤✐➯♥ ❝â ❜❛ ❝❤ú sè ✈➔ ❝❤✐❛ ❤➳t ❝❤♦ ✸❄ ✷✷✳ ❇✐➳t r➡♥❣ A = 717 + 17 · − ❧➔ ♠ët sè ❝❤✐❛ ❤➳t ❝❤♦ ✾✳ ❈â t❤➸ sû ❞ö♥❣ ❦➳t q✉↔ ♥➔② ✤➸ ❝❤ù♥❣ tä r➡♥❣ B = 718 + 18 · − ❝ô♥❣ ❝❤✐❛ ❤➳t ❝❤♦ ổ ú ỵ õ t ự t q✉↔ tê♥❣ q✉→t ❤ì♥✿ ❱ỵ✐ ♠å✐ sè tü ♥❤✐➯♥ ♥✱ ♥➳✉ 7n + 3n − ❝❤✐❛ ❤➳t ❝❤♦ ✾ t❤➻ 7n+1 + 3(n + 1) − ❝ô♥❣ ❝❤✐❛ ❤➳t ❝❤♦ ✾✳ ✭▲♦↕✐ ❜➔✐ t➟♣ ♥➔② ❝❤✉➞♥ ❜à ❝❤♦ s q ợ ữỡ q t ❤å❝✮✳ ✷✸✳ ❛✮ ❈❤♦ ♥ ❧➔ ♠ët sè ❦❤æ♥❣ ❝❤✐❛ ❤➳t ❝❤♦ ✸✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ n2 ❝❤✐❛ ❝❤♦ ✸ ❞÷ ✶✳ ❜✮ ❈❤♦ ♣ ❧➔ ♠ët sè ♥❣✉②➯♥ tè ❧ỵ♥ ❤ì♥ ✸✳ ❍ä✐ p2 +2003 ❧➔ sè ♥❣✉②➯♥ tè ❤❛② ❤ñ♣ sè❄ ✷✹✳ ▼é✐ sè s❛✉ ❝â ❜❛♦ ♥❤✐➯✉ ÷ỵ❝✿ 90; 540; 3675✳ ✷✺✳ ✣✐➲♥ ✈➔♦ ❜↔♥❣ s❛✉ ♠å✐ sè ♥❣✉②➯♥ tè ♣ ♠➔ p2 ≤ a ✿ ❛ ✺✾ ♣ ✶✷✶ ✶✼✾ ✶✾✼ ✺ ✷✶✼ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✷✻✳ ❍❛✐ sè ♥❣✉②➯♥ tè s✐♥❤ ✤æ✐ ❧➔ ❤❛✐ sè ♥❣✉②➯♥ tè ❤ì♥ ❦➨♠ ♥❤❛✉ ✷ ✤ì♥ ✈à✳ ❚➻♠ ❤❛✐ sè ♥❣✉②➯♥ tè s✐♥❤ ✤ỉ✐ ♥❤ä ❤ì♥ ✺✵✳ ✷✼✳ ▼ët ỏ ỳ t tữợ 630 ì 480 ✭❝♠✮ ✤÷đ❝ ❧→t ❧♦↕✐ ❣↕❝❤ ❤➻♥❤ ✈✉ỉ♥❣✳ ▼✉è♥ ❝❤♦ ❤❛✐ ❤➔♥❣ ❣↕❝❤ ❝✉è✐ ❝ò♥❣ s→t ❤❛✐ ❜ù❝ t÷í♥❣ ❧✐➯♥ t✐➳♣ ổ t t tữợ ợ t ✈✐➯♥ ❣↕❝❤ ❧➔ ❜❛♦ ♥❤✐➯✉❄ ✣➸ ❧→t ❝➠♥ ♣❤á♥❣ ✤â ❝➛♥ ❜❛♦ ♥❤✐➯✉ ❣↕❝❤❄ ✷✽✳ ❈â ✻✹ ♥❣÷í✐ ✤✐ t❤❛♠ q ộ ỗ ộ ỗ t số ữớ số ỗ ọ ộ õ ❚➻♠ ❤❛✐ sè tü ♥❤✐➯♥ ❛ ✈➔ ❜ ❜✐➳t t➼❝❤ ❝õ❛ ❝❤ó♥❣ ❧➔ ✷✾✹✵ ✈➔ ❇❈◆◆ ❝õ❛ ❝❤ó♥❣ ❧➔ ✷✶✵✳ ✸✵✳ ❇❛ ❤å❝ s✐♥❤✱ ♠é✐ ♥❣÷í✐ ♠✉❛ ♠ët ❧♦↕✐ ❜ót✳ ữủt ỗ ỗ ỗ t số t tr ữ ❤ä✐ ♠é✐ ❤å❝ s✐♥❤ ♠✉❛ ➼t ♥❤➜t ❜❛♦ ♥❤✐➯✉ ❜ót❄ ✸✶✳ ▼ët ♠↔♥❤ ✤➜t ❤➻♥❤ ❝❤ú ♥❤➟t ❞➔✐ ✶✶✷♠✱ rë♥❣ ✹✵♠✳ ◆❣÷í✐ t❛ ♠✉è♥ ❝❤✐❛ ♠↔♥❤ ✤➜t t❤➔♥❤ ♥❤ú♥❣ ỉ ổ trỗ r ọ ợ ❝→❝❤ ❝❤✐❛ ♥➔♦ t❤➻ ❝↕♥❤ ❝õ❛ ỉ ✈✉ỉ♥❣ ❧➔ ❧ỵ♥ ♥❤➜t ✈➔ ❜➡♥❣ ❜❛♦ ♥❤✐➯✉❄ ✸✷✳ ❚r♦♥❣ ♠ët ❜✉ê✐ ❧✐➯♥ ❤♦❛♥✱ ❇❛♥ tê ❝❤ù❝ ✤➣ ♠✉❛ t➜t ❝↔ ✽✹✵ ❝→✐ q qỵt r ỗ qỵt ❚➼♥❤ sè ✤➽❛ ♥❤✐➲✉ ♥❤➜t ♣❤↔✐ ❝â ♠é✐ ✤➽❛ ❜❛♦ qỵt ố s ởt trữớ ởt số ợ ỡ ỗ ỳ sè✳ ▼é✐ ❧➛♥ ①➳♣ ❤➔♥❣ ✸✱ ❤➔♥❣ ✹✱ ❤➔♥❣ ✺ ✤➲✉ ✈ø❛ ✤õ✱ ❦❤ỉ♥❣ t❤ø❛ ❛✐✳ ❍ä✐ tr÷í♥❣ ✤â ❝â ❜❛♦ ♥❤✐➯✉ ❤å❝ s✐♥❤❄ ✸✹✳ ◆❣÷í✐ t❛ ✤➳♠ trù♥❣ tr♦♥❣ ♠ët rê✳ ◆➳✉ ✤➳♠ t❤❡♦ tø♥❣ ❝❤ư❝ ❝ơ♥❣ ♥❤÷ ✤➳♠ t❤❡♦ t→ ✭♠ët t→ ❝â ✶✷ q✉↔✮✱ ❤♦➦❝ ✤➳♠ t❤❡♦ tø♥❣ ✶✺ q✉↔ t❤➻ ❧➛♥ ♥➔♦ ❝ô♥❣ ❝á♥ ❧↕✐ ✶ q✉↔✳ ❚➼♥❤ sè trù♥❣ tr♦♥❣ rê✱ ❜✐➳t r➡♥❣ sè trù♥❣ ❝❤÷❛ ✤➳♥ ✶✵✵✳ ✸✺✳ ❚➼♥❤ tê♥❣✿ ❛✮ 23476893 + 542771678 ❀ ❜✮ 32456 + 97685 + 238947 ✻ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✸✻✳ ❚➼♥❤ ♥❤❛♥❤ ❝→❝ tê♥❣ s❛✉✿ ❛✮ 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 ❀ ❜✮ + + + + 10 + · · · + 100✳ ✸✼✳ ❚➻♠ ①✱ ❜✐➳t ✿ ❛✮ (x − 78) · 26 = ❀ ✸✽✳ ❚➼♥❤ ♥❤❛♥❤✿ ❜✮ 39 · (x − 5) = 39 · · + · 10 · 12 + · 20 · 24 + · 45 · 54 · · + · · 10 + · 12 · 20 + · 27 · 45 ✸✾✳ ❈❤ù♥❣ tä r➡♥❣ ❤❛✐ sè n + ✈➔ 3n + 4(n ∈ N) ❧➔ ❤❛✐ sè ♥❣✉②➯♥ tè ❝ò♥❣ ♥❤❛✉✳ ✹✵✳ ❚➻♠ sè tü ♥❤✐➯♥ ❛✱ ❜✐➳t r➡♥❣ ✶✺✻ ❝❤✐❛ ❝❤♦ ❛ ❞÷ ✶✷✱ ✈➔ ✷✽✵ ❝❤✐❛ ❝❤♦ ❛ ❞÷ ✶✵✳ ✹✶✳ ❚➻♠ ❤❛✐ sè tü ♥❤✐➯♥ ❛ ✈➔ ❜ ✭❛ ❃ ❜✮ ❝â ❇❈◆◆ ❜➡♥❣ ✸✸✻ ✈➔ ×❈▲◆ ❜➡♥❣ ✶✷✳ ✹✷✳ ❈â ✶✸✸ q✉②➸♥ ✈ð✱ ✽✵ ❜ót ❜✐✱ ✶✼✵ t➟♣ ❣✐➜②✳ ◆❣÷í✐ t❛ ❝❤✐❛ ✈ð✱ ❜ót ❜✐✱ ❣✐➜② t❤➔♥❤ ❝→❝ ♣❤➛♥ t❤÷ð♥❣ ✤➲✉ ♥❤❛✉✱ ♠é✐ ♣❤➛♥ tữ ỗ ữ s ỏ t❤ø❛ ✶✸ q✉②➸♥ ✈ð✱ ✽ ❜ót ❜✐✱ ✷ t➟♣ ❣✐➜② ❦❤ỉ♥❣ ✤õ ❝❤✐❛ ✈➔♦ ❝→❝ ♣❤➛♥ t❤÷ð♥❣✳ ❚➼♥❤ ①❡♠ ❝â ❜❛♦ ♥❤✐➯✉ ♣❤➛♥ t❤÷ð♥❣❄ ✹✸✳ ◗✉➣♥❣ ✤÷í♥❣ ❆❇ ❞➔✐ ✶✶✵❦♠✳ ▲ó❝ ✼ ❣✐í✱ ♥❣÷í✐ t❤ù ♥❤➜t ✤✐ tø ❆ ✤➸ ✤➳♥ ❇✱ ♥❣÷í✐ t❤ù ❤❛✐ ✤✐ tø ❇ ✤➸ ✤➳♥ ❆✳ ❍å ❣➦♣ ♥❤❛✉ ❧ó❝ ✾ ❣✐í✳ ❇✐➳t ✈➟♥ tè❝ ữớ tự t ợ ỡ tố ữớ tự ❧➔ ✺❦♠✴❤✳ ❚➼♥❤ ✈➟♥ tè❝ ♠é✐ ♥❣÷í✐✳ ✹✹✳ ▼ët ❝♦♥ õ ởt tọ õ ởt ữợ õ ởt ữợ tọ õ ởt ữợ t tọ ụ ởt ữợ ọ õ ữợ ♠ỵ✐ ✤✉ê✐ ❦à♣ t❤ä❄ ✹✺✳ ▼ët ❜➔ ♠❛♥❣ ♠ët rê trù♥❣ r❛ ❝❤đ✳ ❉å❝ ✤÷í♥❣ ❣➦♣ ♠ët ❜➔ ❦❤→❝ ✈ỉ þ ✤ư♥❣ ♣❤↔✐✱ rê trù♥❣ rì✐ ①✉è♥❣ ✤➜t✳ ❇➔ ❦✐❛ tọ ỵ ố số trự ọ ❇➔ ❝❤♦ ❜✐➳t tr♦♥❣ rê ❝â ❜❛♦ ♥❤✐➯✉ trù♥❣❄ ❇➔ ❝â rê trù♥❣ tr↔ ❧í✐✿ ✲ ❚ỉ✐ ❝❤➾ ♥❤ỵ r➡♥❣ sè trù♥❣ ✤â ❝❤✐❛ ❝❤♦ ✷✱ ❝❤♦ ✸✱ ❝❤♦ ✹✱ ❝❤♦ ✺✱ ❝❤♦ ✻✱ ❧➛♥ ♥➔♦ ❝ô♥❣ ❝á♥ t❤ø❛ r❛ ♠ët q✉↔✱ ♥❤÷♥❣ ❝❤✐❛ ❝❤♦ ✼ t❤➻ ❦❤ỉ♥❣ t❤ø❛ q✉↔ ♥➔♦✳ ⑨✱ ♠➔ sè trù♥❣ ❝❤÷❛ ✤➳♥ ✹✵✵ q✉↔✳ ❚➼♥❤ ①❡♠ tr♦♥❣ rê ❝â ❜❛♦ ♥❤✐➯✉ trù♥❣❄ ✹✻✳ ❚➻♠ ❜❛ sè tü ♥❤✐➯♥ ❛✱ ❜✱ ❝ ❦❤→❝ ✵ s❛♦ ❝❤♦ ❝→❝ t➼❝❤ 140a, 180b, 200c ❜➡♥❣ ♥❤❛✉ ✈➔ ❝â ❣✐→ trà ♥❤ä ♥❤➜t✳ ✼ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✷ ❙è ♥❣✉②➯♥ ✹✼✳ ❚r➯♥ trư❝ sè✱ ✤✐➸♠ ❆ ❝→❝❤ ❣è❝ ✷ ✤ì♥ ✈à ✈➲ ❜➯♥ tr→✐❀ ✤✐➸♠ ❇ ❝→❝❤ ✤✐➸♠ ❆ ❧➔ ✸ ✤ì♥ ✈à✳ ❍ä✐✿ ❛✮ ✣✐➸♠ ❆ ❜✐➸✉ ❞✐➵♥ sè ♥❣✉②➯♥ ♥➔♦❄ ❜✮ ✣✐➸♠ ❇ ❜✐➸✉ ❞✐➵♥ sè ♥❣✉②➯♥ ♥➔♦❄ ✹✽✳ ❚➻♠ ❝→❝ ❣✐→ trà t❤➼❝❤ ❤ñ♣ ❝õ❛ ❛ ✈➔ ❜✿ ❛✮ a00 > −111 ❝✮ −cb3 < −cba ❜✮ −a99 > −600 ❞✮ −cab < −c85 ✹✾✳ ❚r♦♥❣ ❝→❝ ♠➺♥❤ ✤➲ s❛✉✱ ♠➺♥❤ ✤➲ ♥➔♦ ✤ó♥❣✱ ♠➺♥❤ ✤➲ ♥➔♦ s❛✐❄ ❛✮ ◆➳✉ a = b t❤➻ |a| = |b| ❜✮ ◆➳✉ |a| = |b| t❤➻ a = b ❝✮ ◆➳✉ |a| < |b| t❤➻ a < b✳ ✺✵✳ ❚➻♠ ① ❜✐➳t✿ ❛✮ |x| + | − 5| = | − 37| ✺✶✳ ❚➻♠ x ∈ Z ❜✐➳t✿ ❛✮ |x| < 10 ❜✮ |x| > 21 ❜✮ | − 6| · |x| = |54| ❝✮ |x| > −3 ❞✮ |x| < −1✳ ✺✷✳ ❚➻♠ ❝→❝ ✈➼ ❞ö ❝❤ù♥❣ tä r➡♥❣ ❝→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ❦❤ỉ♥❣ ✤ó♥❣✿ ❛✮ ❱ỵ✐ ♠å✐ a ∈ Z ⇒ a ∈ N ❀ ❜✮ ❱ỵ✐ ♠å✐ a ∈ Z ⇒ |a| > ❀ ❝✮ ❱ỵ✐ ♠å✐ a ∈ Z ⇒ |a| > a ❀ ❞✮ ❱ỵ✐ ♠å✐ a, b ∈ Z ✈➔ |a| = |b| ⇒ a = b ❀ ❡✮ ❱ỵ✐ ♠å✐ a, b ∈ Z ✈➔ |a| > |b| ⇒ a > b✳ ✺✸✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✈ỵ✐ ♠å✐ sè ♥❣✉②➯♥ ❛ t❛ ❧✉æ♥ ❝â✿ ❛✮ |a| ≥ ✿ ●✐→ trà t✉②➺t ✤è✐ ❝õ❛ ♠ët sè ♥❣✉②➯♥ t❤➻ ❦❤æ♥❣ ➙♠✳ ❜✮ |a| ≥ a ✿ ●✐→ trà t✉②➺t ✤è✐ ❝õ❛ ♠ët sè ♥❣✉②➯♥ ❧✉ỉ♥ ❧✉ỉ♥ ❧ỵ♥ ❤ì♥ ❤♦➦❝ ❜➡♥❣ ❝❤➼♥❤ ♥â✳ ✺✹✳ ❈❤♦ |x| = 5; |y| = 11✳ ❚➼♥❤ x + y✳ ✺✺✳ ❚➼♥❤ tê♥❣✿ ✽ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ❛✮ S1 = a + |a| ✈ỵ✐ a ∈ Z ❀ ❜✮ S2 = a + |a| + a + |a| + · · · + a ✈ỵ✐ ❛ ❧➔ sè ♥❣✉②➯♥ ➙♠ ✈➔ tê♥❣ ❝â ✶✵✶ sè ❤↕♥❣✳ ✺✻✳ ❚➼♥❤ ♥❤❛♥❤✿ ❛✮ −37 + 54 + (−70) + (−163) + 246 ❀ ❜✮ −359 + 181 + (−123) + 350 + (−172) ❀ ❝✮ −69 + 53 + 46 + (−94) + (−14) + 78✳ ✺✼✳ ❈❤♦ ✶✽ sè ♥❣✉②➯♥ s❛♦ ❝❤♦ tê♥❣ ❝õ❛ ✻ sè ❜➜t ❦➻ tr♦♥❣ ❝→❝ sè ✤â ✤➲✉ ❧➔ ♠ët sè ➙♠✳ ●✐↔✐ t❤➼❝❤ ✈➻ s❛♦ tê♥❣ ❝õ❛ ✶✽ sè ✤â ❝ô♥❣ ❧➔ ♠ët sè ➙♠❄ ❇➔✐ t♦→♥ ❝á♥ ✤ó♥❣ ❦❤ỉ♥❣ ♥➳✉ t❤❛② ✶✽ sè ❜ð✐ ✶✾ sè❄ ✺✽✳ ❚➻♠ sè ♥❣✉②➯♥ ①✱ ❜✐➳t✿ ❛✮ x + 15 = ❜✮ x − = −8 ❝✮ 12 + (4 − x) = −5 ❞✮ |x| − = ❀ ❡✮ |x − 3| = ✺✾✳ ❚➻♠ ❝→❝ sè ♥❣✉②➯♥ ① s❛♦ ❝❤♦✿ ❛✮ |x| = x ❀ ❜✮ |x| > x ❀ ❝✮ |x| + x = ❀ ❞✮ x + = |x| − 5✳ ✻✵✳ ❈❤♦ ❞➣② sè 1; −2; 3; −4; 5; −6; 7; −8; 9; −10✳ ❈❤å♥ r❛ số rỗ t ỳ ❝→❝ sè ➜②✳ ❚➼♥❤ ❣✐→ trà ❧ỵ♥ ♥❤➜t✱ ❣✐→ trà ọ t t ữủ tự ợ ❜➡♥❣ ❝→❝❤ ❤ñ♣ ❧➼ ♥❤➜t✿ ❛✮ −2003 + (−21 + 75 + 2003) ❜✮ 1152 − (374 + 1152) + (−65 + 374)✳ ✻✷✳ ❚➻♠ ① ❜✐➳t✿ ❛✮ 461 + (x − 45) = 387 ❜✮ 11 − (−53 + x) = 97 ❝✮ −(x + 84) + 213 = −16✳ ✻✸✳ ❱✐➳t ✺ sè ♥❣✉②➯♥ ✈➔♦ ✺ ✤➾♥❤ ❝õ❛ ♠ët ♥❣æ✐ s❛♦ ♥➠♠ ❝→♥❤ s❛♦ ❝❤♦ tê♥❣ ❝õ❛ ❤❛✐ sè t↕✐ ❤❛✐ ✤➾♥❤ ❧✐➲♥ ♥❤❛✉ ❧✉æ♥ ❜➡♥❣ ✲✻✳ ❚➻♠ ✺ sè ♥❣✉②➯♥ ✤â✳ ✻✹✳ ❚➻♠ ① ❜✐➳t✿ ✾ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ❛✮ −12(x − 5) + 7(3 − x) = ❜✮ 30(x + 2) − 6(x − 5) − 24x = 100✳ ❝✮ x(x + 3) = ❞✮ (x − 2)(5 − x) = ❡✮ (x − 1)(x + 1) = ❢✮ (x + 3)(x − 4) = ✻✺✳ ❚➻♠ x ∈ Z ❜✐➳t✿ ❛✮ |2x − 5| = 13 ❜✮ |7x + 3| = 66 ❝✮ |5x − 2| ≤ 13 ❞✮ (x + 1) + (x + 3) + (x + 5) + · · · + (x + 99) = ❡✮ (x − 3) + (x − 2) + (x − 1) + · · · + 10 + 11 = 11 ✭sè ❤↕♥❣ ✤➛✉ t✐➯♥ ✤÷đ❝ ✈✐➳t ❧➔ x − ✈➔ ❦➳t t❤ó❝ ❞➣② ❧➔ sè ✶✶✮✳ ✻✻✳ ❚❤ü❝ ❤✐➺♥ ♣❤➨♣ t➼♥❤ ♠ët ❝→❝❤ ❤ñ♣ ❧➼✿ ❛✮ (−125) · (+25) · (−32) · (−14) ❀ ❜✮ (−159)(+56) + (+43) · (−159) + (−159) ❀ ❝✮ (−31) · (+52) + (−26) · (−162)✳ ✻✼✳ ❈❤♦ S = − + 32 − 33 + · · · + 398 − 399✳ ❛✮ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❙ ❧➔ ❜ë✐ ❝õ❛ ✲✷✵✳ ❜✮ ❚➼♥❤ ❙✱ tø ✤â s✉② r❛ 3100 ❝❤✐❛ ❝❤♦ ✹ ❞÷ ✶✳ ✻✽✳ ❚➻♠ sè ♥❣✉②➯♥ ữỡ s n + ữợ ✶✶✶ ❝á♥ ♥ ✲ ✷ ❧➔ ❜ë✐ ❝õ❛ ✶✶✳ ✸ P❤➙♥ sè ✻✾✳ ❚r♦♥❣ ♥❣➔② ❤ë✐ t♦→♥✱ ✤ë✐ t♦→♥ ❝õ❛ ♠ët ❦❤è✐ ✤÷đ❝ ❝❤✐❛ t❤➔♥❤ ❜è♥ tè♣✳ ◆➳✉ ❧➜② 53 sè ❤å❝ s✐♥❤ ❝õ❛ tè♣ t❤ù ♥❤➜t ❝❤✐❛ ✤➲✉ ❝❤♦ ❜❛ tè♣ ❦✐❛ t❤➻ sè ❤å❝ s✐♥❤ ❜è♥ tè♣ ❜➡♥❣ ♥❤❛✉✳ ◆➳✉ tè♣ t❤ù ♥❤➜t ❜ỵt ✤✐ ✻ ❤å❝ s✐♥❤ t❤➻ ❧ó❝ ✤â sè ❤å❝ s✐♥❤ ❝õ❛ tè♣ t❤ù ♥❤➜t ❜➡♥❣ tê♥❣ sè ❤å❝ s✐♥❤ ❜❛ tè♣ ❦✐❛✳ ❍ä✐ ♠é✐ tè♣ ❝â ❜❛♦ ♥❤✐➯✉ ❤å❝ s✐♥❤❄ ✶✵ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✼✵✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣✿ ❛✮ 211··223 ··523· ····39· 40 = 2120 · · · · · (2n − 1) ❜✮ (n +11)(n = n ✈ỵ✐ n ∈ N ∗ + 2)(n + 3) · · · 2n ✼✶✳ ❚➻♠ ❝→❝ ❣✐→ trà ❝õ❛ ①✱ s❛♦ ❝❤♦✿ −11 x −3 < < ✳ 12 12 ✼✷✳ ỗ rỗ s s số s −789 ✈➔ ❀ ❛✮ −8 31 3131 ❜✮ 22.3114.52 ✈➔ 22.3294.53 ❀ ❝✮ n1 ✈➔ n +1 (n ∈ N ∗) ✼✸✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❝→❝ tê♥❣ s❛✉ ❧ỵ♥ ❤ì♥ ✶✳ ❛✮ M = 83 + 153 + 73 ❀ 29 39 49 ❜✮ N = 19 + + + ❀ 60 100 150 300 31 21 −11 −1 ❝✮ P = 41 + + + + 90 72 40 45 36 ✼✹✳ ❚➻♠ x ∈ Z ❜✐➳t✿ + −1 19 x −1 58 59 −1 + < + < + + 60 120 36 60 90 72 60 ữỡ ủ ỵ t 31 ❛✮ 23 − ( + )❀ 32 23 12 13 79 28 + ) − ( − )❀ ❜✮ ( 13 + 67 41 67 41 17 ❝✮ 38 −( − − ) 45 45 51 11 ✼✻✳ ❚➼♥❤ ❝→❝ tê♥❣ s❛✉ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ủ ỵ t 1 1 A = 1.2 + + + + 2.3 3.4 49.50 2 2 ❜✮ B = 3.5 + + + + 5.7 7.9 37.39 3 3 ❝✮ C = 4.7 + + + + 7.10 10.13 73.76 ✶✶ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✼✼✳ ❚➼♥❤ ❜➡♥❣ ❝→❝❤ ❤ñ♣ ỵ 10 175 31 3❀ 125 17 ❜✮ ❝✮ 11 −5 11 − ❀ 9 33 17 18 19 20 −5 1 + − − + + 28 29 30 31 12 ✳ ✼✽✳ ❚➼♥❤ ♥❤❛♥❤✿ 3 3 − + − + 24.47 − 23 10 1001 13 9 9 24 + 47.23 − + − +9 1001 13 10 ✳ ✼✾✳ ❚➻♠ ①✿ ❛✮ 13 + 23 x = 41 ❀ ❜✮ 34 + 14 : x = −1❀ 5 +x−7 : 16 = 0❀ 24 2 2 1989 + + + + =1 ✳ 12 x(x + 1) 1991 ❝✮ − ❞✮ ✽✵✳ ▼ët ❝❛ ♥æ ①✉æ✐ ❞á♥❣ sổ tứ tr rỗ ♥❣÷đ❝ ❞á♥❣ trð ✈➲ ❆ ♠➜t 12 ❣✐í✳ ◆➳✉ ♠ët ❜➧ ♥ù❛ træ✐ tü ❞♦ ①✉æ✐ ❞á♥❣ t❤➻ ♠➜t ❜❛♦ ♥❤✐➯✉ t❤í✐ ❣✐❛♥ ✤➸ trỉ✐ tø ❆ ✤➳♥ ❇❄ ✽✶✳ ❚➻♠ ① tr♦♥❣ ❝→❝ ❤é♥ sè✿ ❛✮ x7 = 75 ❀ 35 ❜✮ x3 = 47 ❀ x x 112 ❝✮ x 15 = ✳ ✶✷ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✽✷✳ ❚➼♥❤✿ ❛✮ 1 + : 10 − ❀ 12 ❜✮ 185 − 185 151 + 121 ❀ ❝✮ − 71 12 − 8, 75 : 72 + 0, 625 : 23 ✽✸✳ ❚➻♠ ① ❜✐➳t✿ ❛✮ 7, 5x : 9−6 13 13 =2 21 25 − x).5, 25 ❜✮ (1, 16 = 75% 10 − 17 r ố s ợ ởt trữớ tr✉♥❣ ❤å❝ ❝ì sð ❝â 60% sè ❤å❝ s✐♥❤ t❤➼❝❤ ❜â♥❣ ✤→✱ 23 sè ❤å❝ s✐♥❤ t❤➼❝❤ ❜â♥❣ ❜➔♥✱ 40% sè ❤å❝ s✐♥❤ t❤➼❝❤ ❜â♥❣ tr✉②➲♥ ✈➔ 154 sè ❤å❝ s✐♥❤ t❤➼❝❤ ✤→ ❝➛✉✳ ❍➣② t➻♠ sè ❤å❝ s✐♥❤ ❝õ❛ ♠é✐ ♥❤â♠ ❝ò♥❣ t❤➼❝❤ ♠ët ♠ỉ♥ t❤➸ t❤❛♦✱ ❜✐➳t sè ❤å❝ s✐♥❤ ❝õ❛ ❦❤è✐ ✾ ❧➔ ✷✷✺✳ ✽✺✳ ◆➠♠ ♥❛② ❝♦♥ ✶✷ t✉ê✐✱ ❜è ✹✷ t✉ê✐✳ ❚➼♥❤ t➾ sè ❣✐ú❛ t✉ê✐ ❝♦♥ ✈➔ t✉ê✐ ❜è ð ♥❤ú♥❣ t❤í✐ ✤✐➸♠ ❤✐➺♥ trữợ s ❇❛ ♥❣÷í✐ ❝❤✉♥❣ ♥❤❛✉ ♠✉❛ ❤➳t ♠ët rê trù♥❣✳ ◆❣÷í✐ t❤ù ♥❤➜t ♠✉❛ sè trù♥❣ ♠➔ ❤❛✐ ♥❣÷í✐ ❦✐❛ ♠✉❛✳ ❙è trù♥❣ ♥❣÷í✐ t❤ù ❤❛✐ ♠✉❛ ❜➡♥❣ 35 sè trù♥❣ ♥❣÷í✐ t❤ù ♥❤➜t ♠✉❛✳ ◆❣÷í✐ t❤ù ❜❛ ♠✉❛ ✶✹ q✉↔✳ ❚➼♥❤ sè trù♥❣ ❧ó❝ ✤➛✉ tr♦♥❣ rê✳ ✶✸ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✹ ❍➻♥❤ ❤å❝ ✽✼✳ ✶✮ ❚r♦♥❣ ❤➻♥❤ ữợ õ ữớ t ✤✐➸♠ ❝❤÷❛ ✤➦t t➯♥✳ ❍➣② ✤✐➲♥ ❝→❝ ❝❤ú ❆✱ ❇✱ ❈ ✈➔♦ ✤ó♥❣ ✈à tr➼ ❝õ❛ ♥â ❜✐➳t✿ ❛✮ ✣✐➸♠ ❆ ❦❤ỉ♥❣ t❤✉ë❝ ✤÷í♥❣ t❤➥♥❣ ♠ ✈➔ ❝ơ♥❣ ❦❤ỉ♥❣ t❤✉ë❝ ✤÷í♥❣ t❤➥♥❣ ♥❀ ❜✮ ✣✐➸♠ ❇ ❦❤ỉ♥❣ t❤✉ë❝ ✤÷í♥❣ t❤➥♥❣ ♠❀ ❝✮ ✣✐➸♠ ❈ ❦❤ỉ♥❣ t❤✉ë❝ ✤÷í♥❣ t❤➥♥❣ ♥✳ ✷✮ ❱➩ ✤÷í♥❣ t❤➥♥❣ ♣ ✈➔ ❝→❝ ✤✐➸♠ ❆✱ ❇ ♥➡♠ tr➯♥ ♣✳ ❛✮ ◆➯✉ ❝→❝❤ ✈➩ ✤✐➸♠ ❈ t❤➥♥❣ ❤➔♥❣ ✈ỵ✐ ❤❛✐ ✤✐➸♠ ❆✱ ❇❀ ❜✮ ◆➯✉ ❝→❝❤ ✈➩ ✤✐➸♠ ❉ ❦❤ỉ♥❣ t❤➥♥❣ ❤➔♥❣ ✈ỵ✐ ✷ ✤✐➸♠ ❆✱ ❇✳ ✽✽✳ trữợ ởt số tr õ ổ õ ✤✐➸♠ ♥➔♦ t❤➥♥❣ ❤➔♥❣✳ ❱➩ ❝→❝ ✤÷í♥❣ t❤➥♥❣ ✤✐ q✉❛ ❝→❝ ❝➦♣ ✤✐➸♠✳ ❇✐➳t tê♥❣ sè ✤÷í♥❣ t❤➥♥❣ ✈➩ ✤÷đ❝ ọ õ trữợ ✤✐➸♠ ❉ ✈➔ ❊ s❛♦ ❝❤♦ ❉ ♥➡♠ ❣✐ú❛ ❈ ✈➔ ❊ ❝á♥ ❊ ♥➡♠ ❣✐ú❛ ❉ ✈➔ ❋✳ ❛✮ ❱➻ s❛♦ ❝â t❤➸ ❦❤➥♥❣ ✤à♥❤ ✹ ✤✐➸♠ ❈✱ ❉✱ ❊✱ ❋ t❤➥♥❣ ❤➔♥❣ ❜✮ ❑➸ t➯♥ ❤❛✐ t✐❛ trò♥❣ ♥❤❛✉ ❣è❝ ❊✳ ❝✮ ❱➻ s❛♦ ❝â t❤➸ ❦❤➥♥❣ ✤à♥❤ ✤✐➸♠ ❊ ♥➡♠ ❣✐ú❛ ❈ ✈➔ ❋✳ ✶✹ ❙✐❣♠❛ ✲ ữợ rỗ tr ❝➙✉ ❤ä✐ s❛✉✿ ❛✮ ❍➻♥❤ ❝â ♠➜② t✐❛❄ ❈â ♠➜② ✤♦↕♥ t❤➥♥❣❄ ❜✮ ◆❤ú♥❣ ❝➦♣ ✤♦↕♥ t❤➥♥❣ ♥➔♦ ❦❤æ♥❣ ❝➢t ♥❤❛✉❄ ❝✮ ❍❛✐ ✤♦↕♥ t❤➥♥❣ ♥➔♦ ❝➢t ♥❤❛✉ t↕✐ ✤✐➸♠ ♥➡♠ ❣✐ú❛ ❤❛✐ ✤➛✉ ❝õ❛ ♠é✐ ✤♦↕♥ t❤➥♥❣❄ ✾✶✳ ❚r➯♥ ✤÷í♥❣ t❤➥♥❣ ①② ❧➜② ♠ët ✤✐➸♠ ❖ ✈➔ ❤❛✐ ✤✐➸♠ ▼✱ ◆ s❛♦ ❝❤♦ ❖▼ ❂ ✷❝♠❀ ❖◆ ❂ ✸❝♠✳ ❱➩ ❝→❝ ✤✐➸♠ ❆ ✈➔ ❇ tr➯♥ ✤÷í♥❣ t❤➥♥❣ ①② s❛♦ ❝❤♦ ▼ ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ ❖❆❀ ◆ ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ ❖❇✳ ❚➼♥❤ ✤ë ❞➔✐ ❆❇✳ ✾✷✳ ●å✐ ❖ ❧➔ ♠ët ✤✐➸♠ ❝õ❛ ✤♦↕♥ t❤➥♥❣ ❆❇ ❂ ✹❝♠✳ ❳→❝ ✤à♥❤ ✈à tr➼ ❝õ❛ ✤✐➸♠ ❖ ✤➸ ✿ ❛✮ ❚ê♥❣ ❆❇ ✰ ❇❖ ✤↕t ❣✐→ trà ♥❤ä ♥❤➜t❀ ❜✮ ❚ê♥❣ ❆❇ ✰ ❇❖ ❂ ✷❇❖❀ ❝✮ ❚ê♥❣ ❆❇ ✰ ❇❖ ❂ ✸❇❖✳ ✾✸✳ ❈❤♦ ✤÷í♥❣ t❤➥♥❣ ♠ ✈➔ ♥➠♠ ✤✐➸♠ ❆✱❇✱❈✱❉✱❊ ❦❤æ♥❣ t❤✉ë❝ ♠✳ ❛✮ ❈❤ù♥❣ tä r➡♥❣ tr♦♥❣ ❤❛✐ ♥û❛ ♠➢t ♣❤➥♥❣ ✤è✐ ♥❤❛✉ ❜ð ❧➔ ✤÷í♥❣ t❤➥♥❣ ♠✱ ❝â ♠ët ♠➦t ♣❤➥♥❣ ❝❤ù❛ ➼t ♥❤➜t ✸ ✤✐➸♠✳ ❜✮ ❈ù q✉❛ ❤❛✐ ✤✐➸♠ ✈➩ ♠ët ✤♦↕♥ t❤➥♥❣✳ ❍ä✐ ♥❤✐➲✉ ♥❤➜t ❝â ♠➜② ✤♦↕♥ t❤➥♥❣ ❝➢t ♠❄ ✾✹✳ ❱➩ ❣â❝ ①❖② ❦❤→❝ ❣â❝ ❜➭t✳ ▲➜② ❆ tr➯♥ t✐❛ ❖①✱ ❧➜② ❇ tr➯♥ t✐❛ ❖② ✭❆ ✈➔ ❇ ❦❤→❝ ❖✮✳ ❍➣② ❧➜② ♠ët ✤✐➸♠ ❈ s❛♦ ❝❤♦ ❣â❝ BOC ❦➲ ❜ò ✈ỵ✐ ❣â❝ BOA✳ ❛✮ ❚r♦♥❣ ❜❛ ✤✐➸♠ ❆✱❖✱❈ ✤✐➸♠ ♥➔♦ ♥➡♠ ❣✐ú❛ ❤❛✐ ✤✐➸♠ ❝á♥ ❧↕✐❄ ❜✮ ❱➩ ❝→❝ t✐❛ ❇❆✱❇❈ ❤ä✐ ✤✐➸♠ ❖ ♥➡♠ tr♦♥❣ ❣â❝ ♥➔♦❄ ❝✮ ❑➸ t➯♥ ❝→❝ ❝➦♣ ❣â❝ ❦➲ ❜ò ✤➾♥❤ ❇✳ ✶✺ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✾✺✳ ❈❤♦ ❣â❝ ❜➭t xOy✳ ❱➩ ❤❛✐ t✐❛ ❖♠✱ ❖♥ tr➯♥ ❝ò♥❣ ♠ët ♥û❛ ♠➦t ♣❤➥♥❣ ❜í ①② s❛♦ ❝❤♦ xOm = 1200; xOn = a0✳ ❚➻♠ ❣✐→ trà ❝õ❛ ❛ ✤➸ t✐❛ ❖♠ ♥➡♠ ❣✐ú❛ ❤❛✐ t✐❛ ❖②✱ ❖♥✳ ✾✻✳ ❚r➯♥ ♠➦t ♣❤➥♥❣✱ ❝❤♦ t✐❛ ❖①✳ ❱➩ ❤❛✐ t✐❛ ❖②✱ ❖t s❛♦ ❝❤♦ xOy = 1000; xOt = 1500 ✳ ❚➼♥❤ sè ✤♦ ❣â❝ yOt✳ ✾✼✳ ❚r➯♥ ♥û❛ ♠➦t ♣❤➥♥❣ ❜í ❝❤ù❛ t✐❛ ❖① ✈➩ ❜❛ t✐❛ ❖②✱ ❖③✱ ❖t s❛♦ ❝❤♦ xOy = 500 ; xOz = 750 ; xOt = 1000 ✳ ❳→❝ ✤à♥❤ ①❡♠ t✐❛ ♥➔♦ ❧➔ t✐❛ ♣❤➙♥ ❣✐→❝ ❝õ❛ ♠ët ❣â❝✳ ✾✽✳ ❈❤♦ ❜❛ t✐❛ ❖❆✱ ❖❇✱ ❖❈ t↕♦ t❤➔♥❤ ❜❛ ❣â❝ ❜➡♥❣ ♥❤❛✉ ✈➔ ❦❤æ♥❣ ❝â ✤✐➸♠ tr♦♥❣ ❝❤✉♥❣ DOB; BOC ✈➔ COA✳ ❱➻ s❛♦ ❝â t❤➸ ❦❤➥♥❣ ✤à♥❤ t✐❛ ✤è✐ ❝õ❛ ♠é✐ t✐❛ ♥â✐ tr➯♥ ❧➔ t✐❛ ♣❤➙♥ ❣✐→❝ ❝õ❛ ❣â❝ t↕♦ ❜ð✐ ❤❛✐ t✐❛ ❝á♥ ❧↕✐❄ ✾✾✳ ❚r➯♥ ✤÷í♥❣ t❤➥♥❣ ①② ❧➜② ✤✐➸♠ ❖✳ ❱➩ ✤÷í♥❣ trá♥ ✭❖❀ ✸✮ ❝➢t ❖①✱ ❖② t❤ù tü t↕✐ ❆ ✈➔ ❇✳ ❱➩ ✤÷í♥❣ trá♥ ✭❖❀ ✷✮ ❝➢t t✐❛ ❖①✱ ❖② t❤ù tü t↕✐ ❈ ✈➔ ❉✳ ❱➩ ✤÷í♥❣ tr♦♥❣ ✭❉❀ ❇❉✮ ❝➢t ❇❖ t↕✐ ▼ ✈➔ ❝➢t ✤÷í♥❣ trá♥ ✭❖❀ ✷✮ t↕✐ ◆✳ ❛✮ ❙♦ s→♥❤ ❆❈ ✈➔ ❇❉✳ ❜✮ ❈❤ù♥❣ tä ▼ ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ ❖❉ ❝✮ ❙♦ s→♥❤ tê♥❣ ❖◆ ✰ ◆❉ ✈ỵ✐ ❖❇✳ ✶✵✵✳ ❛✮ ❱➩ ABC ❜✐➳t ❇❈ ❂ ✸✱✺❝♠❀ ❆❇ ❂ ✷❝♠❀ ❆❈ ❂ ✸❝♠✳ ❜✮ ❱➩ t✐➳♣ ADE ❜✐➳t ❉ t❤✉ë❝ t✐❛ ✤è✐ ❝õ❛ t✐❛ ❆❇ ✈➔ ❆❉ ❂ ✶❝♠❀ ❊ t❤✉ë❝ t✐❛ ✤è✐ t✐❛ ❆❈ ✈➔ ❆❊ ❂ ✶✱✺❝♠✳ ❝✮ ❍❛✐ t✐❛ ❇❊ ✈➔ ❈❉ ❝➢t ♥❤❛✉ t↕✐ ❖✳ ❉ò♥❣ ❝♦♠♣❛ ✤➸ ❦✐➸♠ tr❛ ①❡♠ ❊ ✈➔ ❉ t❤❡♦ t❤ù tü ❝â ♣❤↔✐ ❧➔ tr✉♥❣ ✤✐➸♠ ❝õ❛ ❖❇ ✈➔ ❖❈ ❦❤æ♥❣❄ ✶✵✶✳ ▼ët ♠↔♥❤ ❣✐➜② ❤➻♥❤ ❝❤ú ♥❤➟t ❜à ❣➟♣ ❧↕✐ t❤❡♦ ❤➻♥❤ ữợ õ ❆❇❈❉ ❧➔ ♠ët ❤➻♥❤ ✈✉æ♥❣✳ DP C ❧➔ ♠ët t❛♠ ❣✐→❝ ✤➲✉✳ ❆❇ ❂ ❉P ✈➔ ❈❉ ❂ ❈P✳ ❚➻♠ ❣â❝ ADP ✳ ✶✵✸✳ ❚r♦♥❣ ❤➻♥❤ ✈➩✱ ❇❊ ❂ ❆❈✱ CAE = 300 ✈➔ AEB = 700✳ ❚➻♠ ❣â❝ ABC r ỗ ữợ ❆❉✳ ●â❝ ABC = 400 ✈➔ ❣â❝ ACD = 800 ✳ ❚➼♥❤ ❣â❝ ❇❆❉✳ ✶✵✺✳ ❆❇❈❉ ❧➔ ♠ët ❤➻♥❤ ✈✉æ♥❣ ✈➔ t❛♠ ❣✐→❝ ❇❈❊ ❧➔ t❛♠ ❣✐→❝ ✤➲✉✳ ❚➼♥❤ ❣â❝ ❆❊❉✳ ✶✵✻✳ ❚➼♥❤ ❣â❝✿ a + b + c + d + e + f ✶✼ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✶✵✼✳ ❚r♦♥❣ t❛♠ ❣✐→❝ ❆❇❈✱ ❆❇ ❂ ❆❈✱ ❆❉ ❂ ❆❊ ✈➔ BAD = 600✳ ❚➼♥❤ ❣â❝ ❈❉❊✳ ✶✵✽✳ ❚r♦♥❣ t❛♠ ❣✐→❝ ❆❇❈✱ ❣â❝ ❆❇❈ ❜➡♥❣ 800✱ ❆❉ ❂ ❆❊ ✈➔ ❈❉ ❂ ❈❋✳ ❚➼♥❤ ❣â❝ ❊❉❋✳ ✶✽ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✺ ▲í✐ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥ ❝❤å♥ ❧å❝ ❇➔✐ ✶✶✿ ❈❤♦ S = + 10 + 13 + · · · + 97 + 100✳ ❛✮ ❚ê♥❣ tr➯♥ ❝â ❜❛♦ ♥❤✐➯✉ sè ❤↕♥❣❄ ❜✮ ❚➻♠ sè ❤↕♥❣ t❤ù ✷✷✳ ❝✮ ❚➼♥❤ ❙✳ ❇➔✐ ❣✐↔✐✿ ❛✮ ❙è sè ❤↕♥❣ ❝õ❛ tê♥❣ ❧➔ ✿ 1003− + = 32 ✭sè ❤↕♥❣✮✳ ❜✮ ●å✐ sè ❤↕♥❣ t❤ù ✷✷ ❧➔ ①✱ t❛ ❝â✿ x −3 + = 22 ⇒ x = 70✳ ❝✮ S = + 10 + 13 + · · · + 97 + 100 ❤❛② S = 100 + 97 + 94 + · · · + 10 + ✭❣✐❛♦ ❤♦→♥✮ ⇔ 2S = (7 + 100) + (10 + 97) + · · · + (100 + 7) ✭✈ỵ✐ ✸✷ ♥❤â♠✳✮ ⇔ 2S = (7 + 100) · 32 (7 + 100) · 32 = 1712✳ ⇔S= ❇➔✐ ✷✷✿ ❇✐➳t r➡♥❣ A = 717 + 17 · − ❧➔ ♠ët sè ❝❤✐❛ ❤➳t ❝❤♦ ✾✳ ❈â t❤➸ sû ❞ö♥❣ ❦➳t q✉↔ ♥➔② ✤➸ ❝❤ù♥❣ tä r➡♥❣ B = 718 + 18 · − ❝ô♥❣ ❝❤✐❛ ❤➳t ❝❤♦ ✾ ❦❤æ♥❣❄ ❇➔✐ ❣✐↔✐✿ ❚❤❡♦ ✤➛✉ ❜➔✐ A = 717 + 17 · − ❧➔ ♠ët sè tü ♥❤✐➯♥ ❝❤✐❛ ❤➳t ❝❤♦ ✾ tù❝ ❧➔ t❛ ❝â (717 + 50) | 9✳ ❚❛ ✈✐➳t sè ❇ ♥❤÷ s❛✉✿ B = 718 + 18 · − = 718 + 53 = 7(717 + 50) − 297 = · (717 + 50) − 33 · 9✳ ❱➻ (717 + 50) | ✈➔ (33 · 9) | B | ú ỵ õ t ự ♠✐♥❤ ❦➳t q✉↔ tê♥❣ q✉→t ❤ì♥✿ ❱ỵ✐ ♠å✐ sè tü ♥❤✐➯♥ ♥✱ ♥➳✉ 7n + 3n − ❝❤✐❛ ❤➳t ❝❤♦ ✾ t❤➻ 7n+1 + 3(n + 1) − ❝ô♥❣ ❝❤✐❛ ❤➳t ❝❤♦ ✾✳ ✭▲♦↕✐ ❜➔✐ t➟♣ ♥➔② ❝❤✉➞♥ s q ợ ữỡ q ♥↕♣ t♦→♥ ❤å❝✮✳ ❇➔✐ ✸✷✿ ❚r♦♥❣ ♠ët ❜✉ê✐ ❧✐➯♥ ❤♦❛♥✱ ❇❛♥ tê ❝❤ù❝ ✤➣ ♠✉❛ t➜t ❝↔ ✽✹✵ ❝→✐ ❜→♥❤✱ q qỵt r ỗ qỵt sè ✤➽❛ ♥❤✐➲✉ ♥❤➜t ♣❤↔✐ ❝â ♠é✐ ✤➽❛ ❜❛♦ ♥❤✐➯✉ qỵt ố ✤➽❛ ♥❤✐➲✉ ♥❤➜t ♣❤↔✐ ❝â ❧➔ ×❈▲◆✭✽✹✵✱ ✷✸✺✷✱ ✺✻✵✮ ❂ ✺✻✳ ▼é✐ ✤➽❛ ❝â sè ❜→♥❤ ❧➔✿ 840 : 56 = 15 ✭❜→♥❤✮ ▼é✐ ✤➽❛ ❝â sè ❦➭♦ ❧➔✿ 2352 : 56 = 42 ộ õ số qỵt ❧➔✿ 560 : 56 = 10 ✭q✉↔✮ ❱➟② ♥❤✐➲✉ ♥❤➜t ✺✻ ✤➽❛✱ ♠é✐ ✤➽❛ ❝â ✶✺ ❝→✐ ❜→♥❤✱ ✹✷ ❝→✐ q qỵt õ q ✽✵ ❜ót ❜✐✱ ✶✼✵ t➟♣ ❣✐➜②✳ ◆❣÷í✐ t❛ ❝❤✐❛ ✈ð✱ ❜ót ❜✐✱ ❣✐➜② t❤➔♥❤ ❝→❝ ♣❤➛♥ t❤÷ð♥❣ ✤➲✉ ♥❤❛✉✱ ♠é✐ tữ ỗ ữ s ❝á♥ t❤ø❛ ✶✸ q✉②➸♥ ✈ð✱ ✽ ❜ót ❜✐✱ ✷ t➟♣ ❣✐➜② ❦❤ỉ♥❣ ✤õ ❝❤✐❛ ✈➔♦ ❝→❝ ♣❤➛♥ t❤÷ð♥❣✳ ❚➼♥❤ ①❡♠ ❝â ❜❛♦ ♥❤✐➯✉ ♣❤➛♥ t❤÷ð♥❣❄ ❇➔✐ ❣✐↔✐✿ ❙è ✈ð ✤➣ ❝❤✐❛✿ ✶✸✸ ✲ ✶✸ ❂ ✶✷✵ q✉②➸♥✳ ❙è ❜ót ✤➣ ❝❤✐❛✿ ✽✵ ✲ ✽ ❂ ✼✷ ❜ót✳ ❙è t➟♣ ❣✐➜② ✤➣ ❝❤✐❛✿ ✶✼✵ ✲ ✷ ❂ ✶✻✽ t➟♣ ❣✐➜②✳ ❙è tữ ì tữ ❱➟② sè ♣❤➛♥ t❤÷ð♥❣ ❧➔ ✷✹✳ ❇➔✐ ✹✺✿ ▼ët ❜➔ ♠❛♥❣ ♠ët rê trù♥❣ r❛ ❝❤đ✳ ❉å❝ ✤÷í♥❣ ❣➦♣ ♠ët ổ ỵ rờ trự rỡ ố t tọ ỵ ố số trự ❜➧♥ ❤ä✐✿ ✲ ❇➔ ❝❤♦ ❜✐➳t tr♦♥❣ rê ❝â ❜❛♦ ♥❤✐➯✉ trù♥❣❄ ❇➔ ❝â rê trù♥❣ tr↔ ❧í✐✿ ✲ ❚ỉ✐ ❝❤➾ ♥❤ỵ r➡♥❣ sè trù♥❣ ✤â ❝❤✐❛ ❝❤♦ ✷✱ ❝❤♦ ✸✱ ❝❤♦ ✹✱ ❝❤♦ ✺✱ ❝❤♦ ✻✱ ❧➛♥ ♥➔♦ ❝ô♥❣ ❝á♥ t❤ø❛ r❛ ♠ët q✉↔✱ ♥❤÷♥❣ ❝❤✐❛ ❝❤♦ ✼ t❤➻ ❦❤ỉ♥❣ t❤ø❛ q✉↔ ♥➔♦✳ ⑨✱ ♠➔ sè trù♥❣ ❝❤÷❛ ✤➳♥ ✹✵✵ q✉↔✳ ❚➼♥❤ ①❡♠ tr♦♥❣ rê ❝â ❜❛♦ ♥❤✐➯✉ trù♥❣❄ ❇➔✐ ❣✐↔✐✿ ●å✐ sè trù♥❣ ❧➔ ❛✳ ❚❛ ❝â a − ❧➔ ❜ë✐ ❝❤✉♥❣ ❝õ❛ 2, 3, 4, 5, ✈➔ a − < 399✳ ❚ø ✤â t❛ ✤÷đ❝✿ a − ∈ {60; 120; 180; 240; 300; 360} a ∈ {61; 121; 181; 241; 301; 361} ❉♦ a | ♥➯♥ ❛ ❂ ✸✵✶✳ ❘ê trù♥❣ ❝â ✸✵✶ q✉↔✳ ❇➔✐ ✹✾✿ ❚r♦♥❣ ❝→❝ ♠➺♥❤ ✤➲ s❛✉✱ ♠➺♥❤ ✤➲ ♥➔♦ ✤ó♥❣✱ ♠➺♥❤ ✤➲ ♥➔♦ s❛✐❄ ❛✮ ◆➳✉ a = b t❤➻ |a| = |b| ❜✮ ◆➳✉ |a| = |b| t❤➻ a = b ✷✵ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ❝✮ ◆➳✉ |a| < |b| t❤➻ a < b✳ ❇➔✐ ❣✐↔✐✿ ❛✮ ✣ó♥❣ ❜✮ ❙❛✐ ✭✈➼ ❞ư✿ ❛ ❂ ✸✱ ❜ ❂ ✲✸✮ ❀ ❝✮ ❙❛✐ ✭✈➼ ❞ö✿ ❛ ❂ ✲✸✱ ❜ ❂ ✲✹✮ ❇➔✐ ✻✺✿ ❚➻♠ x ∈ Z ❜✐➳t✿ ❛✮ |2x − 5| = 13 ❜✮ |7x + 3| = 66 ❝✮ |5x − 2| ≤ 13 ❞✮ (x + 1) + (x + 3) + (x + 5) + · · · + (x + 99) = ❡✮ (x − 3) + (x − 2) + (x − 1) + · · · + 10 + 11 = 11 ❇➔✐ ❣✐↔✐✿ ❛✮ |2x − 5| = 13✳ ❳➨t ✷ tr÷í♥❣ ❤đ♣✿ 2x − = 13❀ 2x − = −13✳ ❱➟② ① ❂ ✾❀ ① ❂ ✲✹✳ ❜✮ |7x + 3| = 66 s✉② r❛ ① ❂ ✾✳ ❝✮ |5x − 2| ≤ 13 ♥➯♥ −13 ≤ 5x − ≤ 13 −11 ≤ 5x ≤ 15 −2 ≤ x ≤ ⇔ x ∈ {−2; −1; 0; 1; 2; 3} ❞✮ (x + 1) + (x + 3) + (x + 5) + · · · + (x + 99) = s✉② r❛ [(x + 1) + (x + 99)] · 50 = (x + 50) · 50 = 0❀ x + 50 = ⇔ x = −50✳ ❡✮ (x − 3) + (x − 2) + (x − 1) + · · · + 10 + 11 = 11✳ ❇ä sè ❤↕♥❣ ✶✶ ð ❤❛✐ ✈➳ t❛ ✤÷đ❝✿ (x − 3) + (x − 2) + (x − 1) + · · · + 10 = ●å✐ sè sè ❤↕♥❣ ð ✈➳ tr→✐ ❧➔ ♥✭♥ ❃ ✵✮ t❛ ❝â✿ [(x − 3) + 10] · n = ❤❛② (x + 7) · n = 0✳ ❱➻ n = ♥➯♥ x + = 0✱ ❞♦ ✤â x = −7✳ ❇➔✐ ✼✶✿ ❚➻♠ ❝→❝ ❣✐→ trà ❝õ❛ ①✱ s❛♦ ❝❤♦✿ −11 x −3 < < ✳ 12 12 ❇➔✐ ❣✐↔✐✿ −9 ❈❤å♥ ♠➝✉ sè ❝❤✉♥❣ ❧➔ ✶✷ t❛ ❝â✿ −3 = ✳ 12 ✷✶ ❙✐❣♠❛ ✲ ▼❆❚❍❙ x −9 ❉♦ ✤â t❛ ❝â✿ −11 < < ✱ s✉② r❛ −11 < x < −9✳ ❱➟② x = −10✳ 12 12 12 ❇➔✐ ✽✵✿ ▼ët ❝❛ ♥æ ①✉ỉ✐ ❞á♥❣ sỉ♥❣ tø ❆ ✤➳♥ ❇ tr♦♥❣ ✸ ❣✐í rỗ ữủ ỏ tr t 21 ❣✐í✳ ◆➳✉ ♠ët ❜➧ ♥ù❛ trỉ✐ tü ❞♦ ①✉ỉ✐ ❞á♥❣ t❤➻ ♠➜t ❜❛♦ ♥❤✐➯✉ t❤í✐ ❣✐❛♥ ✤➸ trỉ✐ tø ❆ ✤➳♥ ❇❄ ❇➔✐ ❣✐↔✐✿ ❱➟♥ tè❝ ❝❛ ♥æ ❦❤✐ ①✉æ✐ ❞á♥❣✿ 31 ❱➟♥ tè❝ ❝❛ ♥ỉ ❦❤✐ ♥❣÷đ❝ ❞á♥❣✿ : 12 = 29 ✳ ❍✐➺✉ 13 − 29 = 19 tố ữợ trổ tỹ tố ữợ trổ tỹ 19 : = 181 ữợ trổ tü ❞♦ tø ❆ ✤➳♥ ❇ ❧➔✿ : 181 = 18 ✭❣✐í✮✳ ❇➔✐ ✽✻✿ ❇❛ ♥❣÷í✐ ❝❤✉♥❣ ♥❤❛✉ ♠✉❛ ❤➳t ♠ët rê trù♥❣✳ ◆❣÷í✐ t❤ù ♥❤➜t ♠✉❛ sè trù♥❣ ♠➔ ❤❛✐ ♥❣÷í✐ ❦✐❛ ♠✉❛✳ ❙è trù♥❣ ♥❣÷í✐ t❤ù ❤❛✐ ♠✉❛ ❜➡♥❣ 35 sè trù♥❣ ♥❣÷í✐ t❤ù ♥❤➜t ♠✉❛✳ ◆❣÷í✐ t❤ù ❜❛ ♠✉❛ ✶✹ q✉↔✳ ❚➼♥❤ sè trù♥❣ ❧ó❝ ✤➛✉ tr♦♥❣ rê✳ ❇➔✐ ❣✐↔✐✿ ◆❣÷í✐ t❤ù ♥❤➜t ♠✉❛ 21 sè trù♥❣ ♠➔ ❤❛✐ ♥❣÷í✐ ❦✐❛ ♠✉❛✳ ❱➟② ♥❣÷í✐ t❤ù ♥❤➜t ♠✉❛ 13 ê trù♥❣✳ ◆❣÷í✐ t❤ù ❤❛✐ ♠✉❛ 13 · 53 = 15 ✭rê trù♥❣✮ P❤➛♥ trù♥❣ ♥❣÷í✐ t❤ù ❜❛ ♠✉❛✿ − 31 + 15 = 157 ✭rê trù♥❣✮ ❙è trù♥❣ tr♦♥❣ rê ❧➔ 14 : 157 = 30 ✭q✉↔✮✳ ✷✷ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ❇➔✐ ✶✵✼✿ ❚r♦♥❣ t❛♠ ❣✐→❝ ❆❇❈✱ ❆❇ ❂ ❆❈✱ ❆❉ ❂ ❆❊ ✈➔ BAD = 600✳ ❚➼♥❤ ❣â❝ ❈❉❊✳ ❇➔✐ ❣✐↔✐✿ ✣➦t DAE = 2x ⇔ BAC = 600 + 2x ABC = ACB = (1800 − 600 − 2x) : = 600 − x ⇔ ADE = AED = (1800 − 2x) : = 900 − x CDE + DCE = 900 − x DCE = 600 − x CDE = (900 − x) − (600 − x) = 300 ✳ ❇➔✐ ✶✵✽✿ ❚r♦♥❣ t❛♠ ❣✐→❝ ❆❇❈✱ ❣â❝ ❆❇❈ ❜➡♥❣ 800✱ ❆❉ ❂ ❆❊ ✈➔ ❈❉ ❂ ❈❋✳ ❚➼♥❤ ❣â❝ ❊❉❋✳ ❇➔✐ ❣✐↔✐✿ A + B + C = 1800 A = 1800 − (1 + 1) ❚ø A + B + C = 1800 B = 80 C = 1800 − (2 + 2) 1800 − × + 800 − 1800 − × = 1800 ✷✸ ❙✐❣♠❛ ✲ ▼❆❚❍❙ 4400 − × (1 + 2) 2600 (1 + 2) EDF + + EP F = 1800 = × (1 + 2) = 1300 ❂ 1800 = 500 ✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ sü q✉❛♥ t➙♠ ❝õ❛ ❜↕♥ ✤å❝✦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ∗∗∗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❍➌❚ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ∗∗∗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ... ❀ 29 39 49 ❜✮ N = 19 + + + ❀ 60 100 150 300 31 21 −11 −1 ❝✮ P = 41 + + + + 90 72 40 45 36 ✼✹✳ ❚➻♠ x ∈ Z ❜✐➳t✿ + −1 19 x −1 58 59 −1 + < + < + + 60 120 36 60 90 72 60 ữỡ ủ ỵ t 31 ❛✮ 23 − (... sè trù♥❣ ❝❤÷❛ ✤➳♥ ✶✵✵✳ ✸✺✳ ❚➼♥❤ tê♥❣✿ ❛✮ 234 768 93 + 54277 167 8 ❀ ❜✮ 324 56 + 9 768 5 + 238947 ✻ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✸✻✳ ❚➼♥❤ ♥❤❛♥❤ ❝→❝ tê♥❣ s❛✉✿ ❛✮ 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 ❀ ❜✮ + + + + 10 +... ❂ ❆❊ ✈➔ BAD = 60 0✳ ❚➼♥❤ ❣â❝ ❈❉❊✳ ❇➔✐ ❣✐↔✐✿ ✣➦t DAE = 2x ⇔ BAC = 60 0 + 2x ABC = ACB = (1800 − 60 0 − 2x) : = 60 0 − x ⇔ ADE = AED = (1800 − 2x) : = 900 − x CDE + DCE = 900 − x DCE = 60 0 − x CDE =