❚❙✳ ◆❣✉②➵♥ ❱➠♥ ▲đ✐ ✭❝❤õ ❜✐➯♥✮ − ◆❣ỉ ❚❤à ◆❤➣ ❉ r❛ ❢t ✶✵✽ ❇⑨■ ❚❖⑩◆ ❈❍➴◆ ▲➴❈ ▲❰P ✺ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✶✵✽ ❜➔✐ t♦→♥ ❝❤å♥ ❧å❝ ❧ỵ♣ ✺ ❱ỵ✐ ❝→❝ t✐➯✉ ❝❤➼ ✤➦t r❛✿ ✲ ởt t t ữỡ tr ợ ✲ ❑❤â ✈ø❛ ✤õ ♥❤÷♥❣ ❦❤ỉ♥❣ t❤→❝❤ t❤ù❝✳ ✲ ❚♦→♥ ♠➔ ❦❤æ♥❣ ❦❤æ ❦❤❛♥✳ ✲ ❳ù♥❣ ✤→♥❣ t↕♦ ♥✐➲♠ t✐♥ ❝❤♦ ❧ù❛ t✉ê✐ ✈ø❛ ①♦♥❣ t✐➸✉ ❤å❝✳ ▼ö❝ t✐➯✉ ✤➲ r❛ ✤➣ ❧➔♠ ❝❤♦ t➔✐ ❧✐➺✉ t❤➔♥❤ ❦➳t q✉↔ t➻♠ tỏ tứ ỗ ữủ s ❦ÿ ❧÷ï♥❣✳ ❈ơ♥❣ ❝❤➾ ❧➔ ✶✵✽ ❜➔✐ t➟♣ t♦→♥✱ ♥❤÷♥❣ ủ ộ ữ õ ỗ t ởt t t❤è♥❣ ♥❤➜t✳ ❈→❝ ❜➔✐ t♦→♥ ♥❤÷ sè♥❣ ❞➟② ✈➔ ❧✐➯♥ ❦➳t ✈ỵ✐ ♥❤❛✉✳ ✳ ✳ ✳ ❍② ✈å♥❣ ♥✐➲♠ ✈✉✐ t❤➔♥❤ ❝ỉ♥❣ ♥❤♦ ♥❤ä ❝õ❛ ♥❤ú♥❣ ♥❣÷í✐ ❜✐➯♥ s♦↕♥ s➩ tr✉②➲♥ ❧û❛ ❍❆❨ ✲ ✣➉P ✲ ❙❆❨ ❝❤♦ ❝→❝ ❜↕♥ ❤å❝ s✐♥❤✱ ❝→❝ t❤➛② ❝ỉ ✈➔ ❝→❝ ❜➟❝ ♣❤ư ❤✉②♥❤ ❦❤✐ sû ❞ư♥❣✳ ❳✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥ sü q✉❛♥ t➙♠ ❝õ❛ ❜↕♥ ✤å❝✦ Þ ❦✐➳♥ ①✐♥ ❝❤✉②➸♥ ✈➲✿ s✐❣♠❛t❤s❣r♦✉♣❅❣♠❛✐❧✳❝♦♠ ✶ ▼Ư❈ ▲Ư❈ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ▼ư❝ ❧ư❝ ✶ ✷ ✸ ✹ ✺ ✻ ✼ P❤➙♥ sè✱ ❤é♥ sè ❙è t❤➟♣ ♣❤➙♥✱ t➾ sè ♣❤➛♥ tr➠♠ ✣↕✐ ❧÷đ♥❣ t➾ ❧➺ t❤✉➟♥✱ t➾ ❧➺ ♥❣❤à❝❤✱ t➾ ❧➺ ❦➨♣ ❙è ✤♦ t❤í✐ ❣✐❛♥✱ t♦→♥ ❝❤✉②➸♥ ✤ë♥❣ ✤➲✉ ❍➻♥❤ ❤å❝ ▼ët sè ❞↕♥❣ ❜➔✐ t♦→♥ ❦❤→❝ ✣→♣ →♥ ✈➔ ❧í✐ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥ ❝❤å♥ ❧å❝ ✷ ✸ ✺ ✼ ✾ ✶✷ ✶✾ ✷✶ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✶ P❤➙♥ sè✱ ❤é♥ sè 35 85 65 75 189 ✶✳ ❘ót ❣å♥ ❝→❝ ♣❤➙♥ sè s❛✉✿ 28 ; ; ; ; ✳ 51 39 100 105 1313 131313 út rỗ s s ❝→❝ ♣❤➙♥ sè s❛✉✿ 39 ; ; 42 1414 141414 ✸✳ ❙♦ s→♥❤ ❝→❝ ♣❤➙♥ sè s❛✉✿ 45 ; 87 ; 89 ; 12 11 ổ q ỗ tỷ sè ✈➔ ♠➝✉ sè✱ ❤➣② s♦ s→♥❤ ❝→❝ ♣❤➙♥ sè s❛✉✿ ❛✮ 45 ✈➔ 32 11 ❜✮ 59 ✈➔ 24 15 ❞✮ 14 ✈➔ 15 16 97 ❡✮ 96 ✈➔ 95 96 17 ❝✮ 13 ✈➔ 27 23 ✺✳ ❚➻♠ sè tü ♥❤✐➯♥ ① ❜✐➳t✿ ❛✮ x9 = 53 ❀ 85 ❜✮ 17 = ❀ x 105 ❞✮ x8 + 32 = 76 ❀ 27 ❡✮ x −3 = 135 ✳ ❝✮ 86 = 15 ❀ x ✻✳ ❚➻♠ ♠ët ♣❤➙♥ sè tè✐ ❣✐↔♥ s❛♦ ❝❤♦ ♥➳✉ ❝ë♥❣ t❤➯♠ ✽ ✤ì♥ ✈à ✈➔♦ tû sè ✈➔ ❝ë♥❣ t❤➯♠ ✶✵ ✤ì♥ ✈à ✈➔♦ ♠➝✉ sè t❤➻ ✤÷đ❝ ♠ët ♣❤➙♥ sè ♠ỵ✐ ❜➡♥❣ ♣❤➙♥ sè ✤➣ ❝❤♦✳ ✼✳ ❈❤♦ ♣❤➙♥ sè 112 ✳ ❍ä✐ ♣❤↔✐ ❝ë♥❣ t❤➯♠ ✈➔♦ tû sè ✈➔ ♠➝✉ sè ❝õ❛ ♣❤➙♥ sè ✤➣ ❝❤♦ ❝ò♥❣ ♠ët sè tü ♥❤✐➯♥ ♥➔♦ ✤➸ ✤÷đ❝ ♣❤➙♥ sè ❜➡♥❣ 74 ❄ ✽✳ ❚➻♠ ♣❤➙♥ sè ❝â tû sè ❧ỵ♥ ❤ì♥ ♠➝✉ sè ✽ ✤ì♥ ✈à ✈➔ s❛✉ ❦❤✐ rót ❣å♥ t❛ ✤÷đ❝ ♣❤➙♥ sè 35 ✳ ✾✳ ❚➻♠ ♣❤➙♥ sè tè✐ ❣✐↔♥ mn ✱ ❜✐➳t r➡♥❣ ♣❤➙♥ sè m n+ n ❣➜♣ ✼ ❧➛♥ ♣❤➙♥ sè mn ✳ ✸ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✶✵✳ ▼ët tr↕✐ ❝❤➠♥ ♥✉æ✐ ❝â 43 sè ❜á ❜➡♥❣ 23 sè ❞➯✱ ❜✐➳t sè ❜á ➼t ❤ì♥ sè ❞➯ ❧➔ ✶✷ ❝♦♥✳ ❍ä✐ tr↕✐ ❝❤➠♥ ♥✉æ✐ ❝â ❜❛♦ ♥❤✐➯✉ ❝♦♥ ❜á❄ ❇❛♦ ♥❤✐➯✉ ❝♦♥ ❞➯❄ ✶✶✳ ❚✉ê✐ ❇è ❜➡♥❣ 89 t✉ê✐ ▼➭✱ t✉ê✐ ▲❛♥ ❜➡♥❣ 14 t✉ê✐ ▼➭✱ tê♥❣ sè t✉ê✐ ❝õ❛ ❇è ✈➔ ▲❛♥ ❧➔ ✹✹ t✉ê✐✳ ❍ä✐ ♠é✐ ♥❣÷í✐ ❜❛♦ ♥❤✐➯✉ t✉ê✐❄ ✶✷✳ ❈â ❜❛ ❜➻♥❤ ữợ ự tờ t tự ❝❤ù❛ ❜➡♥❣ 21 ❜➻♥❤ t❤ù ♥❤➜t✱ ❜➻♥❤ t❤ù ❜❛ ❝❤ù❛ ❜➡♥❣ 23 ❜➻♥❤ t❤ù ❤❛✐✳ ❍ä✐ ♠é✐ ❜➻♥❤ ❝❤ù❛ ❜❛♦ t ữợ ỏ ữợ ũ ✈➔♦ ♠ët ❜➸ ❝↕♥✳ ◆➳✉ ✈á✐ t❤ù ♥❤➜t ❝❤↔② ♠ët ♠➻♥❤ tr♦♥❣ ✺ ❣✐í t❤➻ s➩ ✤➛② ❜➸✳ ◆➳✉ ✈á✐ t❤ù ❤❛✐ ❝❤↔② ♠ët ♠➻♥❤ t❤➻ ❜➸ s➩ ✤➛② s❛✉ ✼ ❣✐í✳ ❍ä✐ ♥➳✉ ❝↔ ❤❛✐ ✈á✐ ❝ò♥❣ ❝❤↔② t❤➻ ❜➸ s➩ ✤➛② s❛✉ ♠➜② ❣✐í❄ ✶✹✳ ❍❛✐ ♥❣÷í✐ ❧➔♠ ❝❤✉♥❣ ♠ët ❝ỉ♥❣ ✈✐➺❝ tr♦♥❣ ✶✷ ❣✐í t❤➻ ①♦♥❣✳ ◆❣÷í✐ t❤ù ♥❤➜t ❧➔♠ ♠ët ♠➻♥❤ 23 ❝æ♥❣ ✈✐➺❝ t❤➻ ♠➜t ✶✵ ❣✐í✱ ❤ä✐ ♥❣÷í✐ t❤ù ❤❛✐ ❧➔♠ 13 ❝ỉ♥❣ ✈✐➺❝ ❝á♥ ❧↕✐ ♠➜t ❜❛♦ ♥❤✐➯✉ ❧➙✉❄ ✶✺✳ ▲ỵ♣ ✺❆✶ ✈➔ ✺❆✷ ❝â ✽✼ ❤å❝ s✐♥❤✱ ❜✐➳t r➡♥❣ 75 sè ❤å❝ s✐♥❤ ❝õ❛ ❧ỵ♣ ✺❆✶ ❜➡♥❣ sè ❤å❝ s✐♥❤ ❝õ❛ ❧ỵ♣ ✺❆✷✳ ❍ä✐ ♠é✐ ❧ỵ♣ ❝â ❜❛♦ ♥❤✐➯✉ ❤å❝ s✐♥❤❄ ✶✻✳ ▼ët ❤➻♥❤ t❛♠ ❣✐→❝ ❝â ❝❤✉ ✈✐ ❧➔ ✶✷✵❝♠✳ ❙è ✤♦ ❜❛ ❝↕♥❤ ❝õ❛ t❛♠ ❣✐→❝ t➾ ❧➺ ✈ỵ✐ 5, 12, 13✳ ❚➻♠ sè ✤♦ ❝→❝ ❝↕♥❤ ❝õ❛ t❛♠ ❣✐→❝✳ ✶✼✳ ▼ët ✤ë✐ ❝ỉ♥❣ ♥❤➙♥ ❧➔♠ ✤÷í♥❣ tr♦♥❣ ✸ ♥❣➔②✳ ◆❣➔② t❤ù ♥❤➜t ❧➔♠ ✤÷đ❝ 72 ❝ỉ♥❣ ✈✐➺❝✱ ♥❣➔② t❤ù ❤❛✐ ❧➔♠ ✤÷đ❝ 38 ❝ỉ♥❣ ✈✐➺❝✱ ♥❣➔② t❤ù ❜❛ ❧➔♠ ♥èt ✺✼ ♠➨t ❝✉è✐✳ ❍ä✐ ✤ë✐ ❝æ♥❣ ♥❤➙♥ ✤â ♣❤↔✐ ❧➔♠ ❜❛♦ ♥❤✐➯✉ ♠➨t ✤÷í♥❣❄ ✶✽✳ ▼ët ỉ tỉ ❝❤↕② q✉➣♥❣ ✤÷í♥❣ ❆❇ tr♦♥❣ ✸ ❣✐í✳ ●✐í ✤➛✉ ❝❤↕② ✤÷đ❝ 25 q✉➣♥❣ ✤÷í♥❣ ❆❇✳ ●✐í t❤ù ❤❛✐ ❝❤↕② ✤÷đ❝ 25 q✉➣♥❣ ✤÷í♥❣ ❝á♥ ❧↕✐ ✈➔ t❤➯♠ ✹❦♠✳ ●✐í t❤ù ❜❛ ❝❤↕② ♥èt ✺✵❦♠ ❝✉è✐✳ ❚➼♥❤ q✉➣♥❣ ✤÷í♥❣ ❆❇✳ ✹ ✷ ❙è t❤➟♣ ♣❤➙♥✱ t➾ sè ♣❤➛♥ tr➠♠ ❙✐❣♠❛ t số s ữợ ❞↕♥❣ sè t❤➟♣ ♣❤➙♥✿ 17 16 132 ; ; ; ; 25 125 ✷✵✳ ❱✐➳t ❝→❝ sè s❛✉ t❤➔♥❤ sè t❤➟♣ ♣❤➙♥✿ 18 26 ; ; ; 25 ✷✶✳ ❱✐➳t ❝→❝ sè t❤➟♣ ♣❤➙♥ s❛✉ ✤➙② ữợ số 2, 15; 5, 022; 4, 6; 0, 324 ✷✷✳ ❑❤✐ ♥❤➙♥ ♠ët sè ✈ỵ✐ ✶✷✹✱ ♠ët ❜↕♥ ❤å❝ s✐♥❤ ✤➣ ✤➦t ❝→❝ t➼❝❤ r✐➯♥❣ t❤➥♥❣ ❝ët ✈ỵ✐ ♥❤❛✉ ♥➯♥ ❞➝♥ ✤➳♥ ❦➳t q✉↔ s❛✐ ❧➔ ✽✽✱✷✳ ❊♠ ❤➣② t➻♠ ❦➳t q✉↔ ✤ó♥❣ ❝õ❛ ♣❤➨♣ ♥❤➙♥✳ ✷✸✳ ▼ët ❝û❛ ❤➔♥❣ ❝â ✽✻✱✺ t↕ ✤÷í♥❣✳ ◆❣➔② t❤ù ♥❤➜t ❝û❛ ❤➔♥❣ ❜→♥ ✤÷đ❝ ✷✻✱✼ t↕✱ ♥❣➔② t❤ù ❤❛✐ ❜→♥ ✤÷đ❝ ♥❤✐➲✉ ❤ì♥ ♥❣➔② t❤ù ♥❤➜t ✻✱✼✽ t↕✳ ❍ä✐ s❛✉ ❤❛✐ ♥❣➔② ❜→♥✱ ❝û❛ ❤➔♥❣ ❝á♥ ❧↕✐ ❜❛♦ ♥❤✐➯✉ t↕ ✤÷í♥❣❄ ✷✹✳ ❈â ❜❛ tê ❝ỉ♥❣ ♥❤➙♥ t❤❛♠ ❣✐❛ ✤➢♣ ✤÷í♥❣✱ tê ♠ët ✤➢♣ ✤÷đ❝ ✷✺✱✼♠✱ tê ❤❛✐ ✤➢♣ ✤÷đ❝ ❤ì♥ tê ♠ët ✺✱✸♠ ✈➔ ✤➢♣ ❦➨♠ tê ❜❛ ✸✱✺♠✳ ❍ä✐ ❝↔ ❜❛ tê ✤➢♣ ✤÷đ❝ t➜t ❝↔ ❜❛♦ ♥❤✐➯✉ ♠➨t ✤÷í♥❣❄ ✷✺✳ ▼ët såt ❝❛♠ ♥➦♥❣ tê♥❣ ❝ë♥❣ ✼✻✱✻✺❦❣✳ ❇✐➳t r➡♥❣ ♥➳✉ ❧➜② ✹❦❣ ð såt t❤ù ♥❤➜t ❝❤✉②➸♥ s❛♥❣ såt t❤ù ❤❛✐ t❤➻ såt t❤ù ❤❛✐ ♥➦♥❣ ❤ì♥ såt t❤ù ♥❤➜t ✶✱✽✺❦❣✳ ❍ä✐ ♠é✐ såt ❝❛♠ ♥➦♥❣ ❜❛♦ ♥❤✐➯✉ ❦✐❧❣❛♠❄ ✷✻✳ ◆❤➔ ❆♥ ♥✉æ✐ ✽ ❝♦♥ ❜á✱ tr♦♥❣ ✤â ❝â ✸ ❝♦♥ ❜á ✤ü❝✳ ❍ä✐ ❛✮ ❙è ❜á ✤ü❝ ❝❤✐➳♠ ❜❛♦ ♥❤✐➯✉ ♣❤➛♥ tr➠♠ tê♥❣ sè ❜á❄ ❜✮ ❚➾ sè ♣❤➛♥ tr➠♠ ❣✐ú❛ sè ❜á ✤ü❝ ✈➔ sè ❜á ❝→✐ ❧➔ ❜❛♦ ♥❤✐➯✉❄ ✺ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✷✼✳ ▼ët ①➼ ♥❣❤✐➺♣ ❝â ✻✵ ❝ỉ♥❣ ♥❤➙♥ ✤÷đ❝ ❝❤✐❛ t❤➔♥❤ ❤❛✐ tê✱ tr♦♥❣ ✤â tê ♠ët ❝❤✐➳♠ 40% tê♥❣ sè ❝æ♥❣ ♥❤➙♥✳ ❍ä✐ ❛✮ ❚ê ♠ët ❝â ❜❛♦ ♥❤✐➯✉ ❝æ♥❣ ♥❤➙♥❄ ❜✮ ❚➾ sè ♣❤➛♥ tr➠♠ ❣✐ú❛ sè ❝æ♥❣ ♥❤➙♥ tê ♠ët ✈➔ sè ❝æ♥❣ ♥❤➙♥ tê ❤❛✐ ❧➔ ❜❛♦ ởt ữớ q s tố ợ ỗ t r ữớ õ ữủ 9% ❣✐→ ✈è♥✳ ❍ä✐ ♥❣÷í✐ ✤â ✤÷đ❝ ❧➣✐ ❜❛♦ ♥❤✐➯✉ t✐➲♥❄ × 5, − 30, × 3, + 14, 58 ✷✾✳ ❚➼♥❤ ♥❤❛♥❤✿ A = 20,14,2 58 × 460 + 7, 29 × 540 × ✸✵✳ ❚➻♠ ❤❛✐ sè✱ ❜✐➳t tr✉♥❣ ❜➻♥❤ ❝ë♥❣ ❝õ❛ ❝❤ó♥❣ ❧➔ sè tü ♥❤✐➯♥ ❧ỵ♥ ♥❤➜t ❝â ❤❛✐ ❝❤ú sè ✈➔ sè ♥➔② ❜➡♥❣ 80% sè ❦✐❛✳ ✸✶✳ ❚r♦♥❣ ♠ët ♣❤➨♣ ❝❤✐❛ sè t❤➟♣ ♣❤➙♥✱ t❤÷ì♥❣ ✤ó♥❣ ❧➔ ✶✵✷✱✺✳ ❑❤✐ t❤ü❝ ❤✐➺♥ ♣❤➨♣ ❝❤✐❛✱ ♠ët ❤å❝ s✐♥❤ ✤➣ q✉➯♥ ✤➦t ♠ët sè ✵ ð t❤÷ì♥❣ ♥➯♥ ❧ó❝ t❤û ❧↕✐ ❜➡♥❣ ❝→❝❤ ❧➜② tữỡ ợ số ữủ ởt số ọ ỡ sè ❜à ❝❤✐❛ ✹✸✷✱✾ ✤ì♥ ✈à✳ ❚➻♠ sè ❜à ❝❤✐❛ ✈➔ sè ❝❤✐❛✳ ✸✷✳ ❈â ❜❛ ①❡ t↔✐ ❝❤ð tê♥❣ ❝ë♥❣ ✶✺✱✻ t➜♥ ❤➔♥❣ ❤â❛✳ ❳❡ t❤ù ♥❤➜t ❝❤ð ❜➡♥❣ 43 ①❡ t❤ù ❤❛✐✱ ①❡ t❤ù ❜❛ ❝❤ð ➼t ❤ì♥ tê♥❣ sè ❤➔♥❣ ❤â❛ ❤❛✐ ①❡ ✤➛✉ ❝❤ð ✤÷đ❝ ❧➔ ✶✱✷ t➜♥✳ ❍ä✐ ♠é✐ ①❡ ❝❤ð ✤÷đ❝ ❜❛♦ ♥❤✐➯✉ t➜♥ õ ữợ ự 4% ố t ữợ ữợ ✤➸ t➾ ❧➺ ♠✉è✐ tr♦♥❣ ❞✉♥❣ ❞à❝❤ ❧➔ 2%✳ ✸✹✳ ởt ữớ ỷ tt tr ỗ ợ s✉➜t 1, 1% ♠ët t❤→♥❣✳ ❚➼♥❤✿ ❛✮ ❙è t✐➲♥ ❧➣✐ s❛✉ ♠ët t❤→♥❣✳ ❜✮ ❙è t✐➲♥ ❝↔ ❣è❝ ✈➔ ❧➣✐ s❛✉ ❤❛✐ t❤→♥❣ ♥➳✉ ♥❣÷í✐ ✤â ❝❤➾ ✤➳♥ rót t✐➲♥ ♠ët ❧➛♥✳ ❝✮ ❙è t✐➲♥ ❝↔ ❣è❝ ✈➔ ❧➣✐ s❛✉ ❜❛ t❤→♥❣ ♥➳✉ ♥❣÷í✐ ✤â ❝❤➾ ✤➳♥ rót t✐➲♥ ♠ët ❧➛♥✳ ✸✺✳ ▼ët ❤➻♥❤ ❝❤ú ♥❤➟t ❝â ❝❤✐➲✉ rë♥❣ ❜➡♥❣ 75% ❝❤✐➲✉ ❞➔✐✳ ◆➳✉ t➠♥❣ ❝❤✐➲✉ ❞➔✐ t❤➯♠ ✷♠ t❤➻ ❞✐➺♥ t➼❝❤ t➠♥❣ t❤➯♠ 12m2✳ ❚➼♥❤ ❝❤✉ ✈✐ ✈➔ ❞✐➺♥ t➼❝❤ ❝õ❛ ❤➻♥❤ ❝❤ú ♥❤➟t ✤â✳ ✸✻✳ ❍➣② s♦ s→♥❤ ❆ ✈➔ ❇✱ ❜✐➳t A = 19, 93 × 19, 99; B = 19, 96 × 19, 96✳ ✻ ✸ ✣↕✐ ❧÷đ♥❣ t➾ ❧➺ t❤✉➟♥✱ t➾ ❧➺ ♥❣❤à❝❤✱ t➾ ❧➺ ❦➨♣ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✸✼✳ ❚ê ✶ ❧ỵ♣ ✺❆ ❝â s trỗ ữủ ọ ợ trỗ ữủ t r ợ ❝â ✹✷ ❤å❝ s✐♥❤ ✈➔ sè ❝➙② ♠é✐ ❤å❝ s✐♥❤ trỗ ữủ ởt ỷ ởt ữủ õ ợ ỹ t tr♦♥❣ ✷✵ ♥❣➔②✱ ♠é✐ ♥❣➔② ❜→♥ ✤÷đ❝ ✸✷✵ ❤➔♥❣ ❤â❛✱ ♥❤÷♥❣ t❤ü❝ t➳ ♠é✐ ♥❣➔② ❝û❛ ❤➔♥❣ ✤â ✤➣ ❜→♥ ữủ õ ọ ợ số õ ♥❤➟♣ ❦❤♦ t❤➻ ✤õ ❜→♥ ✤÷đ❝ ❜❛♦ ♥❤✐➯✉ ♥❣➔②❄ ✸✾✳ út ữợ ởt ỗ ữớ t sû ❞ư♥❣ ✶✺ ♠→② ❜ì♠ ❧➔♠ ✈✐➺❝ tr♦♥❣ ✶✷ ❣✐í✳ ọ út ỗ õ tr t❤➻ ❝➛♥ ♣❤↔✐ sû ❞ư♥❣ ❜❛♦ ♥❤✐➯✉ ♠→② ❜ì♠❄ ❇✐➳t ♥➠♥❣ s✉➜t ❝→❝ ♠→② ♥❤÷ ♥❤❛✉✳ ✹✵✳ ▼ët tê t❤đ ♠❛② ❝â ✶✺ ♥❣÷í✐ tr♦♥❣ ✻ ♥❣➔② ❧➔♠ ✈✐➺❝ ♠❛② ✤÷đ❝ ✶✸✺ ❜ë q✉➛♥ →♦✳ ❍ä✐ ♥➳✉ tê ✤â ❝â ✶✽ ♥❣÷í✐ ❧➔♠ ✈✐➺❝ tr♦♥❣ ✽ ♥❣➔② t❤➻ s➩ ♠❛② ✤÷đ❝ ❜❛♦ ♥❤✐➯✉ ❜ë q✉➛♥ →♦❄ ❇✐➳t ♥➠♥❣ s✉➜t ❧➔♠ ✈✐➺❝ ❝õ❛ ♠é✐ ♥❣÷í✐ ✤➲✉ ♥❤÷ ♥❤❛✉✳ ✹✶✳ ▼ët ✤ë✐ ❝ỉ♥❣ ♥❤➙♥ ❝â ✹✵ ♥❣÷í✐ ✤÷đ❝ ❣✐❛♦ ♥❤✐➺♠ ✈ư ❤♦➔♥ t❤➔♥❤ ❝ỉ♥❣ ✈✐➺❝ tr♦♥❣ ✶✺ ♥❣➔②✳ ❙❛✉ ❦❤✐ ❧➔♠ ✤÷đ❝ ✸ ♥❣➔② t❤➻ ✷✵ ❝ỉ♥❣ ♥❤➙♥ ✤÷đ❝ ✤✐➲✉ ✤✐ ♥ì✐ ❦❤→❝✳ ❍ä✐ ✤ë✐ ❝æ♥❣ ♥❤➙♥ ✤â ❤♦➔♥ t❤➔♥❤ ❝æ♥❣ ✈✐➺❝ ✤÷đ❝ ❣✐❛♦ tr♦♥❣ ❜❛♦ ♥❤✐➯✉ ♥❣➔②❄ ❇✐➳t ♥➠♥❣ s✉➜t ❧➔♠ ✈✐➺❝ ❝õ❛ ♠å✐ ♥❣÷í✐ tr♦♥❣ ♠ët ♥❣➔② ❧➔ ♥❤÷ ♥❤❛✉✳ ✹✷✳ ▼ët ♣❤➙♥ ①÷ð♥❣ ♠ë❝ ❝â ✸✵ ♥❣÷í✐ ✤÷đ❝ ❣✐❛♦ ♥❤✐➺♠ ✈ư ✤â♥❣ ♠ët ❧ỉ ❜➔♥ ❣❤➳ tr♦♥❣ ✽ ♥❣➔②✱ ♠é✐ ♥❣➔② ❧➔♠ ✈✐➺❝ ✽ ❣✐í✳ ❙❛✉ ✷ ♥❣➔② ❧➔♠ ✈✐➺❝ t❤➻ ❝â ✶✽ ♥❣÷í✐ ✤÷đ❝ ✤✐➲✉ ✤✐ ❧➔♠ ✈✐➺❝ ❦❤→❝ ✈➔ sè ❝á♥ ❧↕✐ t➠♥❣ t❤í✐ ❣✐❛♥ ❧➔♠ ✈✐➺❝ ♠é✐ ♥❣➔② t❤➯♠ ✷ ❣✐í✳ ❍ä✐ ♣❤➙♥ ①÷ð♥❣ ✤â ❤♦➔♥ t❤➔♥❤ ❦❤è✐ ❧÷đ♥❣ ❝ỉ♥❣ ✈✐➺❝ ✤÷đ❝ ❣✐❛♦ tr♦♥❣ ❜❛♦ ♥❤✐➯✉ ♥❣➔②❄ ❇✐➳t ♥➠♥❣ s✉➜t ❧➔♠ ✈✐➺❝ ❝õ❛ ♠é✐ ♥❣÷í✐ ♥❤÷ ♥❤❛✉✳ ✹✸✳ ▼ët ✤ì♥ ✈à ❜ë ✤ë✐ ❝❤✉➞♥ ❜à ❣↕♦ ❝❤♦ ✼✵ ♥❣÷í✐ ➠♥ tr♦♥❣ ✸✵ ♥❣➔②✳ ❙❛✉ ❦❤✐ ➠♥ ✤÷đ❝ ✻ ♥❣➔② t❤➻ ❝â ✶✵ ♥❣÷í✐ ❝❤✉②➸♥ ✤✐ ♥ì✐ ❦❤→❝✳ ❍ä✐ sè ❣↕♦ ❝á♥ ❧↕✐ ✤õ ❝❤♦ sè ♥❣÷í✐ ❝á♥ ❧↕✐ ❝õ❛ ✤ì♥ ✈à ✤â ➠♥ tr♦♥❣ ❜❛♦ ♥❤✐➯✉ ♥❣➔② ♥ú❛❄ ❇✐➳t r➡♥❣ ♠ù❝ ➠♥ ❝õ❛ ♠å✐ ♥❣÷í✐ tr♦♥❣ ♠ët ♥❣➔② ❧➔ ♥❤÷ ♥❤❛✉✳ ✼ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✹✹✳ ▼ët ✤ì♥ ✈à ❜ë ✤ë✐ ❝â ✶✷✵ ♥❣÷í✐ ✤❡♠ t❤❡♦ ✶✹✹✵❦❣ ❣↕♦ ✈➔ ❞ü ✤à♥❤ ➠♥ ✤õ tr♦♥❣ ✶✺ ♥❣➔②✳ ◆❤÷♥❣ s❛✉ ✤â ✷ ♥❣➔②✱ ❝â ✸✵ ♥❣÷í✐ ❝❤✉②➸♥ s❛♥❣ ✤ì♥ ✈à ❦❤→❝ ✈➔ ♠❛♥❣ t❤❡♦ ✺✷✽❦❣ ❣↕♦✳ ❍ä✐ sè ❣↕♦ ❝á♥ ❧↕✐ ✤ì♥ ✈à ❝â t❤➸ ➠♥ ✤÷đ❝ ❜❛♦ ♥❤✐➯✉ ♥❣➔② ♥ú❛❄ ❇✐➳t ♠é✐ ♥❣÷í✐ tr♦♥❣ ♠ët ♥❣➔② ➠♥ ❤➳t sè ❣↕♦ ♥❤÷ ♥❤❛✉✳ ✹✺✳ ❇➳♣ ➠♥ ❝õ❛ ♠ët ✤ì♥ ✈à ❜ë ✤ë✐ ❝❤✉➞♥ ❜à ✤õ ❣↕♦ ❝❤♦ ✸✺✻ ❝❤✐➳♥ s➽ ➠♥ tr♦♥❣ ✸✵ ♥❣➔②✳ ❉♦ ♥❤✐➺♠ ✈ö ✤ët ①✉➜t ♥➯♥ s❛✉ ✾ ♥❣➔② t❤➻ ❝â ♠ët sè ❝❤✐➳♥ s➽ ✤÷đ❝ ✤✐➲✉ ✤✐ ❧➔♠ ♥❤✐➺♠ ✈ư ð t➾♥❤ ❦❤→❝✳ ❱➻ ✈➟② sè ❣↕♦ ✤➣ ❝❤✉➛♥ ❜à ➠♥ ✤÷đ❝ ♥❤✐➲✉ ❤ì♥ ❞ü ❦✐➳♥ ✼ ♥❣➔②✳ ❍ä✐ ✤➣ ❝â ❜❛♦ ♥❤✐➯✉ ❝❤✐➳♥ s➽ ✤÷đ❝ ✤✐➲✉ ✤✐ t➾♥❤ ❦❤→❝❄ ✹✻✳ ❇↕♥ ◆❣å❝ ♠✉❛ ✷ ❜ót ❜✐ ✈➔ ✺ q✉②➸♥ ✈ð ❤➳t ✼✹✵✵✵ ỗ t t ởt út t✐➲♥ ♠ët q✉②➸♥ ✈ð✱ ❜✐➳t ❜↕♥ ▼❛✐ ❝ô♥❣ ♠✉❛ ✸ ❜ót ❜✐ ✈➔ ✹ q✉②➸♥ ✈ð ♥❤÷ ✈➟② ❤➳t ✻✾✵✵✵ ỗ ởt ữớ ỹ tứ ❇ tr♦♥❣ ✻ ❣✐í ♥❤÷♥❣ ❞♦ trí✐ ♠÷❛ ♥➯♥ ✈➟♥ tè❝ t❤ü❝ t➳ ❜➡♥❣ 45 ✈➟♥ tè❝ ❞ü ❦✐➳♥✳ ❍ä✐ ♥❣÷í✐ ✤â ✤➳♥ ❇ ❝❤➟♠ ❤ì♥ ❞ü ✤à♥❤ ❜❛♦ ❧➙✉❄ ởt õ ỗ ❤❛✐ tê✿ tê ✶ ❧➔ ❝→❝ ①❡ ❧♦↕✐ ✽ t➜♥✱ tê ❤❛✐ ❧➔ ❝→❝ ①❡ ❧♦↕✐ ✺ t➜♥✳ ❍ä✐ ♠é✐ tê ❝â ❜❛♦ ♥❤✐➯✉ ①❡❄ ❇✐➳t r➡♥❣ ❤❛✐ tê ♥❤➟♥ ❝❤ð sè ❤➔♥❣ ❤â❛ ❜➡♥❣ ♥❤❛✉ ✈➔ ♠é✐ ①❡ ❝❤➾ ởt ữủt õ tũ ữợ ♠➢♠ ♥❤÷ ♥❤❛✉ ✤ü♥❣ ✤÷đ❝ t➜t ❝↔ ✷✶✻ ❧➼t✱ s❛✉ õ t t ữợ ọ ỏ tũ ữợ ❈û❛ ❤➔♥❣ ❝â ✶✺ tó✐ ❜✐✱ ❝û❛ ❤➔♥❣ ❜→♥ ❤➳t ✽✹ ✈✐➯♥ ❜✐ ✈➔ ❝á♥ ❧↕✐ ✽ tó✐ ❜✐✳ ❍ä✐ trữợ ỷ õ ✺✶✳ ❑❤è✐ ❧ỵ♣ ✺ ❝â tê♥❣ ❝ë♥❣ ✶✹✼ ❤å❝ s✐♥❤✱ t➼♥❤ r❛ ❝ù ✸ ❤å❝ s✐♥❤ ♥❛♠ t❤➻ ❝â ✹ ❤å❝ s✐♥❤ ♥ú✳ ❍ä✐ ❦❤è✐ ❧ỵ♣ ♥➠♠ ❝â ❜❛♦ ♥❤✐➯✉ ❤å❝ s✐♥❤ ♥❛♠❄ ❇❛♦ ♥❤✐➯✉ ❤å❝ s✐♥❤ ♥ú❄ ✺✷✳ ❇❛ ♥❣÷í✐ ❧➔♠ ①♦♥❣ ♠ët ❝♦♥ ✤÷í♥❣ tr♦♥❣ ✶✹ ♥❣➔②✳ ❍ä✐ ♠✉è♥ ❧➔♠ ①♦♥❣ ❝♦♥ ✤÷í♥❣ ✤â tr♦♥❣ ✼ ♥❣➔② t❤➻ ♣❤↔✐ ❝➛♥ ❝â ❜❛♦ ♥❤✐➯✉ ♥❣÷í✐❄ ✽ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✼✻✳ ❚➼♥❤ ❞✐➺♥ t➼❝❤ ❤➻♥❤ t❤❛♥❣ ❆❇❈❉✳ ❇✐➳t ❞✐➺♥ t➼❝❤ ❝→❝ ❤➻♥❤ t❛♠ ❣✐→❝ ❆❖❉ ✈➔ ❉❖❈ ♥❤÷ ❤➻♥❤ ✈➩✳ ✼✼✳ ❈❤♦ ❤➻♥❤ t❤❛♥❣ ✈✉æ♥❣ ▼◆P◗ ✈✉æ♥❣ ❣â❝ t↕✐ ▼ ✈➔ ◗❀ P Q = 12 M N ✳ ❑➨♦ ❞➔✐ ▼◗ ✈➔ ◆P ❝➢t ♥❤❛✉ t↕✐ ❆✳ ❛✮ ❙♦ s→♥❤ ❞✐➺♥ t➼❝❤ ❤❛✐ t❛♠ ❣✐→❝ ▼◆P ✈➔ ▼◗◆✳ ❜✮ ❙♦ s→♥❤ ❞✐➺♥ t➼❝❤ ❤❛✐ t❛♠ ❣✐→❝ ❆◗P ✈➔ ❆◗◆✳ ❝✮ ❉✐➺♥ t➼❝❤ ❤➻♥❤ t❤❛♥❣ ▼◆P◗ ❜➡♥❣ 63cm2✳ ❚➼♥❤ ❞✐➺♥ t➼❝❤ t❛♠ ❣✐→❝ ❆◗P✳ ✶✸ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✼✽✳ ❈❤♦ ❤➻♥❤ t❤❛♥❣ ❆❇❈❉ ✭❤❛✐ ✤→② ❧➔ ❆❇✱ ❈❉✮✳ ❚r➯♥ ✤÷í♥❣ ❝❤➨♦ ❆❈ ❧➜② ✤✐➸♠ ▼ s❛♦ ❝❤♦ ▼❆ ❂ ▼❈✳ ứ ữớ t s s ợ ữớ ❉❇ ❝➢t ❉❈ t↕✐ ◆ ✭❤➻♥❤ ✈➩✮✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ✤♦↕♥ t❤➥♥❣ ❇◆ ❝❤✐❛ ❤➻♥❤ t❤❛♥❣ ❆❇❈❉ t❤➔♥❤ ❤❛✐ ♣❤➛♥ ❝â ❞✐➺♥ t➼❝❤ ❜➡♥❣ ♥❤❛✉✳ ✼✾✳ ❈❤♦ ❤➻♥❤ t❤❛♥❣ ❆❇❈❉ ❝â ❆❇ s♦♥❣ s♦♥❣ ✈ỵ✐ ❈❉ ✈➔ ❞✐➺♥ t➼❝❤ ❜➡♥❣ 40cm ✳ ❑➨♦ ❞➔✐ ❆❇ ♠ët ✤♦↕♥ ❇▼ s❛♦ ❝❤♦ ❆❇ ❂ ❇▼✱ ❦➨♦ ❞➔✐ ❇❈ ♠ët ✤♦↕♥ ❈◆ s❛♦ ❝❤♦ ❇❈ ❂ ❈◆✱ ❦➨♦ ❞➔✐ ❈❉ ♠ët ✤♦↕♥ ❉P s❛♦ ❝❤♦ ❈❉ ❂ ❉P✱ ❦➨♦ ❞➔✐ ❉❆ ♠ët ✤♦↕♥ ❆◗ s❛♦ ❝❤♦ ❉❆ ❂ ❆◗✳ ◆è✐ ▼✱ ◆✱ P✱ ◗✳ ❚➻♠ ❞✐➺♥ t➼❝❤ tù ❣✐→❝ ▼◆P◗✳ ✶✹ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✽✵✳ ❈❤♦ ❤➻♥❤ ❝❤ú ♥❤➟t ❆❇❈❉ ❝â ❝❤✉ ✈✐ ❧➔ ✼✽❝♠✱ ❝❤✐➲✉ ❞➔✐ ❆❇ ❤ì♥ ❝❤✐➲✉ rë♥❣ ❇❈ ❧➔ ✶✺❝♠✳ ❚r➯♥ ❇❈ ❧➜② ✤✐➸♠ ▼✱ tr➯♥ ❈❉ ❧➜② ✤✐➸♠ ◆ s❛♦ ❝❤♦ ❤❛✐ ✤÷í♥❣ t❤➥♥❣ ❆▼✱ ❆◆ ❝❤✐❛ ❤➻♥❤ ❝❤ú ♥❤➟t t❤➔♥❤ ✸ ♣❤➛♥ ❝â ❞✐➺♥ t➼❝❤ ❜➡♥❣ ♥❤❛✉✳ ❚➼♥❤ ✤ë ❞➔✐ ✤♦↕♥ ❈▼✱ ❈◆✳ ✽✶✳ ❈❤♦ ❤➻♥❤ ❝❤ú ♥❤➟t ❆❇❈❉✳ ❚r➯♥ ❝↕♥❤ ❆❇ ❧➜② ✤✐➸♠ ▼ ❜➜t ❦➻✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ SDM C = 12 SABCD ✳ ✶✺ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✽✷✳ ❈❤♦ ❤➻♥❤ ❝❤ú ♥❤➟t ❆❇❈❉✳ ■ ❧➔ ✤✐➸♠ ❝❤➼♥❤ ❣✐ú❛ ❝↕♥❤ ❆❇✳ ◆è✐ ❉ ✈ỵ✐ ■✱ ✤♦↕♥ t❤➥♥❣ ❉❇ ❝➢t ✤♦↕♥ t❤➥♥❣ ■❈ t↕✐ ❑ ✭❤➻♥❤ ✈➩✮✳ ❛✮ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ SDIK = 12 SDBC ✳ ❜✮ ❑➫ ■P ✈✉ỉ♥❣ ❣â❝ ✈ỵ✐ ❉❇❀ ❦➫ ❈◗ ✈✉ỉ♥❣ ❣â❝ ✈ỵ✐ ❉❇✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ SDIC = 3SDIK ✳ ❝✮ ❇✐➳t SDIK = 8cm2✳ ❚➼♥❤ ❞✐➺♥ t➼❝❤ ❤➻♥❤ ❝❤ú ♥❤➟t ❆❇❈❉✳ ✽✸✳ ❈❤♦ ❤➻♥❤ ❝❤ú ♥❤➟t ❆❇❈❉ ✈➔ ❝→❝ ✤✐➸♠ ▼✱ ❖✱ ■ s❛♦ ❝❤♦ ▼■ ❂ ■❉ ✈➔ ❆❖ ❂ ❖■ ✭❤➻♥❤ ✈➩✮✳ ❇✐➳t ❞✐➺♥ t➼❝❤ t❛♠ ❣✐→❝ ▼❖■ ❧➔ 25m2✳ ❍ä✐ ❞✐➺♥ t➼❝❤ ❤➻♥❤ ❝❤ú ♥❤➟t ❆❇❈❉ ❜➡♥❣ ❜❛♦ ♥❤✐➯✉ ❤➨❝✲t❛❄ ✶✻ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✽✹✳ ❈❤♦ ❤➻♥❤ trá♥ t➙♠ ❖ t✐➳♣ ①ó❝ ✈ỵ✐ ✹ ❝↕♥❤ ❝õ❛ ❤➻♥❤ ✈✉ỉ♥❣ ❆❇❈❉ ✭♥❤÷ ❤➻♥❤ ✈➩✮✳ ❇✐➳t ❞✐➺♥ t➼❝❤ ❤➻♥❤ ✈✉æ♥❣ ❧➔ 400cm2✳ ❚➼♥❤✿ ❛✮ ❉✐➺♥ t➼❝❤ ❤➻♥❤ trá♥ t➙♠ ❖✳ ❜✮ ❉✐➺♥ t➼❝❤ ♣❤➛♥ ♠➔✉ ✈➔♥❣✳ ✽✺✳ ▼ët ♣❤á♥❣ ❤å❝ tr♦♥❣ ❧á♥❣ ❞➔✐ ✼♠✱ rë♥❣ ✺♠✱ ❝❛♦ ✸✱✺♠✳ ❈→❝ ❝û❛ r❛ ✈➔♦ ✈➔ ❝❤✐➳♠ 16m2✳ ❚➻♠ ♣❤➼ tê♥ q✉➨t ✈ỉ✐ t÷í♥❣ ♣❤➛♥ ❜➯♥ tr♦♥❣ ❝➠♥ ♣❤á♥❣✳ ❇✐➳t r➡♥❣ qt ổ 1m2 tố ỗ ❣❛✐ ❜✉ë❝ t❤➔♥❤ ❤➻♥❤ ❝❤ú ♥❤➟t tr➯♥ t➜t ❝↔ ❝→❝ ♠➦t ❝õ❛ ♠ët ❝→✐ t❤ò♥❣ ❝→❝ tỉ♥❣ ❤➻♥❤ ❧➟♣ ♣❤÷ì♥❣ ✭①❡♠ ❤➻♥❤ ✈➩✮✳ P❤➛♥ ❝→❝ ❝ët ♥ót ❝❤✐➳♠ ✷✱✺❞♠✳ ❚➼♥❤ ❞✐➺♥ t➼❝❤ t♦➔♥ ♣❤➛♥ ❝→✐ t❤ò♥❣✳ ✶✼ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✽✼✳ ❍➻♥❤ ✈✉æ♥❣ ❆❇❈❉ ❝â ❝↕♥❤ ❞➔✐ ✺❝♠ ✈➔ ❝â ▼✱ ◆✱ P✱ ◗ ❧➔ ❝→❝ tr✉♥❣ ✤✐➸♠ ❝õ❛ ❜è♥ ❝↕♥❤✳ ❚➼♥❤ ❞✐➺♥ t➼❝❤ ♣❤➛♥ ♠➔✉ ✈➔♥❣✳ ✽✽✳ ❇✐➳t ❝↕♥❤ ộ ổ ổ tr ữợ ❚➼♥❤ tê♥❣ ❝❤✉ ✈✐ ✈➔ tê♥❣ ❞✐➺♥ t➼❝❤ ❝õ❛ t➜t ❝↔ ❝→❝ ❤➻♥❤ ✈✉ỉ♥❣ ð tr♦♥❣ ✤â✳ ✽✾✳ ❇→❝ ❚÷ ❝â ♠ët t❤û❛ r✉ë♥❣ ❤➻♥❤ t❛♠ ❣✐→❝ ✈✉æ♥❣ ❝â ❤❛✐ ❝↕♥❤ ❣â❝ ✈✉æ♥❣ ❞➔✐ ✻✵♠ ✈➔ ✸✵♠✳ ◆➠♠ ♥❛② ①➣ ✤➔♦ ♠ët ❝♦♥ ♠÷ì♥❣ rë♥❣ ✸♠ ❝❤↕② ❞å❝ t❤❡♦ ❝↕♥❤ ✻✵♠ ✭①❡♠ ❤➻♥❤ ✈➩✮✳ ❊♠ ❤➣② t➼♥❤ ❞✐➺♥ t➼❝❤ t❤û❛ r✉ë♥❣ ❝á♥ ❧↕✐✳ ✾✵✳ ▲➔♠ ①♦♥❣ ♠ët ♠↔♥❤ ✤➜t ❤➻♥❤ t ữợ ữủ ữợ ữủ ỏ ÷ỵ❝ ❧÷đ♥❣ ✤→② ❧ỵ♥ ❞➔✐ ❣➜♣ ❞ỉ✐ ✤→② ❜➨✳ ❈ỉ õ ữợ ữủ s ìợ ữủ ♥❤÷ ❆♥ t❤➻ ❞✐➺♥ t➼❝❤ ♠↔♥❤ ✤➜t t➠♥❣ t❤➯♠ 45m2✱ ÷ỵ❝ ❧÷đ♥❣ ♥❤÷ ❇➻♥❤ t❤➻ ❞✐➺♥ t➼❝❤ ♠↔♥❤ ✤➜t t➠♥❣ t 45m2 ữợ ữủ ữ t õ ú ợ ũ ữủ t➠♥❣ t❤➯♠ ✷♠ ♥ú❛✧✳ ❊♠ ❤➣② t➼♥❤ ❞✐➺♥ t➼❝❤ ♠↔♥❤ ✤➜t ❤➻♥❤ t❤❛♥❣ ✤â✳ ✶✽ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✻ ▼ët sè ❞↕♥❣ ❜➔✐ t♦→♥ ❦❤→❝ ✾✶✳ ❚➻♠ sè ❝á♥ t❤✐➳✉✿ 1+ +3 :4= ✾✷✳ ▼ët ✤ë✐ q✉➙♥ ❦✐➳♥ ✤❛♥❣ ❞✐ ❝÷✳ ❈→❝ ❝❤✐➳♥ ❜✐♥❤ ❦✐➳♥ ♠❛♥❣ ➼t ❤ì♥ ♥û❛ sè t❤ù❝ ➠♥ tr♦♥❣ ❧➛♥ ❤➔♥❤ q✉➙♥ ✤➛✉ t✐➯♥ ❧➔ ✶✷✵❣✳ ▲➛♥ ❤➔♥❤ q✉➙♥ t❤ù ❤❛✐✱ ❝❤ó♥❣ s➢♣ ①➳♣ ✤➸ ♠❛♥❣ ♥❤✐➲✉ ❤ì♥ ♥û❛ sè t❤ù❝ ➠♥ ❝á♥ ❧↕✐ ✶✵✵❣✳ ❱➔ ❧➛♥ ❤➔♥❤ q✉➙♥ t❤ù ❜❛ ❝❤ó♥❣ ♠❛♥❣ t❤❡♦ ✹✽✵❣ t❤ù❝ ➠♥✳ ❈á♥ ❧↕✐ ✷✽✵❣ t❤ù❝ ➠♥ ✈➝♥ ❝❤÷❛ ♠❛♥❣ ✤✐ ✤÷đ❝✳ ❍ä✐ ❜❛♥ ✤➛✉ ❝→❝ ❝❤✐➳♥ ❜✐♥❤ ❦✐➳♥ ❝â ❜❛♦ ♥❤✐➯✉ t❤ù❝ ➠♥❄ ✾✸✳ ▼ët rê t→♦ ❝❤✉➞♥ ❜à ✤÷đ❝ ❝❤✐❛ ❝❤♦ ♠å✐ ♥❣÷í✐✳ ◆➳✉ ♠é✐ ♥❣÷í✐ ❧➜② ✸ q✉↔ t→♦ t❤➻ s➩ t❤ø❛ r❛ ✶✻ q✉↔ t→♦✳ ◆➳✉ ♠é✐ ♥❣÷í✐ ❧➜② ✺ q✉↔ t→♦ t❤➻ s➩ t❤✐➳✉ ✹ q✉↔ t→♦✳ ❍ä✐ ❝â ❜❛♦ ♥❤✐➯✉ ♥❣÷í✐ ❝❤✐❛ ♥❤❛✉ rê t→♦ ✤â❄ ❈â ❜❛♦ ♥❤✐➯✉ q✉↔ t→♦ tr♦♥❣ rê❄ ✾✹✳ ▼ët ♥❤â♠ ❝æ♥❣ ♥❤➙♥ ①➙② ❞ü♥❣ ✤❛♥❣ ❧→t ✤÷í♥❣✳ ◆➳✉ ❧→t ✤÷đ❝ ✷✵✵♠✴♥❣➔②✱ ❤å s➩ ❤♦➔♥ t❤➔♥❤ ❝ỉ♥❣ ✈✐➺❝ sỵ♠ ❤ì♥ ❧à❝❤ tr➻♥❤ ✻ ♥❣➔②✳ ❈á♥ ♥➳✉ ❝❤➾ ❧→t ✤÷đ❝ ✶✻✵♠✴♥❣➔②✱ ❝ỉ♥❣ ✈✐➺❝ ❝õ❛ ❤å s➩ ❜à tr➵ s ợ ọ ữớ ❜❛♦ ♥❤✐➯✉ ♠➨t❄ ✾✺✳ ❚r♦♥❣ ❞➣② sè✿ 4, 7, 10, · · · , 295, 298✳ ❙è ✷✾✽ ❧➔ sè t❤ù ❜❛♦ ♥❤✐➯✉❄ ✾✻✳ ❈â ✸✵ ❤➔♥❣ ❣❤➳ ð ❝→♥❤ ♣❤➼❛ ❇➢❝ ❝õ❛ ♠ët s➙♥ ✈➟♥ ✤ë♥❣✳ ❍➔♥❣ s❛✉ ♥❤✐➲✉ ỡ trữợ ố ũ õ ✶✸✷ ❣❤➳✳ ❍ä✐ ❝â ❜❛♦ ♥❤✐➯✉ ❣❤➳ ð ❤➔♥❣ ✤➛✉ t✐➯♥❄ ❱➔ ❝â t➜t ❝↔ ❜❛♦ ♥❤✐➯✉ ❣❤➳ ð ❝→♥❤ ♣❤➼❛ ❇➢❝❄ ✾✼✳ ✣à♥❤ ♥❣❤➽❛ ♣❤➨♣ t➼♥❤ ♠ỵ✐✿ 3✸4 = + + + 5✸6 = + + + + + 10 6✸3 = + + ❚➻♠ ❣✐→ trà ❝õ❛ ♥ tr♦♥❣ ♣❤➨♣ t♦→♥ n✸8 = 60✳ ✾✽✳ ✣à♥❤ ♥❣❤➽❛ ♣❤➨♣ t➼♥❤ ♠ỵ✐✿ 88 ❚➻♠ 13 15 11 ✳ 12 11 = ✱ 32 = 35 ✈➔ 47 = ✳ 56 ✾✾✳ ▼ët ❝❤✐➳❝ ❤ë♣ ❝â ❝❤ù❛ ♥❤ú♥❣ ✈✐➯♥ ❜✐ ①❛♥❤✳ ❍❛✐ ❝❤✐➳❝ ❤ë♣ ❦❤→❝ ❧↕✐ ❝❤➾ ❝❤ù❛ ✶✾ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ❜✐ tr➢♥❣✳ ◆❤➣♥ tr➯♥ ❤ë♣ ❆✿ ❇✐ tr➢♥❣❀ ♥❤➣♥ tr➯♥ ❤ë♣ ❇✿ ❜✐ ①❛♥❤❀ ♥❤➣♥ tr➯♥ ❤ë♣ ❈✿ ❤ë♣ ❇ ❝❤ù❛ ❜✐ ①❛♥❤✳ ❍ä✐ ❤ë♣ ♥➔♦ ❝❤ù❛ ❜✐ ①❛♥❤ ♥➳✉ ❤❛✐ tr♦♥❣ sè ❜❛ ♥❤➣♥ tr➯♥ ✤➣ ❜à ❞→♥ ♥❤➛♠✳ ✶✵✵✳ ❇è♥ ❤➔♥❤ ❦❤→❝❤ ❆✱ ❇✱ ❈✱ ❉ ✤➦t ❝❤é ð ♠ët ❦❤→❝❤ s↕♥ ✶✽ t➛♥❣✳ ❍å tứ ữợ r ố ự ❈➟♣✳ ❚➛♥❣ ♠➔ ❆ ð ❝â sè ❣➜♣ ✹ ❧➛♥ sè ð t➛♥❣ ❝õ❛ ♥❣÷í✐ ▼➯✲①✐✲❝ỉ✳ ◆❣÷í✐ ✣ù❝ ð ❝❛♦ ❤ì♥ ♥❣÷í✐ ❇ ✹ t➛♥❣ ♥❤÷♥❣ t❤➜♣ ❤ì♥ ♥❣÷í✐ ❆✐ ❈➟♣✳ ◆❣÷í✐ ❆✐ ❈➟♣ ð t❤➜♣ ❤ì♥ ♥❣÷í✐ ❆ ✻ t➛♥❣✳ ❚➜t ❝↔ sè t➛♥❣ ♠➔ ❤å ð ✤➲✉ ❧➔ sè ❝❤➤♥✳ ●❤➨♣ ❆✱ ❇✱ ❈✱ ❉ ✈ỵ✐ ✤ó♥❣ ✤➜t ữợ số t tr s ▼ët ❜→❝ ♥æ♥❣ ❞➙♥ ❝â ✶✹✺ ❝♦♥ ❣➔ ✈➔ ❝♦♥ t❤ä✳ ❈❤ó♥❣ ❝â t➜t ❝↔ ❝â ✹✶✵ ❝❤➙♥✳ ❍ä✐ ❝â ❜❛♦ ♥❤✐➯✉ ❝♦♥ ❣➔❄ ❇❛♦ ♥❤✐➯✉ ❝♦♥ t❤ä❄ ✶✵✷✳ ▼ët trố q t t ỗ ũ ð ❚r✉♥❣ ◗✉è❝ t❤í✐ ①÷❛✮✱ ♠ët ❝♦♥ ❣➔ ♠→✐ ❝â ❣✐→ ✸ q✉❛♥ ✈➔ ♠ët ❝♦♥ ❣➔ ❝♦♥ ❣✐→ ✶ q ữủ ợ ❣✐→ ✶✵✵ q✉❛♥ t❤➻ ❤ä✐ tr♦♥❣ ✤â ❝â ❜❛♦ ♥❤✐➯✉ ❣➔ trè♥❣✱ ❣➔ ♠→✐ ✈➔ ❣➔ ❝♦♥❄ ✶✵✸✳ ❚➼♥❤✿ 1 2+ 3+ 4+ ✶✵✹✳ ❚➼♥❤ ×1 + ×1 + ×1 13 + · · · + 97 ×1 101 ✳ ✶✵✺✳ ❈â ✸ ✤æ✐ t➜t ①→♠✱ ✸ ✤æ✐ t➜t ✈➔♥❣ ð tr♦♥❣ ♥❣➠♥ ❦➨♦✳ ❍ä✐ ❆❧❛♥ ♣❤↔✐ ❧➜② r❛ ❦❤ä✐ ♥❣➠♥ ❦➨♦ ❜❛♦ ♥❤✐➯✉ ❝❤✐➳❝ t➜t tr♦♥❣ ♠ët ❝➠♥ ♣❤á♥❣ tè✐ ✤➸ ❝â ✤÷đ❝ ♠ët ❝➦♣ t➜t ❝ò♥❣ ♠➔✉❄ ✶✵✻✳ ✣✐➸♠ ❝❛♦ ♥❤➜t tr♦♥❣ ♠ët ❦ý t❤✐ ❦➳t t❤ó❝ ♠ỉ♥ ❚♦→♥ tr♦♥❣ ❧ỵ♣ ✺● ❧➔ ✾✾✳ ✣✐➸♠ t❤➜♣ ♥❤➜t tr♦♥❣ ❦ý t❤✐ ✤â ❧➔ ✾✶✳ ❈â ➼t ♥❤➜t ♥➠♠ ❤å❝ s✐♥❤ ❝â ❝ò♥❣ ✤✐➸♠✳ ❍ä✐ sè ❤å❝ s✐♥❤ tè✐ t❤✐➸✉ tr♦♥❣ ❧ỵ♣ ✺● ❧➔ ❜❛♦ ♥❤✐➯✉❄ ✶✵✼✳ ✼✵ ❤å❝ s✐♥❤ ❝õ❛ tr÷í♥❣ t✐➸✉ ❤å❝ ❚❤♦♠s♦♥ t❤❛♠ ❣✐❛ ♠ët ❝✉ë❝ t❤✐ t♦→♥ ❤å❝✳ ✣✐➸♠ sè tr✉♥❣ ❜➻♥❤ ❝õ❛ ❝→❝ ❜↕♥ ❧➔ ✻✻ ✤✐➸♠✳ ✣✐➸♠ sè tr✉♥❣ ❜➻♥❤ ❝õ❛ ❝→❝ ❜↕♥ ♥ú ❧➔ ✼✵ ✤✐➸♠ ✈➔ ❝→❝ ❜↕♥ ♥❛♠ ❧➔ ✻✸ ✤✐➸♠✳ ❍ä✐ t➾ ❧➺ sè ❜↕♥ ♥ú ✈➔ sè ❜↕♥ ✷✵ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ♥❛♠ ❧➔ ❜❛♦ ♥❤✐➯✉❄ ✶✵✽✳ ❈→❝ ♠→② ❆✱ ❇ ✈➔ ❈ tr♦♥❣ ①÷ð♥❣ ❧➛♥ ❧÷đt ♠➜t ✹✱ ✺ ✈➔ ✻ ♣❤ót ✤➸ s↔♥ ①✉➜t ✶✵✵✵ ✈✐➯♥ t❤✉è❝✳ ❈❤õ ①÷ð♥❣ ♥❤➟♥ ✤÷đ❝ ✤ì♥ ✤➦t ❤➔♥❣ ✼✹✵ ✵✵✵ ✈✐➯♥ t❤✉è❝ ✤â✳ ❍ä✐ ❜→❝ ➜② ❝➛♥ ❝❤✐❛ ❝❤♦ ❝→❝ ♠→② sè ❧÷đ♥❣ ✈✐➯♥ t❤✉è❝ ♥❤÷ t❤➳ ♥➔♦ ✤➸ ❝❤ó♥❣ ❜➢t ✤➛✉ ✈➔ ❦➳t t❤ó❝ ð ❝ò♥❣ ♠ët t❤í✐ ✤✐➸♠❄ ✼ ✣→♣ →♥ ✈➔ ❧í✐ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥ ❝❤å♥ ❧å❝ ❇➔✐ ✸✳ ❙♦ s→♥❤ ❝→❝ ♣❤➙♥ sè s❛✉✿ 45 ; 87 ; 89 ; 12 11 < ✣→♣ sè✿ 45 < 98 < 12 11 ❇➔✐ ✶✶✳ ❚✉ê✐ ❇è ❜➡♥❣ 98 t✉ê✐ ▼➭✱ t✉ê✐ ▲❛♥ ❜➡♥❣ 14 t✉ê✐ ▼➭✱ tê♥❣ sè t✉ê✐ ❝õ❛ ❇è ✈➔ ▲❛♥ ❧➔ ✹✹ t✉ê✐✳ ❍ä✐ ♠é✐ ♥❣÷í✐ ❜❛♦ ♥❤✐➯✉ t✉ê✐❄ ❇➔✐ ❣✐↔✐✿ ❚❛ ❝â 41 = 28 ✳ ❱➟② t✉ê✐ ▲❛♥ ❜➡♥❣ 82 tờ tờ ỗ t tờ ỗ tờ ố ỗ tờ ỗ + = 11 ♣❤➛♥✳ ❚✉ê✐ ❝õ❛ ❇è ❧➔✿ 44 : 11 × = 36 ✭t✉ê✐✮ ❚✉ê✐ ❝õ❛ ▲❛♥ ❧➔✿ 44 − 36 = ✭t✉ê✐✮ ❚✉ê✐ ❝õ❛ ▼➭ ❧➔✿ × = 32 ✭t✉ê✐✮✳ ✣→♣ sè✿ ❇è✿ ✸✻ t✉ê✐❀ ▼➭✿ ✸✷ t✉ê✐❀ ▲❛♥✿ ✽ t✉ê✐✳ ❇➔✐ ✶✼✳ ▼ët ✤ë✐ ❝æ♥❣ ♥❤➙♥ ❧➔♠ ✤÷í♥❣ tr♦♥❣ ✸ ♥❣➔②✳ ◆❣➔② t❤ù ♥❤➜t ❧➔♠ ✤÷đ❝ ❝ỉ♥❣ ✈✐➺❝✱ ♥❣➔② t❤ù ❤❛✐ ❧➔♠ ✤÷đ❝ 38 ❝ỉ♥❣ ✈✐➺❝✱ ♥❣➔② t❤ù ❜❛ ❧➔♠ ♥èt ✺✼ ♠➨t ❝✉è✐✳ ❍ä✐ ✤ë✐ ❝ỉ♥❣ ♥❤➙♥ ✤â ♣❤↔✐ ❧➔♠ ❜❛♦ ♥❤✐➯✉ ♠➨t ✤÷í♥❣❄ ❇➔✐ ❣✐↔✐✿ ❙❛✉ ❤❛✐ ♥❣➔② ✤➛✉ ✤ë✐ ❝æ♥❣ ♥❤➙♥ ✤â ❧➔♠ ✤÷đ❝ sè ♣❤➛♥ sỉ♥❣ ✈✐➺❝ ❧➔✿ 37 + = 56 ✭❝æ♥❣ ✈✐➺❝✮ P❤➙♥ sè ❝❤➾ sè ♣❤➛♥ ❝æ♥❣ ✈✐➺❝ ✤ë✐ ❝æ♥❣ ♥❤➙♥ ❧➔♠ tr♦♥❣ ♥❣➔② t❤ù ❜❛ ❧➔✿ ✷✶ ❙✐❣♠❛ ✲ ▼❆❚❍❙ 1− 37 19 = 56 56 ✭❝ỉ♥❣ ✈✐➺❝✮ ❙è ♠➨t ✤÷í♥❣ ✤ë✐ ❝ỉ♥❣ ♥❤➙♥ ✤â ♣❤↔✐ ❧➔♠ ❧➔✿ 57 : 19 × 56 = 168 ✣→♣ sè✿ ✶✻✽♠ ✤÷í♥❣✳ ✭♠✮ ❇➔✐ ✷✻✳ ◆❤➔ ❆♥ ♥✉æ✐ ✽ ❝♦♥ ❜á✱ tr♦♥❣ ✤â ❝â ✸ ❝♦♥ ❜á ✤ü❝✳ ❍ä✐ ❛✮ ❙è ❜á ✤ü❝ ❝❤✐➳♠ ❜❛♦ ♥❤✐➯✉ ♣❤➛♥ tr➠♠ tê♥❣ sè ❜á❄ ❜✮ ❚➾ sè ♣❤➛♥ tr➠♠ ❣✐ú❛ sè ❜á ✤ü❝ ✈➔ sè ❜á ❝→✐ ❧➔ ❜❛♦ ♥❤✐➯✉❄ ❇➔✐ ❣✐↔✐✿ ❛✮ ❙è ♣❤➛♥ tr➠♠ ❜á ✤ü❝ ❝❤✐➳♠ ❧➔✿ : = 0, 375 = 37, 5% ❜✮ ❙è ❜á ❝→✐ ❧➔✿ − = ✭❝♦♥✮ ❚➾ sè ♣❤➛♥ tr➠♠ ❣✐ú❛ sè ❜á ✤ü❝ ✈➔ sè ❜á ❝→✐ ❧➔✿ : = 0, = 60% ✣→♣ sè✿ ❛✱ 37, 5% ❀ ❜✱ 60% ❇➔✐ ✸✹✳ ▼ët ♥❣÷í✐ ❣û✐ t✐➳t ❦✐➺♠ ✶✺ tr✐➺✉ ỗ ợ st 1, 1% ởt t ❙è t✐➲♥ ❧➣✐ s❛✉ ♠ët t❤→♥❣✳ ❜✮ ❙è t✐➲♥ ❝↔ ❣è❝ ✈➔ ❧➣✐ s❛✉ ❤❛✐ t❤→♥❣ ♥➳✉ ♥❣÷í✐ ✤â ❝❤➾ ✤➳♥ rót t✐➲♥ ♠ët ❧➛♥✳ ❝✮ ❙è t✐➲♥ ❝↔ ❣è❝ ✈➔ ❧➣✐ s❛✉ ❜❛ t❤→♥❣ ♥➳✉ ♥❣÷í✐ ✤â ❝❤➾ ✤➳♥ rót t✐➲♥ ♠ët ❧➛♥✳ ❇➔✐ ❣✐↔✐✿ ❛✮ ❙è t✐➲♥ ❧➣✐ s ởt t ỷ ỗ ố t ❣è❝ ✈➔ ❧➣✐ s❛✉ t❤→♥❣ t❤ù ♥❤➜t ❧➔✿ 15000000 + 165000 = 15165000 ỗ ố t t tự ❤❛✐ ❧➔✿ 15165000 : 100 × 1, = 166815 ỗ ố t ố s t ỷ t 15165000 + 166815 = 15331815 ỗ ❙è t✐➲♥ ❧➣✐ ❝õ❛ t❤→♥❣ t❤ù ❜❛ ❧➔✿ 15331815 : 100 ì 1, = 168649, 965 ỗ ố t ❝↔ ❣è❝ ✈➔ ❧➣✐ s❛✉ ❜❛ t❤→♥❣ ❣û✐ t✐➲♥ ❧➔✿ 15331815 + 168649, 965 = 15500464, 965 ỗ số ỗ ỗ ỗ 15000000 : 100 ì 1, = 165000 ✷✷ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ❇➔✐ ✹✸✳ ▼ët ✤ì♥ ✈à ❜ë ✤ë✐ ❝❤✉➞♥ ❜à ❣↕♦ ❝❤♦ ✼✵ ♥❣÷í✐ ➠♥ tr♦♥❣ ✸✵ ♥❣➔②✳ ❙❛✉ ❦❤✐ ➠♥ ✤÷đ❝ ✻ ♥❣➔② t❤➻ ❝â ✶✵ ♥❣÷í✐ ❝❤✉②➸♥ ✤✐ ♥ì✐ ❦❤→❝✳ ❍ä✐ sè ❣↕♦ ❝á♥ ❧↕✐ ✤õ ❝❤♦ sè ♥❣÷í✐ ❝á♥ ❧↕✐ ❝õ❛ ✤ì♥ ✈à ✤â ➠♥ tr♦♥❣ ❜❛♦ ♥❤✐➯✉ ♥❣➔② ♥ú❛❄ ❇✐➳t r➡♥❣ ♠ù❝ ➠♥ ❝õ❛ ♠å✐ ♥❣÷í✐ tr♦♥❣ ♠ët ♥❣➔② ❧➔ ♥❤÷ ♥❤❛✉✳ ❇➔✐ ❣✐↔✐✿ ❙è ❣↕♦ ❝á♥ ❧↕✐ ✤õ ❝❤♦ ✼✵ ♥❣÷í✐ ➠♥ tr♦♥❣ sè ♥❣➔② ❧➔✿ 30 − = 24 ✭♥❣➔②✮ ❙è ♥❣÷í✐ ❝á♥ ❧↕✐ ❧➔✿ 70 − 10 = 60 số số ữớ trữợ s❛✉ ❦❤✐ ❝❤✉②➸♥ ✤✐ ❧➔✿ 70 : 60 = 70 60 ợ ũ ởt số ữủ t số ữớ ✈➔ sè ♥❣➔② ➠♥ ❧➔ ❤❛✐ ✤↕✐ ❧÷đ♥❣ t➾ ❧➺ 70 60 ♥❣❤à❝❤ ✈ỵ✐ ♥❤❛✉ ♥➯♥ t➾ sè sè ♥❣➔② ➠♥ ❧➔✿ − 60 = 70 ❙è ❣↕♦ ❝á♥ ❧↕✐ ✤õ ❝❤♦ ✻✵ ♥❣÷í✐ ➠♥ tr♦♥❣ sè ♥❣➔② ❧➔✿ 24 × 70 : 60 = 28 ✭♥❣➔②✮ ✣→♣ sè✿ ✷✽ ♥❣➔②✳ ❇➔✐ ✹✽✳ ▼ët ✤ë✐ ①❡ ✈➟♥ ❝❤✉②➸♥ ❝â ỗ tờ tờ ❧♦↕✐ ✽ t➜♥✱ tê ❤❛✐ ❧➔ ❝→❝ ①❡ ❧♦↕✐ ✺ t➜♥✳ ❍ä✐ ♠é✐ tê ❝â ❜❛♦ ♥❤✐➯✉ ①❡❄ ❇✐➳t r➡♥❣ ❤❛✐ tê ♥❤➟♥ ❝❤ð sè ❤➔♥❣ ❤â❛ ❜➡♥❣ ♥❤❛✉ ✈➔ ♠é✐ ①❡ ❝❤➾ ❝❤ð ♠ët ❧÷đt✳ ✣→♣ sè✿ ❚ê ✶✿ ✶✺ ①❡ ❧♦↕✐ ✽ t➜♥❀ tê ✷✿ ✷✹ ①❡ ❧♦↕✐ ✺ t➜♥ ❇➔✐ ✼✸✳ ❈❤♦ ❤➻♥❤ t❛♠ ❣✐→❝ ❆❇❈ ❝â ✤✐➸♠ ◆ ❧➔ ✤✐➸♠ ❝❤➼♥❤ ❣✐ú❛ ❝↕♥❤ ❆❈✱ tr➯♥ ❤➻♥❤ ✤â ❝â ❤➻♥❤ t❤❛♥❣ ❇▼◆❊ ♥❤÷ ❤➻♥❤ ✈➩✳ ◆è✐ ❇ ✈ỵ✐ ◆✱ ♥è✐ ❊ ✈ỵ✐ ▼✱ ❤❛✐ ✤♦↕♥ t❤➥♥❣ ♥➔② ❣➦♣ ♥❤❛✉ ð ✤✐➸♠ ❖✳ ❛✮ ❙♦ s→♥❤ ❞✐➺♥ t➼❝❤ ❤❛✐ t❛♠ ❣✐→❝ ❖❇▼ ✈➔ ❖❊◆✳ ❜✮ ❙♦ s→♥❤ ❞✐➺♥ t➼❝❤ ❤➻♥❤ t❛♠ ❣✐→❝ ❊▼❈ ✈ỵ✐ ❞✐➺♥ t➼❝❤ ❤➻♥❤ ❆❊▼❇✳ ❇➔✐ ❣✐↔✐✿ ❛✮ ❍❛✐ t❛♠ ❣✐→❝ ▼❇❊ ✈➔ ◆❇❊ ❝â ❝❤✉♥❣ ✤→② ❇❊ ✈➔ ❝❤✐➲✉ ❝❛♦ ❜➡♥❣ ♥❤❛✉ ✭✈➻ ✷✸ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ❇▼◆❊ ❧➔ ❤➻♥❤ t❤❛♥❣✮ ♥➯♥ SM BE = SN BE ✭✶✮ ❱➻✿ SM BE = SOBM + SOBE ❀ SN BE = SOEN + SOBE ✭✷✮ ❚ø ✭✶✮ ✈➔ ✭✷✮ s✉② r❛✿ SOBM = SOEN ✭✸✮ ❜✮ ❍❛✐ t❛♠ ❣✐→❝ ◆❆❇ ✈➔ ◆❈❇ ❝â ❝❤✉♥❣ ❝❤✐➲✉ ❝❛♦ ❤↕ tø ❇ ✈➔ ✤→② ◆❆ ❜➡♥❣ ✤→② ◆❈ ♥➯♥ SN AB = SN CB ✭✹✮ ❚❛ ❝â✿ SEM C = SM ON C + SOEN = SN CB − SOBM + SOEN ✭✺✮ SAEM N = SAEOB + SOBM = SN AB − SOEN + SOBM ✭✻✮ ❚ø ✭✸✮✱ ✭✹✮✱ ✭✺✮ ✈➔ ✭✻✮ s✉② r❛ SEM C = SAEM B ❇➔✐ ✼✹✳ ❈❤♦ t❛♠ ❣✐→❝ ❆❇❈✳ ▼ ❧➔ ✤✐➸♠ ❝❤➼♥❤ ❣✐ú❛ ❝↕♥❤ ❇❈✱ ◆ ❧➔ ✤✐➸♠ ❝❤➼♥❤ ❣✐ú❛ ❝↕♥❤ ❆❈✳ ❆▼ ✈➔ ❆◆ ❝➢t ♥❤❛✉ t↕✐ ●✳ ❛✮ ❙♦ s→♥❤ ❞✐➺♥ t➼❝❤ t❛♠ ❣✐→❝ ❇●▼✱ ❈●▼✱ ❆●◆✱ ❈●◆✳ ❜✮ ◆è✐ ❈ ✈ỵ✐ ● ✈➔ ❦➨♦ ❞➔✐ ❈● ❝➢t ❝↕♥❤ ❆❇ t↕✐ P✳ ❈❤ù♥❣ ♠✐♥❤ ❆P ❂ ❇P✳ ❇➔✐ ❣✐↔✐✿ ❛✮ ✰ ❍❛✐ t❛♠ ❣✐→❝ ❆▼❇ ✈➔ ❆❈▼ ❝â ❝ò♥❣ ❝❤✐➲✉ ❝❛♦ ❤↕ tø ❆ ①✉è♥❣ ❇❈ ✈➔ ❇▼ ❂ ▼❈✳ ❙✉② r❛✿ SABM = SACM = SABC ✭✶✮ ✰ ❍❛✐ t❛♠ ❣✐→❝ ❇❆◆ ✈➔ ❇❈◆ ❝â ❝ò♥❣ ❝❤✐➲✉ ❝❛♦ ❤↕♥ tø ❇ ①✉è♥❣ ❆❈ ✈➔ ❆◆ ❂ ❈◆✳ ✭✷✮ ❙✉② r❛✿ SBAN = SBCN = SABC ❚ø ✭✶✮ ✈➔ ✭✷✮✱ s✉② r❛✿ SBCN = SACM ❍❛✐ t❛♠ ❣✐→❝ ❆❈▼ ✈➔ ❇❈◆ ❝â ♣❤➛♥ ❝❤✉♥❣ ❧➔ tù ❣✐→❝ ●▼❈◆ ♥➯♥ s✉② r❛✿ SBGM = SAGN ✭✸✮ ✰ ❍❛✐ t❛♠ ❣✐→❝ ❆●◆ ✈➔ ❈●◆ ❝â ❝ò♥❣ ❝❤✐➲✉ ❝❛♦ ❤↕ tø ● ①✉è♥❣ ❆❈ ✈➔ ❆◆ ❂ ❈◆✳ ❙✉② r❛✿ SAGN = SCGN = SAGC ✭✹✮ ✰ ❍❛✐ t❛♠ ❣✐→❝ ❇●▼ ✈➔ ❈●▼ ❝â ❝ò♥❣ ❝❤✐➲✉ ❝❛♦ ❤↕ tø ● ①✉è♥❣ ❇❈ ✈➔ ❇▼ ❂ ❈▼✳ ❙✉② r❛✿ SBGM = SCGM = SBGC ✭✺✮ ❚ø ✭✸✮✱ ✭✹✮ ✈➔ ✭✺✮✱ s✉② r❛✿ SBGM = SCGM = SAGN = SCGN ✳ ❜✮ ❚❤❡♦ ❝❤ù♥❣ ♠✐♥❤ tr➯♥✱ t❛ ❝â✿ SAGC = SBGC ✳ ❍❛✐ t❛♠ ❣✐→❝ ❆●❈ ✈➔ ❇●❈ ❝â ❞✐➺♥ t➼❝❤ ❜➡♥❣ ♥❤❛✉ ✈➔ ❝â ❝❤✉♥❣ ❝↕♥❤ ❇❈ ♥➯♥ ❝❤✐➲✉ ❝❛♦ ❤↕ tø ❆ ①✉è♥❣ ●❈ ❝õ❛ t❛♠ ❣✐→❝ ❆●❈ ❜➡♥❣ ❝❤✐➲✉ ❝❛♦ ❤↕ tø ❇ ①✉è♥❣ ●❈ ❝õ❛ t❛♠ ❣✐→❝ ❇●❈✳ ✷✹ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ❍❛✐ t❛♠ ❣✐→❝ ❆●P ✈➔ ❇●P ❝â ❝❤✉♥❣ ❝↕♥❤ ●P ✈➔ ❝❤✐➲✉ ❝❛♦ ❤↕ tø ❆ ①✉è♥❣ ●P ❜➡♥❣ ❝❤✐➲✉ ❝❛♦ ❤↕ tø ❇ ①✉è♥❣ ●P✳ ❙✉② r❛✿ SAGP = SBGP ✳ ❍❛✐ t❛♠ ❣✐→❝ ❆●P ✈➔ ❇●P ❝â ❞✐➺♥ t➼❝❤ ❜➡♥❣ ♥❤❛✉ ✈➔ ❝❤✉♥❣ ❝❤✐➲✉ ❝❛♦ ❤↕ tø ● ①✉è♥❣ ❆❇ ♥➯♥ ❆P ❂ ❇P✳ ❇➔✐ ✼✻✳ ❚➼♥❤ ❞✐➺♥ t➼❝❤ ❤➻♥❤ t❤❛♥❣ ❆❇❈❉✳ ❇✐➳t ❞✐➺♥ t➼❝❤ ❝→❝ ❤➻♥❤ t❛♠ ❣✐→❝ ❆❖❉ ✈➔ ❉❖❈ ♥❤÷ ❤➻♥❤ ✈➩✳ ❇➔✐ ❣✐↔✐✿ ❍❛✐ t❛♠ ❣✐→❝ ❆❉❈ ✈➔ ❇❉❈ ❝â ❝❤✉♥❣ ✤→② ❉❈ ✈➔ ❝❤✐➲✉ ❝❛♦ ❜➡♥❣ ♥❤❛✉ ♥➯♥ SADC = SBDC ❱➻✿ SADC = SAOD + SDOC ❀ SBDC = SBOC + SDOC ❙✉② r❛✿ SAOD = SBOC = 10cm2 ❍❛✐ t❛♠ ❣✐→❝ ❆❖❉ ✈➔ ❉❖❈ ❝â ❝❤✉♥❣ ❝❤✐➲✉ ❝❛♦ ❤↕ tø ❉ ✈➔ SDOC = 2SAOD ♥➯♥ ❖❈ ❂ ✷❆❖✳ ❍❛✐ t❛♠ ❣✐→❝ ❆❇❖ ✈➔ ❇❖❈ ❝â ❝❤✐➲✉ ❝❛♦ ❤↕ tø ❇ ✈➔ ✤→② ❖❈ ❣➜♣ ✷ ❧➛♥ ✤→② ❆❖ ♥➯♥ SBOC = 2SAOB ✳ ❱➟②✿ SAOB = 21 SBOC = 12 × 10 = ✭cm2✮ ❉♦✿ SABCD = SAOB + SAOD + SDOC + SBOC ♥➯♥ ❞✐➺♥ t➼❝❤ ❤➻♥❤ t❤❛♥❣ ❆❇❈❉ ❧➔✿ + 10 + 20 + 10 = 45 ✭cm2 ✮ ❇➔✐ ✽✵✳ ❈❤♦ ❤➻♥❤ ❝❤ú ♥❤➟t ❆❇❈❉ ❝â ❝❤✉ ✈✐ ❧➔ ✼✽❝♠✱ ❝❤✐➲✉ ❞➔✐ ❆❇ ❤ì♥ ❝❤✐➲✉ rë♥❣ ❇❈ ❧➔ ✶✺❝♠✳ ❚r➯♥ ❇❈ ❧➜② ✤✐➸♠ ▼✱ tr➯♥ ❈❉ ❧➜② ✤✐➸♠ ◆ s❛♦ ❝❤♦ ❤❛✐ ✤÷í♥❣ t❤➥♥❣ ❆▼✱ ❆◆ ❝❤✐❛ ❤➻♥❤ ❝❤ú ♥❤➟t t❤➔♥❤ ✸ ♣❤➛♥ ❝â ❞✐➺♥ t➼❝❤ ❜➡♥❣ ♥❤❛✉✳ ❚➼♥❤ ✤ë ❞➔✐ ✤♦↕♥ ❈▼✱ ❈◆✳ ✷✺ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ✣→♣ sè✿ ❈▼ ❂ ✹❝♠❀ ❈◆ ❂ ✾❝♠ ❇➔✐ ✻✽✳ ❍❛✐ t❤à ①➣ ❆ ✈➔ ❇ ❝→❝❤ ♥❤❛✉ ✺✹❦♠✳ ❆♥❤ ❇➡♥❣ ✤✐ ①❡ ♠→② tø ❆ ✤➳♥ ❇ ✈➔ ❛♥❤ ❉ô♥❣ ❝ơ♥❣ ✤✐ ①❡ ♠→② ♥❤÷♥❣ tø ❇ ✈➲ ❆✱ ❤❛✐ ♥❣÷í✐ ❦❤ð✐ ❤➔♥❤ ❝ơ♥❣ ♠ët ❧ó❝ ✈➔ s❛✉ ✺✹ ♣❤ót t❤➻ ❤❛✐ ♥❣÷í✐ ❣➦♣ ♥❤❛✉✱ ❝❤é ❣➦♣ ♥❤❛✉ ❝→❝ t❤à ①➣ ❆ ✷✺✱✷❦♠✳ ❍ä✐ ✈➟♥ tè❝ ❝õ❛ ♠é✐ ♥❣÷í✐ ❧➔ ❜❛♦ ♥❤✐➯✉❄ ❇➔✐ ❣✐↔✐✿ ❚❛ ❝â ✺✹ ♣❤ót ❂ 109 ❣✐í ❂ ✵✱✾ ❣✐í✳ ❱➟♥ tè❝ ❝õ❛ ❛♥❤ ❇➡♥❣ ❧➔✿ 25, : 0, = 28 ✭❦♠✴❣✐í✮ ◗✉➣♥❣ ✤÷í♥❣ tø ❝❤é ❤❛✐ ♥❣÷í✐ ❣➦♣ ♥❤❛✉ ✤➳♥ t❤à ①➣ ❇ ❧➔✿ ✺✹ ✲ ✷✺✱✷ ❂ ✷✽✱✽ ✭❦♠✮ ❱➟♥ tè❝ ❝õ❛ ❛♥❤ ❉ô♥❣ ❧➔✿ 28, : 0, = 32 ✭❦♠✴❣✐í✮ ✣→♣ sè✿ ❆♥❤ ❇➡♥❣✿ ✷✽ ❦♠✴❣✐í❀ ❛♥❤ ❉ơ♥❣✿ ✸✷❦♠✴❣✐í✳ ❇➔✐ ✾✽✳ ✣à♥❤ ♥❣❤➽❛ ♣❤➨♣ t➼♥❤ ♠ỵ✐✿ 88 13 ❚➻♠ 15 ✣→♣ sè✿ 11 ✳ 12 24 180 11 = ✱ 32 = 35 ✈➔ 47 = ✳ 56 ❇➔✐ ✶✵✵✳ ❇è♥ ❤➔♥❤ ❦❤→❝❤ ❆✱ ❇✱ ❈✱ ❉ ✤➦t ❝❤é ð ♠ët ❦❤→❝❤ s↕♥ ✶✽ t➛♥❣✳ ❍å ✤➳♥ tø ❝→❝ ữợ r ố ự ❆ ð ❝â sè ❣➜♣ ✹ ❧➛♥ sè ð t➛♥❣ ❝õ❛ ♥❣÷í✐ ▼➯✲①✐✲❝ỉ✳ ◆❣÷í✐ ✣ù❝ ð ❝❛♦ ❤ì♥ ♥❣÷í✐ ❇ ✹ t➛♥❣ ♥❤÷♥❣ t❤➜♣ ❤ì♥ ♥❣÷í✐ ❆✐ ❈➟♣✳ ◆❣÷í✐ ❆✐ ❈➟♣ ð t❤➜♣ ❤ì♥ ♥❣÷í✐ ❆ ✻ t➛♥❣✳ ❚➜t ❝↔ sè t➛♥❣ ♠➔ ❤å ð ✤➲✉ ❧➔ sè ❝❤➤♥✳ ●❤➨♣ ợ ú t ữợ ✈➔ sè t➛♥❣ tr♦♥❣ ❦❤→❝❤ s↕♥✳ ✣→♣ sè✿ ❆ → ❉ → ❈ → ❇ → ❚r✉♥❣ ◗✉è❝ ❆✐ ❈➟♣ ✣ù❝ ▼➯✲①✐✲❝æ → → → → ❚❤ù ✶✻ ❚❤ù ✶✵ ❚❤ù ✽ ❚❤ù ✹ ✷✻ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ❇➔✐ ✶✵✸✳ ❚➼♥❤✿ 1 2+ 3+ 4+ 68 ✣→♣ sè✿ 157 ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ sü q✉❛♥ t➙♠ ❝õ❛ ❜↕♥ ✤å❝✦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ∗∗∗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❍➌❚ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ∗∗∗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ... ❧➔✿ 150 00000 + 1 650 00 = 151 650 00 ỗ ố t t tự 151 650 00 : 100 ì 1, = 1668 15 ỗ ❙è t✐➲♥ ❝↔ ❣è❝ ✈➔ ❧➣✐ s❛✉ ❤❛✐ t❤→♥❣ ❣û✐ t 151 650 00 + 1668 15 = 153 318 15 ỗ ố t✐➲♥ ❧➣✐ ❝õ❛ t❤→♥❣ t❤ù ❜❛ ❧➔✿ 153 318 15. .. ❛✮ 45 ✈➔ 32 11 ❜✮ 59 ✈➔ 24 15 ❞✮ 14 ✈➔ 15 16 97 ❡✮ 96 ✈➔ 95 96 17 ❝✮ 13 ✈➔ 27 23 ✺✳ ❚➻♠ sè tü ♥❤✐➯♥ ① ❜✐➳t✿ ❛✮ x9 = 53 ❀ 85 ❜✮ 17 = ❀ x 1 05 ❞✮ x8 + 32 = 76 ❀ 27 ❡✮ x −3 = 1 35 ✳ ❝✮ 86 = 15 ❀ x... 153 318 15 : 100 ì 1, = 168649, 9 65 ỗ ố t ❣è❝ ✈➔ ❧➣✐ s❛✉ ❜❛ t❤→♥❣ ❣û✐ t✐➲♥ ❧➔✿ 153 318 15 + 168649, 9 65 = 155 00464, 9 65 ỗ số ỗ ỗ ỗ 150 00000 : 100 ì 1, = 1 650 00 ✷✷ ❙✐❣♠❛ ✲ ▼❆❚❍❙ ❇➔✐ ✹✸✳