108 bài toán chọn lọc lớp 6

❚❙✳ ◆❣✉②➵♥ ❱➠♥ ▲đ✐ ✭❝❤õ ❜✐➯♥✮ − ◆❣ỉ ❚❤à ◆❤➣

❉

r❛
❢t

✶✵✽ ❇⑨■ ❚❖⑩◆ ❈❍➴◆ ▲➴❈ ▲❰P ✻


❙✐❣♠❛ ✲ ▼❆❚❍❙

✶


▼Ư❈ ▲Ư❈

❙✐❣♠❛ ✲ ▼❆❚❍❙

▼ư❝ ❧ư❝

✶ ❙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✳
❜✮ ❚➟♣ ❤đ♣ ❉ ❝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

❝→❝ ỳ số t ữủ ởt số ợ ọ ❝â t➜t ❝↔ ❜❛♦ ♥❤✐➯✉ sè ❝â ✸ ❝❤ú
sè ♥❤÷ ✈➟②❄ ✭❑➸ ❝↔ sè ❜❛♥ ✤➛✉✮✳
✹✳ ◗✉②➸♥ s→❝❤ ❣✐→♦ ❦❤♦❛ ❚♦→♥ ✻ t➟♣ ♠ët ❝â ✶✸✷ tr❛♥❣✳ ❍❛✐ tr❛♥❣ ✤➛✉ ❦❤ỉ♥❣

✤→♥❤ sè✳ ❍ä✐ ♣❤↔✐ ❞ị♥❣ 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 − 2 = 14✳
❜✮ ❚➟♣ ❤ñ♣ ❇ ❝→❝ sè tü ♥❤✐➯♥ ① ♠➔ x + 5 = 5✳
❝✮ ❚➟♣ ❤ñ♣ ❈ ❝→❝ sè tü ♥❤✐➯♥ ❦❤ỉ♥❣ ✈÷đt q✉→ ✶✵✵✳

✼✳ ❈❤♦ ❆ ❧➔ t➟♣ ❤đ♣ ❝→❝ sè tü ♥❤✐➯♥ ❝❤✐❛ ❤➳t ❝❤♦ ✸ ✈➔ ♥❤ä ❤ì♥ ✸✵❀ ❇ ❧➔ t➟♣ ❤ñ♣

❝→❝ 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♣✳
❝✮ ❉ị♥❣ ❦➼ ❤✐➺✉ ⊂ ✤➸ 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 ❝ư 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 = 7 + 10 + 13 + · · · + 97 + 100✳

❛✮ ❚ê♥❣ tr➯♥ ❝â ❜❛♦ ♥❤✐➯✉ sè ❤↕♥❣❄
❜✮ ❚➻♠ sè ❤↕♥❣ t❤ù ✷✷✳
❝✮ ❚➼♥❤ ❙✳

✶✷✳ ❱✐➳t ❝→❝ t➼❝❤ ❤♦➦❝ tữỡ s ữợ ụ tứ ừ ởt số

25 · 84 ❀

❜✮ 256 · 1253 ❀

❝✮ 6255 : 257

❞✮ 123 · 33 ✳

✶✸✳ ❚➻♠
x ∈ N ❜✐➳t✿
10
x

❛✮ x = 1 ❀
❜✮ 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 = 1 + 31 + 32 + 33 + · · · + 330.

❚➻♠ ❝❤ú sè t➟♥ ❝ò♥❣ ❝õ❛ ❙✱ tø ✤â s✉② r❛ ❙ ❦❤æ♥❣ ♣❤↔✐ ❧➔ sè ❝❤➼♥❤ ♣❤÷ì♥❣✳
✶✻✳ ❚➼♥❤ ❣✐→ trà ❜✐➸✉ t❤ù❝✿

❛✮ (102 + 112 + 122) : (132 + 142)
❜✮ 9! − 8! − 7!16· 822
❝✮ 11 ·(3213· 4· 4· 112 −) 169 ✳
✶✼✳ ❚➻♠ ① ❜✐➳t✿

❛✮ (19x + 2 · 52) : 14 = (13 − 8)2 − 42
✹


❙✐❣♠❛ ✲ ▼❆❚❍❙

❜✮ 2 · 3x = 10 · 312 + 8 · 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✐ ❝ị♥❣ ❝↔ ✹ ❝❤ú sè ✷❀ ✺❀ ✻❀ ✼✱ ✈✐➳t t➜t ❝↔ ❝→❝ sè✿

❛✮ ❈❤✐❛ ❤➳t ❝❤♦ ✹❀
❝✮ ❈❤✐❛ ❤➳t ❝❤♦ ✷✺❀

❜✮ ❈❤✐❛ ❤➳t ❝❤♦ ✽❀
❞✮ ❈❤✐❛ ❤➳t ❝❤♦ ✶✷✺✳

✷✶✳ ❈â ❜❛♦ ♥❤✐➯✉ sè tü ♥❤✐➯♥ ❝â ❜❛ ❝❤ú sè ✈➔ ❝❤✐❛ ❤➳t ❝❤♦ ✸❄
✷✷✳ ❇✐➳t r➡♥❣ A = 717 + 17 · 3 − 1 ❧➔ ♠ët sè ❝❤✐❛ ❤➳t ❝❤♦ ✾✳ ❈â t❤➸ sû ❞ö♥❣ ❦➳t

q✉↔ ♥➔② ✤➸ ❝❤ù♥❣ tä r➡♥❣ B = 718 + 18 · 3 − 1 ❝ô♥❣ ❝❤✐❛ ❤➳t ổ
ú ỵ õ t ự t q✉↔ tê♥❣ q✉→t ❤ì♥✿
❱ỵ✐ ♠å✐ sè tü ♥❤✐➯♥ ♥✱ ♥➳✉

7n + 3n − 1

❝❤✐❛ ❤➳t ❝❤♦ ✾ t❤➻

7n+1 + 3(n + 1) − 1

❝ô♥❣ ❝❤✐❛ ❤➳t ❝❤♦ ✾✳
✭▲♦↕✐ ❜➔✐ t➟♣ ồ s q ợ ữỡ ♣❤→♣ q✉② ♥↕♣ t♦→♥
❤å❝✮✳

✷✸✳ ❛✮ ❈❤♦ ♥ ❧➔ ♠ët sè ❦❤æ♥❣ ❝❤✐❛ ❤➳t ❝❤♦ ✸✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ n2 ❝❤✐❛ ❝❤♦ ✸

❞÷ ✶✳
❜✮ ❈❤♦ ♣ ❧➔ ♠ët sè ♥❣✉②➯♥ tè ❧ỵ♥ ❤ì♥ ✸✳ ❍ä✐ p2 +2003 ❧➔ sè ♥❣✉②➯♥ tè ủ số
ộ số s õ ữợ 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✐ ❝ị♥❣ s→t ❤❛✐ ❜ù❝ t÷í♥❣ ❧✐➯♥ t✐➳♣
❦❤ỉ♥❣ ❜à t t tữợ ợ t ừ ❧➔ ❜❛♦ ♥❤✐➯✉❄ ✣➸ ❧→t
❝➠♥ ♣❤á♥❣ ✤â ❝➛♥ ❜❛♦ ♥❤✐➯✉ ❣↕❝❤❄
✷✽✳ ❈â ✻✹ ♥❣÷í✐ ✤✐ t❤❛♠ q✉❛♥ ❜➡♥❣ ❤❛✐ ❧♦↕✐ ộ ỗ ộ

ỗ t số ữớ ứ ừ số ỗ ọ ộ ❧♦↕✐ ❝â ♠➜② ①❡❄

✷✾✳ ❚➻♠ ❤❛✐ sè tü ♥❤✐➯♥ ❛ ✈➔ ❜ ❜✐➳t t➼❝❤ ❝õ❛ ❝❤ó♥❣ ❧➔ ✷✾✹✵ ✈➔ ❇❈◆◆ ❝õ❛ ❝❤ó♥❣

❧➔ ✷✶✵✳

✸✵✳ ❇❛ ❤å❝ s✐♥❤✱ ♠é✐ ♥❣÷í✐ ♠✉❛ ♠ët út 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 ✈✉ỉ♥❣ ❜➡♥❣ trỗ r ọ ợ
t❤➻ ❝↕♥❤ ❝õ❛ ỉ ✈✉ỉ♥❣ ❧➔ ❧ỵ♥ ♥❤➜t ✈➔ ❜➡♥❣ ❜❛♦ ♥❤✐➯✉❄

✸✷✳ ❚r♦♥❣ ♠ët ❜✉ê✐ ❧✐➯♥ ❤♦❛♥✱ ❇❛♥ tê ❝❤ù❝ ✤➣ ♠✉❛ t➜t ❝↔ ✽✹✵ ❝→✐ ❜→♥❤✱ ✷✸✺✷

❝→✐ ❦➭♦ ✈➔ q qỵt r ỗ qỵt
số t õ ộ qỵt

ố ồ s ừ ởt trữớ ởt số ợ ỡ ỗ ỳ số ộ

❤➔♥❣ ✺ ✤➲✉ ✈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 ❀
❜✮ 2 + 4 + 6 + 8 + 10 + · · · + 100✳
✸✼✳ ❚➻♠ ①✱ ❜✐➳t ✿ ❛✮ (x − 78) · 26 = 0 ❀
✸✽✳ ❚➼♥❤ ♥❤❛♥❤✿

❜✮ 39 · (x − 5) = 39

1 · 5 · 6 + 2 · 10 · 12 + 4 · 20 · 24 + 9 · 45 · 54
1 · 3 · 5 + 2 · 6 · 10 + 4 · 12 · 20 + 9 · 27 · 45

✸✾✳ ❈❤ù♥❣ tä r➡♥❣ ❤❛✐ sè n + 1 ✈➔ 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✐ ♣❤➛♥ t❤÷ð♥❣ ỗ ữ s
ỏ tứ 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❝ ớ t tố ữớ tự
t ợ ỡ tè❝ ♥❣÷í✐ t❤ù ❤❛✐ ❧➔ ✺❦♠✴❤✳ ❚➼♥❤ ✈➟♥ tè❝ ♠é✐ ♥❣÷í✐✳

✹✹✳ ▼ët ❝♦♥ ❝❤â ✤✉ê✐ ♠ët ❝♦♥ t❤ä ❝→❝❤ ♥â ởt ữợ ừ õ

ởt ữợ ừ tọ õ ởt ữợ t tọ ụ
ởt ữợ ọ õ ữợ ợ ờ tọ

ởt ởt rờ trù♥❣ r❛ ❝❤đ✳ ❉å❝ ✤÷í♥❣ ❣➦♣ ♠ët ❜➔ ❦❤→❝ ✈ỉ þ ✤ư♥❣

♣❤↔✐✱ rê trù♥❣ rì✐ ①✉è♥❣ ✤➜t✳ ❇➔ ❦✐❛ tä þ ♠✉è♥ ✤➲♥ ❧↕✐ sè trù♥❣ ❜➧♥ ❤ä✐✿
✲ ❇➔ ❝❤♦ ❜✐➳t tr♦♥❣ rê ❝â ❜❛♦ ♥❤✐➯✉ trù♥❣❄
❇➔ ❝â rê trù♥❣ tr↔ ❧í✐✿
✲ ❚ỉ✐ ❝❤➾ ♥❤ỵ r➡♥❣ sè trù♥❣ ✤â ❝❤✐❛ ❝❤♦ ✷✱ ❝❤♦ ✸✱ ❝❤♦ ✹✱ ❝❤♦ ✺✱ ❝❤♦ ✻✱ ❧➛♥ ♥➔♦
❝ơ♥❣ ❝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➯♥ trö❝ 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| > 0 ❀
❝✮ ❱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❛ ❧✉ỉ♥ ❝â✿

❛✮ |a| ≥ 0 ✿ ●✐→ 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 ❝ơ♥❣ ❧➔ ♠ët sè ➙♠❄ ❇➔✐ t♦→♥ ❝á♥
✤ó♥❣ ❦❤æ♥❣ ♥➳✉ t❤❛② ✶✽ sè ❜ð✐ ✶✾ sè❄
✺✽✳ ❚➻♠ sè ♥❣✉②➯♥ ①✱ ❜✐➳t✿

❛✮ x + 15 = 7
❜✮ x − 5 = −8
❝✮ 12 + (4 − x) = −5
❞✮ |x| − 6 = 5 ❀
❡✮ |x − 3| = 4
✺✾✳ ❚➻♠ ❝→❝ sè ♥❣✉②➯♥ ① s❛♦ ❝❤♦✿
❛✮ |x| = x ❀
❜✮ |x| > x ❀
❝✮ |x| + x = 0 ❀
❞✮ x + 5 = |x| − 5✳
✻✵✳ ❈❤♦ ❞➣② sè 1; −2; 3; −4; 5; −6; 7; −8; 9; 10 ồ r số rỗ t

❞➜✉ ✧✲✧ ❣✐ú❛ ❝→❝ sè ➜②✳ ❚➼♥❤ ❣✐→ trà ❧ỵ♥ t tr 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) = 5
❜✮ 30(x + 2) − 6(x − 5) − 24x = 100✳
❝✮ x(x + 3) = 0
❞✮ (x − 2)(5 − x) = 0
2
❡✮ (x − 1)(x + 1) = 0
❢✮ (x + 3)(x − 4) = 0
✻✺✳ ❚➻♠ x ∈ Z ❜✐➳t✿
❛✮ |2x − 5| = 13
❜✮ |7x + 3| = 66
❝✮ |5x − 2| ≤ 13

❞✮ (x + 1) + (x + 3) + (x + 5) + · · · + (x + 99) = 0
❡✮ (x − 3) + (x − 2) + (x − 1) + · · · + 10 + 11 = 11 ✭sè ❤↕♥❣ ✤➛✉ t✐➯♥ ✤÷đ❝ ✈✐➳t
❧➔ x − 3 ✈➔ ❦➳t t❤ó❝ ❞➣② ❧➔ sè ✶✶✮✳
✻✻✳ ❚❤ü❝ ❤✐➺♥ ♣❤➨♣ t➼♥❤ ♠ët ❝→❝❤ ❤ñ♣ ❧➼✿

❛✮ (−125) · (+25) · (−32) · (−14) ❀
❜✮ (−159)(+56) + (+43) · (−159) + (−159) ❀
❝✮ (−31) · (+52) + (−26) · (−162)✳
✻✼✳ ❈❤♦ S = 1 − 3 + 32 − 33 + · · · + 398 − 399✳

❛✮ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❙ ❧➔ ❜ë✐ ❝õ❛ ✲✷✵✳
❜✮ ❚➼♥❤ ❙✱ tø ✤â s✉② r❛ 3100 ❝❤✐❛ ❝❤♦ ✹ ❞÷ ✶✳

✻✽✳ ❚➻♠ sè ♥❣✉②➯♥ ❞÷ì♥❣ s n + 2 ữợ ừ ❝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
· 3 · 5 · · · (2n − 1)
1
❜✮ (n +11)(n
= n ✈ỵ✐ n ∈ N ∗
+ 2)(n + 3) · · · 2n 2

✼✶✳ ❚➻♠ ❝→❝ ❣✐→ tr ừ s

11
x
3
<
<

12
12
4

ỗ rỗ s s→♥❤ ❝→❝ ♣❤➙♥ sè s❛✉✿
−789
✈➔
❀
❛✮ −8
31
3131

❜✮ 22.3114.52 ✈➔ 22.3294.53 ❀

❝✮ n1 ✈➔ n +1 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 +

−1
19
x
−1 58 59 −1
+
<
+
<
+
+
.
60
120 36
60
90 72
60

ữỡ ủ ỵ t
31
7
8
23
( + )❀
32 23

12 13
79 28
+ ) − ( − )❀
❜✮ ( 13 + 67
41

67 41
8
17
3
❝✮ 38
−( −
− )
45
45 51 11

tờ s ữỡ ủ ỵ ♥❤➜t✿
1
1
1
1
❛✮ A = 1.2
+
+
+ ... +
2.3 3.4
49.50

2
2
2
2
❜✮ B = 3.5
+
+
+ ... +

5.7 7.9
37.39
3
3
3
3
❝✮ C = 4.7
+
+
+ ... +
7.10 10.13
73.76

✶✶




ủ ỵ

1 10 1
175 . −31
. . . 3❀
125 2 17 2

❜✮
❝✮

11 −5 4 11 8
.

− .
. ❀
4 9
9 4 33
17 18 19 20
−5 1 1
+
−
−
.
+ +
28 29 30 31
12
4 6

✳

✼✽✳ ❚➼♥❤ ♥❤❛♥❤✿
3
3
3
3
−
+
−
3
+
24.47 − 23
7 10 1001 13
.

9
9
9
9
24 + 47.23
−
+ −
+9
1001 13 7 10

✳
✼✾✳ ❚➻♠ ①✿

❛✮ 13 + 23 x = 41 ❀
❜✮ 34 + 14 : x = −1❀
5
3
2
5 +x−7
: 16 = 0❀
8
24
3
2 2
2
2
1989
+ +
+ ... +
=1

✳
2 6 12
x(x + 1)
1991

❝✮ 1 −
❞✮

✽✵✳ ▼ët ❝❛ ♥æ ①✉æ✐ ❞á♥❣ sæ♥❣ tø ❆ ✤➳♥ tr ớ rỗ ữủ ỏ tr

♠➜t 4 12 ❣✐í✳ ◆➳✉ ♠ët ❜➧ ♥ù❛ trỉ✐ tü ❞♦ ①✉ỉ✐ ❞á♥❣ t❤➻ ♠➜t ❜❛♦ ♥❤✐➯✉ t❤í✐
❣✐❛♥ ✤➸ trỉ✐ tø ❆ ✤➳♥ ❇❄
✽✶✳ ❚➻♠ ① tr♦♥❣ ❝→❝ ❤é♥ sè✿

❛✮ 2 x7 = 75
❀
35

❜✮ 4 x3 = 47
❀
x

x
112
❝✮ x 15
=
✳
5

✶✷

❙✐❣♠❛ ✲ ▼❆❚❍❙

✽✷✳ ❚➼♥❤✿

❛✮

5 4
1
1
2 +
: 10 − 9
❀
6 9
12
2
❜✮ 1 185 − 185 151 + 1 121 ❀
❝✮ − 71 . 9 12 − 8, 75 : 72 + 0, 625 : 1 23

✽✸✳ ❚➻♠ ① ❜✐➳t✿

❛✮ 7, 5x :

9−6

13
13
=2
21

25

− x).5, 25
❜✮ (1, 16
5
1
2

= 75%

10 − 7 .2
9
4 17

✽✹✳ r ố ồ s ợ ừ 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 ♠æ♥ t❤➸ t❤❛♦✱ ❜✐➳t sè ❤å❝ s✐♥❤ ❝õ❛ ❦❤è✐ ✾ ❧➔ ✷✷✺✳

✽✺✳ ◆➠♠ ♥❛② ❝♦♥ ✶✷ t✉ê✐✱ ❜è ✹✷ t✉ê✐✳ ❚➼♥❤ t➾ sè ❣✐ú❛ t✉ê✐ ❝♦♥ ✈➔ t✉ê✐ ❜è

ỳ tớ trữợ s ✤➙② ✷✽ ♥➠♠✳

✽✻✳ ❇❛ ♥❣÷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✳

✶✸

1
2






ồ

r ữợ õ ✤÷í♥❣ 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❝ ✷✽✳ ❍ä✐ õ
trữợ
s❛♦ ❝❤♦ ❉ ♥➡♠ ❣✐ú❛ ❈ ✈➔ ❊ ❝á♥ ❊ ♥➡♠ ❣✐ú❛ ❉ ✈➔ ❋✳

❛✮ ❱➻ s❛♦ ❝â t❤➸ ❦❤➥♥❣ ✤à♥❤ ✹ ✤✐➸♠ ❈✱ ❉✱ ❊✱ ❋ t❤➥♥❣ ❤➔♥❣
❜✮ ❑➸ t➯♥ ❤❛✐ t✐❛ trò♥❣ ♥❤❛✉ ❣è❝ ❊✳

❝✮ ❱➻ s❛♦ ❝â t❤➸ ❦❤➥♥❣ ✤à♥❤ ✤✐➸♠ ❊ ♥➡♠ ❣✐ú❛ ❈ ✈➔ ❋✳

✶✹




ữợ rỗ tr ớ ❝→❝ ❝➙✉ ❤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 ❦➲ ❜ị ✈ỵ✐ ❣â❝ BOA✳
❛✮ ❚r♦♥❣ ❜❛ ✤✐➸♠ ❆✱❖✱❈ ✤✐➸♠ ♥➔♦ ♥➡♠ ❣✐ú❛ ❤❛✐ ✤✐➸♠ ❝á♥ ❧↕✐❄
❜✮ ❱➩ ❝→❝ t✐❛ ❇❆✱❇❈ ❤ä✐ ✤✐➸♠ ❖ ♥➡♠ tr♦♥❣ ❣â❝ ♥➔♦❄
❝✮ ❑➸ t➯♥ ❝→❝ ❝➦♣ ❣â❝ ❦➲ ❜ò ✤➾♥❤ ❇✳

✶✺


❙✐❣♠❛ ✲ ▼❆❚❍❙

✾✺✳ ❈❤♦ ❣â❝ ❜➭t xOy✳ ❱➩ ❤❛✐ t✐❛ ❖♠✱ ❖♥ tr➯♥ ❝ị♥❣ ♠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 ❤➻♥❤ ỳ t t ữợ ❣â❝ ❛✳

✶✻


❙✐❣♠❛ ✲ ▼❆❚❍❙

✶✵✷✳ ❆❇❈❉ ❧➔ ♠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 = 7 + 10 + 13 + · · · + 97 + 100✳

❛✮ ❚ê♥❣ tr➯♥ ❝â ❜❛♦ ♥❤✐➯✉ sè ❤↕♥❣❄
❜✮ ❚➻♠ sè ❤↕♥❣ t❤ù ✷✷✳
❝✮ ❚➼♥❤ ❙✳
❇➔✐ ❣✐↔✐✿

❛✮ ❙è sè ❤↕♥❣ ❝õ❛ tê♥❣ ❧➔ ✿ 1003− 7 + 1 = 32 ✭sè ❤↕♥❣✮✳
❜✮ ●å✐ sè ❤↕♥❣ t❤ù ✷✷ ❧➔ ①✱ t❛ ❝â✿ x −3 7 + 1 = 22 ⇒ x = 70✳
❝✮ S = 7 + 10 + 13 + · · · + 97 + 100
❤❛② S = 100 + 97 + 94 + · · · + 10 + 7 ✭❣✐❛♦ ❤♦→♥✮
⇔ 2S = (7 + 100) + (10 + 97) + · · · + (100 + 7) ✭✈ỵ✐ ✸✷ ♥❤â♠✳✮
⇔ 2S = (7 + 100) · 32
(7 + 100) · 32
= 1712✳
⇔S=
2

❇➔✐ ✷✷✿ ❇✐➳t r➡♥❣ A = 717 + 17 · 3 − 1 ❧➔ ♠ët sè ❝❤✐❛ ❤➳t ❝❤♦ ✾✳ ❈â t❤➸ sû ❞ö♥❣

❦➳t q✉↔ ♥➔② ✤➸ ❝❤ù♥❣ tä r➡♥❣ B = 718 + 18 · 3 − 1 ❝ô♥❣ ❝❤✐❛ ❤➳t ❝❤♦ ✾ ❦❤æ♥❣❄
❇➔✐ ❣✐↔✐✿

❚❤❡♦ ✤➛✉ ❜➔✐ A = 717 + 17 · 3 − 1 ❧➔ ♠ët sè tü ♥❤✐➯♥ ❝❤✐❛ ❤➳t ❝❤♦ ✾ tù❝ ❧➔ t❛ ❝â
(717 + 50) | 9✳ ❚❛ ✈✐➳t sè ❇ ♥❤÷ s❛✉✿
B = 718 + 18 · 3 − 1 = 718 + 53 = 7(717 + 50) − 297 = 7 · (717 + 50) − 33 · 9✳
❱➻ (717 + 50) | 9 ✈➔ (33 · 9) | 9 ♥➯♥ B | 9
ú ỵ õ t ự t q✉↔ tê♥❣ q✉→t ❤ì♥✿
❱ỵ✐ ♠å✐ sè tü ♥❤✐➯♥ ♥✱ ♥➳✉

7n + 3n − 1

❝❤✐❛ ❤➳t ❝❤♦ ✾ t❤➻

7n+1 + 3(n + 1) − 1

❝ô♥❣ ❝❤✐❛ ❤➳t ❝❤♦ ✾✳
✭▲♦↕✐ ❜➔✐ t➟♣ ồ s q ợ ữỡ ♣❤→♣ q✉② ♥↕♣ t♦→♥
❤å❝✮✳

❇➔✐ ✸✷✿ ❚r♦♥❣ ♠ët ❜✉ê✐ ❧✐➯♥ ❤♦❛♥✱ ❇❛♥ tê ❝❤ù❝ ✤➣ ♠✉❛ t➜t ❝↔ ✽✹✵ ❝→✐ ❜→♥❤✱

✷✸✺✷ q qỵt r ỗ qỵt
số ♥❤✐➲✉ ♥❤➜t ♣❤↔✐ ❝â ♠é✐ ✤➽❛ ❜❛♦ ♥❤✐➯✉ ❜→♥❤✱ ❦➭♦✱ qỵt




❙✐❣♠❛ ✲ ▼❆❚❍❙

❙è ✤➽❛ ♥❤✐➲✉ ♥❤➜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 út ❜✐✱ ✶✼✵ t➟♣ ❣✐➜②✳ ◆❣÷í✐ t❛ ❝❤✐❛ ✈ð✱ ❜ót ❜✐✱

t tữ ộ tữ ỗ ❝↔ ❜❛ ❧♦↕✐✳ ◆❤÷♥❣ 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
sè ♣❤➛♥ t❤÷ð♥❣ ❧➔ ✷✹✳
❇➔✐ ✹✺✿ ▼ët ❜➔ ♠❛♥❣ ♠ët rê trù♥❣ r❛ ❝❤đ✳ ❉å❝ ✤÷í♥❣ ❣➦♣ ♠ët ❜➔ ❦❤→❝ ổ ỵ

ử rờ trự rỡ ố t tọ ỵ ố số trự ọ
❝❤♦ ❜✐➳t tr♦♥❣ rê ❝â ❜❛♦ ♥❤✐➯✉ trù♥❣❄
❇➔ ❝â rê trù♥❣ tr↔ ❧í✐✿
✲ ❚ỉ✐ ❝❤➾ ♥❤ỵ r➡♥❣ sè trù♥❣ ✤â ❝❤✐❛ ❝❤♦ ✷✱ ❝❤♦ ✸✱ ❝❤♦ ✹✱ ❝❤♦ ✺✱ ❝❤♦ ✻✱ ❧➛♥ ♥➔♦
❝ơ♥❣ ❝á♥ t❤ø❛ r❛ ♠ët q✉↔✱ ♥❤÷♥❣ ❝❤✐❛ ❝❤♦ ✼ t❤➻ ❦❤æ♥❣ t❤ø❛ q✉↔ ♥➔♦✳ ⑨✱ ♠➔ sè
trù♥❣ ❝❤÷❛ ✤➳♥ ✹✵✵ q✉↔✳
❚➼♥❤ ①❡♠ tr♦♥❣ rê ❝â ❜❛♦ ♥❤✐➯✉ trù♥❣❄
❇➔✐ ❣✐↔✐✿

●å✐ sè trù♥❣ ❧➔ ❛✳ ❚❛ ❝â a − 1 ❧➔ ❜ë✐ ❝❤✉♥❣ ❝õ❛ 2, 3, 4, 5, 6 ✈➔ a − 1 < 399✳ ❚ø
✤â t❛ ✤÷đ❝✿
a − 1 ∈ {60; 120; 180; 240; 300; 360}
a ∈ {61; 121; 181; 241; 301; 361}
❉♦ a | 7 ♥➯♥ ❛ ❂ ✸✵✶✳ ❘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) = 0
❡✮ (x − 3) + (x − 2) + (x − 1) + · · · + 10 + 11 = 11
❇➔✐ ❣✐↔✐✿

❛✮ |2x − 5| = 13✳ ❳➨t ✷ tr÷í♥❣ ❤đ♣✿ 2x − 5 = 13❀ 2x − 5 = −13✳
❱➟② ① ❂ ✾❀ ① ❂ ✲✹✳
❜✮ |7x + 3| = 66 s✉② r❛ ① ❂ ✾✳
❝✮ |5x − 2| ≤ 13 ♥➯♥
−13 ≤ 5x − 2 ≤ 13
−11 ≤ 5x ≤ 15
−2 ≤ x ≤ 3 ⇔ x ∈ {−2; −1; 0; 1; 2; 3}

❞✮ (x + 1) + (x + 3) + (x + 5) + · · · + (x + 99) = 0 s✉② r❛

[(x + 1) + (x + 99)] · 50
= (x + 50) · 50 = 0❀ x + 50 = 0 ⇔ x = −50✳
2
❡✮ (x − 3) + (x − 2) + (x − 1) + · · · + 10 + 11 = 11✳
❇ä sè ❤↕♥❣ ✶✶ ð ❤❛✐ ✈➳ t❛ ✤÷đ❝✿ (x − 3) + (x − 2) + (x − 1) + · · · + 10 = 0
●å✐ sè sè ❤↕♥❣ ð ✈➳ tr→✐ ❧➔ ♥✭♥ ❃ ✵✮ t❛ ❝â✿
[(x − 3) + 10] · n
= 0 ❤❛② (x + 7) · n = 0✳
2
❱➻ n = 0 ♥➯♥ x + 7 = 0✱ ❞♦ ✤â x = −7✳
❇➔✐ ✼✶✿ ❚➻♠ ❝→❝ ❣✐→ trà ❝õ❛ ①✱ s❛♦ ❝❤♦✿

−11
x
−3
<
<
✳
12
12
4

❇➔✐ ❣✐↔✐✿

−9
❈❤å♥ ♠➝✉ sè ❝❤✉♥❣ ❧➔ ✶✷ t❛ ❝â✿ −3
=
✳
4
12

✷✶


❙✐❣♠❛ ✲ ▼❆❚❍❙

x
−9
❉♦ ✤â t❛ ❝â✿ −11
<
<
✱ s✉② r❛ −11 < x < −9✳ ❱➟② x = −10✳
12
12
12

❇➔✐ ✽✵✿ ▼ët ❝❛ ♥æ ①✉æ✐ ❞á♥❣ sæ♥❣ tø ❆ ✤➳♥ ❇ tr♦♥❣ ớ rỗ ữủ ỏ

tr t 4 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♥❣✿ 1 : 4 12 = 29 ✳
❍✐➺✉ 13 − 29 = 19 ❝❤➼♥❤ ❧➔ tố ừ ữợ trổ tỹ
tố ữợ trổ tỹ 19 : 2 = 181
ớ ữợ trổ tỹ tứ ❧➔✿ 1 : 181 = 18 ✭❣✐í✮✳
❇➔✐ ✽✻✿ ❇❛ ♥❣÷í✐ ❝❤✉♥❣ ♥❤❛✉ ♠✉❛ ❤➳t ♠ët rê trù♥❣✳ ◆❣÷í✐ t❤ù ♥❤➜t ♠✉❛

1
2

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 ❜❛ ♠✉❛✿ 1 − 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) : 2 = 600 − x
⇔ ADE = AED = (1800 − 2x) : 2 = 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
0
B = 80
C = 1800 − (2 + 2)
1800 − 2 × 1 + 800 − 1800 − 2 × 2 = 1800

✷✸


❙✐❣♠❛ ✲ ▼❆❚❍❙

4400 − 2 × (1 + 2)
2600
(1 + 2)
EDF + 1 + 2
EP F

= 1800
= 2 × (1 + 2)
= 1300
❂ 1800
= 500 ✳

❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ sü q✉❛♥ t➙♠ ❝õ❛ ❜↕♥ ✤å❝✦

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ∗∗∗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❍➌❚ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ∗∗∗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✹