Toan bien doi dai so lop 9 chuyen

❚❖⑩◆ ❇■➌◆ ✣✃■ ✣❸■ ❙➮
❇➔✐ ✶ ✭❍❙● ❍➔ ◆ë✐ ✷✵✶✽✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥
a + b + c = 2018

✈➔

1
1
2017
1
+
+
=
a + b b + c c + a 2018

❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ P = b +a c + c +b a + a +c b ✳
√
❇➔✐ ✷✳ ❈❤♦ a, b, c > 0 t❤ä❛ ♠➣♥✿ a + b + c + abc = 4✳ ❚➼♥❤ ❣✐→ trà ❝õ❛
❜✐➸✉ t❤ù❝
Q=

a (4 − b) (4 − c) +

b (4 − c) (4 − a) +

❇➔✐ ✸✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥✿

c (4 − a) (4 − b)

a+b+c=0
a3 + b3 + c3 = 0

✳ ❈❤ù♥❣ ♠✐♥❤

a2019 + b2019 + c2019 = 0

❇➔✐ ✹ ✭❍❙● ◆❛♠ 2✣à♥❤ ✷✵✶✷✮✳
❈❤♦ ❝→❝ sè t❤ü❝ a, b, c ✤æ✐ ♠ët ❦❤→❝
2
2
♥❤❛✉ ✈➔ t❤ä❛ ♠➣♥✿ a

− b = b − c = c − a✳

❈❤ù♥❣ ♠✐♥❤ r➡♥❣

(a + b + 1) (b + c + 1) (c + a + 1) = −1

❇➔✐ ✺ ✭❍❙● ❍➔ ❚➽♥❤ ✷✵✶✹✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥
a2 + b2 + c2 = a3 + b3 + c3 = 1

❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝
M = a2012 + b2013 + c2014

❇➔✐ ✻✳ ❈❤♦ ❝→❝ sè t❤ü❝ x, y, z t❤ä❛ ♠➣♥
x2 + 2y + 1 = y 2 + 2z + 1 = z 2 + 2x + 1 = 0

❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ A = x2019 + y2019 + z 2019✳
❇➔✐ ✼ ✭❙P ✷✵✶✷✱ ✈á♥❣ ✷✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c ✤æ✐ ♠ët ♣❤➙♥ ❜✐➺t ✈➔
t❤ä❛ ♠➣♥✿
2

2
a (b + c) = b (c + a) = 2012

❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ A = c2 (a + b)✳
❇➔✐ ✽✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c ✤æ✐ ♠ët ❦❤→❝ ♥❤❛✉ ✈➔ ❤❛✐ sè t❤ü❝ x, y t❤ä❛
♠➣♥✿
3
3
3
a + ax + y = 0, b + bx + y = 0, c + cx + y = 0
✶


❈❤ù♥❣ ♠✐♥❤✿ a + b + c = 0✳

❇➔✐ ✾ ✭❙P ✷✵✶✼✱ ✈á♥❣ ✶✮✳ ●✐↔ sû x, y ❧➔ ❤❛✐ sè t❤ü❝ ♣❤➙♥ ❜✐➺t t❤ä❛ ♠➣♥✿
1
1
2
+
=
✳
x2 + 1 y 2 + 1 xy + 1
S=

❍➣② t➼♥❤

x2

1

1
2
+ 2
+
+ 1 y + 1 xy + 1

❇➔✐ ✶✵ ✭❙P ✷✵✶✺✱ ✈á♥❣ ✷✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ x, y t❤ä❛ ♠➣♥
0 < x < 1, 0 < y < 1

✈➔

x
y
+
=1
1−x 1−y

❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ P = x + y + x2 − xy + y2✳
❇➔✐ ✶✶ ✭❙P ✷✵✶✽✱ ✈á♥❣ ✶✮✳ ❈❤♦ ♣❤÷ì♥❣ tr➻♥❤✿ x3 − x − 1 = 0✳
●✐↔ sû x0 ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ t➻♥❤ ✤➣ ❝❤♦✳
❛✮ ❈❤ù♥❣ ♠✐♥❤ x0 > 0✳
x20 − 1
❜✮ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ M = x3 . 2x20 + 3x0 + 2✳
0
❇➔✐ ✶✷2 ✭❑❍❚◆2 ✷✵✶✺✱ ✈á♥❣ ✶✮✳ ●✐↔ sû a, b ❧➔ ❤❛✐ sè t❤ü❝ ♣❤➙♥ ❜✐➺t t❤ä❛
♠➣♥ a + 3a = b + 3b = 2✳ ❈❤ù♥❣
♠✐♥❤
r➡♥❣
3
3

❛✮ a + b = −3
❜✮ a + b = −45
❇➔✐ ✶✸ ✭❝❤✉②➯♥ ♥❣ú ✷✵✵✺✮✳❈❤♦ ❝→❝ sè t❤ü❝ x, y, z = 0 t❤ä❛ ♠➣♥✿
(x − y)2 +(y − z)2 +(z − x)2 = (x + y − 2z)2 +(y + z − 2x)2 +(z + x − 2y)2

❈❤ù♥❣ ♠✐♥❤ r➡♥❣ x = y = z ✳

❇➔✐ ✶✹ ✭❑❍❚◆ ✷✵✶✺✱ ✈á♥❣ ✷✮✳ ❱ỵ✐ a, b, c ❧➔ ❝→❝ sè t❤ü❝ t❤ä❛ ♠➣♥
(3a + 3b + 3c)3 = 24 + (3a + b − c)3 + (3b + c − a)3 + (3c + a − b)3

❈❤ù♥❣ ♠✐♥❤ r➡♥❣✿ (a + 2b)(b + 2c)(c + 2a) = 1✳
❇➔✐ ✶✺ ✭❑❍❚◆ ✷✵✶✼✱ ✈á♥❣ ✶✮✳ ✈ỵ✐ a, b ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ t❤ä❛ ♠➣♥✿
ab + a + b = 1✱ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣
a
b
+
=
1 + a2 1 + b2

1 + ab
2 (1 + a2 ) (1 + b2 )

❇➔✐ ✶✻ ✭❆♠s ✷✵✶✻✱
❝❤✉②➯♥
❚✐♥✮ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥
3
3
3
a+b+c=0

✈➔ a

+ b + c = 3abc✳
P =

❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝

a2
b2
c2
+
+
b2 + c2 c2 + a2 a2 + b2
✷


❇➔✐ ✶✼ ✭❆♠s ✷✵✶✻✱
❝❤✉②➯♥
❚♦→♥✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c ✤æ✐ ♠ët ❦❤→❝
3
3
3
♥❤❛✉ t❤ä❛ ♠➣♥✿ a
t❤ù❝

+ b + c = 3abc

✈➔ a, b, c = 0✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉

ab2

bc2
ca2
P = 2
+
+
a + b2 − c2 b2 + c2 − a2 c2 + a2 − b2
❚➻♠ t➜t ❝↔ ❝→❝ sè t❤ü❝ x, y t❤ä❛

❇➔✐ ✶✽ ✭❙P ✷✵✶✻✱ ✈á♥❣ ✷✮✳

√
x2 − 4 y 2 − 4
x−1+
+
+8=4
x
y

♠➣♥

y−1

❇➔✐ ✶✾ ✭❆♠s ✷✵✶✽✱ ❝❤✉②➯♥ ❚♦→♥✮✳ ❱ỵ✐ x, y, z ❧➔ ❝→❝ sè t❤ü❝ t❤ä❛ ♠➣♥
xyz = 1

✈➔

(xy + x + 1)(yz + y + 1)(zx + z + 1) = 0

❈❤ù♥❣ ♠✐♥❤ r➡♥❣

1
1
1
+
+
=1
xy + x + 1 yz + y + 1 zx + z + 1

❇➔✐ ✷✵ ✭❑❍❚◆ ✷✵✶✹✱ ✈á♥❣ ✶✮✳ ●✐↔ sû x, y, z ❧➔ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ t❤ä❛
♠➣♥ ✤✐➲✉ ❦✐➺♥ x + y + z = xyz ✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣

x
2y
3z
xyz. (5x + 4y + 3z)
+
+
=
1 + x2 1 + y 2 1 + z 2
(x + y) (y + z) (z + x)

❇➔✐ ✷✶ ✭❑❍❚◆ ✷✵✶✹✱ ✈á♥❣ ✷✮✳ ●✐↔ sû x, y ❧➔ ♥❤ú♥❣ sè t❤ü❝ ❞÷ì♥❣
♣❤➙♥ ❜✐➺t t❤ä❛ ♠➣♥

2y 2
4y 4
8y 8
y
+
+

+
=4
x + y x2 + y 2 x4 + y 4 x8 − y 8

❈❤ù♥❣ ♠✐♥❤ r➡♥❣✿ 5y = 4x✳
❇➔✐ ✷✷✳ ❈❤♦ ❝→❝ sè t❤ü❝ x, y = 0✳ ✣➦t a = x + x1 , b = y + y1 , c = z + z1 ✳
❈❤ù♥❣ ♠✐♥❤ r➡♥❣ a2 + b2 + c2 − abc = 4✳
❇➔✐ ✷✸ ✭❙P ✷✵✵✻✱ ✈á♥❣ ✶✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c = 0 ✈➔ ✤æ✐ ♠ët ❦❤→❝
♥❤❛✉ t❤ä❛ ♠➣♥✿ a + b + c = 0✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣
b
c
a
+
+
b−c c−a a−b

b−c c−a a−b
+
+
a
b
c

=9

❇➔✐ ✷✸ ✭❙P ✷✵✶✸✱ ✈á♥❣ ✷✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c tọ ỗ tớ
tự

✐✮ (a + b)(b + c)(c + a) = abc✳
✐✐✮ a3 + b3 b3 + c3 c3 + a3
❈❤ù♥❣ ♠✐♥❤ abc = 0✳

= a3 b 3 c 3 ✳

❇➔✐ ✷✹ ✭❙P ✷✵✶✹✱ ✈á♥❣ ✷✮✳ ●✐↔ sû a, b, c, x, y, z ❧➔ ❝→❝ sè t❤ü❝ ❦❤→❝ 0
t❤ä❛ ♠➣♥

❈❤ù♥❣ ♠✐♥❤ r➡♥❣

a b c
+ + = 0 ✈➔
x y z
x2 y 2 z 2
+
+
= 1✳
a2 b 2 c 2

x y z
+ + =1
a b c

❇➔✐ ✷✺ ✭❑❍❚◆ ✷✵✶✸✱ ✈á♥❣ ✶✮✳ ●✐↔ sû a, b, c ❧➔ ❝→❝ sè t❤ü❝ ❦❤→❝ 0 t❤ä❛
♠➣♥✿ (a + b) (b + c) (c + a) = 8abc✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣

a
b

c
3
ab
bc
ca
+
+
= +
+
+
a + b b + c c + a 4 (a + b) (b + c) (b + c) (c + a) (c + a) (a + b)

❇➔✐ ✷✻ ✭❍❙● ◆❛♠ ✣à♥❤ ✷✵✶✻✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ x, y, z t❤ä❛ ♠➣♥
x + y + z = 2, x2 + y 2 + z 2 = 18, xyz = −1

❚➼♥❤ S = xy +1z − 1 + yz +1x − 1 + zx + y 1− 1 − 1 ✳
❇➔✐ ✷✼ ✭t❤✐ t❤û ❑❍❚◆ ✷✵✶✾✱ ✈á♥❣ ✶✮✳ ❱ỵ✐ a, b, c ❧➔ ♥❤ú♥❣ sè t❤ü❝✱
❝❤ù♥❣ ♠✐♥❤ r➡♥❣
2b + c
b + 2c
2c + a
c + 2a
2a + b
a + 2b
.
+
.
+
.
a−b

b−c
b−c
c−a
c−a
a−b

= −3

❇➔✐ ✷✽ ✭t❤✐ t❤û ❑❍❚◆ ✷✵✶✾✱ ỏ ợ a, b số tỹ ữỡ
tọ ♠➣♥ a + b = 1✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣

a
b
2 (ab − 2)
+
=
b3 − 1 a3 − 1
a2 b 2 + 3

❇➔✐ ✷✾ ✭t❤✐ t❤û ❑❍❚◆ ✷✵✶✾✱ ✈á♥❣ ✶✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c ✤æ✐ ♠ët
❦❤→❝ ♥❤❛✉✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣

2a + b 2b + c 2c + a
(2a + b) (2b + c) (2b + c) (2c + a) (2c + a) (2a + b)
+
+
= 3+
+
+
a−b b−c c−a

(a − b) (b − c)
(b − c) (c − a)
(c − a) (a − b)

❇➔✐ ✸✵ ✭t❤✐ t❤û ❑❍❚◆ ✷✵✶✾✱ ✈á♥❣ ✷✮✳ ❱ỵ✐ a, b, c ❧➔ ❝→❝ sè t❤ü❝ t❤ä❛
♠➣♥ a + b + c = abc✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣
a2 − 1 b2 − 1 c2 − 1
+
+
=1+
a2 + 1 b2 + 1 c2 + 1
✹

4
(1 + a2 ) (1 + b2 ) (1 + c2 )


❇➔✐ ✸✶ ✭t❤✐ t❤û ❑❍❚◆ ✷✵✶✽✱ ✈á♥❣ ✶✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c ✤æ✐ ♠ët
❦❤→❝ ♥❤❛✉ t❤ä❛ ♠➣♥ (1 − ab) (1 − bc) (1 − ca) = 0✳ ❈❤ù♥❣ ♠✐♥❤ ❜✐➸✉ t❤ù❝

1 + ab b + c b + c ac + 1 ac + 1 1 + ab
.
+
.
+
.
1 − ab b − c b − c ac − 1 ac − 1 1 − ab
trà ❧➔ sè ♥❣✉②➯♥ ✈ỵ✐ ♠å✐ a, b, c✳
❈❤♦ ❝→❝ sè t❤ü❝ a, b, c ✤æ✐ ♠ët

M=

❧✉æ♥ ♥❤➟♥ ❣✐→

❇➔✐ ✸✷ ✭t❤✐ t❤û ❑❍❚◆ ✷✵✶✽✱ ✈á♥❣ ✷✮✳

❦❤→❝ ♥❤❛✉✳ ❈❤ù♥❣ ♠✐♥❤ ❜✐➸✉ t❤ù❝

c
b
a
c
b
a
.
+
.
+
.
a−b c−b b−c a−c c−a b−a
❧➔ sè ♥❣✉②➯♥ ✈ỵ✐ ♠å✐ a, b, c✳

Q=

❧✉æ♥ ♥❤➟♥ ❣✐→ trà

❇➔✐ ✸✸ ✭❍❙● ❚P ❍❈▼ ✷✵✶✻✮✳ ❈❤♦ ❤❛✐ sè t❤ü❝ ♣❤➙♥ ❜✐➺t a, b t❤ä❛
♠➣♥✿ ab = a − b✳ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝

a b

+ − ab
b a
a, b, c, x, y, z t❤ä❛ ♠➣♥
M=

❇➔✐ ✸✹✳ ❈❤♦ ❝→❝ sè t❤ü❝
❈❤ù♥❣

x = by + cz, y = ax + cz, z = ax + by
♠✐♥❤ 1 +1 a + 1 +1 b + 1 +1 c = 2✳
❈❤♦ ❝→❝ sè t❤ü❝ a, b, c ✈➔ x, y, z = 0 t❤ä❛ ♠➣♥
b
c
a
= =
a + b + c = a2 + b2 + c2 = 1 ✈➔
x y
z

❇➔✐ ✸✺✳

❈❤ù♥❣ ♠✐♥❤✿ xy + yz + zx = 1✳

❇➔✐ ✸✻ ✭❍❙● ❍➔ ◆ë✐ ✷✵✶✹✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c = 0 t❤ä❛ ♠➣♥
✈➔ a + b + c = a1 + 1b + 1c
❈❤ù♥❣ ♠✐♥❤ r➡♥❣ tr♦♥❣ ❜❛ sè a, b, c ❝â ➼t ♥❤➜t ♠ët sè ❜➡♥❣ ✶✳
❇➔✐ ✸✼✳ ❈❤♦ ❝→❝ sè t❤ü❝ a, b, c ✤æ✐ ♠ët ❦❤→❝ ♥❤❛✉ t❤ä❛ ♠➣♥
abc = 1

❚➼♥❤

b−c
c−a
a−b
+
+
= 2019
(a − b) (a − c) (b − c) (b − a) (ac − a) (c − b)
❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ P = a −2 b + b −2 c + c −2 a ✳
❈❤♦ ❝→❝ sè t❤ü❝ a, b, c t❤ä❛ ♠➣♥ abc = 2019✳ ❚➼♥❤ ❣✐→ trà

❇➔✐ ✸✽✳

❜✐➸✉ t❤ù❝

Q=

a
b
2019c
+
+
ab + a + 2019 bc + b + 1 ac + 2019c + 2019
✺

❝õ❛


❇➔✐ ✸✾ ✭❍❙● ◆✐♥❤ ❇➻♥❤ ✷✵✶✺✮✳ ❈❤♦ ❝→❝ sè t❤ü❝ ❞÷ì♥❣ a, b t❤ä❛ ♠➣♥
a100 + b100 = a101 + b101 = a102 + b102

❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ M = a2015 + b2015✳
❇➔✐ ✹✵✳ ❈❤♦ ❝→❝ sè t❤ü❝ x, y, z t❤ä❛ ♠➣♥
x + y + z = 1, x2 + y 2 + z 2 = 1, x3 + y 3 + z 3 = 1

❚➼♥❤ ❣✐→ trà ❝õ❛ ❜✐➸✉ t❤ù❝ Q = x4 + y5 + z 6✳
❇➔✐ 5✹✶✳ ❈❤♦
❝→❝ sè 3t❤ü❝3a, b,3c t❤ä❛
♠➣♥✿ a + b + c = 0✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣
a + b + c a2 + b2 + c2
a + b 5 + c5
=
.
❛✮
5
3
2
7
7
7
4
4
❜✮ 2 a + b + c = 7abc a + b + c4
❝✮ a7 + b7 + c7 = 7abc a2b2 + b2c2 + c2a2
❞✮ 10 a7 + b7 + c7 = 7 a2 + b2 + c2 a5 + b5 + c5

✻