!
!"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A)))
B4%)#%C#()DE#4$%&2:FG&) !
"!
H4"I)1%J%)>K)L!ML)N=O9)P%E)Q)L4RS()/T&1)L4U&4)VE6)
DW&()L"I&X)/Y)Z[).\]^*)
V1US)#4%)()_]*\].*`^)
L4a%)1%E&)$U6)bU%()`_*)@4c#d)24W&1)2e)#4a%)1%E&)1%E")>K)
f%g&)4h)>0&1)23)24"I)489)Q)!"#$%&'()*+,-) ).*.)Q)B4%)#%C#()iiiF6E#4$%&2:FG&))
Bj=)`)k.d*)>%e6lF)#$%!$&'!()!

y = (x + m)(x −1)
2
(1)
*!
"* +$,%!( !(/!0123!.$143!5&!56!78!.$9!$&'!()!:";!5<1!

m = 0
*!
=* #$%!71>'!#:"?@A=;*!BC'!'!7>!:";!DE!$F1!71>'!D/D!.G9!HIJ!(F%!D$%!0F!71>'!HIJI#!.$K3L!
$&3L*!
Bj=).)k`d*)>%e6lF)
F; M1,1!N$OP3L!.GC3$!

2 2 sin x. cos x = 1
*!!
0; BC'!()!N$QD!R!DE!N$S3!.$/D!5&!N$S3!,%!7TU!VOP3L!.$%,!'W3!

z = 5,z
2
+ z

2
= 6
*!!
Bj=)7)k*d^)>%e6lF!M1,1!N$OP3L!.GC3$!

log
2
(x + 3) = log
4
x −1 + 2
*!
Bj=)\)k`d*)>%e6lF)M1,1!$X!N$OP3L!.GC3$!

x
2
+ 3x + y + 2 =
(x +1)(x + 2)
y −1
x
2
+16 − 2 x
2
−3x +4 = y −1−1
⎧
⎨
⎪
⎪
⎪
⎪
⎩

⎪
⎪
⎪
⎪
, x, y ∈ !
( )
*!!!
Bj=)^)k`d*)>%e6lF!BY3$!.$>!.YD$!Z$)1!.G[3!\%F]!Z$1!^UF]!$C3$!N$K3L!L1<1!$_3!0`1!D-D!7Oa3L!

y =
4 + x
2
.ln x
x
, y =
2
x
,x = 2
^UF3$!.GbD!$%&3$*!
Bj=)-)k`d*)>%e6lF!#$%!$C3$!D$EN!c*HJ#!DE!'d.!043!cJ#!e&!.F'!L1-D!5Uf3L!Dg3!._1!c!5&!3h'!
.G%3L!'d.!N$K3L!5Uf3L!LED!5<1!'d.!N$K3L!:HJ#;I!

BC = a 2,ASB
!
= CSA
!
= 60
0
*!BY3$!.$>!.YD$!
Z$)1!D$EN!c*HJ#!5&!Z$%,3L!D-D$!.i!71>'!J!723!'d.!N$K3L!:cH#;*!

Bj=),)k`d*)>%e6lF!BG%3L!'d.!N$K3L!5<1!.GbD!.%_!7j!k\]!D$%!.F'!L1-D!HJ#!5Uf3L!Dg3!._1!#*!
Ml1!m!e&!.GU3L!71>'!D_3$!H#I!n!e&!71>'!.$UjD!7%_3!HJ!.$%,!'W3!

DB = 2DA
I!o!e&!$C3$!
D$12U!5Uf3L!LED!DpF!n!.G43!Jm*!BC'!.%_!7j!D-D!7q3$!HIJI#!012.!n:A=@r;I!

H (−
18
5
;
24
5
)
5&!7q3$!J!
DE!$%&3$!7j!3LU]43*!
Bj=)_)k`d*)>%e6lF!BG%3L!Z$f3L!L1F3!5<1!.GbD!.%_!7j!k\]R!D$%!71>'!s:=@=@?;!5&!'d.!N$K3L!

(P ) : 3x + 2y − z + 4 = 0
*!t12.!N$OP3L!.GC3$!7Oa3L!.$K3L!V!71!^UF!s!5&!5Uf3L!LED!5<1!'d.!
N$K3L!:u;*!BC'!.%_!7j!71>'!m!.G43!V!(F%!D$%!m!D-D$!7TU!L)D!.%_!7j!5&!'d.!N$K3L!:u;*!!
Bj=)+)k*d^)>%e6lF!vLOa1!.F!Vw3L!x!DU)3!(-D$!B%-3I!y!DU)3!(-D$!tz.!e{I!|!DU)3!(-D$!o%-!$lD!
:D-D!(-D$!Dw3L!e%_1!.$C!L1)3L!3$FU;!7>!e&'!N$S3!.$O`3L!D$%!}!$lD!(13$!'~1!$lD!(13$!7O•D!$F1!
DU)3!Z$-D!e%_1I!.G%3L!}!$lD!(13$!3&]!DE!$F1!0_3!vF'!5&!oOa3L*!BY3$!\-D!(U€.!7>!vF'!5&!
oOa3L!DE!N$S3!.$O`3L!L1)3L!3$FU*!
Bj=)`*)k`d*)>%e6lF!#$%!FI0ID!e&!D-D!()!.$/D!VOP3L!.$%,!'W3!

a(a −c )+ b(b − c ) = 0
*!BC'!L1-!.G9!
3$•!3$€.!DpF!01>U!.$QD!


P =
a
b
3
+ c
3
+
b
c
3
+ a
3
+
c
2
+ ab
2
−
a + b
c
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟

⎟
⎟
⎟
2
*!
)
mmm!nLmmm)
!
!"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A)))
B4%)#%C#()DE#4$%&2:FG&) !
=!
M!oV)LpB!)qrV!)fstV)/uM)uV)
Bj=)`)k.d*)>%e6lF)#$%!$&'!()!

y = (x + m)(x −1)
2
(1)
*!
"* +$,%!( !(/!0123!.$143!5&!56!78!.$9!$&'!()!:";!5<1!

m = 0
*!
=* #$%!71>'!#:"?@A=;*!BC'!'!7>!:";!DE!$F1!71>'!D/D!.G9!HIJ!(F%!D$%!0F!71>'!HIJI#!.$K3L!
$&3L*!
"* olD!(13$!./!L1,1*!
=* BF!DE‚!

y ' = (x −1)
2
+ 2(x + m)(x −1); y ' = 0 ⇔

x = 1
x =
1−2m
3
⎡
⎣
⎢
⎢
⎢
⎢
*!
ƒ;!„>!78!.$9!$&'!()!DE!$F1!71>'!D/D!.G9!Z$1!5&!D$q!Z$1!

1− 2m
3
≠ 1 ⇔ m ≠ −1
*!
+$1!7E!.%_!7j!=!71>'!D/D!.G9!e&!

A(1;0), B(
1− 2m
3
;
4(m +1)
3
27
)
I!.F!DE‚!
!


AC
! "!!
= (9;−2),AB
! "!!
= (
−2(m +1)
3
;
4(m +1)
3
27
) //(−9;2(m +1)
2
)
*!
tz]!HIJI#!.$K3L!$&3L!Z$1!5&!D$q!Z$1!

AB
! "!!
, AC
! "!!
Dw3L!N$OP3L!
!

⇔
−9
9
=
2(m +1)
2

−2
⇔
m = 0
m = −2
⎡
⎣
⎢
⎢
(t /m)
*!
HC#)$=;&(!tz]!

m = −2;m = 0
e&!L1-!.G9!DS3!.C'*!!!
Bj=).)k`d*)>%e6lF)
F; M1,1!N$OP3L!.GC3$!

2 2 sin x. cos x = 1
*!!
0; BC'!()!N$QD!R!DE!N$S3!.$/D!5&!N$S3!,%!7TU!VOP3L!.$%,!'W3!

z = 5,z
2
+ z
2
= 6
*!!
F; „1TU!Z1X3!N$OP3L!.GC3$!DE!3L$1X'‚!

sin x > 0

*!
+$1!7E!0C3$!N$OP3L!$F1!52!DpF!N$OP3L!.GC3$!.F!7O•D‚!
!

8sin
2
x.cos
2
x = 1⇔ 2sin
2
2x = 1 ⇔ cos4x = 0 ⇔ x =
π
8
+ k
π
4
,k ∈ !
*!
ƒ;!J1>U!V1…3!.G43!5[3L!.G[3!eO•3L!L1-D!.F!7O•D!D-D!3L$1X'!.$%,!'W3!‚!
!

x ∈
π
8
+ k2π,
3π
8
+ k2π,
5π
8

+ k2π,
7π
8
+ k2π,k ∈ !
⎧
⎨
⎪
⎪
⎩
⎪
⎪
⎫
⎬
⎪
⎪
⎭
⎪
⎪
*!!
0; „d.!

z = x + y.i (x, y > 0)
I!.$†%!L1,!.$12.!.F!DE‚!
!

x
2
+ y
2
= 5

(x + yi)
2
+ (x − yi)
2
= 6
⎧
⎨
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⇔
x
2
+ y
2
= 5
x
2
− y
2
= 3
⎧
⎨
⎪
⎪
⎩

⎪
⎪
⇔
x
2
= 4
y
2
= 1
⎧
⎨
⎪
⎪
⎩
⎪
⎪
⇔
x = 2
y = 1
⎧
⎨
⎪
⎪
⎩
⎪
⎪
(do x, y > 0)
*!
tz]!


z = 2 +i
*!!!
Bj=)7)k*d^)>%e6lF!M1,1!N$OP3L!.GC3$!

log
2
(x + 3) = log
4
x −1 + 2
*!
„1TU!Z1X3‚!

−3 < x ≠ 1
*!
u$OP3L!.GC3$!.OP3L!7OP3L!5<1‚!
!
!"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A)))
B4%)#%C#()DE#4$%&2:FG&) !
‡!
!

log
2
(x + 3) = log
2
4 + log
2
x −1 ⇔ log
2
(x + 3) = log

2
4 x −1
⇔ x + 3= 4 x −1 ⇔ (x + 3)
2
= 16 x −1
⇔
x >1
(x + 3)
2
= 16(x −1)
⎧
⎨
⎪
⎪
⎩
⎪
⎪
x <1
(x + 3)
2
= −16(x −1)
⎧
⎨
⎪
⎪
⎩
⎪
⎪
⎡
⎣

⎢
⎢
⎢
⎢
⎢
⎢
⎢
⇔ x = 5; x = 8 2 −11
*!
HC#)$=;&(!u$OP3L!.GC3$!DE!$F1!3L$1X'!

x = 5;x = 8 2 −11
*!!!
Bj=)\)k`d*)>%e6lF)M1,1!$X!N$OP3L!.GC3$!

x
2
+ 3x + y + 2 =
(x +1)(x + 2)
y −1
x
2
+16 − 2 x
2
−3x +4 = y −1−1
⎧
⎨
⎪
⎪
⎪

⎪
⎩
⎪
⎪
⎪
⎪
, x, y ∈ !
( )
*!!
Phân%tích%lời%giải:!„>!{!N$OP3L!.GC3$!.$Q!3$€.!DpF!$X!DE!

x
2
+ 3x + 2 = (x +1)(x + 2)
V%!5z]!7>!
7P3!L1,3!.F!7d.!

a = (x +1)(x + 2)
I!N$OP3L!.GC3$!.G`!.$&3$‚!

a + y =
a
y −1
⇔
a ≥0
(a + y)( y −1)
2
= a
2
⎧

⎨
⎪
⎪
⎩
⎪
⎪
!*!
„OF!5T!N$OP3L!.GC3$!0zD!$F1!DpF!F‚!!

a
2
−( y −1)
2
a − y( y−1)
2
= 0,Δ
a
= (y −1)
4
+ 4y( y −1)
2
= (y
2
−1)
2
*!
n%!7E!

a =
( y −1)

2
+ (y
2
−1)
2
= y
2
− y;a =
( y −1)
2
− ( y
2
−1)
2
= −y +1
*!
BQD!e&!

x
2
+ 3x + 2 = y
2
− y
x
2
+ 3x + 2 = −y +1
⎡
⎣
⎢
⎢

⎢
⇔
x
2
+ 3x +1+ y = 0
x
2
+ 3x − y
2
+ y + 2 = 0
⎡
⎣
⎢
⎢
⎢
⇔
x
2
+ 3x +1+ y = 0
(x + y +1)(x − y + 2) = 0
⎡
⎣
⎢
⎢
⇔
x
2
+ 3x +1+ y = 0
y = −x −1
y = x + 2

⎡
⎣
⎢
⎢
⎢
⎢
⎢
*!
Lời%giải:%%%%
„1TU!Z1X3‚!

x
2
+ 3x + y + 2 ≥ 0, y >1
*!u$OP3L!.GC3$!.$Q!3$€.!DpF!$X!.OP3L!7OP3L!5<1‚!

x +1
( )
x + 2
( )
≥ 0
x
2
+ 3x + y + 2 =
x +1
( )
2
x + 2
( )
2

y −1
( )
2
⎧
⎨
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎪
*!
!
!"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A)))
B4%)#%C#()DE#4$%&2:FG&) !
r!

⇔
x +1
( )
x + 2
( )
≥ 0
y −1
( )

2
x +1
( )
x + 2
( )
+ y
⎡
⎣
⎢
⎤
⎦
⎥
= x +1
( )
2
x + 2
( )
2
⎧
⎨
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⇔
x +1
( )

x + 2
( )
≥ 0
x
2
+ 3x + y +1
( )
x + y +1
( )
x − y +2
( )
= 0
⎧
⎨
⎪
⎪
⎪
⎩
⎪
⎪
⎪
*!
ƒ!BF!DE!

y >1⇒ x
2
+ 3x +1+ y > x
2
+ 3x + 2 = x +1
( )

x + 2
( )
≥ 0
*!
ƒ!;!v2U!

y = −x −1 ⇒ −x −1 >1 ⇔ x < −2
*!
B$F]!5&%!N$OP3L!.GC3$!.$Q!$F1!DpF!$X!.F!7O•D!

x
2
+16 − 2 x
2
− 3x + 4 = −x −2 −1
⇔ x
2
+16 +1 = 2 x
2
− 3x + 4 + −x − 2
*!
+$1!7E‚!

VT = x
2
+16 +1 ≤6,∀x ∈ −3;−2
⎡
⎣
⎢
⎤

⎦
⎥
,
VP = 2 x
2
−3x + 4 + −x −2 ≥ 2 14 > 6,∀x ∈ −3;−2
⎡
⎣
⎢
⎤
⎦
⎥
*!
n%!7E!N$OP3L!.GC3$!5f!3L$1X'*!
!ƒ!;!v2U!

y = x + 2
.$F]!5&%!N$OP3L!.GC3$!.$Q!$F1!DpF!$X!.F!7O•D‚!

x
2
+16 − 2 x
2
− 3x + 4 = x +1 −1
*!
u$OP3L!.GC3$!3&]!DE!71TU!Z1X3‚!

x ≥−1
*!!
B$/D!$1X3!3$g3!e143!$•N!.F!DE‚!


⇔
−3x
2
+12x
x
2
+16 + 2 x
2
−3x +4
=
x
x +1 +1
⇔
x = 0
x
2
+16 + 2 x
2
−3x +4 = −3 x −4
( )
x +1 +1
( )
(1)
⎡
⎣
⎢
⎢
⎢
*!

M1,1!N$OP3L!.GC3$!:";!0h3L!D-D$!Z2.!$•N!5<1!N$OP3L!.GC3$!7SU!DpF!$X!.F!7O•D!:!ˆ†'!.$4'!
#U)3!“%Bài%giảng%chọn%lọc%Phương%trình%–%Bất%phương%trình%vô%tỷ”!Dw3L! D!L1,;*!

x
2
+16 + 2 x
2
−3x +4 = −3 x −4
( )
x +1 +1
( )
x
2
+16 − 2 x
2
−3x +4 = x +1−1
⎧
⎨
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⇒ 2 x
2
+16 = 13−3x
( )
x +1 − 3x +11

⇔ 2 x
2
+16 −5
( )
+ 3x −13
( )
x +1 − 2
( )
+ 9 x −3
( )
= 0
⇔ x − 3
( )
2 x +3
( )
x
2
+16 +5
+
3x −13
x +1 + 2
+ 9
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥

⎥
⎥
= 0
⇔ x − 3
( )
2 x +3
( )
x
2
+16 +5
+
5 + 9 x +1 +3x
x +1 + 2
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
= 0 ⇔ x = 3
!
B$‰!e_1!.$€]!.$%,!'W3*!cU]!GF!

x; y
( )
= 0;2

( )
; 3;5
( )
*!
!
!"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A)))
B4%)#%C#()DE#4$%&2:FG&) !
"!
HC#)$=;&(!#$%!&'!(&)*+,! /+&!01!&23!+,&3'4!56!!

x; y
( )
= 0;2
( )
; 3;5
( )
7!!!
Cách%2:!89-!

t = x
2
+ 3x + 2+ y ⇒ (x +1)(x + 2) = t
2
− y
7!
:&)*+,! /+&!-&;!+&<-!0=2!&'! >!-&6+&?!
!

t =
t

2
− y
y −1
⇔ t
2
−( y −1)t − y = 0 ⇔ (t − y)(t +1) = 0 ⇔ t = y (do t ≥ 0)
7!
#/!@$%!

x
2
+ 3x + 2+ y = y ⇔
y ≥ 0
x
2
+ 3x + 2+ y = y
2
⎧
⎨
⎪
⎪
⎩
⎪
⎪
⇔
y ≥ 0
( y − x −2)(y + x +1) = 0
⎧
⎨
⎪

⎪
⎩
⎪
⎪
7!
A2!01!BC-!DEF!-)*+,!-G! H+7!!!!
BI=)J)KLM*)>%N6OF!AI+&!-&J!-I0&!B&K3! L+!MN2%!B&3!DE2%!&/+&!(&O+,!,3P 3!&Q+!R>3!0S0!T)U+,!

y =
4 + x
2
.ln x
x
, y =
2
x
,x = 2
DE2+&! V0!&N6+&7!
WX!:&)*+,! /+&!&N6+&!TY!,32N!T3J4?!

4 + x
2
.ln x
x
=
2
x
⇔ 4 + x
2
.ln x = 2 ⇔ x

2
.ln x = 0 ⇔
x = 0(l )
x =1
⎡
⎣
⎢
⎢
7!
#/!@$%

V = π (
4 + x
2
.ln x
x
)
2
−
4
x
2
dx =
1
2
∫
π ln x dx
1
2
∫

7!!!
WX!89-!

u = ln x
dv = dx
⎧
⎨
⎪
⎪
⎩
⎪
⎪
⇒
du =
dx
x
v = x
⎧
⎨
⎪
⎪
⎪
⎩
⎪
⎪
⎪
7!
WX!ZE%!.2?!

V = π(x ln x

2
1
− dx
1
2
∫
) = π(2ln 2− x
2
1
) = π(2ln 2−1)
![T@ X7!
BI=)-)KLM*)>%N6OF!\&N!&/+&!0&1(!Z7]^\!01!49-!RH+!Z^\!56!-24!,3S0!@E_+,!0`+!-Q3!Z!@6!+a4!
N+,!49-!(&O+,!@E_+,!,10!@P3!49-!(&O+,![]^\Xb!

BC = a 2,ASB
!
= CSA
!
= 60
0
7!AI+&!-&J!-I0&!
B&K3!0&1(!Z7]^\!@6!B&NF+,!0S0&!-c!T3J4!^!TC+!49-!(&O+,![Z]\X7!
!
WX!de3!f!56! E+,!T3J4!^\b!-&gN!,3F!-&3C-?!
!

SH ⊥ BC
(SBC ) ⊥ (ABC )
⎧
⎨

⎪
⎪
⎩
⎪
⎪
⇒ SH ⊥ (ABC )
7!
#6!-24!,3S0!Z]^\!@E_+,!0`+!01!

SH = BH = CH =
BC
2
=
a 2
2
7!
A24!,3S0!Z]^!@6!Z]\!01!Z]!0&E+,b!

SB = SC ,ASB
!
= ASC
!
= 60
0
+H+!
Ra+,!+&2E7!
hN!T1!

AB = AC
@6!-24!,3S0!]^\!0`+!-Q3!]b!T9-!


SA = x
7!
i(!jV+,!Tk+&!5l!f64!mK!0_m3+!0&N!-24!,3S0!Z]^b!(3-2,N!0&N!0S0!-24!,3S0!Z]fb!]f^!01?!
!
!"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A)))
B4%)#%C#()DE#4$%&2:FG&) !
n!
!

AB
2
= SA
2
+ SB
2
− 2SA.SB cos60
0
= x
2
+ a
2
− ax,
AH
2
= AB
2
− BH
2
= x

2
+ a
2
− ax −
a
2
2
,
SA
2
= SH
2
+ AH
2
=
a
2
2
+ (x
2
+ a
2
− ax −
a
2
2
) = x
2
⇒ x = a
7!

hN!T1!

AH =
a 2
2
,S
ABC
=
1
2
AH .BC =
1
2
.
a 2
2
.a 2 =
a
2
2
7!
#/!@$%!

V
S .ABC
=
1
3
SH .S
ABC

=
1
3
.
a 2
2
.
a
2
2
=
a
3
2
12
[T@ X7!
WX!A2!01?!

d (B;(SAC )) = 2d(H ;(SAC ))
7!
op!fo!@E_+,!,10!@P3!]\!-Q3!ob!op!fq!@E_+,!,10!@P3!Zo!-Q3!q!-&/!!
!

HI ⊥ (SAC ) ⇒ d(H ;(SAC )) = HI
7!
A24!,3S0!@E_+,!]f\!@6!Zfo!01!
!

1
HI

2
=
1
SH
2
+
1
HK
2
=
1
SH
2
+
1
HC
2
+
1
HA
2
=
2
a
2
+
2
a
2
+

2
a
2
=
6
a
2
⇒ HI =
a 6
6
7!
#$%!

d (B;(SAC )) = 2HI =
a 6
3
7!
BP94).(!AI+&!-&gN!-&J!-I0&!@/!j3'+!-I0&!-24!,3S0!Z]\!-I+&!T*+!,3F+!
A2!01?

S
SAC
=
1
2
SA.SC.sin60
0
=
a
2

3
4
⇒ d (B;(SAC )) =
3V
SABC
S
SAC
=
a
3
2
4
a
2
3
4
=
a 6
3
7!!!
QR&4)$=;&(!\&r!l!,3F!-&3C-!R63!-NS+!-2!-I+&!T)s0!

HA =
1
2
BC ⇒ ΔABC
@E_+,!0`+!-Q3!]7!!!
QS%)#;@)#<T&1)#U)V)\&N!&/+&!0&1(!Z7]^\!01!49-!RH+!Z^\!56!-24!,3S0!0`+!-Q3!Zb!

SB = a

!@6!+a4!
N+,!49-!(&O+,!@E_+,!,10!@P3!49-!TS%![]^\X7!^3C-!

ASB
!
= BSC
!
= CSA
!
= 60
0
7!AI+&!-&J!-I0&!
B&K3!0&1(!Z7]^\!@6!B&NF+,!0S0&!-c! E+,!T3J4!TNQ+!Z^!TC+!49-!(&O+,![Z]\X7!!!
BI=),)KLM*)>%N6OF!A.N+,!49-!(&O+,!@P3! V0!-NQ!TY!tM%!0&N!-24!,3S0!]^\!@E_+,!0`+!-Q3!\7!
de3!u!56! E+,!T3J4!0Q+&!]\b!h!56!T3J4!-&EY0!TNQ+!]^!-&NF!4v+!

DB = 2DA
b!f!56!&/+&!
0&3CE!@E_+,!,10!0=2!h! H+!^u7!A/4!-NQ!TY!0S0!Tw+&!]b^b\!R3C-!h[xyz{Xb!

H (−
18
5
;
24
5
)
@6!Tw+&!^!
01!&N6+&!TY!+,E%H+!
:&)*+,! /+&!T)U+,!-&O+,!hf!56!


x + 2 y −6 = 0
7!
8)U+,!-&O+,!^u!T3!DE2!f!@6!@E_+,!,10!@P3!hf!+H+!01!(&)*+,! /+&!

2x − y +12 = 0
7!
A2!0&;+,!43+&!\bfbh!-&O+,!&6+,!@6!

CH
! "!!
=
3
2
HD
! "!!
7!
hN!

CH
! "!!
=
3
2
HD
! "!!
= (
12
5
;−

6
5
) ⇒ C (−6;6)
7!89-!

CA = CB = a > 0 ⇒ AB = a 2,BD =
2a 2
3
7!
!
!"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A)))
B4%)#%C#()DE#4$%&2:FG&) !
|!
i(!jV+,!Tk+&!5l!&64!mK!\_xm3+!0&N!-24!,3S0!^\h!01!

CD
2
= BC
2
+ BD
2
− 2BC.BD cos 45
0
= a
2
+
8a
2
9
−

4a
2
3
= 20 ⇔ a
2
= 36
7!
de3!^[RzyRW}yX!@P3!R~•!-&EY0!^u!-2!01!
!

BC
2
= 36 ⇔ (b + 6)
2
+ (2b + 6)
2
= 36 ⇔ 5b
2
+ 36b + 36 = 0 ⇔
b = −6(t / m)
b = −
6
5
(l )
⎡
⎣
⎢
⎢
⎢
⎢

7!
ZE%!.2!

B(−6;0),DA
! "!!
=
1
2
BD
! "!!
⇒ A(0;6)
7!
HC#)$=;&(!#$%!][•znXb!^[xnz•Xb!\[xnznX7!!!
BI=)W)KLM*)>%N6OF!A.N+,!B&_+,!,32+!@P3! V0!-NQ!TY!tM%€!0&N!T3J4!q[yzyz•X!@6!49-!(&O+,!

(P ) : 3x + 2y − z + 4 = 0
7!#3C-!(&)*+,! /+&!T)U+,!-&O+,!j!T3!DE2!q!@6!@E_+,!,10!@P3!49-!
(&O+,![:X7!A/4!-NQ!TY!T3J4!u! H+!j!m2N!0&N!u!0S0&!T•E!,K0!-NQ!TY!@6!49-!(&O+,![:X7!!
WX!8)U+,!-&O+,!j!@E_+,!,10!@P3![:X!+H+!+&$+!@-(-!0=2![:X!564!@‚0!-*!0&w!(&)*+,b!@/!@$%!

u
d
!"!
= (3;2;−1)
7!!
hN!T1!(&)*+,! /+&!0=2!j!56!

d :
x = 2 + 3t
y = 2+ 2t

z = −t
⎧
⎨
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
,t ∈ !
7!
WX!de3!

M (2+3t;2+ 2t;−t ) ∈ d
b!-2!01?!
!

d (M ;(P )) =
3(2+ 3t ) + 2(2+ 2t)− (−t )+ 4
3
2
+ 2
2
+1
2
= 14 t +1 ,
MO = (2+ 3t)

2
+ (2+ 2t )
2
+ t
2
= 14t
2
+ 20t + 4
7!
WX!A&gN!,3F!-&3C-!-2!01?!
!

14 t +1 = 14t
2
+ 20t + 4 ⇔ 14t
2
+ 20t + 4 =14(t +1)
2
⇔ 4t + 3= 0 ⇔ t = −
3
4
⇒ M (−
1
4
;
1
2
;
3
4

)
7!
HC#)$=;&(!#$%!T3J4!0ƒ+!-/4!

M (−
1
4
;
1
2
;
3
4
)
7!!!
BI=)+)K*MJ)>%N6OF!„,)U3!-2!j…+,!"!0EK+!mS0&!ANS+b!n!0EK+!mS0&!#$-!5lb!|!0EK+!mS0&!fNS!&e0!
[0S0!mS0&!0…+,!5NQ3!-&/!,3K+,!+&2EX!TJ!564!(&ƒ+!-&)>+,!0&N!†!&e0!m3+&!4‡3!&e0!m3+&!T)s0!&23!
0EK+!B&S0!5NQ3b! N+,!†!&e0!m3+&!+6%!01!&23!RQ+!„24!@6!f)U+,7!AI+&!MS0!mE<-!TJ!„24!@6!
f)U+,!01!(&ƒ+!-&)>+,!,3K+,!+&2E7!
de3!M!56!mK!&e0!m3+&!+&$+!mS0&!ANS+!@6!#$-!ˆl7!
de3!%!56!mK!&e0!m3+&!+&$+!mS0&!ANS+!@6!fNS!&e0!
de3!€!56!mK!&e0!m3+&!+&$+!mS0&!fNS!&e0!@6!#$-!ˆl!
!
!"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A)))
B4%)#%C#()DE#4$%&2:FG&) !
‰!
A2!01?!

x + y + z = 9
x + y = 5

x + z = 6
y + z = 7
⎧
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎪
⎪
⇔
x = 2
y = 3
z = 4
⎧
⎨
⎪
⎪
⎪
⎪
⎩
⎪
⎪

⎪
⎪
7!#$%!0&w!01!y!&e0!m3+&!+&$+!mS0&!ANS+!@6!#$-!5lz!Š!&e0!m3+&!
+&$+!mS0&!ANS+!@6!fNS!&e0z!{!&e0!m3+&!+&$+!mS0&!#$-!5l!@6!fNS!&e0![‹X7!
WX!o&_+,!,32+!4ŒE!56!mK!0S0&!0&32!mS0&!0&N!†!&e0!m3+&!-&gN!T3•E!B3'+![‹X!01!

Ω = C
9
2
.C
7
3
.C
4
4
= 1260
7!
de3!]!56!R3C+!0K!y!RQ+!„24!@6!f)U+,!01!(&ƒ+!DE6!,3K+,!+&2Eb!01!0S0!B&F!+•+,!
X!L(!\F!y!0…+,!+&$+!mS0&!ANS+!@6!#$-!5lb!B&3!T1!|!RQ+!0L+!5Q3!01!Š!RQ+!+&$+!mS0&!ANS+!@6!
fNS!&e0z!{!RQ+!+&$+!mS0&!fNS!&e0!@6!#$-!5l7!ZK!0S0&!(&`+!0&32!56!

C
7
3
.C
4
4
= 35
7!
X!.(!\F!y!0…+,!+&$+!mS0&!ANS+!@6!fNS!&e0b!B&3!T1!|!RQ+!0L+!5Q3!01!y!RQ+!+&$+!mS0&!ANS+!@6!

#$-!5lz!}!RQ+!+&$+!mS0&!ANS+!@6!fNS!&e0z!{!RQ+!+&$+!mS0&!#$-!5l!@6!fNS!&e07!
ZK!0S0&!(&`+!0&32!56!

C
7
2
C
5
1
.C
4
4
= 105
7!!!!
X!7(!\F!y!0…+,!+&$+!mS0&!fNS!&e0!@6!#$-!ˆlb!B&3!T1!|!RQ+!0L+!5Q3!01!y!RQ+!+&$+!mS0&!ANS+!
@6!#$-!5lz!Š!RQ+!+&$+!mS0&!ANS+!@6!fNS!&e0z!y!RQ+!+&$+!mS0&!fNS!&e0!@6!#$-!5l7!
ZK!0S0&!(&`+!0&32!56!

C
7
2
.C
5
3
.C
2
2
= 210
7!
#$%!


Ω
A
= 35 +105 + 210 = 350
7!
ŽS0!mE<-!0ƒ+!-I+&!

P (A) =
Ω
A
Ω
=
350
1260
=
5
18
7!!!!!!!!
BI=)L*)KLM*)>%N6OF!\&N!2bRb0!56!0S0!mK!-&G0!j)*+,!-&NF!4v+!

a(a −c )+ b(b − c ) = 0
7!A/4!,3S! k!
+&•!+&<-!0=2!R3JE!-&;0!

P =
a
b
3
+ c
3

+
b
c
3
+ a
3
+
c
2
+ ab
2
−
a + b
c
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
2
7!
89-!


A =
a
b
3
+ c
3
+
b
c
3
+ a
3
,S = a b
3
+ c
3
+ b a
3
+ c
3
,
!!
Z•!jV+,!R<-!TO+,!-&;0!fN5jg.!-2!01?

A.S
2
≥ (a + b)
3
7!! !!
Z•!jV+,!R<-!TO+,!-&;0!\2E0&%!‘Z0&’2 €!-2!01?!


S = a
2
(b + c).(b
2
−bc + c
2
) + b
2
(c +a)(c
2
− ca +a
2
)
≤ a
2
(b + c) +b
2
(c +a)
( )
b
2
+ a
2
+ 2c
2
− c (a +b)
( )
= c 2(ab(a + b) +c(a
2

+ b
2
)) = c 2(a + b)(ab + c
2
)
!
ZE%!.2?!

A ≥
(a +b)
2
2c
2
(c
2
+ ab)
7!!
#/!@$%!

P ≥
(a +b)
2
2c
2
(c
2
+ ab)
+
c
2

+ ab
2
−
a + b
c
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
2
≥
a + b
c
−
a + b
c
⎛
⎝
⎜
⎜
⎜
⎜

⎞
⎠
⎟
⎟
⎟
⎟
2
≥−2
7!
^>3!@/!-&gN!,3F!-&3C-!-2!01?!
!

c(a + b) = a
2
+ b
2
≥
1
2
(a +b)
2
⇒
a + b
c
≤ 2,c(a + b) < (a + b)
2
⇒
a + b
c
>1

7!!
!
!"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A)))
B4%)#%C#()DE#4$%&2:FG&) !
†!
Cách%2:!A&gN!,3F!-&3C-!-2!01?!

a
2
+ b
2
= c(a + b) ≥
1
2
(a +b)
2
⇒
a + b
c
≤ 2
7!
Z•!jV+,!R<-!TO+,!-&;0!\2E0&%!‘Z0&’2.€!-2!01?!

a
b
3
+ c
3
+
b

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

3
)
=
(a +b)
2
ab(a
2
+ b
2
) + c
3
(a +b)
=
(a +b)
2
abc(a +b) + c
3
(a +b)
=
a + b
abc + c
3
=
a + b
c(ab + c
2
)
7!
hN!T1?!
!


P ≥
a +b
c(ab + c
2
)
+
c
2
+ ab
2
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
−
a +b
c
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠

⎟
⎟
⎟
⎟
2
≥ 2
a +b
c(ab + c
2
)
.
c
2
+ ab
2
−
a +b
c
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟

2
=
2(a + b)
c
−
a +b
c
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
2
= f (t) = 2t −t
2
,t =
a +b
c
∈ 1;2
(
⎤
⎦
⎥

7!
o&FN!mS-!&64!mK!“[-X! H+!+•2!B&NF+,!

1;2
(
⎤
⎦
⎥
-2!01!

f (t) ≥ f (2) = −2
7!
A2!01!BC-!DEF!-)*+,!-G7!
!
!!!!
!
!!
!
!
!
!