Download - Vnmo 30 4-2006-grade 10
បជវ សស ពែកគណតវទ កទ១០
បកែបេយ ែកវ សរ"
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 1
វ អពកបៃពណេវតមេលកទ XII ២០០៦
មខវ$% ៈ គណតវទ() កទ ១០ រយៈេពល ១៨០/ទ
. យម 2 22( 1 3 1)x x xx− + = ++
. គ ABC នប ជង , 2 , 4, ,CA b AB c BBC a C AA= = === , !
ង"ង#$ %ក'()ន%ង R គ* 22 2 2
1 1 1T R
a b c
= + +
+. គ ABC ម,ង# A ង"ង#$ %កក-.ង ABC ប/ជង AB
ង# T , ប*0 # CT #ង"ង#ង# K ផ2ងព T 4ប5 K 6ច!ន.ចក 8 CT
9)យ 6 2CT = ច:គ*ប;<ងប ជងប# ABC
=. យប>? ក#5 ច!@គប#!A86ច!នBនគ#ប# m , នច!នBនគ# n C)មD
3 211 87n n n m− − + ;ចកច#ន%ង 191
E. គ , , 0a b c > យប>? ក#5
( ) ( ) ( ) ( ) ( ) ( )4 4 4
2 2 24 6 6 3 3 4 6 63 3 3 4 6 6 3 33 3
1
a b a c c a a b
a b c
a b b b c c c
+ + ≤+ + + + + + + + +
'()&'()&'()&'()&
ចេលយ
. យម 2 22 3) 1 (1)( 1 xx x x− + ++ =
;Cនក!# ℝ
Fង 2 2 3xx t− + = ច!@ 2t ≥
ព8* ម (1) G6 2( 1) 1 (2)x t x ++ =
2 2(2) 3 ( 1) 2( 1) 0x x t xx⇔ − + − + + − =
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 2
2 ( 1) 2( 1 02
)1
tt x
t xx t⇔ − + + − = ⇔
= = −
ច!@ 2t = , យ)ង,ន 2 22 3 2 3 42 xx xx− + ⇔ − + ==
2 1 2
1 22 1 0x
xx
x
= −
= +− − = ⇔
⇔
ច!@ 1t x= − , យ)ង,ន 2 3 12 xxx − + = −
2 2
0 1
3 12 3 ( 1)
1x xx
x x x
≥ ≥⇔ ⇔ ⇔ ∈∅
=−
= + − −
C:ចន !ន.!Hប#ម គI 1 2;1 2− +
. យ)ងន 2sin sin sin
a b cR
A B C= = =
1 1 1, ,
2sin 2sin 2sin
R R R
a A b B c C⇒ = = =
2 2 2
1 1 1 1
4 sin sin sinA B CS
= + +
⇒
( )2 2 2cot cot1
3 cot4
A g B Cg g+= + +
ក-.ង ABC∆ ន cot .cot cot .cot cot .cot 1gA gB gB gC gC gA+ + =
9)យយ)ងន 2cot
cot 22c
1
ot
gg
g
ααα−= , ក-.ង* 2 1 2cocot t .cot 2gg gα α α= +
( )13 3 2 cot .cot 2 cot .cot 2 cot .cot 2
4S gA g A gB g B gC g C= + + + +⇒
( )16 2 cot .cot cot .cot cot .cot
4gA gB gB gC gC gA= + + +
1(6 2) 2
4= + =
+. K 6ច!ន.ចក 8ប# CT 9)យ L 6ច!ន.ចប/ប#ង"ង#Gន%ងជង BC
* (*)1
2CK CT= Jញ,ន L 6ច!ន.ចក 8 BC ,
2 21
2.CK CTCL CT= = ,
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 3
I 2 / 4 36a = , I 1)12 (a =
Lន.<នMទ%បទក:.ន.ក-.ង BCT , យ)ង,ន
2 2 2 2 . .cosC BT BC BT BCT B= + −
2 2/ 4 144.cos72 aa B⇔ + −=
3(2),cos
2B⇔ = Fង (1)
មO/ងទP, Lន.<នMទ%បទក:.ន.ក-.ង ABC , យ)ង,ន
2 2 2 2 .cos cos / )2 (3c a cb B a ba B= + − ⇔ =
ព (1), (2), នQង (3), យ)ង,ន ( , , ) (12,8,8)a b c =
=. Fង 3 211( 7) 8P x xx x m= − − +
យ)ង,ន 3 191)( ) ( ) (modP bx x a≡ ++
3 2 2 3 3 23 3 1 (mod1 87 191)ax a x a b x x x mx⇔ + + + + ≡ − − +
2
3
11 191) (1)
3 87(mod 191
3 (mod
) (2)
(mod 191) (3)
a
mb a
a
≡ −
⇔−
≡ −≡
2(1) 3 (mod (180 191) 60 191) 19mod 3 1)87(moda a a⇔ −≡ ⇔ ≡ ⇒ ≡
C:ចន m∀ ∈ℤ , នច!នBនគ# ,a b C)មD 3( ) (mod19) )( 1P a bx x≡ + +
ងRម)8 1916ច!នBន;C8នSង 191 3 2k= +
3 3( ) 191) ( ) 1( ) (mod ( ( 1) od )m 9P i iP j j aa+≡ ⇒ ≡ +
Fង , v j au i a= = ++ , *
3 3 3 3191) 191(mod (mod )k ku u vv≡ ⇒ ≡
3 2 3 2 191(mod 191) 1m 91)( odk kv v v vu +≡ ≡ ≡ (ទ%បទ Ferma ) (4)
2 3 3 3 3 19( o 1m d )k kv u v u +⇒ ≡ ≡
3 2 3 3 3 3 1 3 2 3 1 191 3 1. . . . 1(mo 1)d 9k k k k k k ku uv u u u u uu u+ + + + +⇒ ≡ ≡ ≡ ≡
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 4
3 2 191 ( 191m )odku u u+≡ ≡ ≡ (5)
ព (4)នQង (5)Jញ,ន 191) 191(mod (mod )u v i j≡ ⇒ ≡
C:ចន ប) 191, (mo1,2, ... , : 19 )d 1i j ji∀ ∈ ≠ * ( )( ) (mod191)P jP i ≠
Jញ,ន ន 1, 2, ...,191n ∈ V/ង 191(mod 91) )( 1P n ≡ , I ( ) 191P n ⋮
C:ចន ច!@គប#!A8គ#ប# m , ;ងនច!នBនគ# n C)មD 1( 9) 1P n ⋮
E. ( )( ) ( )( ) ( )( )26 6 3 3 6 6 6 3 3 6 6 6 6 3 63 33 32 2b c a b a a ca c a b a a ca c+ + = + + + = + + +
( ) ( )12 3 6 3 9 3 6 6 6 6 6 63 22a b c c b ca aa b ca + + += + +
( )36 6 6 4 2 6 2 4 6 6 2 2 2 2 2 2 23 3 23 2b a b c a b c a ca a ab a c b a c≥ + + + + += =
Jញ,ន ( )( )
4 4 2
4 2 2 2 2 2 2 224 6 6 33 3 a
a a a
a aa a a b a c b cb c≤
+ + + ++ +=
+
C:ចW- ;C ( )( )
4 2
2 2 224 6 6 3 33
b b
a b cc ab b b≤
+ ++ + +
( )( )
4 2
2 2 224 6 6 3 33
c c
a b cb bc a c≤
+ ++ + +
ប:កLងXន%ងLងXAនមYពZង8), យ)ង,ន
( )( ) ( )( ) ( )( )
4 4 4
2 2 24 6 6 3 3 4 6 63 3 3 3 4 6 6 3 33
1a b c
a a a b b b cb c c a ba cb+ + ≤
+ + + + + + + + + .
'()&
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 5
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ១
. យម 2 211 1 (1)2
2x x x− − = −
. ពQនQប ច!នBនពQមQនL< Qជ?ន ;បប[8 1 2 9, , ...,a a a មQនផ8ប:ក() 1
Fង ( )1 2 3 1;6K k k k kaS a ka a+ + ++ == + + , Fង 1 2 3 4 5 6, , ,x ,ma ,S S S SM S S=
ច:ក!#!A8:ចប!ផ.ប# M
+. កគប#Lន.គមនM :f →ℝ ℝ ផ0\ង]0 #8ក_
( ) ( )( ) ( )( )3 2 22 3 , (1)f x y f x x yy f y f xy + + =− ∀ ∈+ ℝ
=. គ ABC ផ0\ង]0 # 1tan tan
2 2 2
A B = យប>? ក#5 8ក_ $!,ច#
នQងគប#Wន#C)មD ABC ;កងគI 1sin sin sin
2 2 2 10
A B C =
'()&'()&'()&'()&
ចេលយ
. ( )22 21 1
1 12 2
x x x x− − = − −
8កខ_ C)មDម (1)ក!# 2 01 1x x− ≥ ⇔ ≤ ព8*
2 21(1) 1 1 2
2x x x=⇔ − − −
ម នH 2 012
21
x x⇔ ≥ ⇔ ≤−
យ)ងន 2 201 2 1 xx x≥ ⇒ − ≥− C:ច*
( )( )2 2 2(2) 1 2 1 1x x x x x x− − = − +⇔ − −
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 6
( )2
2
1
2 1 1
x x
x x
− = − + =
⇔
2 2
2
2
1
2 21 2
21 2 61
1 42 2 2.
1
2
0
0
xx
xx
x x xx x
xx
≥− =⇔ ⇔ ⇔≤
−
= = − − = − = − =
C:ចន ម (1)នHព ( )1 2
2 1; 2 6
2 4x x= = −
. 1 2 3 4 5 6, , ,x ,ma ,S S S SM S S=
9
1 2 3 4 5 6 2 3 4 6 7 81
4 2 3 3 212 4 4ii
S S S S S S a a a aM a a a=
⇒ ≥ + + + + + + + + + + += ≥∑
(@ 9
1
1; 0, 1;9ii
aa i=
= ≥ =∑ ) 1
3M⇒ ≥
>a " "= ក)នព8 1 5 9
2 3 4 6 7 8
1
30
a a
a a
a
a a a a
= = =
= = = = = =
ច!8)យ min
1
3M =
+. ជ!នB 3y x= ច:8 (1) យ)ង,ន
( ) ( )3 2 6 3(0) 2 3 ( ) ( ) (2)x xf f f x fx x+ ++ =
ជ!នB ( )y f x= − ច:8 (1) , Jញ,ន
( ) ( )3 2 2( ) (2 ( ) 3 () ( ) (30 ))f x xf f xf x x f f+ +− =
ព (2)នQង 3 2 3 94 ( ) 3 ((3 ). 0) ;f x f x x x x⇒ − − = ∀
( )( )3 2 3 6( ) ( ) ; (4( )) 4 0x x f xx x xf f x⇒ + ∀=+−
ឃ)ញ5 2 3 6( ) . ( ) 0 ;4 0x x f x x xf + + > ∀ ≠
ព (4)យ)ង,ន 3( ;)f x x x= ∀
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 7
កជ!នBច:8c)ង< Qញ, ផ0\ង]0 #8កខ_ បdន
ច!8)យ 3( )f x x=
=. យ)ងន ( ) tan ( ) tan ( )( )( )2 2
A BS p p a p p b p p a p b p c= − = − = − − −
2
2( )(tan tan
2 2 )p a p b
A B S p c a b c
p p a b c⇒
− −− + −= = =
+ +
2( ) 21
tan tan 32 2 2
(1)A
a b c a bB
a b cc⇔ + − = + + ⇔= + =
មO/ងទP 2. . ( )( )( ) (24 4
)abc r
S pr Sp abc p p a p cR R
b p⇒ = − − −= = =
2(1)
p c
p c c
= − =
⇔
ព (2) Jញ,ន 2 ( )4
ar
b p aa p bbR
− + +=
21 24
(3)r
ab cR
− =
⇒
Fមប!Sប# 1 1sin sin sin
2 2 2 10 4 10
A B C r
R=⇒=
ព (3)យ)ង,ន 220
9ab c= , BមW- ន%ង (1) យ)ងJញ,ន a នQង b 6ប Hប#
ម 2 21 23 0;
20 5 4;
9 3 3ct ct c t c t− + = = =
យក a b≥ , យ)ង,ន 5
34
3
a c
b c
= =
2 2
2 2 2 24 5
3 3c c a A Cb c c B
⇒ + = + = =
⇒ ∆
;កងង# A
ផ0.យមក< Qញ, 4ប5 ABC∆ ;កងង# A យ)ង,ន
2 2 2 2 2 2
2 2 2
(3 ) 4
1 4 ( 2 ) 3
(1
2
)b c b c
a R a R a R
a c b b r
b r p c c rS bc pr
= = = − == + ⇒ = + = = = = =
⇒
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 8
ព (1) យ)ង,ន 2 2 25 2 4sin sin sin
5 5 2 2 2 5
a R r A B Cr R
a r R
== = = =
⇒ ⇒ ⇒
1sin sin sin
2 2 2 10
A B C⇒ =
(ប>e ប#8!f#g<,នយប>? ក#)
'()&
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ២
. គ< Qម 2004
2006 2004
20061
116
1
62
xa x x
a x
+
+ −≤+ ++ −
កច!នBនពQ a ធ!ប!ផ.C)មD< Qម នH;ពប/.i
. គ ,m n 6ពច!នBនគ#< Qជ?ន យម
2 22 1 2 1
1 1sin cos
sin cosn n
m mx
x x+ +=+ +
+. គ I 6ផjQង"ង#$ %កក-.ង មQនម, ABC M 6ច!ន.ចក 8 IC ,
N 6ច!ន.ចក 8ប# AB J 6ច!ន.ចក 8ប# MN Fង , ,x y z 6ប*0 #
បPងW- #Fម , ,A B C 9)យប*0 #នមBយk;ចកប Q ABC 6ព;ផ-ក
()W- យប>? ក#5 4ប*0 # , ,x y z នQង IJ #Fមច!ន.ចBមW- មBយ
=. គ n 6ច!នBនគ#< Qជ?ន ពQនQFSង 2nជBCកនQង 2n ជBឈ បm
នមBយkនច!នBនមBយច:8 ;C8ច!នBន*nQoក-.ង!ន.! 2...1,2,3 , 4, n ,
បmពផ2ងW- ច!នBនផ2ងW- កច!នBន N ធ!ប!ផ.;C8ន8កp
ច!@គប#បPបច!នBនC:ចZង8) នជBCកមBយ IជBឈមBយ ;C8o8)
ជBCក IជBឈ*នពច!នBន ,p q ផ0\ង]0 # | |p Nq− ≥
'()&'()&'()&'()&
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 9
ចេលយ
. c)ង< Qញ ( )22006 2004
2006
160
1
1
a x x
a x
+− −
+ −≤
+ ច!@ 2 1x ≤
ប) 0a < < Qម 20061 0a x+ − < នច!8)យSប#មQនL#
ប) 0a ≥ យ)ង,នម 2004 200616 1a x x= + − −
Lន.គមនMoLងXZង ! 6Lន.គមនMគ: Lន.គមនMក)ន8) [0;1] C:ច*ម នH
ព8 a nQoច*q 3, 17 , Jញ,ន!A8ធ!ប!ផ.ប# a គI 17 ,
ព8* HJ!ងពគI 1, 1x x= − =
. Fង cosin , .s ; 0u vv xx u= = ≠ 9)យ 2 2 1u v+ =
យ)ង,នម 2 22 1 2 1
1 1n nm m
u vu v+ +=+ +
ប) 0uv < LងXង:ច6ង 1, LងXងទPធ!6ង 1, មQនផ0\ង]0 #
Lន.គមនM 22 1
1nm
y xx ++= ច.o8) [ 1;0)− C:ចន ប) , 0u v < ; u v≠ គIមQនផ0\ង]0 #
ពQនQ , 0;v vu u> ≠ យ)ង,ន ( )( )2 12 2 2 1 2 1.mn n m mu u vv u v
+ + +− = −
I ( )( )2 1 2 1 2 3 2 3 1 2 1... ( ) ( ) ( )n n n n n mv uvu u uvv uv u v− − − − − +++ + + + + 2 2 2 1 ...m m mv u vu −= + + +
LងXZងឆ"ង:ច6ង 2 2 2 2 1 2 1 2
..1 1 1 1 1 1 1
12 2 2 2 2 2
. 22
n n m m m− − + + + +
< =
+ + +
LងXZង !ធ!6ង 1
2 2 1
2
m
m mvu−
+ ≥ C:ច* LងXZង !:ច6ងLងXZងឆ"ង
C:ចន sin cosx x= I 4
x kπ π= +
+. Lemma: ABC , Fង ', ', 'A B C PងW- 6ច!ន.ចក 8ប#ប ជង ,BC
,CA AB ប ប*0 # #Fមក!ព:8នមBយប# ' ' 'A B C 9)យ;ចកប Q
' ' 'A B C 6ព បព"W- ង#ច!ន.ច;C86ផjQង"ង#$ %កក-.ង ABC
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 10
ពQ6C:ចន : Fង M 6ច!ន.ចnQo8) ' 'B C V/ង 'A M ;ចកប Q
' ' 'A B C 6ព * ' ' 'MC p b= − 9)យ ' ' 'MB p c= − ;C8 ' ( ' ' ') / 2p a b c= + +
Jញ,ន ( ' ') ( )
'
( ) ( )Ap c pC p b A B A C p b A B
aA
c
aM
′ ′ ′ ′− −=′ ′ ′ ′ ′ ′+ − −′ = =+
) )( (
2
Ap c C AB
a
p c+− −= −
មO/ងទP, យ)ងន
2 A I aA A bA B cp A C′ ′ ′ ′= + +
( ) ( )( ) ( )
2
AB AC BCa b cAB p c Ap b C
+= − − −
+ −= − −
C:ចន ,',A I M #ង#ជBW-
cប#មក8!f#< Qញ Fង G 6ទបជ.!ទ!នង# ABC យ J 6ទបជ.!ទ!ងន#
ប#បBនច!ន.ច , , ,I A B C * , ,I G J ង#ង#ជBW-
Fមចxប#ប!;8ង$!ងផjQ G Fមផ8ធPប 1/ 2− ប!;8ង A G6 ',A B G6 ',B C G
6 'C ប!;8ងប*0 #J!ងប , ,x y z G6ប*0 #J!ងប (Fម Lemma) បព"W- ង# I
ប:កBមJ!ង , ,I G J #ង#ជBW- Jញ,ន , , ,x y z IJ #Fមច!ន.ចBមW- មBយ
=. ពQនQបPបPបC:ចZង ម
22 1n n− + ... ... 22n ... 24n
.
.
.
2 1n +
1n + 2n 22 1n n+ + ... 22 2n n+
1 2 ... n 22 1n + ... 22n n+
M
A
B C 'A
'B 'C
I
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 11
ប)ន ,i j o8)ជBCកV/ង i Nj− ≥ * 22 1nn N+ − ≥
ប)o8)ជBឈ * 22 jn n i N− ≥ − ≥ C:ចន 22 1n nN ≤ + −
ពQនQ 2 12 nN n= + −
Fង 2 2 2 21, ; 32, ..., 1 , 3 2, ..., 4A B nn n n n= =− + + ច!@គប# i nQoក-.ង ,A j nQ
oក-.ង B , យ)ង,ន ( )2 2 23 1 2 1 (**)i j n nn n n≥ − − + + −− =
ច!@ជBCក ;C8នផ0.កd.ប# A , យ)ងy56ជBCបភទទ, ជBឈ;C8
នផ0.កd.ប# A g<,នy56ជBឈបភទទ ច!@ជBCក ;C8នផ0.កធ
d.ប# B , យ)ងy56ជBCកបភទទ, ច!@ជBឈ;C8នផ0.កd.ប# B
យ)ងy56ជBឈបភទទ
Fង ,p q PងW- 6ច!នBនជBCក នQងជBឈបភទទ * 2. 1p n nq ≥ − + , C:ច*
242 4 4 2 1p q pq n nn≥ ≥ − >++ −
Fង ,r s PងW- 6ច!នBនជBC, នQងជBឈបភទទ * 2 1.r s n≥ + , C:ច*
24 4 22r s rs n n+ ≥ + >≥ C:ចន 4 1p q r s n+ ≥+ ++ , Jញ,ន នជBCក IជBឈ;C86បភទទផង នQង6
បភទទផង ព (**) Jញ,ន!A8ធ!ប!ផ.ប# N គIp 22 1n n+ − '()&
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 12
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ៣
. កគប#!A8 , , , , [0,1]a b c d e∈ C)មD
41 1 1 1 1
a b c d eA
bcde cdea deab eabc abcd= + + + + =
+ + + + +
. គ 3 2( )f x ax bx cx d+ + += V/ង 1, [ 1,1( ] (1) )xf x ≤ ∀ ∈ −
កច!នBនថ k :ចប!ផ.C)មD 2 2 , [ 1,1],3 bx c k xax f+ + ≤ ∀ ∈ − ∀ ផ0\ង]0 # (1)
+. ក-.ងបqង#, គ ម|ង2 ABC ផjQ O ប*0 # ( )d < Q8ជ.!< Qញ O #ប ប*0 #
, ,BC CA AB PងW- ង# , ,M N P យប>? ក#5 4 4 4
1 1 1T
OM ON OP= + +
គIមQន;បប[8
=. យប>? ក#5 ច!@គប#!A8គ#ប# m , នច!នBនគ# n C)មD
3 211 87n n n m− − + ;ចកច#ន%ង 191
'()&'()&'()&'()&
ចេលយ
. យមQនធ"),#បង#8កទ:G 4ប5 (*)b c da e≤ ≤ ≤ ≤
ព8* 1 1 1 1 1 1
a b c d e a b c d eA
abcde abcde abcde abcde abcde abcde
+ + + ++ + + + =+ + + + + +
≤
យ [ ], , , , 0,1a b c d e ∈ * :
(1 )(1 ) (1 )(1 ) (1 )(1 ) (1 )(1 ) 0 (1)abc de ab c d e b a− − + − + − − ≥− + − − 4 4(1 ) (2)abcda b c d e b dee a c⇒ ≤ + ≤ ++ + + + .
C:ច* 4A ≤ ប) 4A = *មYពក)នo (1)នQង (2)Jញ,ន
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 13
0
1
1
1
0
0
11
1
a
de
c
d
abc
e
de
a
b c d e
b
⇒ ==
= ⇒
= ∨ ==
= = = = = =
ផ0\ង]0 #c)ង< Qញ ព8 , 10 b c da e= = = == * 4A =
C:ចន ប បព|ន!A8;C8g<ក ( , , , , )a b c d e 6ប ច!~#ប# (0,1,1,1,1)
. Fង ( 1 / 2), (1 / 2), (1( 1 )), B f C f D fA f = − = == − *
2 4 4 2 2 2 2 2,
3 3 3 3 3 3 3 3a A B C D b A B C D= − + − + = − − +
4 4 2 2,
6 3 3 6 6 3 3 6
A D A Dc B C d B C= − + − = − + + −
2 2 24( ) 3 (12 (3
62 8 1) 1)
3
A Bh x ax xbx xc x x+ + = − − − + − −= −
2 24(3 1) (12 8 1)
3 6
C Dx x x x+ − + + −
Fមប!Sប#Jញ,ន 1, , ,A B C D ≤ , C:ច*, ប) [ ]1,1x∀ ∈ − គ,ន
2 2 2 28 1 1 11 4 4 1
( ) 12 3 3 126 3
8 13 6
h x x xx x x xx x≤ − − − − ++ − ++ −+
យ ( )max ,A B A B A B+ = − + *ច!@ [ ]1,1x∀ ∈ − យ)ង,ន
( )2 2 28 1 8 112 12 max 16 , 2 2 224x xx x x x+ − − −+ ≤=−
( )2 2 21 13 3 max 2 , 2 46x x x xx x+ =+ − − − − ≤
[ ]22 16( ) 9, 1,1
6 3h xx + = ∈ −⇒ ≤ ∀
ច!@ 3( ) 4 3f x x x= − * [ ]1,1x∀ ∈ − Fង cosx t= យ)ង,ន ( ) co 1s3f x t= ≤
9)យ [ ] [ ]1,1 ,
2
1
2
1max 3 max 12 92 3bx cax x
− −+ + −= =
Jញ,ន ច!នBនថ k :ចប!ផ.;C8ផ0\ង]0 #!) 8!f#គI 9
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 14
+. Fង ', ', 'A B C PងW- 6ច!ន.ចក 8 , ,BC CA AB * ' ' ' / 2OA OB OC R= = =
Fង i6<. Qចទ|កFប# ( )d Fង ( ) 2
, ,3
i OA xπα′ = =
*
( ) ( ) ( ), , , 2i OB i OA OA OB x kα π= + =′ ′ ′ ′ + +
( ) ( ) ( ), , , 2i OC i OA OA OC x kα π′ ′ ′= + −=′ +
4 4 4
2
1 ' ' '
2
OA OB OCT
OM ON OPR
=
+ +
4 4 44
cos ( ) cos ( )16
cos x xR
α α α+ + + − =
2 2 24
) (1 cos(2 24
(1 cos2 (1 cos(2)) 2 ))x xR
α α α = + + ++ + − +
[ ]4cos(2 2
43 ) cos(2 2 ))2(cos2 x x
Rα α α+ + + −= + +
2 24
2 cos (2 2 ) cos(2 2 )) 14
os )c (xR
xα α α + + ++ −
[ ]cos(2 2 ) cos(2 2 )2sin cos2x x xα α α+ + + − =
[ ]) sin(2 ) sin(2 3 ) sin(2 ) sin(2 ) sin( 3 )si 2n(2 x x x x x xα α α α α α+ − − + + − + + − − −=
3 ) sin(2 3 ) sin(2 2 )sin(2 sin(2 2 ) 0 (2)x xα α α π α π+ − − − == = + −
2 2 22 cos (2 2 ) cosco (s 2 2 )x xα α α+ + + − [ ]cos1
3 cos42
(4 4 ) cos(4 4 )x xα α α+ + −= ++
[ ] [ ]cos(4 4 ) cos(4 42sin 2 cos4 sin) 4 ) sin(4 4 ) ( )(4 30x xx α α α α π α π+ + − −= =+ + −
Fម 4 4
4 3 18(1),(2),(3) 3
2T
R R = + =
⇒ មQន;បប[8
=. (8!f#នg<,នជ)6<Q>a បcង- ម)8ច!8)យo< Q>a បcង) '()&
M B C
A
'A
'C 'B
O N
P
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 15
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ៤
. Fងជ)ងប ក!ព#ក-.ង ABC យ ,',A B C′ ′ គ*ប ម.!ប#
' ' 'A B C 6Lន.គមនMAនប ម.! , ,A B C យប>? ក#5 ម.!ធ!ប!ផ.ប#
' ' 'A B C V/ងfចក()ន%ងម.!ធ!ប!ផ.ប# ABC
)ព8;C8ក)នមYព?
. យប>? ក#5 2 9
7 7
xyzxy yz zx ≤+ + + , ក-.ង* , ,x y z 6ប ច!នBនពQមQន
L< Qជ?នផ0\ង]0 #8ក_ 1x y z+ + =
+. គ , ,a b c 6ប ច!នBនគ#< Qជ?នផ0\ង]0 #8ក_ 1 1 1
a b c− = 9)យ d 6B;ចក
Bមធ!ប!ផ.ប#ពBក យប>? ក#5 abcd នQង ( )d b a− 6ប ច!នBន ,កC
=. យប>? ក#5បព|នម 12
2
xy yz zx
xyz x y z
+ + = − − − =
នច!8)យ;មBយគ#ក-.ង!ន.!
ប ច!នBនពQ< Qជ?ន យប>? ក#5បព|ននច!8)យច!@ , ,x y z 6ច!នBនពQ
ផ2ងW-
'()&'()&'()&'()&
ចេលយ
. ពប ច. $%កក-.ង, យ)ង,ន ប ម.!ប/.នW- C:ច:បZង ម
yប ម.!យ , ,α β γ C:ច:ប
C:ច* យ)ង,ន ' 2 ' 2, , ' 2A B Cα β γ= = =
9)យ , ,A B Cβ γ γ α α β= = =+ + +
Jញ,ន ' ; ,B C A B C A B CA B C A = + − =− ++ −=
'C
A
B C 'A
'B
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 16
c:<យ)ងងRឃ)ញ5 ''A B C AB C A B B A≥ ⇔ ≥ + − ⇔− ≥+
C:ច*, ប) A B C≥ ≥ * '' 'C B A≥ ≥
*! 'C A A B C B CA ≤ ⇔ ≤ + − ⇔ ≥ :ពQ
>a មYពក)នព8 ABC 6 ម|ង2
. ប) 7
9x ≥ , * 9
17
x≤ Jញ,ន 9
7
xyzxy ≤
6ងនGទP 2( )
9y z+ ≤ , * 2 2
9 7xy xz+ < <
C:ច* ក-.ងក ន យ)ងទទB8,ន 2 9
7 7
xyzxy yz zx+ + < +
c:<4ប5 7
9x < , 9)យC:ច* 9
1 07
x− >
យ)ង,ន 2(1 )
4
xyz
−≤
ព* យ)ងWន#;g< យប>? ក#5 29 (1 ) 2
1 (1 )7 4 7
x xx x
− − + −
≤
2(7 9 )(1 ) 28(1 ) 8xx x⇔ −− − + ≤ .
3 2 23 5 19 ( 1)(3 1)0 0x xx x x⇔ + − + ≥ ⇔ − ≥+ .
< QមYពនពQ6នQចj (@ x មQនL< Qជ?ន), ច!@មYព ក)នព8 1
3x =
C:ចន < QមYពg<,នយប>? ក#, ច!@>a មYពក)ន8.F;
1
3x y z= = =
+. Fង , ,a b c
BAd d d
C= = = , C:ច* ,,A B C នB;ចកBមធ!ប!ផ.គI 1
យ)ង,ន 1 1 1
A B C− = , * ( )AB C B A= −
យ)ងយប>? ក#5 ( )B A− g<;6ច!នBន ,កC ប)ផ0.យមក< Qញ, គIg<នច!នBន
បម p មBយV/ង "|យគ.ធ!ប!ផ.ប# p ;ចកច#ន%ង ( )B A− គI 2 1rp + ច!@ r
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 17
6ច!នBនមBយ
c:< ប) 1rp + ;ចកច# A , *ក;ចកច# ( )B B A A= − +
Jញ,ន 2 2rp + ;ចកច# AB ; p មQនច;ចកច# C (@ , ,A B C មQននកF Bម),
Jញ,ន 2 2rp + ;ចកច# ( )B A− មQនម9.ផ8! C:ច* ;C8ចGBច*គI
rp ;ចកច# A , 9)យយ)ងក,ន8ទផ8C:ចW- ច!@ B ព* ;C8ចGBចគI
2rp ;ចកច#ន%ង AB , BចគI;ចកច# ( )B A− មQនម9.ផ8! ក *បe ញ5
( )B A− 6ច!នBន ,កC ; ( ) ,ABC B A ABAB− = * ( )ABC B A− g<;6ច!នBន
,កC, BចគI ( )ABC B AABC
B A
− =−
ក6ច!នBន ,កC;C
C:ច* 2( ( ))b a d d B A= −− នQង 4abcd d ABC= .ទ;6ច!នBន ,កC
=. 6ក#;ង (2,2,2)6ច!8)យមBយ (ក-.ង!ន.!ប ច!នBនពQ< Qជ?ន)
ប)យ)ង$#ទ.ក5 z C:ច6,នC%ង!A8, *គIយ)ង,ន
2
2
2
12 1
11 2
2
1
zx y
z
zxy
z
z
++
− + =
++
=
C:ច* 8កខ_ $!,ច# នQងគប#Wន#C)មD , ,x y z 6ប ច!នBនពQ< Qជ?នគI
2 2 24( 2 12(11 2 )( 1) )) (*z z zz +− > + + .
(d*,ន5 x នQង y .ទ;6ច!នBនពQ) 9)យ 2
11z > (C)មD ,x y នQង z .ទ;< Qជ?
ន) ព8* (*) q យG6
4 3 2 28 69 52 44 0 (2 11)(2 1) 04 ( 2)z z z z zz z+ − + + ≤ ⇔ + + ≤−
ក នg<,នផ0\ង]0 #ព8 11 1
2 2z ≤ −− ≤ នQង 2z =
*នH<Qជ?ន;មBយគI 2z = (Bចព* , 22x y= = , 6ងនGទP យ
;បព|នម នឆq.ច!@ ,x y នQង z )
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 18
ច!@ប H6ច!នBនពQផ2ងទPយ)ងគBពQនQម)8ង#ប !A8nQoច*q
ព 11
2− G 1
2−
ច!@ 1z = − យ)ង,ន,នHមQនផ2ងW- 11( 1, 1, )
2− − − , $8
ច!@ 2z = − *,ន 12 2 21 12 2 21,
5 5x y
+ −= − = − , ពBក.ទ;ផ2ងW-
'()&
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ៥
. គ 3ច!នBន< Qជ?ន , ,a b c ផ0\ង]0 # 1 1 13
a b c+ + = យប>? ក#5
3 3 34 4 43 3 3 2 2 2a b c a b c+ + ≥ + +
. យបព|នម
1 2 3 4
1 2 3 4 1 2 3 4
1 2 3 4 3 4 1 2
1 2 3 4
0
( )( ) 0
( ) ( ) 0
0, 0, 0, 0
x x x
x x x x x x x x
x x x x x x x x
x x x
x
x
+ − <+ + − − <+ − + <
> > > >
+. កគប#ប ច!នBនគ#ធម(6Q;C8ន8ខបខ0ង#C)មDច!នBននមBយk6មធមនព"ន
ប#ប ច!នBន;C8Jញចញពច!នBន*យ< Qធច!~#ប 8ខប#ច!នBន
*
=. គ*ច!នBនជងប#ព9. នQយ|;C8នក!ព:8 4W- , , ,A B C D ផ0\ង]0 #
ទ!*ក#ទ!នង 1 1 1
AB AC AD= +
'()&'()&'()&'()&
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 19
ចេលយ
. Lន.<នM< QមYពក:., ពប!Sប#, យ)ង,ន
331
11 1
3 a a ca b c
bc b= ⇔+ + ≥ ≥
Fង 12 12 12, ,a y bx z c= ==
8!f# q យG6
ច!@ 0, 0
1
0, y z
z
x
xy
> >≥
>
, យប>? ក#5 9 9 9 8 8 8x y z x y z+ + ≥ + +
Lន.<នM< QមYពក:. 9ច!នBន
9 9 8
9 9 8
9 9 8
... 9 (1)
... 1 9 (2)
... 1 9 (3
1
)
x x
y y y
z z
x
z
+ + ≥+ + + ≥+ + + ≥
+
យ)ងន 8 8 8 8 8 833 3 (4)y z y zx x+ + ≥ ≥
ប:កLងXន%ងLងXប < QមYព (2), (3)(1), , (4)យ)ង,ន8ទផ8;C8g<យប>? ក#
9 9 9 8 8 8x y z x y z+ + ≥ + +
. ពQនQបព|ន< Qម 1 2 3 4
1 2 1 3 1 4 2 3 2 4 3 4
1 2 3 1 3 4 1 3 4 2 3 4
1 2 3 4
0
0
0
0
x x x
B x x x x x x x x x x x x
C x x x x x x x x x x x x
D x x x
x
x
A − + + >= − −= −
− − + >= + − − >= >
Fង 1 2 3 4)( )( )(( ) ( )x x x xx x xf x x= − − + +
គ.ព*q យ)ង,ន 4 2 2( ) Axf x Cx Bx x D+ + + +=
យប មគ. , , , 0A B C D > *ម ( ) 0f x = មQនចនH x < Qជ?នទ
C:ច* 1 2, 0x x ≤ , ផ0.យពប!Sប#
C:ចន បព|ន;C8គIមQននH
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 20
+. ច!នBន;C8g<កនSង abc , ច!@ ,,a b c ∈ℕ 9)យ 9, 0 ,1 9a b c≤ ≤ ≤ ≤
Fមប!Sប# 189 81 1082
bca caba c ab b c= ++= ⇔
7 3 4 7 (1)( ) 4( )a b c a b c b= + − = −⇔ ⇔ .
Jញ,ន 4( ) 7 (2)c b− ⋮
7 (3)c b−⇒ ⋮ .
យ 0 , 9b c≤ ≤ * 99 (4)c b ≤− ≤ −
ព (3)នQង (4)Jញ,ន 7,0, 7c b− = − ពQនQប ក
ក ទp 7 7 9b cc b− = − ⇔ = + ≤
0,1, 2 7,8,8 3,4,5ac b = ⇒ =⇒ = ⇒ 370, 481, 592abc⇒ = .
ក ទp 0 cb bc − = ⇔ =
111, 222, ...., 999a b c abc⇒ = = ⇒ = .
ក ទp 7 7 9c bc b ⇔ == + ≤−
7,80,1,2 4,5,9 6,b c a= =⇒ ⇒ = ⇒ .
470, 581, 692abc⇒ = .
C:ចន នJ!ងL# 15ច!នBន;C8g<កគI
481, 592, 581, 692, 222,370 333, 470, 111, ..., ,999
=. 4ប5ព9. $%កក-.ងង"ង#ផjQ O ! R
Fង ( )0 01200AOBα α= < < ង# OH AB⊥ , Jញ,ន 2 2 sin2
AB HB Rα= =
C:ចW- ;C 32 sin , 2 sin
2AC R AD R
αα= =
ជ!នBច:8ប!Sប# 1 1 13sinsin sin
2 2
α αα= +
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 21
C:ច* 3 3sin sin sisin sinn 0
2 2 2
α α αα α − =
+
( )1 5 1 3 1cos cos cos cos cos 0
2 2 2 2 2 2 2cos2
α α α α α α − − − − =
−
I 3 5cos cos2 c c
2o 0sos
2
α αα α + − = +
7cos sin sin 0
4 4 2
α α α =
ច!@8កខ_ 00 120α< < , យ)ង,ន
0
07 7 300cos 0 90
4 4 7
α α α= =⇒ ⇒ =
C:ចន ព9. នជងច!នBន 7 '()&
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ៦
. ).a គ 3ច!នBន< Qជ?ន;C8នផ8ប:ក() 4 យប>? ក#5ផ8ប:កAនពច!នBន
កយក-.ង 3ច!នBន*គIមQន:ច6ផ8គ.Aន 3ច!នBន*ទ
).b គ* 0 0 0 0sin 69 sin183 sin 21s 39 3inS + + +=
. ).a យម 1 31 0
4 2
x
x x
+ − =+ +
).b Fង ,x y PងW- 6" #ម.!J!ងពក-.ងព9. នQយ| 1D នQង 1D យC%ង
5 5 7 0x y− = កច!នBនជងប# 1 2,D D
+. យប>? ក#5 ច!@គប#ច!នBនគ#< Qជ?ន n គ,ន
3 3 3
...1 1 1 1
32 4. (1 )3 2 .. 3 nn
++ + + <+
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 22
=. គ ABC ម,ង# A យC%ង5 L:ង# H ប# nQ
o8)ង"ង#$ %កក-.ងប# * ច:គ* cosA
'()&'()&'()&'()&
ចេលយ
. ).a Fមប!Sប#គ 3ច!នBនពQ ,,a b cនQង 4a b c+ + =
យមQនធ"),#8កទ:G, យ)ងយប>? ក#5 a bb a c+ ≥
ព 2( ) 4aa b b+ ≥ Jញ,ន 2( ) 4( )a b ca b c ≥ ++ +
24( ) 4( )1 ) 66 116(a ba b c a b c abc⇔ ≥ + +⇔ ≥ + ≥ .
aa bb c⇔ + ≥ .
មYពក)នព8 1 2, ca b= = =
).b យ)ងន
0 0 0 0cos15 2sin198 cos152sin54S += .
( ) ( )0 0 0 0 0 02cos15 sin54 2cos15 sin5sin198 sin14 8= −=+
0 0 0
0 0 0 0 0 0 0
0 0
cos36 sin18
cos36 sin18 cos
4cos15
4cos15 2cos18 cos36 sin3615
cos18 cos18
=
= =
0 0
00
cos15cos15
s
s
i 72
i
n
n72= =
6 2
4S
+=
. ).a 8កខ_ 0x ≥
ម ;C8 1 3 4 2 0x x x+ − − + =
3 2 4 1 8 2 (4 1) 3 2x x x x x x x − + = − − = − + +⇔ ⇔
4 1 0
(4 1) 3 2 2 03 2 2
xx x x
x x
− = − + + − = + + =
⇔
⇔
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 23
1* 4 1 0
4x x− = ⇔ =
* 3 2 2x x+ + = យ)ងយ,នH 7 3 5
8x
−=
ន-Q ន ម នHពគI 1 7 3 5;
4 8x x
−= =
).b Fងច!នBនជងប#ព9. នQយ| 1 2,D D PងW- យ n នQង k
8កខ_ ,n k 6ច!នBនគ#< Qជ?ន 9)យ 3 k n≤ ≤
យ)ង,ន " #ម.!នមBយkប# 1D គI ( 2)nx
n
π−= 9)យប# 2D គI ( 2)ky
k
π−=
5( 2) 7( 2)5 7 0
n kx y
n k
π π− −− =⇔=
5 10 7 14nk k nk n− = −⇔ .
75 7
5
nk nk n
nk⇔ ⇔ =+ =
+
357
5k
n⇔ = −
+
យ k 6ច!នBនគ#< Qជ?ន* 35 ( 5)n +⋮
C:ច* ( 5)n + g<()ន%ង 1,5, 7 I 35
;យ 3 ( 5) 35nk ≥ ⇒ + =
C:ចន 30n = 9)យ 6k =
+. ច!@គប#ច!នBនគ#< Qជ?ន k , យ)ងន
3 3
3 3 3
11
1 1)
1
(
k k
k k k k
+ −+
− =+ ( )32 23 3 3 3. 1. ( 1) ( 1)
1
k k k k k k+ + + + +=
Jញ,ន ( )3 3 323 3 3
1 1 1 1
3(1 ).31 . 1. ( 1)k k kk k k k− > =
++ + +
C:ចន 3 3 3
1 1 13
(1 1).k k k k
< −+ +
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 24
*! 3 3 33 3 3 3
1 1 1 1 1 1 1 13 1 3
2 3 4. ( 1)... ...
2 3 2 2 3 1. n nn
+ + + < − + − + <
− + +
+
=. Fង O 6ផjQង"ង#$ %កក-.ង , ន ! r 9)យ K 6ច!ន.ចក 8ប# BC
Fង x 6" #ម.! BHK 0902
Ax =⇒ −
Fង y 6" #ម.! BOK , យ)ង,ន
0 0 0 0 0180 452 180 90 902 2 4
ABy y
A A = =
− = − − + ⇒ = +
BHK ន tan2
BKx
r= 9)យ BOK ន tan 2tan
BKy x
r= =
0 0tan 45 2 tan 90 2cot4 2 2
A A Ag
= = ⇔ + −
2 2 21 tan 1 tan cos sin cos sin
4 4 4 4 4 4
1 tan tan cos sin sin .cos4 4 4 4 4 4
A A A A A A
A A A A A A
+ − + −= =
− −⇔ ⇔
2
sin .cos cos sin4 4 4 4
1 2sin .cos4 4
A A A A A A = − = −
⇔
1 2sin .cos sin
4 4 3 2 3
A A A= =⇔ ⇔
ព*យ)ង,ន 2 8 1cos 1 2sin 1
2 9 9
AA = − = − =
'()&
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 25
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ៧
. យម Zង ម8)!ន.!ច!នBនពQ
4 3 22006 10060 2 2007 0049 00 1x x xxx + + + + + =−
. បe ញ5ប) 1 2,x x 6ប Hប#ម 2 6 1 0xx − + = *ច!@គប# n∈ℕ
ច!នBន 1 2n nx x+ 6ច!នBនគ#មBយ;ចកមQនច#ន%ង 5
+. ពQនQប ច!នBនពQ< Qជ?ន , ,a b c ផ0\ង]0 #8ក_ 2006 2006ac ab bc+ + =
ក!A8ធ!ប!ផ.ប#កនម 2
2 2 2 21 2006
2
1
2 3bP
a b c+ += +
+−
=. គច. @- យ ABCD ន,:ចគI AB ង"ង#មBយ #Fម B នQង C ប/ន%ង
ជង AD ង# E , ង"ង#មBយ #Fម AនQង D ប/ន%ងជង BC ង# F ង"ង#J!ងព
ន #W- ង#ពច!ន.ច M នQង N យប>? ក#5 J!ងព EMN នQង
FMN នកmAផ0()W-
'()&'()&'()&'()&
ចេលយ
. 8កខ_ 200(*)
7
2x ≥ − ម មម:8ន%ង
( ) ( )2 2 2 122. .1003 1003 2007 2 2007 1 0
2x x x xx + ++ − ++ + =
( )2
2 2 12 2007 1
2( 1003) 0
10032 2007
( 100
1 0
3) 0x
xx
x x
x
x
+ −
+ = = −+ − =
⇔
⇔
+ + =
⇔ (ផ0\ង]0 # (*) )
C:ចន Hប#ម គI 1003x = −
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 26
. + C!ប:ងយ)ងយប>? ក#
ច!@គប# n ∈ℕ , ច!នBន 1 2n nx x+ 6ច!នBនគ# (*)
!) ពQច!@ 1 20, ,n nn = == យ)ង,ន
0 01 2 1 1 2xx + = + = .
1 1 2 2 2 21 2 1 2 1 2 1 26; ( ) 2 6 2.1 34x x x x x xx x+ = + = + − = − = .
4ប5!) (*) ពQច!@ 1n k= − ច!@ n k= , យ)ង,ន
( ) ( )1 1 2 21 2 1 2 1 2 1 2 1 2( )k k k k k kx x x xx x x xx x− − − −−+ = + + +
( ) ( ) ( ) ( ) ( )1 1 2 2 1 1 1 1 2 21 2 1 2 1 2 1 2 1 2 (**5 )6 k k k k k k k k k kx x xx x x x xx x− − − − − − − − − −= − =+ +−+ + ++
C:ច*, ប) ( )1 11 2k kx x− −+ នQង ( )2 2
1 2k kx x− −+ 6ប ច!នBនគ#* 1 2
k kx x+ 6ច!នBនគ#
ព* Fម< Q$Lន.នBមគQ < QទO * 1 2n nx x+ 6ច!នBនគ#ច!@គប# n
+ c:<យ)ងយប>? ក# 1 2n nx x+ ;ចកមQនច#ន%ង 5Fម<Qធយផ0.យព ពQ
4ប5 នប ច!នBនគ#ធម(6Q n V/ង 1 2n nx x+ ;ចកច#ន%ង 5
Fង 0n 6ច!នBនគ#ធម(6Q:ចប!ផ.; 0 01 2n nx x+ ;ចកច#ន%ង 5
Fម (**) *ផ8Cក ( ) ( )0 0 0 01 1 2 21 2 1 2n n n nx xx x− − − −−+ + កg<;ចកច#ន%ង 5;C
ជ!នB k យ 0 1n − ក-.ង (**) យ)ង,ន
( ) ( ) ( )0 0 0 0 0 0 0 01 1 2 2 2 2 3 31 2 1 2 1 2 1 25n n n n n n n nx x x xx x x x− − − − − − − −= + +−+ + +
ព*Jញ,ន
( ) ( ) ( )0 0 0 0 0 0 0 03 3 2 2 1 1 2 21 2 1 2 1 2 1 25n n n n n n n nx xx xx x xx− − − − − − − −+ = −+ +− +
កg<;ចកច#ន%ង 5;C ក នផ0.យព 4ប5 0n 6ច!នBនគ#ធម(6Q:ចប!ផ.
;C8 0 01 2n nx x+ ;ចកច#ន%ង 5
C:ច*, ប!Sប# 1 2n nx x+ ;ចកច#ន%ង 5គIមQនន
C:ចន 1 2n nx x+ ;ចកមQនច#ន%ង 5ច!@គប# n
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 27
+. Fមប!Sប#យ)ងJញ,ន 12006 2006
ab bcac + + = ,
យ ,, 0a b c > *ន (, , 0, )A B C π∈ V/ ង A B C π+ + =
9)យយ tan tan tan tan tan tan 12 2 2 2 2 2 2006 2006
A B B C C A ab bcac+ + = = + +
*ប)Fង tan ; tan ; tan2 2006 2 2
A b B Ca c= = = គIយ)ង,ន
2 2 2 2
2 2
2 21 3
tan 1 2cos 2sin 3cos2 2 2 2tan 1 tan 1
2 2
PA A B C
B C
= −+ + = − +
+ +
2 2cos cos 3 3sin 3sin 2sin cos 32 2 2 2
C C C A BA B
−= + + − = − + +
2 2 21 1 103sin 3sin cos 3 3
2 2 3 2 3 3
C C A B−≤ − + + + ≤ + =
ព* 2
13sin cos
2 230
C A B− −
≥
2 213s2sin c in coss
22 2 3o
2
C AC A BB− −⇔ ≤ +
>a " "= ក)ន8.F; cos 1
21
sin3sin cos 2 3
2 2
A BA B
CC A B
− == − ==
⇔
Jញ,ន 1 2; ; 1003 2
22 2c a b= = =
C:ចន 10max
3P = ព8 1 2
; ; 1003 222 2
c a b= = =
=. Fង ; ; ( ); ( )I EF MN K AD BC P EF ADF Q EF BCE= ∩ = ∩ = ∩ = ∩
(( ) )) '( (); BE CEAD O O== .
យ)ង,ន
2 . (1)) ;/ ( KPK AO KDKF == 2 ./ ( )') (2P KB KK O E CK ==
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 28
Jញ,ន 2 2 2
2 2 2
.
.
KF KA KD KF KA
KE KB KC KE KB=⇔= (@ KD KA
KC KB= )
(3)
. . (4)
. .KF KB KA
KF
KEKF KA KD
KKE KB K KC C KE D
==⇔ ⇔
==
យ)ង,ន 2. ( )( ) . ..EA ED KE KA KD KE K KA KD KA KKE EE KD= − += −− −
2. ( )( ) . ..FB FC KF KB KC KF K KB KC KB KKF FF KC= − += −− −
ព (1),(2),(3),(4)Jញ,ន . .EA ED FB FC=
មO/ងទP / ( ). . E OEA ED EP EF= = −Ω
/ ( ). . ′= = −ΩF OFB FC FQ EF
C:ច* . . . (5)⇒= =EP EF FQ EF EP FQ
មO/ងទP, MN 6L|ក2L.:gបប# ( )O នQង ( '),O I MN∈
* / ( ) / ( ) . . .( ) .( )I O I O IF IP IE IQ IF IE EP IE IF FQ′= ⇔Ω Ω = + = +⇔
. .IF EP IE F IEQ IF=⇔ ⇔ = .
ក *បe ញ5 EMN FMNS S= (ប>e g<,នយប>? ក#)
'()&
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ៨
. យម Zង ម8)!ន.!ច!នBនគ# 112 1 3 4 1 2
5x x y y− + = − − +
. កព9.d ( )P x ;C8នមគ.6ច!នBនពQផ0\ង]0 #
( )2 1 0, (2000( ) 7 6, ) 2006x PP x P x + − =+ ∀= ≥
+. ក-.ងបqង#ន 6ច!ន.ចផ2ងW- V/ ង, ប ប*0 #Y? ប#គ:ច!ន.ចនមBយkក-.ង 6ច!ន.ច
នមQននគ:ប*0 #[W- , បW- I;កងW- ទ Fមច!ន.ចនមBយk គង#ប
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 29
ប*0 #;កងន%ងគប#ប*0 #;C8ចង#,ន នQងមQន #Fមច!ន.ច*,
ច:កច!នBនច!ន.ចបព"ច)នប!ផ.ប#ប ប*0 #;កង*
=. គង"ង# ( )O LងR#ផjQ 2AB R= , ប/ន%ងប*0 # ( )d ង# A ច!ន.ច C មBយo8) ( )O
ព C គង#កនqប*0 #;កងនQង # AB ង# D , o8)កនqប*0 #ន គច!ន.ច E
មBយV/ង ,CD DE
នទQC:ចW- 9)យ BC DE= ព E ង#ប ប*0 #
ប/ ,EP EQ Gន%ងង"ង# ( )O , ច!@ ,P Q6ប ច!ន.ចប/ ប*0 # ,EP EQ Fម
8!ប# #ប*0 # ( )d ង# ,N K ច:គ*ប;<ង NK 6Lន.គមនMAន R ព8 C
ច8|o8) ( )O '()&'()&'()&'()&
ចេលយ
. (111
2 1 4 15
)3 2x x y y− + = − − +
8កខ_ 0
1
1
2
4, ,
1y
x y
xx
y
x y
≥ −≥
≥ ⇔ ≥∈∈
ℤℤ
11(1) 3 2 2 1 4 1 (2)
5x y x y− − = + − −⇔
Fង 113 2
5p x y= − − , ព8* (2) q យG6 2 1 4 1p x y= + − −
44 1 2 1 4 1 2 11 2p y pp y x y x+ − = + − =⇔ ⇒ − + ++ .
2 2 4 2 2 1 3)4 (x y pp y− = −⇒ + − .
យ)ងន 4 1y − មQន;មន6ច!នBន ,កC, ពQ6C:ចន 4ប5 24 ,1y n n− = ∈ℕ
LងXZងឆ"ង6ច!នBន * 2 1n k= + , ព8*
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 30
2 24 14 2(4 1 ) 1 0k y yk k k+ + = − ⇒ + − + = , មQនម9.ផ8!
C:ច* (3)ពQក-.ងព8 0p = , C:ចនយ)ង,នបព|នម
11
53 2 05
32 4 2 0
xx y
yx y
=− − = =− + − =
⇔ (ផ0\ង]0 #ម (1) )
. Fមប!Sប#បdន យ)ងន ( )2( ) 7 61 0 (1), xP x P x= ∀ ≥− ++
នQង (2000) 2)2006 (P =
C:ច* ( )22 2(2000 (2000)1) 7 2000 7 (3)6P P+ + +−= =
Fង 21 2000 1x = + *ព (3)យ)ង,ន 1 1 6( )P x x= +
Fង 22 1 1x x= + *ព (1) យ)ង,ន ( ) ( )22 2
1 1 11 ) 6 7 7(P xx P x+ − + = += ,
*! 2 2 6( )P x x= +
យប)< Q$Lន.នBមគQ < QទO Fម8!*!;C8,នជ)C:ចZង8) យ)ងក,ន
B 1 2 3, , , ..., , ...nx x xx ច)នSប#មQនL#;C8ផ0\ង]0 #
1 2 ... ...nx xx < < < < ច!@ 21 1n nxx −= + នQង 6( )n nP x x= +
ព8* ព9.d ( ) ( ) ( 6)Q t P t t= − + នHSប#មQនL# 1 2, , ..., , ...nx xx
Jញ,ន ( ) 0, tQ t = ∀
C:ចន ព9.d;C8g<កគI ( ) 6P x x= +
+. Fមប!Sប# យ)ងនច!នBនប*0 #ក!#ព 6ច!ន.ច;C8,ន6ម.ន , , ,, ,B C D EA F គI
ន 26 15C = ប*0 # Fមច!ន.ចនមBយkន 5ប*0 #, C:ច*ន 10ប*0 #;C8ន
#Fមច!ន.ច* យ)ងពQនQម)8ពច!ន.ចកយ, 4ប5 :,A B ប ប*0 #
;កងទ!~ក#ព A ច.G8)ប ប*0 # #Fម B , #គប#ប ប*0 #;កងទ!~ក#ព B
ក ទp ន 4ប*0 # #Fម B ;C8មQន #Fម A C:ចន ព A គទ!~ក#,ន
4ប*0 #;កងន%ង 4ប*0 #* ប*0 #;កងJ!ងបBនន # 10ប*0 #;កង;C8ទ!~ក#
ព B ង# 4.10 40= ច!ន.ចបព"
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 31
ក ទp ទ!~ក#ព A oន 6ប*0 #;កងទP (ន10ប*0 #មQន #Fម A , មQនគQ
4ប*0 # #Fម B ;មQន #Fម A ), ប*0 #នមBយkន ន%ង # 9ប*0 #;កង ;C8
ទ!~ក#ព B (ក-.ង*ន 1ប*0 #បប*0 #ផ2ងទP), C:ចនន;ថម 6.9 54= ច!ន.ច.
ក-.ងច!មប ច!ន.ចបព" ;C8,នពQនQ នប ច!ន.ចបព"[W- , ; 3
ច!ន.ចបព"បងR),ន6 មBយ ;C8ក!ព#J!ង 3ប#6ប*0 #;កង;C8,ន
ពQនQ, C:ចន L:ង#ប#ប នg<,នង# 3Cង,ច!នBនប
នន 36 20C =
C:ចន ច!នBនច!ន.ចបព"ច)នប!ផ.;C8ចនគI 15(40 54) 40 1370+ − =
=. ∗ យ)ងន
2 2 2 2 2 2 2EQ EO R DP ED O RE = = − = + −
( )2 2 2 2 2 2OBC OC BD DC C BD= =− − − =
C:ច* EP EQ BD= =
* Fង ; ;AK KP x NK y EP EQ BD z= == = ==
យ)ងន ( )O 6ង"ង#$ %កក-.ងម.!ប# ENK
C:ច* 1
.2
( )ENK p KES AD NKR∆ = − =
1(2 )
2 2
x z y ENy R z x z R
+ + + − = − −
⇔
(2 ) ( 2 2 ) 2y R z x z y y x z Rz R yx− = + + + + − ⇔− =−⇔
'()&
C
A
B
O
P
Q
K
E
N
D
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 32
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ៩
. គLន.គមនM * *: \ 1f →ℕ ℕ ផ0\ង]0 #
( ) ( 1) 120 ( 2). ( 3)f n f n f n f n+ + + = + + , ច!@ *n∈ℕ គ* (2006)f
. ប ច!នBនពQ , , ,a b c d ផ0\ង]0 #
0
1 23
22
b c d
d
a
a
b cd
b c
≤ ≤ ≤
+ + ≥
<
+ ≥
យប>? ក#5 4 4 4 4 17a b c d+ + − ≤
+. ប ច!នBនពQ , ,a b c < Qជ?នផ0\ង]0 # 2 2 2 2 1b c abca + + + =
គ*!A8:ចប!ផ.ប# ( )2 2 22 2 2
1 1 1
1 1 1T a
ab c
b c= + + −
− −+ +
−
=. គ ABC∆ 6 [ច Fង 0 0),( ( )BA នQង 0( )C 6ប ង"ង#LងR#ផjQ BC ,
,CA AB ព ,A B នQង C គង#ប ប*0 #ប/ន%ង 0 0),( ( )BA នQង 0( )C ប ប*0 #
ប/នប/ន%ងប ង"ង#Zង8)ង# 1 2 1 2, ; , ;A B BA នQង 1 2,C C យប>? ក#5 6
ច!ន.ច 1 2 1 2, ; , ;A B BA នQង 1 2,C C nQo8)ង"ង# ( )C ;មBយ គ* !ប# ( )C
6Lន.គមនMន%ងប ជងប# ABC∆
'()&'()&'()&'()&
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 33
ចេលយ
. យ)ងន [ ] [ ](2 1) (2 1) (2 1) (2 ) (2 ) (2 1)f k f k f k f k f k f k+ − − = + + − + −
[ ](2 2) (2 3) (2 1)f k f k f k= + + − +
C:ច* (3) (1) (4) (6)... (2 ) (2 1) (2 1)f f f f f k f k f k− = + − −
ច!@ * , 2k k∈ ≥ℕ ប) )(3) (1f f≠ * (2 1) (2 1)f k f k≠ −+
C:ច* 1(3) (1 2) kf f −≥− ច!@ * , 2k k∈ ≥ℕ , (ក នមQនចក)នទ)
C:ចន (3) (1)f f= , Jញ,ន (2 1) (2 1)f k f k a+ = − =
C:ចW- ;C (2 2) (2 )f k f k b+ = = ច!@ *, ; , 2a ab b∈ ≥ℕ នQង *k∀ ∈ℕ ,
9)យFមប!Sប# យ)ង,ន 2120 ( 1)( 1) 121 11a b ab a b+ − − = =⇔+ =
1 1 112
1 12112
1 1122
1 1
1 121
a bb
ab
bb
a
b
− = − = = − = = − = = − = − =
⇔
⇔
C:ចន យ)ង,ន8ទផ8Zង ម (2006) 2
(2006) 12
(2006) 122
f
f
f
= = =
. Lន.<នM< QមYពក:. 4 ច!នBនយ)ង,ន
4 3 4 3 3 4 4 34 , 3 8 , 3 41 3 2a b b d dca c≥ + ≥ + ≥+ .
ប:ក< QមYពJ!ងបZង8)LងX នQងLងXយ)ង,ន
( )4 4 4 4 3 3 34 8 4 )7 (11 3 a db c a b dc+ + ≥ + ++ +
Fង 1 2 2; ;
d d d
a b c b c cα β γ+=+ == + *យ)ង,ន
3 3 3 4 4 48 4 4 (4 ) 4 ( ) 4b dca a b cα β β γ γ+ + = = + − + .
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 34
( ) ( )4 4 4 4 44 4 4aa b c bα β γ+ − + −= .
( ) ( )4 4 4 4 412 8 4a a bb c≥ + − −+ (2)
ព (1)នQង (2) យ)ង,នប>e g<,នយប>? ក#
+. ពប!Sប# * 0 , , 1a b c< <
C:ច*, Fង coscos , , cosa b cα β γ== = ច!@ , ,02
πα β γ< <
គIប!Sប# q យG6 2 2 2cos cos 1 2cos .cos .cos
0
cos
, ,2
α β γ α β γπα β γ
+ + = −
< <
Jញ,ន (1)0 , ,
2
pi
α β γ π
α β γ
+ + =
< <,
ព8* 2 2 22 2 2
1 1 1sin
sin sin ssin sin 3
inT α β γ
α β γ+ ++ + + −=
ច!@ (1)គIយ)ង;ងន 2 2 2si9
s n ni4
sinα β γ+ + ≤ នQង 2 2 2
1 1 1
sin sin n4
siα β γ+ ≥+
Lន.<នM< QមYពក:. 2ច!នBនយ)ង,ន
22
9 3sin
16sin 2α
α+ ≥ , 2
2
9 3sin
16sin 2β
β+ ≥ , 2
2
9 3sin
16sin 2γ
γ+ ≥
នQង 2 2 2
7 1 1 1 7
16 sin sin sin 4α β γ + +
≥
C:ចន 13
4T ≥ Jញ,ន 13
min4
T = ព8 3 1sin sinsin
2 2a b cα β γ= = = ⇔ = = = .
=. ង#ក!ព# 'AA Fង H 6L:ង# ABC∆
យ)ងន 'BDHA $%កក-.ង*យ)ង,ន
2 21 2. ' . (1)AH AA AD AB A AA kA = == = .
ពQនQម)8ប<Qធច!S#
:kAN 1 1AA ֏ .
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 35
2 2A A֏ (2) .
'A H֏ .
យ)ង,ន 5ច!ន.ច 1 0 2, ' ,, ,AA AA A nQo8)ង"ង# 0( )AA
នLងR#ផjQ 0AA ;មBយ (3)
ព (2)នQង (3)* 1 2 0(: )kAN AA AA֏
Jញ,ន 1 2H A A∈ 9)យ 1 2. . (4)HA HB HA HA′=
C:ចW- ;C Fង ', 'B C 6ជ)ងក!ព#គ:ព ,B C ប# ABC∆
យ)ង,ន 1 2. . (5)HB HB HB HB′= នQង 1 2. . (6)HC HC HC HC′=
យ H 6L:ង# ABC∆ * . . .HA HA HB HB HC HC′ ′ ′= =
ព (5)(4), , (6)យ)ង,ន 1 2 1 2 1 2. . . (7)HA HA HB HB HC HC= =
0 0 0, ,BBA CCA 6ប ប*0 #មCOទ|ប# 1 2 1 2 1 2, ,A B B C CA បBកBមJ!ង (7) *គI
6ច!ន.ច 1 2 1 2 1 2, , , , ,BA CA B C nQo8)ង"ង# ( )C ;មBយ;C8នផjQ G (G 6ទបជ.!ទ!ងន#
ប# ABC )
Fង R 6 !ប# ( )C យ G nQo8) 0AA *ប)ទ%បទ Stewart
យ)ង,ន 2 2 2
21 18
aG
bA
c+ +=
ព* 2 2 2
1
1
3 2
aR GA
b c+ +==
'()&
B
A
C 'A 0A
2A
1A
2D
H
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 36
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ១០
. គបច!នBនពQ< Qជ?ន , ,a b c ផ0\ង]0 #8ក_ 1a b c+ + = ក!A8:ចប!ផ.
ប#កនម 1 1
1 2( )M
ab bc ca abc= +
− + +
. គព9.dCIកទ 4: 4 3 22004 2007 2003 2( ) 2 50 00 6 0x xP x x x+ + + +=
យប>? ក#5 ( ) 0x xP > ∀ ∈ℝ
+. គច. @- យម, ( )AA CD DB BC= $%កក-.ងង"ង#ផjQ O , ប*0 # ∆
មBយបន%ង AB #ង"ង#PងW- ង# ,M N យប>? ក#5 L:ង#ប#
ប , ,AMD MCB CAN nQo8)ប*0 #;មBយ
=. គច. ABCD ន AB # CD ង# ,E BC # AD ង# F
យប>? ក#5 ប L:ង#ប# J!ងបBន , , ,ABF ADE BEC DCF
nQo8)ប*0 #;មBយ
'()&'()&'()&'()&
ចេលយ
. Fមប!Sប# 2 2 2 2( )1 1a b c ab bc caa b c ⇔ + + + + ++ + ==
2 2 21 2( )ab bc ca b ca⇔ ++ + +− = .
M g<,នc)ង< Qញ 2 2 2
1 a b cM
cca abb
+ += ++ + 2 2 2
1 1 1 1
a ab bcb c ca= + +
+ ++
យ)ងន ( ) 1 19
1ab bc ca
ab bc ca + + + +
≥
(Fម< QមYពក:.)
1 1 1 9
ab bc ca ab bc ca+ +
+ +⇒ ≥
យ)ង,ន 2 2 2 2 2 2
1 9 1 2 7M
a ab bc ca a ab bc ca abb c b bc cc a≥
+ + ++ = + +
+ + + + + ++
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 37
Lន.<នM< QមYពក:.ច!@បច!នBន< Qជ?ន, យ)ង,ន
( )( )3 22 2 2 2 2 23
1 2 1
a ab bc ca ab c b ab c cac b≥
+ + + + +++
+ +
9)យ ( )( ) ( )22 2 222 2 23
2 2 2 1
3 3 3
a b ca b ca ab bc ca
ab bc cab c
+ ++ + + + ++ + ≤ = =+ +
C:ច* 2 2 2
1 29
a a c cac bb b+
+ +≥
+ +
យ)ងកន 2 2 2a b c ab bc ca+ + ≥ + +
2 2 2 2 2 2 3( )b c ab bc ca ab ba c ca⇔ + + + + + ≥ + + .
( )2 3( ) 1 3( ) ab bca b c ab bca c ca+⇔ ≥ + + ≥+ + +⇔
21
37
1ab bc ca ab bc ca
⇔ ≥ ⇔+ +
≥+ +
យ)ង,ន 9 21 30M ≥ + = >a " "= ក)នព8 1
3a b c= = =
C:ចន min 30M = ព8 1
3a b c= = =
. ក ទp ប) 0x ≥ * ( ) 0p x >
ក ទp ប) 0x < យ)ង,ន
4 3 2 4 3 2 2 211 1 1
4 21
4x x x xx x x x xx
+ +
+ + + + = + +
+
+
2 2
2 21 1 1
2 201
2x x x x = + +
+ + >
យ)ង ( )P x c)ង< Qញ
( )4 4 3 2 22004 1( ) 2 3 1xP x x xx xxx + ++ += + + − + .
( )4 2 4 3 23 ( ) 1 2004 012 x xx xxx x x+ + − + + + + + >+ ∀= ∈ℝ .
C:ចន ( ) 0,x xP > ∀ ∈ℝ
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 38
+. Fង 1 2 3, ,H H H PងW- 6L:ង#ប#ប , ,MBA D CM CAN ឃ)ញ5
J!ងបន .ទ;$ %កក-.ងង"ង#BមW- មBយ;C8$%ក'ច. @- យ ABCD .
Fម8កប#L:ង#, យ)ង,ន
1
2
3
(1)
(2)
(3)
OH OA OM OD
OH OM OC OB
OH OC OA ON
= + +
= + +
= + +
ព (1)នQង (2)Jញ,ន 1 2OH OH OA OB OD OC− = − + −
2 1 (4)H H BA CD⇔ = +
C:ចW- ;C, យ)ង,ន 2 3 (5)H H NM AB= +
យ || ||AB DC MN * ,CD AB NM ABα β= =
C:ចន ព (4),(5)Jញ,ន 2 1 2 3 2 1 2 3( 1) , ( 1)H H AB H H AB H H tH Hα β= − = + ⇒ =
C:ច* 1 2 1 3||H H H H
, Jញ,ន 1 2 3, ,H H H #ង#ជBW-
=. C!ប:ងយ)ងយប>? ក#5 ង"ង#$ %ក' J!ងបBននច!ន.ចBម P មBយ
Fង P 6ច!ន.ចបព"ប# ( )EBC នQង ( )CDF , C:ច*គI
B A
C D
M N
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 39
( )0 0180 180CPE CBE BAF BFA= − = − +
0 0180 180CPE CPD EAD EPD EAD⇒ + = − ⇒ + = .
Jញ,ន ច. ADPE 6ច. $%កក-.ង, C:ចន ង"ង# ( )ADE #Fម P
C:ចW- ;C, យ)ងយប>? ក#ង"ង# ( )ABF ក #Fម P ;C Fង , , ,M N R S 6ជ)ង
ប ក!ព#ទ!~ក#ព P G8) , , ,AB CD BC AD Jញ,ន , , ,M N R S #ង#ជB (ប*0 #
Simson) Fង 1 2 3 4, , ,HH H H PងW- 6L:ង#ប , , ,ADFA F BECB នQង
DCF Fម8កខ_ ប#ប*0 # Simson, Jញ,ន ច!ន.ចក 8ប# 1 2, ,PH PH
3 4,PH PH nQo8)ប*0 # Simson , MNRS ព* 1 2 3 4, , ,HH H H #ង#ជBW-
(ប>e g<,នយប>? ក#)
'()&
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ១១
. គ , ,x y z 6ប ច!នBនពQផ0\ង]0 # 0;; 00 x z xy yzx y zx+ + > + + >>
យប>? ក#5 2 2 2. .. . 4x a xyy b yz zx Sz c+ + ≥ + + ( , ,a b c 6ប;<ងជងJ!ងប
ប# , S 6កmAផ0ប# *)
. គ ABC ផ0\ង]0 # cot 2cot 23cot 02 2 2
A B C+ − =
ក!A8:ចប!ផ.ប# cosC
+. គd 2 2 1,1 4
( ) ;2 5
f mx x mx − + ∈ =
យម
( ) ( )22 1mff x x x + = − (1)
=. គង"ង#ផjQ O ! R នQងច!ន.ច A o8)ង"ង#*, o8)ប*0 #បន%ងង"ង#ង# A
គច!ន.ច M V/ង MA R= Fម M ង#ប*0 # #មBយ;បប[8 ;ង
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 40
# ( )O ង# B នQង C ( B nQo8)ច*q M នQង C )
កទF!ង ,B C C)មD ABCS∆ ន!A8ធ!ប!ផ.
'()&'()&'()&'()&
ចេលយ
. យ)ងន ( ) ( ) ( )2 2 2 2 2 2 2 2. .. y b z c xx a a x y bb c x z c+ + = − ++ + +−
2 22 .cos ( ) ( )xbc A x xb cy z+ += − + + .
2 .cos 2 ( )( ).bcx A x y x z bc≥ − + + + (ក:.)
I ( )2 2 2. ( )( ) .c s. o. 2y b z c bcx a x y x z x A+ + −+ + ≥
មO/ងទP
( )22 20 ( )( )( )( cos).cos ( )( )2 .cos 0x y x z A xx x y x z A x x y x z A≥ ⇒ + ++ + + +−− + ≥
ព* : 2 2 2( )( ) cos ( )( ).cos ( )sin2 0x y x z x x y x z A xy yz zxA x A+ + + + + − + +− ≥
( ) ( )2
2( )( ) .cos .sinAx y x z x xy yz x Az⇔ ≥+ + + + +
( )( ) .cos ( ).sinx y x z x A xy yz zx A+ +⇒ ≥+ + − .
C:ច* 2 2 2. . 2. .sinx a xy yz zy b z c bc x A++ + ≥ + , ; 2 .sin 4bc A S=
C:ចន 2 2 2. .. . 4x a xyy b yz zx Sz c+ + ≥ + + (ប>e g<,នយប>? ក#)
. ពប!Sប#យ)ង,ន cot 2cot 23cot 02 2
1)2
(A B C
g g g+ − =
យ)ងប!;8ង (1) G6Sង
cot cot cot cot cot cot 02 2 2 2 2 2
B C C A A Cx g g y g g z g g + + + + + =
យផ0%មប មគ.យ)ង,ន 11, 12, 13y zx = == −
C:ច* 11 cot cot 12 cot cot 13 cot cot2 2 2 2 2 2
B C C A A Bg g g g g g
+ + + = +
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 41
sin sin sin
2 2 2 2 2 211 12 13
sin sin sin sin sin sin2 2 2 2 2 2
B C C A A B
B C C A A B
+ + + ⇒ + =
11cos sin 12cos sin 13cos sin2 2 2 2 2 2
A A B B C C=⇒ +
11.sin 12.sin 13.sin 11 12 13A B C a b c+⇒ + =⇒= .
( )2 2 2 2 2 2(11 12 ) 121169 144 2.132 169 2 cosc b ab ba b a a ab C⇒ = ⇒ + + = + −+ .
2 22 (132 169cos ) 48 3.25 2.20bab C a ab⇒ + ≥+ = .
20 3 132cos
169C
−⇒ ≥ , >a ()ក)ន8.F; 4 3 5a b=
C:ចន 20 3 132min cos
169C
−=
+. ( ) ( )22 22(1) 1 01 2 2 1mx m x xx xm⇔ − + ++ −− =−
( ) ( ) ( )22 2 2 24 (2 12 1 1 1 0 ( )) 2x m x x m xx x⇔ + − + + + − =+ +
ពQនQLងXZងឆ"ងប# (2)6dCIកទពAន ,m x 6,/ S/ ;ម/, យ)ង,ន
( ) ( ) ( ) ( )2 222 2 22 11 4 1 1 1x x x x x′∆ = + − + + + −
+
( )24 2 3 2 24 1 4 2 4 2 1x x x x xx x+ + + − − = + −=
Hប# (2) គI 2 2
1 2
1
( 1)
2;
2 2
x xm m
x x
x x− + +=+
+=
យ)ងc)ង< Qញ ( )2 2
24 . 02 2( 1)
1 2x xx m m
xx
x
x x− − + +− = +
++
2 2(2 1) .1 (2 1 2 2 0)m x m x mx x⇔ − + + − = − + +
2
2
(2 1) 1 0 (3)
(2 1) 2 2 0 (4)
m x
x m m
x
x
− + + =⇔
− − + − =
យ)ងន 23
1 3(2 1)
24
24m m m
= + − +
− =
∆
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 42
នQង 24
1 2 2 1 2(2 1) 4(2 2 ) 4
2
2 2m m mm
∆ = + − − =
+ −+ +
ច!@ 3
4
1 4;
02 5
0m
∆ >∈ ⇒
∆ < , C:ច* (4)W( នH9)យ (3)នHព
2 2
1 2
2 1 (2 1) 2 1 (2 1),
2 2
4 4m m m mx x
+ − + + + +− −= =
C:ចន ម (1)នHព 1 2,x x
=. Fង I 6ច!ន.ចក 8ប# BC , ង# ( )AH HBC BC⊥ ∈
4ប5 ម.! HMA α=
Fម O ង# OP AH⊥ , Jញ,ន OAP α∠ =
យ)ង,ន 1. .
2ABCS AH BC AH IC==
2 2 2 2 2 2sin sin sinR OC ROI OI PR R R Hα α α=− −= =−
យ)ងន (si cos )nHP AH AP R α α−= − =
Jញ,ន 2 2 2 2sin (sin cos ) sin 2sin cosABC R R RS Rα α α α α α= − − =
2 32 sin cosR α α= .
យ)ងន 32
3 6 2 2.cos .cos cossin
sin sin 273
αα α α α α =
=
42 2
4
sin cos 1 3 327 27.
4 4 16
α α +≤ = =
(Lន.<នMក:. 4ច!នBន)
C:ច* 2 3 2 2.cos3 3 3 3
2 sin 2.16 8
ABC R RS Rα α == ≤
C:ចន 4
22ax
8
7m ABCS R= >a " "= ក)ន8.F;
22sin
3cos
α α=
2 3tan tan 3α α⇔ = ⇔ = (α 6ម.![ច6នQចj) I 060α =
C:ចន ប ច!ន.ច ,B C ;C8g<ក6ច!ន.ចបព"ប#ប*0 # # uM ផX.! MA ,នម.! 060
B
A
C
H
M
P
α
O
u
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 43
6មBយន%ង ( )O
'()&
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ១២
. ក x ប) 3 45
x x + =
. គប ច!នBនពQ< Qជ?ន , ,a b c យប>? ក#5
2 2 2
2 2 2 2 2 2
(2 ) (8
( ) (
2 ) (
) (
2 )
2 )2 2
a b c b c a c a b
a b cb c c a a b
+ + + + + ++ + ≤+ + + + + +
+. យម 3 3 2xx x= +−
=. គ ABC ម, ង"ង#$ %កក-.ង ប/ន%ង AB ង# T CT
#ង"ង#ង# K 4ប5 K 6ច!ន.ចក 8 CT 9)យ 6 2CT =
ច:គ*ប;<ងប ជងប# ABC
'()&'()&'()&'()&
ចេលយ
. យ)ងពQនQម)8ប ក
)a ប) 0x < * 3 3 30 0
x x x < ≤
⇒ ≤
C:ចW- ;CH យ)ង,ន 40
x
≤
C:ចន ម W( នH
)b ប) 0x > * 3 4 3 4
x x x x <
⇒
≤ ព8កន នQងនQយមន|យ;ផ-កគ#,
យ)ង,នបក
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 44
30
x ∗ =
នQង 45
x =
ព 30
x =
Jញ,ន 30 1
x≤ < Jញ,ន 3x >
យ 45
x =
Jញ,ន 45 6
x≤ < Iក 2 4
3 5x< ≤
ប ច*q ន មQនន x ;C8ផ0\ង]0 #ទ
31
x ∗ =
នQង 44
x =
ព 31
x =
Jញ,ន 31 2
x≤ < Jញ,ន 3
23x< ≤
យ 44
x =
Jញ,ន 4
51x< ≤ , កយ)ងន:<ក មQនម9.ផ8;C
32
x ∗ =
នQង 43
x =
យ)ង,ន 41
3x< ≤
C:ចន ច!8)យ 41
3x< ≤
. ពQនQឃ)ញ5 (1)ពQច!@ , ,a b c កពQច!@ , ,ka kb kc
C:ចន យ)ងច4ប5 3a b c+ + =
យ)ង,ន 2 2 2
2 2 2 2 2 2
( 3) ( 3) ( 3)(1)
2 28
(3 ) (3 ) 2 (3 )
a b c
a b ca b c⇔ ≤
+ − + − + −+ + ++ +
យ)ងពQនQម)8 2 2
2 2 2 2
6 9
(3 ) 2
( 3) 1 1 8 6.
3 22 3 3 31
x x x
x x
x
x x xx
+ + = =+ ++ −
+ − + − +
2
1 8 6 4 4. 1
3 ( 1) 32 3
xx
x≤
+ += + + −
C:ច- LងXZងឆ"ង 4( ) 8
34 a b c+ +≤ =+
>a " "= ក)ន8.F; a b c= = (ប>e g<,នយប>? ក#)
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 45
+. C!ប:ងយ)ងពQនQឃ)ញ5 ប)ម នH x *g<ន 22 x− ≤ ≤
ព* យ)ងFង 2cosx t= ច!@ 0 t π≤ ≤ ម ;C8 q យG6
3 6cos8cos 2cos 2t t t− = + I 2cos3 2cos2
tt =
ព* យ)ង,នH 4 40, ,
7 5tt t
π π===
C.ចន !ន.!Hប#ម ;C8គI 4 42, 2cos , 2cos
5 7T
π π =
=. Fង K 6ច!ន.ចក 8ប# CT 9)យ L 6ច!ន.ចប/ប#ង"ង#ន%ងជង BC
C:ច* (*)1
2CK CT=
យមQនធ"),#8កទ:G យ)ងពQនQពក
• ក ទp AB AC= I b c= Jញ,ន L 6ច!ន.ចក 8ប# BC
2 21
2.CK CTCL CT= =
គI5 2 / 4 36a = I 1)12 (a =
Lន.<នMទ%បទក:.ន.ក-.ង BCT , ច!@ ABCβ =
2 2 2 2 . .cosC BT BC BT BCT β= = − .
2 2/ 4 144.cos cos (72 2)3
4a a β β⇔ + ⇔ == − (Fម (1) )
មO/ងទP Lន.<នMទ%បទក:.ន.ក-.ង ABC , យ)ង,ន
2 2 2 2 .cos cos / 2 (3)c a ca ab bβ β= + − ⇔ =
ព (1),(2) នQង (3)យ)ង,ន ( , , ) (8,8,12)a b c =
• ក ទp AC BC= I a b= *គI T 6ច!ន.ចក 8ប# AB
Lន.<នMទ%បទប*0 #ប/p 2 21
2.CK CTCL CT= = (Fម (*) ) I 2( / 2 36)a c− =
Jញ,ន 6 / (4)2a c= + Lន.<នMទ%បទពF9X|ច!@ :BCT
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 46
2 2 / 4 72 (5)a c= + ព (4)នQង (5) : 6c =
C:ចន ( , , ) (9,9,6)a b c =
'()&
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ១៣
. យម 2 22 3) 1 (1)( 1 xx x x− + ++ =
. គ 2 2( 1) 1bf x a xx cx= −+ + ≤ ច!@គប# x nQoច*q [ 1;1]−
យប>? ក#5 2
| | 4
).
).
3
a
ax
a
b bx c
≤
+ + ≤
+. គ ABC , នប ជង , ,Aa C B cB bC A= == 9)យ , ,a b cm m m
PងW- 6ប;<ងប#ប មCOន;C8គ:ចញពប ក!ព:8 , ,A B C
យប>? ក#5 ABC ម|ង28.F; 2 2 22 a b ca b c m m m+= ++ +
=. គ!ន.! 1,2,3,4,5,6,7,8,9,10,11,12D = កច!នBន!ន.!ង D C)មDម
13x y+ = W( នHo8)!ន.!ង*
'()&'()&'()&'()&
ចេលយ
. ;Cនក!#ប#ម គI D = ℝ
Fង 2 3 ,2 txx − + = ច!@ 2t ≥
ព8* ម (1) q យG6 2( 1) 1 (2)x t x ++ =
2 2(2) 3 ( 1) 2( 1) 0x x t xx⇔ − + − + + − =
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 47
2 ( 1) 2( 12
)1
0tt
t xx
xt
⇔ − + + − = ⇔=
= −
iច!@ 2t = , យ)ង,ន 2 22 3 2 3 42 xx x x− + ⇔ − + ==
2 1 2
1 22 1 0x
xx
x
= −
= +− − = ⇔
⇔
i ច!@ 1t x= − , យ)ង,ន 2 3 12 xxx − + = −
2 2
0 1
3 12 3 ( 1)
1x xx
x x x
≥ ≥⇔ ⇔ ⇔ ∈∅
=−
= + − −
C:ចន !ន.!Hប#ម គI 1 2;1 2− +
. ).a ពប!Sប#យ)ង,ន
3 1 3 31 8,. 3 2 3 4
2 2 4 2
af b c a b c
= + + +≤ +
≤
⇒
3 1 3 31 3 2 3 4
28
2 4 2
af b c a b c
− = − + − +
≤ ⇒ ≤
នQង 6 3 3 2 3 2 3 4 4 8a a a b b c c c= + + − + + −
3 2 3 4 3 2 3 8 8 8 244 8a b c a b c c≤ ≤ + ++ − + + =+ +
Jញ,ន 4a ≤
).b ∗ ពQនQម)8 3;1
2x
∈
, យ)ង,ន
0 2 3 3 3 3 2 32 3 2 3 3x x− −≤ ⇒ ≤< − −
20 2 3 4 3 4 2 3 4 2 3 4x x x≥ ⇒ − ≤> − − −
នQង 2 22 3 2 3 2 3 2 3a bx cx ax bx c+ = ++ +
( ) ( ) ( )3 2 3 4 2 3 3 2 3 4a b c x ax x c x= + + + − + −
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 48
3 2 3 4 2 3 3 2
8 4 2 3 3
4
2 6
3
3 3 3
a b c x a x x c x≤ + + + − +
≤ + + − =
−
−
Jញ,ន 2 3bx cax + + ≤
∗ ពQនQ 31;
2x
− −
∈ , យ)ង,ន
2 3 3 3 3 0 2 32 3 2 3 3x x− + + < + −≤ ≤ +⇒ ,
4 2 3 3 0 44 2 32 2 3 4xx≤ + ⇒ ≤− + < + − +
នQង 2 2 22 3 2 3 3 2 3 2 32 2 3ax ax bx c abx x cc bx+ + + −= + = − −
( ) ( ) ( )3 2 3 4 2 3 3 2 3 4a b c x ax x c x= − + − + − +
3 2 3 4 2 3 3 . 2 3 4
2 3 3 4 28 4 3 6 3
a b c x a x x c x≤
≤
− + + + + +
− + + − + =+
Jញ,ន 2 3bx cax + + ≤
ពQនQម)8 3 3;
2 2x
−
∈ , យ)ង,ន
2 2 23 3 1 1 11
2 21
4 2 3x xx x≤ ⇒ ≤ ⇒ − ≥ ≥− >⇒
Jញ,ន 2 13
1
3
bx cax + + ≤ =
C:ចន [ ]2 3, 1;1ax bx c x+ + ≤ ∀ ∈ −
+. i ABC∆ 6 ម|ង2 2 2 22 a b ca b c m mm⇒ + + = + +
យ ABC∆ 6 ម|ង2 * a b c= = នQង 3
2a b cm ma
m = = =
ព8* 3a b c a+ + = , នQង 2 2 2
2 2 2 3 3 32 2 3
4 4 4a b c
a a am mm a= ++ =+ +
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 49
C:ចន 2 2 22 a b ca b c m m m+= ++ +
i 2 2 22 a b cm ma b Ac BCm+ + ⇒ ∆= ++ 6 ម|ង2
យ)ង,ន ( ) ( ) ( )2 2 2 2 2 2 2 2 2
2 2 22 2
, ,4 4 4a b c
c ab a c b a cm m m
b+ + += = =
− − −
C:ច*
( )
( )
( )
2 2 2
2 2 2
2 2 2
32
23
223
22
b c a
c a b
a b c
m
m
a m m
b m m
c m m m
= − = −
+
= −
+
+
Jញ,ន ( ) ( ) ( )2 2 2 2 2 2 2 2 23( ) 2 2 2
2 b c a c a b a b ca b c m m mm mm mm m+ + ++ + = − + − + −
Lន.<នM< QមYព Bunyakovski, យ)ង,ន
( ) ( ) ( ) ( )2 2 2 2 2 2 2 2 2 2 2 22 2 2 3. 3b c a c a b a b c c a bm m m m mm m m m m m m− + − + −+ + + ≤ + +
I ( ) ( )2 2 2 2 2 233. 3 2
2 c a b a b ca b c m a b c m mmm m+ + + + =≤ + + ⇔ + +
>a មYពក)នព8
( ) ( ) ( )2 2 2 2 2 2 2 2 22 2 2b c a c a b a b cm m m m m mm m m− = −+ −+=+
2 2 2a b c a b cm mm m m m⇔ = = ⇒ = = I ABC∆ 6 ម|ង2
i C:ចន ABC∆ 6 ម|ង2 2 2 22 a b cma b m mc= = + +=⇔
=. យ)ងន 1 12 2 11 3 10 4 9 5 8 6 7+ = + = + = + = + = +
Fង 1 |1 AA A D⊂= ∈ នQង 12 A∈
2 | 2AA D A= ⊂ ∈ នQង 11 A∈
3 | 3A A D A= ⊂ ∈ នQង 10 A∈
4 | 4AA D A= ⊂ ∈ នQង 9 A∈
5 | 5AA D A= ⊂ ∈ នQង 8 A∈
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 50
6 | 6AA D A= ⊂ ∈ នQង 7 A∈
Fង B 6!ន.!គប#ប !ន.!ងប# D Fង C 6!ន.!គប#ប !ន.!ងប# D
V/ង គប#!ន.!ង*មQននផ0.កJ!ងងន:<បព|នមBយក-.ង 6 បព|នច!នBន
(2;11), (3;10), (4;9), (5;8)(1;12), , (6;7)
ព8* 1 2 3 4 5 6\ ( )C B A A A A A A= ∪ ∪ ∪ ∪ ∪
1 2 3 4 5 6\ (1)C B A A A A A A∪ ∪ ∪⇒ = ∪ ∪
ពបPបFងប ( 1,2,3,4,5,6)iA i = នQងC , យ)ង,ន ប) A C∈ *ម 13x y+ =
W( នH, C:ច* C គI6ច!នBន ប !ន.!ងប# D ;C8g<ក
យ)ងន
6
1 1 1
6
6 6 611n n i j i j k i j k l
i j i j k i j k lnn
A A A A A A A A A A A≤ < ≤ ≤ < < ≤ ≤ < < ≤== <
= − ∩ + ∩ ∩ − ∩ ∩ ∩ +∑ ∑∑ ∑∪
1 2 3 4 5 661
i j k l mi j k l m
A A A A A A A A A A A≤ < < < < ≤
+ ∩ ∩ ∩ ∩ − ∩ ∩ ∩ ∩ ∩∑
យប ( 1,2,3,4,5,6)i iA = នB*ទ()W- *
6
21 6 1 2 2
61 116 ; 15n i j
i jn
A A A A C A A A A≤ < ≤=
= ∩ = ∩ = ∩∑∑i
36 1 2 3 1 2 3
1 6
(20 3)i j ki j k
A A A C A A A A A A≤ < < ≤
∩ ∩ = ∩ ∩ = ∩ ∩∑
46 1 2
13 4 1 2 3 4
6
15i j k li j l
A A A A C A A A A A A A A≤ < < ≤
∩ ∩ ∩ = ∩ ∩ ∩ = ∩ ∩ ∩∑
1 2 3 41
56
6i j k l mi j k l m
A A A A A A A A A A≤ < < < < ≤
∩ ∩ ∩ ∩ = ∩ ∩ ∩ ∩∑
យ)ងន
i 13 409 (4)2 6B ==
i យ 1 2 3 4 5 6A A A A A A D∩ ∩ ∩ ∩ ∩ = * 1 2 3 4 5 6| 1 (5)A A A A A A∩ ∩ ∩ ∩ =∩
i d.នមBយkប# 1A នSង 11;12 Y∪ , ច!@ 1Y 6!ន.!ងមBយប# \ 1;12D
C:ច* 1A គI()ន%ងច!នBន!ន.!ង \ 1;12D , ព* 101\ 1;12 10 (6)2D A⇒= =
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 51
id.នមBយkប# 1 2A A∩ នSង 21;12;2;11 Y∪ , ច!@ 2Y 6!ន.!ងមBយប#
\ 1;2;11;12D C:ច* 1A គI()ន%ងច!នBន!ន.!ង \ 1;2;11;12D ព*
81 2\ 1;2;11;12 8 (7)2D A A= ∩ =⇒
C:ចW- ;C, យ)ង,ន
61 2 3 2 , (8)A A A∩ ∩ =i
41 2 3 4 , (92 )A A A A∩ ∩ ∩ =i
21 2 3 4 5 2 , (10)A A A A A∩ ∩ ∩ ∩ =i
ព (2),(3),(5),(6),(7),(8),(9),(10) យ)ង,ន
10 8 6 4 21 2 3 4 5 6 15.2 20.2 15.2 6.2 1 3367 (11)6.2A A A AA A − +∪ ∪ ∪ ∪ = − + − =
ព (1)នQង (11), យ)ង,ន ច!នBន!ន.!ងប# D ;C8g<កគI 4096 3367 729− =
'()&
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ១៤
. កព9.d ( )P x ផ0\ង]0 # ( )( ) ( ) ( ) ( )222 2 3 11 2P P x p x P x x x
+ = + +
+ +
. គ n ច!ន.ច 1 2, , ..., nAA A o8)បqង#, មQននបច!ន.ច#ង#ជBW- , មQនន
បBនច!ន.ចបងR),ន6ប8c: ម Fង 1 2, , ..., mII I 6គប#ប ច!ន.ច
ក 8;C8ក)ចញពប LងR#;C8នច.ងJ!ងព6ពច!ន.ច ,i jA A ;C8
,1 )( i j n≤ ≤ Fង M 6ផ8ប:កប;<ងLងR#;C8នច.ងJ!ងព6ពច!ន.ច ,i jA A
ក,ន ,1 )( i j n≤ ≤ Fង N 6ផ8ប:កប;<ងគប#LងR#;C8នច.ងJ!ងព6
ពច!ន.ច ,i jI I ក,ន ,1 )( i j m≤ ≤ យប>? ក#5 2 3
.4
2nnN M
− +≤
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 52
+. គ ABC [ច$ %កក-.ងង"ង#LងR#ផjQ O , ,M N P 6ច!ន.ចក 8
ប ធ-::ច , ,BC CA AB o8)ជង AB គពច!ន.ច 21,A B , o8)ជង BC
គពច!ន.ច 21,B C , o8)ជង CA គពច!ន.ច 1 2,C A V/ ង
1 2 1 2 1 2A AA BB BB CA CC C= = = = =
3A 6ច!ន.ចបព" 1PA ន%ង 2 3;NA B 6ច!ន.ចបព" 1MB ន%ង 2PB 3;C 6ច!ន.ច
បព" 1NC ន%ង 2MC យប>? ក#5 3AA 3 3, ,BB CC បព"W- ង#ច!ន.ចមBយ
=. គ , ,a b c 6ប ច!នBនពQមQនL< Qជ?នផ0\ង]0 # 3a b c+ + = ក!A8ធ!ប!ផ.
ប#កនម 9 10 22A ab ac bc= + +
'()&'()&'()&'()&
ចេលយ
. 22 2 2( ) 2 ( )( ( ( 3 1) () 1) 1 )P P x P x P x x x+ + + ++ =
យ)ងន ( ) 0P x = មQនផ0\ង]0 #, C:ចន ( )P x នSង
11 1 0( ) ; (... 0)n n
n n nx a x aP a axx a−−+ + + + ≠=
ព8ព*q LងXJ!ងពប# (1) *Bធ!ប!ផ.ប# ( ( )) 1P P x + គI 21( )n n n nn n na x aa x+=
9)យBធ!ប!ផ.ប# 22 2 2( ) 2 ( ) ( 3 1)x P x xP x + + + + គI
( )( )( )( ) ( )
22 4 4
22 22 4 2 8
8
;
1 ; 2
;
2
2
n nn n
n n
a n
a a
x a x
x x x n
x n
= >
+ = + =
<
C:ចន ប) 2n ≤ * 2 8 2 8n xx n= ⇒ = (មQនម9.ផ8)
C:ច*, 2n > 9)យ 21 4 4 4 ; 1n n nn n nx a x na a+ = ⇒ = =
C:ចន ( )P x នSង 4 3 23 2 1 0( ) a x a xP x xx a a+ + + +=
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 53
Fង ( )22( ) ( 13 1)G x P x x x+ += − + , យ)ង,ន
22 2 2( ) 3 ( ) 1 ( ) ( 3 1)(1) ( ( )) 1 1x P x P x x xP P x P⇔ + + + = − + − + +
( )( )
22
22 2
2
( ( )) 1
( ( ) ( ) 2
( ( )) (
( ) 3 ( ) 1 ( )
( ) 3 ( ) 1 1 ( ) ( ) 3 (
)
) 1
(( ) 2 ) 3 ( ) 1
x P x G x
x P x G
P P x P
P P x P G x P
G P x
x x
G
P x
xx G x P P x
+ = ⇔ +
− −
=
+ −
⇔ + + + = + +
⇔ + +−
ប) ( ) 0G x ≠ , Fង deg ( )G x k= * 3k ≤ *យ)ង,ន
( )2
deg ( ( )) 4
deg ( ) ( ( ) 3 ( )2 81)
G P x k
G x G x P kx P x
= − = + +
+
4 88
3k k k⇒ = + ⇒ = (មQនម9.ផ8)
C:ចន ( ) 0G x = I ( )22 3 1 1 ( 1)( 2) 3( )( )x x xP x xx x+ + − = + += +
. ច!@ ABC នមBយk នច!ន.ចក 8ប ជង , ,AB BC CAគI , ,M N P ,
យ)ងនមYព 1( )
2MN NP PM AB BC CA+ + = + + (*)
ច!@ច. ABCD (ច #យខqBនង IមQន,/ ង) នច!ន.ចក 8ប
LងR# , , , , ,AB CD BC DA BD AC PងW- គI , , , , ,M N P Q R S យ)ងន< QមYព
1( ) (*
2*)MN PQ RS AB CD BC AD BD AC+ + + + + + +≤
P
A
B C
M N
R A
B
C D N
M
Q S
P
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 54
ពប ច!ន.ច 1 2, , ..., nAA A , យ)ងបងR)គប# , ច. ;C8នប ក!ព:8
nQក-.ងប ច!ន.ចន, BចបងR)គប#មYព (*) , < QមYព (**) ;C8g<W- ច!@
, ច. 9)យប:កLងXន%ងLងXJ!ងL#បj: 8W- យ)ង,ន< QមYព (***)
យLងR# i jI I នមBយknQo8) ;មBយ Iច. ;មBយZង8) *
ន<ន;មងគ#oLងXZងឆ"ងប#< QមYព (***) ,
C:ចន LងXZងឆ"ង (***) N=
LងR# i jA A នមBយkន<នក-.ង 2n − នQងក-.ង ( 2)( 3)
2
n n− − ច.
*មគ.ប#ព8ព*q LងXZង ! (***) គI
21 ( 2)( 3)
22 2
3
4
2n n nn
n− − − + =
− +
C:ចន LងXZង ! 2 3 2
(***) .4
n nM
− +=
C:ច* 2 3
.4
2nnN M
− +≤
+. ពQនQម)8ង"ង#ផjQ I $%កក-.ង ABC ច!@ប ច!ន.ចប/ 1 1 1, ,M N P ប#ប
ជង , ,BC CA AB Fង K 6ផjQប!;8ង$!ងផ8ធPប< Qជ?ន ប#ង"ង#ព ( )I នQង ( )O
យ 1 ||IM OM (;កងBមW- ន%ង BC ) *យ)ង,នចxប#ប!;8ង$!ង
: ( ) ( )R
rK IV O→
1
1
1
M
N
P
M
N
P
→→
→
យ 1 2AA AA= នQង 1 1AP AN=
*យ)ង,នចxប#ប!;8ង$!ង
1
11 1:
AA
APAV P A→
1 2N A→ M
P
A
B C
N
K I
O
1M
1N
1A 2A
3A
1P
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 55
Fងចxប#ប!;8ង$!ងផjQ H 6ផ8គ.Aនចxប#ប!;8ង$!ងJ!ងពZង8) * , ,H A K
#ង#ជBW- 9)យ 1, ,H P A #ង#ជBW- , 2, ,H N A #ង#ជBW-
C:ចន H [W- ន%ង 3A , I 3AA #Fម K
C:ចW- ;C 3 3,BB CC #Fម K * 3 3 3, ,BBA CCA បព"W- ង#ច!ន.ចមBយ
!W8# !W8# !W8# !W8# បកជនចប)ទ%បទ Menelauis Ceva− ជ!នB ប)ផ8គ.Aនចxប#
ប!;8ង$!ង
=. យ)ងន 3a b c+ + =
9 10 22 9 10( ) 12A ab ac bc ab a b c bc= + + = + + + .
2 2
9 10(3 ( ))( ) 12 (3 ( )
30( ) 1
)
10( 2 36) 3
ab a b a b b a b
a b a b b b ab
= + −+ +
+ + + − += − − −+ +
.
ពQនQម)8 2 3( ) tf t t= − + ច!@ 0 3t≤ ≤ យ)ង,ន 9max ( )
4f t = , ,នព8 3
2t =
Jញ,ន 10 ( ) 12 ( )99
22max ( )2
3A f a b f b ab f t= + + − ≤ =
C:ចន !A8ធ!ប!ផ.ប# A គI 99
2, ក)នព8 3
0,2
ba c= ==
'()&
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ១៥
. Fង K 6ផjQង"ង#$ %កក-.ង 1 1,, BA C CB Fម8!ប#6ច!ន.ចក 8ប#
ប ជង ,AC AB ប*0 # 1C K #ប*0 # AC ង# 2B , ប*0 # 1B K #ប*0 #
AB ង# 2C V/ងកmAផ0 ABC ()កmAផ0 2 2AB C
គ*ម.! CAB
. យប>? ក#5 ក-.ង ABC កយ គ,ន
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 56
5sin sin sin sin sin sin
2 2 2 2 2 2 8 4
A B B C C A r
R≤+ + + , ក-.ង* ,R r Fម8!ប#
6 !ង"ង#$ %ក', $ %កក-.ង ABC
+. គ n ច!នBនពQ 1 2, , ..., nxx x ផ0\ង]0 #8ក_ 2 2 2 21 2 3 ... 1nx x xx + + + + =
យប>? ក#5 1 22 2 2 2 2 21 1 2 1 2
...... 21 1 1
n
n
xx
xx xx
n
x
x
x+
+ + + ++ + <
+ + +
=. យបព|នម
( )( )( )
2 2 2 2 2
2 2 2 2 2
2 2 2 2 2
( ) 1
( )
3
14
5( ) 1
y z x z
z x y
x x y
y y x
z z x
z
y z z y
+ = + +
+ = + +
+ = + +
'()&'()&'()&'()&
ចេលយ
. Fង 2 2, ,, ,CA b AB c AB x AB a yC C= = = = =
ប*0 #ព. BK #ជង AC ង# D យ)ង,ន
1KB c a a c
KD AD CD b
+= = = >
Jញ,ន bcAD
a c=
+ នQង D nQoច*q A នQង 2B
Lន.<នMទ%បទ Menelaus ច!@ ABD នQងប*0 # 2 1B KC យ)ង,ន
2 1
2 1
. . 1B C KB
B C
A
A B KD
D =
Jញ,ន . 1 (1)
bcx a c bca c
x bx
b a c⇒
+=
− ++ =−
C:ចW- ;C យ)ង,ន (2)bc
ya b c
=+ −
Fមប!Sប# កmAផ0 ABC = កmAផ0 2 2AB C , Jញ,ន (3)xy bc=
A
B
2C
1C
C
2B
K
D
1B
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 57
ព (2)(1), , (3)យ)ង,ន 2 2 2 (4)ba c bc= + −
Fមទ%បទក:.ន.យ)ង,ន 2 2 2 2 cos (5)b c bca A= + −
ព (4), (5)Jញ,ន 1cos
2A = − នQង 060CAB =
. យ)ងន tan (sin sin ) cos cos2
AB C B C+ = +
tan (sin sin ) cos cos2
BC A C A+ = +
tan (sin sin ) cos cos2
CA B A B+ = +
នQង 4 sin sin sin
2 2 2 cos cos cos 1
A B CRr
A B CR R
= = + + −
Lន.<នM< QមYពក:. tan sin tan sin tan sin tan si2 2 2
2 n2
A B A BB A B A≥+
1
sin sin tan sin tan sin2 2 4 2 2
A B A BB A
⇔ +≤
C:ចW- ;C យ)ង,ន 1sin sin tan sin tan sin
2 2 4 2 2
B C B CC B+≤
នQង 1sin sin tan sin tan sin
2 2 4 2 2
C A C AA C+≤
ប:កLងXន%ងLងXAន< QមYពZង8)បj: 8W- យ)ង,ន
sin sin sin sin sin sin2 2 2 2 2 2
A B B C C A+ + ≤
1
tan (sin sin ) tan (sin sin ) tan (sin sin )2 2 2 2
cos cos cos cos cos cos cos cos cos 1 1
2 4 4 4
A B CB C C A A B
A B C A B C A B C
≤ + + + + +
+ + + + + + −= = + +
យ)ងន 3cos cos cos
2A B C ≤+ + , Jញ,ន
5sin sin sin sin sin sin
2 2 2 2 2 2 8 4
A B B C C A r
R≤+ + +
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 58
+. < QមYព;C8មម:8ន%ង
2
1 22 2 2 2 2 21 1 2 1 2
...1 1
1. 2
( )..1
n
n
x x x n
x x xx x x
+ + + <
+ + ++ + ++
Fម< QមYព Bunyakovski យ)ង,ន
LងXZងឆ"ង 22 2
1 22 2 2 2 2 21 1 2 1 2
. .(1)1
.. (2)...1 1
n
n
x x x
x x xn
x x x≤ + + +
+ +
+ + +
+ +
យ)ងន 2 21 1
2 2 2 21 1 1
11
(1 1 1)
x x
x x x= −
+ +≤
+
( )
22
2 2 2 22 21 1 21 2
1 1
1.
1.
1.
x
x xxx x≤
+ +++−
+
( )
2
2 2 2 2 2 22 2 21 1 1 21 2
1
... ....
1
1 .. 1 1n
n nnx
x
x x xx x xx −
≤+
−+ + + + ++ + + ++
Jញ,ន LងXZង !ប# (2):ច6ង
2 2 2 2 2 2 2 2 2 21 1 1 2 1 2 1 1 2
1 1 1 1 1. 1
1...
... ...1 1 1 1n nx xn
x x x x xx x x−
− + − + +
+ + +− + + + + + + + + +
=
11
2 2
nn = − =
(>a < ច#Z @នV/ងQច 2 1,2,, ...,0ix i n> = )
< QមYពg<,នយប>? ក#
=. ∗ . ក នLaQមBយក-.ងច!មបLaQ , ,x y z () 0
:0x =i យ)ង,នបព|នម 2 2
2 2
2 2
0
0
0
z
zy
z
y
y
==
=
Jញ,ន 0, z ty = = ∈ℝ I 0, y tz = = ∈ℝ ,
ក-.ងក នបព|ននច!8)យ ( , , ) (0,0, ), (0, ,0),x y z t t t∈ ∈ℝ
:0y =i C:ចW- ;C យ)ង,នច!8)យ ( , , ) ( ,0,0 (0,0,), ) ,x y t t tz ∈ ∈ℝ
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 59
0 :z =i C:ចW- ;C យ)ង,នច!8)យ ( , , ) ( ,0,0 (0, , ,), 0)ty t tx z ∈ ∈ℝ
∗ . ក 0xyz ≠
;ចកម នមBយkក-.ងបព|ន ន%ង 2( )xyz យ)ង,ន
2
2
2
2
2
2
31
4
5
1 1 1
1 1 1 1
1 1 1 1
z y x x
x z y y
y x z z
+ +
+ + + +
= +
= +
= +
FងLaQជ!នBយ 0 0 0
1 1 1, ,x
y zy z
x= = = យ)ង,នបព|នZង ម
2 20 0 0 0
2 20 0 0 0
2 20 0 0 0
) 3 (1)
( ) 4 (2)
( ) (
(
5 3)
y x x
x z y y
y x z z
z + = + +
+ = + +
+ = + +
ប:កម J!ងបប#បព|នន យ)ង,ន
20 0 0 0 0 0) 12( *)( ) (y z x yx z+ + = + + .
យម CIកទព យ)ង,ន 0 0 0 4 (4)y zx + + = I 0 0 0 3 (5)x y z+ + = −
i ព (1),(4)យ)ង,ន 0
13
9x = , ព (2),(4) យ)ង,ន 0
12
9y = , ព (3),(4)យ)ង,ន 0
11
9z =
i ព (1),(5)យ)ង,ន 0
6
5x = − , ព (2),(5)យ)ង,ន 0 1y = − , ព (3),(5)យ)ង,ន 0
4
5z = − .
ន-Q ន 9 9 9 5 5( , , ) , , , , 1, (0, ,0), (0,0, ), ( ,0,0),
13 12 1,
1 6 4t t tx y z t
− − −
∈
∈ ℝ
'()&
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 60
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ១៦
. ក!A8ធ!ប!ផ.ប#កនម 3 3yP x y x+= , ច!@ ,x y ∈ℝ 9)យ ,x y ផ0\ង]0 #
8ក_ 2 2 1xyx y+ + =
. កគប#ប ABC នប;<ងជងJ!ងប6ប ច!នBនគ#< Qជ?ន, 9)យ
មQននB;ចកBម នQងផ0\ង]0 #មYព 2
2 2 2 6cot 4cot 9cot
2 2 2 7
A B C p
r + + =
,
ច!@ ,p r PងW- 6កនqប Q នQង !ង"ង#$ %កក-.ង ABC
+. គយកប ជង , ,BC CA AB ប# ABC ធ")6,, ង#oZង'ន:<
;កងម,ប , ,MBC NCA PAB យប>? ក#5ប*0 #J!ងប , ,AM BN
នQង CP បព"W- 9)យ J!ងព ,ABC MNP នទបជ.!ទ!ងន#;មBយ
'()&'()&'()&'()&
ចេលយ
. Fង a 6!A8មBយប#កនម P , នន|យ5បព|នZង មនច!8)យ
2 2
3 3
1 (1)
(2)
xy y
x y y a
x
x
+ + =+ =
Fង 2 2;u x y v xy+= = (8កខ_ 2u v≥ ) បព|នZង8)មម:8ន%ង 1u v
uv a
+ = =
Jញ,ន ,u v 6Hប#ម 2 0X X a− + =
Hផ0\ង]0 # 2
92vu a≥ ⇔ ≤
C:ចន 2max
9P = ព8 1
3x y= = ±
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 61
. យ)ងន cot cot cot2 2 2
p A B Cg g g
r= នQង
cot cot cot cot cot cot2 2 2 2 2 2
(*)A B C A B C
g g g g g g= + +
ពប!Sប#, យ)ង,ន
2
2 2 249 cot 4cot 9cot 36 cot cot cot2 2 2 2
1)2
(2
A B C A B Cg g g g g g
+ + = + +
Lន.<នM< QមYព Bunyakovski ,
2
2 2 236 cot cot cot cot 4cot 9cot492 2 2 2
2)2 2
(A B C A B C
g g g g g g + + + +
≤
>a ()ក)ន cot 2cot 3cot
2 2 26 3 2
A B Cg g g
=⇔ =
ប:កBមន%ង (*) , យ)ង,ន
7sincot 7
2527 56
cot sin2 4 65
7 63cot sin
2 9 65
AAg
Bg B
Cg C
⇔
==
= = = =
Lន.<នMទ%បទ.ន. 40 1325 65 65
45 137 56 63
a ba b c
a c
== = =
⇔
យ)ងយក 4013, , 45a b c= = =
ន-Q ន ABC∆ នជងJ!ងប ( ; ; ) (13 ;40 ;45 )a b c k k k= ច!@ *k ∈ℕ
+. )a ង#oZង' ABC∆ , ប 1 1 2 2 2 1, ,B ACC AC ABBB AC * , ,M N P PងW- 6ផjQ
ប# J!ងបZង8)
ពQនQម)8ចxប#បង"Q8 0902:BQ A B→
1 1 2C ABB B C→ ⇒ = នQង .
Fង 6ច!ន.ចក 8ប# , យ)ង,ន PងW- 6,មធមប# 1 2 (1)B BA C⊥
D AC ,DM DN 1ACB∆
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 62
ប:កBមន%ង , យ)ង,ន នQង
ពQនQចxប#បង"Q8
នQង
6ក!ព#ប#
ធ")C:ចW- ;C, យ)ងយ,ន5 ,BN CP 6ប ក!ព#ប# MNP∆
C:ច* ,AM BN ,CP បព"W- ង#L:ង#ប# MNP∆
Fង ,Q R PងW- 6ច!ន.ចក 8 ,PN AM , 9)យ G MQ BD= ∩
Fម!SយZង8) 090 :DQ PN MA→ , * 090 :DQ Q R→
DQ DR⇒ = នQង (2)DQ DR⊥
2ACB∆ (1) 1 2
1 1
2 2DP DM AB B C== = DP DM⊥
090 :DQ P M→
A P MAN N→ ⇒ = PN MA⊥
MA⇒ MNP∆
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 63
យ DR 6,មធមប# ||DC R MCAM∆ ⇒ នQង 1
2DR MC=
9)យ MC MB= នQង MC MB⊥ * DR MB⊥ នQង (3)1
2DR MB=
ព (2)នQង (3) , យ)ង,ន ||DQ MB នQង 1
2DQ MB=
Fមទ%បទF8 1
2
QD QG DG
BM GM BG= = = នQង ,BD MG G Q∈ ∈ 9)យ ,BD MQ 6
មCOនប# ,ABC MNP∆ ∆ , Jញ,ន G 6ទបជ.!ទ!ងន# ABC∆ នQង MNP∆ ,
(ប>e g<,នយប>? ក#) '()&
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ១៧
. យបព|នម
( )( )( )
2 2
2 2
2 2
(1)
(
6 1 9
6 1 2)
6 1 9
8
(3)
x x
y y
y
z
z x z
= + + =
= +
. គ , , 0a b c > ផ0\ង]0 # 1a b c+ + = ក!A8:ចប!ផ.ប#
( ) ( ) ( )
ab bc caT
c b c a c a b a b= + +
+ + +
+. គ ABC នប ជង , ,a b c នQងប ម.!ផ0\ង]0 # 2 , 4B A C A= = .
គ* 22 2 2
1 1 1S R
a b c
= + +
, ច!@ R 6 !ង"ង#$ %ក' ABC
=. គ ABC , ង#oZង' *ន:< ម,បគI
1 1 1, ,AC B BA C CB A∆ ∆ ∆ នប ជង,គI , ,AB BC CA9)យម.!o,() α
យប>? ក#5 1 1 1, ,BBA CCA 6ប*0 #បបព"W- ង#ច!ន.ចមBយ
'()&'()&'()&'()&
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 64
ចេលយ
. យ)ងន 2 2 2(1 9 ) 3 (2 3 ) (6 *)y x y xx y= + ⇔ = −
3
:2
y = មQន;មន6Hប#បព|នម
3
:2
y ≠ ព (*) យ)ង,ន 2
3(2 3 )
yx
y=
−
ម ន នH 20
3(2 30
) 3y
y
y⇔ ≥ ⇔ ≤ <
−
C:ចW- ;C 2 20 , 0
3 3x z≤ < ≤ <
យ)ង,ន 0x y z= = = 6HមBយប#បព|នម
ច!@ , , 0x y z > , ព 2
: 16
(1)1 9
y xy
x xx=
+≤ ⇔ ≤
ព (2) : z y≤
ព (3) : x z≤
1
3y x z y x y z⇒ ≤ ≤ ≤ ⇒ = = =
C:ចន ច!8)យប#បព|នម គI 1 1 1(0,0,0); , ,
3 3 3
. យ)ងន
2 2
. . .( ) ( ) ( )
ab bc ca ab bc cab c c a a b
c a b c b c a c a b a b
+ + + + + + + + + +
=
( ) 2( ) ( ) ( )
ab bc cab c c a a b T
c b c a c a b a b
+ + + + + + + = + + +
≤
យ 2 2 2 23( )( )a b ab bc ca a b c cac ab bc≥ ++ + + ⇔ + + ≥ + +
* 2
3( ) 3ab bc ca
ca b c
a b
+ + ≥ + +
=
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 65
2 3T⇒ ≥ ⇒ !A8:ចប!ផ.ប# T គI 3
2
!A8:ចប!ផ.ន ទទB8,នព8 1
3
ab bc ca
c a ba b c⇔ = = == =
+. យ)ងន 2sin sin sin
a b cR
A B C= = =
1 1 1, ,
2sin 2sin 2sin
R R R
a A b B c C⇒ = = =
2 2 2
1 1 1 1
4 sin sin sinA B CS
= + +
⇒ ( )2 2 2cot cot1
3 cot4
A g B Cg g+= + +
ក-.ង* ABC∆ ន cot .cot cot .cot cot .cot 1gA gB gB gC gC gA+ + =
9)យយ)ងន 2cot
cot 22c
1
ot
gg
g
ααα−= *Jញ,ន 2 1 2cocot t .cot 2gg gα α α= +
( )13 3 2 cot .cot 2 cot .cot 2 cot .cot 2
4gA g A gB g B CS gC g+ + + +⇒ =
( )1 16 2 cot .cot cot .cot cot .cot (6 2) 2
4 4gA gB gB gC gC gA= + + + = + = .
=. យមQនធ"),#8កទ:G, 4ប5 ˆ ˆˆA B C≥ ≥
ក ទp 0ˆ 180A α+ < Fង , ,M N P PងW- 6ច!ន.ចបព"ប# 1 1 1, ,BBA CCA Gន%ង
, ,BC CA AB ង# 1 1 2 1,A AB A CHH A⊥ ⊥ , យ)ង,ន
1
2
MB BH
MC
MB
MC CH
−= = − 1
1
AB
ACA
S
S= −
1
1
. .sin(
. .
.
si
sin( ) )
.sin( ) )n(
B
C
c BA c B
b CA b C
α αα α
++
+= − = −+
C:ចW- ;C, .sin( .sin(,
.sin(
) )
) .sin )(
NC PA
NA PB
a C b A
c A a B
α αα α
+ += − = −+ +
1 1 1. ,. ,1MB NC PA
AA BB CCMC NA PB
= ⇒−⇒ បព"W- ង#ច!ន.ចមBយ
ក ទp 0ˆ 180A α+ = 1 1 1, ,BBA CCA បព"W- ង#ច!ន.ច A
ក ទ+p 0ˆ 180A α+ >
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 66
.sin(
.sin(
.si
)
)
n(
.sin(
si
)
)
)
)
n(
.sin(
MB c B
MC b C
NC a C
NA c A
PA b A
PB
MB
MC
NC
a B
NA
PA
PB
αα
αα
αα
+= − = −+
+= = −+
+= = −+
. . 1MB NC PA
MC NA PB= −⇒ 1 1 1, ,BBAA CC⇒ បព"W- ង#ច!ន.ចមBយ
'()&
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ១៨
. គ ABC នប;<ងជងJ!ងបPងW- គI , ,a b c O 6ផjQង"ង#$ %ក'
H 6L:ង# R 6 !ង"ង#$ %ក' 4ប5 OH # CB នQង CA ង# P នQង
Q យប>? ក#5 8ក_ $!,ច#នQងគប#Wន#C)មDច. ABPQ $%ក
ក-.ង,នគI 2 2 26b Ra + =
. គ p 6ច!នBនបម យប>? ក#5ច!នBន
....1 .... 2 ... .... 911 22 99 123456789p p p
−
;ចកច#ន%ង p
+. Fង , , ,s t u v 6ប ច!នBនnQoក-.ង 0;2
π V/ង s t u v+ + + = π
យប>? ក#5 2 sin 1 2 sin 1 2 sin 1 2 sin 1
cos cos cos0
cos
s t u v
s t u v
− − −+ + + ≥−
'()&'()&'()&'()&
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 67
ចេលយ
. យ)ងន ងRC:ចZង ម
• ក-.ង ABC , យ)ងន os ;2 .cCH R C BOCH A= = −
• • OH PO QC AB⊥ ⇔ $ %កក-.ង
យប>? ក#
• Fង K 6ច!ន.ចក 8ប# AB , យ)ង,ន 2 2 .cos 2 .cosBOKR CH K RC O= ==
យ)ងន ( ) ( )0 090 90 B AOCH HCA OCA A B= − − −= − = −
• • ង#ប*0 #ប/Gន%ងង"ង# ( )ABC ង# ;C # AB ង# T , ព8*យ)ង,ន
,TCB CAB OC CT= ⊥ *ច. ABPQ $%កក-.ង,
C:ច* ||CTCPQ CA PQB TCB CO OH= = ⇔ ⇔ ⊥
Lន.<នMច:8Gក-.ង8!f#;C8
ច. ABPQ $%កក-.ង
.cos( ) 2 .cos .cos( )CO OH R R C B ACO CH HCO⇔ ⊥ ⇔ ⇔ = −= .
2 2 2 2 22sin 2sin 3cos2 cos 1 62 A aA B b RB+ = + ⇔ +−⇔ ⇔ = =
. ពQនQម)8 3:p = យន%ងយ,ន5ពQ6ផ0\ង]0 #
ពQនQម)8 3:p ≠ យ)ងន
1 1 1
8 7
0 0 0
10 2 10 ... 9 10p p p
p k p k k
k k k
n c− − −
+ +
= = =
+ −= + +∑ ∑ ∑
( )( )
( )
8 7
9 8
1 2.10 ... 8.10 9
10 ... 1
110 10
9
010 91
9
p p p p
p p p
c
c
− + + += −+
+ + −= −+
p 6B;ចកប# n 8.F; 9p 6B;ចកប# 9n (@យ)ងពQនQម)8 p ខ.ព 3)
យ)ងន%ងយប>? ក#5 9 810 ... 10 910 9p p p c+ + + − − ;ចកច#ន%ង 9p
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 68
យ 9 89 9 111111111 10 ... 100 10c+ = = + + + *យ)ងន%ងយប>? ក#5
( ) ( )9 8 9 810 ...1 10 10 .. 100 10 .p p p + +−+ + + + ;ចកច#ន%ង 9p
Fមទ%បទ Fermat , យ)ងន
( )1 10 (mod )0 10pmp m m p≡ ≡
101 o )0 (m d 9mp m≡ .
យ (9; ) 1p = * 101 o )0 (m d 9mp m p≡
Jញ,ន ប>e g<,នយប>? ក#
+. Fង tan ; tan ; tan ; tana s b t c u d v= = = = ព8* , , , 0a b c d >
យ)ង,ន tan( ) tan( ) 0 01 1
a b c ds t u v s t u v
ab cdπ + ++ + + = + + + = +
− −⇒ =⇒
a b c d abc abd acd bcd+ + + = + + +⇔ .
Jញ,ន 2( )( )( ( )) a b c d abc abd aa b a c a d a cd bcd+ + + + + ++ ++ + =
( )( )2 1a a b c d= + + ++ .
2 ( )(1 )
( )
a a c a d
a b a b c d
+ ++
⇒+
+ =+ +
Lន.<នM< QមYព Bunyakovski :
[ ]2 2 2 2
22( ) ( ) (1 1 1 1
) ( ) ( )a b c d
a b c d a b b c c d d aa b b c c d d a
+ + + + + + + ++ + + + + + + + + + + + +
=
( )2 2 2 21 1 1 1a b c d+ + + +≥ + + +
Jញ,ន 2 2 2 2 21 1 1 1 ( )a b c d a b c d+ + + ++ + +≤ ++ +
គI5 ( )1 1 1 12 tan tan tan tan
cos cos cos coss t u v
s t u v+ + +≤+ + +
នគI6ប>e ;C8g<យប>? ក#
'()&
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 69
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ១៩
. យម 33 6 11 48x x x−= −+
. គប ច!នBនគ# , ,x y z ផ0\ង]0 # 9x y z
y z x− + =
យប>? ក#5 3 xyz ∈ℤ
+. គ ABC∆ ន 2
A C< < π នQងប ច!នBនពQ , ,m n p ផ0\ង]0 #
0cos cossin
2
m n pBA C
+ + = យប>? ក#5ម 2 0nx pmx + + = នH
( )0;1x∈
=. គ ABC∆ ផ0\ង]0 # 3cos .cos .cos
8A B C = −
).a យប>? ក#5នម.!មBយប# ABC∆ ន" #:ចប!ផ.() 0120
).b Fង , ,M N P 6ច!ន.ចឆq.ប# , ,A B C ធPបន%ង , ,BC CA AB
យប>? ក#5 , ,M N P #ង#ជB
'()&'()&'()&'()&
ចេលយ
. < Qធទp 3 33 36 16 1 8 4 1 6 1 (2 ) 2 (1)x x x xx x x+ = − − ⇔ + = ++ +
ម នSង ( ) ( )3 1 26f x f x+ = ច!@ 3( ) tf t t= + 6Lន.គមនMក)នក-.ង ℝ
C:ចន ម (1) មម:8ន%ង 33 6 1 2 8 6 1x x x x+ = ⇔ − =
ងR
ប) ( )2 3 21 8 6 4 23 24 1 3x x x xx x⇒ − > ⇒ − −> = >
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 70
*Hប#ម (1) ប)ន g<nQoក-.ង [ ]1;1−
Fង [ ]cos , 0;x t t π= ∈ (1) q យG6
3 1 1 24cos cos3 ,3c
2os
2 3(
9)tt tt kk
π π− = ⇔ ⇔= +± ∈= ℤ
Jញ,ន (1)ន!ន.!ច!8)យគI 5 7cos ; ;
9 9cos cos
9S
π π π =
< Qធទp Fង 3
3
3
6 186 1 2
8 4 2 1
xx y
x x y
y = ++ = ⇒
= + +
ព 2 ម យ)ង,ន ( )3 38 2( )y x yx x y− = − ⇔ =
C:ចន (1) 3 18 6x x⇔ − =
ប ជ!fនបនទP យC:ច< QធទមBយ
. B;ចកBមធ!ប!ផ. 0 0 0( , , ) ; ;x dx y dyx y z dzz d ⇒ = = == , ច!@ 0 0 0, ,x y z ∈ℤ
9)យB;ចកBមធ!ប!ផ. 0 0 0, , 1( )y zx = ព8*
0 0 0 3 3 30 0 0 0 0 0
0 0 0
9 (1); xyz d x y z x y zx y z
y z x= ∈ ⇔ ∈− + = ℤ ℤ
ងR បព|នច!នBន 0 0 0 0 0 0( ) :, , 1y z y zx x = ± មQនចផ0\ង]0 # (1) * 0 0 0 1y zx ≠ ±
Fង p 6B;ចកច!នBនបមមBយប# 0 0 0x y z
យ 2 2 20 0 0 0 0 0 0 0 09 (2( ) )1 z y x z y x y zx⇔ − + =
នQង 0 0 0, , 1( )y zx = * p 6B;ចកប#ច!នBនពក-.ង 3ច!នBន 0 0 0, ,x y z
យមQនធ"),#8កទ:G, 4ប5 p 6B;ចកប# 0 0,x y 9)យមQន;មន6
B;ចកប# 0z Fង ,m n PងW- 6ច!នBននQទ2នប# p ក-.ង ប!;បកង#ប#
0x នQង 0 )( ,y m n +∈ℤ
យ)ងពQនQពក
)a ប) 2 1:n m≥ + យ)ង,ន 2 1mp + 6B;ចកប# 2 20 0 0 0 0 0, , 9y y x x y zz *ព (2)Jញ,ន
2 1mp + 6B;ចកប# 20 0x z , ;យ 2
0x ;ចកមQនច#ន%ង 2 1mp + * p 6B;ចកប# 0z
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 71
(ផ0.យព ពQ)
)b ប) 2 1 2 :1n nm m⇔ +≤ − ≤ យ)ង,ន 1np + 6B;ចកប# 2 20 0 0 0 0 0 0, 9,z y x yx zx
*ព (2) Jញ,ន 1mp + 6B;ចកប# 20 0z y , ;យ 0y ;ចកមQនច#ន%ង 1np + *
p 6B;ចកប# 0z (ផ0.យព ពQ)
C:ចន 2n m= * 3m 6ច!នBននQទ2នប# p ក-.ង ប!;បកង#ប# 0 0 0x y z
Jញ,ន 30
10 0
imi
k
i
pzx y=
= ∏ (ច!@ , 1,ip i k= , 6ប B;ចកបមប# 0 0 0x y z )
C:ចន 30 0 0x y z ∈ℤ
ច!!ច!!ច!!ច!! យ)ងចបe ញ,ន5 នបព|នបច!នBន ( , , )x y z ច)នSប#មQនL#;C8ផ0\ង
]0 # 9x y z
y z x− + = , 4J9M 3 , 9, y tx yt z= = = ច!@ t 6ច!នBនគ#ក,នខ.
ព 0 , នQង 3 3xyz t=
!W8#!W8#!W8#!W8# @កប!;បកង#oក-.ង8!f#ន គIបកFម@កយBន;Cគ5
Phân tích tiêu chuẩn : ប!;បកង#??
+. C!ប:ង, ច!@ ABC∆ ផ0\ង]0 # 2
A Cπ< < យ)ង,ន
( ) ( )( ) ( ) 21 1cos .cos cos cos 1 cos sin
2 2 2
BA C A C A C B= − + + < − =
0 2 0 sin cos2 2 2 2 2
B BA C A B A A
π π ππ< < < + < < < − < <⇒ ⇒ ⇒
Fង 2cosc (os ; sin ; 0 1, 02
1)B v
u A v uwu
w C v= ⇒= = < < < <
នQង 2( )f x mx nx p= + +
ប) 0p = ប!Sប#បdនG6 0m n
u v+ =
ប) 0m = * 0 ( ):n f x= 6 " ព9.d:ន " *នH ( )0;1x ∈
ប) 0 ( ) ( )fn v
mu
x x x nm
m−⇒ ⇒ = +=≠ នH (0;1)n v
xm u
∈= − =
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 72
ប) 0p ≠ , យ)ងប!;8ងប!Sប#G6
2 2
2 2 20 0
m n p u v v uw vm n p p
u v w v u u v w
−+ + = + + − =
⇔
2
(0)v uw v
f fu uw
− =
⇔
C:ចន [ ]2
2. (0) (0) 0
v uw vf f f
u uw
− =
<
(@ (1)នQង 0(0)f p= ≠ )
Jញ,ន ( )f x នH 0; (0;1)u
xv
⊂
∈ * ( )f x នH ( )0;1x ∈
=. )a B*ទប# , ,A B C គIC:ចW- 9)យពប!Sប#Jញ,ន ABC∆ 6 J8, *
យមQនធ"),#បង#8កទ:G, 4ប5 090 BA C> > ≥ យ)ង,ន
( )3 3cos .cos .cos cos . cos( ) cos
8 4A B C A B C A= − − −⇒ = −
( ) 2 3cos 1 cos
3 1cos coscos 0
244A A A AA⇒ ≤ − − ⇒ − ⇒ ≤ −− ≥
C:ចន 120A ≥ មYពក)នព8 030B C= =
(ផ0\ង]0 # 0 0 0.cos30 .cos303
cos1208
= − )
)b Fង , ,H K L PងW- 6ច!8;កងប# , ,A B C o8)ជងឈម (យ)ងo;4ប
5 A 6ម.!J8) Fង ,AB u AC v= = , យ)ង,ន
( ).cos .cos. .
c BAH AB BH u BC
c B
a au v u= + = + = + −
.cos .co.cos os s.c b C c Bu v
a c B c B
au v
a aa+ = +−= (@ .cos .cosa b C c B= + )
,.cos .cos
BK BA AK u v CL CAc A b A
b cAL v u= + = − + = + = − +
, ,M N P PងW- 6ច!ន.ចឈមប# , ,A B C ធPបន%ង , ,BC CA AB *យ)ង,ន
2 cos 2 cosb C c BAM u
a av= +
, 22 ;
.cosAN AB B
c A
bK u v= + = − +
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 73
2 .cos2AP AC CL v u
b A
c= + = − +
Jញ,ន cos cos2
2 cos1
A BMN AN AM c v
b
b a au
C − + = − = −
cos cos2
2 cos1
A C c B
c aMP AP AM b u
av
− +
= − = −
C:ចន , ,M N P #ង#ជBW- ,MN MP⇔
នទQC:ចW-
ក នមម:8ន%ង
cos cos cos cos 2 cos 2 cos2 .2 1 1
A B A C b C c Bc b
b a c a a a − − = + +
( )( ) ( ) ( )4 cos cos cos .cos 2 cos 2 cosa A b B a A c C a b C a c B− − = + +⇔
( ) ( )4 cos cos cos cos 2 cos cosa A b B c C A a b C c B− + = + + ⇔
( )4 cos cos cos cos 3a A b B c C A a− + = ⇔
sin sin4 cos cos cos cos 3
sin sin
B CA B C A
A A
− + =
⇔
( ) ( )4 cos cos cos 3B C B C A⇔ − + − − =
3cos .cos .cos
8A B C = −⇔
មYពច.ង យ,នក-.ងប!Sប# *យ)ង,នប>e g<,នយប>? ក#
'()&
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 74
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ២០
. កគប#Lន.គមនM :f + +→ℝ ℝ ( +ℝ 6!ន.!ប ច!នBនពQ< Qជ?ន) ផ0\ង]0 #ពមW-
ន:<8ក_ ពZង ម
,(1) x y +∀ ∈ℝ , ប) x y≤ * )( ()f x f y≤
( )(2) 200( 6. ). , ,
f y
xf x y f x y +
= ∀
∈
ℝ
. គ ,m n 6ពច!នBនគ#< Qជ?នផ2ងW- ន ( , )m n d=
គ* ( )1,202006 06 1m n+ + (>a ( , )m n បe ញពB;ចកBមធ!ប!ផ.Aន m នQង n ).
+. ក!A8ធ!ប!ផ.ប#កនម3 3 3x
Fy
xy
z
z
+ += ច!@ , ,x y z nQoច*q
[1003;2006]
=. គ [ច ABC $%កក-.ងង"ង#ផjQ O ប*0 # AO #ជង BC ង# D
o8)ជងJ!ងព AB នQង AC PងW- គច!ន.ច M នQងច!ន.ច N V/ង
DB DM= នQង DC DN= CM នQង BN #W- ង# E Fង H នQង K 6L:ង#
ប EBM នQង ECN យប>? ក#5 HK ;កងន%ង AE
'()&'()&'()&'()&
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 75
ចេលយ
. 4ប5Lន.គមនM :f + +→ℝ ℝ ផ0\ង]0 #ពមW- ន:<8កខ_ J!ងព
,(1) x y +∀ ∈ℝ , ប) x y≤ * )( ()f x f y≤
( )(2) ( ). 200 ,6,
f yf xy f
xx y + = ∀
∈
ℝ
ព (2)យក 1x = យ)ង,ន 2006( ( ) (3) ,
))
(f f y
fy
y+∈= ∀ ℝ
,x y +∀ ∈ℝ , ប) x y> *ព (1) យ)ង,ន ( ) ( ( ) ( )( ) ) ( )f y f f x f ff x y≥ ⇒ ≥
2006 2006( ) (( )
( ) ((
)) )f f x f xy
f x f yf y⇒ ≥ ⇒ ≥ ⇒ =
a +⇒ ∃ ∈ℝ V/ង ( ) , xf x a +∀ ∈= ℝ (4)
ព (3)នQង 2006(4) 2006,aa x
a+⇒ = ∀= ⇒ ∈ℝ
( ) 2006, xf x +⇒ ∀ ∈= ℝ ផ0.យមក< Qញប) :f + +→ℝ ℝ នQង ( ) 2006, xf x +∀ ∈= ℝ *ចx##5 (1)នQង (2)
.ទ;ផ0\ង]0 #
. Fង ,m n
rd
sd
= = នQង 2006d b= , យ)ង,ន
( , ) 1, 20 ,06 2006m r n sr s b b= = = , b 6ច!នBនគ:
យ)ងក 1,2006 1) ( 1, 1)(2006m n r sg b b+ + = + += , យ)ង,ន
(mo1 1 )(1)
1 1(mod )
dr r
s sg g
b bg g
b b
+ ≡ −
⇒+ ≡ −⋮
⋮
1 1 ( ) 1 ( , ) 1 (2)r r rg b kg k kg b b gb + ⇒ + = ∈ ⇒ − = ⇒ =⋮ ℤ ( , ) ,1s u vr ⇒ ∃= ∈ℤV/ង (3)1ru sv− =
( , ) 1r s = * ,r s 6ច!នBនC:ចW- Iន8កគ:ផ2ងW-
ក ,r s .ទ;
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 76
ព8* 1rb + នQង 1sb + ន 1b + 6B;ចកBមមBយ, Jញ,ន ( 1 (4))g b +⋮
ព (( 1) )
( 1)
mod
(mod )(1)
ru u
sv v
g
b
b
g
≡ −⇒
≡ −
យ ,r s .ទ; 9)យ 1ru sv− = * ,u v ន8កគ:ផ2ងW-
C:ច* ( 1) ( 1) 0 (mod )ru sv u vb b g+ = − + − ≡
1 1( 1) (5)sv sv svb b bb bg g+ +⇒ + = + ⇒⋮ ⋮ ព (4)នQង (5) 1g b⇒ = +
ក ,r s ន8កគ:ផ2ងW-
( 1) )2. ( 1) ( 1) 0 (mod )
( 1) )
(mod(1)
(mod
rsrs r s
sr r
gb g
b g
b ≡ −⇒ ⇒ ≡ − +
− ≡≡ −
22. (6)rs g gb⇒ ⇒⋮ ⋮ យ b 6ច!នBនគ: 1, 1r sb b⇒ + + 6ប ច!នBន g⇒ 6ច!នBន
ព (6)នQង (7)យ)ង,ន 1g =
C:ចន 1,2006 1) 2006 1(2006m n d+ + = + ប) ,m n
d d6ច!នBនC:ចW- ,
1,2006 1) 2006 1(2006m n d+ + = + ប) ,m n
d dន8កគ:ផ2ងW-
+. យ , ,x y z នB*ទC:ចW- *យ)ងច4ប5 20061003 x y z≤ ≤ ≤ ≤
Fង y kx= នQង z hx= 21 )( k h≤ ≤ ≤ យ)ង,ន
3 3 3 3 3 3 31
(1). .
x kA
x kx
k x h x
hx kh
h+ + ++= =
យ)ងន%ងយប>? ក#5 3 3 3 32
(2)1 1
2
k kh
hk k
+ ++ +≤
3 3 3 3 3 2 2(2) 2 2 (2 .2 (2 ) 2 ( 2 )) 02k h kh h hk h h h h+⇔ + ≤ + + ⇔ − + −+ ≤− .
3 22 ] 0 (*(2 )[1 4 )h h k h⇔ − ≤− − + យ 0,12 4 0hh ≥ − ≤− នQង 3 2 2 22 2 2 0h k hk − ≤ − ≤ * (*) ពQ
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 77
មO/ងទPយ)ងន 3 3 3 21 ( 1)(
52 2 2
2 10 9 9)0 (3)
k k k k
k k
k k
k
+ − + + − ≤+ −− = =
ព (1),(2)នQង (3)យ)ង,ន 5A ≤ មO/ងទPព8 1003 2006, zx y == = * 5A =
C:ចន !A8ធ!ប!ផ.ប# 5A =
=. Fង P នQង T PងW- 6ច!ន.ចក 8ប# BM នQង AB
ADP B⇒ ⊥ (@ DM DB= )នQង OT AB⊥
22 1||
MB PB AP
AB AO
AT
B BDP
=⇒ = −
⇒
2 2AP AD
AT AO= − = −
C:ចW- ;C, យ)ងន 2NC AD
AC AO= −
C:ច* MB NC
AB AC=
||MN B AC E⇒ ⇒ #Fមច!ន.ចក 8 J ប# MN
Fង 1( )γ 6ង"ង#ផjQ I ,LងR#ផjQ AE 9)យ
2( )γ 6ង"ង#ផjQ J ,LងR#ផjQ MN
Fង ,F G PងW- 6ច!8;កងប# E
G8) AB នQង AC Fង U 6ច!8;កងប# M G8) BN 9)យ V
6ច!8;កងប# N G8) CM
យ)ង,ន 1)/ ( .H HE HFP γ = នQង 2)/ ( .H HM HUP γ =
យ , , ,M F U E nQo8)ង"ង#LងR#ផjQ EM C:ចW- *
1 2. . / ( ) / ( )H HHE HF HM H PU Pγ γ= ⇒ = .
C:ចW- ;C, យ)ងន 1 2/ ( ) . . / ( )K KKG KE KN KV PP γ γ= = =
Jញ,ន Jញ,ន Jញ,ន Jញ,ន KH 6666L|ក2L.:gបប#ង"ង#J!ងL|ក2L.:gបប#ង"ង#J!ងL|ក2L.:gបប#ង"ង#J!ងL|ក2L.:gបប#ង"ង#J!ងព 1 2),( ( )γ γ
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 78
Jញ,ន IJ EK AH HK⊥ ⇒ ⊥
!W8#!W8#!W8#!W8# L|ក2L.:gប គIបក;បFម@កបចjកទ<Pម;C85
Trục ẳng phương : L|ក2L.:gប??? '()&
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ២១
. យម
( )( )2 242 3 200 2005 4 4 37 0 20061x x x xx x x x− − − = ++ + + −
. ).a គ , 0, 1yx y x> + ≥ ក!A8:ចប!ផ.ប#កនម
9 4851 23
7P x y
x y= + + +
).b គ , , 0a b c > យប>? ក#< QមYព
( ) ( ) ( ) ( )2 2 2 2 2 2 3ab b bc c ca aa b c ab bc ca+ + + + + + ≥ + +
+. គ , ,a b c 6ប;<ងជងJ!ងបប# មBយ9)យ R6 !ង"ង#$ %ក'ប#
* យប>? ក#5 3 3 3
24a
a b c
b cR
+++
+ ≤
=. គ!ន.! 1 2 3 2006, , , ...,a a aA a= គបងR)!ន.! , ,B C D C:ចZង ម
1 2 3 2006, , , ...,b b bB b= ច!@ 12007 1(1 1,2,..., 2006),
2i i
ia
ba
a a++= = =
1 1 3 2006, , , ...,c c cC c= ច!@ 12007 1(1 1,2,..., 2006),
2i i
ib
cb
b b++= = =
1 2 2006, , ...,dD dd= ច!@ 12007 1( 1,2,..., 2006),
2i i
ic
i cc
cd ++= = =
គC%ង5 A D= នQង 1 1a = ក 2 3 2006, , ...,aa a '()&'()&'()&'()&
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 79
ចេលយ
. ម ;C8មម:8ន%ង
( ) ( )222 241 2005 30 1 01x x xx x x+ −+ − + + + − =
2 1
1 0
1 5
2
x
x x
xx
− − =+ −
⇔ ⇔+ − =
. )a Lន.<នM< QមYពក:.
( ) 9 482 49 21
72 42 24 68P x y y
x y
= + + + + + ≥ + +
=
Jញ,ន !A8:ចប!ផ.ប# P គI 68, ព8 3 4;
7 7x y ==
)b Lន.<នM< QមYពក:."
( )22 2 3 3) (1)( )( aab bc ca ab b ab b c ca+ + + + ++ ≥
( )( ) ( )22 2 2 2 3 3 (2)bc c c cab a ab bc ac+ ++ + + + ≥
( )( ) ( )23 3 3 3 (3)ab b a ab bc ac ab bc ac c c+ + + + + +≥
ព (2)(1), , (3)យ)ងJញ,ន< QមYពg<,នយប>? ក#
+. យ)ងន ( )2
( ( )) ( ) 0OC b c OA c aa OBb + + + + ≥+
( ) ( ) ( ) ( ) ( )2 2 2 2 2 .OC Oa b b c c a R a b b c A⇔ + + + + + + + +
( )( ) ( )( ). 22 . 0b c c a cOAOB OBa a OCb+ + + + ++ ≥
( ) ( ) ( ) ( ) ( )( )2 2 2 2 2 22a b b c c a R a b b c R b + + +⇔ + + +
−+ +
( )( )( ) ( )( ) ( )2 2 2 22 2 0b c c a R cc a a b R a− −+ + + + + ≥+
( )2 2 2 2 24 ( )( ) ( )( ) ( )( )R a a b a c b b c b a ca c c c bb a+ +⇔ ≥ + + + + + + + +
( ) ( )2 2 3 3 34( )a b c a bR b c cc a ab+ +⇔ ≥ + + ++ + 3 3 3
24b c aba
a b c
cR
++
++ +⇔ ≤
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 80
=. យ)ងន 2 222 2 2 22 2
2 3 2006 1 2 3 2006 11 2 1 2
2 2 2 2 2 2... ...
a a a aa aa a a aa a + +
+ + + ++ +
≥ +
+ +
+
2 2 2 2 2 21 2 2006 1 2 2006... ...a ba a b b⇒ + + + ≥ + + +
យប>? ក#C:ចW- ;Cយ)ង,ន
2 2 2 2 2 2 2 2 21 2 2006 1 2 2006 1 2 2006... ... ...b b c c d d db c+ + + ≥ + + + ≥ + + + .
Jញ,ន 2 2 2 2 2 21 2 2006 1 2 2006... ...aa a d d d+ + + ≥ + + +
យ A D= (Fមប!Sប#) *មYពក)ន
ព*Jញ,ន 1 2 2006... 1aa a= = = =
'()&
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ២២
. កH6ច!នBន< Qជ?នប#បព|នម
1 2 31 2
2 3 42 3
2006 2007 12006 2007
2007 1 22007 1
2007
2007
...............................
2007
2007
x xx
x xx
x xx
x
xx
xx
xx
xx
xx
+ − =
+
− =
+
− =
+ − =
. ក!A8ប# , ,m a b C)មDម 5 33 0mxx − − = នHព 1 2,x x ;C8 1 2,x x
6HJ!ងពប#ម 2 0axx b+ + = , ក-.ង* a នQង b 6ប ច!នBនគ#
+. គ ABC Fង ; ;a b cm m m PងW- 6ប;<ងមCOនJ!ងប9)យ ; ;a b ch h h
PងW- 6ប;<ងក!ព#J!ងប;C8គ:ចញពក!ព:8 , ,A B C ប# ABC
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 81
យប>? ក#5 1
4sin sin sin2 2
1
2
a b c
a b c
m m mA B Ch h h
+ + ≤ +
=. គ ABC នQង O 6ច!ន.ចមBយnQoក-.ង ABC Fម O គង#
ប*0 #បផ2ងW- 1 2 3; ;d d d , ពBកPងW- #ជង ,AB BC ង# ,M N , #ជង ,BC
CA ង# ,P Q នQង #ជង ,CA AB ង# ,R T Fង 1 2 3; ;S S S ; S PងW- 6កmAផ0
; ;OPN ORQ OMT នQង ABC យប>? ក#5 1 2 3
1 1 1 18
S S S S+ + ≥
'()&'()&'()&'()&
ចេលយ
. Fង 3
, 1;20072007
kk
xky = = យ 0kx > * 0ky >
ព8*បព|នម ;C8 g<,នc)ង< QញC:ចZង ម
1 2 31 2
2 3 42 3
2006 2007 12006 2007
2007 1 22007 1
1
1
1
1
.
(*)
.
.
yy
yy
yy
y
y yy
y yy
y yy
y yyy
+
− =
+ − =
+ − =
+ − =
យ)ងឃ)ញ5 1 2 2007... 1yy y= = = = 6ច!8)យមBយប#បព|នម (*)
យ)ងន%ងយប>? ក#5បព|នម (*) នច!8)យ;មBយគ# 1 2 2007... 1yy y= = = =
យ)ងន 1 21
(1), 1;1
.2007k k k
k k
y y ky
yy+ +
+
+ − = ∀ =
(ន( 2007k kyy + = ) ប:កគប#ប ម ប#បព|នម (*) FមLងXន%ងLងX
1 2 20071 2 2 3 2007 1
1 1... ..
1. (1)
. . .y
yy y
y yy yy++ + = +++
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 82
ព*យ)ង,ន ងRឃ)ញC:ចZង ម
ងRទp ប) 0
1ky > * 0 1 1ky + < (ច!@ 0 2; ... ; 01; 20 7k ∈ )
ពQ6C:ចន ,@ប) 0
1ky ≥ *យ)ងJញ,ន
0 0 0
0 0
2 11
11
.k k kk k
yy
y yy+ +
+
= + − >
ព 0
1ky ≥ នQង 0 0 0 0
0 0
2 3 1 21 2
1.
11k k k k
k k
y y yy
yy+ + + +
+ +
> ⇒ = − >+
ច;បនC:ចន, យ)ងJញ,ន 0 01, ...; ; 2; ... ; 22 071; 0;k k ky k> ∀ ∈ + នQង 0 1 1ky + ≥
ព (1)Jញ,ន LងXZងឆ"ង 2007> 9)យLងXZង ! 2007< (ផ0.យW- )
យធ") បកយC:ចW- ន%ង ងRទ;C, យ)ង,ន ងRទC:ច
Zង ម
ងRទp ប) 0
1ky < * 0 1 1ky + > (ច!@ 0 ...1 ;;2 07; 20k ∈ )
cប#មក8!f#យ)ង< Qញយ)ង,ន
4ប5 0 ...; 20071;2;k∃ ∈ C)មD 0
1ky >
Fម ងRទ នQង ងRទ, យ)ងJញ,ន
0 0 0 0 01 2 3 4 20071; 1; 1; 1; ... ; 1k k k k ky y yy y+ + + + +< > < > < (ផ0.យព ពQ@
0 02007k ky y+ = )
C:ចW- J!ងង;C យ)ងJញ,ន5 មQនន 0 ...1 ;;2 07; 20k ∈ C)មD 0
1ky < ទ
C:ចន 1 2 2007... 1yy y= = = = 6ច!8)យ;មBយគ#ប#បព|នម (*)
ព*Jញ,ន 31 2 2007... 2007x xx = = = = 6ច!8)យ;មBយគ#Aនបព|នម ;C8.
. យម 5 3 0mxx − − = នHព 1 2,x x 6Hប#ម 2 0x ax b+ + =
*ព9.d 5 3 0mxx − − = ;ចកច#ន%ង 2 0axx b+ + =
Lន.<នMប<Qធ;ចកព9.d 5 3 0mxx − − = នQង 2 0axx b+ + = យ)ង,ន
( ) ( )5 2 3 2 2 323mx ax b axx x x a x ab b a− − = + + − + −− +
( )4 2 2 3 23 2 3a b b m aa x a b b− + − −+ −+
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 83
4 2 2
3
3 0 (1)
2 3 0 (2)
a b b m
a b
a
ab
− + − =⇒
− − =
ព ( )2 2 (3( )2) 3b aa b⇒ − =
យ 2 2,a b a b∈ ⇒ − ∈ℤ ℤ ព (3) a⇒ 6B;ចកប# 3
នបBនក
i ក ទp 1a =
2(3) (1 2 ) 3 2 3 0b bb b− =⇔ ⇔ − + = ម W( នH ($8)
i ក ទp 1a = −
2(1 2 ) 01
(3) 2 3 3
2
0b b bb
bb
⇔ − − = ⇔ − −= −
== ⇔
, oទន យ)ង$8 1b = −
ព (1) 5m⇒ =
i ក ទ+p 3a =
23 (9 2 ) 3 9 1 09 73
(3) 24
bb b b b±⇔ − = ⇔ − + = ⇔ = ($8)
i ក ទ=p 3a = −
2 9 89(3) 3 (9 2 0) 13 2 9
4bb b bb
±⇔ ⇔ − − = ⇔− =− = ($8)
ផ0\ង]0 #c)ង< Qញ, ឃ)ញ5 1, 51,b ma = −− == គI 2 2 1ax b x xx + + = − − នQង
( )( )5 5 2 3 23 5 3 1 2 3x x xmx x x x x x− − = − − = − − + + + ,
Jញ,ន ម 2 0axx b+ + = នHពផ2ងW- 1,2
1 5
4x
±=
9)យម 5 3 0mxx − − = នHព 1 2;x x 6Hប#ម 2 0axx b+ + =
(ផ0\ង]0 !) ប#បdន8!f#)
C:ចន 1, 51,b ma = −− ==
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 84
+. យ)ងន 1
4sin sin si1 (1
n2 2 2
)a b c
a b c
m m mA B Ch h h
+ + ≤ +
1 (2)a b c
a b c
m m m R
h h h r⇔ ++ + ≤ (@ 4 sin sin sin
2 2 2
A B Cr R= )
គ.LងXJ!ងពប#< Qម ន%ង ABCS∆ យ)ង,ន
1 1 1(2)
2 2 2a b c ABCam bm cm S pR∆⇔ + + ≤ +
) )1 1 1
( ( (2 2 2
) (3)a b c ABCa m b m c mR R R S∆⇔ − + − + − ≤
យ)ងន a R AM OAm OM− = − ≤
C:ចW- ;C bm R ON− ≤ នQង cm R OP− ≤
ព (3)យ)ង,ន LងXZងឆ"ង 1 1 1. . .
2 2 2a OM b ON c OP+≤ +
⇔ LងXZងឆ"ង OBC OCA OAB ABCS S S S≤ + + = =LងXZង !
C:ចន (1) g<,នយប>? ក#
=.
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 85
Fង 4 5 6, ,OMQ OTP ORNS S S S S S∆ ∆ ∆= = =
4 5 6 4 5 6
1. . . . . .s. . .sin .
8sininOM ON OP OQ OR OT OS S S O O⇒ =
1 2 3 1 2 3
1. . . . . .sin .sin .sin
8. .OM ON O O SOP O S SQ OR OT O= =
យ)ងន 3 6
1 2 3 1 2 3 1 2 3 4 5 6. .
1 1 1 3 3
S S S S SS S S S SS S+ ≥+ =
មO/ងទP 1 2 3 4 5 661 2 3 4 5 6 6 6
S S S S SS S S S S S
S S+ + + + +≤ ≤
1 2 3
1 1 1 18
S S S S+ +⇒ ≥ , >a " "= ក)នព8 1 2 3 4 5 6 6
S S S S SS
S = = = = == ,
I 1 2 3; ;d d d 6ប*0 #មCOនJ!ងប 9)យ O 6ទបជ.!ទ!ងន#ប# ABC∆ '()&
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ២៣
. គ ;កង ABC ;C8នL.ប/:ន. BC នប;<ង() a ;ចក BC 6101
;ផ-ក()W- ព8* ABC g<,ន;ចក6101 :ច 9)យ
oច!ក 8នម.!ង#ក!ព:8 A() α Fង h 6ប;<ងព AG BC
យប>? ក#5 101tan
2550
h
aα =
. កគប#ប ច!នBនពQ m C)មDបព|នម Zង មនH6ច!នBនពQ , , :x y z
1 1 1 1
1 1 1 1(*)
x y z m
x y z m
− + − + − = −
+ + + + + = +
+. ផ8ប:កAន m ច!នBនគ:< Qជ?នផ2ងW- នQង nច!នBន< Qជ?នផ2ងW- ()ន%ង 2001
ក!A8ធ!ប!ផ.ប#កនម 5 2A m n= +
=. យប>? ក#5ប) , ,a b c 6ប;<ងជងJ!ងបប# មBយនប Q() 1
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 86
* 2 2 213
274
1
2a b c abc≤ + + + <
'()&'()&'()&'()&
ចេលយ
. Fង AMN 6 oច!ក 8នកmAផ0 S , យ)ង,ន
50,
101 101
a aBM CN MN= = = នQង 1 .
.2 202
h aS MN h= =
Fង I 6ច!ន.ចក 8 BC ក-.ង AMN , យ)ងន
2
2 2 222
AN AN
IM
AM + = + ច!@ 2
2 2 5101
2 10201
a aAI ANAM + == ⇒
C:ចន 2 2 2
sin 2 2 . 101tan
cos . 2550
S AM
AN MN
AN h
AM AN AM a
ααα
= = ==+ −
. 8កខ_ , , 1x y z ≥
យ)ង,ន ( ) ( ) ( )( ) ( ) ( )
1 1 1 1 1 1 2(*)
1 1 1 1 1 1 2
x x y y z z m
x x y y z z
+ + − + + + − + + + − =
+ − − + + − − + +⇔
− − =
Fង 1 1; 1 1; 1 1u x x y yv w z z= + + − + + − = + + −=
យ , , 1x y z ≥ * , , 2u v w ≥ ផ0.យមក< Qញប) , , 2u v w ≥ យ)ង,ន
2
2 1 2 1 21 1
2 411x x x u u
u u ux + − − = + = + +
=
⇒
⇒ −
22
41
1
4u
ux
= +
⇒ ≥
C:ចW- ;C 1,y z ≥ C:ចន 8!f#;C8 q យG6p កគប#ប ច!នBនពQ m C)មD
បព|នZង មនច!8)យ : (2
, 1 ), 2 1 11
u v w mu v w
u v w
I+ + =
+ + =
≥
8កខ_ $!,ច#
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 87
4ប5 ( )I នច!8)យ ( , , )u v w Fម< QមYព Bunyakovski , យ)ងន
1 1 1 92 ( )
29m u v w
u wm
v = + + + + ≥ ⇒
≥
8កខ_ គប#Wន#
4,ម5 9
2m ≥ , យ)ងយប>? ក#5 ( )I នច!8)យ យក 3w = (ផ0\ង]0 # 2w ≥ )
2 3
( ) 3(2 3)
2
u v mI m
uv
+ = − −=
⇔
ព8* ,u v 6Hប#ម 2 3(2 3)2 3) 0
2( m X
mX − − + − =
(2 3)(2 9) 02 3 (2 3)( 9
2,
2 )m m u v
m m m±∆ = − − ≥ ⇒
−=
− −
យ)ង,ន 2 2(2 3)(2 9) ( 6) ( 3) ( 2 2)6m m h h h h<+ +− −− = < +
2 9 2 3 2 2 (2 3)(2 20 ,9)h um m m m v≥= − − − > − −⇒ ⇒ ≥
C:ចន ( )I នច!8)យ , , 2u v w ≥
.បមក 9
2m ≥ 6គប#ប !A8ផ0\ង]0 #!) 8!f#
+. ផ8ប:កAន m ច!នBន< Qជ?នគ: ផ2ងW- មQន:ច6ង
2.... 2 2.( 1)
2 42
mmm
mm
++ ==+ ++
ផ8ប:កAន n ច!នBន< Qជ?នផ2ងW- 21 3 ... (2 1)n n≥ + + + − =
C:ចន ពប!Sប#Jញ,ន 2
2 2 21 12001
2 4m n nmm ≥ + + = ++
−
2
21 1
2 42001 (1)nm⇒ + ≤ + +
យ)ងន ( )2
2 2 221 5 1 5
5 2 5 2 52 2
( )2 2
2nA m n m n m = + = + + − + −
≤ + +
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 88
ព (1) នQង (2)Jញ,ន 238,407 (1 5
29 320 1 )04 2
A A + − ⇒ ≤
≤
យ ,m n 6ច!នBនគ#< Qជ?ន* 5 2A m n= + 6ច!នBនគ#< Qជ?ន
C:ច* ព (3)យ)ង,ន 238A ≤
>a " "= ក)នព8 2 2
(5)
2001 (
2
6
2
)
5 38
m m
m n
n
+ =
+ + =
យ 0 0: , ) (0,119(5) )(m n = 6HមBយប# (5)
C:ច* (5)នច!នBនH ( , )m n ច)នSប#មQនL#ក!#យ
2
119 5( )
t
nt
m
t
= =
∈−
ℤ
យ , 0m n > *យ)ង,ន 0
2 0119
119 5 ,0 0 25
3t t
tt
t t
>> − > <
⇔⇔ ∈ < <
ℤ
ជ!នBច:8 (6) យ)ង,ន 2 2 22 (119 5 ) 2001 29 11884 12160 0tt t t t+ + − = ⇔ − + =
20t⇔ = , ព8* 40
19
m
n
= =
C:ចន !A8ធ!ប!ផ.ប# A គI 238 ព8 40, 19m n= =
=. យប>? ក#5 2 2 2 1
24a b c abc+ + + <
Fង S 6កmAផ0 , យ)ង,ន
1 1 1 1
2 2 2 2S a b c
= − − −
I 2 (1 2 )(11 2 )(1 2 ) (1)6 a bS c= − − −
យ 216 0S > *ព (1)Jញ,ន
(1 2 )(1 2 )(1 2 1 4( ) 8 () 00 2 )ab bc ca abc a b ca b c ⇔ + + + − − +− +− > >−
( )2 2 2 21 4( ) 8 0 1 2 ( ) 8 0ab bc ca abc a b c a b c abc ⇔ − + + + − > ⇔ − + + + − + + − >
( )2 2 2 2 2 2 18 01 2 4
2b c a b c aa a c bb c⇔ − + + ⇔ + + + <− >
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 89
យប>? ក#5 2 2 2 13
274b cT a abc+ + + ≥=
យ)ងយន%ងយ,ន5 ( )( )( ) (2)a b c b c a c ab ba c ≥ + − + − + −
ព (2)យ)ង,ន
(1 2 )(1 2 )(1 2 ) 1 2( ) 4( ) 8a b c a b c ab bc ca abcabc abc⇔ ≥ − − − ⇔ ≥ − + + + + + −
9 1 4( )1 4
( )9 9
abc ab bc c abc ab bc aa c⇔ ≥ − + + + ⇔ + + +≥ −
C:ច* 2 4 16( ) 2( ( ))
9 9a b c ab bc cT aba bc ca≥ + + − + + − + + + 5 2
( )9 9
ab bc ca= − + +
យ 2 3() )( aa b cc b b ca≥ + ++ + * 25 2 13( )
9 27 27T a b c− + =+≥
>a " "= ក)នព8 1/ 3a b c= = =
'()&
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ២៤
. ក!A8:ចប!ផ.ប#កនម ( ) ( ) ( )
88 81 2
2 2 22 2 2 2 2 21 2 2 3 1
... n
n
aa aA
aaa aa a+
+ + += + + ,
ច!@ 1 2, , ..., naa a 6ប ច!នBន< Qជ?ននQង 1 2 2 3 1... na a a a a Ka + + + = ( K 6ច!នBន
ថមBយ)
. កគប#ប ច!នBនគ#ធម(6Q n C)មD 2005 2006 2 2n n nA n + + + += 6ច!នBនបម
+. 1).យប>? ក#5ម Zង មនHបផ2ងW- 3 3 1 0 (*)xx − + =
2).Fង 1 2 3, ,x x x 6HJ!ងបប#ម (*) 9)យបច!ន.ច 1 2 3, ,M M M នប#
.PងW- គI 1 2 3, ,x x x nQo8);ខ2 ង ( )C នម 4 26 4 6xy x x− + +=
យប>? ក#5 គ8#!.យ O 6ទបជ.!ទ!ងន#ប# 1 2 3M M M
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 90
=. យប>? ក#5 3
33 3 32 4 6 5 3.
cos cos cos7 7 2
7
7
−π+ + =π π
'()&'()&'()&'()&
ចេលយ
. Lន.<នM< QមYព Bunyakovski យ)ង,ន
( ) ( ) ( )
244 41 2
2 2 3 2 2 288 8 21 2 2 3 11 2
2 2 22 2 2 2 2 21 2 2 3 1
...
...
n
nn
n
aa a
aa a
a a a
a a a a a a B
n na a a
+ + + + + + +
+ + + +
=+
≥
យ)ងកន ( ) ( ) ( )44 4
2 2 2 2 2 2 1 21 2 2 3 1 2 2 2 2 2 2
1 2 2 3 1
... ... . nn
n
a aaa
aa a a
aa a a
a a a+ + + + +
+
+ + + + + +
( )22 2 21 2 ... na aa≥ + + +
Jញ,ន 2 2 21 2 ...
2na a
Ba+ + +≥
C:ច* ( ) ( )2 22 2 2 2
1 2 1 2 2 3 1
4
... .
4 4
..n na a a a a a aa a K
An n n
+ + + +=
+ +≥ ≥
C:ចន 2
min4
KA
n= , ទទB8,នព8 1 2 ... n
Ka
na a= = = =
. ព8 1n = * 6A = 6ច!នBន
ច!@ 1n > យ)ង,ន
( ) ( )2005 2006 2 2 2 2A n n n nn n n n− − + ++ ++ +=
( ) ( ) ( )2004 2 2004 21 1 12 nn n n n n= + +− − + +
យ ( )6682004 31 1n n− = − ;ចកច#ន%ង ( )3 21 ( 1) 1n nn n− = − + + * 2004 1n − ;ចកច#
ន%ង 2 1n n+ + Jញ,ន A ;ចកច#ន%ង 2 1 1n n+ + >
C:ចន មQននច!នBនគ#ធម(6Q n C)មD A 6ច!នBនបមទ
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 91
+. i Fង 3( 1) 3f x xx −= +
យ)ង,ន ( 2). (0) 0, (0). (1) (1). (2 00, )f f f f f f< < <− *ម នHប 1 2 3, ,x x x
9)យ 1 2 32 0 1 2x xx < < < <− <<
i Fង 0 0)( ,G x y 6ទបជ.!ទ!ងន# 1 2 3M M M
Fមទ%បទ;< 1 2 3 00 0xx x x+ + = ⇒ =
1 2 2 3 3 1 3x x x xx x+ + = − .
ច!@ 1,2,3i = , យ)ង,ន
3 4 23 1 0 (3 1) 3i i i i i i ix x x x xx x− + = ⇒ = − = − .
( ) ( ) ( )4 4 4 2 2 2 2 2 21 2 3 1 2 3 1 2 3 1 2 33 3x x x x x x x x xx x x⇒ + + + + + + + += − =
យ)ងន ( ) ( ) ( )4 4 4 2 2 21 2 3 1 2 3 1 2 3 1 2 36 4 18y y x x xy x x xx x x+ + = + + + + + +− + +
( )2 2 21 2 33 18 0x xx + += − + =
Jញ,ន 0 0y =
=. ពQនQម cos4 cos3x x=
3 24co(cos 1 s 4cos 1) 0)(8cosx x x x−⇔ + − − =
3 24cos 4cos 1
cos 1
8co 0s x x x
x⇔
+ − − ==
ងRឃ)ញ5 1 2 3
4 62cos 2cos , 2co
2s
7 7,
7tt t
π π π= = = 6HJ!ងបប#
ម 3 2 2 1 0t tt + − − =
Fម;< 1 2 3
1 2 2 3 3 1
1 2 3
1
2
1
t t
t t t t t t
t t t
t + + = −+ + = −
=
Fង 3 3 31 2 3t tA t+ += , នQង 3 3 3
1 2 2 3 3 1t t t t t tB + +=
យ)ង,ន 3 3 4A AB= − នQង 3 3 5B AB= − Jញ,ន 3 3 (3 4)(3 5)B ABA AB= − −
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 92
3 3( 3) 7 0 3 7AB AB⇒ − + = ⇒ = − 3 33 35 3. 7 5 3. 7A A⇒ = − ⇒ = −
'()& វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ២៥
. ប*0 #ព.ក-.ងនQងព.'ប#ម.! C Aន ABC #ប*0 # AB ង# L នQង M
យប>? ក#5ប) CL CM= * 2 2 24BCAC R+ = ( R6 !ង"ង#$ %ក' ABC∆ )
. គ , , 0a b c > យប>? ក#5
( ) ( ) ( )
20062006 2006 2006 2006 2006 2006
1 1 14
2 2 2
1 1 1
a b c a b c a ba b c c
≥ + + + + + + + +
+ +
+. គ ABC 6 [ចផ0\ង]0 #8ក_
( )1 cos cos cos cos cos cos cos cos cos 2cos cos cosA B B C C A A B C A B C+ + + − + + =
យប>? ក#5 ABC 6 ម|ង2
=. កគប#ប H6ច!នBនគ# ( ; )x y ប# ( ) ( ) ( )32 2x x y xy y+ = −+
'()&'()&'()&'()&
ចេលយ
. ប) CL CM= * CML ;កងម, (@ CL ;កងន%ង CM , Fម8ក
ប#ប*0 #ព.J!ងពប#ម.!មBយ) ជ) )!.យC:ច:ប (O 6ច!ន.ចក 8 )ML ,
( ;0), ( ;0), ((0;0), ( ;00; ), ( 0 ); ),A a B b C McO cc L −
Fម8កប#ប*0 #ព.យ)ង,ន AL AC
LB CB=
2 2 2 2 2 2 2
2 2 2 2 2
( );0
( )
cAL AC c a a c c
LB CB b c b ac ab B
+⇔ ⇔ −= = −
=+
⇒
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 93
( )24 2 2
2 2 2 2 22
(1)c c
BCa
AC aa a
c c
+ =
+
+ = + +
Fង I 6ផjQង"ង#$ %ក' ABC , យ)ង,ន
2 2 2 2
2 222
2 2 2 2 2
( )( )
( )
yx aAI CI AI
cAI BI
x y cCI
BAI x a xIa
y y
+ = + −=
⇔ ⇔=
−=
= − −
+
=
+
2 2
2 2
2 2
2 2;
22
ax cy aa
ca axa
cc
Ic
− =
=
−+⇔ ⇒+
ព* 22 2
2 24 (2)4c
Ia
Ra
C +
= =
Fម (1)នQង (2) ⇒ ប>e g<,នយប>? ក#
. Lន.<នM< QមYពក:., យ)ង,ន
2007
20062006 2006 2006 2006 2006
1 1 1 1 1.
( )2
2
2. (1)a
a b a b a ba b+ =
++
≥ ≥
C:ចW- ;C 2007
2006 2006 2006(
1 2
(2
1)
)b c b c+≥+ នQង
2007
2006 2006 2006(
1 2
(3
1)
)c a c a+≥+
C
y
A B LO M x
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 94
ប:ក (2)(1), , (3)LងXន%ងLងX, យ)ង,ន
20062006 2006 2006 2006 2006 2006
1 1 1 1 1 1
( ) ( )2 (4)
( )a b c a b b c c a
+ + + +
≥ + + +
C:ចW- ;C, Lន.<នM (4)យ)ង,ន
20062006 2006 2006
2 (1 1 1
5)( ) ( ) ( )
Aa b b c b c c a c a a b
+ + + + + + + + + + +
≥
ច!@ 2006 2006 2006
1 1 1
( ) ( ) ( )A
a b b c c a= + +
+ + +
ព (4), (5) ⇒ ប>e g<,នយប>? ក#
+. បdន;C8,នg<,នc)ង< QញG6
(1 cos )(1 cos )(1 cos ) cos cos cosA B C A B C− − − =
យ)ងយប>? ក#5 1 cos 1 c1
os 1 cos
cos cos cos(1)
A B C
A B C
− − −
≥
Fង tan , tan , tan ( , , 0)2 2 2
A B Cx z x y zy === >
ជ!នB 2 2 2
2 2 2
1 1 1cos , cos , cos
1 1 1
x y zA B C
x y z
− − −= = =+ + +
ច:8 (1) យ)ង,ន
2 2 2
2 2 2 1
1 1 1
x y z
x y z xyz
≥
− − −
cot cot cotan tan t2 2
t n2
aA B C
g gA B C g⇔ ≥
cottan tan tan cot cot2 2 2
A B CgA B C g g≥ + ++⇔ +
tan tan ta tan tan tan (2)2 2 2
nB C
A B CA C A B+ +≥ + ++ ++⇔
មO/ងទP 0 , tan0 2 tantan2 2 2
xy
yx
y xπ π< < + +< < ⇒ ≥
>a " "= ក)នព8 x y= យ)ង,ន
tan ta a2
n 2 t nA BA B+≥+ , tan ta a
2n 2 t nB C
B C+≥+ , tan ta a2
n 2 t nA CA C+≥+
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 95
ប:កLងXន%ងLងXយ)ង,ន (2) មYពo (2)ក)នព8 A B C= =
C:ចន ABC ផ0\ង]0 #8កខ_ ;C8,ន 6 ម|ង2
=. 8!f#;C8g<,នc)ង< Qញ6 ( ) ( )2 2 23 (12 3 0 )y y x y x xx + + = + −
ប) 0y = * x យក!A8ក,នnQក-.ង ℤ
ប) ( ) ( )2 2 220 1) 3: ( 3 0y y x xx y x≠ ⇒ + − + + =
( )3 2 26 15 8 ( 8)( 1)x x x x x x x∆ = − − − = − +
C)មDម នH6ច!នBនគ#* ∆ g<;6ច!នBន ,កC, គI5
( )( )2( 8) 4 4 16x x a x a x a− = − − − + =⇔
យ 4 4
4 4 2 4
x a x a
x a x a x
− + − −
− + + − − = −
≥
*យ)ង,នប ក C:ចZង ម
i4 16
4 8,54 1
x ax
x a
− + = − =− − =
⇒ ($8)
4 8 9
4 514 2
x a xx
xx a
− + = = − = = −− − = ⇒ ⇒i
4 4 8
4 404 4
x a xx
xx a
− + = = − = =− − = ⇒ ⇒i
ជ!នBច:8ម , យ)ង,នប H6ច!នBនគ#គI
(( ;0) )t t ∈ℤ , (9; 21), ( 1;(9; 6) 1), (8; 1, 0)− − − −−
'()&
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 96
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ២៦
. កHប#បព|នម
( ) ( )2
2 2
1
3 4 2 ( 0 , )
28
3
40
xy
y xy
x y z
x x
x
z yz x y z
y z
+
+ + + + =
+ =
+ + =
< <
−
. កប គ:ច!នBនគ# ( , )x y ផ0\ង]0 #ម ( ) ( )23 2 3 33 3 1 1 0x y xx y− + + + =−
+. គ ABC , Fង , ,a b cm m m Fម8!ប#6ប;<ងមCOន;C8គ:ពប
ក!ព:8 , ,A B C 9)យ , ,a b cr r r Fម8!ប#6 !ង"ង#$ %កក-.ងម.! g<W- ន%ងប ម.!ន
ក!ព:8 , ,A B C យប>? ក#5 2 2 2 2 2 2a b c a b cr r r m m m+ + ≥ + + ,
)មYពក)នព8?
=. Fង I 6ផjQង"ង#$ %កក-.ង ,ABC ង"ង#នប/ន%ងប ជង , ,BC CA AB ង#
, ,K L M PងW- Fង B គង#ប*0 #បន%ង MK , ប*0 #ន # ,LM LK ង#
S នQង R យប>? ក#5 ម.! RIS 6ម.![ច
'()&'()&'()&'()&
ចេលយ
. Fង , , 2v x y y zu x t= + == + Jញ,ន u v t< <
1
83
, ,4
0
uvt
uv vt tu u v t
u v t
= + + = −
+
⇒
+ =
⇒ 6Hប#ម 3 31
42
X X− =
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 97
Fង cos cos1
0 )2
3 (X α α α π⇒ = < <=
2
9 3
kπ πα +⇒ = @ u v t< <
5cos
7cos , cos,
99 9u v t
π π π= = =⇒
7cos
95 7
cos cos9 9
5 7 5cos cos cos cos
9 9 9 9
x
y
z
π
π π
π π π π
= = − = − − = −
⇒
C:ចន
7cos
95 7
cos cos9 95
cos9
x
y
z
π
π π
π
= = − = −
6HJ!ងប;C8g<ក
. ក ទp ប) 1y = − *យ)ង,នម
3 2 03 0xx x− = =⇔ I 3x =
C:ចន 0 3;
1 1
x x
y y
= = = − = −
ក ទp ប) 1y ≠ − *គ.LងXJ!ងពប#ម ន%ង 3( 1) 0y + ≠
3 3 2 3 3 2 3 31) 3 ( 1) 3 1 1) 0( ( ) (x y yy x y x+ − + + + − + =
3 3 3 3 3( 1) 0 1y x y xx yx y⇔ + − − = ⇔ + −=
3
21
11
yx y
yy= =
++⇔ − +
C:ចន Hប#ម C!ប:ងគI 20
;1
1yy
y
x x
y
= = = − ∈
− +ℤ
+. យ)ងន ( ) (( ) ( )( )) ( )a b cp b r p cS p a r p p a p br p c= − =− − −− −==
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 98
C:ច* 2 2 2
2 2 22 2 2( ) ( ) ( )a b c
S S Sr
p a p b p cr r+ +
−+ +
− −=
( )( ) ( )( ) ( )( )p b p c p a p c p b p ap
p a p b p c
− − − − − −= + + − − −
( )2 2 2 2 2 23
4a b cm mm a b c=+ + + +
Fង x p a
y p b
z p c
= − = − = −
, Jញ,ន x y c
x z b
y z a
+ = + = + =
នQង x y z p+ + =
យ)ងg< យប>? ក#5 ( ) 2 2 23( )
4( ) ( )
yz xz xyx y z y z x z
x y zx y≥ + + + +
+ + + + +
យ)ងន ( )2 2 2 2 2 22y z x z x y
x y z x yz y z x
zy x
+ + + + +
≥ +
+
2 2 2( ) 2yz xz xy
x y z x xy yz zxy zx y z
+ + + + + + + ≥ + +
Fម< QមYពក:. ( ) ( )2 2 21 1
2 2x z y zy xy z x+ + ≥ + +
C:ច* LងXZងឆ"ង 2 2 2 2 2 23 3(( ))
4)
2(y z xy yz zx z z xx x yy = + ≥ + + + + + + + + +
មYពក)នព8 a cy bx z ⇒ == == , I ABC 6 ម|ង2
=. ឃ)ញ5 BI 6L|ក2ឆq.ប# MK I MK SI RB BI⊥ ⇒ ⊥
ក-.ង BKR∆ , យ)ងន 0cos
2,2 2 2 cos
2
90
CBKC B C
BRK BKRA
BR+= = − ⇒ =
C:ចW- ;C, យ)ង,ន cos
2
cos2
ABM
BSC
=
2. .BR BS BM BK BK= =
2 2 2 2 2 2 22 ( )IS RS BI BR BSIR BR BS⇒ + − = + + − + .
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 99
2 2 2 2. ) 2(2 )( 2 0BR BS BIB BK II K− = − = >=
Fមទ%បទក:.ន., យ)ងJញ,ន 090RIS <
'()&
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ២៧
. គ ABC នប ម.!6ម.![ច $ %កក-.ងង"ង#ផjQO ! R Fង 1 2 3, ,R R R
PងW- 6 !ង"ង#$ %ក'ប , ,OBC OAC OAB ; p 6កនqប Qប#
ABC យប>? ក#5 7
1 2 3 4
729
16R R
RR
p≥
. យបព|ន< Qម ( ) ( )2 3 2 3
2
. 6 5 2 6 4 (6 1)
2 21 (2)
x x
xx
x x x
x
x− + + − +
+ ≥ +
=
+. យម 4
62
cos 23 1
cos4 tan 7x
x
x +
+ =
=. កH6ច!នBនគ#< Qជ?នប#ម
7 4 2 4 3 35 7 2 5 7 20062 x y x x y y yx + + − − − =
'()&'()&'()&'()&
ចេលយ
. Fង , , , ABCAC b AB c SC SB a = == = 9)យ p 6កនqប Q ABC
យ)ង,ន 2
1 1
. .
4 4OBC
OB OC BC R
R R
aS ==
2
1
.
4 OBC
a RR
S⇒ =
C:ចW- ;C 2 2
2 3
. .;
4 4OAC OAB
b R c RR
S SR= =
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 100
B
A
O
C
A
O
C
6 6
1 2 3 3
.
. .6464
3OBC OAC OAB OBC OAC OAB
abR c abcR R R
S S
R
S S S S⇒ = ≥
+
+
6 7
1 2 3 3 2
27 27
64 16
R abc
S SR
RR R⇒ ≥ =
យ 3 4
2 ( )( )( )3 27
p p ap a p b p c p
S p b p c p− + − + −
= − − − ≤ =
* 7
1 2 3 4
729
16R R
RR
p≥ , មYពក)ន8.F; ABC 6 ម|ង2
. ព (2)យ)ងJញ,ន 321 0 4 0
xx xx ⇒ > ⇒ +> >+
C:ច* 2 2 61 7
7 11
07
xx
xx x
> − +−+ − > ⇔ ⇒
>
< − −
ប:កBមន%ង8កខ_ 3 6 5 0xx − + ≥ យ)ង,ន 1 21( )
23x ≥ − +
ព8* យ)ងនប ងRC:ចZង មន
3 2 2 21 1( 1)( (6 5 5 1 (
2 2) 5) 2 6)x x x x xx x x xx− + + − ≤= − − + + − = + −
( )3 3 3 3
32 6 331 1
. .4 4 42 2 3 2 2 3
x x x xx xx
= = ≤ + + = +
ព*យ)ងJញ,ន ( )( )2 3 2 36 46 5 2 6x x x xx x− + ≤ + − +
C:ចន (1)មម:8ន%ង 2
3
5
42
12
x xxxx
− = =
−⇔
=
+ , ផ0\ង]0 # (2)
C:ចន < Qម នច!8)យ 2x =
+. Fង 2
cos 21
cos
xu
x= + នQង 2tanv x=
ម នSង 4 34 73 vu + = យ)ង,ន
2
22 2 2
cos2 cos2 1 2cos(1 tan 2
cos cos co)
s
x xu
xx
x x xv
+= ==+ + + =
B
A
O
C
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 101
ប)< QមYពក:., យ)ង,ន
( )( )
444 3
3 3
3 121 1 1 43 4 7
1 1 3 4 2 12
3 uuu v
v
uu
v v v
+
≥+ + + ≥⇒ ⇒ + ≥
+ + ≥ + ≥
>a ()ក)ន 4
31
1tan
41
1x
v
uv x k
π π=
⇔ ⇔ ⇔ = ± ⇔ ± = =
= +
C:ចន ម នH 4
( )x k kπ π+± ∈= ℤ
=. ម ;C8មម:8ន%ង 4 2 3( ).(5 2 7) 1.2.17.59y xx y− + + =
យ ( )4 0 2y xx − > ⇒ ≥ 2 32 7 5 16 7 285y x⇒ + + ≥ + + =
( ) 4 2;17 31; ; 4; 59yx⇒ − ∈
ប) 2x = * 16 1 15
16 2 14
y y
y y
− = = − = =
⇒
ន;គ: (2;14);;C8ផ0\ង]0 #ម
ប) ( )43: (859 1 ) 22y y yx x≥ ≥ − ≥ − ⇒ ≥
2 35 2 7 2006y x+ + > , មQនផ0\ង]0 #ម
C:ចន ម នច!8)យ;មBយគ# (2;14)
'()&
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 102
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ២៨
. ).a កH6ច!នBនគ#ធម(6Qប#ម
( ) ( )22 2 4 4 24 28 17 14 49x xy y y+ + = + + +
).b កប !A86ច!នBនគ#< Qជ?នផ2ងW- 1 2, , ..., nxx x V/ង
2 2 21 2
...1 1 1
1nx x x
++ + =
. ).a 4ប5ម 4 3 2 1 0bx cxx bx+ + + + = នH
បe ញ5 2 2( 2) 3cb + − >
).b យម 3 3 3 0xx + − =
+. កប Lន.គមនM ( )f x នQង ( )g x ក!#យបព|នZង ម
( 1) (2 1) 2
(2 2) 2 (4 7) 1
f x g x x
f x g x x
− + + = + + + = −
=. ).a គច. ABCD $%កក-.ង ( , )O R ង# Ax ;កងន%ង AD # BC ង# E ,
ង# Ay ;កងន%ង AB # CD ង# F យប>? ក#5 EF #Fម O
).b គ ABC oន%ង, ង#ច. () BCDE ព ,D E ង#ប*0 #;កង
AB នQង AC , ប ប*0 #ន #W- ង# M ក!ន.!ច!ន.ច ច!ន.ច M '()&'()&'()&'()&
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 103
ចេលយ
. ).a Lន.<នM< QមYព Bunyakovski , យ)ង,ន
( ) ( ) ( )2 22 2 2 2 4 24 7. 4 71 1x y x y+ + ≤ + + +
( )4 4 217 14 49x y y≤ + + +
C:ច* 2 24 (2 )(27 ) 7x x y xy y+ −+ ⇔ ==
យ ,x y ∈ℕ * 22 0x yx y ≥ − ≥+
យ)ង,ន 2 7 2
2 1 3
x y x
x y y
+ = = − = =
⇔
, C:ចន (2,3)S =
).b យ)ងន 1 2, , ..., nxx x មQន:ច6ង 2 *
2 2 2 2 2 21 2
1 1 1 1 1 1 1 1 1
2 3 ( 1) 1.2 2...
.3 ( 1)... ...
nx x x n n n+ ≤ ++ + + + < +
+ +++
1 1 1 1...
1 11 1 1
2 2 3 1 1n n n< − + − + − = − <
+ ++
C:ចន 2 2 21 2
...1 1 1
1nx x x
++ + <
C:ច*, មQននប !A86ច!នBនគ#< Qជ?នផ2ងW- ;C8ផ0\ង]0 #
2 2 21 2
...1 1 1
1nx x x
++ + <
. )a យ)ងន 4 3 2 22
11
0 0b
x bx cx bx x b cx x
x+ + + + = ⇔ ++ + =+
22
1 10x b x c
x x + + + =
⇔ +
យ)ងន 2 2 0btt c+ + − = ច!@ 2 22
1 12t x tx
x x⇒ += + = − នQង 2t ≥
2 (2 ) tt c b⇔ = − − .
Lន.<នM< QមYព Bunyakovski , យ)ង,ន
[ ] ( )24 2 2 2(2 ).1 . (2 ) 1t c b t c tb= ≤ + − − +−
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 104
4
2 22
(2 )1
bt
tc≤ − +
+
យ 2 2 42
11
2 4 6x t tx
+ ≥ ⇒ ≥ ⇒ ≥
* ( )4 2 4 23 1 3 3 16 12 31 0t t tt=− + − − ≥ − − >
( )4
4 22
3 1 31
tt t
t> + ⇔ >
+
C:ចន 2 2( 2) 3cb + − >
).b Fង 1x y
y= −
យ)ង,ន 3
33
1 1 13 0 3 03y y y
y y y
− − − = − = ⇔ −
−
Fង 3t y= , យ)ង,ន 13 0t
t− − =
( ) ( )2 31
3 11
3 1 0 33 122
3t yt t =− − = ⇔ ± ⇒ = ±
C:ចន ( )3 31
3 12 3
32
13x −
±= ±
+. Fង 21 2 2 2 43 6,x ux u x u− = + == + +⇒
(( 1) (2 1 2 2) (4 7)) 2 4 6f x g x f u g ux u⇒ + + ++ =− + +=
(2 2) (4 7) 4 6f x g x x+ + + = +⇒
ព8*យ)ង,ន (2 2) (4 7) 4 6 (2 2) 7 13
(2 2) 2 (4 7) 1 (4 7) 3 7
f x g x x f x x
f x g x x g x x
+ + + = + + = + + + + = + = − −
⇒−
Fង 2 72 2 6
2(
2)
uu x x xx f⇒ ⇒ =−+ == +
Fង 7 3 74 7
4( )
4 4
tt x xx g x⇒ = ⇒ = −−= + −
=. )a . Lន.<នM :EFD A M֏ (មQនចx#ពចxប#នទ!! )
យប>? ក#5 ( )M O∈
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 105
យ)ងន 0180BCD BAD+ =
; 0180EAF BAD+ = * BCD EAF=
យ EAF EMF= * BCD EMF=
Jញ,ន ច. EFCM $%កក-.ង, Jញ,ន
, ,MCE MFE MFE EFA EFA MAB= = =
C:ច* MCE MAB ABMC= ⇒ $%កក-.ង, Jញ,ន EF nQo8)LងR#ផjQ
I EF #Fម O
).b Fង H 6L:ង#ប# ABC , Fង BC a=
HBC MED CH DM∆ = ∆ ⇒ =
:CHT D M֏ , ;យ ( , )D C a∈ * ( , )M H a∈
'()&
O
B A
E
D F C
M
E
A
B C
D
H
M
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 106
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ២៩
. កH6ច!នBនគ#ប#ម 3 327 2009xyx y+ + =
. យប>? ក#5 ផ8ប:កប !A8ធ!ប!ផ.នQង!A8:ចប!ផ.ប#Lន.គមនM
3cot
cot 3
xgy
g x= 6Lន.គមនMនQJន, ក-.ង* 0
2x< < π
+. គ ABC 6 [ច យប>? ក#5
cos cos cos cos cos cos 3
4cos cos cos
2 2 2
B C C A A B
B C C A A B+ +
− − −
≤
=. គ ABC មBយ o8)ជងJ!ងបប# គង# ម|ង2ប, V/ ង
ម|ង2នមBយk*oZងផ2ងW- ព ABC ធPបន%ងជងBម
W- Fង ∆ 6 ;C8នប ក!ព:86ផjQប#ប ម|ង2*,
C:ចW- ;C o8)ជងJ!ងបប# ABC គង# ម|ង2ប, V/
ម|ង2នមBយk*nQoZងC:ចW- ន%ង ABC ធPបន%ងជងBម
W- Fង ′∆ 6 ;C8នប ក!ព:86ផjQប#ប ម|ង2
;C8ទ)ប;ង# យប>? ក#5 ABC S SS ′∆ ∆= −
E. គ , , 0a b c > យប>? ក#5
( ) ( ) ( ) ( ) ( ) ( )4 4 4
2 2 24 6 6 3 3 4 6 63 3 3 3 4 6 6 3 33
1a b c
a a a b b b cb c c a bc ca
+ + ≤+ + + + + + + + +
'()&'()&'()&'()&
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 107
ចេលយ
. Fង ,y x a a= + ∈ℤ ម q យG6
2 2 3(27 3 ) 2009 0 (*(27 )3 ) a aa ax x+ − − + =−
8កខ_ C)មD (*) នHគI
2 2 3) 4(27 3(2 )( 2009 07 3 )a aa a − − − + ≥− I 23( 14)( 9)( 41 574) 0aa a a +− +− ≥−
យ 2 41 574 0aa + + > ច!@គប# a *9 14a≤ ≤
យ a ∈ℤ * 9,10,11,12,13,14a ∈
ព8 9a = , យ)ង,នម 1280 0= (W( នH) C:ចW- ;C, ព8 a PងW- ន:<
ប !A8 10,11,12,13 ,ម .ទ;W( នH
ព8 14a = , យ)ង,នម 2 210 735 0 0715x x x− − = ⇔ −−
ច!@ 14, 7a x= = − , Jញ,ន 7y =
C:ចន ម ;C8នH6ច!នBនគ#;មBយគ#គI ( 7,7)−
. យប>? ក#5 ផ8ប:កប !A8ធ!ប!ផ. នQង!A8:ចប!ផ.ប#Lន.គមនM
3cot
cot 3
xgy
g x= 6ច!នBននQJនមBយ, ក-.ង* 0
2x
π< <
យ)ងន ( ) ( )3 2
2 3 2 2
3tan tan 3 tan
61 3tan tan 1 3tan tan
x x
x x x x
xy x
π− − = = − − ≠
Fង 2tant x= , យ)ង,នម 2 ( 1) 3 0 (1)3 y tyt − + + =
ម (1)នHព8
2 236 0 34 1 0 17 12 17( 1) 2 212y y y yy − ≥ ⇔ − + ≥ ⇔ − ≤ ≤ ++
Jញ,ន min 17 12 2y = − ព8 3 2 2 tan 2 18
t x xπ⇔ ⇔− = == −
max 17 12 2y = + ព8 33 2 2 tan 2 1
8t x x
π+⇔ ⇔= == +
C:ចន min max 34yy + = 6ច!នBននQJន
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 108
+. យ ABC 6 [ច* cos 0, cos 0, cos 0A B C> > >
Lន.<នM< QមYពក:., យ)ង,ន
22cos cos 4cos cos.sin cos .cos .sin cos cos sin
cos cos 2 2 2cos2
2
B C B C A A AB C B C
B C B C = + − +
≤
≤
យC:ចW- ;C, យ)ង,ន
22cos .coscos cos sin
2cos2
C A BC A
C A + −
≤
នQង 22cos coscos cos sin
2cos2
A B CA B
A B≤ + −
ប:កប < QមYពZង8)LងXន%ងLងX, យ)ង,ន< QមYព
2 2 2cos cos cos cos cos cos2 sin sin sin
2 2 2cos cos cos2 2 2
B C C A A B A B CB C C A B
+ + + + + − − Α −
≤
(cos cos cos cos cos cos )A B B C C A+ + +
[ ]21 1(cos cos cos ) 3 (cos cos cos )
3 2A B C A B C≤ ++ + − + +
( ) ( )1 3 1 3. . cos cos cos 3 cos cos cos
3 2 2 2A B C A B C+ + + − + + = ≤
C:ចន cos cos cos cos cos cos 3
4cos cos cos2 2 2
B C C A A BB C C A B
+ +− − Α −
≤
=. យន%ងយ,ន5 , ′∆ ∆ 6ប ម|ង2
Fង , ,Bc C A bA aB C= == ពQនQ 1 2O AO យ)ងន
2 2 21 2 1 2 1 1 22 .cosO O A O A O A OO AO= + −
( )2 2
03 3 3 3. .co2. 6s
3 30
3 3c b bc α
=
+ − +
2 2 02 .cos( 601
)3
b bcc α+ − + =
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 109
C:ចន ( )2 2 2 01 2
3. 2 .cos( 60 )
4
3
12S cO O b bc α∆ = + − +=
C:ចW- ;C, យ)ងយប>? ក#,ន5 ( )2 2 02 .cos( 63
0 )12
b bcS c α′∆ = + − −
Jញ,ន ( ) ( )0 03.2 6. cos c 0os0
126S bcS α α′∆ ∆− += −−
03
.2sin 60 .sin6 ABCbc Sα ==
E. យ)ងន ( )( ) ( ) ( )26 6 3 3 6 6 6 3 3 63 3 2a a ab c b a c ca+ + = + + +
( ) ( ) ( )12 3 6 3 9 3 6 6 6 6 6 63 22a b c c b c b aa a ca= + + ++ +
( ) ( )33 6 6 6 4 2 6 2 4 6 6 2 2 2 2 2 2 2 233 3b a b c a b c a c b a c a b aa a c≥ + + + = + = +
Jញ,ន ( )( )
4 4 2
4 2 2 2 2 2 2 224 6 6 3 33 a
a a a
a aa a a b a c b cb c≤
+ + + ++ +=
+
C:ចW- ;C ( )( )
4 2
2 2 224 6 6 3 33
b b
a b cc ab b b≤
+ ++ + +
នQង ( )( )
4 2
2 2 224 6 6 3 33
c c
a b ca bc c c≤
+ ++ + +
ប:កប < QមYពZង8)FមLងXន%ងLងX, យ)ង,ន< QមYព
( )( ) ( )( ) ( )( )4 4 6
2 2 24 6 6 3 3 4 6 6 3 3 4 6 6 3 33 3 3b c c
a b c
a a a b b b ca a bc c+ + + + + + + + ++ +
2 2 2
2 2 21
a
a
b c
b c
+ +≤+ +
=
'()&
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 110
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ៣០
. យម 2006 20052005 20 6 10x x+− =−
. គ ABC នម.!J!ងបផ0\ង]0 #8ក_ max , ,2
A B Cπ≥
ក!A8:ចប!ផ.ប#កនម cos .cos .cos
2 2 2
sin .sin .sin2 2 2
A B B C C A
TA B C
− − −
=
+. គ!ន.! ABមនប ច!នBនគ#ធម(6Q;C8ន8ខ 2006ខ0ង# ផ0\ង]0 #ពមW-
ន:<8ក_ ពZង ម
( )i . ច!នBនគ#ធម(6QនមBយkប# AមQនន8ខ 0ក-.ងប ខ0ង#ប#,
( )ii . ប) 1 2 2006...aa a a= nQoក-.ង A* 2006 2006
1 1i i
i i
a b= =
=∑ ∑
1).យប>? ក#5 ផ8ប:កប 8ខFមខ0ង#ប#ច!នBនគ#ធម(6QនមBយkប# A
;ង;6ច!នBនថមBយ
2).កច!នBនគ#ធម(6Q:ចប!ផ.ប#!ន.! A
=. គច. ,/ ង ABCD នកmAផ0 S មQន;បប[8 $ %កក-.ងង"ង# ( )O នQង
V/ងផjQ O ;ងnQoក-.ងច. * Fង I 6ច!ន.ចបព"ប#LងR#
ទgងJ!ងព AC នQង , , ,; M NBD Pl Q Fម8!ប#6ច!ន.ចឆq.ប# I ធPបន%ងប
ប*0 # , , ,AB BC CD DA ក!#!A8ធ!ប!ផ.ប#កmAផ0ច. MNPQ
'()&'()&'()&'()&
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 111
ចេលយ
. ងRឃ)ញ5ម នប HគI 2005x = នQង 2006x =
ប) 2006x > , *ម W( នH@
2005 1x− < − *LងXZងឆ"ង 1>
ប) 2005x < , *ម W( នH@
2006 1x− > *LងXZងឆ"ង 1>
ប) 2005 2006x< < , * 0 2005 1x< − < , 0 2006 1x< − <
C:ច* 20062005 2005 2005x x x− − = −<
20052006 2006 2006x x x− − = −< ,Jញ,ន LងXZងឆ"ង 1<
. យ)ងប!;8ងកនម T G6Sង sin sin sin
sin .sin .sin
A B CT
A B C
+ +=
Lន.<នMទ%បទ.ន.ច!@ ( )( ) ):
(a b b c c aABC T
abc
+ + +=
យមQនធ"),#8កទ:G, 4ប5 max , ,c a b c=
យ)ង,ន 2 2 2 2 22 cos 22
a b ab C ac
abb ac b= + − ≥ + ≥ ⇒ ≤
32 .2 2 3 . .a b a b c c a b a b c c
Tb a c a b b a c a b
+ + = + + + + + +
≥ +
34 3 4 3 2(2,5)2c ab
ab≥ + ≥ +
C:ចន min 3 24T = + 8.F; ABC∆ ;កងម,
+. ).a ពQនQប ច!នBនZង មAន!ន.! 1 1 2006: ...A a a aa = , ច!@ 2006
1i
i
a S=
=∑
1 2 2006...bb b b= , ច!@ (mo3 10)di iab ≡
1 2 2006...cc c c= ,ច!@ (mo3 10)di ibc ≡
1 2 2006...dd d d= , ច!@ (mo3 10)di icd ≡ 1,2 6)( 00i =
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 112
Fមប!Sប#យ)ងន 2006 2006 2006 2006
1 1 1 1i i i i
i i i i
a b c d S= = = =
= = = =∑ ∑ ∑ ∑
យ 1 (9 1,2006)i ia =≤ ≤ *យប)< Qធជ!នBយ)ង,ន
20i i i ib c da + + + =
ព* 2006 2006 2006 2006
1 1 1 1
4 i i i ii i i i
a b c dS= = = =
= + + +∑ ∑ ∑ ∑ ( )2006
1
20 0062i i i ii
a b c d=
+ + ×=+= ∑
Jញ,ន 10.030S = (ប>e g<,នយប>? ក#)
).b ពQនQច!នBនគ#ធម(6Q 1 2 2006...aa a a= , ក-.ង* 1 2 1003... 1aa a= = = = នQង
1004 1005 2006... 9a aa = = = =
ឃ)ញ5 a A∈ យ)ងយប>? ក#5 a 6ច!នBនគ#ធម(6Q:ចប!ផ.ប# A
ពQ6C:ចន, 4ប5 c A∃ ∈ ផ0\ង]0 # c a≤ នQង 1 2 2006...cc c c=
ព8*, យ (, 0 1,2006)ic ia c =≤ ∀ ≠ * 1 2 1003... 1cc c= = =
2006 1003 2006
1 1 1004
10.030i i ii i i
c c c= = =
⇒ = + =∑ ∑ ∑ (Fម!នB a )
2006
1004
10.030 1003 1003 9ii
c=
⇒ = − = ×∑
9 ( 1004,2006)ic i c a⇒ = = ⇒ = (ប>e g<,នយប>? ក#)
=. Fង 1 2 3 4, , ,HH H H PងW- 6ច!8;កងAន I G8)ប ប*0 # , , ,AB BC CD DA
យ O nQoក-.ងច. ABCD * 1 2 3 4, , ,HH H H PងW- nQo8)ប
LងR# , , ,AB BC CD DA
យ)ង,ន 1 22 . .
1. sin.sin
2MIN MIN IH IS IM HI BN∆ = = (1)
2 2
1 2
. ). .
4. ( 1
. . . 2sin (2)AIB BIC AIB BIC ABC
ABC
S S SIH S
AB BC AB BC A
S SI
B BCH B∆ ∆ ∆ ∆ ∆
∆+= ≤ = =
(1)នQង (2) 2.sinMIN ABC ABCS SS B∆ ∆ ∆⇒ ≤ ≤
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 113
C:ចW- ;C ,NIP BCD PIQ CDAS SS S∆ ∆ ∆ ∆≤ ≤ នQង QIM DABS S∆ ∆≤
ព* MNPQ MIN NIP PIQ QIMS S S SS ∆ ∆ ∆ ∆= + + + 2ABC BCD CDA DABS S S S S∆ ∆ ∆ ∆≤ + + + =
>a មYពក)ន8.F;
sin sin sin sin 1
AIB BIC CID DIASABCD
A B C D
S S S∆ ∆ ∆ ∆= = =⇔
= = = =
6ច. ;កង
C:ចន max 2MNPQ SS = , ទទB8,នព8 ABCD 6ច. ;កង
'()&
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 114
វ1េសរបឡងអពចេវតម) កទ១០ េលកទ XII
វ ទ៣១
. កH6ច!នBនគ#ប#ម 2( 1)( 7)( 8) (1)x x x x y+ + + =
. យម ( )44 3 2048cos 768 (*)16cos x x+ = −
+. គ , ,x y z 6បច!នBនពQ< Qជ?នផ0\ង]0 # 1x y z+ + = ក!A8ធ!ប!ផ.ប#
xyzx yP
x yz y zx z xy= + +
+ + +
=. គ ABC∆ $%ក'ង"ង#ផjQ I , ន ; ;Ca A B cB bC A= ==
យប>? ក#5 2 2 2
. . 1
3. 3. .b IB c I
AI BI CI
a I CA≤
+ +
'()&'()&'()&'()&
ចេលយ
. Fង 4t x= + , យ)ង,ន 2 2 29)(( 16) (2)1) ( tt y⇔ − − =
Fង 2 2 , (2)25
, (2 2 )(2 2 ) 492
u u y u yut − ∈ ⇒= + − =ℤ
ក ទp
2 2 49 2 2 1 2 25 2 25
2 2 1 2 2 49 12 12
u y u y u u
u y u y y y
+ = + = = = ∨ ∨ − = − = = = −
⇒
2 25 5 1t xu ⇒ = ± ⇒ == I 9x = −
ព* 12)( , , () (1 9; 1 ); 2x y ± − ±=
ក ទp
2 2 49 2 2 1 2 25 2 25
2 2 1 2 2 49 12 12
u y u y u u
u y u y y y
+ = − + = − = − = − ∨ ∨ − = − − = − = − =
⇒
4 ( ; )2 25 0 ( 4; 12)xu x yt⇒ ⇒ = − ⇒ = − ±= − =
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 115
ក ទ+p
2 2 7 2 2 7 2 7 2 7
2 2 7 2 2 7 0 0
u y u y u u
u y u y y y
+ = + = − = = − ∨ ∨ − = − = − = =
⇒
2 6 47 12 tu t⇒ = ⇒ = ±=
0x⇒ = I 8x = −
2 9 32 7 tu t⇒ = ⇒− = ±= .
1x⇒ = − I 7x = −
( 1,0),( , ) ( 7,0), ( 8,0)(0,0),x y⇒ − − −=
.បមក, ម នប ច!8)យ6ច!នBនគ#គI
( 1;0), ( 7;0), ( 8;0), (1;12(0;0 ), (1), ; 12),− − − − ( 4;12), ( 9;12( 4; 1 ), ( 9; 122 )), − − − −− −
. Fង 2cosu x= , 8កខ_ 2u ≤ , ព8*
( )44 43 4 (4 3(* )) u u⇔ + = −
4 43 4 4 3u u⇔ + = − ច!@ 3
4u ≥
Fង 4 4 3v u= − ច!@ 0v ≥ , ព8*យ)ង,នបព|នម
4
4
3 4 (1)
3 4 (2)
u v
v u
+ =+ =
យកម (1)Cកម (2)LងXន%ងLងX
4 4 2 2( )[4( ) ) 4]( 0( )u u v u vvv u uv − +− = − ⇔ + + =
u v⇔ = (@ 3
4u ≥ នQង 0v ≥ )
ព8 u v= * 4 42) 3( uu⇔ + =
2 2
2( 2 3) 0
2 3 0 (*
1( 1
)) u
uuu
u u⇔ + + = ⇔
+ + =
=−
ម (*) W( នHទ
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 116
11 2cos 1 c s 2
2,o
3u x kx kx
π π= = = +⇔ ⇔ ⇔ = ± ∈ℤ
+. 1 1
1 1 1
xy
zPyz xz xy
x y z
= + ++ + +
Fង 2 2tan , tan2 2
yz A zx B
x y= = ច!@ 0
0
A
B
ππ<
<<
<
យ)ង,ន 1 . . .xy xz yz xy zx yz
x y zz y x z y x
+ + + = + +
1 tan .tan
2 21 . cot tan2 2tan tan
2 2
A Bxy zx yz zx yz xy A B C
gA Bz y x y x z
− ++ = − = = =
⇔ ⇒+
(@ ( ) 0; );2
(A BB C Aπ π π+ < = − + ∈ )
2 2
2 2 2
tan1 1 sin2 cos cos2 2 21 tan 1 tan 1 tan
2 2 2
CA B C
PA B C
= + + = + ++ +
⇒+
11 (cos cos cos )
2A B C= + + +
មO/ងទP 3 3cos cos sin sin 2cos .cos 2sin .cos3 2 2 2 2
C CA B A BA B C
π ππ + −+ −+ + + = +
3 32cos 2cos 4cos 4cos 2 32 2 4 6
C A B CA Bπ π
π− + + −+≤ + ≤ = =
C:ច* 1 3 3 32 3 1
2 21
4P
− = +
≤ + , មYពក)នព8
62
33
A B A B
CC
π
π ππ
= = = + =
⇔ =
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 117
tan 2 3; tan 312 3
yz zx xy
x y z
π π= = = − = =⇒
2 3 3; 7 4 3x y z= =⇒ = − −
=. Lន.<នMទ%បទ.ន.ក-.ង ,BCI យ)ង,ន
.sin2
sinsin sin cos cos2 2 2 2
BaIC BC BC a
B B C AIC
BIC A= = =+ ⇒ =
C:ចW- ;C .sin .sin
2 2;cos cos
2 2
C Ab c
IAC
IBB
= =
C:ច* . . . tan .tan .tan2 2 2
A B CIA IB IC abc=
យ 2 2 231 tan tan tan tan tan tan tan tan tan2 2 2 2 2 2 2 2
32
A B B C C A A B C≥= + +
2 2 2 1 1tan tan tan tan tan tan
2 2 2 27 2 2 2 3 3
A B C A B C⇒ ≤ ⇔ ≤
Fម< QមYពក:.ច!@បច!នBន, យ)ង,ន
32 2 2 2 2 2. 3 . . .. .a b IB c IC abc IAI IB CA I+ + ≥
2 2 2 3 2 2 2
1 1
. 3. . . . .b IB c IC abc IA IB ICa IA⇒ ≤
+ +
3
2 2 2 2 2 2
1 1
. 27 ... . .b IB c IC IB Ia IA ab Cc IA ≤
⇔
+ +
3
2 2 2
. . . . 1
. 27 27 .3 3 27. .3. 3
IA IB IC IA IB IC abc
a IA ab IB bC c bcc I a⇔ ≤ =
≤
+ +
2 2 2
. . 1
3. 3.. b IB c
IA IB IC
a I ICA⇔ ≤
+ +
មYពក)ន8.F; 2 2 2.. .b IB c
A B C
I Ia A C
= =
= =
បជវសសពែកគណតវទកទ១០
2012-11-17បកែបេយៈ ែកវ សរ Page 118
A B C ABC= =⇔ ⇔ ∆ 6 ម|ង2
'()&
លកហសឆ#ង េកើត'នេ)យអេចត-េ.ក/0ងេសៀវេ2េនះ សមេម56ខន8អភយេ;ស ល<ក=បេច>កេទស នង ?របកយេ.ក/0ងេសៀវេ2េនះ គB?របកែប េហើយេ)យមនCចរកៃដគក/0ង?រជយតGតពនត= េ-ះកហសឆ#ងHកដ
BមនCចរIលងHនេឡើយ េហតេនះ ខK0ពតBសមេ;សទកBមនលកហសឆ#ងែដលេកើត'នេ)យប?រLមយ េ)យមនHន<ងទក ។ សមឲ=េសៀវេ2េនះ ?OយBឯករវBវ នងBមត8ដលRរបសអ/កសកSគណតវទគបរប។
ជនពរសLងលR បTងៃវ េBគជយក/0ង?រសកS!!
បកែបចបេ.VណWយ, ៃថZទ ១១ ែខ វច[? \ ២០១២.....