vnmo 30 4-2006-grade 10

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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⇔ − + − + + − =



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= = ,



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+ + + + +⇒ ≡ ≡ ≡ ≡



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+ + ≤

+ + + + + + + + + .

'()&



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− − = − +⇔ − −



( )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= ∀



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

= = = − == + ⇒ = + = = = = =

⇒



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#)

'()&


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− ≥

'()&'()&'()&'()&



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



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



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+ − '()&




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



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



+. 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




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



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



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 )



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-

'()&


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= +

'()&'()&'()&'()&



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



+. 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= +



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 '()&


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

++ + + <+



=. 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

− = − + + − = + + =

⇔

⇔



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

< −+ +



*! 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 = − = − =

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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 = −



. + 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



+. 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 ==



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#)

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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



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*



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"



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− = + + + + − ⇔− =−⇔

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C

A

B

O

P

Q

K

E

N

D




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∆

'()&'()&'()&'()&



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+ + = = + − + .



( ) ( )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 ֏ .



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




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≥

+ + ++ = + +

+ + + + + ++



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



+. 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



( )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#)

'()&


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



# ( )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

+ + + = +



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

= + − +

− =

∆



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



6មBយន%ង ( )O

'()&


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



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#)



+. 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



2 2 / 4 72 (5)a c= + ព (4)នQង (5) : 6c =

C:ចន ( , , ) (9,9,6)a b c =

'()&


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⇔ − + − + + − =



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= + + + − + −



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= ++ =+ +



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∈



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⇒= =



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− =

'()&


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

− +≤



+. 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+ + + +=



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



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



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= ==

'()&


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



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



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≤+ + +



+. < 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



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

'()&




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= = ±



. 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∆



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∆



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#) '()&


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

'()&'()&'()&'()&



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

+ + ≥ + +

=



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+ >



.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

'()&


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

− − −+ + + ≥−

'()&'()&'()&'()&



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



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#

'()&




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⇒ − > ⇒ − −> = >



*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



(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

∈= − =



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= + = − +



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#

'()&




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

'()&'()&'()&'()&



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;



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



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



Jញ,ន IJ EK AH HK⊥ ⇒ ⊥

!W8#!W8#!W8#!W8# L|ក2L.:gប គIបក;បFម@កបចjកទ<Pម;C85

Trục ẳng phương : L|ក2L.:gប??? '()&


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 '()&'()&'()&'()&



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

++

++ +⇔ ≤



=. 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= = = =

'()&


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



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++ + = +++



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− + − −+ −+



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 = −− ==



+. 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#

=.



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∆ '()&


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



* 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#



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 = + = + + − + −

≤ + +



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⇔ − + + ⇔ + + + <− >



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= = =

'()&


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



=. 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



+. 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= − −



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

+⇔ ⇔ −= = −

=+

⇒



( )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



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+≥+



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)− − − −−

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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− =



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= − =− − −− −==



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⇒ + − = + + − + .



2 2 2 2. ) 2(2 )( 2 0BR BS BIB BK II K− = − = >=

Fមទ%បទក:.ន., យ)ងJញ,ន 090RIS <

'()&


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= =



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



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)

'()&




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 '()&'()&'()&'()&



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= ≤ + − − +−



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∈



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




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

+ + ≤+ + + + + + + + +

'()&'()&'()&'()&



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



+. 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+ − + =



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

+ +≤+ +

=

'()&




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

'()&'()&'()&'()&



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 =



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∆ ∆ ∆⇒ ≤ ≤



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

'()&




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⇒ ⇒ = − ⇒ = − ±= − =



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



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

= = = + =

⇔ =



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

= =

= =



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.....

vnmo 30 4-2006-grade 10

Education