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បេងៀនេយេកគ ហ៊ត សុីត www.keoserey.wordpress.com Page 1 បជុំលំត់គណិតវទ . ខងងឆងនចននគជនមយគខ 4 បខងន!គ"ប#ច$ងង%& នចនន នញ(ចននថ* +,នបង- "ងន.* 1 4 នចនន/ម 0កចនន2ចបផ$នចនន/ម 4. ABCD 5ទ+78ច$9:;កង<យ 30 AB m = នង 40 BC m = ប$0. =(កឈ0?ទ+@ង ខ$.ABC?កC $ ងទ+78 ;/ចD យ0Eងប$0.ព+0(ក?ជBC 5Gប0 0កចD យន H. 0កចននគធម*5 k ;/ 1, 001 k k នម8Gប0 =. មន$.J K(កបង- 50ងងមយ C កLបយកចននមយ Mយ,បខ(#GCក;/ ?;កN0B 4(កកC $ ង0ងង ប( បមក C កAគ:( នងប9.ព+ចននមធOមPគនGCក;កN0 ងBQង4 ទ20ខ,នបRS ញT មធOមPគ;/,នប9.<យមន$.JC កA (មន ;មន ចនន/ម;/មន$.JC កA,ន0.) 0កចនន;/,ន0.<យមន$.J,នប9. មធOមPគ 8 W. ចននន+មយAនចនន 2; 3; . 1; ..; 25 !,នគ.0.0?កC $ ង@Xង 5 5 × ;/ចនន?កC $ ជ0/កន+មយA !.Y កC $ ង<បកន 0កGប0នផប2កចនន?កC $ ងជ0ឈ0ទ+H Z. នប[$(*នចននព+ 1 / 2008 ;/នផប2កខង0ប.E;ចក<ចន\ង 5 ]. +9: ABC 5+9:.ម^ងJនងចន$ច P មយ;បប_?កC $ ងប8ង;មយ5មយ+9: ABC ;/ ; PAB PBC នង PAC 5+9:.ម, នចន$ច P .Y ?ទ+@ង ខ$.ABCប[$(*ន? a. b%យបcក.មPព 2 3 2! 3! 4! ( 1)! ... 1 2 3 ( 2)! 2 2 2 2 2 2 n n n n n + + + = × × × + + - × d. b%យបcកTប , , mnp 5ចនន.នQននង 1 mn np mp + = (
19

Teacher hort siath maths exercises

Jun 29, 2015

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Serey Keo

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Page 1: Teacher hort siath maths exercises

បេ�ងៀនេ�យេក�គ� ហ៊ត សីុ�ត

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�បជំុលំ�ត់គណិតវ�ទ�

�. ខ�ង��ងឆ�ង�នច�ន�នគ��� �ជ��នម�យគ��ខ 4� ប�ខ�ង�ន� �!�គ �"ប�#ច$ង�ង%& ��នច�ន�ន

ន�� �ញ(�ច�ន�នថ*+,នបង-��"�ងន�.*� 1

4�នច�ន�ន/�ម� 0កច�ន�ន�2ចប�ផ$��នច�ន�ន/�ម�

4. ABCD 5ទ+78 ច�$9:;កង<យ 30AB m= ន�ង 40BC m= � ប$0. =(ក�ឈ0?ទ+@�ង

ខ$.ABC ?កC$ងទ+78 ;/�ច�D យ0Eងប$0.ព+0(ក�?ជ��BC 5G��ប0�� 0កច�D យន� �

H. 0កច�ន�នគ��ធម*5�� k ;/� 1,001k

k �ន��ម8G��ប0� �

=. មន$.J �K(ក�បង-��50ង�ង�ម�យ� �C ក�Lប�យកច�ន�នម�យ M�យ ,ប��ខ(�#GCក;/�

?;កN0B�� 4(ក�កC$ង0ង�ង�� ប(� ប�មក �C ក�Aគ:( ន�ង ប9.ព+ច�ន�នមធOមPគ�នGCក;កN0

�ងB��Q�ង4� ទ20�ខ,នបRS ញT មធOមPគ;/�,ន ប9.<យមន$.J�C ក�A (ម�ន

;មន ច�ន�ន/�ម;/�មន$.J�C ក�A,ន0 �.) � 0កច�ន�ន;/�,ន0 �.<យមន$.J,ន ប9.

មធOមPគ 8�

W. ច�ន�នន+ម�យA�នច�ន�ន 2; 3; .1; ..; 25 �!�,នគ.0.0?កC$ង@Xង 5 5× ;/�ច�ន�ន?កC$ង

ជ�0/កន+ម�យA �!�.Y��កC$ង��<ប�ក�ន� 0ក����G��ប0��នផ�ប2កច�ន�ន?កC$ងជ�0ឈ0ទ+H�

Z. ���នប[$(* នច�ន�នព+ 1/�� 2008;/��នផ�ប2កខ�ង�0ប.�E;ចក<ច�ន\ង 5�

]. �+9: ABC 5 �+9:.ម̂ងJន�ងច�ន$ច P ម�យ; ប ប_�?កC$ងប8ង�;�ម�យ5ម�យ �+9:

ABC ;/� ;PAB PBC∆ ∆ ន�ង PAC∆ 5 �+9:.ម,�� ���នច�ន$ច P .Y��?ទ+@�ង

ខ$.ABC ប[$(* ន?

a. b%យបc� ក�.មPព 2 3

2! 3! 4! ( 1)!...

1 2 3 ( 2)!2

2 2 2 2 2n n

n nn ++ + + =× × × ++ −× �

d. b%យបc� ក�Tប� , ,m n p 5ច�ន�ន.ន�Qនន�ង 1mn np mp+ + = (�

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2 2 2)( )(1 1 )(1T nm p+ += + 590�នម�យច�ន�ន.ន�Qន �

�K. ក) 0ក��ម8�2ចប�ផ$��នផ�ប2ក 6 6c d+ ប� 2 2 1c d+ = �

ខ) 0ក��ម8ធ�ប�ផ$��ន 3 22 4

1

5n

nN

n n+ −=

−+ �

��. ច�ន�នគ��ធម*5�� A�នខ�ង�ច$ង 9យខ$.ព+ 0ន�ង 5�

��ច�ន�ន 5 3( 5)AA × + �ន�ខខ�ង�ច$ង 9យប[$(* ន?

�4. 90 ABCD ម�យ�ន ជeង 12 �!�;ចក5 14490fក@� 0ង�ង�ម�យ9�.*� 6 �!�,នគ2.ប[�

ន\ង ជeងQ�ងប�ន�ន90 ABCD � 0កច�ន�ន90fក@ ;/�.Y��?កC$ង0ង�ង�Q�ងb.eង�

�H. ក) 0ក , ,a b c ;/� (5) (8)abc cba= កC$ងន�ន�ម��&.cg (5)abc 5ច�ន�ន.0.0កC$ង បព̂នh

B� 5ន�ង (8)cba កC$ង បព̂នhB� 8�

ខ) ��កC$ង បព̂នhB�i;/� 4 212! 11! 10 10! 10− − +=

�=. 0កច�ន�នគ���2ចប�ផ$� �ន�ខច$ង 9យ 1986M�យ;ចក<ច�ន\ង 1987 �

�W. b%យបc� ក�T កនjម�ង 9មម�នG� �ជ��នk

2 )9( ) 24(ab bc ca cd ad bQ a c db − + + + + += + +

�Z. គlO 101ច�ន�ន� �ជ��ន 0 1 100; ; ...;aa a ;/� 1 0a a> ,

2 1 0 3 2 1 100 99 983 2 ; 3 2 ; ...; 3 2a a a a aa a a a= − = − = − �

b%យបc� ក�T 99100 2a > �

�]. ABCD 5ច�$9:L0\កកC$ង0ង�ង�ន�ង M 5ច�ន$ចកi& � [ ]CD M�យGង-�� ទ!ង9��BC �ង�

ច�ន$ច P � 0ង�ង�9��@ម P ប[�ន\ង [ ]CD �ង� M ជ�ប AC ន�ង BD ម&ងទn� �ង� R ន�ង Q �

ច�ន$ច S ?�� BD M�យ BS DQ= ប(� ��9��@ម S b.ប AB ជ�ប AC �ង� T �

បRS ញT AT RC= �

Page 3: Teacher hort siath maths exercises

បេ�ងៀនេ�យេក�គ� ហ៊ត សីុ�ត

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�a. ក) b%យបc� ក�Tk 5 102 7 8 2+ <

ខ) គlO ,a b∈ℕ b%យបc� ក�To[ង��ចច�ន�នម�យកC$ងច�(ម 2ច�ន�ន ;b aa b ម�នធ�5ង 3 3 .

�d. ក) 0កp.គ���ន.ម+90 2 0mtt n+ + = ប� 198m n+ =

ខ) 0ក �+7�$5ច�ន�នគ��� �ជ��ន ( , , )a b c ;/� 1 1 1 4

5a b c+ + = �

4K. ក) ��ច�q���ម8គ��i�ន , ,x y z ;/�បi& �+7 2 25; 3 3x yx y− − + + ន�ង

2 5 5zz− − − �ន��ម8.*�BC �

ខ) 0កp.គ��� �ជ��ន�ន.ម+90 2010u v+ = �

4�. ប;�ងម/rនQ�ងH , ,AD BE CF �ន ABC∆ ម�យ.*� 9,12,150nងBC � 0ក�ផ� កs ABC∆

44. tប� xន�ង y 5ព+0ច�ន�នគ��ខ$.ព+ 1− ;/� 3 1

1

x

y

++

ន�ង 3 1

1

y

x

++

5ច�ន�នគ��ម�យ�

b%យបc� ក�T 2010 1 1yx − +⋮ �

4H. 0ក គប�ច�ន�នបuម�នXង ( 1)( 2)1

6;

n n nn

+ + + ∈ℕ �

4=. tប� 1 2;x x 5p.ព+0�ន.ម+90 2 4 1 0xx − + = � b%យបc� ក�T ច�q� គប�ច�ន�នគ��

� �ជ��ន n (� 2 21 2

n nx x+ vចបង-��,នផ�ប2ក90�ន 3ច�ន�នគ���BC �

4W. គlO90 ABCD ជeង.*� 3� ?�� [ ]AD ន�ង [ ]CD wព+0ច�ន$ច M ន�ង N ;/� ជeង

9ច� MDN �ន ប;�ង.*� ជeង�ន90� យក E 5ច�ន$ច ប.ព��ន AM ន�ង BN �

0ក ប;�ង�ន ME <យ/\ងT 4NE = �

4Z. ម$ខQ�ងG.��នគ2បម�យ �!�គ<ក�x8 ក<យច�ន�នគ��� �ជ��នផJងABC ;/�ច�ន�ន?��

ម$ខ5ប�BC ព+0ខ$.BC o[ង��ច 2� 0ក��ម8GបNប0��នផ�ប2ក ច�ន�ន��ម$ខQ�ងZ�

4]. ទ�ព̂0�ន.n�yម�យ �!�គបង��ខព+ 1/�� n� .�.J�C ក�ប2ក�ខទ�ព̂0Q�ងG.� M�យ

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ទទ��,ន�ទhផ� 1922� គ/\ងT E�នក�M$.<យ,នប2កច�ន�នខ8�ព+0/ង,

ច200កច�ន�ន(��

4a. @ង 3 20( ) 9 9 4x x x xP = + + + � ច�q�ច�ន�នគ�� 1n ≥ គក�:��

1( ) ( )n nP x P x n−= − � 0កមគ$:�ន 2x កC$ង 2008( )P x �

4d. , ,A B C ន�ង D 5ក�ព2�Q�ងប�ន�នច�$9:L0\កកC$ង កC$ង0ង�ង�<យGង-�� ទ!ង AC 9��

BD �ង� E � ប� 4; 6B D EC C A= == M�យ BE ន�ង DE 5ច�ន�នគ��� 0ក ប;�ង BD �

HK. 0កច�ន�នព�� x ;/� 1 1 1 x x+ + + = �

H�. ច�ន$ច ( , )A a b ?��ប8ង� ( )xOy 5ច�ន�នគ�� Q�ង a ន�ង b 5ច�ន�នគ��� ក�:��ច�ន�ន�ន

ច�ន$ចគ�� ;/�ព̂ទhជ$�� �ញ<យ90 (0�មQ�ង ព�;/�5 ជeង90) បង-��<យច�ន$ច (100,0),

( 100,0);(0,100), (0; 100)− − �

H4. 0កច�ន�នគ��� �ជ��ន�2ចប�ផ$� ;/�vចប�;�ង5ផ�ប2កផJងBC �នច�ន�នគ��� �ជ��ន ,a b ន�ង c

;/� ,a cb a+ + ន�ង b c+ 590 ,ក/�

HH. b%យបc� ក�T Gង-��P� ប�0nងBC បi& ផz�� 1 2 3; ;O O O 4;O �នបi& 90;/�,ន.ង�?

�� ជeង�ន ប�"2 9ម ABCD #�ង {ផ&$�,ន590ម�យ�

H=. tប�T .�$+� 1 2 3; ; ; ...a aa ប�ពញទ�(ក�ទ�នង 2 1n n na aa + += − ន�ង 2na = �

ប�ផ�ប2ក�ន 2004��/�ប2ង.*� 2008, 0កផ�ប2ក�ន 2009��/�ប2ង�

HW. �ខ;/�,ន ប�.�Xប�បង��ខ ទ�ព̂0�ន.n�yម�យ �!�,នគXប� M�យច�ន�ន.0$ប�ន�ខ

;/�,ន ប�គ� 2007� 0កច�ន�នទ�ព̂0កC$ង.n�y �

HZ. =ច�ន�នគ��គ2�BC �!�,នគ/កចញព+.�$+�ច�ន�នគ�� 1,2,3,...,n M�យមធOមPគ�នច�ន�ន

?.��.*� 51,5625� 0កច�ន�នគ�� ធ�ប�ផ$�;/�,ន/កចញព+.�$+��

H]. ក) p.�ន.ម+90 2 1axx b+ + = 5ច�ន�នគ��� �ជ��ន�

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បRS ញTច�ន�នគ�� 2 2a b+ 5ច�ន�ន.�.�

ខ) ច�ន�នគ��� �ជ��ន n5ច�ន�នម�យ;/�ធ�5ង90 ,ក/ម�យ�

b%យបc� ក�T 2n 5ផ�ប2ក�ន 290 ,ក/�

Ha. កC$ង �+9: 0 0ˆ ˆ90 , 30,ABC C A= = ន�ង 1AB = � @ង �+9: ABD , ACE ន�ង BCF

5 �+9:.ម̂ងJ<យ , ,D E F ? { �+9: ABC �

@ង DE ប.ព� AB �ង� G � 0ក�ផ� កs �+9: DGF �

Hd. គlOច�$9:,[ ង ABCD � 0ក.�:$� ច�:$ ច �នច�:$ ចកi& � P �នបi& Gង-��;/� 2

ច�:$ ច M ន�ង N @ម��<ប�?�� ជeងព+0 AB ន�ង CD �

=K. គlOច�ន�ន , ,x y z ផ�|ងx� �� 2 2

2 2

3

16

xy y

y y z

x

z

+ + =

+ + =

b%យបc� ក�T 8xy yz zx+ + ≤ �

=�. 0ក គប�ច�ន�នគ�� n /�មN+lO 2 1997 31 0999P n n+ += ;ចក<ច�ន\ង 6n �

=4. ក) 0កបi& ច�ន�នគ�� ,x y ផ�|ងx� �� 3 3 1995yx + = �

ខ) 0កព+0ច�ន�ន a ន�ង b <យ/\ងTផ�គ$: ( )( ) 21903a ba a bb =++ �

=H. ABCD 5ច�$9:,[ ង <យ AD b.ប ,BC AC ;កងន\ង BD ន�ង 5AC = �

CF 5ប(� ��ព+ C # AD M�យ 4CF = � 0ក�ផ� កsច�$9: ABCD �

==. យក 1 2 8; ; ...;a a a 5ច�ន�នគ��� �ជ��ន ;/�k

2 2 2 21 2 3 8(2 ) (3 ) ... (8 ) 204a a a a+ + + =+ � 0ក���� 1 2 8...a a aA a + + += �

=W. Aន�ង B 5ព+0ច�ន�នបuម� �ជ��ន;/� 2 2

1 1 192

20 0005 2 4A B −− = � 0ក���� B �

=Z. គlO 1

2 3x =

−� 0ក�����ន 6 5 4 3 22 4 23 3x xA x x x x− − + − + −= �

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=]. យក ,a b ន�ង c 50R� .� ជeងQ�ងប+�ន �+9:ម�យ� tប�T a ន�ង b 5p..ម+90k

2 4( 2) ( 4)c c xx + + = + ន�ងម$�ធ�5ងគ�ន �+9:គ� 0x � 0ក�����ន x�

=a. គព�ន��Oម��.�$+�ច�ន�នព�� { }na ក�:��<យ 1 1n

nn

aa

na+ =+

ន�ង 1 1a = ច�q� 1n ≥ �

0ក�����ន 2005

12007004A

a= −

=d. ច�q�ច�ន�នគ��� �ជ��ន k គ.0.0k

20 1 2... (1 ) ...(1 )(1 2 )(1 3 ) k

kkx x x x a a x a x a x+ = + + + ++ + + ;/� 0 1; ; ... ; kaa a 5

មគ$:�នពM$7� 0ក�����2ចប�ផ$��ន k ប� 0 1 1... 2007ka aa −+ + + ⋮ �

WK. 0កគ2�ន��<ប��នច�ន�ន ( , )x y ;/�ប�ពញ.ម+90k 2 2 2( )y xx y xy+ = + +

W�. កC$ងច�$9:;កង ;ABCD Pន�ង Q 5ច�ន$ច?�� [ ]AB ន�ង [ ]BC 0nងBC ;/� �+9:

,APD PBD ន�ង QCD �ន�ផ� កs.*�BC � គ:( AP

PB �

W4. f 5Gន$គមន}ក�:����.�:$� ច�ន�នគ�� M�យផ�|ងx� ���ក~ខ:� �ង 9មk

�- 0(0)f ≠

4- (1) 3f =

H- ( ). ( ) ( ) ( )f x f y f x y f x y= + + − ច�q� គប� ,x y ∈ℤ� គ:( (7) 1164f +

WH. ក) គlOបi& ច�ន�ន , , ,a b c d ផ�|ងx� ���ក~ខ:� 2 2 2 225; 16b c da + = + = ,

20ac bd+ ≥ � 0ក����ធ�ប�ផ$��ន T a d= + �

ខ) គlO 2 2 1a b+ = � 0ក����ធ�ប�ផ$��ន 3 315 3 4820 6aP a b b− + −=

W=. កC$ងប8ង��ន �+9: ABC M�យ?�ង { �+9:ន� គ.ង� �+9: ; ;CQB P AC ABR

;/� � � � �0 045 ; 30PBC CAQ BCP QCA= = = = ន�ង � � 015ABR BAR= = �

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b%យបc� ក�T � 090QRP = ន�ង QR PR= � (IMO 1975)

WW. យក n5ច�ន�នគ��� �ជ��ន;/�0 \.ម�យ�ន.ម+90/� កទ+ព+0

( )24 4 3 4 3 24 0x x n+ + − =− 5ច�ន�នគ��� 0ក�����ន n�

WZ. គlO បព̂នh.ម+90 255

224

xy xz

xz yz

+ = − =

� 0កច�ន�ន �+7�$�ន��<ប� �នច�ន�នគ��� �ជ��ន

( , , )x y z ;/�ប�ពញlO បព̂នh.ម+90�ង�� �

W]. គlO.�$+�ច�ន�នព�� ( )nU ;/� 0 2U = − ន�ង 131

2n nU U −= − , គប� *n∈ℕ �

ក) ក�:����ទ2# nU 5Gន$គមន}�ន n

ខ) គ:( 0 1 ...n nS U U U= + + + �

Wa. បRS ញTប� m5ច�ន�ន.ន�Qន� �ជ��ន(� 1m

m+ 5ច�ន�នគ��;�ម�យគ��ប� 1m = �

Wd. ក) បRS ញT ម�ន�នច�ន�នគ�� ; ;a b c ;/� 2 2 8 6b ca + − =

ខ) @ង a ច�ន�នគ�� .111..a = ( m �ខ 1)

. 0100 .. 5b = ( 1m − �ខ 0?ច(8 ��ខ 1ន�ង�ខ 5)�

បRS ញT 1ab + 590ច�ន�នគ��� 0ក0 \.90�ន 1ab + <យយកទ�0ង�/2ច a ន�ង b �

ZK. @ង n5ច�ន�ន�ន�ខ 5ខ�ង� (;/�ខ�ង�ទ+ម�យម�ន 0) ន�ង m5ច�ន�ន�ន�ខ 4ខ�ង�បង-��

ចញព+ n <យ�ប�ខ�ង�កi& �� ក�:�� គប� n ;/� n

m5ច�ន�នគ�� �

Z�. គlO 4 3 2( 1) x xf x x x= + + + + � 0ក.�:��ព�;ចក ( )5f x ន\ង ( )f x �

Z4. ក) បRS ញT ច�q� គប�ច�ន�នព��� �ជ��ន ; ; ;p q r s ;/�

2 2 2 21)( 1)( 1)( 1) 8( 1p q q r r s s pqrsp + + + + + + + + ≥

ខ) b%យបc� ក�T ច�q� 2, 3, ..1 , .n =

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2

( 1) 2 . !

). ( 1) (2 1) 6 ( !

).

)

n n

n n n

n n

b n

a

n n

+ ≥

+ + ≥

ZH. ច�q� គប�ច�ន�នគ��� �ជ��ន n@ង 1 1 1( ) 1

2 3...h n

n= + + ++ �

5tQM0:} 1 1 1(1) 1; ;

2(2) 1 (3) 1

2 3h hh = + = += +

b%យបc� ក�T ... ( 1) . ( ); 2,3,(1) ( .2 (3 ..) ) h n nn hh nh h n+ − = =+ + + +

Z=. គ.ង-�ឃ�ញTk 2 2 25 11; 56 16 45 111− = − =

2 2 2111111; 5556 44455 5 111111116 445 = − =− .

0ក��ទ2#@មtQM0:} �ង�� 0�ចb%យបc� ក�E�

ZW. គlO �+9: ABC � 0ក.�:$� ច�ន$ច M ;/� 2MAB

MAC

S

S= �

ZZ. ច�q�B� b 5ច�ន�នគ��� �ជ��នi;/� 1367631?ព�ន� �!�,នគ.0.05គ2ប

�នB� b �

Z]. ប� x5ច�ន�នព��� �ជ��នម�ន.*� 1ន�ង n5ច�ន�នគ��� �ជ��ន� b%យបc� ក�T

2 11

(2 1).1

nnx

n xx

+− = +−

Za. គlO.�$+� Fibonacci { }nU ក�:��/2ច�ង 9មk

0

1

1 1

0

1

; 1n n n

U

n

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ខ) b%យបc� ក�T 1 2 2... 1n nU U UU ++ + + −= គប� n∈ℕ

Zd. គlO.�$+� { } 1,2,3, ...;nU n = ក�:��/2ច�ង 9មk

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1

12 2 ; 2,3, ..) .

3

4

(2 1 nn nnUU n

U

n −

+

=

= + =

b%យបc� ក�T 0 2 1

knn

nk

CU

k==

+∑

]K. ក) 0ក�ខព+0ខ�ង�ច$ង 9យ�ន 2 3 20072005 ... 22005 20 00505 + ++ +

ខ) <�b%យ បព̂នh.ម+90k

3 2

3 2

3 2

6 12 8 0

6 12 8 0

6 12 8 0

x x

z y y

z

y

x z

− + − =

− + −

=

− + − =

គ) ប|បធnបព+0ច�ន�ន 22222A = ន�ង 2 3 20063 ... 20061 2B + + += + �

ឃ) យក f 5Gន$គមន}�ន����ព�� ;/�k

( , ) ( , ) 2 ( , ) 2f x y f x z f y z z= − − ច�q� គប�ច�ន�នព�� ,x y ន�ង z

គ:( (2006,1000)f �

]�. ក) 0ក��B* ន xកC$ងទ�7ញ2�$នk 17

4 3

3 20

1x x

x

+

ខ) 0កp.គ���ន.ម+90 22x x=

]4. គlO90 ABCD �ន AB a= ន\ង� M 5ច�ន$ចម�យច�̂�?��Gង-�� ទ!ង AC � គ2. ME

;កងន\ង AB ន�ង MF ;កងន\ង BC � ក�:��ទ+@�ងច�ន$ច M �� AC ;/� កs�ផ� �+9:

DEF �2ចប�ផ$�� 0ក�����2ចប�ផ$�(��

]H. កC$ង02ប�ង%& � AD AB= , � � � 090DAB DCB AEC= = =

ន�ង 2006AE = �

0ក កs�ផ�ច�$9: ABCD � E

A

B

C D

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]=. ក) 0ក គប�ច�ន�នគ��ធម*5�� ( , )x y ;/�ប�ពញ.ម+90 2 2 2( )y xx y xy+ = + +

ខ) 0កច�ន�ន n�ន�ខ Hខ�ង��2ចប�ផ$�;/� ប�ខ�ង�Q�ងប+គ� , ,a b c (�

n a b c ab bc ca abc= + + + + + + �

]W. Q�ប+5 ផ�ប2កគlO�ង 9មvច�ចចញយក590,នក�<យ បRS ញT @ម90ព��

�ទhផ�ទទ��,នម�ន �\ម �!�ទ ប�គជ�ន�.�ខ�នខ�ង�ន+ម�យAផJងBC .�Xប�GកJ0ន+ម�យA

<យធ��90G�:�G�iងo[ង �\ម �!��

THREE

FIVE

EIGHT

+

]Z. ក) គlOច�ន�ន 11...11;/�?ព�ន� �!�គប�;�ង5 បព̂នhB�4�

0ក900ប.�E ;/� �!�.Y��?កC$ង បព̂នhB�4 ;/0�

ខ) ផ�|ងx� ��T 1 1 1 1 1 1 1 11

2 5 8 11 20 41 110 1640= + + + + + + +

]]. 0ក គប�Gន$គមន} :f →ℝ ℝ ផ�|ងx� ��k

[ ]3 2 2) 2 ( )( 3 ( )y y xf f xyx f y f = +

− + + គប� ,x y ∈ℝ �

]a. <�b%យ.ម+90�ង 9មកC$ង.�:$� ច�ន�នព��k

ក) 4 3 22006 10060 2 2007 0049 00 1x x xxx + + + + + =−

ខ) 2 22( 1 3 1)x x xx− + = ++

]d. គlO �+9: ABC �ន ជeង , ,Ca A B cB bC A= == ន�ងម$� 2B A= , 4C A= ,9�0ង�ង�

L0 \ក {គ� R � គ:( 22 2 2

1.

1 1S R

a b c

= + +

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aK. គlO , , 0a b c > � b%យបc� ក�Tk

( ) ( ) ( ) ( ) ( ) ( )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

+ + ≤+ + + + + + + + +

a�. lOប+ច�ន�ន� �ជ��ន ; ;a b c ផ�|ងx� �� 1 1 13

a b c+ + = �

b%យបc� ក�T 3 3 34 4 43 3 3 2 2 2a b c a b c+ + ≥ + + �

a4. គlO �+9: ABC .ម,� �ង� A� <យ/\ងTG0�2.ង� H �ន �+9:?��0ង�ង�

L0 \កកC$ង�ន �+9:� គ:( cosA�

aH. គlOពM$7/� កទ+ប�នk 4 3 22004 2007 2003 2( ) 2 50 00 6 0x xP x x x+ + + += �

b%យបc� ក�T ( ) 0P x > គប� x∈ℝ �

a=. គlO , ,a b c 5បi& ច�ន�នព��ម�នG� �ជ��នផ�|ងx� ��k 3a b c+ + = �

0ក����ធ�ប�ផ$��ន 9 10 22A ab ac bc= + + �

=W. <�b%យ បព̂នh� �.ម+90k

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

+ − − <+ + − − <+ − + <

> > > >

aZ. 0ក គប�បi& Gន$គមន} :f + +→ℝ ℝ ផ�|ងx� ���ក~ខ:� ព+0�ង 9មk

)a គប� ,x y +∈ℝ ប� x y≤ (� )( ()f x f y≤

( )). ( . ). 2007

f yb f x y f

x =

, គប� ,x y +∈ℝ

a]. <�b%យ.ម+90k

42 2 2)( 3 2007) 2005( 4 4 30 1 200. 6x x x xx x xx− − =+ + − + − + �

aa. ក) គlO , 0x y > ; 1x y+ ≥ � 0ក�����2ចប�ផ$��នកនjមk

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9 4851 23

7P x y

x y= + + +

ខ) គlO , , 0a b c > � b%យបc� ក�� �.មPពk

2 2 2 2 2 2 2)( )( ) ( )( ab b b bc c c ca a ab ca bc a+ + + + + + ≥ + + �

ad. 0ក�����ន , ,m a b ;/�.ម+90 5 3 0mxx − − = �នp.ព+0 1 2;x x <យ 1 2;x x 5p.ព+0

�ន.ម+90 2 0axx b+ + = , កC$ង(� a ន�ង b 5ច�ន�នគ�� �

dK. គlO �+9: ABC ន�ង O 5ច�:$ ចម�យ?កC$ង �+9: ABC � @ម Oគ2.ប(� ��ប+

ផJងBC 1 2 3); ( )( ; ( )d d d 9�� ជeងព+00nងBC ,AB BC �ង� ,M N , 9�� ជeងព+0 ,BC CA �ង�

,P Q , ន�ង9�� ជeងព+0 ;CA AB �ង� ,R T � យក 1 2 3; ; ;S S S S 0nងBC 5�ផ� កs�ន �+9:

; ;ORO N QP OMT ន�ង ABC � b%យបc� ក�T 1 2 3

1 1 1 18

S S S S+ + ≥ �

d�. ក) 0កp.5ច�ន�នគ��ធម*5���ន.ម+90k ( ) ( )22 2 4 4 24 28 17 14 49x xy y y+ + = + + + �

ខ) 0ក����គ��� �ជ��នផJងBC 1 2; ; ...; nxx x ;/� 2 2 21 2

...1 1 1

1nx x x

++ + =

d4. ក) tប�.ម+90 4 3 2 1 0bx cxx bx+ + + + = �ន0 \.�b%យបc� ក�T 2 2( 2) 3cb + − > �

ខ) <�b%យ.ម+90 3 3 3 0xx + − = �

dH. គlO.�$+� ( )nU ;/�ក�:��<យ ( )1

21

5

19 ; ; 1

5n n nU U U

U

n n+

=

= − + ∈ ≥

@ង 1 2

1n

nkk

VU=

=+∑ ; ; 1n n∈ ≥ℕ � គ:( lim n

nV

→+∞

d=. 0ក គប�Gន$គមន} :f →ℝ ℝ ផ�|ងx� ��k ( ) ( ).f x f y x f y x xy y + + = + + គប� ,x y ∈ℝ

dW. យក K 5ផz��0ង�ង�L0 \កកC$ង �+9: ABC � 1 1;B C @ម��<ប�គ�5ច�ន$ចកi& ��ន ជeង

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[ ]AC ន�ង [ ]AB � ប(� �� 1( )C K 9��ប(� �� ( )AC �ង� 2B , ប(� �� 1( )B K 9��ប(� �� ( )AB

�ង� 2C ;/��ផ� កs �+9: ABC ន�ង 2 2AB C .*�BC � គ:(ម$� �CAB �

dZ. ក) 0ក *n∈ℕ /�មN+lOច�ន�ន �2

11...1 77 ... 7n n

A −=��

590 ,ក/ �

ខ) 0ក គប�បi& ខ�ង� b 5�ខ;/� �

2

11...1 ...n n

bb bB −=��

590 ,ក/ �

d]. យក 1 1 1

2 31 ...nS

n+ += + + ច�q� n5ច�ន�នគ��ធម*5��ម�ន.2នO�

b%យបc� ក�T 2 2 2 21 2 3

1 1 1 12

2. 3. ....

nS S S n S++ + + <

da. ក) ប|បធnប 4ច�ន�ន 5555555553

5555555557A = ន�ង 6666666664

6666666669B = ម�ន ប��[.$+នគ���ខ

ខ) ប|បធnប 4ច�ន�ន 20062006!C = ន�ង 2007

2D =

គ) ប|បធnប 4ច�ន�ន 3 32006 2004E += ន�ង 32. 2005

dd. <�b%យ.ម+90 2 211 1 2

2x x x− − = − �

�KK. គlO �+9: ABC ផ�|ងx� �� 1tan . tan

2 2 2

A B = � b%យបc� ក�T �ក~ខ:� L�,ច�ន�ង

គប� Bន�/�មN+lO �+9: ABC 5 �+9:;កងគ� 1sin .sin .sin

2 2 2 10

A B C = �

�K�. ក) គlOប+ច�ន�ន� �ជ��ន �នផ�ប2ក.*� 4� b%យបc� ក�T ផ�ប2កព+0ច�ន�នiក�,នកC$ង

ច�(មប+ច�ន�ន(�ម�ន�2ច5ងផ�គ$:�នប+ច�ន�ន(� �

ខ) គ:( 0 0 0 0sin 69 sin183 sin 21s 39 3inS + + +=

�K4. ក) <�b%យ.ម+90 1 31 0

4 2

x

x x

+ − =+ +

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ខ) យក ,x y 0nងBC 50R� .�ម$�កC$ងព+0�នពM$9:ន�យ̂�ព+0 1D ន�ង 2D � <យ/\ងT

5 7 0x y− = � 0កច�ន�ន ជeង�ន 1D ន�ង 2D �

�KH. b%យបc� ក�T ច�q� គប�ច�ន�នគ��� �ជ��ន n យ�ង,នk

3 3 3

...1

32

1 1 13

2 4 (1 )3 n n++ + + <

+

�K=. បi& ច�ន�នព�� , , ,a b c d ផ�|ងx� ��

0

1 23

22

b c d

d

a

a

b cd

b c

≤ ≤ ≤

+ + ≥

<

+ ≥

b%យបc� ក�T 4 4 4 4 17a b c d+ + − ≤ �

�KW. គlOប+ច�ន�នព��� �ជ��ន , ,a b c ផ�|ងx� ���ក~ខ:� 1a b c+ + = �

0ក�����2ចប�ផ$��នកនjម 1 1

1 2( )M

ab bc ca abc= +

− + + �

�KZ. គlO �+9:.ម,� ABC � 0ង�ង�L0 \កកC$ង �+9:ប[�ន\ង ជeង AB �ង� T � ( )CT 9��

0ង�ង� �ង� K � tប�T K 5ច�ន$ចកi& � [ ]CT ន�ង 6 2CT = �

គ:(0R� .� ជeង �+9: ABC �

�K]. ផ�ប2ក�ន mច�ន�នគ2� �ជ��នផJងBC ន�ង nច�ន�ន..� �ជ��នផJងBC គ� 2001� 0ក����ធ�ប�

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ខ) 0ក��;ចក0�មធ�ប�ផ$��ន 1 3 5 20052006 2006 2006 2006; ; ; ... ;C C C C �

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��K. ធ�� ប�:��ធ+�ង 9ម ;/�GកJ0ន+ម�យA@ង<យ�ខព+ 0/�� 9ខ$.ABC �

o y s

o y

b

b

s

s

i l l y

+

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( )f x 5Gន$គមន}ក�ន ន�ង 0 (1)

1( ). ( ) 1

10

(2

( ) ,

)

x

f x f f xx

f xx

+ >

+ =

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��4. គlOច�$ម$ខ OABC �ន , ,OA OB OC ម�យគ2;កងBC ; ;Ox B C zO yA O= == ន�ង

3x y z+ + = � 0ក����ធ�ប�ផ$��ន 9�;.��L0 \កកC$ងច�$ម$ខ OABC �

��H. tប�T a 5ច�ន�នគ��� �ជ��ន;/� 2 2a − ;ចក<ច�ន\ង a � .�$+� { }nu ក�:��/2ច�ង 9ម

1

1 2 1nun

a

u

u

+

=

= −

, ច�q� គប� 1,2 .., .n =

b%យបc� ក�T ច�q� គប� 1,2 .., .n = គ,ន5ន�ចz 2 2nu − ;ចក<ច�ន\ង nu �

��=. គlOច�ន�ន 2006 2005

00 ... 0100 ... 51 0A x=�� ��

� 0ក���� x /�មN+lO A ;ចក<ច�ន\ង 37�

��W. <យច�ន�ន 20082006 បង-��,ន5ផ�ប2កច�ន�នគ��� �ជ��ន� ច20យកផ�ប2កគ2ប�ន គប�បi&

��Q�ង(�;ចកន\ង 3 ��គ,ន.�:��.*�ប[$(* ន?

��Z. @ង ... (21 1.3. )5 ) (2 1 !!n n− = − ន�ង ... (2 )2.4.6 (2 )!!n n=

b%យបc� ក�T (2005)!! (2006)!!+ ;ចក<ច�ន\ង 2007 �

��]. គlO0ង�ង� ( )O <យ�នGង-��ធC2ព+0 AB ន�ង CD 9��BC �ង� M � @មច�ន$ចកi& � S �ន

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Page 16

[ ]BD គ2. ( )SM 9�� [ ]AC �ង� K � b%យបc� ក�T 2

2

AM AK

CKCM= �

��a. ច20ធ�� ប�:��ធ+�ង 9ម;/�GកJ0ន+ម�យA@ងlO�ខខ$.ABC ព+ 0/�� 9�

r c s

r a

a

b

c

s

r a s s

+ � �� ? ?? ?r ac s= = ==

��d. 0ក គប�Gន$គមន} :f →ℝ ℝ ផ�|ងx� ���ក~ខ:� k ( ) 1( ) ( ). f yf x y f x e −+ = គប� ,x y ∈ℝ �

�4K. គlO n5ច�ន�នគ��� �ជ��នផ�|ងx� ���ក~ខ:� ( , 1) ( , 2) 55C n n C n n− + − = �

ច200ក��5ច�ន�នគ�� កC$ងទ�7ញ2 �$ន ( )7 38 5n

+

�4�. គlO.�$+� { }nx ក�:��<យ 1

12

1

; 12 3

nn

n

xx n

x

x+

=

= ≥+ +

ក) ច20ក�:����ទ2#�ន { }nx

ខ) b%យបc� ក�Tច�ន�ន 2

22

2n

n

x

x− vចបង-��,ន5ផ�ប2ក90�នHច�ន�នគ���BC ( 1)n ≥ �

�44. គlO 1 2 2006.. 10 .a a a≤ ≤ ≤ ≤ ≤ � ច20 ប|បធnបk

( )1 1 2 2005

2005

..log .A a a a+ + += ន�ង ( )1 1 2 2006

2006

..log .B a a a+ + += �

�4H. គlO �+9: ABC ;កង �ង� C �ន BC AC< � យក O 5ផz��0ង�ង�L0 \ក { ន�ង I 5ផz��

0ង�ង�L0 \កកC$ង� <យ/\ងT �+9: BIO ;កង� b%យបc� ក�Tk

ក) � 090BIO =

ខ) r p c= − ( r 59�0ង�ង�L0 \កកC$ង p កន8�ប0 �� � �+9: ABC )

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គ) 2 2 22 2 3b c ac bca + + = +

�4=. ក) គlO �+9: ABC �ន tan atan , , t nBA C បង-��,ន5.�$+�នព�ន&ម�យ�

0ក�����2ចប�ផ$��នម$� B

ខ) គlO , , 0a b c > � 0ក�����2ចប�ផ$��នកនjមk

a b c b c c a a bP

b c c a a b a b c

+ + += + + + + ++ + +

�4W. គlO900 ABCD ជeង.*� 3� wច�ន$ច M ?�� ជeង BC � ប(� �� AM 9�� ជeង DC

��ប(8 យ �ង� P � ប(� �� DM 9�� ជeង AB ��ប(8 យ �ង� Q � BP 9�� CQ �ង� I �

ក) គlO 1CM = � គ:( ,BI CI

ខ) ព� M ច�̂��� BC , 0ក.�ន$�ច�ន$ច I �

�4Z. ក) គlO , , ,a b c d 5ច�ន�នគ��� �ជ��នផ�|ងx� ��k

0

0

0

0

0

a b c d e

a b c d e

a b c d e

a b c d e

a b c d e

− + − + > + − + − >− + + − + > − + + − >

− + − + + >

0ក����ធ�ប�ផ$��ន ( )b dM a c += + <យ/\ងT 2006a b c d+ + + = �

ខ) 0ក��ធ�ប�ផ$�កC$ងទ�7ញ2�$ន 100(1 0,2)+ �

�4]. កC$ងGន$គមន}5ប� :f →ℝ ℝ ផ�|ងx� ���ក~ខ:� k

(1) 1f = − ន�ង ( ) ( ) ( ) 2f x y f x f y xy+ = + + គប� ,x y ∈ℝ

�4a. 0ក គប�Gន$គមន} :f →ℕ ℕ ផ�|ងx� ���ក~ខ:� k

( 1) ( ) 2f mn mf n+ = + ច�q� គប� ,m n ∈ℕ �

�4d. ក) b%យបc� ក�T 1 1 1 11 3

1! 2! 3.

! !..

n++ + + + < �

ខ) គlO.�$+� ( ) ;mS m∈ℕន�ង 4m > ក�:��/2ច�ង 9មk

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4 11; 1( 2) 2( 3) 3( 4) ... ( 2).1m mS S m m m mS += = + − + − + − + + − �

b%យបc� ក�T 4m mS C= �

�HK. គlOGន$គមន} f ក�:���� ℝន�ងផ�|ងx� ���ក~ខ:� k

). ( 3) ( ) 3fa f x x≥ ++ .

( 2) 3. ( ))b f x f x+ ≥ + ច�q� គប� x∈ℝ

b%យបc� ក�T Gន$គមន} ( ) ( )g x f x x= − 5Gន$គមន}ខ�ប�

�H�. Gន$គមន} f ក�:���� ℝ ផ�|ងx� �� ( 4) ( 4) ( )f x f x f x+ + − = ច�q� គប� x∈ℝ�

b%យបc� ក�T f 5Gន$គមន}ខ�ប�

�H4. �+9: XYZ �នម$�;កង �ង� Y ន�ង 2 06228, 0YY ZX == � ប�ម$� Y �!�;ចក5 Hច�;:ក

.*�BC /� ជ�ប XZ �ង� P ន�ង Q <យ ; ; ;X P Q Z .Y��?@ម��<ប��� XZ �

គ:(���� ( ) ( )PY YZ QY XY+ + �

�HH. �នពM$7 P �ន/� កទ+W ;/��ន�ក~:k/2ច�ង 9មk ប� Z 5ច�ន�នក$�ផ8�ច;/�

5 2006 1Z Z+ = (� ( )2 0P Z = � គ:(ផ�ធnបk (1)

( 1)

P

P − �

�H=. ប� , ,x y z 5ច�ន�នព��� �ជ��ន;/� 2 2

22 2

3x y x y

k ky xy x

= + + +

� 0ក����G��ប0��ន k .

�HW. គlO �+9: ABC �នម$�b._ចQ�ងប+� b%យបc� ក�Tk

ក) 9a b ch rh h+ + ≥ , ;/� , ,a b ch h h ក�ព.� �+9: ន�ង r 59�0ង�ង�L0 \កកC$ង

ខ) 1 1 1 2

a b cm m m R+ + ≤ , ;/� , ,a b cm m m ម/rន �+9:ន�ង R 9�0ង�ង�L0 \ក {

�HZ. ក) យក 1 2 3, ,x x x 5p..ម+90k 3 2 2 3(2 3) (2 9) 2 3 7 0m x m m xx m m− + + − + − + − =

0ក����ធ�ប�ផ$� ន�ង�����2ចប�ផ$��ន 2 2 21 2 3 1 2 3x x x xx xP + + +=

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Page 19

ខ) គlO , , 0a b c > ផ�|ងx� ���ក~ខ:� 1abc =

0ក�����2ចប�ផ$��ន 2 2 2 2 2 2

bc ca abP

a bb a c c b a a c bc+ + += + + �

�H]. គlOច�$9: ABCD @មច�ន$ចកi& � K �នGង-�� ទ!ង BD .ង�ប(� ��b.ប5ម�យ AC

ប(� ��ន�9�� AD �ង� E �

b%យបc� ក�T ( )CE ;ចកច�$9: ABCD 5ព+0;ផCក�ន�ផ� កs.*�BC �

�Ha. គlO.�$+� 1 2 2; ; ...; nu u u ផ�|ងx� ��k

12 2 1

1

2

1& ( )

200

0 1,

.

2 1

1

61

i

n nn x

n n

nn

i

i n

u uu

u u

i

u

f x

S nu

++ −

+

≠ ∀ =

=− +

=

=

=

=∑

� គ:( 2

1

1n

ii

Au

f=

=

∑ �

�Hd. គlO.�$+� { }nU ក�:��<យk 0

21

2

3 8 1 ;n n nU U U

U

n+

=

= + + ∈

� 0ក��ទ2#�ន nU �

�=K. 0ក គប�បi& ច�ន�នគ�� n;/� 2 2006n + 590 ,ក/ �

��ម��គជ �យក��ង����ក�!!��ម��គជ �យក��ង����ក�!!��ម��គជ �យក��ង����ក�!!��ម��គជ �យក��ង����ក�!!