teacher hort siath maths exercises
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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$ច S ?�� BD M�យ BS DQ= ប(� ��9��@ម S b.ប AB ជ�ប AC �ង� T �
បRS ញT AT RC= �
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�a. ក) b%យបc� ក�Tk 5 102 7 8 2+ <
ខ) គlO ,a b∈ℕ b%យបc� ក�To[ង��ចច�ន�នម�យកC$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
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3
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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:
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[ ]AC ន�ង [ ]AB � ប(� �� 1( )C K 9��ប(� �� ( )AC �ង� 2B , ប(� �� 1( )B K 9��ប(� �� ( )AB
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ខ) យក ,x y 0nងBC 50R� .�ម$�កC$ងព+0�នពM$9:ន�យ̂�ព+0 1D ន�ង 2D � <យ/\ងT
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0ង�ង�L0 \កកC$ង� <យ/\ងT �+9: BIO ;កង� b%យបc� ក�Tk
ក) � 090BIO =
ខ) r p c= − ( r 59�0ង�ង�L0 \កកC$ង p កន8�ប0 �� � �+9: ABC )
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Page 17
គ) 2 2 22 2 3b c ac bca + + = +
�4=. ក) គlO �+9: ABC �ន tan atan , , t nBA C បង-��,ន5.�$+�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 + + +=
បេ�ងៀនេ�យេក�គ� ហ៊ត សីុ�ត
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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+
=
= + + ∈
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�!!
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