WholeIssue 33 7 - CMS-SMC · 2019. 8. 2. · 7 / * ,p,!
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Transcript of WholeIssue 33 7 - CMS-SMC · 2019. 8. 2. · 7 / * ,p,!
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Qy1
Q 9C8�93A<. T 2-8<.m = 1
2(x5+y5)
k� T
x5 < y5QR9C8L*
x5 < m < y5k~FH* T =wID2c8 � 2-=D.�g3*�=BgR9C=7= T AC,h2rg:*-4JI12p�<; � T ,�8 � [L*
g:*-=]o:2�8:9CnLAC4J2-=]o:2|,{*@o T =1.D4 T A<;C. T 9C8�2-8<.D4J2x5
2 .y5
k-e T 83= Tx5 = y5
kR�<8p;C. T , T =7*�8<.\WK`j1Q39C8E93A<. T 2-8<.
x5 = −2x5 + 24Q3o � 9"�;
x5 = 82�.zo:9C8:I
y5 = 8k
��S ³
�e�I12�=D.J*�o:2<Q�8:9�.D4J2�o T =1.D4 T A<;C. T 9C8+= � �2-I14 T .1Q4Qx3
Qx4
Q8Q22
Q16
Qy3
Qy4
Q8k
±a*rnE9-[C2a838:2ro:2�,h*ro T =D.D4 T AC;C. T 9C8E2-=D.x1 + x2 + x3 + x4 + x5 + y5 + y4 + y3 + y2 + y1
10
=1 + 4 + x3 + x4 + 8 + 8 + (−2x4 + 24) + (−2x3 + 24) + 16 + 22
10
=107 − x3 − x4
10
k�C8�.J2-83*-8<.�I19Cn�gC.J2wo32-=�d}* T .J=6�C;R2�,{*BnE9-[C2a838:2w2-=D.
9, 42 .
x4 = x3+1Q-9C8r93A�. T 2a8<.
−2x3 +106 = 94�-* T 83= T Q
x3 = 62 .
x4 = 7k<�]8�o �2�o<; T .6�:83*-,h2-nE2-8<.6,h2-= � *-,h2a;:4m=
y3 = 122�.
y4 = 10k~±a*ro T =D.D4 T AC;C. T 9C8E9C4 T x T 83*a,}2@2a=1.zo39C8:I
1Q4Q6Q7Q8Q22
Q16
Q12
Q10
Q8k
~ k�\m±a*EA32a,P,}2�=�9CnEnE2�o32�´ T ,}A32-4b.�k j]��9C83= T o �2-4J9C83=w,}2a=6d}* �I79C83=�gR9C=7= T AC,h2-=�o32�gR2a41q
n�;C.J2-4]\�I � 2-=D.1q%�*�q�o T 4J2rn$�2 ,h*-8:xa2-4�jy,h2-=BI70 T � 4J2-=wo<;�8:9CnLAC4J2123
2�.]*�o:o T . T 9C838:9C83=�,}2.D9<;C.�k�±a*�=�9CnEnE2w.D4J9<; � �2�2|= � �2�I14 T .
123 + 132 +213 +231 + 312 +321 = 1332k
�z;R2a,�4 �2-=1;3,P.m*a.c*a;:4 T 9C83=Xq18:9<;:=E93A�.D2a8C;�= T 8:9<;:=�* �CT 9C83=�d}* T .r,{*"=�9CnEnE2"o32-=5040
d}* �I79C83=Bo:2�gR2a4Dn�;C.D2a4f,}2a=�I70 T � 4J2-=�oC;E8:9CnLAC4J21234567 �
����� ����������������������k�*�83=�,h2-=
5040gR2a4Dn�;C.J* . T 9C83=�Q~I70C*��<;R2LI70 T�� 4J2<Q�g:*-4Dn T
1Q2Q3Q4Q5Q6Q7Q
*�g3g:*-4m* � �/.5040/7 = 720
d�9 T =�ID9CnEnE2p;:8 T . �2<QR,}2pn��2anL2p8:9CnLAC4J2�o:2pd�9 T =�I19Cn�nL2o T ¢1* T 8:2CQ3ID9CnEnE2rID2-8<.m* T 8:2CQ32�.DI k�±-2|.D9�.m*-,�2a=1.?o:9C8:I
720 · (1 + 2 + 3 + 4 + 5 + 6 + 7)
·(1 + 10 + 100 + 1000 + 10000 + 100000 + 1000000)
= 720 × 28 × 1111111 = 22399997760k
� k<\m±-2 � 9a[<*�xa2 �* �z;"�2�A32�I k jR51;3, T 2�.m.D2|2 .] R0 T , T g3gR2wg:*-4m.J2-8<.y2-8cn��2anL2B.J2-n�g:=?2 .6g:*-41qI19<;:4J2a8<.c,}2a=
250Y�n �C; T = �2-g3*�4J2a8<.LF_9C8<.D4 �2-*a,Bo32 �z;"�2�A32�I�o3*-83=Eo32a;�u � 9 T .J;:4J2-=
T o32-8<. T �C;R2a=-kz R0 T , T g:g:2Eg:*-4JI19<;:4b.|,h*Eg:4J2an T �2-4J2EnL9 T . T �2EoC; .D4m*�£W2 . �*80
Y�n���0"2�.w,h*=X2-I19C8:o32|nE9 T . T �2&�*
120Y�n���0�kC�C8rd}* T .XQ T ,C*-4D4 T/� 2|2-8rn��2anL2�.J2-n�g:=?�C;R2]51;3, T 2�.m.D2|�<; T
*+4J9<;3,!�2�.D9<;C.|,}2E,h9C8:x$�* �<T .J2-=7=X2LI19C83=D.J*-8<.D23k]±a*+I19C83=�9CnEn�*a. T 9C8�o � 2-=7=X2a8:I12�o:2�I12.�[CgR2ro32 � 9 T .J;:4J2ro �2agR2a8:oEo:2�,{* �CT .D2a=�=�2@oC; � �2-0 T ID;3,h2:k���,{,h2@2-=D.zo39C838��2�2@g3*�4f,{*�d�9C41qn�;3,h2
c = 10+v
20
Q�9"�;v2-=D.�,{* �<T .J2-=7=X2w2-8pY�n���0c2�.
c,{*�I19C83=�9CnEn�*a. T 9C8c2-8r, T .D4J2a=
g:*-4100
Y�n k6�-*�I70C*�8<.��C;R2LI12p£W9<;:4Dq1,!�*3Q�,h2�, T .D4J2Lo � 2a=�=�2-8:ID2 � *-;C.��CQ/�����CQ~I19CnEA T 2-89C8<.1q T ,�o �2agR2a83= �2r2a83=X2anLA<,}2�gR9<;:4y,}2 � 9-[<*�xa2 ������ ����������������������k
±-2-=yI19C83=�9CnEn�*a. T 9C83=64J2-=7gR2-IJ. T/� 2a=fo:2@ R0 T , T g:g:2wo<;:4m*�8<.6I70C*�ID;:8:2|o32-=6nL9 T . T �2a==X9C8<.
c1 = 10+ 8020
= 142�.
c2 = 10+ 12020
= 16\}2-8�, T .D4J2-=�g:*-4
100Y�ncj�k-��9Cn�nL2
T ,�g:*-4JI19<;:4b.BoC;:4m*-8<.BI70C*�ID;:8:2po:2a=wnL9 T . T �2a=125
Y�nGQ:=�*pI19C83=�9CnEn�*a. T 9C8�.D9�.m*-,h2c2-=D.1, 25(c1 + c2) = 1, 25 × 30
kw 3; T =X�<; � ;:8_, T .D4J2 � *-;C.0, 80
�CQ6,}2GI19 �;C.�.D9�.m*-,]o:2 R0 T , T g3gR2@2-=D.
0, 80 × 1, 25 × 30 = 30�:k
��S��
��*a,}ID;3,h9C83=rnE* T 8<.J2-83*-8<.�,}2LI19 �;C.w.D9�.m*-,?o32�51;3, T 2 .m.J2:kfU�,yd}*-;C.�o:9C8:I �2�.J*�AC, T 4�=7*�CT .D2a=�=�2:k���9CnEnE2p2 ,{,h2c*pg3*�4JID9<;:4J;�,[email protected]*�£W2 .�o3*-83=B,}2rn��[email protected]�g:=|�<;R2L R0 T , T g3gR2<Q=�* �<T .J2-=7=X2+2-=D.��2-x *-,h2 �*+,h* �<T .J2-=7=X2�nL9a[C2-838:2Go:2H R0 T , T g3gR23k?��9C8<.D4m* T 4J2-nE2-8<. �* I12�C; � 9C8"gR9<;:4J4m* T .�I14J9 T 4J2CQ�,{* �<T .J2-=7=X2�nE9-[C2a838:2L=1;:4w, � 2a83=X2anLA<,}2Lo<;�g3*�4JID9<;:4m=r8 � 2-=D.g:*-=&�2�x *a,}2 �*@,h*rnE9-[C2a838:2po32-= �CT .D2a=�=�2-=]=7;:4f,}2@g:*-4JI19<;:4m=ak�±a* �<T .J2-=7=X2@nE9-[C2a838:2c2-=D.o:9C838 �2-2wg:*-4
v = 250/(t1 + t2)Q�9"�;
t12�.
t24J2ag:4 �2-=�2-8<.J2-8<.~,h2-=�.J2-n�g:=fo32wg3*�4JID9<;:4m=
\�2a8L032 ;:4J2-=Jjyo:2rI70C*�ID;:8:2ro:2a=]nL9 T . T �2a=-k3�]8�*
t1 =125 ���80 ������� =
125
80
0 2 .t2 =
125 ���120 ������� =
125
120
0~k
�9C8:I`Q
v =250
Y�n125
80
0+
125
120
0 = 96Y�n���0rk
±a*LI19C83=�9CnEn�*a. T 9C8 o:2c51;3, T 2 .m.J2E2-=D.|o:210 + 96
20= 14, 8
, T .D4J2a=�g:*-4100
Y�n k��-*I19C83=�9CnEn�*a. T 9C8p.D9�.m*-,h2ro � 2-=7=X2a8:I12@*a;:4m*po39C8:I��2�.��2ro:2
14, 8 × 2, 52 .f,}2rID9 �;C.f.J9�.J*a,
o:2|51;3, T 2�.m.D2@2a=1.0, 8 × 14, 8 × 2, 5 = 29, 60
�3k�C83=�2-nEAC,h2CQa,h2 � 9a[<*�xa2w*a;:4m*�ID9 �;C.��2|2-8cIJ*�4DAC;:4m*-8<.
30, 00+29, 60 = 59, 60�:k
� ( , �$#�%4�;& I#)( ,$0=0 #)('(+*,),+/ *, &)*, �$#�� 5 � , M�� \b±-2-= �*�xa2-=yn�;3,P. T g3,h2-=ak j � ;���Kw*�9 �;C.yK-S���S|*a;���nE* T \ T 8:ID,{;:= T/� 2anL2a8<.bj~K-S�S��3Q�5�2-*-8*c2 ;
5d�9 T =z, � �*�xa2@o:2�=�*��3,{,h2@�R,h* T 4J23k � ;L�rn�* T *a;����@*�9 �;C.z\ T 8:ID,P;:= T � 2-nE2-8<.bjfK-S�S��CQ
5�2-*-8E*r2 ;4d�9 T =z, � �*�xa2ro32@=7*��:,P,}23kCZ:4J9<; � 2a4f,h*roC*a.J2ro:2�83* T =�=7*�8:ID2@o:2rI70C*�ID;:8~k
¡|9�.J2#�|g3*�4 �*�xa2CQ@9C8'2-8<.J2-8:o�,h*�o �2a�:8 T . T 9C8�;:=1;R2a,P,}2��<; T 2a=1.+,h2�8:9CnLAC4J2o � *�838 �2-2-=BID9CnEgC,!�2�.D2a=��2-I19<;3,!�2�2a=�o:2ag3; T =],{*r83* T =7=�*-8:I123k����� ���������������������5�����6����7(8���:9;� $N��<$68���=���;���K�$`k
�]8"o �2-= T x 8:2a4m*Lg3*�4J
, � *�838 �2-2Lo32�83* T =�=7*�8:ID2Eo:2c5�2a*�8 2 .wg:*-4C
, � *-838��2�2Eo:283* T =�=7*�8:ID2ro:2@��,{* T 4J2:k�¡w9<;:=�*-,P,}9C83=�IJ*-,hID;3,}2a4?=7;:4?o:2 ;�upo3* .D2a= Q<,h*@g:4J2an T �2-4J2rI7039 T = T 2*�4DA T .D4m* T 4J2-nE2-8<.�2-8<.D4J2�,}2L��K�*�9 �;C.�K-S���S�2�.w,}2���nE* T K-S�S��3Q�=�9 T .B,}2L��KLo �2�ID2-nEA34J2K�S���SCQ32 .y, � *-;C.D4J2@I7039 T = T 2�*�4DA T .D4m* T 4J2-nE2-8<.?2-8<.D4J2�,}2��@nE* T 2 .y,}2@���@*�9 �;C.]K-S�S��CQC=�9 T .,}2rK�2a46£}; T ,{,h2 .]K-S�S��:k
U�,R2-=D.?ID,{* T 4f�<;R2�,h2-=z*-838 T/� 2a4m=�* T 4J2a=]o:2B5�2a*�8�2�.?�R,h* T 4J2�=�9C8<.6,}2���nE* T 9<;�,h2@��K*�9 �;C.XQ:=�*-83=|�C; � 9C8+gC; T =7=X2rg:4 �2�I T =X2a4z,}2-�C;R2 ,�2a=1.z,h2��<;R2a,bk�¡w9<;:=w* � 9C83=|o32a;�u�ID*-=X2a=&�*I19C83= T o �2-4J2a4�kD� ����k�5�2a*�8E2-=D.?8 �2�;:8L�@nE* T 2�.z�R,h* T 4J2<Q<;:8���K@*�9 �;C.�k
±-2p��Kco �2�ID2-nEA34J2pK�S���SCQ3, � �*�xa2co32�5�2a*�8L2a=1.1989 − J
2�.?, � �*�xa2co:2r��,{* T 4J2r2-=D.1989 − C
k�¡w9<;:=]* � 9C83=
1989 − J = 5(1989 − C)k \WK`j
±-2�K�2a4�£}; T ,P,}2�.?K-S�S��CQ�, � �*�xa2�o:2�5�2a*�8p2-=D.1992 − J
2 .�, � �*�xa2�o:2���,{* T 4J2�2a=1.1991 − C\}g3*�=
1992 − Cg:*-4JI12��<;R2c, � *�838 T � 2-4m=7* T 4J2�o:2��R,h* T 4J2c8 � 2a=1.�g:*-=�2a8:I19C4J2c*�4J4 T � �2p2-8
K�S�S<�`j�k�¡w9<;:=�* � 9C83=1992 − J = 4(1991 − C)
k \W�`j
��S�M
�C8v=X9<;:=D.D4m* [<*-8<.p\�K�j@o32"\���j1Q?9C8H.D4J9<; � 23 = C − 1981
�fo39C8:IC = 1984
2�.J = 1964
k�C8E2 � 2 .XQ�,}2r��Kro �2�ID2-nEA34J2cK-S���S3Q�5�2a*�8�* � * T .
25*-83= QC=�9 T .?I T 8:�pd�9 T =z, � �*�xa2ro:2
��,{* T 4J2:k�±-2rK-2-46£}; T ,{,h2 .zK�S�S<�<Q�5�2a*�8�* � * T .28
*-83= QC=�9 T .?�<;:*a.D4J2�d�9 T =z, � �*�xa2@o:2@��,{* T 4J2:kD� ���:k�5�2a*�8E2-=D.?8 �2�;:8���K@*�9 �;C.z2 .z�R,h* T 4J2<Q<;:8E�rn�* T k
±-2G��K+o �2�ID2-nEA34J2GK-S���S3Q~,}2a=��*�xa2-=co:2�5�2-*-8H2 .���,{* T 4J2�=�9C8<.�,h2-=rn��2-nE2-=c�<;R2o3*-83=],h*rIJ*�=BK<Q32 .?8:9<;:=]* � 9C83=z, � �2��<;:*a. T 9C8E\WK`j�k
±-2GK�2a4|£}; T ,{,h2 .rK�S�S<�<Q6, � �*�xa2+o:2�5�2-*-8_2-=D.1991 − J
2�.�, � �*�xa2Go32+��,{* T 4J2+2-=D.1992 − C
k�¡w9<;:=]* � 9C83=
1991 − J = 4(1992 − C)k \W�`j
Z:9<;<£W9<;:4m=@2-8G=X9<;:=D.D4m* [<*-8<.�\�K�j�o:2E\���j1Q�9C8 *2 = C − 1977
��I � 2-=D.1q%�*�q�o T 4J2<QC = 1979
2 .J = 1939
ke T 83= T Q�,}2r��Kro �2-I12anLAC4J2cK�S���SCQ�5�2a*�8E2 .?��,{* T 4J2r9C8<.?4J2a=�g:2�IJ. T � 2-nE2-8<.
502 .
10*�83=�Q�.J*-8:o T =B�<; � *a;+K�2a46£}; T ,{,h2 .zK�S�S<�<Q T ,{=�9C8<.52
2 .13
*-83=-kU�,3[�*co39C8:I]o:2 ;�uc=�9<,{;C. T 9C83=\�
¡w* T =7=�*-8:I12@o32|5�2a*�8 � �@nE* T K�S�Ma³RQ¡B* T =�=7*�8:ID2@o:2r�R,h* T 4J2�� ��K@*�9 �;C.]K�S��a³RQ
9<;¡w* T =7=�*-8:I12@o32|5�2a*�8 � ��K@*�9 �;C.]K�S���SCQ
¡B* T =�=7*�8:ID2@o:2r�R,h* T 4J2�� �@nE* T K�S<�-S:k� ( , �$#�%4�;& I#)( ,$0=0 #)('(+*,),+/ *, &)*, �$#�� 5 � , M
� � \m±a*cgR9<;3,h2px#�2-9Cn$�2�.D4J2:k jf��8:2r�Rx�;:4J2cg3,{*�8:22-8Hd�9C4DnE2Go � ��;3dw2a=1.ro �2 , T n T .��2�2+g3*�4r�C;:* .D4J2*�4JIJ=�o:2@I12a4JID,}2a=�o �2a= T x 8��2-=]g3*�4
ABQBF
QFE2 .
EAn T =~A39<;C. �*]A:9<;C.�o32?,{*zd}* �I79C8 T 8:o T �C;��2-2
g:*-4~,{*B�Rx�;:4J2&�*wo34J9 T .D23k��-*�I70C*�8<.6�<;R2�,}2B4m*�[C9C8AO
2a=1.�o32z,h9C8:x�;R2a;:41Qao �2 .J2-4Jn T 8:2-4�, � * T 4J2]o:2
,h*���x�;:4J23k
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����� ����������������������k¡|9�.J9C83=|�<;R2r,}2r=�2�x nE2-8<.
AC =√
2k�± � * T 4J2co:2r, � ��;3d
A2a=1.��2-x *-,h2 �*c, � * T 4J2
oC;�o32-n T q7I12a4JID,}2ABO
\�o32|4m* [C9C81j��G, � * T 4J2|oC;�=�2�IJ.J2a;:4
BAF\}4m*�[C9C8
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n =a2 + b2
5= 5k2 +8k +5t2 +6t+5 = (k +2t+2)2 +(2k − t+1)2
Qd}4J9Cn�>r0 T I70|>r2�=�2�2z.10C* .
n = c2+d2 Q�>r032-4J2c = k+2t+2
*-8:od = 2k−t+1
kZ�032a4J2ad�9C4J2<Q T d
(a2 + b2)/5 = nQ:>r032a4J2
aQbQn ∈ � Q:.1032-8 n = c2 + d2d�9C4?=�9CnL2
cQd ∈ � k
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/ � / � � 5 # $ &�A #<�('�� /�0�, ��Mp { �^~ k ·1$)�,9<� �"��� ���,+���¼+6� &"2 �D4�����, �+¸ ��¼+ � �0� ��� � �K�#��$-�)�������+¸ ��¼+���b¹�� $ ���"� �-����2
´ T � 2-81001
4J2-*a,:8C;:nLA32-4m=zg3,{*�ID2�op*�4J9<;:8:op*@I T 4JID,h2�=7;[email protected]*a.?2a*�I70c8C;:nLA32-4T =6.1032w*�4 T .10CnE2 . T IynE2-*-8p9<d T .m=f8:2 T xa03A:9<;:4m=z9C462 ,h=�2 T .J=6.b>r9@8:2 T xa03A39<;:4m=?*�4J2|2��<;:*-,WQg:4J9 � 2�.10C*a.?*a,{,3.1032�8C;:nLA32-4m=�*�4J2@2��<;:*-,bk����� ������� � �.-<2�/+� 9�9 �"��� �6�����6�¥¸ �����$'������� %%º�� � � � � ���6���07�� � � � �*��� � ��������$`k
±-2 .x1
Qx2
Q. . .
Qx1001
o32-8:9�.J21001
4J2a*-,:8C;:nEA:2a4m=]g3,{*�ID2�op*�4J9<;:8:o�*@I T 4JID,}2T 8+*p�<u�2-oGo T 4J2-IJ. T 9C8~kp�C9C4BID9C8 � 2-8 T 2-8:ID2CQR>r2p4J2-x *�4JoL.1032 T 8:o T I12a=
1Q2Q. . .
Q1001*�= T 8<.J2�xa2a4m=�nE93oC;3,h9
1001\ T 8�g:*-4m. T ID;3,h*-4XQ
x0 = x1001*-8:o
x1002 = x1j�k
�C9C462-*�I70iQ<2 T .1032-4
2xi = xi−1 +xi+1\ T d
xiT =y.1032|*-4 T .10CnL2�. T IynE2-*-8�9<d T .m=
8:2 T xa03A:9<;:4m=Jjz9C4]2a,{=X2xi−1 = xi+1
� T 8�2 T .1032-4]ID*-=X2<Q |xi+1 − xi| = |xi − xi−1| k��9C83=�2��<;R2-8<.J, [�Q T d<>r2B,}2�.δ = |x2 − x1| Q-.1032a8@>r2|0C* � 2 |xi+1 − xi| = δ
d�9C4�*-,P,ik
¡|9�.J2�.10C*a. 1001∑i=1
(xi+1 − xi
)= x1002 − x1 = 0
Q:= T 8:I12x1002 = x1
kw�C*�I70.D2a4Dn 9<d6.10 T =@=1;:n T =@2 T .1032-4
δ9C4 −δ
k+U�dδ 6= 0
Q�.1032a8G.1032�8C;:nLA32-4�9<dfgR9C= T . T � 2.D2a4Dn�= T 8+.1032�=1;:n n�;:=D.�2��<;:*-,~.1032�8C;:nLA32-4|9<df8:2-x *a. T � 2�.J2-4JnE=@\h= T 8:I12p.1032�=7;:nT =
0j7kcµ:;C.?.10 T = T = T n�gR9C=7= T AC,h2CQRA32�IJ*-;:=�2r.1032c=1;:n 0C*�=w*-8+93o3oL8C;:nLA32-4B9<d�.J2-4JnE=ak
)G2rID9C8:ID,{;Ro32�.10C* .δ = 0
k
³3��K
Z�032a4J2ad�9C4J2<Qy*-8�[G.b>r9 8:2 T xa03A39<;:4 T 8:xE8C;:nLA32-4m=r*�4J2L2��<;:*-,WQ6*�8:oG.10<;:=�Q6*-,P,f9<d.1032@8C;:nLA32-4m=
x1Qx2
Q. . .
Qx1001
n�;:=1.zA32r2-�C;:*a,mk��3 ,$0�, A ,$0�, &�A #< (?2=# 0=0�, 2�&?�$#�%��'& �#)('� � ��� 5 I&�& , >"M
p { � � k ·1$ �,9<� ��� � ���r¸ $�K�)� �����=�:��� $�� � �����$$���" � �K�#��$-�)����� � .)& ��� .��.��;�������;��� � ��2 �"��� �� � � &�62
´ T � 2-8�.D4 T *-8:x�,}2ABC
*-8:oDE ‖ BC
Q�> T .10D ∈ AB
*-8:oE ∈ AC
k�4J9Cg
gR2a4Dg:2-8:o T ID;3,{*�4m=fd}4J9CnD
*-8:oE
.J9BC
Q-nL2-2 . T 8:xBC
*a.F
*-8:oK
Q�4J2-=7gR2-IJ. T/� 2 ,/[�kU�d [ABC]
[DEKF ]=
32
7
Q3o32 .J2-4Jn T 8:2�.1032@4m* . T 9 |AD||DB|
k����� ������� ��� D,�?�0���� /�$$���+¼�� $�� � �K�#��$-�)�#�=�=��» ���6» ���,¸ ������ � �D�0$ �5� � �'$ ����� �������3k
±-2 .AH
A:2�.1032_*-,/. T .J;Ro32Hd}4J9CnA
.D9BC
Q�nE2�2�. T 8:xDE
* .Lk �<4J9Cn
[ABC]
[DEKF ]=
32
7
Q T .fd�9<,P,}9�>@=f.10C*a. 12
AH · BC = 327
DE · LH��.10C* . T =�Q
AH
LH=
64
7· DE
BC
k \WK`j
� T 8:ID2DE ‖ BC
Q�.D4 T *-8:x�,}2a=ABC
*�8:oADE
*-4J2@= T n T ,{*�4`k�Z�0<;:= Q
AD
AB=
DE
BC
k \W�`jZ�032rID9C8:o T . T 9C8
DE ‖ BC*-,{=X9 T nEgC, T 2-=?.10C* .
AH
LH=
AB
DB
k \W�`j
�a;RA3=D. T .J;C. T 8:xp\W�`jz*-8:oG\���j T 8<.J9�\WK`j1Q:>r2p93A<.m* T 8 AB
DB=
64
7· AD
AB
k���9C8<. T 8C; T 8:[email protected]=X9<, � 2�d�9C4 AB
DB
,h2-*�o3=?.J9c.1032�d�9<,P,}9�> T 8:x�2-�C; T � *-,h2-8<.?=D.J* .D2anL2a8<.J=��AB2
DB2=
64
7· AD
AB· AB
DB=
64
7· AD
DB
Q(
AD + DB
DB
)2=
64
7· AD
DB
Q(
AD
DB+ 1
)2=
64
7· AD
DB
Q(
AD
DB
)2− 50
7
AD
DB+ 1 = 0
Q
4J2-=1;3,P. T 8:x T 8 AD
DB=
1
7
9C4 AD
DB= 7
k
...
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A
B C
D E
F H K
L
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p { ��� k ·1$ �,9<� ��� � ���r¸ $�K�)� �����=�:��� $�� � �����$$���" � �K�#��$-�)����� � .)& ��� .��.��;�������;��� � ��2 �"��� �� � � &�62
evg:* T 4?9<dR.b>r93q7o T x T .f8C;:nLA32-4m=B0C*-=z.1032�d�9<,{,h9�> T 8:x|g:4J9Cg:2-4b. T 2a= �K3kwZ�032@=1;:n�9<dR.1032�d�9<;:4zo T x T .J= T =
17k
�3kwZ�032@=1;:n�9<dR.1032|.b>r9�8C;:nEA:2a4m= T =89
k�3kwZ�032@g34J93oC;RIJ.z9<dR.1032�d�9<;:4?o T x T .m= T =
210k
³�kwZ�032@g34J93oC;RIJ.z9<dR.1032�.b>r9p8C;:nLA32-4m= T =1924
k
�2 .J2-4Jn T 8:2�.1032�.b>r9p8C;:nLA32-4m=ak
����� ������� � �� ������$)�� ��2�/ ��� �=�?�0�)�<� � � ���6�b» � � �<� �|k� T 8:ID2
1924 = 22 · 13 · 37 Q�.1032a4J2�*-4J2�21u-*�IJ.J, [12
gR9C= T . T � 2�o T/�<T =�9C4m=z9<d1924
Q83*�nE2a, [
1Q2Q4Q13
Q26
Q37
Q52
Q74
Q148
Q481
Q962
Q�*-8:o1924
k�Z�0<;:= Qz.10329C8C,/[pd}*�IJ.D9C4 T ¢1* . T 9C83=]9<d
1924*-=?.1032�g34J93oC;RIJ.z9<d�*@g3* T 4?9<d:.b>r93q�o T x T .f8C;:nLA32-4m=]*�4J2
26 · 74*-8:o
37 · 52k?U}.fd�9<,{,h9�>@=?d}4J9Cn&*-8�[E9C8:2@9<dR.1032��R4m=D.y.10C4J2�2�g:4J9Cg:2-4b. T 2a=z.10C* .
.1032@8C;:nLA32-4m=?>r2�=�2�2-Yc*-4J237
*�8:o52
kº ���� $�¼��yU�dR>r2@*�=7=7;:nE2r9C8C, [�g:4J9Cg:2-4b. T 2a=w�r*-8:o�³RQ�>r2@8:2-2�o�8:9�.zn�*�Y�2@*-8�[
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y=�* . T =1dP[
x + y = 89*-8:o
xy = 1924Q T .fd�9<,{,h9�>@=f.10C*a.y.1032@8C;:nEA:2a4m=�*�4J2
37*-8:o
52k
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� T 8:o�*_�C; T 8<. T IEg:9<,/[C8:9Cn T *-,f(x)
=7;RI70v.10C*a.XQ T dn
T =L*_g:9C= T . T/� 2 T 8<.J2�xa2a4I19C83= T =D. T 8:x+9<dz.1032 o T x T .
74J2-g:2-* .D2-o
k. T nE2-=�Qy.1032-8
f(n)I19C83= T =D.J=�9<dz.1032 o T x T .
74J2agR2a*a.J2�o
5k +3. T nL2a=-k�\��<9C4y27u-*-nEgC,}2<Q
f(77) = 7777777777777k j���9Cn�g:*-4J2
> T .10�Fv����M��m������M �<��M��?�<k����� ������� � ���'$�¼ �������D���� � ��� �"� ����b» � � � � � �;���6����7 ��� ��� �*��� ����������$`k
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A:2r*pg:9<,/[C8:9Cn T *-,:> T .10�.1032po:2a= T 4J2�oEg:4J9Cg:2-4b.�[�k�U�dn
T =B*cgR9C= T . T � 2T 8<.J2�xa2a4cID9C83= T =1. T 8:x+9<d?.1032Go T x T .
74J2-g:2-* .D2-o
k. T nE2-=�Qy.1032a8
n = 79(10k − 1)
k)G2 4J2-�C; T 4J2
f(n).D9vID9C83= T =1.c9<d].1032"o T x T .
74J2-g:2-* .D2-o
5k + 3. T nE2-=��z.10C* . T = Q
f(n) = 79
(105k+3 − 1
) k� T 8:ID2
n = 79(10k − 1)
Qa>r2�0C* � 210k − 1 = 9
7nQ<*-8:or.10<;:=
10k = 97n + 1
kZ�032a8
f(n) = 79
(1000 · (10k)5 − 1
)= 7
9
(1000 · (9
7n + 1
)5 − 1) k
Z�0<;:=�Qf(x) = 7
9
(1000 · (9
7x + 1
)5 − 1) k
³3���
)G2rIJ*�8c>@4 T .J2f(x)
T 8�*r=�9CnL2`>r0C*a.f8 T ID2-4yd�9C4Dn �f(x) = 7000
9
(97x + 1
)5 − 79
= 70009
((97x + 1
)5 − 1)
+ 70009
− 79
= 777 + 70009
((97x + 1
)5 − 1) k
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T = T 83=XI14 T A32�o T 8*rI T 4JID,h2
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Γk]± T 8:2-=
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Y*-8:o
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6
A B
CD
P
W
X
Y Z
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kH�]4�o:9G.1032X[ � )G2L8:2-2�o .J9"g3;C.@9<;:4�= T nEgC, T ��IJ*a. T 9C8�0C*a.@9C8A3*�I7Ya>@*-4Jo3=B032a4J2r\{, T Y�2|>r2@o T oE2a*�4m, T 2-4�j6*-8:o�g3;3,P,:.10 T =?.D2a4Dns*�g3*�4b.�k6)G2@xa2�.
1
k(k + 1)=
(k + 1) − k
k(k + 1)=
k + 1
k(k + 1)− k
k(k + 1)=
1
k− 1
k + 1
k
³3��S
¡|9�>�QCg34J93I12-2�o T 8:x�*�= T 8E9<;:4z21u-*�n�g3,h2@*�A:9 � 2CQ�>r2@xa2 .2007∑
k=1
1
k(k + 1)=
2007∑
k=1
(1
k− 1
k + 1
)= 1 − 1
2008=
2007
2008
k
eV= T n T ,{*�4 T o32-*r*-g:gC, T 2-=?.D9�g:4J93o<;RIJ.J=akfZ�032r=D.J*-8:o3*-4JoE8:9�.m*a. T 9C8�d�9C4?g34J93oC;RIJ.m=T 8�n�*a.1032anE* . T IJ= T = n∏
k=1
akQ~>r0 T I70"4J2ag:4J2a=X2a8<.J=�.1032�g34J93oC;RIJ.
a1a2a3 · · · ank�e?8
27u�g:4J2a=�= T 9C8�, T Y�2 20∏k=1
k + 1
k
IJ*�8EA32|>@4 T .m.D2a8E9<;C.?*-=
20∏
k=1
k + 1
k=
2
1· 3
2· 4
3· · · 20
19· 21
20
k
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21k
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k=1
F (k + 1)
F (k)
9C4 n∏
k=1
F (k)
F (k + 1)
Q
>r032-4J2<Q3*-=BA32ad�9C4J2<QF (k)
T =]=X9CnE2�dh;:8:IJ. T 9C8E9<dkkzU�d:>r2|>@4 T .J2r9<;C.y.1032@g34J93oC;RIJ. T 8
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U � ,P,RID,}9C=�2�> T .10c*|d�2�>�g:4J93A<,}2anE=fd�9C4~[C9<;[email protected][@[C9<;:4m=X2 ,{dJk6\hZ�032|,h*-=1.f9C8:2�> T ,P,4J2��<; T 4J2L*a.|,h2-*-=1.�9C8:2�.D4 T xa9C8:9CnE2 .D4 T I T o:2a8<. T .W[~k j ��9<;�n T xa0�.�2 � 2a8",h9393YE.10C4J9<;Rxa0.10 T =]nL9C8<.10 � =�FH*�[:032an&g:4J93A<,}2anE=?.D9396¹
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n7 − nn�;:=D.z*-, >@* [C=
A:2�*@n�;3,/. T gC,}2@9<d7k
�3k@� T 8:op.1032�=7;:n1√
1 +√
3+
1√3 +
√5
+1√
5 +√
7+ · · · +
1√2005 +
√2007
k
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k=1
k2 + 4k + 4
k2 + 3k + 2
k
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cos 256
k
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T 8<.J2�xa2a4nk
{ kL±-2 .S
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o T =1. T 8:IJ.�=7;RAC=X2�.J=A1
QA2
Q. . .
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iQ1 ≤ i ≤ k
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9C4Bi = S − Ai
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=1;RI70 .10C*a.|>r2�ID*-8H*-, >@* [C=I703939C=X2
Bi=X9r.10C*a. ⋃
1≤i≤k
Bi = Sk
� kL±-2�.ABCD
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ABQBC
QCD
Q�*-8:oDA
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ABCDky��039�>�.10C*a.
EH*-8:o
FG*-4J2@g3*�4m*a,{,h2a,bk
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T 83= T o:2r.1032�I T 4JID,h2CQ�A<;C.w8:9�.B* . T .m=�ID2-8<.D4J2<Q��:8:ogR9 T 8<.J=
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ABCDk
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f(x) < f(y)T d
x < yQ<*�8:o
\ TPT jf
(2xy
x + y
)=
f(x) + f(y)
2
d�9C4?*a,{,xk
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d�9C4?=�9CnL2 � *a,{;R2@9<dxk
� k�eHx 4m*-gR0G
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D=X9L.10C* .�.1032-4J2 T =�*-8"2-o:xa2EA32 .b>r2-2-8�.b>r9 � 2-4b. T ID2-= T dy*�8:o 9C8C, [ T d6.1032
o T�� 2a4J2-8:ID2EA32 .b>r2-2-8�.1032 T 4�,{*�A32a,{= T = T 8D
kp��039�>�.10C*a. T dn
T =|=7; � I T 2a8<.J,/[�,{*�4Jxa2<Q>r2rIJ*�8�*a,/>@*�[3=z�:8:o�*rx 4m*�g:0c> T .10
n� 2-4b. T ID2-=f>r0 T I70 T =]8:9�.zx 4J2-* .�k
³:K�K
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0C*�=]o T =D. T 8:IJ.y4J2-g34J2-=�2-8<.m*a. T 9C83=]*-=z*�=7;:n9<d�.b>r9�=��C;:*-4J2-=B9<d�g:9C= T . T/� 2 T 8<.J2�xa2a4m= �
n = a2 + b2 = c2 + d2 kB :4J9 � 2�.10C*a.n
T =*cID9CnEg:9C= T .D2�8C;:nEA:2a4�k{ k?UW8�*|.D2�.D4m*�032-o34J9C8
ABCDQ�,h2 .
rA:2|.1032�4m*�o T ;:=�9<d:.1032 T 83=�I14 T A:2-o�=�g:032-4J2<QC*-8:o
,}2�.rA
QrB
QrC
Q6*-8:orD
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1√r2
A − rArB + r2B
+1√
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C
+1√
r2C − rCrD + r2
D
+1√
r2D − rDrA + r2
A
≤ 2
r
k�C9C4�>r0C*a.zY T 8:o�9<dR.D2�.D4m*�032-o34J9C8Eo:932a=B2-�C;:*a, T .�[�039<,}o �� k@Z�032�I T 4JID,}2a=
Γ1*-8:o
Γ2T 8<.J2-4m=�2�IJ.w2a*�I70+9�.1032a4B* .
M*�8:o
Nk�e�=1.D4m* T xa0�.�, T 8:2
g:*-=�= T 8:xc.10C4J9<;Rxa0M
T 8<.D2a4m=X2-IJ.J=|.1032EI T 4JID,h2Γ1
* .A 6= M
*-8:o+.1032LI T 4JID,}2Γ2
* .B 6= M
Qf=7;RI70".10C* .M ∈ AB
kvZ�032 T 8<.D2a4D83*a,�A T =�2�IJ.J9C4r9<d?.1032+*�8:x�,h2AMNnL2-2 .m=r.1032GI T 4JID,h2
Γ1T 8�.1032+gR9 T 8<.
DQ?*�8:o".1032 T 8<.J2-4J83*-,]A T =�2�IJ.J9C4r9<d?.1032G*-8:x�,}2
BMNnE2�2�.J=z.1032cI T 4JID,}2
Γ2* .
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CD*-=Bo T *�nE2 .J2-4
g:*-=�=�2-=?.10C4J9<;Rxa0p.1032�n T o:q1gR9 T 8<.z9<dR.1032�=X2-x nL2a8<.AB
k~ kG±-2 .
nA32E*Eg:9C= T . T/� 2 T 8<.J2�xa2a4|*-8:o
A = {a1Qa2
Q. . .
Qan} A:2�*E=X2�.|9<df4J2a*-,
8C;:nLA32-4m=akz� T 8:o�Q T [email protected]�=z9<dnQ-.1032B.J9�.J*a,38C;:nLA32-4f9<d3dh;:8:IJ. T 9C83=
f : A → A> T .10
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) − f(f(y)
) ≥ x − yd�9C4f*�8�[
xQy ∈ A
> T .10x > y
k�/�� � ����/�/�P�� �� � � �������%�
� k]±-2 .n
A32@*@gR9C= T . T � 2 T 8<.J2�xa2a4XQC*-8:o�,}2�.A = {(x1, x2, . . . , xn) | xi ∈ ∗
+
Qi = 1
Q2Q. . .
Qn} k
e_dh;:8:IJ. T 9C8f : A → T =�o32a�:8:2�o�*-=zd�9<,{,h9�>@= ��d�9C4f*-,P,
(x1, x2, . . . , xn) ∈ AQ
f(x1, x2, . . . , xn) =1
x1
+1
2x2
+1
3x3
+ · · · +1
(n − 1)xn−1
+1
nxn
k��039�>'.10C* .
f(C1N , C2
N , . . . , CnN) = f(2n−1, 2n−2, . . . , 2, 1)
Q<>r032a4J2Ck
n
T =].10328C;:nLA32-4z9<d
kq72a,h2-nE2-8<.?=1;RA3=�2 .m=�9<d�*�8
nq72a,h2-nE2-8<.?=�2 .XQ�d�9C4
k = 1Q2Q. . .
Qnk
³:K��
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f(x3)− f
(y3)
=(x2 + xy + y2
)(f(x) − f(y)
)
d�9C4?*a,{,R4J2a*-,R8C;:nLA32-4m=x*�8:o
yk
� k�±-2 .ABC
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*-8:o�I T 4JID;:nLqI12a8<.D4J2
Ok Z�032 T 83=�I14 T A:2-o'*�8:o�I T 4JID;:nE=�I14 T A:2-o�I T 4JID,}2a=H0C* � 2�4m*�o TPT
r*-8:o
RQ
4J2-=7gR2-IJ. T/� 2 ,/[�k�U�dP
T =�*-8V*-41A T .D4m*-4W[Vg:9 T 8<.p9<d�.1032 =�2�x nE2-8<.[OH]
Q]g:4J9 � 2G.10C* .6r ≤ PA + PB + PC ≤ 3R
k� k_e?8 T 8<.J2�xa2a4
nT =p=�* T o".J9_A32 �K� �6� T d |n| T =p8:9�.�.1032+=X�<;:*�4J2G9<d�*�8 T 8<.J2�xa2a4�k
�2 .J2-4Jn T 8:2�*a,{, T 8<.D2-xa2-4m=
m> T .10�.1032wd�9<,P,}9�> T 8:xBg34J9CgR2a4m.W[X�
mID*-8�A32|4J2-g34J2-=�2-8<.J2�o
T 8 T 8C�:8 T .J2a, [|n�*�8�[�>@*�[3=�*-=~*z=7;:n�9<da.10C4J2�2]o T =D. T 8:IJ.1��� �6� T 8<.J2�xa2a4m=�>r039C=�2zg34J93oC;RIJ.T =?.1032@=��C;:*-4J2r9<d�*-8E93o:o T 8<.J2�xa2a4�k
� � �E� ��/�/�P�� %�%� � �������%�� k?�C9C4f*-,P,RgR9C= T . T � 2�4J2a*-,R8C;:nLA32-4m=
aQbQ3*-8:o
cQCg:4J9 � 2�.1032 T 8:2��<;:*-, T .W[
∣∣∣∣∣4(a3 − b3
)
a + b+
4(b3 − c3
)
b + c+
4(c3 − a3)
c + a
∣∣∣∣∣ ≤ (a− b)2 +(b− c)2 +(c−a)2k
� } k�2�.D2a4Dn T 8:2w*a,{,�.1032wg:9<,/[C8:9Cn T *-,{=
P (X)> T .10@4J2-*a,3ID932 � I T 2a8<.J=�>r0 T I70@=�* . T =1dP[
.1032@4J2a,{*a. T 9C8(x3 + 3x2 + 3x + 2
)P (x − 1) =
(x3 − 3x2 + 3x − 2
)P (x)
d�9C4z2 � 2a4W[�4J2-*a,�8C;:nEA:2a4xk
�;� k�±-2�.ABC
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Q?*�8:oH,h2 .�U|A32 T .m=T 8:ID2-8<.D4J23k�±-2 .
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AIB, [ T 8:x
T 83= T o32p.1032c.D4 T *�8:x�,h2ABC
kcZ�032�=1.D4m* T xa0�.B, T 8:2a=w.10C4J9<;Rxa0P
g3*�4m*a,{,h2a,�.J9CA
*-8:oCB
nL2-2 .AB
* .D
*-8:oEQ:4J2a=�g:2�IJ. T � 2a, [~kfZ�032@, T 8:2�.10C4J9<;Rxa0
Pg:*-4m*-,P,}2 ,R.J9
ABnL2-2 .m=CA
*�8:oCB
*a.F
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DF*�8:o
GET 8<.D2a4m=X2-IJ.z9C8p.1032rI T 4JID;:nLI T 4JID,}2r9<d:.1032�.D4 T *-8:x�,}2
ABCk
� { kz±-2�.ak
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\h* jn ∈ [0, 10k)
�R.10C*a. T =�Qn
0C*�=�27u-*�IJ.J,/[k
o T x T .m=�\ T 8Go:2-I T n�*-,�8:9�.m*a. T 9C8�jy> T .10,}2a*�o T 8:xw¢72a4J932-=]*a,{,h9�>r2�o��
\�A�jp.1032�o T x T .J=z9<dn
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A
A1A2
B CD
G
I
MN
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A1A2T =L.1032Vg:2-4JgR2a8:o T ID;3,h*-4+A T =X2-IJ.D9C4�9<d�.1032
*-,/. T .J;Ro32AD
kBe?=w*�I19C83=�2��<;R2-8:ID2A1A2
T =�.1032c, T 8:2�£W9 T 8 T 8:x�.1032pn T o:q1gR9 T 8<.J=M*�8:o
N9<d�.1032@= T o:2a=
AB*-8:o
ACk�e?8L2-*-=m[c>@*�[p.D9�=X2-2�.10 T = T =z.J9�8:9�.J2�.10C*a.f.1032
I T 4JID,h2]>r039C=X2wo T *�nE2 .J2-4 T =AC
\/> T .10rI12a8<.D4J2M
*-8:or4m*�o T ;:=MA
j�g3*�=7=X2a=�.10C4J9<;Rxa0A2
*-8:oD
��.1032�A T =X2-IJ.D9C4w9<d∠ACD
nL2-2 .m=w.1032�gR2a4Dg:2-8:o T ID;3,{*�4|A T =�2�IJ.J9C4w9<dAD\}83*-nL2 ,/[
MNj�* .y.1032@g:9 T 8<.
A29<d:.1032rI T 4JID,h2:k
³:K��
¡|9�>#>r2�.J;:4J8 .D9 =�9<,{;C. T 9C83=�d}4J9Cn 9<;:4�4J2a*�o32-4m=�.J9 g:4J93A<,}2anE=cx T � 2-8 T 8 .1032¡|9 � 2-nEA:2a4f�-����M�8C;:nEA:2a4y9<d<.1032g»+��$����$�*-8:o�.1032wg:4J93A<,}2anE=?9<d:��* .D2-xa9C4W[�e�9<dC.1032µ�2a,{*�4m;:=|FH* .1032-n�*a. T ID*a,��z, [3n�g T *�oE�-�����CQ:� T 83*-,,?�9<;:8:o�Q3x T � 2-8 �J�-����M � ³:�-M"� ³:���"�<k{ k
(- 2³¸ �1¹ �?��,
)±-2�.
P (x) = (x +1)p(x − 3)q = xn + a1xn−1 + a2xn−2 + · · · + an−1x + anQ
>r032-4J2p*�8:o
q*-4J2@g:9C= T . T/� 2 T 8<.D2-xa2-4m=ak
\h* j�´ T � 2-8p.10C* .a1 = a2
QCg34J9 � 2�.10C* .3n
T =]*@gR2a4Jd�2-IJ.?=X�<;:*�4J23k\�A�jG :4J9 � 2|.10C* .y.1032-4J2@27u T =D. T 8C�R8 T .D2 ,/[�n�*�8�[�g:* T 4m=
(p, q)9<d�gR9C= T . T � 2 T 8<.D2-xa2-4m=
p*�8:oq=7;RI70c.10C* .f.1032@2��<;:*-, T .W[
a1 = a2T = � *a, T o�d�9C46.1032@g:9<,/[C8:9Cn T *-,
P (x)k
���� �� � ���r· ��� $ $)�g¸ ��$�����¹��������6� ���"�0�����6� � ��?� �$���5� ��$-���+�)�������;� !������ ����� / � ��� �i¼��0� �� �*�����6� � �$)�����)��2 �:9;$)����"���D�*�+������$�������� � .¥¸ ��$��6��¹����'�=��2
\}*�j�Z�032E4J939�.J=@9<dP (x)
*�4J2L2 �<T o32-8<.J, [ −1*-8:o
3Q�> T .10G4J2a=�g:2�IJ. T � 2En�;3,/. T q
g3, T I T . T 2a=p*-8:o
q\/>r032-4J2
p + q = nj�k�2a8:9�[email protected]=�2c4J939�.J=|A-[
r1Qr2
Q. . .
Qrn
Q>r032-4J2
r1 = r2 = · · · = rp = −1*�8:o
rp+1 = rp+2 = · · · = rn = 3k
��= T 8:xf.1032f>r2a,P,}q7Y�8:9�>@8�4J2a,{*a. T 9C83=6A:2�.b>r2�2a8�4J939�.m=�*�8:o�I1932 � I T 2-8<.m= Q`>r2�0C* � 2n∑
i=1
ri = −a1*�8:o ∑
1≤i<j≤n
rirj = a2kB�]8�.1032c9�.1032-4]0C*�8:o�Q:;:= T 8:x�.1032cY�8:9�>@8
� *-,P;R2-=�9<dR.1032@4J939�.m= Q->r2@93A<.m* T 8 n∑i=1
ri = −p + 3q*-8:o
∑
1≤i<j≤n
rirj =
(p2
)(−1)2 + pq(−1)(3) +
(q2
)(3)2
= 12p(p − 1) − 3pq + 9
2q(q − 1)
kZ�0<;:=�Q
a1 = p − 3q*-8:o
a2 = 12p(p − 1) − 3pq + 9
2q(q − 1)
kZ�032a4J2ad�9C4J2<Q
a1 = a2T d�*�8:o�9C8C,/[ T dp2 − 3p − 6pq + 9q2 − 3q = 0
k \WK`j�<4J9Cn \�K�j1QR>r2�o32�o<;RI12c.10C*a.
pT =�o T/�<T = T A<,}2�A-[
3QR>r0 T I70 T 8L.J;:4D8+d�9C4JID2-=
q.D9+A:2�*
n�;3,/. T gC,}2@9<d3k]±-2�.
p = 3a*-8:o
q = 3bk
Z�0<;:=�Qa1 = a2
T d?*�8:o�9C8C, [ T da2 − a − 6ab + 9b2 − b = 0
Q�>r0 T I70 T =2��<; T/� *a,}2a8<.�.D9
(a − 3b)2 = a + bk�U}.�d�9<,{,h9�>@[email protected]* .
a1 = a2T d]*�8:o_9C8C, [ T d
3n = 9(a + b) = 9(a − 3b)2QC*�8:oc>r2�*-4J2ro39C8:2:k
\�A�j6��4J9Cn�*�A:9 � 2CQa1 = a2
T d3*-8:oc9C8C,/[ T d(a−3b)2 = a+b
Q�>r032-4J2p = 3a*�8:o
q = 3bk]±-2 .
a − 3b = xk�Z�032-8
a1 = a2T d�*�8:oE9C8C, [ T d
(a, b) =(34x(x + 1), 1
4x(x − 1)
) k \W�`j� T 8:ID2
a*�8:o
b*�4J2+g:9C= T . T/� 2 T 8<.D2-xa2-4m=�Q6>r2GI703939C=�2�*�8�[Hg:9C= T . T/� 2 T 8<.J2�xa2a4
t*-8:o�Q
,}2�.m. T 8:xx = 4t
T 8"\W�`j1Q�>r2Eo:2-oC;RID2�.10C*a.XQ~d�9C4(p, q) = (36t2 + 3t, 12t2 − 3t)
Q.1032@gR9<, [38:9Cn T *a,
P (x)=�* . T =1��2a=
a1 = a2k
³:K-�
~ k(� 2%/<��#�.��
) C9C= T . T/� 2�8C;:nLA32-4m=
a1Qa2
Q. . .
Qan
*�8:ob1
Qb2
Q. . .
Qbn
=�* . T =1dP[.1032rI19C8:o T . T 9C8
a1 + a2 + · · · + an = b1 + b2 + · · · + bn = 1k
� T 8:op.1032@=7nE*a,{,h2-=D.fgR9C=7= T AC,h2 � *a,{;R2@9<dR.1032@=1;:na2
1
a1 + b1
+a2
2
a2 + b2
+ · · · +a2
n
an + bn
k
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±-2 .S
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T = 12
k� T 8:ID2
S = 12
d�9C4ai = bi = 1
n
\i = 1
Q2Q. . .
Qn)
Q�.1032�g34J939<d:> T ,P,�A32rID9CnEgC,}2�.D2 T d>r2@=�039�>�.10C* .
S ≥ 12
k±-2 .
T =a1b1
a1 + b1
+a2b2
a2 + b2
+ · · · +anbn
an + bn
k¡|9�.J2�.10C* .XQ<d�9C4
i = 1Q2Q. . .
QnQ
aibi
ai + bi
≤ ai + bi
4
Q= T 8:I12
(ai + bi)2 − 4aibi = (ai − bi)
2 ≥ 0k�Z�032-4J2 d�9C4J2CQ
T ≤ 1
4
n∑
i=1
(ai + bi) =1
2
k
¡|9�>S =
n∑
i=1
(ai − aibi
ai + bi
)= 1 − T ≥ 1
2
k
� k(��2³º �����������/¼�� � -<2 �����'�W¼���
)\h* jre�gR9C= T . T � 2 T 8<.D2-xa2-4 T =?ID*a,{,h2�o �K����� T d T .yID*-8�A32�4J2ag:4J2a=X2a8<.D2-op*-=?*�8c*�4 T .10CnE2 . T I
nL2a*�8E9<d�=�9CnL2@\}8:9�.z8:2-I12a=�=7*�4 T , [Eo T =D. T 8:IJ.bj6g:9C= T . T/� 2 T 8<.J2�xa2a4m=B2-*�I70E9<dR>r0 T I70T =]*@8:9C8:q18:2�x * . T/� 2@g:9�>r2-4?9<d
2k
:4J9 � 2|.10C* .?*-,P,RgR9C= T . T � 2 T 8<.J2�xa2a4m=�*�4J2�8 T ID2:k\�A�jce�g:9C= T . T/� 2 T 8<.J2�xa2a4 T =_IJ*-,P,}2-o ,�?�� T d T .�IJ*�838:9�.�A:2�4J2ag:4J2a=X2a8<.D2-o�*-=_*-8
*�4 T .10CnE2 . T IBnE2-*-8G9<d�g3* T 4b> T =X2������?�������;��g:9C= T . T/� 2 T 8<.D2-xa2-4m=�2a*�I70�9<d�>r0 T I70 T =*@8:9C8:q78:2-x *a. T � 2@gR9�>r2a4z9<d
2k
:4J9 � 2|.10C* .y.1032-4J2r21u T =1. T 8C�:8 T .J2a, [�n�*�8�[p;Rx�,/[�gR9C= T . T � 2 T 8<.J2�xa2a4m=-k����� ������� � �r· ��� $ $)�g¸ ��$��6��¹����'�=��� ���"�0�����6� � ��?� �$���5� ��$-���+�)��2
\}*�jy�C9C4y2-*�I70cgR9C= T . T � 2 T 8<.J2�xa2a4kQ�,h2 . Nk
A32w.1032�=�2 .f9<dR*a,{,<.1032|8 T ID2�gR9C= T . T � 2T 8<.J2�xa2a4m=+>r0 T I70�IJ*�8�A:2V4J2-g34J2-=�2-8<.J2�o�*�= *-8�*-4 T .10CnL2�. T I+nL2a*�8�9<d
2kgR9C= T . T � 2
T 8<.J2�xa2a4m=�2-*�I70+9<d�>r0 T I70 T =w*�8:9C8:q18:2�x * . T/� 2pg:9�>r2-4�9<d2kc±-2 . N =
⋃k≥1
Nkk|)G2
> T ,P,Rg:4J9 � 2�.10C*a. N = � ∗ k
³:K-S
F=���g� ��k?�<9C4z2a*�I70k ≥ 1
Q�>r2r0C* � 2 Nk ⊆ Nk+1k
·1$ � � . ��Z�0 T =]d�9<,P,}9�>@=?d}4J9Cn#.1032@2��<;:*-, T .W[∑
2i
2k=
∑2i +
∑2i
2k+1
Q>r032-4J2r2a*�I70�9<dR.1032�=7;:n�=�0C*�=�27u-*�IJ.J,/[
2k =7;:n�nE*-8:o3=akF=���g� �:k?U�d
xQy ∈ N .1032a8
x + y ∈ N k·1$ � � . �~��= T 8:xp±-2-n�nE*cKCQ�>r2�nE*�[��:8:o
k ≥ 1=7;RI70c.10C*a.
xQy ∈ Nk
k�±-2�.
x =
∑2i
2k
*-8:oy =
∑2j
2k
Q>r032-4J2r2a*�I70�9<dR.1032�=7;:n�=�0C*�=�27u-*�IJ.J,/[
2k=7;:n�nE*-8:o3=ak6Z�032-8
x+y =
∑2i +
∑2j
2k=
∑2i+1 +
∑2j+1
2k+1∈ Nk+1 ⊆ N k
)G2r0C* � 21 = 20
1∈ N kf��= T 8:xp±-2-n�nE*p�<Q�>r2@o:2-oC;RID2�.10C* . T d
x ∈ N Q�.1032-8x + 1 ∈ N k�Z�032-8�Q3A-[ T 8:o<;RIJ. T 9C8�Q N = � ∗ k
\�A�j]±-2 . U A:2|.1032@=X2�.z9<d�*-,P,:;Rx�,/[�g:9C= T . T/� 2 T 8<.D2-xa2-4m=akF=���g� �:k?�<9C4?*-8�[�g:9C= T . T/� 2 T 8<.D2-xa2-4
nQ�>r2@0C* � 2
n ∈ U T d�*-8:oE9C8C,/[ T d2n ∈ U k
·1$ � � . ��)G2|> T ,{,:g:4J9 � 2�.10C*a.n 6∈ U T d�*-8:oE9C8C, [ T d
2n 6∈ U k� T 4m=1.XQ6=7;:g3gR9C=�2
n 6∈ U k Z�032-8n =
1
k
∑i∈I
2i d�9C4�=X9CnE2L=X2�.I ⊆ �
> T .10|I| = k
k�Z�0<;:=�Q2n =
1
k
∑
i∈I
2i+1 Q=X9r.10C*a.
2n 6∈ U k¡|9�>#=7;:g3gR9C=�2
2n 6∈ U kcZ�032-82n =
1
k
∑i∈I
2i d�9C4B=X9CnE2�=X2�.I ⊆ �
> T .10|I| = k
kE� T 8:I12p.1032E21u�g:9C8:2-8<.m=�*�4J2Eo T =D. T 8:IJ.XQ T .Bd�9<,P,}9�>@=w.10C* .w*a.wnL9C=D.|9C8:2 T =0�
032-8:ID2CQ�*a.�nE9C=1.�9C8:2�9<d�.1032p8C;:nLA32-4m=2i >r0 T I70�*-g:g:2-*-4 T 8E.1032p=7;:n T =|93o3o~kcµ:;C.
.1032|>r039<,}2�=7;:n�0C*-=z.J9�A:2r2 � 2a8~kyZ�0<;:= Q38:9C8:2@9<d�.1032r21u�g:9C8:2-8<.m= T =0kyZ�032-4J2 d�9C4J2CQ
2-*�I70E9<d:.1032r21u�g:9C8:2-8<.m= T =]* .f,h2-*-=1.1k�Z�032a8
n =1
k
∑i∈I
2i−1 Q<*�8:o n 6∈ U kU}.wd�9<,P,}9�>@=wd}4J9Cn ±-2-n�nE*L�p.10C*a.]>r2E9C8C, [ 0C* � 2p.D9��:8:o 9C8:2�;Rx�, [G8C;:nLA32-4�Q
=�*�[xQ�A32�IJ*-;:=�2cd�9C4�=1;RI70L*�8C;:nLA32-4?>r2�o32�o<;RI12r.10C*a.
2pxT =B;Rx�,/[Ld�9C4�2-*�I70�8:9C8:q
8:2�x * . T/� 2 T 8<.D2-xa2-4pk
)G2@8:9�>�=X039�>13
T =z;Rx�,/[�ke?=�=1;:nL2<Q�d�9C4y*@I19C8<.D4m*�o T IJ. T 9C8�Q�.10C* .
13 6∈ U k�Z�032-8c.1032-4J2�27u T =D.k ∈ � ∗ *-8:o
I ⊂ �> T .10 |I| = k
=1;RI70p.10C* .13k =
∑
i∈I
2i k \WK`j
U}.zd�9<,{,h9�>@=].10C* .13k ≥ 1 + 2 + · · · + 2k−1 = 2k − 1
k�µ:;C.XQR*c=D.D4m* T xa0�.Jd�9C4b>@*-4JoT 8:o<;RIJ. T 9C8�=�039�>@=?.10C* .
2n − 1 > 13nd�9C4f*-,P, T 8<.D2-xa2-4m=
n ≥ 7k�Z�0 T =�x T � 2-=
k ≤ 6k
³:�-�
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13kkc�]8:2�ID*-8+8:9�> � 2-4 T dP[E.10C*a.XQ�d�9C4B2-*�I70
k = {1Q. . .
Q6} QR.1032�A T 83*-4W[
27u�g:*-83= T 9C8�9<d13k
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(�12�¸ ��$��������+��
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f : → =7;RI70.10C*a.?.1032c27u�g:4J2a=�= T 9C8
f(x + y) − f(x) − f(y).m*�Y�2a=w27u-*�IJ.J,/[�.b>r9 � *-,P;R2-=
0*�8:o
1d�9C4 � *�4 T 9<;:=]4J2-*a,
x*�8:o
y ������ ������� � � "� ���+�.��¥·.��i¼�»+����� ��K�����+»+�����5���5� ��� ��� $:�<�=��"��� � ,� � � ��2
��2-=ak?�<9C4?27u-*-nEgC,}2<Q<,}2�.
f(x) =
{x
T dx ≤ 0
x − 1T d
x > 0k
Z�032a8fT =�ID,h2-*-4J, [�=7;:4�£W2�IJ. T � 2:kz¡|9�>�.J*�Y�2@*-8�[
x ≤ yk
\ T j?U�dx ≤ y ≤ 0
Q�.1032-8f(x + y) = x + y = f(x) + f(y)
k\ TPT jyU�d
x ≤ 0 < y*-8:o
x + y ≤ 0Q�.1032a8
f(x + y) = x + y = f(x) + f(y) + 1k
\ TPT{T jyU�dx ≤ 0 < y
*-8:ox + y > 0
Q�.1032-8f(x + y) = x + y − 1 = f(x) + f(y)
k\ T � j?U�d
0 < x ≤ yQ�.1032a8
f(x + y) = x + y − 1 = f(x) + f(y) + 1k
� k(!�2 �.��$ ���+��������
)� T 8:o�.1032�*-4J2-*E9<d�.1032EI19C8 � 21uLg:2-8<.m*�xa9C8
ABCDEQ~x T � 2-8
.10C*a.AB = BC
QCD = DE
Q∠ABC = 150◦ Q
∠CDE = 30◦ QC*-8:o BD = 2k
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A
B
C
D
E
p p
q
q
r
s
2
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..................................
150◦.....................
15◦........................................θ
.........................................................
75◦
...............................
30◦
±-2 .p = AB = BC
*�8:oq = CD = DE
k@±-2 .r = AC
Qs = CE
Q:*�8:oθ = ∠ACE
k+�<4J9Cn .1032Lx T/� 2a8 T 8Cd�9C4Dn�*a. T 9C8�Q�>r2�=X2-2�.10C* .∠ACB = 15◦ *-8:o
∠DCE = 75◦ k�Z�032a8 r = 2p cos 15◦ *�8:os = 2q cos 75◦ = 2q sin 15◦ k6Z�0<;:= Q
rs = 2pq ·2 sin 15◦ cos 15◦ = 2pq sin 30◦ = pqkzµC[@*�g3g3, [ T 8:x].1032w��9C= T 8:2@±a*�>
.D9 4BCDQ�>r2@xa2�.
4 = p2 + q2 − 2pq cos(θ + 90◦) = p2 + q2 + 2pq sin θk
³:��K
��9C83=�2��<;R2-8<.J, [�Q
[ABCDE] = [ABC] + [CDE] + [ACE]
= 12p2 sin 150◦ + 1
2q2 sin 30◦ + 1
2rs sin θ
= 14(p2 + q2 + 2pq sin θ) = 1
4(4) = 1
k
¡|9�>�>r2�.J;:4D8�.J9�=X9CnE2r=X9<,P;C. T 9C83=z.J9Eg34J93AC,h2-n�=�d}4J9Cn���* .D2-xa9C4W["µ~Q�µ�2a,{*�4m;:=FH*a.1032anE* . T IJ*-,��z,/[CnEg T *�oL�-�����CQR� T 83*-,�?�9<;:8:oEx T/� 2a8�*a. �m������M �a³:����� ³:�-���<k~ k
(! 2 �.��$ ���+��������
) <* T 4b> T =X2co T =1. T 8:IJ.�g:9C= T . T/� 2 T 8<.J2�xa2a4m=
aQbQcQdQeQfQgQhQ
*�8:on
=7*a. T =7dP[c.1032@2��<;:*-, T . T 2-=n = ab + cd = ef + gh
k� T 8:op.1032@=7nE*a,{,h2-=D.fgR9C=7= T AC,h2 � *a,{;R2@9<d
nk
���� �� � ��� !$����� �K�0�&/;���0�)�W¼���� � � �*�+�"��� � �$)�����)��kZ�032@=7nE*a,{,h2-=D.fgR9C=7= T AC,h2 � *a,{;R2@9<d
nT =
47k
Z�032 T o32-* T =6.D9@;:=X2w8C;:nLA32-4m=z*-=?=7nE*a,{,C*-=?gR9C=7= T AC,h2C� .10C* . T = Qa.1032|8C;:nLA32-4m=1Q
2Q3Q4Q5Q6Q7QR.D4W[ T 8:xr.D9L�:8:o+8C;:nLA32-4m=B>r0 T I70+ID*-8 A32�21u�g34J2-=7=X2-oL.b> T ID2 T 8�.1032
d�9C4Dn&9<d�*rg34J93oC;RIJ.z9<dR.b>r9cd}*�IJ.J9C4m=-k)G2r0C* � 2
12 = 2 · 6 = 3 · 4*-8:o
35 = 5 · 7 = 1 · 35Z�0<;:=�Q T d:>r2�.m*�Y�2a = 2
Qb = 6
Qc = 5
Qd = 7
Qe = 3
Qf = 4
Qg = 1
Q3*�8:oh = 35
Q<.1032a8�Q<d�9C4n = 47
Q�>r2@0C* � 2n = ab + cd = ef + gh
k� k
(! 2 �.��$)�����=������
)Z�032c�C;:*�o34 T ,{*a.J2-4m*a,
ABCDT =wI�[CID, T I]*�8:oL0C*-=].1032rg34J9CgR2a4m.W[
.10C*a.AB = BC = AD +CD
k�´ T/� [email protected]*a.∠BAD = α
*-8:[email protected]* .~.1032|o T *�xa9C83*a,AC = d
Q<�R8:oc.1032@*-4J2-*r9<dR.1032�.D4 T *-8:x�,}2ABC
k���� �� ��� � � �0� ��'�'���"��� ��?o»+�=��� �@»6�"�� �?�0� ��$��,�<� �?����$)�-�,� � 9;�"�=� � �7�������'¸ � �0�?��4���Fº<�.����6� ��$-���+�)��� ��� �� ������$)�� ��2�/ ��� �=�?�0�"�<��� � �"���#» � � �<� ��2 ��?� �� �*���(�"��� ������� � �r¸ �����"�4���5�����6���07�� � � � �*��� � ��������$52
)G2�> T ,{,R=X039�>�.10C* .[ABC] = 1
2d2 sin α\�032a4J2r*-8:o T 8c>r0C*a.fd�9<,P,}9�>@=�Q
[·] o32-8:9�.J2-=]*�4J2a* j7k±-2 .a = AB = BC = AD + CD
*-8:oθ = ∠ABC
k�Z�032-8
[ABCD]
= [DAB] + [DCB]
= 12AB · AD sin α + 1
2BC · CD sin α
= 12a(AD + CD) sin α = 1
2a2 sin α
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A
B
C
D
d
a a
θ
*�8:o
[ABCD] = [ABC] + [ADC] = 12AB · BC sin θ + 1
2AD · CD sin θ
= 12
(a2 + AD · CD
)sin θ
³:���
Z�0<;:=�Qa2 sin α =
(a2 + AD · CD
)sin θ
k \WK`jµC[p.1032�±a* >'9<d���9C= T 8:2a= Q
d2 = AB2 + BC2 − 2AB · BC cos θ = 2a2(1 − cos θ)�
.10C*a. T = Q2(1 − cos θ) = d2/a2 k�ef,{=X9�Q
d2 = AD2 + CD2 + 2AD · CD cos θ
= (AD + CD)2 − 2AD · CD(1 − cos θ) = a2 − AD · CD(d2/a2
) Q>r0 T I70 T n�g3, T 2a=].10C* .
a2 + AD · CD = a4/d2 k]��= T 8:x�.10 T =B4J2a=7;3,/. T 8L\WK`j1Q<>r2cxa2�.a2 sin α =
(a4/d2
)sin θ
��.10C* . T =�Qd2 sin α = a2 sin θ
k� T 83*-,P,/[�Q
[ABC] = 12AB · BC sin θ = 1
2a2 sin θ = 1
2d2 sin α
k���6� 2 ��)G0C*a. T =?.1032@*-83=D>r2a4 T dR.1032r�<;:*�oC4 T ,h* .D2a4m*-, T =]8:9C8:q7I19C8 � 21u � �
¡|21u-.�>r2�,}9393Y@* .?=�9<,{;C. T 9C83=?d}4J9Cn�9<;:4f4J2a*�o32-4m=?.J9�g34J93AC,h2-n�=�9<d���* .D2-xa9C4W[E��Qµ�2a,{*�4m;:=�FH* .1032-n�*a. T ID*a,��z,/[CnEg T *�oL�-�����CQR� T 83*-,�?�9<;:8:o�Q3x T/� 2a8E* . �m������M �a³:�-S��<k� k
(��2(�7�0$ � �����
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k ≥ 3gR9C= T . T � 2 T 8<.J2�xa2a4m=E=1;RI70H.10C*a.p8:9H.b>r9
9<d?.1032-n *-4J2GID9Cg:4 T nE2E>r0 T ,}2�*�8�[ .10C4J2�2G9<d?.1032an *-4J2GID9Cg:4 T nE2:k�2 .J2-4Jn T 8:2+*-,P,
gR9C=7= T AC,h2 � *-,P;R2-=�9<dkk
\�A�j�[email protected] T =D.@*�8 T 8C�:8 T .J2G=�2 .r9<d]gR9C= T . T � 2 T 8<.D2-xa2-4m=p=�* . T =1dP[ T 8:xE.1032
=�*-nL2@I19C8:o T . T 9C8 ������ ������� � �r· ��� $ $)�g¸ ��$��6��¹����'�=��� ���"�0�����6� � ��?� �$���5� ��$-���+�)��2
\}*�j�Z�0 T = T =]g:9C=�= T A<,}2�d�9C4?2-*�I70k ≥ 3
k±-2 .
p1Qp2
Q. . .
Qpn
Q. . .
A:2|.1032�=�2��<;R2-8:ID2@9<d�g:4 T nE2-=�Q T 8 T 8:I14J2-*-= T 8:x�9C4Jo32-4`k±-2 .
k ≥ 3A:2�*�8 T 8<.J2�xa2a4�k
)+4 T .D2 T 8G*�4J9�>s*-,P,�.1032�g:* T 4m=(i, j)
d�9C41 ≤ i < j ≤ k
Q�*-8:o+,{*�A32a,�.1032an1Q2Q. . .
Q 12k(k − 1)
d}4J9Cn ,}2 d{.w.J9 4 T xa0�.�k �C9C4i ∈ {1
Q. . .
Qk} Q�,h2 . xi
A32�.1032g:4J93o<;RIJ.]9<d�*a,{,3.1032�g:4 T nE2-=
pnd�9C4�>r0 T I70
nT =?.1032�,h*�A:2 ,�9<d�*rg3* T 4?I19C8<.m* T 8 T 8:x
ik
U�diQj ∈ {1
Q. . .
Qk} > T .10
i < jQ-.1032a8
xi*-8:o
xj0C* � 2B.1032�I19Cn�nL9C8rd}*�IJ.D9C4
prQR>r032a4J2
rT =B.1032�,{*�A32a,69<d�.1032�g:* T 4
(i, j)��.10<;:= Q
xi*-8:o
xj*-4J2�8:9�.|I19Cg34 T nL23k
�]8�.1032p9�.1032a4]0C*�8:o�Qpr
o T/�<T o32-=B8:9C8:2c9<d�.1032c9�.1032-4z8C;:nEA:2a4m=x1
Q. . .
Qxk
�<.10<;:= Q*�8�[p.10C4J2-2r9<dR.1032a=X2�8C;:nLA32-4m=�*-4J2rID9Cg:4 T nE2:k
\�A�jz¡|9�Q�.1032a4J2 T =]8:9 T 8C�:8 T .J2@=X2�.?=7*a. T =7dP[ T 8:xB.1032rID9C8:o T . T 9C8~ke?=�=1;:nL2<QRd�9C4]*�I19C8<.D4m*�o T IJ. T 9C8�Q:.10C* . {x1
Qx2
Q. . . } T =w=1;RI70L*�8 T 8C�:8 T .J2p=�2 .�k
Z�032�8C;:nEA:2a4x1
0C*�=r*L�R8 T .D2L8C;:nLA32-4@9<d?g:4 T nE2+o T/�<T =�9C4m=-kH�C9C4�* .�,h2-*-=1.�9C8:2�9<d.1032-=�2�o T/�<T =�9C4m= Q]=�*�[
dQ?>r2 n�;:=D.�0C* � 2
gcd(x1, xj) = dd�9C4 T 8C�:8 T .J2a, [vnE*-8�[
j\}A:2-ID*a;:=X2
x1T =�8:9�.rI19Cg34 T nL2E> T .10H*-8�[v9<dz.1032G9�.1032-4
xj=Dj7kV) T .1039<;C.@,h9C=�=p9<d
xa2-8:2a4m*-, T .W[�Q3>r2c*�=7=7;:[email protected]*a.d = gcd(x1, x2) = gcd(x1, x3)
k@U}.]d�9<,{,h9�>@=�.10C* .gcd(x1, x2, x3) = d > 1
Q:>r0 T I70+ID9C8<.D4m*�o T IJ.m=B.1032�I19C8:o T . T 9C8L.10C* .B*-8�[L.10C4J2-2�9<d.1032
xi=]*�4J2@I19Cg34 T nL23k
³:���
{ k(!�2 �,� �¼
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y = x2=�9".10C* . T .J=�0�[3g:9�.D2a8C;:=X2 T =�g3*�4m*a,{,h2a,f.D9".1032
x��*`u T = T d�*�8:ov9C8C, [ T d].1032
*-,/. T .J;Ro32Hd}4J9Cn .1032H4 T xa0�.E*�8:x�,h2 T =+2-�C;:*a,w.J91k \/e#.D4 T *-8:x�,}2 T = T 83=�I14 T A:2-o T 8�*
g:*-4m*�A39<,h* T d�*a,{,3.10C4J2-2 � 2-4b. T ID2-=�9<dR.1032|.D4 T *�8:x�,h2@*-4J2r9C8p.1032�g:*-4m*�A39<,h*:k j���� �� � � � � �0� ��' �'���"��� ��?�»+�=��� �H»6�?0� ����� K� $-��� � �?����$)�-�,� �9;�?��� � ��� ��7�������'@¸ �����"�4���5�Fº �.K�"��� ��$-���+�)��2 � �"�#��P¸ � �0�?��4�� � � ����$���������� ���6����7 ��� � ��*��� ����������$52
±-2 .ABC
A:2@*�.D4 T *-8:x�,}2|> T .10∠A = 90◦ Q3g3,{*�ID2�o�=�9c.10C*a.y.1032c0�[CgR9�.J2-8C;:=�2
BCT =@g:*-4m*-,P,}2 ,6.J9+.1032
xq7*`u T =@*-8:oG.1032 � 2a4m. T I12a=
B*-8:o
C, T 2�9C8G.1032Eg:*-4m*�A39<,h*
y = x2k��a;:g3gR9C=�2r.10C* .z.1032�I1939C4Jo T 83* .D2a=|9<d
B*�4J2
(b, b2)Q3>r032a4J2
b > 0k|Z�032-8
.1032rI1939C4Jo T 83* .D2a=�9<dC
*-4J2(−b, b2)
k]±-2 .(x0, y0)
A:2|.1032rID939C4Jo T 83*a.J2-=�9<dAk
¡|9�.J2w.10C*a. −→AB = [b − x0, b2 − y0]
*-8:o −→AC = [−b − x0, b2 − y0]
k6� T 8:ID2∠A = 90◦ Q->r2r0C* � 2 −→
AB · −→AC = 0
��.10C* . T =�Q
x20 − b2 + (b2 − y0)
2 = 0k \WK`j
U�d 4ABCT = T 83=�I14 T A:2-o T 8_.1032"g:*-4m*�A39<,h*
y = x2 Qz.1032-8 A, T 2-=E9C8v.1032
g:*-4m*�A39<,h*CQc>r0 T I70 T nEgC, T 2-= .10C* .y0 = x2
0
k Z�0 T =�Qp.J93xa2 .1032a4�> T .10�\�K�j1Q�x T � 2-=y0 − b2 + (b2 − y0)
2 = 0Q39C4
(b2 − y0)(b2 − y0 − 1) = 0
ky� T 8:ID2b2 6= y0
Q->r293A<.m* T 8
b2 − y0 = 1Q�>r0 T I70�nL2a*�83=?.10C*a.y.1032�*-,/. T .J;Ro32�d}4J9Cn
AT =�2-�C;:*a,:.D9
1k
��9C8 � 2a4m=X2 ,/[�Q T d?.1032+*-,/. T .J;Ro32�d}4J9CnA
T =�2-�C;:*a,f.D91Qy.1032a8
b2 − y0 = 1k
��2 .m. T 8:xb2 = y0 + 1
T 8G\WK`j1Q:>r2�93A<.m* T 8x2
0 − (y0 + 1) + 1 = 0Q~9C4
y0 = x20
kZ�0<;:=�Q
A, T 2a=�9C8p.1032@g3*�4m*�A:9<,{*r*-8:o 4ABC
T = T 83=XI14 T A32�o T 8p.1032@g3*�4m*�A:9<,{*Rk� k
(! 2���� �¼
)Z�032�o T *�xa9C83*-,{=
A1A4QA2A5
Q�*-8:oA3A6
9<d�.1032�ID9C8 � 27uE0321u-*�xa9C8A1A2A3A4A5A6
nE2�2�.w*a.w*EgR9 T 8<.K
kE´ T/� 2a8+.10C* .A2A1 = A2A3 = A2K
QA4A3 = A4A5 = A4K
Qa*-8:oA6A5 = A6A1 = A6K
Qag34J9 � 2?.10C* .�.1032�0321u-*�xa9C8T =�I�[CID, T I k����� ������� � �� ������$)�� ��2�/ ��� �=�?�0�)�<� � � ���6�b» � � �<� ��2
)G2��R4m=D.H93A3=�2-4 � 2�.10C* .A2A4
T =�.1032gR2a4Dg:2-8:o T ID;3,{*�4+A T =X2-IJ.D9C4�9<d
KA3Q
A4A6T =
.1032�g:2-4JgR2a8:o T ID;3,h*-4�A T =X2-IJ.D9C4"9<dKA5
Q�*-8:oA6A2
T =.1032 g:2-4JgR2a8:o T ID;3,h*-4&A T =�2�IJ.J9C4#9<dKA1
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RQC4J2a=�g:2�IJ. T � 2a, [~k�Z�032-8
4A2PA3∼= 4A2PK ∼ 4A6QK
kZ�0<;:=�Q
∠A3A6A4 = ∠A5A2A4 = ∠A3A2A4k
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A1A2
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A0 A1 A2 An−1 An
B0
B1
B2
Bn−1
Bn
Pr a a a
s
bb
b
X1X2
Xn
Y
· · ·
· · ·
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Qa = A0A1
Qs = PB0
Qb = B0B1
QY = [PA0B0]
Q3*-8:oXn = [An−1AnBnBn−1]
Q�>r032a4J2[·] o:2a8:9�.D2a=]*�4J2a*3Q<*�=?;:=7;:*a,mk?��;:4m.1032a4DnE9C4J2CQC=�2 .
α = a/r*-8:o
β = b/sk6)G2�0C* � 2
Xn
Y=
[PAnBn]
Y− [PAn−1Bn−1]
Y
=(r + na)(s + nb)
rs−(r + (n − 1)a
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rs= (1 + nα)(1 + nβ) − (1 + (n − 1)α)(1 + (n − 1)β)
= α + β + (2n − 1)αβ�
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kZ�0<;:=�Q
X1QX2
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T =]*�8�*-4 T .10CnL2�. T Izg34J93x 4J2-=7= T 9C8~kUW8G.1032EID*-=X2c>r032-4J2
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2�
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k� k
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0C* � 2A = C
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T =�ID9C8 � 27uRk�2-8:9�.J2�*-4J2-*A-[
[·] k )G2�> T ,P,+=X039�> .10C*a.[ABC] = 1
2d2 sin α
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A
B
C
D
� T 4m=1.XQ
[ABC] = 12BA · BC sin ∠ABC = 2AD · CD sin ∠ADC = 4[ACD]
�.10<;:= Q
[ABC] = 45[ABCD]
k \WK`j¡|9�>�Q
[ABCD] = [ABD] + [DCB] = 12AB · AD sin α + 1
2CB · CD sin α
=(AD2 + CD2
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= 4AD2 + 4CD2 + 8DA · DC cos ∠ADCQ
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.10 T = T n�nL2-o T *a.J2a, [r[ T 2a,ho3=
[ABC] = 45
· 58d2 · sin α = 1
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k
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AOnL2-2 .m=y.1032�I T 4JID;:nEI T 4JID,h2Γ9<d 4ABC
*|=X2-I19C8:o@. T nE2CQ�*�8:or,}2�.P
A32w.1032|g:9 T 8<.y9C8.1032�, T 8:2
EF=1;RI70�.10C* .
A′P ⊥ EFk� :4J9 � 2p.10C* .�.1032p, T 8:2-=
EFQBC
Q�*�8:o�.1032.J*-8:xa2-8<.f, T 8:2�.J9
Γ*a.
A′ *�4J2@I19C8:ID;:4J4J2-8<.XQ3*-8:op.10C*a.∠BPA′ = ∠CPA′ k
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±-2 .a1
Qa2
Q. . .
Qan
A32@g:9C= T . T/� 2�4J2a*-,R8C;:nEA:2a4m=-k� :4J9 � 2�.10C* .( n∑
k=1
an+1
2
k
)n
≤n∏
k=1
( n∑
k=1
akk
) k
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±-2 .A(z)
A:2�*'g:9<,/[C8:9Cn T *-,�9<d+o:2-x 4J2�2n
> T .10#I19Cn�g3,h27u�ID932 � I T 2a8<.J=ak�a;:g:g:9C=X2�.1032�¢72-4J932a=
z1Q
z2Q
. . .Q
zn9<d
A(z)*�4J2 o T =D. T 8:IJ.r8:9C8:qD¢72a4J9_I19Cn�g3,h27u
8C;:nLA32-4m=ak� :4J9 � 2|.10C*a.n∑
k=1
ezk
z2k
n∏
j=1j 6=k
1
zk − zj
= 0k
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n!gR2a4Dn�;C.J* . T 9C83=
σ9<d
(1, 2, . . . , n)Q|d�9C4E039�> nE*-8�[ T =
σ3 .1032T o32-8<. T .�[�g:2-4Jn�;C.m*a. T 9C8 ��;{ � ~ k ·1$ �,9<� ��� � ��� / 24�62�¸ ����� ��� � ��2�&"2 - �5� ��9;��� � �� !����?����� ��� � .
�������"�+�)�5� �)&��,� 9N�$��)!!�;���0�:k±-2 .
xQyQ�*�8:o
zA:2pg:9C= T . T/� 2p4J2a*-,~8C;:nEA:2a4m=B>r0 T I70�=�* . T =1dP[
x2 + y2 = z2 k��9C83=D.D4J;RIJ.|*�, T 8:2�=X2-x nL2a8<.AC
> T .10�,}2a8:x`.10zkG±-2 .
BA32�*�8�[Gg:9 T 8<.w=7;RI70�.10C* .
BC = x*-8:o
90◦ < ∠ABC < 180◦ k ±-2 .M
A32E*EgR9 T 8<.�9C8AC
=1;RI70+.10C* .∠MAB = ∠MBC
kz±-2�.D
A32|.1032�g:9 T 8<.?9C8p, T 8:2BM
9C8c.1032@9Cg3gR9C= T .J2�= T o32�9<dAC
d}4J9CnB
=1;RI70c.10C*a.AD = y
ky��039�>�.10C* .∠ADM = ∠DCM
k�;{ � � k ·1$ �,9<� ��� � ���� �$ �5����$�� �k¼+"��= ���"� �� ��� ���� ��� �� $ � ���*� � ./º<��"�=�6���6º<��"�=�6���� /fk
��9<, � 2|.1032�d�9<,P,}9�> T 8:x|d�9C4x�
x
(√3 − 2x +
√5(1 − x2) +
√3
2
)=
√2
3
k
³:�-�
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f : [0, 1] → =7;RI70c.10C*a.
f(x) = 1 + x
∫ 1
0
f(t) dt + x2
∫ 1
0
[f(t)
]2dt �
�;{ � � k¨·1$)�,9<� �"��� ��� � �0$-�"�43& ���'?0���¥¸ K������$ �5�?���¥º ���������0�:k±-2 .
xQyQC*�8:o
zA32@gR9C= T . T � 2�4J2-*a,R8C;:nLA32-4m=�=�* . T =1dP[ T 8:x
xy + yz + zx + xyz = 4k
:4J9 � 2|.10C*a.\h* j
(x+2)(y+2)+(y+2)(z+2)+(z+2)(x+2) = (x+2)(y+2)(z+2)�
\�A�jp.1032-4J2 T =�*z.D4 T *-8:x�,}2y>r039C=�2z= T o:2a=�0C* � 2z,h2-8:x`.10C=(x+2)(y+2)
Q(y+2)(z+2)
Q*�8:o
(z + 2)(x + 2)k
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±-2 .n
A32@*@gR9C= T . T � 2 T 8<.J2�xa2a4�k?� � *a,{;:* .D2|.1032@=1;:n �bn−1
2 c∑
i=0
(n − i − 1
i
)2n−2i−1
n − 2i
Q
>r032-4J2 bxc T =?.1032 T 8<.J2�xa2a4?g:*-4m.z9<dxk
k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k�;{ � � k�·1$ �,9<� �;8�:9;��$�& �,��"� ���$$� ��-�����,� $)��� �b»6$)��� �����<k
�-*�I70C*-8<.?�C;R2�,{*�=1; T .J2 {an}∞n=0
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an+3 = an+1 + ang:9<;:4
n ≥ 0QC= T nEgC, T ��2a4
√a2
n+5 + a2n+4 + a2
n+3 − a2n+2 + a2
n+1 − a2n
k�;{ � � k#·1$)�,9<� �;8� 9;� $ %'�� ���$��.�$ �+"�0�(8��� =������� ��� ��"���$����7��������� ��� � �K�#��$-�)�������/;�"�������¹�� � �)� !*� 8� � �fk
±-2-=�8:9CnEA34J2a=@o:2G± ;RID*-=Ln
=7*a. T =7d�9C8<.B,{*E4J2 ,h* . T 9C8 o:2�4 �2-ID;:4D4J2a8:I12L0 = 2
QL1 = 1
Q�2 .Ln+2 = Ln+1 + Ln
gR9<;:4n ≥ 0
k���9 T .k;:8 2a8<. T 2a4Bg:9C= T . T d�g:* T 4`k
Z:4J9<; � 2-4lim
n→∞
({k
√Ln
}− {k
√Ln−k + k
√Ln−2k
}) Q9"�; {x} 2-=D.f,h*rg3*�4b. T 2�d}4m*�IJ. T 9C8383* T 4J2ro:2
x\}I � 2a=1.1q%�*�q7o T 4J2CQ {x} = x − bxc Q:9"�; bxc2-=D.f,h*@g3*�4b. T 2@2-8<. T �2-4J2@o:2
xj7k
³:��K
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P;:8 gR9 T 8<.�oC*�83=�,h2�g3,{*�8"o<;G.D4 T *�8:x�,h2
ABC.J2a,y�<;R2
PC = PB2�.
PA = ABk:��9 T .
x,{*rnE2-=1;:4J2ro32�, � *-8:x�,}2
PBCk�F_9C8<.D4J2-4z�<;R2
sin(B − C) = 2 sin C cos(B + 2εx)Q
9"�;ε = 1
= T ,{*ro34J9 T .D2BC
= �2-g3*�4J2�,}2a=]gR9 T 8<.J=P
2 .AQ32�.
ε = −1= T 8:9C8~k
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OQIQR
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Qy2�.aQb2�.
c,}2a=r,}9C8:x�;R2 ;:4m=�o:2a=�I�9�. �2a=
oC;H.D4 T *-8:x�,}2ABC
9Cg3gR9C= �2-=c*-;�u *-8:x�,}2a=AQB
2 .Ck��2a= T x 8:9C83=pg:*-4
M2�.
N,}2a=rg:9 T 8<.m=c9"�;IO
ID9<;:gR2E,h2-=ro34J9 T .D2a=AB
2 .AC
k?F_9C8<.D4J2-4��<;R2L,h2-=@gR9 T 8<.J=BQ
CQM
2 .N
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\�9"�;ha
2-=D.|,h*�0C*-;C.J2a;:4�*�AC* T =�= �2�2E=1;:4|,}2LI�9�.��2BC
jw2�.@�<;R2CQ�o3*-83=cI12LID*-= Q�9C8"*�*a;:=�= T1
MN=
1
a+
1
b + c
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��9 T .|4J2a=�g:2�IJ. T � 2-nE2-8<.O
2�.R
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ABCk3��9 T .
E=7;:4
AB2 .
F=1;:4
ACo:2 ;�urg:9 T 8<.m=?.J2a,{=]�C;R2
O=X9 T .6,}2
n T , T 2a;�oC;�=�2�x nE2-8<.EF
kR��9 T .A′ ,h2rgR9 T 8<.�9��;
AOID9<;:gR2@,}2rI12a4JID,}2cI T 4JID9C83=XI14 T .
ΓoC;L.D4 T *-8:x�,}2ABC
;:8:2�o32a;�u T �2anL2rd�9 T = QR2�.]=X9 T .P
,}2rgR9 T 8<.�=1;:4z,h*�oC4J9 T .J2EF
.J2a,�C;R2
A′P ⊥ EFk~F_9C8<.D4J2a4��C;R2r,h2-=woC4J9 T .J2-=
EFQBC
2 .z,{*@.J*-8:xa2-8<.J2 �*Γ
2a8A′=X9C8<.zID9C8:I19<;:4m*-8<.D2a= Q32 .?�C;R2
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��9 T .a1
Qa2
Q. . .
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k=1
an+1
2
k
)n
≤n∏
k=1
( n∑
k=1
akk
) k
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��9 T .A(z)
;:8pg:9<,/[C8 �9CnL2|o32�o32�x 4 �2n
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z1Qz2
Q. . .
Qzn
o:2A(z)
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n∑
k=1
ezk
z2k
n∏
j=1j 6=k
1
zk − zj
= 0k
³:���
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n!g:2-4Jn�;C.m*a. T 9C83=
σo:2
(1, 2, . . . , n)Q�ID9CnLA T 2a8G["2-8"*�qJ.1q T ,fo:2E=X9C4b.D2
�C;R2σ3 =�9 T .y,{*@gR2a4Dn�;C.J* . T 9C8 T o32-8<. T .��2 �
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��9 T .xQy2�.
z.D4J9 T =f8:9CnLAC4J2-=?4 �2�2 ,h=fgR9C= T . T d}=?=7*a. T =7d}* T =7*�8<.
x2 +y2 = z2 k<�]8o:2a=�= T 8:2@;:8�=X2-x nL2a8<.AC
o:2r,h9C8:x�;R2a;:4zk���9 T .
B;:8�g:9 T 8<.?.D2 ,~�<;R2
BC = x2�.
90◦ < ∠ABC < 180◦ kC��9 T . M;:8�g:9 T 8<.y=1;:4
AC.J2a,:�C;R2
∠MAB = ∠MBCk
��9 T .��:83*-,h2-nE2-8<.D
,h2]g:9 T 8<.�=7;:4R,h*�o34J9 T .D2BM
o<;rI�9�.��2]9Cg:g:9C= �2��*AC
g3*�4�4m*-g:g:9C4m.�*
Bo32@=�9C4m.J2@�C;R2
AD = yk�F_9C8<.D4J2a4z�C;R2
∠ADM = ∠DCMk
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Z:4J9<; � 2-4?g:9<;:4z�C;R2 ,x�
x
(√3 − 2x +
√5(1 − x2) +
√3
2
)=
√2
3
k
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f : [0, 1] → .D2 ,{,h2@�C;R2
f(x) = 1 + x
∫ 1
0
f(t) dt + x2
∫ 1
0
[f(t)
]2dt �
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xQy2 .
z.D4J9 T =]8:9CnLAC4J2-=�4 �2�2 ,h=]g:9C= T . T d}=z=�* . T =1d}* T =�*-8<.
xy + yz + zx + xyz = 4k
F_9C8<.D4J2-4z�<;R2\h* j
(x+2)(y+2)+(y+2)(z+2)+(z+2)(x+2) = (x+2)(y+2)(z+2)�
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Q(y + 2)(z + 2)
2�.(z + 2)(x + 2)
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��9 T .n
;:8E2-8<. T 2-4fgR9C= T . T dJkR� � *a,{;R2a4f,{*r=�9CnEnE2��bn−1
2 c∑
i=0
(n − i − 1
i
)2n−2i−1
n − 2i
Q9"�; bxc 2a=1.f,{*@g:*-4m. T 2r2a8<. T �2a4J2ro32
xk
³:���
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A32c*�8�[��<;:*�oC4 T ,h* .D2a4m*-,WQ�*-8:oE,}2�.M
A:[email protected] T o3q7g:9 T 8<.�9<dAB
k�]8w.1032z= T o32-=
CBQDC
Q *-8:oAD
Qa2-�C; T ,h* .D2a4m*-, .D4 T *-8:x�,}2a=CBE
QDCF
Q�*�8:oADG*�4J2wID9C83=1.D4m;RIJ.D2-oc21u-.J2-4J83*-,P,/[�kz±-2�.
NA:2].1032Bn T o3q7g:9 T 8<.�9<d
EF*�8:o
PA32].1032Bn T o:q
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−→CA,
−−→CB)
b ���a�8�1�a��������/�� b �a��§U���/�xbyb �a��§
z���W�%���G�������a�#����§©���������a�∠(
−−→BC,
−→U )
b∠(
−→CA,
−→U )
b ����§∠(
−→AB,
−→U )
b ���a�8�1�a��������/�� Xª �L��� �%���4� b �m��§#�����2π
b �%����#�a�F�1� �#�x��� ���U�%���G��/� {α, −α, π − α, α − π}
b ���%���4�
cos2 α = cos2 x����§
sin2 α = sin2 xX ª ��� b ���/�a���_�m��§#�����
2πb �U�?������
y = ∠(−→CA,
−−→BC) + ∠(
−−→BC,
−→U ) = x + C + π����§
z = ∠(−→AB,
−−→BC) + ∠(
−−→BC,
−→U ) = x + π − B
X�z�#��� b
X = cos2 x cos2(x + C) cos2(x − B) + sin2 x sin2(x + C) sin2(x − B)
= 18(1 + cos 2x)[cos(2x + C − B) − cos A]2
+ 18(1 − cos 2x)[cos(2x + C − B) + cos A]2
= 14cos2(2x + C − B) + 1
4cos2 A
− 12cos 2x cos(2x + C − B) cos A
= 18[1 + cos(4x + 2C − 2B)] + 1
4cos2 A
− 14cos A[cos(4x + C − B) + cos(C − B)]
= 18cos(4x + 2C − 2B) − 1
4cos A cos(4x + C − B)
+ 18
+ 14cos A[cos A − cos(C − B)]
X
³:�-M
��= T 8:xE.1032G2-*-=m[:qJ.D93q1g:4J9 � 2+4J2a,{*a. T 9C8cos A − cos(C − B) = −2 cos B cos C
Q>r2r93A�.J* T 8
8X = cos(4x + 2C − 2B) − 2 cos A cos(4x + C − B) + 1 − 4p
= M cos 4x + N sin 4x + 1 − 4pQ
>r032-4J2M = cos(2C − 2B) − 2 cos A cos(C − B)*�8:oN = 2 cos A sin(C − B) − sin(2C − 2B)
k)G2r2a*�= T , [p�:8:op.10C*a.
M2 + N2 = 1 − 8pk�Z�032a8
8X =√
1 − 8p cos(4x + ϕ) + 1 − 4pQ
d�9C4?=�9CnL2ϕkzU}.fd�9<,P,}9�>@= T n�nL2-o T *a.J2a, [p.10C* .y.1032@n T 8 T n�;:n&*-8:o�nE*`u T n�;:n � *-,P;R2-=
9<dX
*�4J2 18(1 − 4p − √
1 − 8p)*�8:o 1
8(1 − 4p +
√1 − 8p)
QC4J2a=�g:2�IJ. T � 2a, [~k& � ������U�d 4ABC
T =�2��<; T ,{*a.J2-4m*a,�Q�.1032a8p = 1
8
Q~*-8:o�.1032-4J2 d�9C4J2CQ�.1032 � *a,{;R29<d
XT = T 8:o:2agR2a8:o:2a8<.E9<dB.1032 .D4m*-83= � 2-4m=7*-,B*�8:o�I19C83=D.J*-8<.J,/[�2��<;:*-,].D9 1
16
k�Z�0 T =g:*-4m. T ID;3,h*-4�IJ*�=�2�>@*-=�.103293A�£W2�IJ.�9<d *�g34J93AC,h2-n g:9C=X2-o�A-[ �|kEZ�0 �2�AC*-;3,/. T 8�"�. $��6�? � ����� �*�:8� ���������+K�� 8�' 8����"�����"��$)���QRK�S������<K-S<��KCQ3g~k����3k
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a�8�1�8�©�%���4�
(√a + 3
) 15 +
(√5 − 2
) 13 =
(√a − 3
) 15 +
(√5 + 2
) 13
X
� gL�Ah!g�i8q,�,j�g�� i8q��pq v�u fGi�g v ����q,gLx�iml�nrwHfOq u x����Ew�j u i v j�n4| d1f8j=i4l�n��£j/fOq u x � g v�v j#j�|� v q�x �£gLx�| ��� |�� � ¢� u xzk � s�q h � ��f'��q�i| orq�i8i/g1�afOq � g1����sHj/f�x � � u �,j���x#q,}aj�fOi%q���na|� g4h v q�x�|!o���|�� � ¢ �
� �/�α =
√5 + 1
2
�a��§β =
√5 − 1
2
X �H���a�α − β = αβ = 1
X � �������
α3 =√
5 + 2����§
β3 =√
5 − 2b �U�?������ (√
5 + 2) 1
3 − (√5 − 2
) 13 = 1
X �a���x =
(√a − 3
) 15
X �H���a�A�%���U��� ��a�����#���4�����������a�%�����a�
(x5 + 6)15 = x + 1
X � ��� �������
x5 + 6 =√
a + 3 > 3b �U�?������
x + 1 > 1 � �U���a�����bx > 0
X���#���4������� � �� ��� ���#����#�4���4�#� ���
x5 + 6 = (x + 1)5X � �����a��§������©�a��§¡�%��� �
���F��� #����� ������4�x4 + 2x3 + 2x2 + x − 1 = 0 � �%���4� ���
b
(x2 + x + α)(x2 + x − β) = 0X
³:���
��,h2-*-4J, [�Qx2 + x + α
0C*�=w8:9E4J2a*-,�4J939�.m= Q:= T 8:I121 − 4α < 0
kBZ�032p9C8C,/[EgR9C= T . T � 24J939�.z9<d
x2 + x − βT =
x =−1 +
√1 + 4β
2=
−1 +√
2√
5 − 1
2
Q>r0 T I70p,}2a*�oC=?.D9r.1032r9C8C,/[�=�9<,{;C. T 9C8X�
a = (x5 + 3)2 =
(
−1 +√
2√
5 − 1
2
)5
+ 3
2
=7 + 5
√5
2
k
���6����Z�032w,{*�=D.6=D.D2ag�IJ*�8pA32 � 2-4 T �R2�oc2 T .1032-46A-[pAC4J;C.J2wd�9C4JI12|I19Cn�g3;C.m*a. T 9C83=z9C4A-[�;:= T 8:x�*cID9CnEgC;C.D2a4?*-,hxa2�AC4m*r=m[C=1.J2-n k �
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,$5 � / ("> C , ���.E /;6), & 3 , / ('��A ,$0./ �a = 9 +
(1+
√5
2
)−5 :�A.3; % ,DC #;A /)0 > / (?> 1 #)3�('�$#)(E /;6),
a = 9 +(
−1+√
52
)5 M
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n ≥ 2b �%�����a�����/�������
xn + x−n = 1 + x�����S�U�����L�2���A�%���G���#���a� #�4� (1, 1 +
1
n
) X
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f(x) = xn + x−n − x − 1X �����O�����U� �'����� �%���4�
f(1 + 1
n
)> 0
X �����n = 2
b �U�A������f(1 + 1
2
)= 7
36
X �#���n ≥ 3
b �%��� ���������p���4�H�z���a�����a� ���_���F���4��%���4�(1 +
1
n
)n
> 1 + n
(1
n
)+
n(n − 1)
2· 1
n2= 2 +
n − 1
2n≥ 2 +
1
n�
�%�#��� bf(1 + 1
n
)>(1 + 1
n
)n − (1 + 1
n
)− 1 > 0X
ª � �a� b �U�?���/�g(x) = xnf(x) = x2n − xn+1 − xn + 1
X �z���4�m� � ��� �4���� ��� ��4���F� A�%���/�
g(x) = (x − 1)h(x)b �U���a���
h(x) = x2n−1 +x2n−2 + · · ·+xn+2 +xn+1 − xn−1 − xn−2 − · · · − x −1X
� �������f(1 + 1
n
)> 0
b �U�?§��a§��1��� �%���/�g(1 + 1
n
)> 0
�a��§h(1 + 1
n
)> 0
X�����
h(1) = n − 1 − n = −1 < 0 � �%�#���b �%��� ¦ �#���a���m�a§����4���©¨��a�F�1�p�H���������4�
���_���F���4�S�%���4� �%���a���©� �a�������r ∈ (
1, 1 + 1n
) �8�1�8�p�%���/�h(r) = 0
X ¦ � �%���a�����#�F�����?��%���4�
f(r) = 0X
³:�-�
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f(1) = 0/ (">
f ′(1) = −1&�#<> , >��'2 , &I3 / &
f(1+ε) < 0$ # 0 � ��� 2� , ("& % B � 5 / % %
ε > 0M
/ �D��3"#;A , >9& 3 / &f3 / �<#)('% B #)( , 0 #'#'& ,�� 2 ,), >) (;E #)( , : / (?>Z�N#'# , �$& / �)% ��3 , > & 3 , �$& 0 #)('E ,$0
0�, � ��%I&�&I3 / &" ( $ / 2!& :r ∈
(1, 1 + 1
n(n−1)
) M
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�a�/�abbb �a��§
c��� ��� ?�1���%�F��� ��W���a�4�����������4�O� b ����§?���/�
p���W�Z���4�a�#�����m���a�
�8�1�8�©�%���4�0 < p < 1
X� � ������ ��Z�%���/�
a
(b + c)p+
b
(c + a)p+
c
(a + b)p≥ 1
2p
(a1−p + b1−p + c1−p
) X
� ������� ��Z�%���/� b ���p = 1/3
b �%���a�
a
(a + b)p+
b
(b + c)p+
c
(c + a)p≥ 1
2p
(a1−p + b1−p + c1−p
) X
� � � ? ���� ��?��� §��������� ��
a√a + b
+b√
b + c+
c√c + a
≥ 1√2
(√a +
√b +
√c) X
� g v ����q£gLx¤�£gAh u f'� � �� l�n � u g�orq�x�s �E� u x 4|�� ���nLj/xrw�q�x�s�� s�q£j/��� q �s � y�szg�g v |� q�x�s�#gLx a| � q£jL��x u � X
� �������p�%�����������1�����§������a�����4��� � ����?���������4�a���a�#��� b �U���_�� �8�����1����p�%���/�a + b + c = 1
X �z�#��� #���/��§�� �%���U�a����� #�a���a�#� �����a�����4��� �
a
(1 − a)p+
b
(1 − b)p+
c
(1 − c)p≥ 1
2p
(a1−p + b1−p + c1−p
) X
� �F���#���W������G�#� �4�4���a�O�4��� � b �U�p���� ����8�8�����©�%���4�a ≥ b ≥ c
X_¦ �E�%���a�����#�F�����?��%���4�
1
1 − a≥ 1
1 − b≥ 1
1 − c
X
³:�-S
��= T 8:x���032-A-[3=�032 � � =�UW8:2-�C;:*a, T .�[�*-8:op.1032|eBF ��´]F UW8:2-�C;:*a, T .�[�Q�>r2@0C* � 2a
(1 − a)p+
b
(1 − b)p+
c
(1 − c)p
≥ a + b + c
3
[1
(1 − a)p+
1
(1 − b)p+
1
(1 − c)p
]
≥ 1((1 − a)(1 − b)(1 − c)
)p/3≥ 1
[1 − a + 1 − b + 1 − c
3
]p =3p
2p
k
Z�0<;:=�Q�>r2�8:2-2�oE9C8C, [�g34J9 � 2|.10C*a.a1−p + b1−p + c1−p ≤ 3p k±-2 .
f(x) = x1−p kf� T 8:I12 f ′′(x) = −p(1 − p)x−p−1 *-8:o0 < p < 1
Q�>r2=X2-2�.10C* .
f ′′(x) < 0d�9C4
0 < x < 1kzl@2a8:I12<Q
fT =�*rI19C8:IJ* � 2�dh;:8:IJ. T 9C8L9C8
(0, 1)k
��= T 8:xB5�2-83=�2-8 � =|UW8:2��<;:*-, T .W[�Q�>r2@xa2 .a1−p + b1−p + c1−p ≤ 3f
(13(a + b + c)
)= 3f
(13
)= 3p k
���6� 2 �6�3�<;:*-, T .W[E039<,ho3= T d�*-8:oE9C8C, [ T da = b = c
k ������ ������� ����9;� $'�R\�A�j ��� �.���9K� �k¸ �#»6���6��»6��� � ��� � ��� �� $ � ���*� � .�� � �������=�+�(��� �·1����$ �����-����»6��� � ��� � � �����!�6� � k
)G2|> T ,{,3�R4m=D.?g:4J9 � 2�.10C* .XQ<d�9C4?*a,{,x > 0
Q�.1032�d�9<,{,h9�> T 8:x T 8:2��<;:*-, T .W[E039<,ho3=4x3 3
√2
3√
x3 + 1≥ 5x2 − 1
k \WK`jZ�0 T = T 8:2��<;:*-, T .W[ T =?.D4 T �CT *-, T d
x ≤ 1/√
5k?�C9C4
x > 1/√
5QCo:2 �R8:2
f(x) =128x9
(x3 + 1)(5x2 − 1)3k
:4J9 �<T 8:x T 8:2-�C;:*a, T .�[�\WK`j T =�2-�C; T � *-,h2-8<.y.J9�=X039�> T 8:xw.10C*a.f(x) ≥ 1
k�)G2@I19Cn�g3;C.J2
f ′(x) =384x8(1 − x)(2x2 − 3x − 3)
(x3 + 1)2(5x2 − 1)4Q
d}4J9Cns>r0 T I70p>r2c=�2�2�.10C*a.f ′(x) > 0
d�9C41 < x < 1
4
(3 +
√33) *-8:o
f ′(x) < 0d�9C41/
√5 < x < 1
*-8:o�d�9C4x > 1
4
(3 +
√33) k�Z�0<;:=�Q
f(x) ≥ min{
f(1)Q
limx→∞
f(x)}
= min{1Q 128125
}= 1
d�9C4?*a,{,x > 1/
√5k�Z�032-4J2 d�9C4J2CQ T 8:2-�C;:*a, T .�[�\WK`jy039<,}oC=]d�9C4f*a,{,
x > 0k
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k� T 8:ID2
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n
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n
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2
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2
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2
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k ≤ −1k
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2
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A
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C′
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D′M
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�z���4�QP ⊥ MN
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) �
³�³:�
.10C*a. T = Q(QB2 − QC2
)+(QD2 − QA2
)= (PB2 − PC2) + (PD2 − PA2)
k� T 8:ID2
QD′ ⊥ BCQQC′ ⊥ AD
QPD ⊥ BC
Q�*-8:oPC ⊥ AD
Q�.1032�*�A39 � 2 T =2��<; T/� *a,}2a8<.f.J9(D′B2 − D′C2) + (C′D2 − C′A2) = (DB2 − DC2) + (CD2 − CA2)
�.10C*a. T = Q
C′D2 − (DB2 − D′B2) = D′C2 − (CA2 − C′A2)k
� T 8:ID2 4D′BD*�8:o 4C′AC
*�4J2E4 T xa0�.1q1*�8:x�,h2�o�Q�>r2EID*-8 ;:=�2�.1032 <[�.10C*�xa9C4J2a*�8Z�032-9C4J2-n .J9 4J2`>@4 T .D2�.1032�,h*-=1.�2-�C;:* . T 9C8"*-=
C′D2 − D′D2 = D′C2 − C′C2k
Z�032a8�Q�= T 8:ID2 4C′D′D*-8:o 4C′D′C
*�4J2�4 T xa0�.1q7*-8:x�,}2-o�Q�>r2 93A�.J* T 8�.10322��<; T/� *a,}2a8<.]2-�C;:* . T 9C8
D′C′2 = D′C′2 Q�>r0 T I70 T =?.D4J;R23k��% �$# �$#�% 6), > � B � ���.L��"HIF �?J #�� �J�����J�F�� J�: H!M KM�� 5 % �# �',$0=0�/)0 : / % % / >�#�% I>):<F -"/ ( �@ �.L � �� � #�'L<H � J�: *J�F��)� � �"H � � ��KH : / (">%�'@ H� ;H=J � ����L *�.L��� �$:� '#�E 0 #��(?#�:�F -"/ ( ��� C H � � � �!� H�F :@ ���$#�� 0 �F;#��'& 3 ,$0 (GF'& / & , � (; 6�,$0 �! �& B : 1 # - % (':�@ J�: � F$��� 1 J C L�8�M C � � ����?:K8 0�/ ("> , � 0�/ 0 , :�� ����� �� C � � 1 �.L J � F : � 0 � � % ( , (;E B�5 ( / �! 4� 5 :�HI('(��(� 0 �;2 � :;�����$& 0 / �� M ��H���� 1 J C L F�J�L+:� ,$/;6),$0 &�#)(':
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QSR VZV?X � �a�������%�4�m�4�#� XWY�[a\L\#�Z^ \ ® b L [4c d1f8g4h!g�i�j�k?l�n � g�iztj ���/q�i � t� u {��8w u f�f%j/f%g�|��x#q,}aj�fOi%q�� u �2d1g v q,���j�y�x#q,y u k�j � u � u4v ��x�n u |2w u f8y�j v gLx u | � h u q�x X�a�/�
z1
bz2
b. . .
bzn
���Z�%��� �8�4�����a�E�#�1�%���?�%���_����� �U�1�#� ������p���4�
A(z) = zn + an−1zn−1 + · · · + a1z + a0
b�U���a���
a0 6= 0X ���� ��Z�%���4�
det
n z1 z2 · · · zn
z1 1 + z21 1 · · · 1
z2 1 1 + z22 · · · 1
......
.... . .
...zn 1 1 · · · 1 + z2
n
= a21
X
� g v ����q£gLx lLn � g�iztj � � �Wq£jL�£g�| ��x#q,}aj�fOi%q,k u k k�j v ��� v q u |po u f u y u q,l�g�| � j�xzj{/��j v�u |��g�k�q � j�k©i v q as � v nplLn_��sHjUjak�q��£gLf=Xª �L���G�%���/�2�%���?�_�4���O� �?���A�%���?�����������a� �����/���4�m�4�#� �L��� ��G���
A2n
b �U���4���
An =
0 1 1 · · · 11 z1 0 · · · 01 0 z2 · · · 0...
......
. . ....
1 0 0 · · · zn
X
³�³C�
Z�032�o32 .J2-4Jn T 83*�8<. T 8L.1032�g34J93AC,h2-n T =det
(A2
n
)= D2
n
Q:>r032a4J2Dn = det(An)
k�-u�g:*-8:o T 8:x
Dn*a,}9C8:xw.1032�,h*-=1.f4J9�>�Q�>r2��R8:oc.10C*a.XQ<d�9C4
n ≥ 2Q
Dn = (−1)n+2 det
1 1 1 · · · 1 1z1 0 0 · · · 0 00 z2 0 · · · 0 00 0 z3 · · · 0 0...
......
. . ....
...0 0 0 · · · zn−1 0
+ (−1)2n+2znDn−1
= −z1z2 · · · zn−1 + znDn−1k
� T 8:ID2D1 = −1
QC*�8E2a*�=m[ T 8:oC;RIJ. T 9C8E=�039�>@=?.10C*a.
Dn = −n∑
k=1
(z1z2 · · · zk · · · zn)Q
>r032-4J2zk
T 8:o T IJ*a.J2-=?.10C*a.y.1032�d}*�IJ.D9C4zk
T =]n T =7= T 8:xCk�Z�032a8Dn = (−1)na1
QC= T 8:I12a1 = (−1)n−1
n∑k=1
(z1z2 · · · zk · · · zn)A-[ � T 2 .m* � =L�<9C4Dn�;3,h*:k�U}.rd�9<,P,}9�>@=r.10C* .
det(A2
n
)= a2
1
k¡|9�.J2�.10C* .y.1032rID9C8:o T . T 9C8
a0 6= 0T =�=1;:gR2a4 �3;R9<;:=-k
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