A 9C c A c V aA A q E A a SX`TFYy}vEYhga#@HrdqIfASwChuShg'vEc
sYR A 9Cy9 V q T A VA a c A ER i VAyA VR X`TFYUFCBcIAWYH#Shga
IIFfAUHSy dca hcYW%iQISGq bb V A a c A a s R q E A A BASwwIfbbr @wH
C @HbPX`Td9@Hc fRa bdRVtuEzIIFfAUHShgYWfTShga5@p 9i qC A VR P c s
fRa AHAP A @@D@w5t hga ut~zEIIFfAUHwSy dcDFA@fH@b@rdAwC @wC sr| c
VA a c A aC xHRH V PH fA@IAXYFGWFbHBcb5ShgvHrdqIfAa v @wC wv w
Shg@YqWfT~Shg}vE'DY H TAE 9C c VR A a c E V PH A asR E A a c aCg
fRDF@YFvEcrdqIfAa dazE@'ShgIYqWfTE'Shga zEI@WTYUsYfA@IXYFGiHe a9CTA
E V V A asR A VAp q H R HA T AE C YhA`hc}Shga@HrdqIfAaSdAw@YGC vEc
sR E A A c V A VCT E PH 9H a 9 V a c A C @wC
BA@wvCcfVuFzCIfADvCv9@TYF HwvEc HR HA bIUCaUIWi V 9T C wvC@FdAVuvEc
q V c A a 9CT wCzDUvCbPSghHi{F@YFA YhgG@IBA@BPhcEa
UYb9vF%@wUF@YF}DBA@HwEa cUsYbv9FFCYg E aH 9 A E csR 9C PHC9CTA E 9
C R 9C E hgBY5t#dAWWrF@BPhcEdFSY%vEbIUCaUIW}wvC@TFdAl cg VT PA xAy
C c HR HAi V 9 V sR 9C A a c aCg Y%AX`TFYyShg'vE}DY5t PH @wC rs s V
aA VAi A aCg PH @wC YR#IfAqcIWShga#vEc#DY5t rs fR~wvC@TBb@IWdiIWvEc
aDYhga a V 9 cPHA VAi C hgXfTE bR~UHYRWrShg~vEc H ac iA a
BA@wvCcfV|iHw 9H a uw uruh}~reheue5fu
FEuwBRv9YRygh@IdAYvSlgGvEvc@b5Shg#Sy c t 9 s H V Ag 9H A a A
adzE@r|{FvCcIfADCrwShg5@HbdPwCu{tbRs#bRsubbBRdca
dcWwbyEBdcDrIShgDwvCbUbHwX V A 9 V a q A a c V V V EH aAi qR aC qA
aC H cPC C
ShgbdRtVIdW@GI@@wCewrC@wCHS@icVl5IfASfRIw@HwYhgaSI@IABv9FYg A a q s
g V TVAPH H PH ci V aA8 a R E C Q AH 9C
EBdcDrIShgDIAWywvCbUbwCXYUwShgavEcXfThEEvvEYhgaldaDfAUbfRvFdAWTDDAlW'
aC qA aC H i H cPCH A A c c E aH P V aC sVTx+y =4 m P EA n(2a + 1)
(a + 1)
H a 9 VAHA C VR H a 9 E T cHT C A c
bBRdcDdC`vcFzCI@IIvbbRBdcb`TYRGuAXYBbW%Sy DfAFdAV A t c V aH 9C cT
V VAP V AP V c aHA q s qA a A c V 5SvsbR h UIBAFYy @YFAbR'I@@HWTebR
@UzCDwHEUI@PXTdaEbdRVtIShgSy DfAFdA~A5t vsQ bbR GvEcIfAdaFwyhga
cu5FUfwrq}rIBAbSdR@yfR}bbRBdcb`TYRG'aUHc@Wv9v9u# c 9H V EC HC t PA
VC E q 9p Vi a EH a 9 E Vi c t A {Fb@vEcbfRa bF5SghSsYR 9CH V cPA A
a
bBRhcvEdcF@Sgl5S@HvcbUD@PrShg%dAbRD@p%bbBRdcb`TYGH~b@IAGS5}I@UDdA|{Qa
chzEISy bWT H AP A A 9PCA A a V sA EH a 9RE c EPH E Rg t VAPC V VA
cH FdA@y@bfRwFfAUIGDdA@v@pev9v9u5serqwoenWf'lbihYWYsYRUbfRu x Vi
AHR a P aHAE Vi A c t p m e k j gd f ed Qi
XUSWFSIeywp#XfThEvEcevEYhgubdRtVsrIBAbSdR@iShgfRebbRBdcb`TYRGXWTUS#@IGAGFCDBA@8
x Q A c a q E q 9p V A a a EH a 9 E V RQ PHAE E 9 (6 4 2 0 ( &
$ " 7%531)'%#! 10
5 8 = 5 + 6 + 7 + 8 = 26 OP EN PF RF = 5 P QR FQ = 9 P QR F
3
(2a 1) (a 1)
xy = 12
T REN D
4
QR
m
5
QR P R = 13
n
EAR
m < n m a n2
x + 5xy + y
P
O
Q
n
... ..... ........... .. .. .......... ...... . .. .......... ..
.. ..... ..... .. .. . ... .. . .. .... .. .. ...... ... .. ......
... .. .... .. . .. .........................
........................ . . .. .. . . . . .. . . .. . . . . .. ..
. . . . . . . . . . . . . . .. .. . . . . .. . . .. . .
..................... . .. .. ........................ . . ...... .
...... ...... ... ........ ..
. .. ... .. .. .. .. .. .. . .. . .. . . .. . . .. . .. . . . .
.. . .. .. . .. .. . .. . .. . .. . . .. .. .. . . . .. . . .. . .
. . .. . . .. . . . . . . .. . .. . . . . . . .. . .. . . . ... .
.. .... . .. . . . . ..........................................
.......................................... . . .
A
E
9
N
2
5n2 +6nF 5 P D R 13 R T
m n
Y5ABSidca @5RGvE9FvEc sR 9 9T q C C aC a Ay V sR 9 9T C
DYhgSFdR@i Y5BASidca @q%vEc PHC VA a c @wEzIIFfAUH~dAVwC @wC rs
vcBc PH Y5ABSidca @5vEc sR 9 9T q C aC DYhga Ay V VA a c qRE V
SFdR@i IIFfAUHyAbGbRs P@wuzEIIFAfUHydAwC @HwC rs Bc HC VA a c V P
IIFfAUHc~bGEbRs VA a A qR V vsc Y5BASidca @5~@fRyPBFhCvEc IIFfAaUH}
p sR 9 9T q C Ap a c E VA c H
zEVFSFdAI@dPb5BAbBphchEYRWWURsuShgybBRdcDCWvdcda@Tthga
u#@HrdqIfAa@P Qwp cCi P VA VR 9 E i VT A a H a E c t A c V A
BAbBphcvEby b'vEc @wC zEIIFfAUHa BbhPFx FdAbhgaSghUYzaXYSdR@#ShgUsr
Shc y @`Td9@Hc 9 c cP PH VA a c c c A V A asR TP Vi A a A E PH HAA
aAp VA a V PH c aC c cP A P a @wC IF5zt@#zEIIFfAUHcudAwC @wC
vE}DYghva#Ea Bb5@p @HwC A BC'
bBRdcDCWvdcda@tTUIBAdc|fT'w@vEcyhgfAUGE3`I5dtbwC H a E aH E aRH a
VA EH WUSvvQvsdcda@T @wC
IXUdRbhga@Hvc9~ShgzaFGzEVIfAUHABd9dVdc#vEYgh'DS}SdAw@YGE VTRQ E PH
g T V A A a A a c c a EAR VCT
ShgaeYvD@BPhc#WURShgafRa#UII@wCavEc~DYhga dAfVaUIfAhga
u|%DCzS5vEABd9dVdc P A sR EA E VT s A aHAH aC H c t H t VP c
fRwvC@TBb@IAWdiIWi IXTUdRbhg@Hvc~Shg@sYR a V 9 cPH VA g V a A 9 A a
bBRdcD@YFrShg5@HrdqIfAr H aCTA A a A c V aA
UHYRWShg#vEvdAw@YGySghYrdAfVaUIfASl a c i A a c VCT E A asR H Ag
@wC IXTUdRbhg@Hvc~Shg@YR PH g V a A 9 A as bBRdcD@YFShgavElDY5t
UHYWiShgXDC EzaFGzEIfAUHc CBa{wSi bdRVt}@Hvc}SghIrs p H aCTA A caCg
a cR A aa A V a VC q s A 9 A a @wC PH gT V A 9 A a9 9 VC g T V A 9
A a R H aC A A a aCg IXUdRbhga#@HvcShga~fRbDBAv9FzCwSi
IXUdRbhga#@HvcShgbsYwbRBdcD@TYFuShgvEcDY BC @BPhcvbvEc A E HR
UHYRW~S P H A E H R ac iA P Ea R ahg%@wC @BPhcvb~dADwfAVE C
Bdc{aIS'FCYg @HwC dAVw@YUHc Y Wi Sgle A @wC PH VAy E CTw
w
sYR A 9C 9 ER qT cC q A A aA
X`TFYy%BAbBphchEYWiWqFryShgau@HcrdqIV fA@P rsu vEc sYbRGvEcby cbP
RV C aC cpCp Vi A yvEc DYhga{Qa vcv9bUbSdR@Shga aG|Shga
bdRVtsIGYShg AE A q HAER aC a A RiiT Q rbR@@wzCrvEc
vs5DYhg{Q{aVIWbdR@iShgahga cu~IIFfAUHc3wyvEc DYhgwGEYW@WD 9 q PH V
c aC a Ai V A t VA a HC 9 C A hga r y c Dqdc~FdAbhga}vEc rs
Y|@I@wYhghxvaDBAuShg'DC uY@%dcHRs AXU`T vCFSYrwSw#fA@zaA
HXzTdqfVIbafAEVda Hb@fRA P EA a A V sR PHA PHC s 9 A a a C 9i Qp F
P vEc IIFfAUHua BbhPU hcE@bfRFGy YR@IP@wYghhxwIBcwShgDC @HdcUvCSQwp
VA a c c c x PH AE s PHA HC ag V A a a C 9i FAfzaXzTfVdabbfRwvEc
VIIFfAUHcva cBbhPU hc3IIFfAUH|Sy dca hcYWya BbPhFSYvEc P EH A a c x
E VA a c A ERi c c xAy C wD@5Shg5wwFABv9DSDzCfVa#fA@wCdavEcbP qCAp
A a Qp P 9Ay H FCwfRShga@HrdqIfA@P hgaG@IBA93Hc|dAfVAq hgaw@dAwC 9
a a A A c V aA H a Rp V @wC rs# fR%hUbDbfRqwD@vSgha bbBRdca PH a Cp
EA q CAp A EH hxFAWdA'YuVI@qWY|IAfva'r%twSgG5FC UHYWuaDC VsR Ap TH
C V s H R E E a cRi
VbdRdVVcrqeIShgw#ShgaeDAUBcfVda%@wCXbdRdVVr#@buRfwDBAv9FzCwSi VA aR
A E E PH V c q AHR a9 9 VC vEDSDzCfVa fAdWUGEbdRVts awIBvcbYwD@p3X
UHcYWi 9Ay VTR C q g 9sR qCA a R A a a sR 9HC HC aC AA q V c q R t
Shg%DC YBA@wwvDyaFyzEbdRdVVr5b M N V A 1 M 11 100
CD
36
ABCD P (0, 5) Q (0, 3)
c
a
F
b a 2b A n
d
B
M
{1 2 3 . . . 999 1000} M 1000
H
F
1 N
7
G
A (0, 0) B (0, 8) C (8, 8) D (8, 0) AB F (8, 1)
S
5c + 4d H A
10a + b = 7m 7 AD SA 30 S N 7 Q B
H (4, 4) FG
AV
n = 7k
N
2A5
S
G
F
.... .... .... .... .... .... .... ...
.........................................
........................................ .. .. .... .. ..... ..
..... .. ..... .... .... . . .... .... .... ........ .... .
........................................................
........................................................ .. .
7
P
P
13B
G
x
A
m
(A, B)
30
0
4c d x k Q QV
P
9
A H R c A EA V a EA A q q E 9 EA 9 T FA SHb@P%Afa
@fTE5vC9~@PDdqIfAa A D@PuwbRG5vC}daD5ABv9DAX' C BAFCFA
daED#DAdqIfAa}zEIBAcrqIdA@i D@5qwbGvvC9 DdASb@5@vfAa @fT5@WbwC} 9 A
E V V V EAP A qRE E Vp qRH AP c E AH T EH AB@wC ~@%dAWTfEDq 9H 9 AP
V A vCI@HrdqIfAA WSA dVwCWH daDA 9 VA c V a V T E alFCzIA
DvCv9@TYFA A9 V a c BA@wvCcfVaW%daDA 9H HT E V 9 V AHR aHA HT
IBAvc@TFA @bIUCUIWi3W%daEDA A q q V cP 9 EHC TC aH E E a
wwzCDUvCb|AB5bw}}WYIFA UbRG%DA wR E 9 ET a aA TC aH E E 9HC E 9 ET
a HRP E a DBAu@UfRYIFA UbRG5DAB@w%DBA~@UfRaUb@5DA wR AHR 9R HT EA V
9T V AHR HAi H @Hdc|C @bISQ YWiW#daD~IBAvc@FA w@bIUCaUIW A VT 9Cy 9
EA 9 T @P}WBAFYuvC'daDvBAv9DAXY aA I c BA@wvCcfVTYudAfVaA qcIA i 9
H a P V BA9udaED}DAXY A9 T aA c I C dAVFvC@TBb@IWdiIWya YGE AXYdAIA
bwrq@P c 9 cPHA VAi cR T V cHC A V E a cRi HT E WfTUHYWW}daDA
AB@HwvCcfV~AB#bw} 9 a 9 EHC hw uruh}~reheue5fu P QR F PF P R = 13
RF = 5 FQ = 9 P QR
cRi H FEaUHYW'Bvcd9S{Q@bfR%dAVwC PH @wC DYhgSFdRW} DC
EazFGzEIfAa aC a Ay V8 a A V PH a A V a x c PAPH a @HF@@IfADfA @wC
DC EazFGEzIfAUHc PAPH a a V P F@@IfADfA DC EzaFAGzEIfAaUHc Y#D@@wTH
Ca a DC EzaFAGzEIAfUHc V a sR EAX P Bxdca FCdAwC @wC DYhgaGE Q DSy
dczaFAW7E 9 V PH aC R 9A i FdAV @HwC bRdAwC @wC EUHYS8 x P H V PH a
cR V aA q c c E IfAwvCbP~Ea FC hga u%DzCS~vEc BAd9 c t H t VP
fxVdc@wC IfAawvCbyEa 5FC hga ut|%tDzCSP PH V A q cP c E c H V
AB@HwvCcW 9 V vEc BAd9dVcXSfAbTUrvEc a C GYBRvfRGbbSgWi' AE 9 a
AEHR 9 fAdbR@wvIShgISfRafb5IFBAv9YR@wCbiQIAlwC YRDX`TFYvDCY5%bb V s
PHC VA aA VR t HA 9 PH 9 V H s EA 9Cy a g t VR V E A a c cC C AQ 9i
c A C Ag {9uw@p~Shg%H'DBAbfpwCrq hga cut@Hdc{awCdaqdcvEYhga
HFU5PFSDvCS}vE#qw~Sl p tR A a c E 9 V
GYBRvfRAGbbSWi'fAdbR@HwIShgaISfRa AE 9 a EHRg 9 V s P C VA A
fb5tuIFBAv9YRv@wwbiQIAlwC 9uw@p#Shga~HDBAbfpwrq hga
uE{aVwCdaw#ShgaUrs BC VR HA 9 PHC9 V H tR A c E 9 VC c t E A qC A
VAH c t A a V FEzI@SHu#Shg#dAwC zEISDvCS5zt'IShgwShgw@wC
IGYBRvShgavEl9uw@Shga bdRVtvABbfpwr#daFvCvShgDUUCa VAQ 9i R a VA aR
A a PH VAE 9 A c tRp A q s 9 VC q E 9 A a EA Q i D A g 5 H E P C U
A R W 9 H S a F R IA a 9 V A C g A A H SdaR 5@tVgzE }I@SHAV YvCIS9R
GSlSWg ibRhgGaR @w9H uw@FpcR H SyahgC GbbbdRHqE SDE i 'bwwIrhgu5ItH
zlABbv9V 9Y@ShgHbFbiQIAlbIdHH ShgTaa c A bbR '# cut t A tVs BA9 fpV
C q R a #A bR @ c y RIdAV E W EH 9 R H q V UUCa#hgUDvIFBAv9Y%@wC
bQiIl GbbSWi'W{9uw@~HDBAbfpwrAbG}dAw}dAISgl A CA HA 9R PH 9 VA
AEHRg 9 tRp C c E 9 VC q qRE VC VA
S
C2
DE Y Z DF DE EY FZ DEF EY C1 P FZ C2 Q EY C1 R FZ C2 S P Q R
x+y =4
DEF DF
xy = 12
N
5
P EA
OP EN EAR
C1
T REN D
QR QR
n 5
#
!
5n2 + 6n
3
4
"
"
"
... .. .. .. . ...... ... .. .. ... .. . ... ..... ... ... .. ..
... ...... .. .... ...... .. .. .. .. ... ... .. ... .. .... . . .
... . .. ... . .. .... . . .. . . . .. . . . .. . . . .. . .. . . .
. .. . . . ... .. ... . . . .. . . . .. . . .. . .. . .. . . . . ..
. .. . . .. . . . . . . . .. . . . . .. . . . ... . . . . .. . . .
.. . . . . . .. . . . . .. . . . .. . . . . . . . .. . . .. . .. .
. . . . .. . .. .. . .. .. .. .. .. . .. . . .. .. .. .. .. .. ..
.. .. .. .. .. . .. .. .. .. .. .. .. .. ... ... .. .. ... ... . ..
.... .. .... .
......................................................................
. .
.....................................................................
.. .. .. .. ..... ...... .. . . ...................... .........
.... .... ................... .......... ....................
..................
C2...................................................D.....................................................................C1
...... ..... . .
E
N
x2 + 5xy + y 2
P
O
Q
. . ...... ......... . .. .......... .. . .......... ......
...... . ... .. ... .... ... . .. . .. .. . ...... . ...... ... .
.......... ... .. .... .. .. .. .. ........................
.................... .. . . .. . . . . .. .. . . . . .. . . .. . .
. . . . . . .. . . . .. . . .. . . . . .. . . ..
..................... . ........................ . .. .. ...... .
...... ...... ... ........ .. .. .. ... ..... . .. .. .. . .. . ..
. . .. . . .. . .. . . . . . .. . .. .. . .. . . .. .. . . . . . ..
.. .. . . . . . .. . . .. . . . . . .. . . .. . . . . . .. . .. . .
. . .. . .. . .. ... . . . .... . .. . .. .. . .
..........................................
.......................................... .
A
E 9
N
F 5
P
D R F
13
R
T
#
#
#
SGbbWIA 5hC}IWvdcda@T FDABbBpchhEYWi AEHR i V E VA E E 9 ER E
9iT aCT E VA V a DBASWUfR#dAfVD@Y'DBA9I@HcrdqIfAA P wSi#BAbBphcvEby
cbdaDA aA DdAVl cYhgvEYdRfVa VC 9 c P E E c @DdASpb@5Y@D@'a
@YSdR@iuBA~AXYUwYhgUI EzIABdcUIDAB~dAVcfVFA WTUWDA vEvcv9dcbT AP E
V qRH TAP EAP cTP V 9 T aHC C V aHA E 9 V Ri E a C
@vDBAY@pSQvD@P%@b@5UbGE A aA DdAVl A P c w AP E 9R q E EA HRP aHR a
E R cYSafAhg UYwhCTH @5a fTYbRBdcBCa Ew E H UCWvdcda@tT @WT#wbfR%A
IA BPhcbbfR#FS}daDA wHhchED@Pw @SAGbbWIA vCh'IWvdcda@T x E A H A q
q V E H E C i E H E A H E H R i V E V A E a VC A ECicT c V 9 x iT A
wSiGhEFSY@YfAa YdRSP~vCUv9dchxzaFAWWUfR#BAd9dVIfAwSA dVwCY'DA wR
dAfVD@Y A V TP E a aCT aH AP V T c EH H YIrUIAI@wCva YGEv9c YdAIA
bwr@P dAfVUIfA@ABd9dIfAHWva @zTfVdabbfR P TC aH H a cR T V cHC q A
C dAVFvC@TBb@IWdiIW}daDl@TYA wS%AGhEFSW@Y c 9 cPHA VAi EAc a VCi
ECicT fAa YdRSvC@%bBRdcD@YFA WI@HrdqIfAA wSA
dVwC%YPdAfVaUIfABA9ldaDA c VP 9 AP H aCT 9VA c V a V T H E a cRi
UHYW'A a Ci ECicT c VP 9 A A VwS%AGhEFSW@YufAa YdRSyvC~@P
bBRdcD@YFA 9#daDAwBAv9DAXY UHYRWIC AWWUfR V C 9 @ S vC9 I H aCT E 9
T a c i T iT A V c BCrABadcA {awSiwSvCC yUwyCPH hfAhEaFYSdRCcE
~PafA YSP p V i w a i c dRV vCuC BADA v9FzCwSidaDAW@Y'A
wSuGhEFSS@YfAa YdRS5vC#@PwbBRdcD@TYFA 9daD%ABv9DAX' BC 9 9 9 VC E
cT a VCi A ECicT c VP 9 A H aC EA 9 T A wR %BA9 a EUHYW}EDA a cRi V
E E WfTdaDA UHYWi BAA A wR uBAVWfTUbGE a cR 9a a 9 EaHR aA a A
EwbRGWURWvC aA q q E VT i A dVwC V H hw AP 9 E i VT 9Cy AP V E 9i 9
VA c V @5BAbBphchEYRW'WABFY#@@HwzCD#@`TS5vCI@HrdqIfAaA P T VTRi
cpCp V c 9 C E A P V T A c c P H c I C BA FFA daDA @}WGvEby bWTa
YRGE AXY}WUW~A a vcv9bUbSdR@i vC9 BAbIGbIA bwbP
dASpb@WydPVwhCFCYI}a chvEYShg 9p qAEH 9 EHC V qRH H T E g TC cR bRI
fAUwYy @fTE%A A c@bdR@rvC5WvcIA yW@Y vsdca chYWIABdcUIHWudAIA
PBhcbbfRHX H c aHC c a Vi Vi 9 A V cT ERi V aHA T V EH
@}VWBAFYyuvCI@HrdqIfAA P vEYRvEYdRfVa~C FFA daDA AXYaUwYghUI @P AP
T 9C 9 VA c V a c s c 9C E T HC C A TaH 9i H E c E AP ERiV HA A c A
ShgXIA a rIA fVDfA 9C HWUwdC vCSuIA DdAVl Yhg# hc}@bvsdca
hcYWIBAdcaUIqIA FYT@P A TC c q a HW%a @zTfVdabbfR3' @|fAa YdRSA a
rIA fVDfA 9C W5aUwdC vCS3HIrDdAl Yhg hcE@P T c EH H AP c VP c q a
HT H 9i A E V c A vsdca hcYWiIBAdcUIAWTa @zTfVdabbfRHeFDdAVl
Yhg@Hdcr@vsdca hcYRWIABdcUIWT#daDA ER V aH H c EH E c AP E i V aHA
H E TA c 9 HRQ V 9 VC WY@HrqW`TybSDzC%ABwSi
AXzTWUfRdwSiBAFCwfRwfA@wCdavEbPvCuI@HrdqIfAA P q @P VT VC 9 a a H c
9 VA c V a A VT TH A T H C aH a c a Ri T WAX@bBR9@HW5WTUYhgUbR A I
UHcYWIC UIBAbGdAVY@HcrW`TbSDCzAB9 bbRBFfWA |@|dASqb@H aH cy TA q 9
HRQ V EH c A V AP Vp R HFC{aIfArWDA @i}I UHYWwIeVYdRVr~dAfVbIC
W@iwzCtVs c V HT E V a cRi TC c c q aT 9 Ai @YvA~VYdRVcrW|C ABDA
v99FzCwS5daD5Ai CzfVABuUb@P cT a c q HT 9 VCi EA a a 9 aHR
UHYWiI|daD3fAdWUGvC9Ub@YA@HrW`TbSDzCVuWT a cR TC EA V TRE aHRP T c
q 9 HRQ H dAfVda@`Tv9v9cwwzCDUvCbA aUHcYWIC @AB@wC E A q q V cP Ri
T AP 9H WTWIdqbR#C dAIA bwr@PzEVYdRVr#Y@#A UvCS#uF H VA V s V cHC q
A c c q TAP 9i C H AP AT V T @5XYBcIA qWYH VT 9Cy a VA c V a q q 9
E WBAFY#fAzafA}I@HrdqIfAA lSAIA vvC'daDA T 9Cy ERi aHA q V aHA TC V
R VWBAFYvC9 vsdca hcYWUIIfAzaBcfVdaEIBAdcUI#AXAYPUYhgV eWTUH SYE8 A
P A q q E 9 H a A q BASwIwf'wYl a A Uw@`Td9@Hc C
@zEIBAdcaUID@%wbRG%vC}aUwCA wqbfR 9i qA A VC8 aHC A a b@DA 3bR cH P
H hc%wBEvsdcDFA b@EzIBAdcaUID@vUbRGE aA cI E aA aC H HRH V HA EAP
aH # #
m
n
5 8 = 5 + 6 + 7 + 8 = 26
(2a 1) (a 1)
(2a + 1) (a + 1)
S
S
#
N
1
30
SA
AV
V
1
A
m
n
m < n m
S
.... .... .... .... .... .... .... ...
.........................................
........................................ .. .. .. ... .. ...... ..
...... .. ..... .... . .... .... .... .... ........ .. .
........................................................
....................................................... .. ..
N
Q
#
N
P
M
x
Q
{1 2 3 . . . 999 1000} 1 M 1000 100
ABCD P (0, 5) Q (0, 3) CD
x
A (0, 0) B (0, 8) C (8, 8) D (8, 0) AB F (8, 1)
H (4, 4) H
2A5
13B P
(A, B) ! #
A
F
F
B A
G
AD
H
B
M
36
M
FG
Q
G
1
n
A
a
30
F
N
m n
0
G
V
P
9
A IC R 5 E A c 5t V c DC v V R E A a A y E C t 7A R g y 1 FICRE
5UzE DUSIdVdAQHAaC {QW{aV VTYE78A IDSy Dhgy YYgcC B 5t
IBAbbr@HwbYyCcH hgYa9 | I@A PB9YDqdcSYFu8I5 UC w c gl V S A V Rs w
A R SDYy5rISy YBRdcDzCyShgbdRYuIUzEwDS{Q{aVcYhg%IUE IABbDu@p aA
AyCg A t HA c E a V A a q V RC VCAQ a A C V cyC 1 A s p h f C B b`
a `SY FT TSQ yth(w`rqD`zieTbrDCbEFgeTSd`"TEcB@8QGfC"&FYFG4"RVU
W U0RP H @ B SXUIVFA U'aUIGIdAFGE@bECi D i8A95 C H A D @ 7A I E V R
vEDY} vE5IUCE b@wvCYR 3 fRau3 IUCE IBAbD18YRRBdcDzC5Shga DbS@ c aCg
c A CPH 9 A V cyC s a V A QCPR BRdcDzCShgwHvdAI5%8I56@wC b@wvCYR 4
IBAbD12Y%DIUvShga IU#EzwDSrQ{{aVYl~ a V A a c V A t 7A R P H C P H
9 V c y C s R E A C A R C V C A Q c g 0'( $" ) % ! uh}~reheue}u~eeD
Y UwHFdAwDW@w'aDYhga c P VCAiiC C AH 9C aC qA aC 9CH a cy A a a EH
a 9 E A a A c A t a I@IBAv9FYg%EBdcDrIShgD'FbbBRdcDCa bUHi
UwwShgfR#bbRBdcb`TYRGvShg%Sy 5DfA
#
#
V A q q HT EE E H a S BAd9dIfA%IA wWVWfTDA Ta hc'aUbRGE A
EUHYW%DAByXY#IdAfVUbRIA UHcYWi a cRi E 9 AT V aH q a R T AiT IC
WWUfR @}UIAII@bBRYdR@uAB'A AP aH qAH 9 Vi 9 a UHYWIC WWUfR
@5UIII@bBRYdR@i a cRi T AiT AP aHA qAH 9 V A UHYWIC WWTUfR A UHcYWi
a cRi T Ai a a R T A T IC WiWUfR AB@wvCcfVY~zEWTfAa 9H a TP V
xXIYrD@rUIBAYRGE aA XTYdAIA bwrq TCg EAP aH c A V cHC @P A
WfTE|UbGE aA EBvsdczaFWi A a V aHR A xE V a c i E GDdAEUHYRW'DA
dAfVaA wvCb#@P ABd9dVIfA q cP A HWA dAfVaA wvCb@P ABd9dIfAW5A
UzCfVa Ta q cP A V HT C H 9H a C E 3SBA@wCbTUydaDA AB@wvCcfV|A 9H a
A VAi A c H T @dPIWu@HYfRaU} Y V V E E V TR i 9 qAEHA V cCy aHAyTA
V aA c A IdAWfThEFC
~WUW|BAbpIGbI5IBAv9v9FYDzCfVaUISbWivC9wCXv|fAaza BcVXXTYfAhhxdaDA E
Ay H q FSDC IA b H AP VT 9Cy E 9AT @wzEWBAFYyDBAv9DXY5 R 8 9 HCP
VW@UdPTA SIiuY@@HRA9{ pfRBAUbE wbIDdAEV BAWv9Vv9 fTbhEc p FC
VWUWABfAb@AGqwI}fRA FIdAV ABV3Fy p V W Y a } Y E E T R 5 9 p I bH
v9v9c Y A c H T i q E C UzCfVaDBAv9DhAhxUISbAWyvC9wCXvfAzaa BcX
bYFA PICWY@yABbwb~DABv9v9bp r@v9c BC x E 9 aHAyT i V aA c V aTp T
9Rp 9 EHCP E c C Q E HCHC aH FDfAaUwbDUUbRGE E aTC TAP E aA aHC V i
E c V c VAP 9 A 9HAc AHH V i c
DdAfVbIY@}DBA9e#fAUwbdPIAWdaDA~BAv9v9bp#dAIA bdHI@uvCSyA
bIb@TY#@SbRGzEIAW~C SfAa @fTE @PhcbHF}C IDSbU@H%@vAXUiR
@HYfRUH}}vE@bi IA hcvEYdRfV'daDvC9wX IA FY@daEDA A E c aA TCAyTR AP
T A c a cT A q c a EA VC A q cTAP fAzaa BcVX
IBArIdA@idaD@HYfRU}h{Y@vYDBAv9v9b#Y@5U@WHWTUYghaUISA bIvC9VwX c V c
q V EA A c aH 9Rp TP E cp TAP TR AHT C HAy 9HA C a c A c aH 9 AP T
a 9R HT EHCP E cp AP Vp qRH HT C Q A~fAzaa BcVX @HYfRU} BAYR
V@VWUfR {Y@pWbwbDABv9v9b~@#dASb@W5Y9rb @5BASidca @TrHWWhchE@I'daDA
AP 9 9 q T ETC E AT XY AXYvEDfAEzIABdcUID@eaUbRGE A I vcBc T 9 a V
aHA EAP H a c VIdAfVUbIA P @BASidca @quWTldaDA aHR q AP 9 9T H E
@5BASidca @rHW'daDA AP 9 9T q T E ATH A V H q XTY'IdAfVaUbRIA P
XY@bfR BxDAXY#IBAdcUIAW'WUWi 9 T V aH HT VTR AT 9 a V HA EAP HR
XY~vEDfAEzIABdcaUI~D@#aUbGE aA I Bc c XY@bfRBDXTY ATH 9A V aHA H TR
IBAdcUIWTuVWUWi hcE @BASidca @3WT5daDA IABdcaUIWT Yya b p AP 9 9T q
H E V HA H T cP H
a
k
b m
#
c
d 4c d
5
n
N
DEF C1
R
S S
x x + 30 = 1:2:5
R
P
C2 Y
6 7
FZ
(2x + 30) Z EY
FZ DEF EY C2 EY C1 FZ
DE
DF
DF
a 2b
DE
C1 Q
C2 P Q
5c+4d 7
x
10a + b = 7m
... .. . . ..... ....... ... .. .. ... ... ..... ... ..... ...
.. .. .. .... ...... .. . ... .. .. .. .. .... .. . ... ... .. ..
.... . .... ... . . . . .. ... . . .. ... .. . . . . . .. .. .. . .
. . . . ... . . . ... . . .. . . .. . . . . .. . . .. . . . ... . .
. . . ... . . . . .. . . ... . . . . . .. . . . . .. . . . .. . . .
. . . .. . . .. . . . . . . . .. . . .. . . .. . . . . . . . .. . .
.. .. .. .. .. . . .. . . .. .. .. .. .. .. .. .. .. .. .. . .. ..
.. .. .. .. .. .. .. .. .. .. .. .. ... .. .. ... . . .. .. .. ..
.. . ... . .. .. . . . ...
.......................................................................
................................................................ ..
.... . .... . .. .. ..... .... ....................... ........
.... ..................... ......... .......................
....................
7
C2...................................................D.....................................................................C1
...... ..... . .
E
x = 6
5 6 + 30 = 60
n = 7k
7
6:7 7
F
N
EAR PH DSI@wC bdRdVVrrShgwfRaDBAv9FzCwSi%vEc fA@HhcEB V c q A a
9 9 VC dAIS5t VAg UIADFG~@Hvc9~Shg%bhgw@DBAvc9
DYhgafAwtFrwD@rFfAazFWdAr@w'aUI@PBdc@HShga aH qAE A A a HR aRp E aC
aR E qCAp P A V PHC HA c A HA bRsriQfVqwSQEeYR%vEFC|wD@BPbFdR @i
UIDFGy@HvcvSl~ FC aUIqDFAGE V aA q s c HC EA cy V aHA qAE A 9 Ag E
@HvcShgaYR~UHYWfi@BPrrShgaD@UC UHYRW#DC aUIDFAG@HvcSghazFGzEIfAa A
9 A s a cR x c q A 9Ap ac i Ca HA q E A 9 A a A V x c t c VR A a g
T V aV c q A a a9 9 Ci A t V A t @Hv9v9cu# ca
HBcbwShgIXUdRbhg}bdRdVVrwShgwfReDABv9FzCVwSw@Hvc9wC DzCSPw5ers s p
h f C B b` a `SY FT WFTSQ PH B@
yth(w`rqD`zieTbrDCbEFgeTSd`"TEcB@8QGfC"&FYFG4"RVUSXUVU0RIFGE@CDbiA9
UHcYWi a R ShglY~DfADbHbdPbSfRyShga%@HcrdqIfAAvFvEFC ShgrbR
UHYRWyShgyfRwPFfAzaFAWdAvEc A asR E aC c VR A A V a c A a H a c i A
a a V PH @wC aC V c q A a EA E c VR A a q CAp V E 9
DbdRdVVr%Shg}DUBcfVdaHBcbvShgbdRVts qwD@IAGFvC
@HvcShgbRb@wCda@wC@wvCSi ShgfR%wvC@TBb@IAWdiVIWwvEubdRdVVrDS A 9 A
a H EPH E PH AH 9 A a a V 9 cPH Ai c V c q aC P @HwC
dAw#EGEbBRdcb`TYGEr5ztauSgl VC aA H a 9R R A V bR EA c H aCTA a DSy
bRBdcD@YFfRUHc @wC bR5DX`TFYyShg@HdcbTa dcdabXD PH V s EA 9Cy A a a
EpT V bR cc V sA9 @Hby bRrSy YRGE PH H aCT V E t 9 PH EH aCTA a V s
EA C @wC rbBRdcD@YFAbfRzaUBCs|fRa~@T~EuwBRv9FC r@wC
ybbRBdcD@YFfRUHc bRuDX`T9FYy c VP SU c c AEA a E T VR GDShga@HdcbTa
dcdabpXDBdcDzCSU@YbyShgabBPDBAbDQEc SA y Hc vC@9 bbRHVdqT RBDadcs
Bdc@DYzCTaC FAy@@TC YHdcbSaTAa hgdc#bvCEdaa by D hgBYg5t a VPCT A E
9 pXT s p h f C B b` a `SY FT WFTSQ PH B@
yth(w`rqD`zieTbrDCbEFgeTSd`"TEcB@8QGfC"&FYFG4"RVUSXUVU0RIFGE@CDbiA9
r EH aCTAsR q E E A a A R bbRBdcD@YFY)IfAdabQuShgSy 9Yw vEc A q E A
VC aC a c cP R a C A E c aH whCuShgadAw}DYhg'Ea Bbr5ztyQ
d9zaUIf#bHFCUbfR HC a E 9 VA c A R wYhghEDAB'IIFfAaUH~Sy dca hcEYWi
C aC a cpCp Vi Ag A q A a c cP R a C A Ay g a }DYhg5{Qa
vcv9bUbSdR@%SlSwhCEuShg'Ea Bbr5ztQ d9azUIf%SDCYDCYhga H a
wCYhg}hEEDBA9 zEIIFfAUHcuSy dca hcYRWi VA a A E SDY55t a
@fTDdA@URBbGdA@%Shg5fR}vEYhg@Hb@ AyCg A 9 E V ET cy Vi A a a c a
cPP A q E A a c cP R a c aH aC SwhC#ShgEa Bbv5zt%HFCUbfR'DYhga
DYhghEDAB'EzI@WYH dAwC#dAISgl p aC a E 9 VAp qT V VA VA qT
FEzI@pWYH sYX9FCawfR~@Sy c R a CET A EAE A V 9 VA q T
DGFCrFdAbhgalv9'FzEI@pWYH YReFCwfRybRs WUdRDFdAS@WTUhgUDAHc s 9 a a
C V iT V P VPH g C A VAp q H EAP c AEC VAp T dAw5dAIShgrhgBY5XYR
DfA UBvc9#zEI@WTY~D@X`Td9@H5 GFXFEzI@qWYH VC VA a cg ts a F U D SP
H T g UC A 9 a T V V C V A c g F a t UBvc}zEIY@wpRAsV CWwYfRTaq
HDCE l@bPVA XRs `Td9@i WHc~dR5%FFXdACPE @zEWIU@5hgWD5Hc
dAVw}dAIShgY#WhgRas F9 BYw5YRCs bDR s fA WUdRVD A 9 G F V A pq YH A
T Ca fRa V i T P V P H g C I@ H A c VAp a R H b p a FAdS@WUvcTB
zEhgIU@DWHq Y}zEH D@X`Td9W@YHTq 3 hgXGfTA FEE XdAIw@|hcBPCAVV
dAbIbSfRAHVE glaR f DAGFC#w@by}YdAT Vw'DdA fISAAaV gl CB U V A p T
EA P c C A 9 % E E FdAV hga A C s p h f C B b` a `SY FT WFTSQ PH B@
yth(w`rqD`zieTbrDCbEFgeTSd`"TEcB@8QGfC"&FYFG4"RVUSXUVU0RIFGE@CDbiA9
A q E A a VC a a c cP R a SwhC~Sgh#dAw}DCYhg'Ea Bbr5tzvQ d9zaUCIfA
wYhga'hEDBAIIFfAUH#Sy dca hcYRW%DYhgu{Qa vcv9bpUbSdR@~Sgh#@HrdqIfA
p HC E 9 VA a c A E i C aC a c Cp Vi A a A c V aA E aH bHcFCUbfR A
q E A VC aC a c cP R a C SwhC~Shga#dAw}DYhg'Ea Bbr5ztvQ
d9azUIfAHcFCaUbRfDYhga Sy hc@`Td9@Hc H aC A E PH @wC HAA aA VA a c
s VAp qT A a A c V A IF5zt@p zEIIFfAUHYR I@WYH Shgn@HrdqIfAa CB
131
y = 1 z = 1
122 133
252 28 = 999 111
1000
y x
243
9
19 x = 2 x = 12 z =1 z = 1 4 2 x = 19 y = 3 z = 1 12 4
111 9 9 = 81 9
1000
x+y+z x y2 z2
x 3y 2 + z
2
112 113 9 9 = 81
100
y = 1
9
xy
y = x+4
Q
x
OR
Q
OR
y = x G
PQ
OR
R PG
= 2 = 2
= 0
243 + 9 = 252
G
y=
2 3
y = x+4 100
3y 2 + y 2 = 0
1000
P (1, 3)
121 99 = 81 9
x y x=2 999 100
dAw@YGEWYRhgG@HIBA~@BPchuShg#@HrdqIfAa VCT s a 9 A E A a A c V R
E aH 9 A t A FA#Q DSy dczaFW7DdAV @wC dAwC @wC
sY~YhgG@IABuSgl'wwBR9D@p x 9A A iE P H V PH H tRg E %wSGE~FC
dAw@YGEeYR @wC D@PBhc|brdAwC @wC EaUHYSr VCT s PH EA E HR V PH cR8
vEc%EUIfAUbfRShgYAX`TFYyFCwfRyShga fA@I5XBv9rwfAa Y5YvDdAfVa vc9
@HwBv9rq aH aH A a sR 9C 9 a a A HA c q cg t sR E P C c
fADvCYfRSShgUYDdAfVa vc9 @pda@q}dAIShga wDdAfVUYDdAfVa vc9
dAw'dAIShgfA@HI p a 9 R sRE A ET VA qC sR E VC VA a c vE{Qa
vcv9bBphchEYRWrQ bb%Shgy@wC c c E i 9HR A a PH AyCg ET q A q
SDYda@%5t wCDdAfV sR aHTR q A a Qp 9 c cP VA A a c q a 9 R PHC c
YvUWUwC|Shg%vEc rs wBAbBphcvEby
bvEcuIShgaISfRBv9r|fADvCYfRSShg3@wfAa Yg5t s H UR q A c a 9 R T c q
c t XT q E C a Y@UUgaT lW% wCShga Bv9rqfADvCYfRvESc ~SzER hgIV
wFFUBaCCcEH v9brrSfAA hga Y59v9g FYhgs r`TFY|XfAy BFuCc9
wv%cvEafRaazt lVdAg I SA ghCBfA@HhcE E A @ H fR al e R W 9 S C A q
R A S T H FG B @ zEI@HFCUbfRv hcuhg@`a DY9 EUfAGUPaS Gb`H fR5ShgPA
IY%CRE@Dbi`TA9y V A c a H E S sYR aUI @s X 9FY A H a A C
FCwfRShg}vE}DYSdAfVa vcIWi vE5Bv9 rq%fAa Y5tShg@Y5fABc@%Sghv@wC
dAfVa vcIWi 9 a a A a c aCg 9 VA c c cg A asR Vi A a PH 9 VA
vEwBv9rfADvCY S Yr @rA c cq a9 A sR Vi 9V C Vf YR V i A gl as
fRSwSDhgCRq dAwwhghgacauyFBfAP BAcv9SvEShgDaa YhgdAfVa IaV
vc@HIWUiaAC bRf5ScA vE hg@qswDdAwSYhg9A `T SrhgfA'ca B vE@YS pBC fV
t 9 c} C A F H a YuA W R y S 'DC g c R q A ca
XBv9r%fADvCYfRSSghF5Bv9rufAa cY55ghXTq c q a 9 R EC c q g t
FCwfABcuzt%FYFFuIAdqwUXBv9rfADvCYfRSShg%bRBv9rq|fAa Y5IShga DAhga
uFBAv9S E a ECg ECC V VC c q a 9 R V c cg t VA c c t P 9 dAwCuSY}
IShgwa vS hg@P wwDdAfVhga utrFBAv9SvEzEI@HFCaUbfRyGDShgWYv@FDdAfVa
vc9 V Ay VA R A a HC qC c P 9 c VA c H AEA a sR AH E PH @wC sR E Ci
t VA c H E E g ECC VA VC YDBAdca dcUSwCwhga cu}zEI@HFCaUbfR
hcFCY#FFuIdqwU ) $" % ! uh}~reheue}u~eeD x
HFCUSfRa c apR a EH A A AGwC%5t vEcFC Shga bRvEc fA@HI H c vEc
YRbBRdcD@YF5Shga @wC YDfADbHbdPbSfR PH sR E aC c VR s H aCTA A
Shg)bdRY5 A a q V AyCg A SDY5t fA@HI c dAbRDdAISl V s VAg g T V a E
A E C i P H IXUdRbhgDGhEFSy@wC V c q A a a V 9 cPHA VAi
bdRdVVrShg~fRewvC@TBb@IWdiIW}vEc fA@HhcBE dAIS5t
UIDFG5@HvcuShgbR@wC h HBcb5ShgvIXUdRbhga VAg aHA qAE A 9 A a H PH c
VR A a g T VPG y = x + 2 y = x+4 P (1, 3) G = (1, 1) OG = GR R =
(2, 2) P R PR y = 1x + 8 Q 3 3 y=0 Q = (8, 0)
20 66 $3 20 + $2 33 + $1 66 = $192
31
PG
2
X
Y
c 2 (mod 3)
Y
D........................................................................................................................................................................................
C .
A
XY AX
. . . .. . . . . .. . . .. . . . . . .. . .. . . . . . . . .. .
.. . . . . . . .. . .. . . . . . . . . . .. . . . .. . . . . . . .
.. . . . .. . . . . . . . .. . . . .. . . . . . . . .. . .. . . . .
. . . . . .. . .. . . . . . . . . . .. . . . .. . . . . . . . .. .
. . .. . . . . . . . .. . .. . . . . . . . . . .. . .. . . . . . .
. . . .. . . .. . . . . . . . . .. . . .. . . . . . . . . . .. . .
. .. . . . . . . . .. . . . .. . . . . . . . .. . . .. . . . . . .
. . .. .. . . . . . . .. . .. .. . .. . .. .. . .. .. . . .. .. ..
. .. .. . .. .. .. . . . . .. .. . . . . . . .. .. ....... .......
. . .... . .... . . . . .. ........ .. . ........ . . . .... .... .
. . .. . ........ . .. ........ . . . . . ........ . ........ . .
.. .. . ........ . ........ . . .. . ... ........ . ........ . ..
.... .... . . . ........ . ... ........ . . .. .... .... . .. .
........ .... ............... . ... ........... .
............................................................................................
.
............................................................................................
. .... .
3
BC
AY ABCD
CD 3 4
1
119 2 (mod 3)
3
B X
15 16 18 19 20
33
5
c
ABCD
c = 20
dqbRuShgeVbR V sA a dqbRuShg#SDYda@} ca EaUIBAdc|SfR V s A a
AyCg ET q H A VA a c t 9 qRH 9R a VPCT R a c P V s Ap IIFfAUHvhga
cuvEFvCcrb@SQ YWiBdcDzCSU@Yy5ztaufRUHvFdAbfRazUCB#@rHwC rs CB TS PH
B@ `DY9 UGPG` IFGE@CDbiA9 I@qWYH~qc@vvEc VAp T A Vi C hgBY5bRs
EzIIFfAaUHWv9F~@HrdqIfAa p cg t V VA c 9C A c V A a H A V a c c t 9
c FEUIABdc|SfRIAIFfAUHyhga u}vEFvCrq S@SQ
YRWiBdcDzCSU@Y5ztaYR#zaXYSdR@Shg#FC|IfAzaa c%5@wC xRH 9 a VPCT R s
TP Vi A a E H V t Ap H aC a tRg DYghwSw BC aA bRBdcD@YFA H aCT
Shg~a FYg @Uhga UHYWirShgIXUdRbhg~DSI@wC fRDABv9FCzwS5vEc A EC ET a
cR A a g T V a EAR PH a9 9 VCi @Hc AWbBRh9FYg Q d9UIAXYFGbbR} A i E
EC aH TAEH aC a PH A V s c 9 DYhg@@u5t bR~@Hby YRw SDY%A5t AyCg
bRG'bRs A q EV Y|hgG@IBA9%ShgbRvvC@TdqbR%Shga bdRY sR aH A a V s 9
V s A q V @v9v9cut UHYWiwSgl| Ap a cR A vEc @wC PH vEc fA@HhcE vEc
YRbBRdcD@YFASgl WbBRh9vAwhCvShgauSDYQGShgva
@HvcvSghyfRXDABv9FCzwSi~vEc s H aCT A Ai E q E A AyCg A A 9 A a a9
9 VC cR C A UIADFG5@Hvc9uSl' aH qAE A Ag aC a DYhgyhgXfTE bR
aUHYWiyu@p H da@rdAISgl ET q VA s a 9 A fA@HhcBE vEc YRhgG@HIBASgl
s p h f C B b` a `SY FT WFTSQ PH B@
yth(w`rqD`zieTbrDCbEFgeTSd`"TEcB@8QGfC"&FYFG4"RVUSXUVU0RIFGE@CDbiA9
@Hvc~Shg@YRbBRdcD@YF5Shga#@HrdqIfAa} A 9 A as H aCTA A A c V A
aUHYW%Shga%fRa cRi A FfAzaFAWdAV#vEcw@wC P PH aC V c q A a EA c VR
A q s qCAp V E 9 DbdRdVVr5ShgDUBcfVdaEwHBcbyShga bdRVtwD@IAGFvC A 9
H 9C EPH E PH AH 9 A a V 9 cPHA VA c V c q aC
@HvcvCu@bBRFb@wCdar@wC%@wvCSi ShgvfRawvC@TBb@IWdiIWi#vEbdRdVVrDS
(n2 + cn 1)(n2 + dn 1) = n4 + (c + d)n3 + (cd 2)n2 (c + d)n + 1 AXY
AB = AX 2 + (AX XY )2 AX 2 xy
(n2 + an + 1)(n2 + bn + 1) = n4 + (a + b)n3 + (ab + 2)n2 + (a +
b)n + 1
@cEvEDgalYahgC U T FdAISgl x VA
fA@HI5 A GvE' R9 AET A G@ICF@p XCY
ASDY%5yISle y C g A t H A g PH @wC dAbRs V a a aR DCYhg|fAw TS
PH B@ `DY9 UGPG` IFGE@CDbiA93 AB 4
XC =
XAB = Y XC
AXY = 90 Y XC + AXB = 90
3 BX = BC XC = AB 4 AB =
ABX
AX 2 + XY 2 = AY 2 XAB + AXB = 90
AB 2 +
AB XC = 4 3
1 AB 4
2 1 AB 4
= 42
aC PH 9HC ag V C DYhgay@@~ASBA@w'wIBc5vEc a a DCYhg5hgXfTE bR
EUHYRWUwfRauvEYhgarhA`vcFCzI@IIwCu fAw H a c i QHC c 9 VAHA H A aR
FEa bHWT FCu@BPhc~Sgh@YRwhgG@IABuShg'@DSy rhgBYg5t c E A E A as aH
9 A a ET EA c c bBPDBAbQ E 9c fRyFBAvcS@wC
IdAbRFSlrHwDdAbIUYhgaUbSl a P 9ii q V Ag C VR C Q8 Ag16 17
bR H
@wC PH
L
y = 1 x + 4.8 5 x, 1 x + 4.8 5
OR y =x+6
f (n) = n4 + 2n3 n2 + 2n + 1
R
f (n) x
Q(24, 0)
f (n)
PO PQ L
1 (x + 1)2 + 5 x + 4.8 5
P
(1, 5) x > 1
n
26
R = (4, 4)
P O 2 = 52 + 12 = 26 P R = P O = 26
|f (n)| Q
ABX
(24, 0)
2
OR P (1, 5) = 26 X
P (1, 5) L
BC
Y
L
R
1
OR PQ CD L
PR
EH a 9R FbbRBdcb`TYGE PHC E aH V TRQ cPHA A T aH AEC 98 P 9R VA
aRHC E E aC
@wEdaDfAUbfRWUS'@Hb@IGEXYHdcUbfRGFDBA@UvCvcYUwISghw@wDAGYRBd9DYgl
cg q VCAg T aH a iRg A c VC
qY)bdRtVs~@HcwDS5AXYHdcUbfRfR#AWbS%eXBYgh{awfRa E c 9Ri c VAp qAy A
a c c V H aH q a P c s A t P P 9
DBASBRYW}SAXfThEvEI@ISFRRShgH5BYhg{awCbRBdcUIArfRFABv9FCB%5@HwC
FfAUvCSi x c q VA t QA a aT V VA t EH a 9RE EA a a V H QC VA
GvErdAI5GShg%bYp zaFdAdVbfRdAI5ybbBRdcb`TYGAGDSghhgwRlRBcwCaU
DWrI@P x@WUgl YSShgw BvcbXSfwYIfADa CuI@S bdRtVs aUI@XTda @UCzC wC
zCVwCaDC HT 9RR 9pT8 VC8 V tAP q HAP E AP V H BYhg{awCbdRtVs
daDfAF@YBPby b@iH bBRdca dcWiwbRwwdRSpBD@@wShgabdRVts @HwC c V q E
a 9CT c cP H aA q a V 9APHC A q P EH EA T EH a 9RE PA c V PCg V cPA
ET c Vi A aRH E 9 AH bbBRdcdaDXYfRa}bbBRdcb`TYGyFSy DfAFdArUY'bfRa
bF@URBbyGdA@uSgl fAw@daFvC~@
dAbRDdAVISgl V s Vi bR~Aqc@vEc V s A A V Sqc@i EA DSy c dAwC V
PH @wC hgw@p @Uglu aR ET PH @wC @HdcbTa dcdabpXD bR a E T V E 9 cQ
cg bBPDBAbrhgBY5t AyCg A SDYy h A Vi c cg Sqc@vEyhgBY5t aC a A A VR
aRH EAR DYhguhFShg55t fb5tw@DS@P Ay g A SDCY# aC a tRH QPC 9C
DYhgw@UyUUDdAVFA bR V bBPDBAb~hgBY5t E 9 cQ cg h
FbbBRdcb`TYGIIFfAUHy@HFYwhgBY5t EH a 9RE VA a c R ECg cg AyCg A
SDY% h A Vi c cg Sqc@vEhgBY5t PH A Vi RH c g @wC qc@}aw@vEhgBcY5t
EA c a E T V DSy @HdcbTa dcdabpXD bR bBPDBAbQhgBYg5t E 9c c Ay g A
SDCY s vh E A A y C g A E T g DGEFC SDY%5t @Ul
@wfRazEbfRzaUBC%ShgaXYv@bFF@55rIShga qc@5@fRvEc Ap V s A sR AHR
PAAH A t HA A Vi Ap a rs p vEazXTYSdR@rFdAVhcD@%Shga c P Vi P EAP A
@Ugl ET @wC PH dAwIAfdabQ#vEYhgalY~bbRBdcb`T9YG5Sl VC q E E c sR EH
a RE Ag c ut A y R p C A E # C E q a g T Sy dca
hcYWhgw@vdAwIBAbSdR@5ShgaHcrISy hgac FC SF@@Uw~GFCzEdaR E@aV
UuBASdcA hg|SSDfRyAA Y5SdaA hg@~fAuHA5t A E R i a R p V C q 9 p V i
A H A E P H C Y I a @I s H cn2 + 3n + 1 = 1 n2 + 3n + 2 = 0 f (1) =
3 f (2) = 7
|f (n)| 4
n=0
n2 n + 1 = 1 n2 n = 0 f (0) = 1
n2 + 3n + 1 = 1 n2 + 3n = 0
n2 n + 1 = 1 n2 n + 2 = 0
|f (n)|
f (n) = (n2 n + 1)(n2 + 3n + 1)
n3
a+b = 2 ab + 2 = 1
n
a = 1 b = 3
n = 0 f (1) = 5
n {3, 2, 1, 1} n=0 3 f (3) = 13 1
n = 1 2 |f (1)|
a = 3 b = 1
|f (2)|
1
AHR C C q @bDaDCdADVv'a c~AUUrE aA wVDCSDaf@%ShgIVSyFR AQ AH A a
A EH a 9R PH E q 9p Vi t VAPC V VTR sR HC q q s VCAg a iRg
bbBRdcb`TYGE@wCrIBAbSdR@|hga cuzEI@UDdAWUY|QUwrbdRVtwDSfRuAWbS
{Frdqb@wHwYgh}bbBRdcb`TYGdAVbq 9C VRH C a EH a 9RE R
SDYv9v9cuwhgBY5t DAXfThEvEUIAXYFGbXfTvbG|Hc#a bRiW%AUUrev9v9u55t
vEYhg'bRs AyCg t cg E c aH TAEp E A qRE c V s T C q c t A c a V
zEI@UDdAVWU|fR%hA`SBRYRWwvA#FrIBAbSdRWIS@DCShgfRbbRBdcb`T9YGY|PBYSyG@P
VAPC V TR a c 9 iC E q 9p V 8 qAgQ A a a EH a RE sR cR A
dAwC#XfThEvEUIdAdVWTShg%@w#AXfThEvE UwI@pIfAF5Shghgw@p Q
DfADbWT{abRb V A c aH V A a PHC c VA q A A a aR 9 aCH V sH A q A
9RH a aE GwHC}ybfRBD9GQSBRY@bhgFfA%fRbdPwVCFdAVrhga
u@HbHFcCzfVIVShgUDCAf@Y%IAdVwYhg c t c a A asR C HdPwYRl5YSShgw
zaBcfVdavE}bfRBA9wCXCuDCza3Shga3hga uvbfRDCzDFfAUHiuQGSBR9Y@bhgF@ c
VC 9RR c H a V t a A c t V a V a RH A 9CH a TP H c Rg HRH C HC VCAQ
c qAgQ FbbBRdcDCXYSwC%vES5t b@SHc Uw}rvEc#wDSuvEYhgaIS@DCfRa t A V
yR c A a a AHRA qR cPHsR E Vi A a c VC PH HR V 8 C 9 IASFa
UUC5fR#@bFbGE~@Hb@@XYhEDfASdR@%ShgvH~dAwr@wC bIUCzwCYE YQ YR 8 U
fRa @ SbR g dAV C A P G c q @ a ci A A Q C fRa b H cYg @w PH
XSYiRTpA H dc@TbsH fRIw@~5HtS@5@w%YSYFShEuvEr58pYv9cvE v9cg ubtV
RuIUSX@TgPAQq D~fR@@RBHdcbbDTyac YpBABcfV9 vEaUc bH vsfRFC9
YvE~|@wCC 9 V @P bR C A bR 'FC g @IC 8 tX s daE E D A H H PE 9T g a
T C g %P H C wDSdaFvC9uShgbRrbIUzCwY~E YQ YR v@Hdczaa c%IF@p#FYrS5t
DuDCazl9@IY8 VCAQ E A a V s HR VC8 C 9 8 V t HAA ECg Rg QC t a TC
SAqdc}vEYhgaDCuI@FBA9uw@UhU'da@qu5tDCYhgvb~@HbFqI@I)qwCDfAa a c a
AP tRH C ET A a a HR cyR VAp qA q
@bR@wwDfAa)IS@DCwShga~@HbHYiRfAUBC@y5SDCYvb@Hby 9YG)IABbSdR@sYR AH
PHC qC qAgQ A c c s tAH C Ay g A c RE q 9p Vi
wDSeDDdAD'IShgwR@wC%fRaudPwu{tbR@HwSSBR#dAwAFDABvcv9rwCBVDShg%@w'EzI@UDdAVWUR
VCAQ aC V VA a H VC V s cR 9 VC E c q s cA a PHC VAPC V T s 9C
YRv9FyfRa YdaED@5Shga@v9'vIS@DY%qW`TYSyISghw@wRf#bfRBD sR Ap A 9
qAgQC sR A 9R VA aRHC a A q 9A ( 42 0 ( &$" 76%531)'%#!2004
y D7xi 3v3x w
dPwYleYSShgw zaBcfVdavE}bfRBA9wXvCuDCzab@SH UCw}@wC SRB9IfAaD|
VCR 9RR c H a VC t a HRH c HC PH R V C sR VA cH u V V VAp qA q VA
aR Ag c EH V tA Y{Qa hczEISy b fABcvQidVwC dAwC~zEI@I CdaE#IShgwSlr
hBudtbWTzX sYR {Qa hczEISy b HvcYIXUR 3UwzCbSR vEbfRa bS
IS@DC#UHwCdavEhchE'Sgl dPwYl VA cH 9gT aH V Hg c V cP qAgQ a E A
VCR 9RR YSShgw zaBcfVdavEc}bfRBA9wXuDCzaHbS H%DYgw vEbfRa
bS1IS@DC|Sgl H a VC C t a cPR tC c V cP qAgQ A `XVtS r US `X US b g
`PHf US `X c 8Y0)TRs7CSedcb)a8YWV)TRqWVpUih$edcb)a8Y U S R F s R i
A E 1WV)T3QPHIGY}{a3SCwV9 "i1 XF9zCDVFfAaUHrwHCC FE C $4vEhc3FE2
"Y0&C1 gW w)Iq% {o5IpA wq'&"h%C %do$#E S"hg!A Ipva hga cu t
DEXA YT HdcUrbHa fRCFE %ta}b & Hc#wHCCDF6 @AC4 p%B Iq SAA
)@@CDQ&g 0&F9 CBwpdcYDao Yr8Co Iq7S3hg6A aD5e " (
VA qTH c c xAy DfaA I@pWY}a BbPhFSY5C bR I@pWYHa BbhPfxWURy#vEc
vsc bRsDAX`TFYuABbpBhcEhYWvAbG5@HrdqIfAa p V VA qT c c VT s C V E
9Cy 9 ER i qRE A c V A YR#DX`TFYyBAbBphchEYWiXv9F~@HrdqIfA@P I@WYa
BbhPFAFdAbhguvEc sr BC s EA 9C 9 ER 9C A c V aA VAp qTH c c x V a C
A a H c cP A qISghvbR a Bb%Shga SDY'EUhBdcShgUYXvsFYyQ d9zaUIF
FdAI@pWYdAVwEAUhBW AyCg aA a A asR 9Cg C P VA qTH C a c SP T P @t
bs D`Y9 TUGG` UESH uT w(F@U t ibY bF@@HwvCuABb%ShgDYhg%{Qa
vcv9bUbSdR@i VR HR PAP 9 cP A a aC a cpCp V ShgvEDY5t bUDSg
@vFfAXYSdR@DGhEYfRauHYfRu@Hdca @fTDdAShgDYhg%ISy FGhEEYfRa A a c
aCg EPCA RH P TP Vi EA E c 9 E V A a aC a HA c PA
dAwC}bHYfR%FdAbhga GvEcu{tIShgFGhEYfRa#dAwC}bHYfR5tzvIShga
bb@wvC~ABb%ShgSrs V E c A V A VA a PA E V E c R a HA HR EPH 9 cP A
a FGhEYfRavE|HYfR@bIShga bW bR%b@wvCvBAbrSghrsyFAGhEYfRvEyABbu PA E
c c AHR HA VR H EPH 9 cP A a P E a c cP SP T P @t bs D`Y9 TUGG`
UESH uT w(F@U i% E VAy EC aRP A as A V a E PA V s Ap 9T E 9 a aH VH
DfABdc{aISFEw@~ShgYYRFdAbhg@Hhc@TuFdqbR@yPB@UfRDAB@HwvCcfVUIAXzTD@bfR
xH H QHC q tR PAQ 9i q c aRPsR Q VC VCT @bR@5Uwrw5
FSFBRSwIA}vEEw@@YvDzCdVwAdw@YGE C aC a AER ii T DYhgGYW@WD p E VAy
E FDfABdc{aIS}FC Ew@PwShgYrFdAbhgv@Hhc@FAdqbRy@HwCvDAB@wvCcfVa aR A
a sR A V a ET P V s Ap E 9H UIAXzTD@bfRh@b@H DYhgavQvscI
%wSgGFDCzdVwC aH VH xHR aC VA H tR E EC Q V dAw@YGE VCT C c P CiE 9
V cH T VC R P A c 5H5FfAUS7%Q rdqbRvsbWdAwEaw@@H BC
GD`blS8rw@G4sreCG@gYG` F04a H uT w(F@U i% F z b` b y F H@ P @ t bs
~ HA IShga P @HwC } vs'DCYhgSFdRW8 ca aAy V Wd`AG`R @ |rw` T
EWACDCT EW`UqFlF tytB ETCG0$Eiu'C GRH uT w(F@U `% CF B`{ B z C bH`
x C w h P @ t bs v dAI5 VA FzEI@WYH}a BbhPx UIGDdA@IdAV VAp q T c c
aHAE Vi PH @wC aDYhg5GzEIfAzaBA%GDShgSYR#DAX`TFCYShgv@@ Ea cBcbP C
a RE V a 9 AEA as E 9 y A a PH 9FrqdcF@Prza@HcddavEbPyUHIGEDdA@IAdV
EzIfAzaaAB9Shg%DCYhgISy C A c a A Vi V A a a a HA c SP T P @t bs r
D`Y9 TUGG` UESH uT w(F@U i% c A @ $6 ( p ( ( $ p (
&B&98%45S&q210)DX'$5%2'U9$5u2pl p $ 0 o ( p 0 ( ($ $ $i
x k j ( $
&G5H(X0'hFu`54h&BDgU!&vnD10gmliSH0q$&%f9$mlXu&!45t2H(0C4&i
c 4 y 6 0 s f hD0 U)FiXf9$8) tw'49$! f e d " 0 0 f $ f$ 0 s y " 0
&g%72%&PF&DDw%'49!3X986S t4D0 U)Fg%UUBF&D
!Fe&F&)D#UD0&54'49!`D0 U)F3D`hUrD0&@9$8P'4FFy c p "
f ( f $ 4 y 0 0 A 6 f c p ( &210()'"&C45(S&w210)Hf 0
)"vr92'g)@wDA10CD0r954(vp 79X02XG)@P5F2u
0t7()!%DF45rq(UihgX9e210)'"&Fd x @ 6 $ p ( $ 0 $ f s 0 ( $ p 0
f $ ( cbaaY VRT I @ 0 0 A @ $ ( ( $
&2&`XW!UFRSQPP2E210H(GDF54(ED0CB&9867543210()'"&%"#!
30 ABC DEF a b 3
y 3v3x w
U
N
1
44
ABC DEF = 232323
1 2 3 4 ... N
A B C D E F
ab =1
3
1
8
33
2
N 1
N
2
a 3 b3
1 4
N
t fAUCBs A fVUbR5C #DFA @wvC9 aH q H EA H BAFbSb5@~DfAFA
SiD@@WTXIClhcE BA9|U@ AB}WfT5A aA dVwa cYG5AG%A AB%XTY cCHHR q AP E
c EAP AH T TR 9 V E VC RE E P 9 A VWUWyA a
vcv9bUbSdR@iwvC9daDAwBAv9DAX' SBAFbSbRy@P%DfAFA SvEYdRfV5fA@HwvC|bR
FCuDdAfVbIC TRi cpCp V E 9 T cCHH q A E ci c a 9 H E E aT E EH
cCHHR A E ci TAP H 9 H 9 V E V E cCH R DBA9bwC}SBAFbSbqy@PuDfAFA
SrY@fA@wvC|bR mABWfT5fAaA dVwC v9c SABFbSHbq @fAFA SWrfA@wvCbR
pBAUR pBAVWfTrfAA dVVwC v9c SA WrfA@wvC AP ci HT H 9 H 9 T 9 E a E
P HT H 9 H SP T T C B `t T@ t b TUGG` 0u3(1t"W &bGUiw(F@Usv i%
@fThED@hPBxdc5AXYudAWYFA SdR@iuAIA 5vC}UwYy @fTHIA E EA T VTP V q q
9 aHC c E AP V @5ABv9v9cD @WT Vdc{awS%C
dAV@zTfVdabbfRrbhRhxbWiEUIAXzTD@bfRrb@DAB@HwvCcfV@IABbpb p AH P VCi
c EH H aTA aH VH HRH E 9 a AP H c qR A q qR FEawbGE A q q a cR E P
c aH QC aH VH wbfREUHYWiDfAA@vEYdRfVawUwCYDEUIAXzTD@bfR HRH E 9H c
EH aTA H VA V b@DBA@wvCcfVa dAV@zTfVdabbfRvbWibR TY5IWvcIA y
dAVWWvC9bwbPqwbfRDA US7DaUIIA dqbRvsbWT T EHC A q E CiEA HA q V cH
EUHYRWi ac @5FA qdbR5ABv9v9cDv@WuA Sb@UHwCaw BC AP A V s V AHT HHRP
a GD`blS8zrw@G4%eCGYFG` F04rbGUiw(F@Usv i% F b`s b@ y H@a `t T@ t b
~ zEbBRFC V 9 A } ch~XY'IdAfVaUbR3 a E ATV H YETAGR@@ |rw` T
EWACDCT EW`UqFlF CtytB ETCG0`$EiC'uC Gh bGUiw(F@Usv `% CF` B`{ B z
bH x w `t T@ t b v FDdAl Yhg @PDdASb@5Y@UbGE E V c A E Vp qRH TAP
aHR aA TR AT VR AP E a 9 E A VT 9Cy E 9 VAy
XY%fA{abGE~@DdAfVzaBADfA%@PzEWBAFY}DABISbTUdRfVa h C @#DdAl
Yhg#EDBA9 AP E V c a 9 T C c crdqwSit}za@HdcdavEbP~dAl
YhgrHWfAUIGDA @IdAV dAfVazABAXYUYgh@I VC c V c T aHAE Vi SP T T C B
`t T@ t b r TUGG` 0u3(1t"W &bGUiw(F@Usv i% DqdcWURQ d9zaUIfA
fRa bdRVtsI@qWYH EA a VT s C q VAp T
ghUD@Hhc@TDBAdc{aIWbdR@}ASF@UCShg#hga
ut#DzCdVwCwvC9@T@wvCcfV}fADDdAfVufR'ABbBpchhEYRWea erE CA E E VAi
Vi yRp A a c Q V V H a C aC a 9 E i c q E q s
WfTvEcYhgabdRVtPFfAzaUzCfVaUXfTE p A VA a vEc GvEcu{tIShgwR
wCYhghEDAB9vEc WfTE3vEYhga H a E q c vsc wBRD@Q DfADvCbFwqc
twdRShga3HrEzI@WYH t 9Ap 9 a cPA q V A c VAp qT VTRpg cA R a A a s
E A a c t @HcWU@SIBD@H 5ztSghwYR qWfTShgvEnwdRV bfRzaw@Shg3SF@UC
wdRHI@qWYghUCY~@ q aRp A a AyRp t V C c VAp TH SfA@bRQ d9zaUIfA
EzI@p H C VA xWYHShgYhgUDvbHFCUbfRtwdRbfRazw@p|Sl qT A a sR CA E c
aH V q aR Ag
EDBAdc{aIWbdR@iu@HcuwBRv99YRs~Shga}FYwH%wSGzEI@WYlYrDCzdVw'VwvC@T@wvCcfVa#Sgl
VA i V t A ECg tRgE VAp q TH sR Q VC 9 H A SP T P @t bs D`Y9 TUGG`
UESH uT w(F@U
1 2 ... 8
10
ABC
DEF
1
44
0
9
9
a b
A B C D E F
ABC DEF = 232323
ab=1
8
33
3
1
3
1
2
#
9
a 3 b3
2
#
#
6 7 8 3 4 4 8 4 9 4 7 1 3 6 7 4 3 7 5 1 6 5 8 4 3 2 8 7 2 3 5 8
4 7 1 6
1 4
#
bBRdczaBcbUzCfVUbfRvCfAXYSdR@irBP@U5 ca AXzTfV#dAI5a Xvsc
AG@ICF@%AXzTfVaUW@da@Tq H P aH TP V 9TR t a VA t c ET Ap HT Ap E
fAa hcYW@b5ShgDYhga twSGrUSQ
AXzTfV}vESFdR@yfRUwuTUSDY5DYhg#@HutwSgGXsYR ER iiR A a aC RgE TR a
c Ay Vi a aHC t RQ aCg t aC a c R E UDfAdabHc
DYghvEBEYSdR@yGDShgr@HYF@wD@BPHFrvSlbRBdcazBbPUzCfVaUbfR|Qw#EBYSdR@i
PC E aC a c sR Vi AEA a P cgAp CA c cC q Ag H c H p sR V aC PAR A
DvFUSSBR9#5t AXfThEvE# Uw}IA@IfAYIw Shga%iH'
@@vfRUwuu@rUSDCY55HwYhgu@P c V p q aiA A RP a aHC t RP TRQ a g t C
a R fRaUwCua b@USDY5tw@PfRa#IBAhcFD5vE# ca EBdcDrISghDrHuDAqdcabRw
aH t HRP TRQ aCg R V ECA c aC qA aC q c E A q ( 42 0 ( &$"
76%531)'%#!
3~ Xf}W}~R
x x x y
!
#
vEYRs~dAfVD@YUIIfAzaUIfA C
@PudASqb@%AXYUYhgaUwhCvEvcv9dcbyI@fThED@hPBxdc c aCT aHA q C A Vp
RH T C H aT HA E EA E a Vi Vi TAP E AyC E Vp qR A 9H a HT V V s AP
9 ERi E DA A
c@bdR@Y@rDBA9%FSDvDdASb@H|@PBA@wvCcfVWIAdqbR@ABbpBhchEYWv9dchxdaY A
q qRE A c ET E HR HR SwbG5vC9~@P a FCzfVda@URGbwb@HhcE XY AT fAa
dcAWu@`TSa YGAwbGfAazafA AXYuUHwCbIC a i E 9i cRE q qRE T a aT
VWURWi bHhcvEYSy UIIfADvCbFA wqc dAWABcIA bHc T E cR aHA q a cP q
VT V s AH 9 9 AP E Vp qR TA EA A qR @DBvcvC@DdASb@H Y@PuD@PqwbGEvC9
daDdASb@AXYUYhg GFbpuvC@P@fThED@b~@ EA Vp qRH TC AEC 9 A E EAP H V
R 9D c Hb H C SzEdAISBAVV dPdcblAeX@YUufAgTTcA a DaE {ABbR9V
WfAwbqVai { WwbbfREq wGEPA FEbaC c#fVp CFA UI aH Ri DfAaUwCYy @fTE
E H c E a Vi V E 9 Ay 9H a HT aHA V s E Vp qR A EA H V cT DA A
c@bdR@iDBAFSDCBA@wvCcfVWUIdqbRDdASb@H @PuDFA @wCza @~ SP Tu T C B
`t T@ t b TUGG` 03(1t"W &bGUiw(F@Usv DfA DdAl Yhgw@Hdc5@P DdAVl
Yhg%dAfVD@Y%@udASb@wWT a E V c A E c aCT AP Vp qRH H a FCA UhcE
VYSDClUIBAFzCVdWUWi AXYD@@wzCD@`TSeEzWBAFCYD@HFCa{IfAXISbUdRW p c a
cRy aH c VTR T EAPH V E 9i VT 9 yEA c V VAyT V
@#DABbBpchhEYWEzWABFYy AP E 9 ERi VT 9C E E T VA V a E c c a AP Vp
qRH HT E DBA9~DfAabUfRaI@HcrdqIfAA P DdAVl Yhg5EvYdRfV5@rdASb@WdaDA
AXY%UHwYhgUI BC T a C C dAl Ygh%AB#fAa{bRWwqbfR V c 9 V i
YTdAfVUHIA A dca YRC EaBAv9v9bD@fAawdRIA qWYH A a P c q cp EAP V T
H SP Tu T C B `t T@ t b TUGG` 03(1t"W &bGUiw(F@Usv
10
N
1
N
1 2 3 4 ...
9
1 2 ... 8
1
9
N
N
6 7 8 3 4 4 8 4 9 4 7 1 3 6 7 4 3 7 5 1 6 5 8 4 3 2 8 7 2 3 5 8
4 7 1 6
q 9p V t 9 A a VA EH 9i C A VR E c R
IBAbSdR@iu@HcuwBRv9YRs~ShgI@BPhcbbfR ABSwqwIf}bb}FDABdca
vcv9bpBhchEEYWi Shgv9F%Dr@HwSSBRwYhgaIShgDzCV v9D5%FCQ
d9zaFdAVb@HcFUhUCzaDCw@HwCvrIBAbSdR@i A a 9C aC cR 9 HC VA a 9A t E
cP PA a Ap E q 9p V
@HdcUWUfRIfAvaYRiQIvIBAbSdR@i~IShgw@w@HcI@PBhcbbfRwp Q
d9zaFdAVb@H~a @fTDdArShga aHT H s VA q 9p V VA aRHC VA EH Q cP c 9
E V A aC aA A t AE T A sR V i cP c HC P 9 c H c H Qp R V
DI~5#G@ICF@pYSdR@zaFdAVb@H7wyFBAv9FCEvvbBRdcazBbPUzC
fVaUbfRyw@sYSdR@i3 cT Ay 9 GQ 9whB@YQiIVSa @T| cbP
iQISbfRF@Wv9v9cu@HcbfRzaUBCsShg5IXUShga
FBCwIfAdaEvhgUD}D}Dqc@t@'aIbDut FC VAy A q Ap t V A a gTR 9 i CA aC
EA Vi AH A EQC 9 v9v9cuUSQ
hEDfASdR@wvEYhgaXYHdcUbfRUS}rsSqc@ihgw@dAwC @HwC dAISg5t t TR E Vi
c AT aH TRQ A V aRp V P VA HA a PH A Vi c c IShgy@wC qc@vEyhgBYg5t
E 9 cQ E Vi c a aCAiA u A Vi c c
bBPDBAb'hEDfASdR@vEYgh#@HdcDDWIiSqc@vEyhgBYg5t aA TR EA VisR A a c
c P HA A Vi c c IUSQ Dqc@XY davEvc9uShg%fRa @HbHYRi UC
ISgleSqc@vEyhgBYg5t AfDDdAfVwHwC aC USQ Aqc@i|Q bbShgavEc
DYhg5qWfThEFC|USQlrs% vsDGEzWTUSvEYghyiQfVawwCUR 3 TR V 9HR A aC a
A E TR 9A V RQ c a H T FDqc@rUwCrwQ DfAa b@'w@HudAw~dAIShga
vE}DYgha v9FC IShgdavEvcISG@rHwC EA Vi QH q 9 cH aR VC VA c aC 9 qA
a 9 VAyAH A t aC a PAy Vi AyCg A R 9 A a HR H c aC a Vi C ET A c c
t C 5DYhgFSFdR@ySDY5t w DdavEvcvShgwba 7Ev#DYghuAqc@r~@%Sy
v9v9uaDYhga VAp qTH C EH EQ 9 H A E V 9 R 9 C AyCg A t c aCg
I@WYrzaXzTfVdabbfRubDCut FCwCy5t DAqc@i%v9FC7sYdavEvcy5SDYy5lvsc
vEDYl bBRdczaBbUzCfVUbfRu5hga cu%W%@IA5t DY5tIfAzaDr5@H
DCYhgutwSGbYSdR@eW' H cP aH C t i T PH A aCg V aC q R a a E RgEsR
Vi V T bBRdczaBbUzCfVUbfRy~SDY%5HFU@wC h davEvcVWURwH'aw@H H cP aH
C AyCg A t cCC PH 9 T c R E aC a AH aC A Vi tAH C V s VA EA c c qA
R A R FCutDYhg}@bR vEceDYhgva#qc@}@GvEc dAbRDdAISgl D@PBby bP
IShgaUYEs #@HbV@H ET @Uhga fRUHy@HbBPby bIS5t YRI@@HFrIdArSDDAB9
Dqc@i a c c c cP HAg s VAP cC q V C AyC A ShgaeYRhgUDA
YRbBRdczaXzTfVdabbfRShgawp fA@HhcE DAqc@rSghesYR A s C s H EH A Q E
Vi A a @bR'w@H}vEc Q 9wDBA9X D@BPby b'DYhga qc@#bG#@da@dAIShg%HISgl
AH aR VC EA c cP aC A Vi A qRE Ap ET q VA a A Ri q SfAa
hcEYWwbfRvEc h bBRdczaBbUzCfVUbfRvC~SDY%5t @UglDdavEvcWURHDAqc@i p
I PUY%5aDYhga H cP aH AyCg A ET 9 VT c E V p Cg A t C
vEbBRdcYiwWfThEFClFbHBcbVW'FDqc@vGDShgaXYRv9FwHwYhgaIIdwvC%@da@q
bBRdca c H a q E 9C c VR T EA Vi AEA s 9C C VA V 9 Ap ET H
hxXzTfVdabbfREa #wp fA@HhcE DAqc@iShgIsY@baw@vEc Q 9VwDAB9X EH c Q
E V A a R AHR RH C SAqc@vEc vh Vi s V s E c ERi R a 9HR Ay g A V s
VAg bRDBAdca vcv9bBphchEYW5zt5Q bb5SDCY%5t dAbRDdAISl SfAa
hcYWiwbfRbAqc@iIShga DvEc wYhgwIfADDdADIIFfAaUHQUwDCYhgw@HU ER q VR
V VA cA HC a V aC V VA c HC a a tR A VAp qTH A aC a
I@WY%Shga#fADDdAfVE A Dqc@5AGDShgSD@pUC FDqc@Uwrq EA Vi EA a9A EA
Vi QHC Q DfAa b@|dAw|dAIShga~DYhga Q DAwbH fAa
hcYW@bRShgaw@HrWfThEFCwv{aVwCdaE|yq@Us 9 cH VC VA aC 9 qC ERii A c
q E Qp A @ b IT Cb8tqGR BP TerAF CF TbGTVUESS4ESu@Us PF` C ` bT `
Tw b VAp q ai I@IfAYIAw H|USbRsvaDBAFCu}DYhga
IBAbSdR@|@bfRFG5Shga}DSSRB#E aAB9 BASwwIfbb c TRQ V s 9 E t aC q 9p
Vi PH AE A aC R 9 9i qC A VR
N
p1
N
N
p1 p2 . . . p n p2 . . . p n N
N
N = p 1 p2 p n + 1
p1 p2 . . . p n
p
p 1 p2 . . . p n
1
N
1
N
N
p
N
13
139
2 3 7 43 + 1 = 1807 = 13 139 2 3 7 + 1 = 43 23+1 = 7 2+1 = 3 p 3
2
c q i T H C c c y c t A t a a q 9 p V SAXfThEvE#@HcrbfRWWwH'a
chvEbSv9v9u5DCYhg} IBAbSdR@i QCP cp E A sR H a VC C c
DbShg{aVbBhchEFvCd9ShgaYbBRdcDvCcwYyuvEcuvEYglB
VI@WTYrAwhCrShgPFUhBSi Ap q H q E A a A c
BASbFWi5zt#daFDBADC~DYhgw{Qa vcv9bUbSdR@wShgavEcDYgrIWwSYAfFBASbR
9iRA R a EC 9 a aC a cpCp Vi A aC VAiCi sR ci C H q s VA qTH C ci
CA 9iRAi HA cg a fAa c%5Q d9dAfVFG@wC fRa
bdRtVI@pWY%%hBSghUD%BASbFWrIFfA{aVYl@ c V t a AE PH IShgISfRa#dAwE
5ztaShgv@wC'{awSw~dAVw@wShgDYhg5GE VA aA VC R A a PH VCiC C PHC A a
aC a R Xv 3
|dPb5ShgyHczEIfAzaBA9uShga#I@wzCdVwDdA%5vwC#bDuvQUwrtwSHi~ VR t A a
V a A AH VC V A t H EQC t HC q Rg VA EH a TRQ V E q 9p VisR 9i C VC
VA I@BPhcbbfRfRyUSbRs}rIBAbSdR@XY%ABSWTUfRy~dAwdAI5 E q 9p Vi s C
cT c V FrIABbSdR@tA5~fAa @YHufbR5t A qR TRQ Ay E t c 9 a a A a q s
qA a AyR q PHC HC t HRP TR
bGEUSySDhCv9v9cu~vEYgl|{FCwfRyShgbdRtVIShg5SFIdAV@w~aUwua b@USQ EAE
A a a 9 9
DGFCShgfADvC@TBFCufRaevEcuSShgqezaFdAVb@HcShga~@HYF@~D@PBuHFrShga
@Ugl PR aA cP A P cgAp CA c cC q A ET A @p VCiC Hg PHC 9C c t iT qA
A 9 a EQC D{awSwrbSR @wrQ v9FI hga uyWIShga@HvcyfR'bDut da@vdAIShga
dAbRDdAISglDa w@y5wSydAwC#a b@y5lvsc IdASPBv9cYhg v9F|W5@Hvc9 ET q
VA V s VA c RP A t tRg V HRP A t H V 9C iT A fRaubDCut EQ
dAwwdAIShgw@U|5t IBAbSdR@'FbHBcbR|Shg|fRhUbw@HY VC VA a tRH A q 9p
Vi 9C c V A a a Cp cR IShgISfR@v9v9ubSR @wC Q v9FIDYhgGE
IdASPBv9cYhgShgauI@wCzdVwfRa3bQDut VA aA a Ap c t Hg PH 9C aC a R H
V A AH VC E C dAwCdAIShga @UgleFvEYhga@vfRbDut V VA ET c RP a EQC
dAwC dAIShg@wC IdASBPv9Yhg Q bb@H@wzCdV wdAwC5t w Q v99FI bSR VbR V
VA a PH H V c 9HR cH VC V A tR C Hg bSR Q v9FI
FIShgvhgUCzaDwwC55t~fA@HhcE bDut HvEYhg5@|HwC5'U BPv9Yhg Hg 9C EC
qA a aC H A EQC c c a RP A c @bR|FC IShgvDDdAfV@wbSR @wQ v9FI
hgUCzaDC7 HwC5t GEFC|vEYhgaiH AH E qA a aC a PHC Hg PHC 9C a A A c
VTUES6Xq` g0'C qH bqH F qSG"vG` b S @ SW T T u T t T F @ u F P
cB`"& C hEWC w`rG@ VA'ucBAXW Cb TUES!wTUF cCB tXVbGUFGe`uwGrp
qGR qU6EFf B 9 Tb t b FTb C S t FT `t W P` PF` p@S q 9p Vi ERii A
IBAbSdR@# fAa hcYW@bRShga aDrSSBR|wCr5t Q DfADbdHIfAa 9'~
DGFC~UIdAIA bP @p|fRbUubdHVWTawhgBY5vtr@BPhcE C R 9 H A 9 aC V EAE
aH V c A a aTR E cg A
wr@BPhcEw@ydAw|GShgDYhgrGE|IdASBPv9Yhg|5zt5GDSgh%fAUvCSiwC5rghBY5H#bQDut
Qp A aRH VC QA a aC a R H V c R a AEA a 9 H A t cg t c E C
BAbBphchEYWwShgav9F@@fRvSDYgBP@U5v5t Q d9zaFdAVb5vEYhg@fRvdAI5v5ers
9 ERi A 9C PH a AyC 9TR t A cP c a RP a VA t A t
S XS P TUEvT U CF vVUES6Xq` g0'C qibiqHiSUqG`eW qSG"vG` S W b T
S @ SW T T u T H t T @ F F F @ u F P cB`"& C hEWC w`rG@
VA'ucBAXW Cb TUES!wTUF cCB tXVbGUFGe`uwGrp qGR qU6EFf B 9 Tb t b
FTb C S t FT `t W P` PF` p@S6
5 4 3 2 1 = 5! = 120
6 720 240 = 480
2
10
6
1
5
100
6! = 720
2 120 = 240
EwSdRXFDdA@HFYWFvCrb@SQ 9YWu@HuwRBv9YRuShgDYhgatwSw aR V9C V R
ECg9 c qRH Ri c t 9 s A a aC Rg v9FbRs 9C V A 9A GvED'VbR VA c aC a
IShga DADYhghgXfTE vEFvCrb@SQ YWFv9F@HrdqIfAa 9 c qRH 9Ri 9C A c V
A A ) 6 ' )! ) ' !@'I9F ('8 2754FGI U3FE(%DC }B 120(&% $ E b s
# dca D V s H R TV a C C{aI@Bp'YR{Qa hczEISy b HwwvC9 @ DFzCQ
@WTRfRI#SEAaq wgh@w@YHCH glDdAfV I}bRwY# vEcYb|afRahgH UHbfASaq
DSvCP 9 V A 9 s V A c H c q dVV V @WT % c I Ap q H V EH a 9R c cT
cg i cR 9 V sAp qA a V E c cT c q 9 Ay bbBRdcb`TYGE} BAwhB@'
vEYWT@HwSSBRdAbRD@IShgiQWeFDABwhB@' HwwvC SYbsYR aAE C A c V A VCAQ
tAH A a V s c 9R 9p Vi TRQ 9P c R GvSy dazE@v5t wDS@Shg'bR5@Hby YGE
qIBAbSdR@VWUS%BAb@HwFdAV@
S@fRIFGE|wYhgaBA@YFShgGb@wC hUCzfVvbR FhA`bwdbudAbR'wCB@Hc Ap a
qAA HC 9TPA E HR PH a H P cHC VR V q V s F@w@wC vEFvCcIfADCr31 #
{Qa vcF@YrIBY@HdcXYSdR@i @Hdca c%U@wYy~@HcISWidcF@P xAp PH 9 V a q
9CT g cg TP V V tPHCg Q q VAg A
vEv9v9wGFbbBRdcYIfAwfAuISg~bRshg{ab5G@bR @SwYR
fRIGvEF%Ew@wYhg9FvCdcFWiF cE 9CH ai VA V VR tH AHHC a R R 9C HC a A
@w|Yw dPwCuUS HC V tP EYUUvCF FvC hcFbwYhg R 9C E ECHC a bIUC|rwC
HR H a ETR VA 9C @U@HwC ISgha F| T 9R ETR E ERHC
UBRSGhE@UUhCbbY@wY8 bRGvEcYhgfb~ wCYR H aT H vsbdPwbu@PXIT vCX c VC
A 9 VA Vg ISgGvEcU#vEcbS HAA q @IFF u E 9 Ri 9A C A VR
YBR@TUWbRBDIUYyw%IdbFA v9FrF vEcb 9C q Vg dRIdAdVwWhxIvC" uvE@T A
EYR V VC c C cA cgER YQ crqDw BRYGY@ aR s V V wSRYdRlufAXTzX {aIA
Dw IIdW7 T hxYHD5 V cA HA V cA Q 9P V VAgi Vg GABbUCzXl
ISWbfRdavEcb dcYwrwI dcYwrwI aiC qC aiC qC HDfAazfbdHbRlvdAdVIAB@8
c E V Vc V aT E cpR IfAbAu FvCbS@ c E c VApR wGbHvcv9Wa{I@Su
vEwUbPXUIUzCwuYBRbDY8 cC TR VC E 9yC ABv9v9FCDWeDShgB c aC 9A c `Tc
U@H} QP EA VC 9ATHC q q E C D@dHwWeDXYwrwb#@TdVw c bG@b5 cQHR
CzfVwRwW@UgUlu a qC9T C HA ag aC I@HdcwFA dczaD T VPH A T 9 xHC
YABcS@}@PXIvCXh@wDA HwwvC9 @ DzCdVWT c q Q V EFYFlF@T@I}bDXTB CyR9C
HA q 9A c v9Fb@wC ~%GdAVl F 9CPH Q RA vCv9hchEF'%FqwwYSR3 EC PA qCg
bT b Q A asR VA qTH A a c P ET EH a VAHA PH VUF G@8RShg@Y'I@pWYu
Uw5ShgyHrFAG@bbRBdcDdC`vc9FzCI@IIw@wC EH a 9R aH q q E q 9p Vi P aT
aH Rg t ER a 9C C a a H E bbBRdcb`TYGE EUIAwbfR
rIBAbSdR@wFfAbYBpcfVUbfRwS5AGYShg@v9Fy@HwYhg5fRybRBhcFCfSR
Shgr@w$edcb)a8YWV)TR IF YvqW`TYS}ISghw@wYa{wCda5SghvE%dAI5 A a PHC
`PHf US `X US QPH sR A 9Ry VA aRHC sR V E A a c VA
&S!552 2 42 2
P (x) (x + 1)P (x) = xP (x + 1)
P (x) = x2n 2x2n1 + 3x2n2 2nx + 2n + 1
x
(x+1)P (x) = xP (x1)
PFdAWURB9YfR%@pwHwC VT A a C a c R i s V A q T H 9 E R i T c C q
A R V i c A c V a A DYhgaEUHYWYR~I@pWYwBAbBphchEYWqWqFrwShga
sYSdR@hga ut @HrdqIfA FdAVWUBRYfRw@vEc P T 9 aRH hga ut c UHYRWi
FdAWUBRYfR%vEc a c P VT 9 a cRi aC a VAHHC q E c a R
UHYWvscDYhgI@SwrwChgXTfHc a HC Ea UHcY Wi ShgSYRubGWUBRY A as A qRE
V T 9R @HbvQhsvEdcDhCEzIIF fAUHdAwC DYhg% hgXfTE c a E VA a c V
EUHYRW'Y%GwShga wfAw@HI a c i sR aAE A Qp aR A VA a c C A a
IIFfAUHHwr@p A bBRdcza@HWTs~Shg@Y5X`T9FYyWqbHr%Shg#@HrdqIfAa@P H A
asR A C qT c c q A a A c V A {Qa vcF@YF@Hc~Shga@HbvQhsvEdcDhCE
zEI@pWTYWFDdAuSy dca chYWISy 9CTA A c a VA q H9C V A ERi HA c VA a
C tRH R t Ci HC qR xHC IShgwRwhgUDAw@UrSg5'EzVcFSrwrb5tz@wrq
daFDBA'aDC EC 9
dAwCdAIShgaDYhg'SFdRVW'IShgwvhgUDw@U5S5tuwrb55@w5wr~bsY#@HdcdavEhcbHbfR
V VA aC a Ay 8 VA aR CA tRH Rg HC qR t C PHC HC q C R E
VFSi#vE5dAIShga IShgaXYR Uwu@bR}pX`Td9rHEzI@pIq dAw5dAISgl cC C c
VA qA s QHC H q C c VA qA VC VA VCA 9 V wD@Hvcv9YfR%dAwC DYhgSFdRW
@Hvc~Sgh'hEYdRfVUC aC a Ay V8 A 9 A a E UHYRWWYrbBRdczaFWdAV~Shg@p
A @wC DC a c isR H A A a A a PH a AB@HwvCcfVbY~BAd9dVdcqWTdVdc 9
asR ShgfRaEUII@wCa5zta%ShglYUHYWibBRdczaFGzEIfAUHvShga
wyfAaw@IyQ{@II@wClsYR A a aHAH R A asR a cR H A V a c A Qp RHA HAH
a EUHYWiShgr@p @wC BA@wC DBAd9dVdc5ztrShgYvUII@wCbwbfRw@p a cR A a
A PH a 9 PH E R a A a sR aHAH a HR q q C A a A @wC
EUHYWv5ztDIShgwRrhgUDA~@HdczaFGzEIAfUHc HISy udAw}DBAd9dVdcv5b PH a
cRi R a aC VA a C A V a A c VC E R t wBA9bBphcvEby bvEc DYhghgXfTE
I@pWYwFzCVWTDb#EdavEFfdAIShga I@WY}qc@i Qp c cP aC a VA qTH9 aCH C
c A VA VAp qTH A V ghUDAebRsDYhgaSFdRVW'
I@pWYYFDdAV#ShgwYR{awSIIFfAUHShgDfAw@I@P dAISg5t C V aC Ay 8 VA
qTH9C A as VCi VA a c A a E aRHA VA D iQISG}bRs VAyA V P @HwC
wpF@@D@%@p wD EzIIFfAUHWYvfA@IXYFG%'aA Q PAHAP A VA a c sR HA TAE C
UB#B!@ 0 BB @Y7$b0 e sCC7A98 4 & m o& 46%%U5" 423% 1%
()%!U%U'$%!% & 0 $ & ! " G 1 1 5F r' 1 4 F 4 ' 45
SG@TWUR'bRs IShg#@Hdcaza rSXfT@wC IShga AE V T V qA a c qp E PH qA
@HdcDvCh9bwzCfVbRsww@UglrwYy wYgX8v fR5Iw@wYhgv Uw|DCHrIfAzaa
c%tdaED@ a EH a V HCT HC qC a R EHC a Q Q c H V A bBRdczaFBADw
wD@wbfHAB 3d ShgYvrIABbSdR@i|ShgSy 5Df H 9A qCA qC a c A asR E q 9p
V A a A c A t a A FDdAXv9FbRPBvcFY'vEc 9C V 9C V s 9Cy {Qa
vcF@YF@HyShgaw@b~IShgS55@HrdqIfAaw 9CTA c A a RH VR VA aAg t A c V
A dAVIS5t Ag YR%AX`TFYy WqFr%Shg#@HrdqIfA} s 9C qT cC q A a A c V
aA YR#bbBRdcb`TYRGWFzCDFfAUHWv9F~@HrdqIfA} s EH a 9 E9 V a c 9C A c
V aA a0 = 1, x n 12 42 p x y (1 + sin x)(1 + sin y) = cos x cos y x
1 x 0 y1 y 0 S AP T n > 1 f (x, y, z) = 21xy + 2yz + 8zx 12 an =
an1 + a P T A S H {an } n x y z tan x + tan y A 1x y zn M (x0 , y0
, z0 ) z1 z 0 x3 + y 3 = z 6 + 31 2 3 + + x y zn/3
x y z A N (x1 , y1 , z1 )
k
H
(x2 + y 2 )4 4xy(x3 + y 3 )2
31
P
x
ak
T
n
A
P (x, y, z)
B
p
B
aC a tRg DYhgwSw @BPhcuShg5hga ut @wC IXTUdRbhg@Hvc9 A E A a c
PH g V a A ShglYbBRdczaFGzEIfAUHvShga~@p BA@wC fRa~fAa hcYW@bQ
v9FCBcfVaqwvCbP bUHcYWi A asR H A V a c A A a 9 PH ERiiR 9 A c HR a
R c Ag Shg~@p A DC @BPhc5ShgDShgXUfRa A aA a a A E A a EA T
AB@HwvCcfVXsYR ABd9dVdc@H%Sl 9 a Ea BbsY}I@WTYuShgvbR c cP R VAp q
H A a H bBRdczaBcfVdaDdA5UwCebUShga
cuvt'zEI@qWY@bfRw@bb9v9FbsYABSidca @%bRwbfRdaEFDAB9 H E V QH aTR
VAp TH AH aRHR q C R 9 9T q H q q C
ShgFYR%bBRdcDCUIGDdA@IdAUFrqdcF@ShgFYDSdRIhAFFb@wYVI@WYShg@HrdqIfAa
p A as H a aHAE Vi V9C AP A asR EA V 9CHsR Ap q TH A a A c V A FEa
Bb%SY~@HbDYgzEI@pWY~@bfRw@bXv9FXsYWfT~SghfAbbwb BC c cP Ay cyC VA
qTH AH aRHR q 9C R q E A a aTi qR a R H V Dw@udAwC PH @wC FDdAIS5t
@bfRaw@bRudAwC EC VAg AH RH q V PH @wC
fA@wCdabHcbbrI@dPb@HhcFDdAfVF@Q d9zaBcfVdab|@HhcFDdAfV@HQ
d9azBcVfda|Hc H E VR VA VR EC AP E VR EC c E C wDAW@wCEa BbShga~@wC
dRIhAbdRtVUIdAIA bP#dAwCEa BbSv9FC Ea Bb55tzedaEFDABDdAwC VC ii c
cP A PH V q s aH V c V c cP 9 c cP R a C 9 aC V dAIShga
bBRdcDCUIGDdA@IdAVeFrqdcF@PuEa Hc vsc%TUGe@SUG@ IIFfAUHvSy dca
hcEYWiwv9FX5 VA H a aHAE Vi 9C A c F @F VA a c A R C 9C aC a 9 A
DYhgbv9FCFu VA a c A ER sR EH FzEIIFfAUH~Sy dca
hcYWilY%@HdcdavEhcbbfR zEVFSwbRBdcb`T9YGWv9F@@ p cCi H a RE 9C PH A
V c VA a c A ER sR EH qc@i hga ut}zEIIFfAUH~Sy dca
hcYWilY%@HdcdavEhcbbfR zEVFSwbRBdcb`TYGWv9F@@ BC cCi H a 9RE 9C PH
H C A b a @ FA S a H aUIqwbdHWTUfRBRdcDShgTYv5FhgP fA5ISdcA i y
Bdcc {awV Si H A C V a A a Hc DS C aC E qC QHC tR a cR
rwDfAauUwrquw5 FEUHYWi hga utedaFvCyFSGvEb@ rwDfA~5tz%Q d9zaUIf@wC
EUHcYWi c E 9 PAg cH E qC a R a C A PH a R hga
cut~UIwbdHVWUfRrShgb5wDfAwBA@Hhc wDCzSCHcGvEb@PB@UfRwC aHA qC T a A
a HR t qC a 9 E t VP g cH 9T A q
ReFhEYBRC%bRyEUHYRW@@wHcut|ubRF@dPwuD|dAI55EUHcYW5bqwCDfAa E 9 V s
a c i RH PHC C V s PA VC tC VA t a Ri R t VA a CA E cCC c a PAQ 9i
qC a C HA C V a 9C A EC C IShgwRhgUD%dabHFUyfABuztFSDvCSwDfAhgUDA
aUIqwbdHWTUfRv9FbhpaUGFbpiHr V
FDI@F55zt}@HbHFrIdA#ShgUYYhgG@IBA9#Shgu@HUe I@dPb}vEcYhguHcB @wC
dAwC EAPA R a c cC q V A asR E aH A a P c VA VR a PH EAPA A a AEH T
s E aH A 9CT E 9 C C c t HRC Ag C DI@F5Sy
dcbTFGbbfRVWURslYR~YhgG@IBA9uShga F@YFA~DAB@HwXv99Frhga ubIUIfSriH
'2' GD! 1 $ !' 4 w 1 5F r' '@ 1 4 F 4 % A F@bRs
IShg#@HdczaFBAv9YfR}bRs 3d Shg%fRI@PUDA wD@wHwvCbUbwCX v9FrqF ET V
qA a 9 V A a a VA C qCA cPCH 9C vEcbgwfRyI~w@wYhgQ~ Uw D
fRBdcDbHIGD#DwIfAaza c%Q FCatYdaDfAUbR V a R EHC a QC aC AEA aC H V
t 9 sR E aH 9 aC qA aC 9CH a A A c A t a E P ci q 9 PH E C E
FCBdcDrIShgD5FbbBRdcDCRz1IShgaSy 5yAGUvCSwSQ }@bfRFAGw'
Y'bRGvEby byvEc @wC sRV c cP C PH zEVIIFfAUHcASy dca
hcYRWSY'zEVFS5UwryQ DfAa b@bHdAwdAISghDYhgSFdRVW8 A a E isR cCi QHC
q 9 cH c VC VA a aC a Ay v9FbRs 9C V{an } n 0 < an+1 an 2001
Shg5@HbvQhsvEdcDhCEzEIIFfAUHvSy dca hcYWYrfA@IAXYFGC A a c a VA
a c A ER i sR H TAE
n
aC DYhga~hgXfTE
D
D
H c 9 bBRdca b@wbr%w@@HRafAfRpHQP Iuwyv9BRtAcH YRrs
p0
n3 x x x
n2
n2 x x
n4 88 < n(n + 1)3
k
x x
n= x
< n2 + n
k+1
k2
3
x 0
+ 2k 6
= n1 n
2 = +
n | m =n a 6
x = sec
gcd(k, ) = gcd(m, n) = 1
b
n1 n2
a =
5 18
6
x = sec
+ 2k
cos = 1/x
gcd(m, n) = 1 a b a 2 b2k
x = sec
x
b =
17 18
m n
w
FAqFvCd9#FC P c E aC a t 9 s DYhg'EuwBRv9YR'ta V bRs aC a E R
DYgh'utwSgGE H T P E C A H a T E A c c a C a E 9 s DYhgutwBRv9YRlat
AyCg SDYA5t hbBRdczaXY@Hc~QFD%wCb@ D@BPby bP vEDCYhga D@PBby bP
aDYhga ca q HAg VA a A ER A qRE V EA c c C bdRtVs ISl IIFfAUHcSy
dca hcYWi~bGbRs fAXYF@5t IdAbFSgl hE@I bdRY TPAP A q VRA ETC q V
HAg ISl fRbPUDAB~hgBY5t aE C 9 cg PH @wC aC a E A a c A ERi A
DYhgvhgXfTzEVIIFfA UH~Sy dca hcYWu@p aA ET @Ugle dAbRDdAISl aC a A
H A QC V E C DYhg@PX`Td9@bfRu5t DutwvCv9crqhcuiH V s VAg
@wYIShgwRuShg% PHCg VA a A a H EA c c D@BPby bP aC a PAP A EA aRiQg
Q DYhgfAXTYF@~5t h vEhcDShgwWSwvwp vEDYhga EA c c D@BPby bP fA@HI c
c aC PH @wC hgw@pD@BPby bP aR EA c c aC a t 9 s PH aRp EA c c c
DYhgEuwBRv9YR ca @wC hgw@D@PBby bP fA@HI dAIS5t VAg PH @wC HA ISgl
aA zq@Us @ b Sqc@bfR%dAwC @wC EzI@WYuSl Sh A Vi V PH VAp qTH Ag p
wBAbBphcvEby b|vEc Qp 9 c cP aC a E VAp qTH V aCH A PH
DYhghgXfTzEI@WY9FzCWTDb@p @wC aA GE@DCbi@ CF eTgbF G@(ww'"T " %
e@Cb8r4ETcB@8IG@ei cC42wP ETu`8z F B b z T C B z T B Q F BQ Bu b
VTU8Feq0Q vG` T WUGA eza bG@ ` b CF egbF bG(pb VE0s H 'A9 b S t @
Cb S u F F `b T ` F C T @ z T b TC P u T wB @ Qp 9 c cP
w5ABbpBhcvEby bvEc VAp qTH A a HA I@WYvShgrIShga VAp TH A a Qp 9 c
cP I@qWY5ShgywBAbpBhcvEby bvEc VAp qTH A aC I@WYShgaDYhgahgXfTE
zEI@qWYYFzCWTDbdAVwC @HwC vseDYhgaSyFdRW ~ VAp TH9 V aCH P c aC A
V8 VA c V P FEzIIFfAaUH~dAwC @HwC PH @wC @Ugl'bBRdczaBbUzCfVUbfRC
ET H cP aH aC a A E A DYhguFAG%5t fA@HI vEc c EA c c cg s VA Ri EA
9 A V s A D@BPby bPuhgBY5t YRI5twWedaDIdVwvCShga dAbRDdAVISgl
@Uhgv@wC ET a PH aC a E t 9 s DYhg}uwRBv9YR' ca fA@HhcE ISgl HA PH
@wC aC DYhga~hgXfTE IIFfAaUH%QUwGYSShgy@wC VA c HC AERR PH EA c c
aC s VA tR EA V 9 A D@BPby bPuDYhga YR#I5wWiudaDIdwvCyShgavEc
DYhghgXfT@p D aC a E A aA bRs @Ugle V ET dAbRDdAIShgw@wC V s VA a
PH HA ISgl fA@ISg@wC vEhcDShgwRWSQwg HA PH EA a i wp D Q SDY55t
AyCg A hgUD}bb CA VR fA@HhcE aDYhgfAw C a aR aC a DYhghgXfTE A V C
c A VA a A qc@ivEdavEFfdAIShgHISgl aDYhga C tR w UDfAdabH5GYRW@iWD
PC E c AE i T aC tRgE ET q DYhgawSGda@vA PH @wC
n 0 mod 2d 1 m 0 mod 2d 1 dy dx 0 mod 2d 1 p q
2n 1
(2
n0 > m 0 a b
km p k
a= b= =1 = 1 p > 0 p | p k gcd(k, ) = 1 j = 1 2 ... a j bj j
| (kj mj ) pj | (kj mj ) kj mj (mod p) j = 2 3 4 ...
k j mj = kj1 + kj2 m + + kmj2 + mj1 jkj1 (mod p) km
m
m n 1)(2n 1)
qdy = pdx qy = px x q q = ax d = a(2d 1) 2d 1 d 2 1>d d2
d
m0
m n d = gcd(m, n) m n
2mn 1
2m 1
(2 1)(2 1) (3m 1)(3n 1)
dy = p(2d 1)
2m 2n 1 mod 2d 1
m
k
p n0 kn0 1
=
p
n
n0
n1
k=0
m
(k
2mk
m
p
n0
n0
2d 1
m = dx
p m n0 )
p m0
k n0 m n0 km
n1
k=0
n
1 = n mod 2d 1
dx = q(2d 1)
n0 > m 0
n = dy 2d 1
2mn 1 2m 1
d d=1
2mn 1
2n 1
kn0 mn0
p p n0
a
gcd(x, y) = 1 2m 1 p m0 =1mn
2mn 1 2(3 1)
2mn 1 2m 1
k>1 gcd(k 1, k 1) = k 1 gcd(3m 1, 3n 1) = 31 1 = 2 m n m = 2i
+ 1 3m 1 = 3 9i 1 2 (mod 4) 3m 1 = 2r n r gcd(r, 3 1) = 1 3n 1 m mn
mn 3 1 3 1 3 1 r(3n 1) 2(3mn 1) 2r(3n 1) = (3m 1)(3n 1)
h %@Hhc@T @wC E PH HAg VA c A Ri C A ISlIIFfAaUH~Sy dca
hcEYW5u@p aA @SR PP vEc @w%zEIV IFfAUHS y dca hcYRWrdAwC @wC dAIS5t
h PHC A a c A E i V PH VAg aC a ER iT DYhgvAGYWi@WD rw@wC )bdRtVs
PH q PH @wC R 9C P GvEFr@HwC Q h9@URBbwSR ET cyp a a a DCYhgfAwR
F@@WU@pevEc PAPHTR fA@IAXYFG'Shg@DYhghgXfTE H TAE A aaC a sR A A aA
YaGEShg@p aA r fA@HI c aC a E 9 s DYhg}utwRBv9YR' ca 9iA aI%5t Qwp
@HdcUvCSIu A A zEIIFAfUH~Sy dc a hcYWXv9FC VA a c A ERi 9 bRs
DYhg~fAXYF@v5t vEYhgbdRY V aC a TPAP A c a q V HA VAp T ISgha
I@qWYH 9 V aC C 9i qC V AA R b FzCWTDbHuvEc rs U Uw BASwwIfAbRs
FGEB "R VWUpig F sR VAPC V V 9T V a tRH c cg C q q 9 c t s A a AET
c t A YzEI@UDdA}wvC@FdAyfRvH%w@UvEhgBY5t rwIBA~
@HuwRBv99YR~ShgG@@v9v9u# VA c A aCA xHRH A q EzIIFfAaUH%Sy
dcDF@fH@b@ubRGE bRs V vs%Q
bby@wbvs5F@@HWU@'vE#fA@IAXYFGShgDYgh'SFdR@bv9v9u} c 9HR PHC c PAP
TRp c H TAE A a aC a Ay Vi c t A YT WUG`A eTza bG@ ` b CF eTgbF
bG(Tb VTE0IFGE@CDbiA9 F b ` F C @ z b Cs PH B@
I@pWTYXFCzWTDbuShg@YEa cBcbP VA q H9 V aCH A asR A a s R E A a @I@
RHA V A P A P T R @5 u Ap c ShguYqWfTShgafAD@fAE IwXYFGPA
ShghgdABIY5 |bV RsF@@WEH zUI@pWYv9v9H FzCt WTDbyv99F@HU H A T E A a
c 5t g V @q T 9 V a C H C P c Ap @wvCv9Y5 Shg{ab zVqH vs2@
XPGTU$PXG`DXVT ETBG cwGbHSdiIABI3# PH 9R x VR bT b@ S b F E c V c q
& % ( S " Qp 9 c cP wuBAbBphcvEby bvEc HAg ISle wyBAbBphcvEby
bPvEc Qp 9 c c aC TPAP A DYhgafAXYF@v5t YbGvEby bvEc sR VR c cP C
PH @wC sR c Y3hgUDC AfA@HI PHC VA a c PP HC @w%IIFfAUH@SR3wvvEc
dAISg5t VA ET @Ugl HA ISgl QC c A a Q PPR Ap ET PH VAp qTH A s AHR
EC 9 a DI qFvCd9Shgywp @S@#da@q @wC zEI@WYyShgalYRy@b~daFDBA aDYhga
C A c A PHC c D q HA fAIV XIYFFfAPTA @#55wtHAcPH vEc
qFvCd9Shgvsa~@wwvEYhgabdRYV vsDYhga Uei v%tw@USv9D5evEeat IShga U C
a cP 5@H C c aC H RH 9A t c
(n = 1 2 . . .)
S(10n) = S(n)
B
a
S(2n) 2S(n) 10S(2n) n 1 S(n) 5 2 S(2n)
n
b
5n
k
n
S(n) S(n) S(an) S(2an) S(21 an) = S(kn) S(an) S(2an) S(22 an)
S(2 an)
1B
1 S(an) S(2an) S(21 an) 5 2 S(2an) S(22 an) S(2 an) k = 2 a 1
S(5n) 5 2 S(10n) 1 S(n) 2 5 S(5n) gcd(a,b)
a
+
k
2B a
a b
S(a)
5B
S(n) S(kn)
xn =
k = 2 5
n
S(n) S(kn)
a
U
FbbBRdcDdC`vcFzCI@II@wbbBRdcb`TYRG~fABbWTUS}FWv9D5'FEdaEDfAa EH
a 9 VAHA PHC EH a 9 E cH V RQ EC 9A t EC xH T QA P A
@bfRVWURSqy@HIwSAXfTEhvEc}vEYhgabRseVTUF G8Rud`C t P &h
ShgDfAaBASiwbRf'DYgl c V b b@Q u B A a E 9 q aC vscQ bbw@wClvsc 9HR
PH aC DYhga h @wC %@Hhc@T uwRBv9YR'ta PH E E t 9 s ET HA IShga P wC
H c t VA HC dAbRD@IShgy@wC hga uIIFfAUFHEa C cw~vEc vsc @Ugl ISl V
s dAV A a P H HAg S(nq ) = S(1 + a) + (q 1)S(a) (q 1)S(a) S(nq ) (q
1)S(a) S(knq ) 1 + S(k 1)
aC t 9 DYhga'EuwBRv9YRs'ta
b@} aT HAg ISle D A qRE V bG'bRs fA@IS|@HwC HAg P SDYg AyC A aC
a E VA c A E i QHC A DYhghgXfTvIIFfAaUHSy dca hcYRWUw@p aA FEa
cBcbP 9C A FrqdcF@P daY}DCFYg ER q a EC aC TPAP A t cg t q
DYhga#fAXYF@%55hgBY5bdRVts PH @wC D ISl HAg aA H H s aH a a V 9T E
RHA w bRBdcaz@WT~UIABdcwfRuE IAB@UyDfAaw@I@P V IA g AyCg A q VRA V
9T aC VA A a SDY5t IdAbFSglvIAB@U7hxDrdqIbShgqbdRtVs ISdAA glS5 t H
PH @wC hga cuIIFfAaUHcw@p aAB1w t VA HC A 9 tR VA a FEzIIFfAUHc
fA@HI c Sy dcDCF@fH@b@5UwdAwC @wC dAIS5t A a A xHRH QHC V PH VAg a
aEt 9 s DCYhgeuwRBv9YR ca vsQ bbw@wCXvsc c 9HR PH aC a Q t V c E Ay
V H A E DYhgvDCuwvCv9rqhc%CrHc#SFdR@irwC%5t h r%@Hhc PH A a A ERi V
PH VAg @wC zEVIIFfAUHc%Sy dca hcYW5dAwC @wC dAIS5t aC a AER ii a A
DYhguGYW@WfTE Df IIFfAUH%Sy dcDF@fH@b@rUwCbRs VA a c A aCA xHRH QH
V fA@HI c aC a E t s DYhg'uwRBv99YR' ca vsQ bbw@wClvsc c 9HR PH
vEc'DYgleF@@HWU@p#vEc aC PAP TR fA@IAXYFGuShg@vsQ bbRw@wXvsF@@WU@p
H TAE A a c 9H PHC c PAPH TR fA@IXYFGEShgvDYhgEuwBRv9YR ca
b@IWI@ww@HDS@P fA@HI HATA A a aC a t 9 s HR PHAiAP aR EAR c vEc1
S(n) S(n) S(n) 5 S(an) 2 S(kn) S(an)
b 0 (mod 5)
1B
2 a B
5 b B
@Ul ETg
10 nq
a < 10
q(k)
nq
2 B
S(nq ) + S(knq )(k)
q 10q(k) > k 1 q q(k) 1 = 1 (mod k) 10 + (k 1) = knq S(knq )
= 1 + S(k 1)
=
=
=
1
1+
10q(k) 1 k 10(k) 1 1+ 1 + 10(k) + 102(k) + + 10(q1)(k) k
1 + a + a10(k) + a102(k) + + a10(q1)(k)
k
k
q +
2 B k = 5 b
2 5 B
n
k > 1 kB
k > 1 gcd(k, 10) = 1 10(k) 1 (mod k)
S(n) S(an)
10(k) 1 a = k
1+a
gcd(k, 10) = 1 k = 2 5 bB aB b kB /
(k)
a
S(n) S(kn)
k B /
dAbRDdAISl V s VAg
FvEv9FC@wH}daYBR'IfRXvsDAGrqYgwRBv9F'w@DS@P c aAP c E 9 aA a 9 E
c t 9C aRH EAR bShgbICuShgbYp
U@wYwvCdcBdcDrIShgDrvrIFUrfRWUIBcrIXTU@IvE~dAIShgal QDut VR aT A a
aT QiiCg H aC qA aC q C iAA a VTR V g RHA c VA C
@HbHFC{aIfAUIyGvEFbY@UBRcIGE~vHvIfAzaa
cV%t@HwCEBdcDrIShgDrDCYhgDfADCzfVda@`Tv9v9c c c V aHA R 9C aTp ET
VA C c H Ap aC qA aC q a a E a E bBRdca hcYRWwfSlFSy
YGE@yBP@USGrIBAbSdR@5hgXfTEuwS5fRF'EawIBchbH'bHFy@wC H E i A Ag PA
9R Ap 9TRgE E q 9p Vi tRg a EC g E c E cC PH E q 9p V E V a c QH q
P TP a c c VA C V Ag A rIBAbSdR@i~@HdcdaDdA
IfAUHyUwCrvfRa%FfAXYSdRfVUHvEI@PUDdA~SleSSy dczaXzTfVdabHyQ
bHFC{aVIfA Ec 9c vEc#SS@p#ShgbYp bBRdczaXzTfVdabHSy
%fR}@UrYwqIfAazD~@'EFu}dAVISgheSiwYfgIS8 RR A aaT H E c A c a APC q
ai aC RH C t A a E C VA P H V s VR HAy Vi VA cA C 9 E H tRH VA E 9
A FfA@IdAIADdAVbISFdR@}IShga D%dAVwEa @fTDdAV%w@UhfxIGhEDAB~Shga PH
H cP T PAPH a c A a a H tRH 9A t VC RRp A a c AET 9 E V ER
@wC%fA@IBAbXICrF@@IfAUH%Shg5fRv%w@Uv9D5#dAw5SS@5ShgyHwPFG@}Ea
@fTDdAdaY3 FEBdcDrIShgDryShga bdRtVszaUzCfV@PBP@U5)IShgHFvC
Siwf|Rfuqdc~@HcwUb#FEzI@UCDdAV a C q A a C q A q a A 9 T R t q A a
c 9 A a A a C V A P daYw3SSfRdazEI@@HW|@v9v9ut@wC iQwCUIIBADA
{awSvdaYqSghbRs dAw3PFG@T ER q Qp PR VAP T Ap c PH V aHA q 9 VCi ER
A a V VC AE E V Ag t 9 s aC a qA A a a 9p 9iiC cP VC A a c EA T A
rdqIfAauSlFEuwBRv9YRDYhg II ShgvfRuBAbUCBvcS@wwQ
d9zaFdAVbvdAwwQGShglvs#DGE@%Sg Ea @fTDdADfADCdabrdqIfAa}D@@D@rQ
bb5SDYhg'bHwDuS#vEcYhg%SU DDbRBdczaXzTfVdaE 9 E V E a E VR E V
EAHAP 9HR Ag aC a E CA q Ag a Q H x c A c a ai q aC RH c VAg aC a E
a VR aTC A s Vi A a @H5Sy |fR'YwIfAzaDw@~vE%dAISl7
DYhgDfADCdaEbShgbIvShga fAUCBDdA@vSghriH VRA VAp T PH VpA 9 V aC p
qR A qR 9 V aC cp q QibFSglI@qW @wC zCSFIB' EBcbfRDbHcbb iQfVabFA
FvCcbfRDbHbbR PHC HCA T q s q P VAy ci A asR Q EA C V s ECp A
@wwD@BPvcd9XU
bdRtV~AbfRrFdAISFfREBSbfRa~Shg@YdaEYR3#DhChEDHw'bRvEchFb5Shga
FCFG@IF@v9FvSDYbBRdcDbHUrqcE bShgbIvSgh#Fv9D5#FC h
bwvCdcBdcDrIShgaDrq E PAET HAAp 9C AyCg H aC cC VR aTC A a EC 9A t
E EH aC qA C 9CH E s i V E AH aR 9iR i VA aR PHC VR aTC A a HAA Ap
EH FbbBRhchEDAYdR@V Q v9cwhChEDfA
F@%w@tH|BASbFAW~IShgw@w~bShgbIwSghIF5zta@ybbRBdcaDC BxbWwbfR
EdaDfAUbfR wfb5tXFbbBRdcDDdAfVFdAV@wEDffAhgw@p vE9FbdHWTUiR
P@w#wSSl cHT q q E aH E VR 9CH aC PHC a a aR C V HC ERR FDfAdWUT
GE%UwrbdRtVvF@wDBAyIF@#SDY}rIIGDSglU II vEv9FCbSghbIShga E V R QHC
q q s PAHC 9 HAAp AyCg E qA AEA qA C 9 VR aTC A aCg 9 E VR q 9p Vi
a q9 aTCAp C E c 9 A 9 s V Q EA C DY5t a
@fTDdAV}b)IBAbSdR@rhgDCr@TvsdcbID@y#bHFvCSiwfQ
v9@TDdAwC|DChhEDwhgUY DSGBA9~fAD@YUzCDdI@@WblYSShgGIBcY%ShgIShga
DDCBwiWUdRVD 9Ay aCTP V VAPHT VR9RR E g g A a VA cA a T @Hby
YGIBAbSdR@bX`Td9EBdcDrIShgDrq%@YIhA`bwdbrw%fRABbpU@`TFCYvQ
v9FvCdcFWi7EDA c 9RE q 9p Vi VR p aC qA aC CsR V cHC VR HC a 9 C 9
y 9 A
@pBP@U5@taDwIXUShg@wCFD@BPIFCBdcDrIShgDrqHc#FfAdaDdAVIfAUH'@bSUw@bR@HbUCDdAV
A 9TR t agTR a PH ECA c9 aC qA aC P E a c AHRQHCV s cP BAbUYFRi
ISS@Shg5DUUrEBSbfRw@HdcdaDdAIfAUHYR3Q{ABcwCYShga@wC3bBRdca hcYWiwfA
9pCQ HA RRp A a EAC q ci a E V a c s a V y A PH H ER wDBAd9wShga
EUI@PXTdayfAD@YUzCDdI@@W|@wCYSShgGEIBcYFfAUIABFC|Q
v9FCBdcDrIShgaDrq VC A aHA E aCTP V VAPHT PH 9RR g g P aH 9 a 9 aC
qA C fRa}BAbBphchEDfAfUCIShgaDUUrqeDYhgDDSGBACDC~IfAzaa
c%'dAVwbDChhEDSgh#IXTUSgha ' 9 E qA EAC aC a9Ay 9 a H V t C EQ EA A
a g R 9 EQ FbDhChEEDA 9 a qA a q 9 a CAp s H 9 F C y B ` T
FCBdcDCrIShgDCr@TvsdcbTID@YRbBRdczaFBAv9YfRC~vEc~UuvG@
d`tC`eEWCDSRUESS` CBcI} VA q sR H R E 9 aC qA aC A a q E VAE EH ER
9 aC qA a c YwbBRdcaDvCdcSGhE'bFCBdcDrIShgDyShgbdRtVsDBAcIGbbRBdca
chYWiwFFCBdcDrIShgDCbHwvCdc BxYR%ShgauHwSSR@~@HbHeIShgw'E IIdI@Gbb5
bGEhDAYdRWYvbRBdca bUCzfVSghuiH 9 A c ER p A c VA aR VA VApEHR VR E
s V 8 sR H cP a A a zws F bTu b C F pT bT bGe@hEWCV0TA U WEtp bGb8z
gs @ P rC zV'wAC F2% l$87$64320 (&$" H uETC w)T | % '5 1) '%#!
P T p @Ac@G DIUSi% G @WTU@fpIWwSi xwU wA xU(b EAC PH R VAiC x Uw
CBcIA}bYR|bBRdcDvCdcSGhE'whgDShg5wFSgGvEvcbpXbi IIdVI@Gbb5 EhYRu
Qwp V q s H a R E aC A a Qp PA 9 T VA ApEHR E F Cy ` UuvG@
d`teBe`EWCDSRTUESS` ( " $ 0 (" 7f62(#Y2552
a a A a c PA c a DDffA#ShgyHF@HFCUHbfR EzaFiA
dR@ihgdwDGDdAV@wC~rIBAbSdR@iu@UdRIWYwShg5@Hvc9whUCaFiR
IPB@USgGI@UCDdAV V VCAE PH E q 9p V ET VA q TH A a Q HA 9TR E VAP
ShgrIShga v9dc@IW@w'IWwSwhga
cuFAdqwCvE'I@UDdAvShgXrsFDfAaw@fPF@wCXFCBcbfRdavEcYg A a HA HAi PHC
VAiCi t P V c VAPC V A a E R AH 9 V hga
cu}IShgISfRriQbFShg}I@qWY@wC#zEI@pWYYuEazFW7FvbGEdHwDABv9v9u5SS@p t
VA aA a VRA a VA p TH P H VA q TH sR A iEC A qR VC 9 c t RR
vEYhgaYRI@UDdAu~DUI@PXTdaEeYSShgGdESIBYgFfADYy cdwQ
v99FCBdcDrIShgDrrfRw@Iq c s VAPC V aHA 9RR xg c P aC aR q aC qA aC
q C a PHA x q V 9TR t aC a RR C R 9
bfRFdABP@U55DYhgSS@pGvEFC%vEctaSFdADF@vEBdcDrIShgDrq@H@fTzEVWb~aw@H
A VAP aC qA aC C c Ti R
dAwCbY%EBdcDrIShgDryUBvcS5EUI@PXTdabRF@DBhcD@yAGzEWTUfReDSGABd9@{Qa
chzEISy bWT V aTp aC qA aC q A 9 Rg t aHA E V s PAH EAP V 9Ay x VA
cH C V P c E 9A t c aC a RR a a C c bT s B T t rbRsFfAa
@fTv9D5vEDYhgSS@hpDffAvEzVqH vu` T | USEuYF CbGE@B F
FEUI@XTdaEXYSShgGdESIBYrFfADYy dcwRyfRABbBphcEhDfAfUC aHAP 9RR xg
cg P aC a q a 9 E dAwCuDYhgEzaFiA
dR@ihgdwDGDdAF@BP@|@bhR@{QUI5tzubHFCUbfR%IfAYiwYhgyvEcYl %twR V aC
a V VCAE V PA cT AH x aHA a E c aH V a C g H VDShgbRzEI@WYHEBSbfRa
UXTdavIShg{aVWT3fRa{Qa bWTa{bW@bRwrEzI@UDdAShga cA a H VAp qTH c ci
QP E VA s cH VRii HC VAPC V A A cy Vi EA VR aTC A EA E VPPC RRp A
sR V a C E 9 A @BPbFdR@GfRadAVhcD@PE
bShgbIuShgaDGhEDdAS@U%SS@vShgaWY}IfAYiwYhgdaFvCuSgl FzEI@p VA x T 9
aHAPH EH HR H E cP aAH s A a V C PH EH EH EC
qWYHFCUI@@IfAGbwzCfVabbBRhchE@TGvEb#Bdc@IdAtV}VIShgDCzyw@wC
bbRBdcazXTzfVdabbfRhEFSi x q PHCV 9 VAp qT c VpA 9C AP V a C c P E
cP ci aV a bfRu@wlIBA@zTV zEI@WYHBFzCSFIBF@X`Td9@Hc
IfAYiwYhg%H~FAGhE@TGvEbEBSbfR@IAShg
FzEIIFfAUHc~Shga%fRvHFC{aIW}DYhgrIBAbSdR@}I5dtbwwC%@bR
zEI@WYWFDdA@HcVbBRSi VA a A a c VAi aC a E q 9p Vi VA EHC H AH VAp
qTH9C V 9 Uf|wp DYhgGSUbfR|fRa#DIUbwCrq DBhcD@bUI@PBdcfUQw#IShgS5t
bShgbI5Sl x A Q aC a QAyH EACH H EAP VR aHA C p VA aAg VR aTC Ag
FbbBRdczaUzCtVshgXfTEYRG@T%ShgwpbBRdcD@YS5E v9DSwSy YRGrfRtwSFGv5t
IfAYwYg EH s AE A a Q H aCT 9A8 A 9 E a Rg AAE A V aiC
iHFbbBRdczaUzCtVFAXYHdcUbfRYbBRhchE@TGvEbyvE|
IfAaYwYYwIBvc9YIBYgSgl EH s P T aH sR H E cP C c V iCg sR ag g c A
V aiC a AH A asR H AE H s PA T aH A a a PAHR R
IfAYwYhgDf@%ShgSYbBRdczaFGbBRdczaUCzVtrFXYHdcUbfR%Sgh5fR5F@bWfidaEYWi
R gr F C q S E 9 A H @ C b A P c T FC C hga IA aD c V @EA wUBPzap
zC@fV9 UvETda SbG3S@SwhgHRPP C vCIvEVcq fAYDhgac r%3hgbaH BRdc@za
HXbYFSRPQTc i dRfIUHcwwdAH T@DHa r@IwdAHqP @daVC |IS@hgAAVEi z @aP
WH SHA cyG5btE wPB5HwXYS@pShgShgzCVa c P a A YR as Va W C V V I A R
SR A R sR E A a P aT A aC EH a 9RE E cPH R PR A q V 9Tp
Y%daDdAVwShgHcFhA`vcv9dcbv9D9
5t~vEc~DYhgaubbBRdcb`TYGhgXfTr@Hb@@sYSShga~wvC@YUCa C E TP a c VR
aTC A q VR 9 HCA T A a cy EH aCT A A a Cgi c
}DfAXYSdRfVUHbShgbIShga bhga
cbIB'vwD@BPvcd9XU5ShgvCb}bbRBdcD@YF%@HdcUHwYWbBR} @Hby
YGHiyFGhE@TGvEbGvEFvdAwCv9DS@wFzCbIUYhgaUb'Y5bbRBdcD@TYFSgluDBdcaq
c 9RE PA E cP R 9C V 9A8 PHC E VRC Q8 sR EH aC A A A x VC V 9TPR q
PHC q VR 9 HCA T A a Rp cy A 9 E c H aC A Shga cwwvC@YS@wbhga
cbIB'vwD@BPvcd9XU%Shg%hgaw@5vCb%PFSy YRGvEybRBdcD@TYFA%@Hdca xHC i
c VCA 9 EH aCT A aHC i c c t 9 cC q 9CAP V i g
@wYgWbBR}}wD@HvcFbbBRdcD@YFA@HdcUwYgWbBR}hga uuQ
bHFr'vEFD@#IfAaYwCY FzEI@WTYXFvCdcFW7EIShgwDAGhE@TGvEbw@wC VAp q H9
Ai VA aR E E cP PH DA@bvhdwDG7Sha'D@BfGD@'bShbICSha Bdabha c}bYR
IdbFSlSFCUIq E qc V i V Rs g V C A E A g E A pc V E A P V R g a T A
g c A q g V s q A V R A g 9 a H A Ub@HWwyShga@HbFdR@i}IfAva'bhga
cbIB'rwD@BPvcd9XTUyShg}DfAXYSdRfVaUHcw@wEzI@qWYH xCP T A cy V V s q
VR 9 HCA A a E TP PHC VAp T 9 V aCH A FzCWTDbShgaHc{Qa
vcv9bBphcvEby bY%bbBRdcw@H#BhcFbhga uvEFD@#IfAaYwCYhgdaEz@vSgl c c
cP sR EH aR ECp c t 9CAP V i V A 9 yHw T vA Q 9 A
zEIfAYwYhgyShgaEzaF@SbfR#DYhgUDdAbhg5C#vE5dAISA ahg bq hgIaV
cSbGI5ARyVA B'5DF@EAC PBBvcSd9c bXfRi U~fASdAhgfVa GvEbDWwUCTqV
bRHs V aiC A AHH aC a PC V a c V t wR a a c P
FCuzEI@pqWYFDdArShg|@wC zEI@pWYFbbBRdcDzCVwSgha zEIIFfAUHrShga
EzI@pWYFzCWT E VA TH 9C V A a PH VA qTH 9CH a A VA a c A VA qTH 9 V
hxDbyShgarhga utvEFD@P#bShgbIvShgaDYhg%SGBAvcD@|fRwFABvEv@bR
EaUIfAUbYABbUb aCH A c 9CA VR aTC A aC a Ay 9Ap a P 9 c AH H aHR sR
9pC Sghy@HbUCDdAVbRW U
zEI@WYFDdAV'YUXTdawShgbdRVtwAGvEcwChgBY5~EBSbfRav@Hc A a cP H i VAp
qTH 9C sR QP E A a q s V cg t ci hxdaDAdIfAUH7 UIAGDdA@rfRavEc
bShgbI5ShgvfR#@HbdPbfRfUC SS@vEYhgXYqF%@ E V a c aH E Vi VR aTC A a
a c V RRp c asR cC AH @gs$Uq@8zVTUF G@8&4ETcB8&Q cBB T
VTb2TDb cBC #b V$i1 $! RuETC w)T | a @ b b b Q T B @ F x u " C 9 #
P H T p WUvS DIUSiXFA ISFfR@dPwYg x Y hxw xU(b EAC c VAy VC 9C aH 8
Qp PAg 9p T HREHAy a VA V Uw v9FufABdcUIdAVWwFSGvEvcbXbi
bGbISGfA35hBcI@PFdAYrQwp zVqH v$` T wUESrYF CGE@B F bT sB | T b
t
G
R!Q0C %F$)P1F5G %FBE1C$B0BA@)5 ""$)6 )3 "2 01)' %&$" GIH 05
D ( 5 9 7 %8575 4 3 ( #! 4 = n nc
HA A Vi PP HC A ISgleSqc@@SRw~@p aA )F $d %o Qwp g T V a c 9 9T
HA a PH EH H aiQ sR E V a IXUdRbhg%@HbQ Sidca
@qIShg@wCbbBRdczaUzCtVswvCdcYSGS'Y~rdqIfAHc @Hb@wSiwfA cPHC V s H
ER i q AP H s a Q V x V a wFP @HFCUSRw@pDrq bRbBRdc a
hcYWwbfRF@bRBdczaUzCVtHwvCdcYSi GSdqIfAazFFA dAbhgC Qp A c ap A QC
PH @wC VAg HA IShga Aqc@i@SwuvEc dAIS5t V PPR HC vsc dAVbq R VA VT
VA a A ER QH @wC PH AGYSShgDr55@wC ERR QC q A t PH
fxIShg{aWwIIFfAUHcuSy dca hcYWiUwC b s H a i S % 9 C c T FA H A a
HA V GvEcu{tIShgdc BR Sza UzCqtVcrwvCvEdcc YSGDSYvahgCaQ hgWfRq
fThEa UFI5BACHEa FDYy @Y5 Evc Q{a vcFzCI@IIShgXa YhEYRBbbRR s c vsc
A V A a @} A Vi A Q rqA t 9 V A H I s R E 9 a C A TUShg a | dAISg5t
VA E FC FGhEDdA@wfAu@fRa
dqbRs~ShgSYwbBRdczaUzCtVIWbdRV@5uwRBv9F'DYhg~vC@dqbRuAB@HhcE PA E
Vi Ap V A a s R H s V A i i C E t 9 C a C a 9 T V s 9 C TP a c A 9
Ai yfAXYSdRfVUHy5t Q v9FCWvdcFWF dAVIS5t bR|bBRdca
chYWwbfRF@bRBdczaUzCtVs Ag V s H ERi q AP H
wvCdcYSGS)dqIfAzaFFdAbhgUFhCzEISy bWT#UIGDdA@5fRvEfAaw@}vEYhgYY~Sy
dcazFiA SSgl H aiQ V xA V a9 VA cH C aHAE Vi a c RH c asR A pR A
SSy dcDF@vEbbRBdczaUzCtVs A aCAH c EH
wvCdcYSGS5ShgSYRu@bdaFDBA9DCWYRwDBhc'BFzCSFIBFShg#dAIS5bbRBdcazUzCVtywvCdcYSQGSsYR
H aiQ A as AHR EC a s H E c VpA 9C A a VAg t EH s H ai q E A a EC
PA E Vi A Ap H WfTvShgFFGhEDdA@wf@wC DYhgwH%w@UuvEat5SSy dca
hcYRWy@p aDYhga aC a tRH c A E i A C bBRdca
b@bfRShgFwvCDdAVyFYDhAYHYhgw~DYhgaFfADCBb@HvvE% ca HGvE' H cPH A a
PA 9 ECg 9 c aC P a cP c c c R9 dAIS5vQ bBphchEYW}YIAfwfA
v99FbRuXTzfV#@fRyIFAGE VAg t 9 ERi ai C V s A a Ap a HA
vEcdAWTzaFAi bfR%Shga FhA`cwrqwWfTdAwC IBAbSdR@ivEYhg5bRWdcDChYs
@wC fRwwrqwC 3 V H A P VC q E V q 9p V c a H a ERu PH aR qC hgDC
bX}u HDfAdabdHIAl YEa @fTDdAvShgIS5t DCYhgDfAw@wbe bRBdca b@PUwiH a
9p c E V sR 9 E V A a HAg a a E aRH QT H cP C V s A c E H V Qp PA
VAy HAAp E g q 9p Vi A bRv@HYhgGDfA@wzCYlv
wFWvcISrIF@~FCYIABbSdR@vSgha be }u QT fRa%@HbdPbfRfwFbbBRdczaUzCtVy
wvCdcYSGS FBAv9FCdAwC vEFCSdR@idcFdAV SF@UCw@BPch~wIBcV c V E H s H
a i Q P 9 V 9 V A y R p A E a g ShgbybbBRdczaUzCtVsSgl A a HR EH A
V VA c A ER 9 C 9RE bRs zEIIFfAaUHwSy dca hcYWiHcABbpUYy YG5vEc
bBRdcD@YFuShgDYhgDfADCda}@TIzCfVw@wC'YdPb@lYudAWTzaFAi
bfR%ABbUCw@H3 H a C T A A a a C a E a E E a P H E R V 8 s R V H 9 p
a Rn 4 = x 1 n {4 5 . . . } + y 1 + z 1 n < 108
$ 2 " S $ (552 0 ix 7 Da 3 x y v x 4/n
x y z
x y z
n = 2k
1/r
n = rp
4/p
k2
n 1 121 169 289 361 529 (mod 840)
+
n1 2
4/n x = 1/2k2 p
1
n
4/n
+
1 1 1 4 = + n x y z
n+1 2
1
y=z
n > 3
4/n
n
n1 2
4/n = (1/r) (4/p) 4/n
n
1
4/n
n+1 2
1/2k2
n = 109
109
4
=
1
109 1 2
1
1
+
109 + 1 2
1
109 1 109 2
n = 1001
1001
4
=
=
54
7 11
1
+
55
13
4
323730
=
1 1 1 + 462 539 42042
13 + 1 2
=
77
1
1
13 1 2
1
[email protected]
UGw 4 wCrdqISw 1 H VAg A 9 ET IFBAv9YRwHdcda@ fA@IBAdcwIfAbbwbq
REBdcDCrIShgDeYUIAfq{awSI H V aTi a qA aC sR aH VCiA CrbdRwC w HbSR
q V g G fVbR 3 UvC9IA tA V VA Vi P PH b@ S bT II@Hc@F S%@F XPGTU
VqH vsuF C$rET0@Uu'wB@ b2% Qbe dPwCYhgBu BH bs T F T V c & % (
b "
13
+
13 1 2
13 + 1 2
1 1 1
109 + 1 2
@
p b Y
0 f
YR%dAfVUIfA@H uShg'vEc DYhgSFdR@i D@PBhcEWURs s aH c A a aC a Ay
V EA VT v9FfRUII@wCuBAd9dVdc@HwC%SDYhgUDA 9C a aHAH a c H AyCg C
vE9FzCIfADvCv9cSU@TY V a VPC q s V 9 cPHA VAi A a FdAbhgrShgrs3Q
DSy dczaFW7DdAV fRa bdRVtvEzwvC@TBb@IWdiIW| ShgsYR A V a A a 9A A
iE aA s A a A FAShgv@p A a YR~bBRcIfAUHShgH#aUHYW@p aA s V V a c A
a c cRi C A ~G9(%(} UuG@ T " frCtq@UF 0`$P ETBAG`DY9 H uT w(F@U ui
b p CH | F P @t bs XuU#XeWwq)9 VeTwT T
@Hh@TUIAqIWwfrwPFfdDIIXfFutIABbSd@vEYli u ` t S T b @ x cE Ea H cVA
i A Q p AaEA TE EC q 9 p RV i c g UHYWuShg5IXUdRbhg}DGhEFSi a cRi A
a g T V a EA EC aC Ay V8 aHAH 9C V a c HR q q VA aR A a A
DYhgauSFdRW|DUII@wCaFbdHIfAUHbwbfRIShgwwSgh5@p aA @wC DBAd9dVdc PH
E SghvIF5zt@pEUII@wCYFbdHIfAUH5bRwbfR 5ztSghbYu@bvEc @HvcSgl Q DSy
dca A a H A A a A a H A H a 9 C V a c H q q R a A a s R A H R A 9 A
9A A E c i A EA a X E hxFWi7DdAV Hc%F@BpcfVGbUcaEE H HYRW
#ShgaDDBAaE C d9dVdcSl D@PBYhcs dASfVA ghaUfAhg@HTUafRc Shg'@p aA
PA Ag R I A a A H ~G9(%(} UuG@ T " frCtq@UF 0`$P ETBAG`DY9 H T
w(F@U i b p CH | F P u @t bs 3
sRsR Vi c V A C HC A c YlYSdR@BFCzSpFIB9Fw~Sy EPA A 9H AHR ER aC
cg t bFFfAwfvBA@wC~@bdaYqDhgBY5yHc sR EA E A a A YD@PBhcuShg#@p aA
~G(%(}IstVytwP 7u s d`"WT {mH T w(F@UbRBdczaFdAdVb ``i 9 bTw@ b Bu
P u @ t bs H VR ~ ~ ~ 8a2 b2 c2 + a b c a2 + b2 + c2 6R2 b 2 + c 2
a2 ABC 3abc cos A a b 2 + c 2 a2
a C a A y E P CzV c c P H EbFFfAwfA~BA@wC#@bdaYD5hgBY5tuHc
DYhgSFdR@YisV @@bPcA`T BUwhgqWdTa dAVwdcV CEa vE c rs vs@wC PA 9H
AHR ER q aC cg R D hcE S E A ~G9(%(}IstVytwP 7u s d`"WT {mH uT
w(F@UbRBdczaFdAdVb `i bTw@ b Bu P @ t bs H VR v ~ ~a b c R ABC
3
c A @ $6 ( p ( ( $ p (
&B&98%45S&q210)DX'$5%2'U9$5u2pl p $ 0 o ( p 0 ( n ($ $
$i x k j ( $
&G5H(X0'hFu`54h&BDgU!&vD10gmliSH0q$&%f9$mlXu&!45t2H(0C4&i
h210)'"&i9"q)'Gq$BDAH08%45Sp&GF&")D%45( c ( $ ( p $ 6 (
D0P)DUG)e 0ts &98654(e210q210)'"&G54gGh4D0 U)FBXf988 t'49$!
( ( @ A @ $ (f ( ( c y $ 6 0 s f f e d " 0 0 f $ f$ 0 s y " 0
&g%72%&PF&DDw%'49!3X986S t4D0 U)Fg%UUBF&D
!Fe&F&)D#UD0&54'49!`D0 U)F3D`hUrD0&@9$8P'4FFy c p "
f ( f $ 4 y 0 0 A 6 f c ( h210)"'&G )#54H(0t`2h&986v%s
&98C#)54%)HfDh)3h7(qU jDH0U`t&2a&Y (" s 6 $ A @ $6 ( (
( 0 0 $@ A" $ p $ ( cba W I ( $ ( $$ " 4 " 0 0 ( 0 A @ $ 6 ( ( "
Ug54`)DwhH0q U#BF&DGDF54DB&98wB210)'&Fd
IBC
d
I
P
ICA
f
ABC P BC CA AB AEP F BF P D CDP E P ABC IA( )
E
ABC AB BC CA
d e f
D E F
D E F
IAC
bI Shga HR A sYDdAwrShga~DfAw@I@P R C VC A E aRHA w aC a Ay V
DYhg%SFdRW8 PH @wC dAIS5t VAg PH @wC DCYhgahgXfTEQ
v99FbdHIfADfPFfAzaXzTfVdaE a C V a A xH V @bfRvdAwC E 9H
DBA@wvCcfVa AB@wvCcfValY~D@PBhc5Sgh 9H sR EA E A a H GUF` `w`rq`8 w
CP C A9 E@AS w H uT w(F@U `i F`t C p ` T C @ P @t bs ~ HR A asR C V
A a E RHA bI Shg@YyDdAwC~Shg}DfAaw@I@P @wC PH sR HCA a qRA A a
YwDq}BcfVAbFI5Shg'vEc DCYhgSFdRW8 a aAy V Q DSy dczaFW7DdAV @wC DC
@HwC 9A A iE P H a P EzaFGzEIfAUHc DYhgau@wC Q DSy dczaFW7DdAV @wC
DC @HwC EUIDFGEShga A V a aC P H 9A A iE P H a P aHA qA A aHA H C
aC AER i i T 9A A iE EFq fRa}UII@wCa%DYhgaGYW@WD Q DSy dczaFW7DdAV
@wC DC @HwC aAA PH a P EA EA T D@BPhcE#DShgXUfRa BAd9dVX
dAfVUIfA|hga ut BAd9dVdc@H#FCYg c aH c cE ETp qRg @YbSeu GUF`
`w`rq`8 w CP C TA9 E@CAS w H uT w(F@U i F`t C p ` @ P @t bs v P
@HcU D DYhg#AGYWi@WD aC a ER iT CF` | b` W z b` @w C P @ t b s r
Wd`AGR @ rTG4XS4%FGG4FG#Y rV H uT w(F@U `i 9H SAB@wC
BxcfV5Shg5hA`cIfAzaUCzwYgyDUHYWwvCbFwSQShgwRGvEF~vE#aUHcYWuE
HdczaFDCYhg5GEYWi a A a V VC a cRi H cPA q q E A a 9C c Ri A a a A
R xWWD UHYWi'E
HdczaFDBCUIdAdVWT@bfR#dAwC#BA@wCvcfVbY}D@PBhcShgXISFyQ
v9FbdHIfAaDfA iT a cR A a aH V H V 9H a CsR EA E A aVAyR 9C V
FfAzaXzTfVdabbfRDAdw@YGEShgIYbBPYdRfVUIfAShgfRDfABdca{ISShg@HbHcYiR
D@Hvc}Sgl P EH E VCT A asR E c aH A a aE VAy A a c EA 9 A CF` | b`
W z b` @w C P @ t b s Wd`AGR @ rTG4XS4%FGG4FG#Y rV H uT w(F@U i
k=1 n
aC a Ay 8 VAp T 9C V DYhgSFdRVW}FzEI@qWYHXFDdAVudAwC
DYhg#AGYWi@WD aC a ER iT YT WUG`A F{@qVw`EbY92wd`cBC hwtT 4G@ H uT
w(F@U i F b b@H ` u `S P @ t bs H%w@UhBxv9D5tw@}bYp a
@fDdr%tw@UvvEvEYli tR H 9A aR H a T 9 TEAV H R H C c c g A a aH H R
9 V A a aC Ay V H V H V DUIdAdVVWT@bfRwGvEFC~dAwC ShgDYhgaSFdR@i
aUIdAdVWT@bfR%dAwC Shg@rs @wC PH dAwC P@wC
fREUII@wCXFbdHIfAUHrbRwbfRySl#IShgwRQUwrfRabRBcIfAUHc V H a aHAH
a9C V a c H q q Ag VA a HC V V a @HcD@p@b@H @wvCSShga~H%ISy dAwC
@wC DABd9dVdcUHYRi vEbFdAbgl A AHR AH 9i A c HA c V PH E a c cP A V
~G9(%(} UuG@ T " frCtq@UF 0$P ETBAG`DY9 H uT w(F@U i b p CH`| F P
@t bs
jk
x2 k
jk
1jkn
k
x1 > x 2 > > x n
ln(xj xk )
A1 A2
min
BC
AB
[AM ON ] [Z1 Z1 . . . Zn ]
ABCD M N AM AN CD P Q
AD
E
n
F
n(n 1) 1 (1 + 2 ln 2) 4 2
2 n2 + n k
A3
n
n
CAE = BAF = ACB + < 180
[ABDE] = [CAF D] Z 1 Z2 . . . Z n
ABC DBC ECA F AB DBC = DCB = ECA = F BA = BAC + < 180 ABC +
< 180 jk O [AEF ] [CP Q] Z 1 Z2 . . . Z n Aj EFk=1 n
[Z1 Z1 . . . Zn ]
Ak
k ln k
jk
a loga b+
rs aDYhgaSFdR@i C Ay V PH @wC C ` w FT z W`dAFGR @ |tD@ w`b8z%T
qWUV(gP B ` CS H uT w(F@U i P @t bs VA c A Ri C IIFfAaUH~Sy dca
hcEYW5vEc dAISg5t VA DCYhgSFdRW8 a aAy V CF` w z FT z Wd`AGR @ |tD@
w`b8%T qWUV(gP B ` CS H uT w(F@U %i P @t bs aC a Ay V DYhgSFdR@i
y}k=1 n
rs CB aC Ay V DYhgaSFdR@i P @HwC Wd`AGR @ |tD@ w`b8%T qWUV(gP B
` CS H uT w(F@U i CF` w z FT z P @t bs DUIdAdVWT@bfR%dAwC DCY a A y
V aH V H V a hgSFdRW8 A i C 9 a I FG arbR~ U i A Daa Yhg 5a hgXfTE
Q DSiA 7y dcazFW7EDdAV H A H a DA@BPhcEEYUvAUDHqaE YWv@SBPAxfiE
hgrqShgEH@YpcA RW v@p aAB9 A Q DA9 BSA9 @dcywaH zvCFc WafVA DiHE dA
V E sR cR c A a GUF` `w`rq`8 w CP C A9 E@AS w H uT w(F@U ~i F`t C p
` T C @ P @t bs bI Shg@YyDdAw~Shga HR A asR C VC A EDfAw@HI@P aR A
a a Ay V DCYhgSFdRW8 Q DSy dczaFW7DdAV DC EaUIqDFAGE 9A A iE a HA
EAFq a A fRa~DBAv9FzCwS@HvcC 9 9 VCi A 9
FzCIfADvCv9cSU@TYfSyUbfRuISy 9 V a VPC A H C HA c GUF` `w`rq`8 w CP
C TA9 E@CAS w H uT w(F@U `i F`t C p ` @ P @t bs AB AC BD CD z w (n
1) n AM EN F P [EBCF ] : [GBCH] = EF : GH Z 1 Z2 . . . Z n ABC D E
F BC CA M N P F D F B DC F M : F D = F N : F B = DP : DC |z| = 1
w+zk
AB
x
n
ABCD E F G H
n1
(n 1)
1 lim loga n n n!
k=1
|cos(kx)|
an
(1 + k)a
k=1
n1
k=1
(n k) |sin(kx)|
(n k) 1 z k
n
a2
a b c>1
loga c
+b
>0
logb a+
k=1
logb c
1 1 1 abc a 2 + b 2 + c 2 +c logc a+
= e
[Z1 Z1 . . . Zn ]
logc b
rs p
AD
#
!
SdfA 9~@%BAbIG@v9c Y|YSbf#@`TS~dfA a FCzIfa cIA DCa @fDA uz AVa
A H 9 p qAE T T H H R E 9 i AVa P c VA V q a 9 TE V A
{hchE@IC'UbGEuBA9 DBAzEbBRFC DfAUwzCVWUfR@bfRUbGE
DABWch~XTY'IdAfVaUbR3 ET aHR E 9 V 9 E aH T H aHR E 9 E A V H aA C
D@dHIfAUH}D@WwqbfR EA V a c EAHT q E aHAH a E DfAUII@ wCvDBA9 A
wSwbb@DBhcDA FDdAfVbIY@YI5WBAcIA aUH5aUwCaA H a VCi EHRH E E aTC
TAP TC VT V c H H T WTXIC A a EUHYiR vEbDBAd9dIfAvEYdRfV@Sb@PbR
wvCSiABbHw} a c cP E V c a AHHR H H 9 9 E C % b C p@ CH` | F `t T@
t b ~G9((} UuG@ T " frtqUF 0$gP ETBAG`DY9 bGUiw(F@Us i
BA@HwvCcfVTYa cVfGbH~ABd9dIfATY%dAVfUHIfA 9 a P E c V P a BAudaDA
UHYRWwBA9%zEbV BRFC DA wR 5DdAfVD@YYI~UII@HwCa cfVGbHABd9dIAfW@PFA
hEEYWi 9 E a c i 9 E a E aCT TC aHA a E c V H T A R a A E V a VPCT
c a EAP H C AT V HR DdAIA
DvCv9cSU@YyvEYdRfV%D@WTUYgh}hcEwXY%IdAfVaUb3 A V AP EA 9 A i E A P
bP c i E 9 a E UIAaH ISy dczaFW7DdAWfTa E Yw @vDFA
hEvEBAFb@UwCHpC9c ucvCDfVE dA3FYvCTcVa @TVWb@cAPTcB BIIWAAVH
diaUIHV WuDY@W%FABT S%DABa YRw A a q A A i E yc R V P c yaUHc R i W
c H % b C p@ CH` | F `t T@ t b ~G9((} UuG@ T " frtqUF 0$gP
ETBAG`DY9 bGUiw(F@Us ui Xu`U#SXeWwq)r VeTwT T x u t T 9 b @
9DBAdcBSBRyBAFSD~DFA Iq%DfA@IBAcIA wfA rfAa @fTvCrC A IA IXfTyA A
5IA bSdR@yz c 9 9 AyC EA HA E H V i P c E 9 V E a C A q 9p Vi A
UHYRWuBAwSi%GhEFSi XY'IdAfVaUb3XS@WwbfR5@dHIfAUHc a c i 9 VC A EC
AT V HR AHT q q A V a fAUHII@wCdAfVbIC 9 a Yw A
DBAd9dIfAY@YI5D@WwbfRD@dHIfAaUHvDfAaUII a AH a aT cR a E V TAP TC
EAHT q q EA V c E HA xH TAP EAP AH E @wCa%Y@#D@5@WT 9daDA fAa
YdRSyC A EaUHYWyYIC A c VP a cRi T a E a E 9 aHA qA A iE V aHA T DA
wR DBAUIISy dcza FW7DdAUIShgXUfRa A DAB@wvCcfVD@P A a E 9H a EA a
Ea cfVGbHc~DBAd9dIfA%DA E E V E BA@wvCcfVa|YP~a
cfVGbHyABd9dIfATYdAfVUIfArAB9 a YRw 9H T E c V
