matharound the world part 1

8/9/2019 matharound the world part 1

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eroberogeday lwm pl:nbriBaabR&tKNitviTa nig BaNiCkm

H U N S E N

S T R O N G

3

Problems and Solutions


2/159


3/159

GkcUlrYmRtYtBinitbeckeTs

elak lwm qun

elak Esn Bisid

elakRsI Tuy rINa

elak Tit emg

elak RBwm sunitelak pl bunqay

GkRtYtBinitGkraviruT

elak lwm miKsir

karIkMuBT&rkBaa lI KuNaka

GkniBn nig eroberogelak lwm pln nig elak Esn Bisid


4/159

Garmkfa

esovePA KNitviTaCMuvijBiPBelak EdlGksikakMBugkan

enAkgden xMJTneroberogeLIgkgeKalbMngTukCaksar sRmabCaCMnYydlGksikaykeTAsikaRsavRCavedayxng nig mageTotkgeKalbMngcUlrYmelIksyvis&yKNitviTaenARbeTskm


5/159

CaTIbBabeneyIgxMJGkeroberogsUmeKarBCUnBrdlGksikaTaMgGs[mansuxPaBmaMmYn nig TTYlC&yCMnRKbParkic .

tdMbgfTI07 mkra 2009GkniBn lwm pln

Tel : 017 768 246


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lwm pln nig Esn Bisid

H U N S E N

3

S T R O N G

Problems and Solutions


7/159

rkasiTiRKbyag


8/159

KNitviTaCMuvijBiPBelak

EpkRbFanlMhat

- 1 - PaK1


9/159


- 2 - PaK1

EpkRbFanlMhat1-BIcMnYnBitviCman nigx y epgpatTMnakTMng 11y3x4 = cUrkMntrtmGtibrmanGnuKmn_

6 )3)(7y)(x()y,x(f y2x + 2 enacUrkMntRKbtm kgcx [2

;0] edaydwgfa 2

xcos

13

xsin

13 ++ 4 mYy-enAkgRtIekaN Edl cAB,bAC,aBCABC = cUrbgHajfa cb a2Asin + rYcsresrTMnakTMngBIreTotEdl

tag[ceRsedogKa .4-eKyk I na ]4

;4

[ . cUrkMntGnuKmn_

os(s

fkMntelI[ ebIeKdwgfa cxsin)x2in]1,1 xf + 2 cMeBaRKb kgcena .jfa trYcsRmYl f )x(tan x I5-cUrbgHa tanatana2tana3an atana2tana3 Edl

2

ka

RKbcMnYnKtrWTLaTIhV .k


10/159


- 3 - PaK1

6-cUrbgHajfa ( )2ab21xcosb1xsina1 + + + cMeBaRKb2

xb,0a 0,0< -e e mYyEdlK[ct kaNEkg ABCD AB3BC = .aBIrcMnucnRCug Edl QB =P nig Q C PP]BC[ QC.cUrbgHajfa DPDBCDQC C .CacMnYnkMpcBIr .-eK[

1z nig

2zcUrRsayfa ( )2

2

2

1

2

21

2

22|zz||zz|

1|z||z| + 9-eK[

2Canig cMnYnkMpcBIrEdl

1z z 1|z||z|

21=

.nig z.z 21 1cUrRsayfa21

21

zz1+ CacMnYnBzz + tmYy .10-KNnalImItxageRkam 4x

4x1212...121212

lim4x man n skaer .


11/159


- 4 - PaK1

11-eK[sItncMnYnBit kMnteday =

)u( n0

nig1u1u4 n +u6u4

uu

2

n

3

n

4

n

1n + RKb INncUrbgHajfa 4

n1nu

1u ++ 111 RKb INn CaGnuKmn_n .KNna

1)(1n

2 ++ .nmM kat gcMnuc .

rYcKNna nu n12- x1).....(xxxx1(P n2422n )x13-eK[RtIekaN ABC mYymanRCugRbEvg c,ba ,knbnatBu ] RtC AB[ DcUrRsayfa

2

Ccos

ba

ab2CD + .KNnaplbUk4-

1Sn

!n.n.....!3.3!2.2!1. + Edl n(n!n 1.2.3)....2n)(1 .KNnatm=15- )29tan3)...(3tan3)(2tan3)(1tan3(A oooo +


12/159


- 5 - PaK1

16- itEdlK[ CacMnYnB epgpatlkx&N z,y,x ycosxcos 0zcos = cosnig x3cos 0z3cosy3 = cx2scUrbgHajfa 2cosy2osco 0z .

k g.cUrRsayfa

17-eK[RtIekaN mYy .ABCbnatBumMA atRCC,B, ]AB[,]AC[,]BC[erogKaRtg ,'A 'C,'B

0'2

'AA2 =

CC

)BA

sin(

'BB

)2

ACsin()

CBsin(

+

+

18-eK[BhuFa n)asasinx()x(P co ndlnwg *INcUrrksMNlnviFIEckrvag )x(P 1x2 + .

&n9-edaRsayRbB smIkar yyx4 + bax bayx 4 3CacMnYnBitEdlEdl a nig b 0ba .0-cUrbgHajfa 9

)1nn)(1nn(

19999

1n44

=

+=


13/159


- 6 - PaK1

21- IK[GnuKmn_ kMntRKbf }0;1{Rx eday 1x)

x

1(f)x(f)1x2(x + + .cUrKNna )2009(f....)3(f)2(f)1(fS + 22- ncMnYnKtrkMntRKbKU viCmanebIeKdwgfa )n;m(

)m(13nm 22 n .23-eK[RtIekaN C m gA .AB yEkgRtk nigeKy F]AC[E ]AB[ Edl ABCAEF nig erogKaCaeCIgcMeNalEkgn nig.ayfa

nig ACBAF .EeKtag E 'F ' E FelIRCug BC[ ]cUrRs BC'FFEFE'E ?24-eK[ CabIcMnYnBitviCman . cUrRsaybBaakfa c,b,a

2)bb)cb(4a 33 33 a(4c ba)ac(4 accb 3 33 33 ++++++++ t =5-eK[sItcMnYnB = 1n,2a2a 1aa 1n1n 10 an


14/159


- 7 - PaK1

cUrRsayfa1nn

aaa +1n 2 + .26-kgRtIekaN mYycUrRsaybBaakfa ABCr

2

Csin

2

Bsin

2

Asin

R4

111 gdl r n R C ekaN .

CabIcMnYnBitviCmanEdlakaMrgVgcarwkkg nig carwkeRkARt

27-eK[ b,a c, 1cbaabc4 + cUrbgHajfa )cabcab(2

c

ba

b

c

a

cb + a 222222 + 28-eK kMnteday 101n

[sIt }a{n

1a0

= nig 4na....a.aa + aRKb .a cMeB n 1cUrbgHajf 2a1n= cMeBaRKbn a

n .CabIcMnYnBitviCmanEdl cab129-eK[ ,a bc,b 1ca = .rbgHajfa 16)

a

1c()

c

1b()

b

1a(

222 30-eK[ 432 222,2cos222,2cos22 2cos2 === cUrrkrUbmnTUeTA nig RsaybBaakmnenapg .BITahrN_xagelrUb


15/159


- 8 - PaK1

31-edayeFIvicart dacnwg CaniccMeBaRKbCatin .n,7

amkMeNIncUrRsaybBaakfacMnYn 1n1n6

n92A

++ Eck 11cMnYnKtFm32-eK[ N61009743831E nnnnn

I

acnwg CanicRKb .cUrbgHajfa nE Eckd 187 INn33-cUrRsaybBaakfa 1n21

1 .ehtuen x2sin1)x2(sinf + tag x2sinz = dUcen z1)z(f + cMeBa 1z1 < .mageTotRKb ]

4,

4[x

eKman 1xtan1


34/159


naM[ .1xtan0 2 dUcen xcos1xcos1xtan1)x(tanf 222 = .lMhatTI5cUrbgHajfa atana2tana3tanatana2tana3tan = Edl

2

ka

RKbcMnYnKtrWTLaTIhVk .dMeNaRsaybgHajfa atana2tana3tanatana2tana3tan = eKman )aa2tan(a3tan +

atana2tanatana2tana3tana3tan

atana2tan)atana2tan1(a3tan

atana2tan1

atana2tana3tan

++ +=

dYcen atana2tana3tanatana2tana3tan = .

- 27 - PaK1


35/159


- 28 - PaK1

lMhatTI6cUrbgHajfa ( )2ab21xcos

b1

xsin

a1 + + cMeBaRKb

2x0,0b,0a

< dMeNaRsaybgHajfa ( )2ab21

xcos

b1

xsin

a1 + + + eKman

xcosxsin

ab

xcos

b

xsin

a1)

xcos

b1()

xsin

a1( + tamvismPaB eKman GMAM

xcosxsin

ab2

xcos

b

xsin

a

eKnxcosxsin

ab

xcosxsin

ab21)

xcos

b1)(

xsin

a1( +

22

2

)ab21()x2sin

ab21()

xcos

b1()

xsin

a1(

)xcosxsin

ab1()

xcos

b1)(

xsin

a1(

++

BIeRBaRKb2

x0< eKman 1x2sin .


36/159


- 29 - PaK1

lMhatTI7eK[ctuekaNEkg mYyEdl BCABCD AB3 .P nig Q CaBIrcMnucnRCug Edl]BC[ QCPQBP = .cUrbgHajfa DPCDBCDQC .dMeNaRsaybgHajfa DPCDBCDQC

tag DQC;DPC;DBCeyIgman 1tan,2

1tan,

3

1tan =

eday +

== tan16

11

3

1

2

1

tantan1tantan)tan(

dUcen DPCDBCDQC .

A

B C

D

P Q


37/159


- 30 - PaK1

lMhatTI8eK[ nig CacMnYnkMpcBIr .1z 2zcUrRsayfa ( )22

2

1

2

21

2

21|z||z|2|zz||zz| +

dMeNaRsayRsayfa ( )2221221221 |z||z|2|zz||zz| + eKman )zz)(zz(|zz|

2121

2

21+

( )1|z|zzzz|z||zz|| zzzzzzzz|zz

2

22121

2

1

2

21

22212111

2

21 ++

ehIy )zz)(zz(|zz|2121

2

21

( )2|z|zzzz|z||zz| zzzzzzzz|zz| 222121

2

1

2

21

22212111

2

21 ++ bUkTMnakTMng ( nig1 ( )2 eKn ( )2

2

2

1

2

21

2

21|z||z|2|zz||zz| + .


38/159


- 31 - PaK1

lMhatTI9eK[ nig CacMnYnkMpcBIrEdl1

z2

z 1|z||z|21

= nig z .1z.21

cUrRsayfa21

21

zz1

zz++ CacMnYnBitmYy .dMeNaRsayRsayfa

21

21

zz1

zz++ CacMnYnBitmYytag21

21

zz1

zzZ ena++=

21

21

z.z1

zzZ + +=

eday 1|z|z.z 2111 = ena 11 z1z = ehIydUcKa 22 z1z = eKn Z

1zz

zz

z

1.

z

11

z

1

z

1

Z12

12

21

21 =++=++=

eday ZZ = ena 21 21 zz1 zzZ ++= CacMnYnBit .


39/159


- 32 - PaK1

lMhatTI10KNnalImItxageRkam 4x

4x1212...121212lim

4x

man n skaer .dMeNaRsayKNnalImIttag

4x

4x1212...121212limL

4xn

= ( man skaer )neKn

4x

4x1212...1212limL

4x1n = ( man skaer )1n

sntfa 4x1212...121212)x(fn nig 4x1212...121212)x(f n +


40/159


eKman )x(f4x1212.....1212)x(f).x(f

16x1212...121212)x(f).x(f

1nnn

nn

eKTaj)x(f

)x(f)x(f

n

1n

n

= eKn

)x(f

1lim.

4x

)x(flim

)x(f)4x(

)x(flimL

n4x

1n

4xn

1n

4xn

)4(f

1.LL

n

1nn eday 8)4(fn = eKn 1nn L

8

1L naM[ CasItFrNImaRtmanersug)L( n

8

1q = nigtY

4x

4x12limL

4x1 = .tamrUbmntYTIn eKn 1n1n qLL eday

8

1

4x12

1lim

4x

4x12limL

4x4x1 =+ = nig8

1q = ena

nn8

1L = .dUcen

n4x 8

1

4x

4x1212...121212lim =

.

- 33 - PaK1


41/159


- 34 - PaK1

lMhatTI11eK[sItncMnYnBit kMnteday )u( n1u0 = nig

1u4u6u4

uu RKb

n

2

n

3

n

4

n

1n + INncUrbgHajfa 4

n1nu

11

u

11 RKb ++ INn rYcKNna u CaGnuKmn_n .n n dMeNaRsay

bgHajfa 4n1n u

11

u

11 ++eyIgman 4

n

n

2

n

3

n

1n u1u4u6u41

u11 ++

4

n

4

n

4

n

4

n

n

2

n

3

n

4

n

u

11

u

)1u(

u

1u4u6u4u

+=

+= dUcen 4

n1nu

11

u

11

+

+.


42/159


KNna CaGnuKmn_nn

u n

eyIgman 4n1n

u11

u11 ++ eKn + + + n1n u11ln4u11ln tag + nn u11lnv naM[ + ++ 1n1n u11lnv eKn .n1n v4v =TMnakTMngenbBaakfa CasItFrNImaRtmanersug)v(

n4q =

nigtY 2lnu

11lnv

0

0=

+ ( eRBa 1u0 = ) .

tamrUbmn 2ln4q.vv nn0n = .eday + nn u11lnv eKTaj 2ln4u

11ln n

n

=

+ naM[ n4n

2u

11 =

dUcen12

1u

n4n

= .

- 35 - PaK1


43/159


lMhatTI12KNna)xx1).....(xx1)(xx1(P

1nn22422

n

++

dMeNaRsayKNna)xx1).....(xx1)(xx1(P

1nn22422

n

++ eKman )aa1)(aa1(a)a1(aa1 2222242 + eKTaj 2

42

2

aa1

aa1aa1 +

+= yk k2xa = eKn 1kk 2k1k1kk22

22

22

xx1

xx1xx1 +

++++=

2

22n

0k22

22

nxx1

xx1

xx1

xx1P

2n1n

1kk

2k1k

++=

++= ++

+= dUcen 222n xx1 xx1P 2n1n ++= + .

- 36 - PaK1


44/159


- 37 - PaK1

lMhatTI13eK[RtIekaN mYymanRCugRbEvg .ABC c,b,aknbnatBunmMC kat[ RtgcMnuc D .]ABcUrRsayfa2

Ccos

ba

ab2CD + .

dMeNaRsayRsayfa2

Ccos

ba

ab2CD +eyIgman S

CDBACDABCSS +

eday Csinab21SABC = 2

CsinCDb

2

1SACD = nig

2

CsinCDa

2

1SCDB = eKn

2

CsinCD.a

2

1

2

CsinCD.b

2

1sibCab

2

1 + eKTaj

2

Ccos

ba

ab2

2

C

sin)ba(

CsinabCD ++

= .dUcen

2

Ccos

ba

ab2CD + .

C

A D B


45/159


46/159


lMhatTI15KNnatm)29tan3)...(3tan3)(2tan3)(1tan3(A

oooo + dMeNaRsayKNnatmA

)29tan3)...(3tan3)(2tan3)(1tan3(Aoooo + eKman

oo

o

oo0

1cos60cos

61sin1tan60tan1tan3 =

oo

o

oo0

oo

o

ooo

oo

o

ooo

29cos60cos

89sin29tan60tan29tan3

3cos60cos63sin3tan60tan3tan3

2cos60cos

62sin2tan60tan2tan3

=

==

eKn29ooooo29o

oooo

)60(cos

1

29cos...3cos2cos1cos)60(cos

89sin....63sin.62sin.61sinA =

eday2

160cos

o = dUcen 5368709122A 29 = .

- 39 - PaK1


47/159


- 40 - PaK1

lMhatTI16eK[ CacMnYnBitEdlepgpatlkx&N z,y,x0zcosycosxcos = nig cos 0z3cosy3cosx3 = cUrbgHajfa cos 0z2cosy2cosx2 .

dMeNaRsaybgHajfa cos 0z2cosy2cosx2 tamrUbmncos acos3acos4a3 3 eKTaj )1(x3cosxcos3xcos4 3 + )3(z3coszcos3zcos4

)2(y3cosycos3ycos4

3

3

++

bUkTMnakTMng eKn )3(,)2(,)1(0zcos4ycos4xcos4

333 = 0zcosycosxcos 333 = (eRBa 0zcosycosxcos = nig 0z3cosy3cosx3cos = )eKman 0zcosycosxcos = eKn zcosycosxcos


48/159


elIkCaKUbeKn

zcosycosxcos3xcosycosxcos

zcosycoszcosycosxcos3xcos

zcosycos)ycosx(cosycosxcos3xcoszcos)ycosx(cos

333

333

333

33

=

eday 0zcosycosxcos 333 = eKTaj 0zcosycosxcos3 = naM[ 0xcos = 0ycos = 0zcos = .edaysntyk 0xcos = ena zcosycos eKn

0)1zcos2(z2cosy2cosx2cos

)1zcos2)(1ycos2)(1xcos2(z2cosy2cosx2cos

22

222

dUcenbBaHaRtUvnRsaybBaak .

- 41 - PaK1


49/159


- 42 - PaK1

lMhatTI17eK[RtIekaN mYy . bnatBumM katRCugABC C,B,A

]AB[,]AC[,]BC[ erogKaRtg .'C,'B,'AcUrRsayfa 0

'CC

)2

BAsin(

'BB

)2

ACsin(

'AA

)2

CBsin( =

+

+

dMeNaRsay

eyIgman SB'CC'ACCABC

SS + eday Csinab21SABC =

2

Csin'CCb

2

1S 'ACC = nig

2

Csin'CCa

2

1S B'CC = eKn

2

Csin'CC.a

2

1

2

Csin'CC.b

2

1sibCab

2

1 +

C

A 'C B


50/159


eKTaj )1(2

Ccos

ba

ab2

2

C

sin)ba(

Csinab'CC +

+=

eKman2

Ccos

2

Csin2

2

BAcos

2

BAsin2

Csin

BsinAsin

c

ba

+==

eday2

Csin

2

C

2cos

2

BAcos = =

eKn2

Ccos

2

BAsin

c

ba= naM[ )

2

BAsin(.

ba

c

2

Ccos

tam eKn)1( )

2

BAsin(

ba

abc2'CC

22

eKTaj

abc2

ba

'CC

)2

BA

sin( 22 = .

dUcKaEdrabc2

ac

'BB

)2

ACsin(

;abc2

cb

'AA

)2

CBsin( 2222 =

=

dUcen 0'CC)

2

BAsin(

'BB

)2

ACsin(

'AA

)2

CBsin( =

+

+

.

- 43 - PaK1


51/159


- 44 - PaK1

lMhatTI18eK[BhuFa P n)acosasinx()x( + Edl *INn cUrrksMNlnviFIEckrvag nwg x)x(P 12 + .dMeNaRsaysMNlnviFIEcktag )x(R CasMNlnviFIEckrvag nwg)x(P 1x2 +-ebI ena1n = acosasinx)x(P + dUcen acosasinx)x(R + CasMnlnviFIEck .-ebI eKn2n )x(R)x(Q)1x()x(P 2 + Edl CaplEck nwg)x(Q BAx)x(R + eKn BAx)x(Q)1x()acosasinx( 2n + ebI enaix = BAi)acosasini( n + A.iB)nasin(.i)nacos( + eKTaj nig)nasin(A = )nacos(B = dUcen )nacos()nasin(x)x(R + .


52/159


- 45 - PaK1

lMhatTI19edaRsayRbBnsmIkar + byaxyx bayx 44 3 Edl nig b CacMnYnBitEdl aa 0b .dMeNaRsayedaRsayRbBnsmIkar

+byaxyx

bayx

44

3 smmUl1

y

yxbyax

)yx(ba

44

3

+

443

44

3

yx)yx(y)yx(a

yxbyax

)yx(ybyay

+

+ eday ena0ba 0yx eKTaj 23443 xy3x

yx

yx)yx(ya ++ = ehIy 323 yyx3a)yx(b +

eKnRbBn naM[ == byyx3 axy3x 32 23 +33 bayx bayx dUcen2

babay;

2

babax .3333 ==


53/159


- 46 - PaK1

lMhatTI20cUrbgHajfa 9)1nn)(1nn( 199991n 44 = += dMeNaRsaybgHajfa 9

)1nn)(1nn(

19999

1n44

=

+= eKman n1n)n1n)(n1n( 4444

n1n

1)n1n)(n1n( 4444 + eKTaj 44

44n1n

)1nn)(1nn(

1 + 9110000

)n1n(

)1nn)(1nn(

1

4

9999

1n

449999

1n

44

=

+ = dUcen 9

)1nn)(1nn(

19999

1n44

=

+= .


54/159


- 47 - PaK1

lMhatTI21eK[GnuKmn_ kMntRKbf }0;1{IRx eday

1x)x

1(f)x(f)1x2(x + .cUrKNna )2009(f....)3(f)2(f)1(fS +

dMeNaRsayKNna )2009(f....)3(f)2(f)1(fS + eKman )1(1x)x

1(f)x(f)1x2(x +

CMnYs edayxx

1 kgsmIkar ( eKn )1)2(

2x

xx)x(f

2x

x)

x

1(f

1x

1)x(f)

x

1(f)1

x

2(

x

1

22

++=++

dksmIkar nig Gg:nwgGg:eKn )1( )2(

2x

2x2)x(f

2x

)1x(x2

2x

xx1x)x(f

2x

x)1x2(x

2

22

++=++++

+ eKTajn

1x

1

x

1

)1x(x

1)x(f ++


55/159


- 48 - PaK1

eKn [ ] = = +20091k20091k 200920082009111k1k1)k(fS dUcen2009

2008)2009(f....)3(f)2(f)1(fS = .

lMhatTI22cUrkMntRKbKU ncMnYnKtviCmanebIeKdwgfa )n;m()nm(13nm 22 + .

dMeNaRsaykMntRKbKU )n;m(eKman )1()nm(13nm 22 + -krNITI1 nm =eKn naM[n26n2 2 = 13n = dUcen .13nm =-krNITI2 nm eyIgBiniteXIjfaebI( CaKUcemyrbs enaeKn)n;m )1(

)m;n( kCaKUcemyrbs Edr .)1(sntfa m . tamvismPaB Cauchy Schwarzn


56/159


eyIgman )nm(2)nm( 222 + 26nm

)nm(26)nm(2

< +eday enaeKnnm < 26nmm2 < 13m


57/159


-cMeBa eKn3m = 0)2n)(15n(30n13n2 = naM[ .15n =-cMeBa eKn10m = 0)2n)(15n(30n13n2 = naM[ .15n =srubmkeKTTYlnKUcemyMKUKW

})10;15(;)3;15(;)13;13(;)15;10();15;3({)n;m( = .

- 50 - PaK1


58/159


- 51 - PaK1

lMhatTI23eK[RtIekaN mYyEkgRtg . eKykABC A ]AC[Enig F Edl]AB[ ABCAEF nig ACBAFE .eKtag nig F erogKaCaeCIgcMeNalEkgn E nig F'E 'elIRCug [ .]BCcUrRsayfa E BC'FFEFE' ?dMeNaRsayRsayfa E BC'FFEFE'

tag BC cAB,bAC,a = .RtIekaNEkg nig manmMABC AFE ABCAEF vaCaRtIekaNdUcKa .

A

B C

F

E

'F 'E


59/159


eKnpleFobdMnUc kBC

EF

AC

AF

AB

AE = Edl 0k> tag[pleFobdMnUc .eKTaj akEF,bkAF,ckAE = .RtIekaNEkg nig manmMABC C'EE ACB rYmvaCaRtIekaNdUcKa . eKn

EC

BC

'EE

AB = naM[BC

EC.AB'EE =

eday ckbAEACEC eKna

)ckb(c'EE

= .dUcKaEdreKTaj

a

)bkc(b'FF

= eKn a )bkc(baka )ckb(c'FFEFE'E +=

a

bc2

a

)cba(kbc2

a

kbbckakcbc

222

222

===

BIeRBakgRtIekaNEkg eKmanABC 222 cba + (BItaKr)tamvismPaB eKmanGMAM bc2cba 222 eKTaj BCa

a

bc2 = .dUcen BC'FFEFE'E .- 52 - PaK1


60/159


- 53 - PaK1

lMhatTI24eK[ c,b,a CabIcMnYnBitviCman . cUrRsaybBaakfa

2)ba(4c

ba

)ac(4b

ac

)cb(4a

cb

3 333 333 33++++++++

dMeNaRsayRsayfa sayfa

2)ba(4c

ba

)ac(4b

ac

)cb(4a 3 +cb 3 333 3333 ++++++++ )cb( 3

+eKman b(bc3cb 33 )c PaB AM an+tamvism M eKmG bc2cb eKTaj 2

2cbbc + naM[ 3)cb(43)cb(bc3 + eKn 33333 )cb(

4

1)cb(b

4

3)cb(c + ajKT 3 33 )cb(4cb + 3 33 )cb(4acba + + naM[ ( )1

cba

cb

)cb(4a 3

cb33 ++++ +


61/159


- 54 - PaK1

dUcKaEdr ( )2cba

ac

)ac(4b

ac

3 33 ++++ nig ( )3c)ba(c

ba

3 33 ba

ba

4 +++edaybUkTMnakTMng eKn ++ .

)3(,)2(,)1(

2)ba(4c

ba

)ac(4b

ac

)cb(4a 3

cb

3 333 3333++++++ .+

lMhatTI25eK[sItcMnYn t = = 1n,2a2a aa 1nn1n 10

2 ++

a

1

cUrRsayfa a1nn1n

aa .dMeNaRsaya aa 2 +sayf 1nn a +1n

a2a + aa RK

2aa2(cnn1nn1n

eKman a 1tag c bn

nn1n

1nnneKn c 1n1n1n + aa a) 2c +


62/159


- 55 - PaK1

TMnakTMngenbBaakfa CasItnBVnmanplsgrYm)c(n

2d = 01gtY 0c1 11aa = .eKn 2n2)1n(2d)1n(cc

1n

1nnKTaj a 2n2a

)

( ) (

)1n(n1a

)12aa

1k2aa

n

0n

n

1k

n

1k1kk

=

=2n(n

eKTaj 1nna 2n

+ nig 1)1n()1n(a 21n

+ 2 + niga

1n+ 1n21nn 1nna24 ++

nn(a day )(1a 222221nn n)1n()1nn = 1nnaa 24

1nn+ .dUcen

1nn1naaa 2 ++ .


63/159


- 56 - PaK1

lMhatTI26kgRtIekaN ABC mYycUrRsaybBaakfa

r

R4

111 2

Csin

22Edl r nigB

sinA

+sin

R CakaMrgVgcarwkkg nig carwkeRkARtIekaN .dMeNa

Rsayfasay ( )1

r

2

Csin

2

Bsin

2

Asin

R4

111 + tag ,bAC,aBCag A,bAC,aBC

cB cBA =

sbTkUsInUskgRtIekaN eKman 222 edaytamRT ABCAcos.bc2cba +

2

Asin21Acos

2 ena )

2sin21(bc2 Acba 2222

eKTajbc4

)cba)(cba()cb(22

bc4

a

2

Asin

2 += tag

2

cbap = + ( knbrimaRtnRtIekaN )Kn )cp(2cba nig )bp(2cba


64/159


- 57 - PaK1

eKnbc

)cp)(bp(

2

Asin2

= naM[

bc

)cp)(bp(

2

Asin = . dUcKaEdreKTaj

ab

)bp)(ap(

2

Csin;

ac

)cp)(ap(

2

Bsin = = eKn

abc

)cp)(bp)(ap(

2

Cs sin

2

Bsin

2

Ain

= eKman R4abcpr =)cp)(bp)(ap(pS = eKTaj S.R4ab nigc = S.r

p

S)cp)(bp)(ap( 2 =

eKnR4

r

S.R4

S.r

2

Csin

2

Bsin

2

Asin = .vismPaB smmUleTAnwg )1

( )222

Csin

2

Bsin

2

Asin

2

Bsin

2

Asin

2

Csin

2

Asin

2

Csin

2

Bsin

2

Csin

2

Bsin

2

Asin4

14

2

Csin

1

2

Bsin

1

2

Asin

1

eday2

Asin

a

ap

bca

)cp)(bp()si

2

=ap(2

Csin

2

Bn

2=


65/159


- 58 - PaK1

eKTaj ap = a2Asin

2

Csin

2

Bsin

. dUcKaEdreKTajn b

bp2

Asin

2

Csin

2

Bsin

= nigc

cp

2

Csin

2

Bs

2

Asin in =

smmUleTAsmPaB ( )2 nwg 2

c

cp

b

bp

a

ap ++ aB eKn amvismP GMAM a)ap(2a)ap(p naM[

p

)ap(2

a

ap dUcKaEdr

p)bp(2

bbp nig

p)cp(2

ccp eKn

Bit2c

cp

b

bp

a

ap

)cp()bp()ap(2

c

cp

b

bpap

++

++ padUcen

r4

Csin

Bsin

R2

1

2

1

2

Asin

.1


66/159


- 59 - PaK1

lMhatTI27eK[ CabIcMnYnBitviCmanEdl abc4c,b,a 1cba + rbgHajfa bgHajfa

)cabcab(2c

ba

b

accb 222 ++a

222 ++ dMeNaRsayRsayfa

)ab(2 + cabcc

ba

b

a

a

22 ++aB eKman ccb

2222 +tamvismP GMAM

4

abcbaabc4 + naM[41c eKTaj 1abc )1(abc31abc4cba ( eRBa )1abc tamvismPaB GMAM eKn ( )2

c

ab2ca2cccb 222 +ba

b2

c

ba

b

a

a

222 ++ mPaB Cauchy - Schwarz eKman amvis [ ]2222 )ca()bc()ab(3)cabcab( + +eday ])ab()ca()bc([

abcca= 2ab2

b

ca2 222 + bc2


67/159


- 60 - PaK1

eKTaj ( )3)cabcab(abc3

2

c

ab2

b

ca2

a

bc2 2+ tam nig eKTajn)2( )3(

( )4)cabcab(abc3

2

c

ba

b

ac

a

cb22 + 22222 ++

22 +eyIgman

bca2)ab()ca(

abc2)ca()bc(

cab2)bc()ab(

222

222

222

eKn )ab([2 2 )cba(abc2])ca()bc( + )cba(abc)ca()bc()ab(

222 + aMgBIrnwg )()(ab(2fmGg:T )caab(2)bc )ca)(bc(2 a(abc3)cabc 2eKn bab( )c

( tamKman a abc3cb ( )1 )

naM[eKTaj 2222 cba9)cabcab( abc3cabcab

( )5 )cabcab(2)cabcab(abc3

2 2 +eKTajn am )4( nig )5()cabcab(2

c

baa

a

cb22222 +

b

c2 + .


68/159


- 61 - PaK1

lMhatTI28eK[sIt a{ }n kMnteday= nig1a

04a....a.aa

n101n+ cMeBaRKb .1n cUrbgHajfa 2aa

1nn=+ cMeBaRKb .1n

dMeNaRsaybgHajfa 2aa1nn=+ g;teXIjfaRKbyIgse INk eKman .eKman n 0ak > .eKman 0ak >4a....a.aa

n101n4a....a.aa

n101n+

1n102nKn a 4a.....a.a ++ 2

n102n

n0n102n

)2a.....aa(

4)a....a.4)a.....aa(

4)4a......a)(a.....a.a(

++ 1a(a aa

2

n102n

n10

a

++++

eKTaj 2a... n..aaa 102n + 2)4a(a

1n2n++ 2aa 2n1n =+ dUcen 2aan 1n = .


69/159


- 62 - PaK1

lMhatTI29eK[ CabIcMnYnBitviCmanEdlc,b,a 1cabcab = .cUrbgHajfa 16)

a

1c()

c

1b()

b

1a(

222

dMeNa RsayagHajf 16)a1c()c1b()b1a( 222 tag 222 )

a

1c()

c

1b()

b

1a(A = +

+a + accbba2c1b1a1cb 222222 eyIgBinit22

a

cabcab

a

1 +=2

a

bc

a

c

a

b + 22

ca

b

1

b

bcab +=2

b

ac

b

c

b

a + + 22

c

cabcab

c

1 +=2

c

ab

c

a

c

b + PaB eKman amvism GMAM 3c

ab

b

ca

a

bc

,ab 2ac

c

a

,2b

c

c

b

;2

ba222 eKn 93222

c

1

b

1

a

12+

22=


70/159


- 63 - PaK1

ehIy 3amageT c2c

.c

b.

b

a3

a

c

c

b

b

a3 =

ot b,ab2ba 22222 ca2ac;bc2 + 222naM[ a2 ca2bc2ab2c2b2 + aM[ 1cabcabcba 222 = eKn 1A 16)3(29 = .+

dUcen 16)acb 1c()1b()1a(222 .

lMhatTI30eK[432

2cos2222,

2cos222,

2c22 os

== cUrrkrUbmnTUeTA nig RsaybBaakapg .

=BITahrN_xagelrUbmnendMeNaRsayeKmanrkrUbmnTUeTA ,

2cos222,

2cos22

32==

42

cos2222=


71/159


- 64 - PaK1

tamlMnaMTahrN UeTAdUcxa eyIgGacTajrkrUbmnT geRkam1n

)n(2

co222 s2.........2 += + .eyIgtagRsaybBaakrUbmnen

)n(

n2......222A +

RK n *IN

eyIgman 21 2cos22A = BitfavaBitdltYTI yIgbma Kp1p

)p(

p2222A

2cos2...... +

=

BitayfavaBitdltYTI

+

eyIgnwgRs 1p + KW2p1p

2

cos2A += BiteyIgman p1p A2A + aytamkarbmad

1pp 2cos2A +

= eyIgn

2p2p

2

1p1pA

2cos2

2cos4

2cos22 +

==+ BitdUcen 1ncos22.........222 +=

)n(2 .


72/159


- 65 - PaK1

lMhatTI31edayeFIvicartamkMeN acMnYn ncUrRsaybBaakf

2 + ckdacnwg CtFm1n1

n9A ++ E aniccMeBaRKbn6 11cMnYnK Catin .

dMeNaRsayRsaybBaakfacMnYn 1n6A 1nn 92 ++ Eckdacng11-ebI 0n = eKn 1192A

0= Bit ( Eckdacnwg11 )-ebI eKn 091n = 1911292A 27

1 Bitg ).

favaBitdltYTI KW( Eckdacn 11eyIgbma p 1p1p6p 92A ++ eyIgnwgRsayfavaBitdl YTI KWEc

p9

Eckdacnwg t11 1P +2p7p6

1p92A ++ + kdacnwg11

2peyIgman1

2A 7p6 ++ + 11p1p66

1p

+ )9.29(A p62p)92(2 ++ 9.55A64

9A64 1p

p

A

)649(A

+p1p+

1p

p1

++


73/159


- 66 - PaK1

edaytamkarbma Eckdac wg ehIy.np

9.55A64A

pA n 11 1p9.55 +

Eckdacnwg11eKTaj 1pp1

++ Eckdacnwg t .n

A

11 BdUcencMnYn n1n6 92 1+ kdacnwg11 eBaEc CaniccbcMnYnKtFmCatin .RKlMhatTI32

97831 nnn

K[ INn,617043E nn 0 ac cR

cUrbgHajfan

E Eckd nwg 187 Can Kb INn .dMeNaRsaya Eckdacnwg Canic bgHajf RK

nE 187 INn

Edl nig CacMnYnbfmrvagyfa Eckdacnwg eyIgEtUvRsay[eXIjfa Eckdac nig .

ab1nn

eday 187 = Ka .1711 11 17dUcenedIm,IRsa nE 187

nE 11 17eKman ba)(ba(a 2n1nn N.)ba()b....+


74/159


- 67 - PaK1

Edl 1n aaN 1n2n b.......b + n 610709743831E nnnnn K , INn ( ) ( )

*INq,p;)q9p8(11q99p88

q)610709(p)743831(

610709743831 nnnn

709743831E nnnnn

eKTajn Eckdacnwg 11 .nEmageToteKman

INn,610 *IN;;)213(1734221

)743709()610831(

)743709()610831(nnnn

Tajn Eckdacnwg .nwg CanicRKbeK nE 17dUcenn

EckdacE 187 INn .


75/159


- 68 - PaK1

lMhatTI33cUrRsaybBaakfa

1n21n

1.....

3

11

21 ++

adMeN RsayybBaakfa sa

1n21n

1.....

3

1

2

11 < ++ man+eyIg

1k2

1

1k1kk1kk1k

11 +++ k1k1k2 1 + ebI 12

22

1:1k=

ebI 2332

1:2k=

ebI 34421:3k ------------------------ebI--------- n1n1n2

1:nk +


76/159


77/159


- 70 - PaK1

-bmafavaBitdltYTI KW kk k321k321 aaa + a......a.a.aka.......... BiteyIgnwgRsayfavaBitdltYTI 1k+ KW

1k1kk3211kk32

a.a......a.a.a)1k(aa..........aaa + ++ 1tag*INk,0a,0x;x

)a......aaax(

)x(f k

1k

k321 =+ eyIgn

2

k

k

x

ax(x)a()x('f

1k

k2121 )a.....a....aax)(1k++ =

[ ]( )2 k21

k

k21

2

k21

k

k21

xa....aakx)a....aax(

x

)a....aax(x)1k()a....aax(

=+=

ebI naM[0)x('f =k

a.....aax k21

+= cMeBa

k

a..... kaax 21+ eKn +=

k

k211kk21

k2

k

a +11k

k

k21

k21

k

a.....aa)1k()

k

a......aa(f

a....a

)....k

a....aa(

)k

aa

(f

+++

+21 aaaa....+

=+

+ +


78/159


- 71 - PaK1

taragGefrPaBnx

)a.....aax()x(f

k

k21+=

x ok

..aa 21 a. k )x('f . +.

)x(f m

PaBxagelIcMeBaamtaragGefr 0x eyIgTajn k

k3211k

1k

k21

ka.....aaa)1k(

x

)a......aa

x(

)x(f

+++

+

+

ma=

tamkarb kk321k321

a......a.a.aka..........aaa smmUl k21ka.....+ k21 a.........a.a

k

aa +

eKTaj k211k1kk21 a......aa)1k(x )a.......aax( ++ + aM[ (*)xa.......a.a)1k(a.....aax 1kk21k21

++ yk CYskg0ax1k>+ ( )*


79/159


- 72 - PaK1

eKn1k

1kk3211kk321 aaa + a.a......a.a.a)1k(aa.......... + ++ BitdUcen n n321n321 .....aaa + a......a.a.ana..... .lMhatTI35edaRsayRbBnsmIkar ytanxtan3

==34ytanxtan

28ytanytanxtan6xtan

3

4224 Edl

2yx0

< .dMeNaRsayedaRsayRbBnsmIkar

+ tanxtantanxt 33

==34yyan

28ytanytanxtan6xtan4224

smmUl = = )tanxtan4y 2(316ytanxtan4)1(28ytanytanxtan6xtan

33

4224


80/159


- 73 - PaK1

bUksmIkar nig eKn )1( )2(424 4(3162)tax(tan = )31()328yn + eday

2yx0

< enaeKn ytanxtan0 < eKTaj )3(31ytanxtan + eFIplsg 1( nig 2() eKn )424 )31()324(1628x(tan 3)ytan eKTaj )4(1tanxtan eRB3y a ytanxtan0 < ar nig nksmIk )4( eK tan2)3( 2x = KTaj 1xtan = naM[

4x

= .plsgsmIkar nig eKnF )3( )4( 32ytan2 = TajK 3ytan = naM[

3y

= .ceneKnKUcemy )

3y,

4x(

== .


81/159


- 74 - PaK1

lMhatTI36eK[ CaRCugrbsRtIekaNmYynigtagnRtIekaN .a

c,b,a pCaknbrimaRtcUrbgHajf + c1b1a12cp 1bp 1ap 1dMeNa

RsayajfagH + c1b1a12cp 1bp 1ap 1 nYnBit > eKman 2

cMeBaRKbc > 0B;0A0A)BA( 22 BB.A2 +

2 n

= A 2 B.A4BB.A2 B.A4)BA( 2 eKTaj

BA

4

B.A

BA + )1(BA 4B11A + yk B,apA bp nK cbap2BA = CYskg )1(eKn )2(

c

4

bp

11 dUcKaEdreKTajap .


82/159


- 75 - PaK1

)3(a

4

cp

1

bp

1 nig )4(b4ap 1cp 1 aB Gg:nwgGg:eKn Gg:nwgGg:eKn kvismP )4(,)3(,)2( )4(,)3(,)2(c

4

b

4

a

4

cp

2

bp

2

ap

2 + dUcen +p 1 c1b1a12cp 1bp 1a .lMhatTI37kcUrRsaybBaakfa

3n4

1n4

1n4

1n4

1n4

3n4 +


83/159


)3n4)(1n4(

1

1n4

1

)3n4)(1n4(1n4

)3n4)(1n4(22n8

)3n4)(1n4(2)3n4()1n4(

+++++

KuNGg:TaMgBIrnwg )1n4( eyIgn

)1(3n4

1n4

1n4

1n4 ++ tamTMnakTMng nig eKTajn )1( )2(3n4

1n4

1n4

1n4

1n4

3n4 +


84/159


tamsRmayxagelIeyIgman3n4

1n4

1n4

1n4

1n4

3n4 +


85/159


lMhatTI38eK[ a CabIcMnYnBitviCman .c;b;cUrbgHajfa

2

3

ba

c

ac

b

cb

a +++ dMeNaRsaybgHajfa 23ba cac bcb a +++ tag

===

pba

nac

mcb

eKn ( ) ( ) ( ) pnmbaaccb + naM[ 2 pnmcba += eKTaj

=

+==

2

pnm

c

2

pnmb

2

mpna

eKn

- 78 - PaK1


86/159


- 79 - PaK1

p2

pnm

n2

pnm

m2

mpn

ba

c

ac

b

cb

a ++=+++ ++ +++ n

m

m

n

ac

b

cb

a ++ 3nppnpmmp21ba c amvismPaB eKn GMAM 2n

p

p

n;2

p

m

m

p;2

n

m

m

n + eKn ( ) 2332221ba cac bca =2b ++++ dUcen

2

3

ba

c

ac

b

cb

a ++ .+lMhatTI39kKNna n)1n(3.2

1

2.1

1

Edl 2n > 1

....

4.3

1 + xedayeRbIvismPaB GMAM n n( )1 cMnYnxageRkam n)1n(

1;

4.33.22.1

1;

1;

1 cUrbgHajfa 2n )!n(n < .dMeNaRsaykKNna = n 2k k)1k( 1n)1n( 1....4.313.212.11


87/159


- 80 - PaK1

eKmank

1

1k

1

k)1k(

)1k(k

k)1k(

1 = eKnn

11

k

1

1k

1

k)1k(

1 n

2k

n

k

=2 = dUcenn

11

n)1n(

1...

2.1

1 + ..3

1

3.2

1 4 .n(n <

tamvismPaB x bgHajfa 2)!n

GMAM n

n321n321a......a.a.ana.....aaa + eKn 0a; k

)1n(

n)1n(

1......

3.2

1.

2.1

1)1n(

n)1n(

1....

3.2

1

2.1

1

2n

2

1n

)1n(2

)1n(

)1n(

)!n(n

1 > n)!n( n)!n(

n

n

1

!)1n(!n

1)1n(

n

1n

!)1n(!n1)1n(

n11

dUcen 2n )!n(n < .


88/159


- 81 - PaK1

lMhatTI40eK[ ;b;a c CabIcMnYnBitxusBIsUn .rbgHajfa

a

c

c

b

b

a

a

c

c

b

b

a222 +++

dMeNaRsay

bgHajfa accbbaacba ++ cb 222 ++ tamvismPaB eKman GMAM 3

b ac.cb.ba3accba 3 222222222 =+

eKTaj )1(3accbba 222 ++ mageToteKman

ac21

ac

c

b21

c

b

b

a21

b

a

2

2

2

eKTajn )2(a

c

c

b

b

a23

a

c

c

b +++ ba222 +


89/159


- 82 - PaK1

bUkvismPaB nig Gg: nig Gg: eKn )1( )2( + +++ accbba23accbba23

222 b +

++ accbba2accba2222

cena

c

c

b

b

a

a

c

c

b

b

a222 +++ .

lMhatTI41eK[ ;y;x z CabIcMnYnBitviCmanminsUn .rbgHajfa222222 zyx

9

z

1

y

1

x

1 + dMeNaRsaybgHajfa

222222yxzyx + z9111 + vismPaBenGacsresreTACa

9z

zyx

y

zyx

x

zyx2

222

2

222

2

222 ++ 6

z

y

y

z

z

x

x

z

y

x

x

y

9z

y

z

x1

y

z

y

x1

x

z

x

y1

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2


90/159


eday 2y

x.

x

y2

y

x

x

y2

2

2

2

2

2

2

2 = 2

z

x.

x

z2

z

x

x

z2

2

2

2

2

2

2

2 = nig 2y

z.

z

y2

y

z

z

y2

2

2

2

2

2

2

2 = eKn 6222

z

y

y

z

z

x

x

z

y

x

x

y2

2

2

2

2

2

2

2

2

2

2

2 = BitdUcen

222222zyx

9

z

1

y

1

x

1 + .lMhatTI42eKman xlog1 1 aay = nig ylog1 1 aaz = cUrRsayfa zlog1

1

aax= .dMeNaRsay

Rsayfa zlog1 1 aax = eKman xlog1 1 aay = naM[

xlog1

1ylog

a

a

eKTaj )1(ylog

11xlog

a

a

- 83 - PaK1


91/159


dUcKaEdr ylog1 1 aaz = naM[ylog1

1zlog

a

a eKTaj )2(zlog

11ylog

a

a

ykTMnakTMng eTACYskg eKn )2( )1(zlog

11

11xlog

a

a 1)z(log

zlog1xlog

a

a

a

zlog1

1xlog

1)z(log

zlog1zlogxlog

a

a

a

aa

a

= eKTajn zlog1 1 aax = .

- 84 - PaK1


92/159


- 85 - PaK1

lMhatTI43eKBinitsItncMnYnBit ( kMnteday)u

n

==+

1u2

4u5u

4u

n

n

1n

1 cUrKNna u CaGnuKmn_n .

nn

dMeNaRsayKNna u CaGnuKmn_n n nsmIkarsMKalnsIt1u2

4u5u

n

n

1n = KW 1r2 4r5r = mans04r6r2 2 = 2a

cr;1r

21=

tagsIt2u

1uv

n

n

n = eKn n

n

n

n

n

n

n

1n

1n

1nv3)

2u

1u(3

21u2

4u5

11u2

4u5

2u

1uv ==

== +++ TMnakTMngenbBaakfa CasItFrNImaRtmanersug)v( nnigtY3q =

2

3

2u

1uv

1

1

1== .


93/159


tamrUbmn2

33

2

3qvv

n

1n1n

1n=

eday2u

1uv

n

n

n = eKTaj1

2

3

13

1v

1v2u

n

n

n

n

n ==

dUcen23

)13(2v

n

n

n = .lMhatTI44eKBinitsItncMnYnBit kMnteday)u(

n

==+

1u

4u3u

3u

n

n

1n

1 cUrKNna CaGnuKmn_n .nu ndMeNaRsayKNna CaGnuKmn_n

nu nsmIkarsMKalnsIt

1u

4u3u

n

n

1n = KW 1r 4r3r = 0)2r(4r4r 22 = eKTajs 2rr 21 = .tagsIt2u

1v

n

n - 86 - PaK1


94/159


- 87 - PaK1

eKn2u

1v

1n

1n ++ eday 1u 4u3u nn1n = eKn2u

1u

2u24u3

1uv

n

n

nn

n

1n =+= eKman 12u

1

2u

1uvv

nn

n

n1n== efrnaM[ CasItnBVnmanplsgrYm)v(

n1d =

nigtY 123 12u 1v 11 = .tamrUbmn n1n1d)1n(vv 1n = eday2u

1v

n

n eKTaj n 1n2n12v12u nn += dUcenn

1n2un

+= .


95/159


- 88 - PaK1

lMhatTI45eKBinitsItncMnYnBit kMnteday )u( n = 4u1 *INn,uu

n1n

cUrKNna CaGnuKmn_ .eNa

nu n n

d Rsay aGnuKmn_nNna nu C neKmann1

uu = n+eKn n1n ulnln

2

1u =

genbBakfa (l n CasItFrNImaRtmanersugnakTMn un ) 21q = tamrUbmn ( ) 1n2 11n

1

n 4ln4ln1

ulnuln = n12

q

dUcen ( ) 1n2 1n 4u = .


96/159


97/159


- 90 - PaK1

tamrUbmn0

qv nn

v eday 6n2vbanvu nnn + eKn 0eBa n = 6v00 u +

u0

aM[ 56v0

16 Knnnn

2

1

2

11v .

dUcen 6n22

1u

nn+ .

lMhatTI47edayeRbIGnumanrYmKNitviTacUrKNnalImItxageRkam 2x

2x22.....222limL

n2x = ( man n raDIkal ) .

eNad Rsay ageRkam NnalImItxeyIgman 2x 2x22.....222limLn 2x =eyIgn


98/159


- 91 - PaK1

4

1

2x2

1lim

)2x2)(2x(

4)x2(lim

2x

2x2limL

2x1= 2x2x =++=

212x2x2

2x2x2

4

1L

4

1

2x

2x2lim

2x22

1limL

)2x22)(2x(

2x2lim

2x

2x22limL

= +=+= =

eyIgsntfavaBitdllMdabTI KWk

kk4

1L =

TyIgnwgRsayfavaBitdllMdab I )1k( + KW 1k1k4

1L += eyIgman

2x

2x2.....222 2limL

2x1k knigPaKEbgnwg

+=KuNPaKy2x2..2222)x(M + . + eKn)x(M).2x(

4x2....2222limL

2x1k =

Bit1kkkk1k

2x2x1k

4

1

4

1

4

1L

4

1L

)2(M

1L

2x

2x2....222lim

)x(M

1limL

+

=

dUcenn2x

n4

1

2x

2x22.....222limL = =


99/159


100/159


- 93 - PaK1

dUcen2nn

2cos2U +

= .aplKuN 0n U....UUP KNn 1U n2 naM[amrUbmn 2a2sin = acosasin

an

a2n

si

siacos2 =

2n2n

n 1k2

sin +0k

n

0k

2k

n

0k2kkn

2sin

1

2sin

2sin

)

2sin

()2

cos2()U(P

+

= = +=+

=

==

dUcen2n

n

2sin

1P

+ .

hatTI49l nYnBit kMntelI eday K[sItnc )U(

nIN

2

2U0 = nig INn,211U 2n1n + U= KNna nuKmn_n .

nCaGU n


101/159


- 94 - PaK1

dMeNaRsayKNna CaGnuKmn_n nU neyIgman

4sin

20

=2 U

8sin

2

4sin11

2

U11U

1=

22

0 ==

favaBitdltYTI KWbma p 2pp 2sinU += yIgnwgRsayfavaBitdltYT )1p( + KW3p1p

U +2

sin + BiteyIgman

2

U11U

1p

= 2p karbma2pp

2sinU

= eyIgn

2

2U

2p2 + sin11

1p+=

3p

3= p2

2p

2sin

2

2sin2

2

2cos1

++ =

=

Bitcen 2nn 2sinU += .


102/159


- 95 - PaK1

lMhatTI50eK[sItncMnYnBit )u( n kMnteday = au0 > 2a,IN,2uu 2n1n n rRsaybBaakfa nn 2222n

2

4aa

2

4aau

+

=

dMeNaRsayRsaybBaakfa nn 2222n

2

4aa

2

4aau

+

=

bI 0n = eKn a2

4aa

2

4aau

22

0=+= Bit

avaBit KWbm dltYTIk kk 2222k

2

4 aa

2

4aau

+

=

tdltYTyIgnwgRsayfavaB I 1k+ KW1k1k

22

22

1k

22

u += 4aa4aa

+

eyIgman 2uu 2k1k


103/159


- 96 - PaK1

Ettamkarbma kk 2222k2

4aa

2

4aau

+

=

eyIgn 22

4aa

2

4aau

22

22

2

1k

kk

+

= 2

4

4aa2

2

4aa

2

4aau

222

22

2

1k

1k1k

+

=+

+ 1k1k 2222

1k2

4aa

2

4aau

+

+

= BitdUcen nn 2222n

2

4aa

2

4aau

+

= .


104/159


- 97 - PaK1

lMhatTI51kcUrkMntelxnGBaat ncMnYnd,c,b,a abcdebIeKdwgfa

dcba9abcd = xcMeBatm ,b,a EdlnrkeXIjxagelIcUrbBaakfad,ccMnYn abcd nig dcba suTEtCakaerkd .dMeNaRsaykkMntelxnGBaat d,c,b,aeKman )1(dcba9abcd =

kTMng eKTajnt EtmYyKtKW .amTMna m)1( a 1a =cMeBa 1a = eKn )2(1dcb9bcd1 = tamTMna g )2( eKTMn Tajn 9d = eRBa = es d 819manelxxagcug 1 .cMeBa 9d = eKn )3(1cb999bc1 = kTMn TajnamTMna g )3( eK 0b = nG T )eRBa 9b m acmanRtaTuke


105/159


- 98 - PaK1

cMeBa b eKn0 )3(01c999c10 = kTMn TajnamTMna g )3( eK 8c = eRBa 72989c = Efm 8 [elxxagcuges0 .cMeBa 8c = eKn 91089 9801 .acen 0b,1 9d,8c, = .facMnYnbBaak abcd nig dcba suT aerkd

0EtCakcMeBa 98c,b,1a d, = eKn

2331089abcd = = nig 2991980dcba = suTEtCakaerkd

.


106/159


- 99 - PaK1

lMhatTI52sntfa HUNSEN CacMnYnmYymanelxRMmYyxgEdleBlEdleKKuNcMnYn lTplCacMnYnmanelxN,E,S,N,U,H Caelx .ennwg 3 eKTTRMmYyxgKWSTRONG Edl G,N,O,R,T,S CaelxcUrrkcMnYn HU gNSEN n STRONG = 8NeBa N = nigcMeBa S,3S,6 9 .sKatageday ).Gkrx elxxusKa

STRONG

3

HUNSEN dMeNaRsayrkcMnYn HUNSEN nig STRONG


107/159


- 100 - PaK1

eKman STRONG

3HU NSEN

kcMeBa 3S,6N = G6TRO3

3

6E63HU tamtaragRbmaNviFIKuNenbBaakfa 1H = cMeBa 1H = taragviFIKuNeTACa G6TRO3

3

6E36U1 tamtaragviFIKuNeneKTajn 8G = eRBaeTACa

36manelxagcuges8 .cMeBa 8G = taragviFIKuN68TRO3

36E63U1

tamtaragviFIKuNeneKTajn 5E = eRBa Efm3E 1


108/159


- 101 - PaK1

manelxcugeRkayesnwg 6 .cMeBa 5E = taragviFIKuN TACa

68TRO3

3

63U1 56 tamtaragviFIKuNeneKTajn 0O = eRBa Efm

33 1

manelxcugeRkayesnwg 0 .cMeBa 0O = taragviFIKuNeTACa068TR3

3

6U1 356 tamtaragviFIKuNeneKTajn 9R= eRBa Efm36 1manelxcugeRkayesnwg 9 .cMeBa 9R= taragviFIKuNeTACa

9068T3

36U1 356


109/159


- 102 - PaK1

tamtaragenbBaakfa minGacFMCag eTehIyedayGkrxusKatagedayelxxusKaenaeKTajn

U 2

2U = EtmYyKtBIeRBa ehIy minGacyktm 1H,0O = U 0 1dEdleTotneT .cMeBa 2U = taragviFIKuNeTACa 90T3 68

56

tamtaragviFIKuNeneKTajn3

1263

7T = BIeRBafa

3790683

126356

edayeRbobeFobtaragviFIKuN

STRONG

nwg3HUNSEN379068

3

126356


110/159


- 103 - PaK1

eKTajn 126356 nigHUNSEN = 379068STRONG = CacMnYnEdlRtUvrk .cMeBa 9S,8N = tamrebobdUcxagelIeKn yIgedaRsay

956784

3318928

edayeRbobeFobtaragviFIKuN

STRONG

3

HUNSEN nwg956784

3318928

eKTajn 318928HUNSEN = nig 956784STRONG = CacMnYnEdlRtUvr k .


111/159


- 104 - PaK1

lMhatTI53cUrbMeBjRbmaNviFIbUkxageRkam

YENOM

EROM dMeNa

DNES

+

(GkrxusKatagedayelxxusKa )RsaybMeBjRbmaN UkxageRkam FIb YENOM EROM

+ DES

N

cemyrbstaragviFIbUkenKW 25601

5801

7659+

.( sisedaRsayedayxng )


112/159


- 105 - PaK1

lMhatTI54eK[ACacMnYnmYymanelxRMxgEdlelxxgvaerobtamlMdab

x;1x;x +x;2x;1 + ehIy CacMnYnmYymanelxBRMmYyxgEdlelxxgvaerobtamlMdabx;1x;1x;2x;1x;x .

adMeNa+

cUrbg jfaebIA CakaerRkdena B kCakaerkdEdr .RsayeyIgmanelxTaMgRMxgn A erobtamlMdabkarbgHaj

x;1x2x;1x;x + naM[ 92x1x0 ; x + 7x0 .aebI 1x = en 211112321A = kd .-ebI ena A = mnCakaerkd .e minEmnCakaerkd .

= minEmnCakaerkd .

-ebI 2x = ena 23432A = minEmnCakaer3x = 34543 minE-eb na4x = 45654A =-ebIx ena5 56765A =


113/159


- 106 - PaK1

-ebI ena minEmnCakaerkd .6x = 67876A =-ebI ena minEmnCakaerkd .7x = 78987A =srubmkeKn x Edl[ A Cakaerkd1 Catm .vaerobtaml

;1

eday elxxg dab Bx;1;2xx;x x;1x

eBa eKn+

enac 1x = 2351123201B = Cakaerkd .CakaerkdEdr .cen ebIA CakaerRkdena B klMhatTI55eK[cMnYn )elxn2( 11.....11A 11 nig )elxn( 444.....444B = cUrbgHajfa 1B + CakaerRkd .AdMeNaRsaybgHajfa A 1B + CakaerRkd eyIgman

9

110111A

n2 =

.....111n2(

=

)elx


114/159


115/159


- 108 - PaK1

eyIgn bb10a100a1000aabbN + + )ba100(11N b11a1100N + [ ] )1()ba(a9911+ day 9a0 nig b0 9 ena 18ba0 ng edIm,I[ GacCakaerRkdluRtaEthuKuNn ehIytamTMnakT )1( Nba + CaB 11 18ba0 enaeKRtUv[

Etm .Ga11ba = yKtTMnakTMng )1( csresr ( ) 1a9(1111a9911 2 )N + ehtue k a N Cakaer dkalN 1a9 + Cakeday aerRkd .9a naM[ 101 811a9 eKTaj )

= (K g )161a9 = (Kanskg IN 1a9 = (Kanskg IN )25 1a9 + ansk36 IN(Kanskg )491a9 = IN

naM[ 7a641a9 = = e y 4711b = cencMnYnEd KW 4(Kanskg I ) .811a9 = NdU lRtUvkMntena 774 288 .


116/159


- 109 - PaK1

lMhatTI57eK 0889333,1089 2man 33 22 = 111088893333,11 .agelIcUrrkrUbmnTUeTA nigRsaybBaakrUbmnenpg111108888933333

2 =BITahrN_xdMeNaRsayeKman 111088893333,110889333,108933 222 = .gGacbeg;trUbmnTUeTAdUcxageRkam

...)1)1n(

)

2

=

1111088889333332 =tamlMnaMeney

9888.....8880111.....111333..333n(

n(

.

eyIgtagkarRsaybBaakrUbmn 9888.........8880111..........111A)1n()1n(

=


117/159


)n(

2

2n2nnn2

n1nn2

1n1n1n

)1n(

1n

)1n(

333............333

3

110

9

)110(

9

110.210

9

818010.81010

910)110(9

810)110(

9

1

910.888.........88810111.........111

=

===+=

++

+

+

dUceneKnrUbmn 9888.....8880111.....111333.....333

)1n()1n(

)n(

2

= .

- 110 - PaK1


118/159


lMhatTI58eKman 999800019999,998001999,980199 222 = .9999800001999992 =BITahrN_xagelIcUrrkrUbmnTUeTAnigRsaybBaakrUbmnenpgdMeNaRsaytambMrabeKman

999800019999

998001999

980199

2

2

2

==

= 9999800001999992 = tamlMnaMeneyIgGacbeg;trUbmnTUeTAdUcxageRkam

1000.....0008999.....999999.....999)1n()1n()n(

2

= .

karRsaybBaakrUbmn eyIgtag 1000.........0008999..........999A)1n()1n(

=

- 111 - PaK1


119/159


)n(

22n

nn2

n1nn2

n1n1n

n1n

)1n(

999............999)110(

110.210

110.81010

110.810)110(

110.810999.........999

=+

+ ++

++

+

dUceneKnrUbmn1000.....0008999.....999999.....999

)1n()1n()n(

2

= .

lMhatTI59eKman 444355566666,443556666,435666 222 = .4444355556666662 =BITahrN_xagelIcUrrkrUbmnTUeTAnigRsaybBaakrUbmnenpgdMeNaRsaytambMrabeKman

444355566666,443556666,435666222 =

4444355556666662 = .- 112 - PaK1


120/159


- 113 - PaK1

tamlMnaMeneyIgGacbeg;trUbmnTUeTAdUcxageRkam 6555.....5553444.....444666.....666

)1n()1n()n(

2 = .karRsaybBaakrUbmn eyIgtag

6555.........5553444..........444A)1n()1n(

=

)n(

2

2

)n(

2

)n(

2n

nn2

nn2

nn1nn2

1nn1n1n

)1n(

n1n

)1n(

666............666)999.........999(3

2

)999.........999(9

4)110(

9

4

9

410.810.4

9

4)52740(1010.4

9

545010.510.2710.410.4

610)110(9

510.310)110(

9

4

610555.......55510.310444........444

=

=

==+

=

+=+

+

+

+

dUceneKnrUbmn

6555.....5553444.....444666.....666)1n()1n()n(

2

=


121/159


- 114 - PaK1

lMhatTI60eKman

111111445556,11114556,115622 2222 = .BITahrN_xagelIcUrrkrUbm nigRsaybBaakrUbmnenpg=

111111114445555622 = nTUeTA

dMeNaRsayrkrUbmnTUeTA nigRsaybBaak

22

2

==

1111111144455556 = tamlMnaMEdleK[eyIgGacsresrrUbmnTUeTAdUcxageRkam )n(

nigRsaybBaak

111111445556

11114556

11

22

2

==

1111111144455556 22 = tamlMnaMEdleK[eyIgGacsresrrUbmnTUeTAdUcxageRkam )n2()n()n(

eKmanKman111111445556

11114556

1156

2

22 =562

22 =

22

22 111.......111445.....444556....555 =

22 111.......111445.....444556....555 =)n2()n(

sRmayepgpatrUbmnen

)n(

2445...ag

)n(

2

n..444556....555T


122/159


- 115 - PaK1

( )

( )9

999.........999

9

110)

9

110(110

1)110(9

41)110(

9

51)110(

9

41)110(

9

5

1)110(9

4

1)110(9

5

)1444......444(1555.....555

)n2(n2n

n

nnnn

2

n

2

n

22

==+ +

+ ++

Bit .dUcen

)n2()n(

2

)n(

111.......111445.....5....555 =

)n2(n

111..........111T =

244456 .

lMhatTI61eK[ 011nn aa..........aaA = nig 011nn a2a.........aaB fa Eckdacnwg t Eckdacnwg .say luRtaE

A 7 B 7

dMeNaRsay nwg enanaM[manbIB Eckdac 7 INq Edl q7B =eKn q7a2a........aa 011nn = .. naM[ )1(a a2q7a..........a 011nn


123/159


- 116 - PaK1

eyIgman)2(a10a...........aaaa.........aaA 1nn 011nn01 + yk CUskg eKn

q7

)1( )2(

00a)a2(A 10+

)a3q10(7a21q07aa20q70A0000

+ naM[ Eckdacnwg . RtaEt Eckdacnwg .A 7dUcen Eckdacnwg luA 7 B 7lMhatTI62eK[cMnYn 0121nn aaa......aaA a.....,,a, Caelx .yfacMnYn Eck alNaEdl 0 a,a n21cUrRsa dacnwg 6 kA

0n21a)a..aa(4y . +

dMeNa+ Eckdacnwg 6 .

RsayeyIgman 0121nn aaa......aaA = n

2

10a10.....a10a10aA

2

n+


124/159


- 117 - PaK1

eyIgman )6(mod410

eyIgn )6(mod410)6(mod410)6(mod410

n

3

2

)6(mod)a.....aa(4aAn210

+ Eckdacnwg luRtaEt

n1

edIm,I[ A 6 (mod0)a....aa(4a20

)6 cencMnYn Eckdacnwg kalNa2

dU A 6n1

a)a...aa(4y0Eckdacnwg 6 .

lMhatTI63cUrkMntelx a nig edIm,I[cMnYnb abba CaKUbncMnYnKt .dMeNa

Rsaynigntelx a btag ab10b100a1000abbaN +

b110a1001 ++ NN )b10a91(11


125/159


- 118 - PaK1

edIm,I[ N CaKUbncMnYnKtluRtaEt*INk,k121k1110a91 332b eday 9b0,9a0 ena 909b10a910 eKn 9091210 3 mmUl

121

627 +

121

909k0

3 = naM[ 1k= KTajn 121b10a91 = naM[ 10 a9112b 1= enaday 0b 0

10

a91121

91

301a +

91

121 = naM[ 1a = ehIy 310

91121b == .enc nig3b,1a = 3111331abba = .


126/159


- 119 - PaK1

lMhatTI64eKtag r nig R erogKaCakaMrgVgcarikkg nigcarikeRkAnRtIekaN AB mYy .kcUrRsayfa

R

r1CcosBcosAs +co + .xcUrRsayfa r2R

dMeNaRsaykRsayfa

R

r1CcosBcosAcos +

2

BBcosAcos

Ccos

2

CBcos2

2

Asin21Ccos 2

+ +

2

Csin

2

Bsin

2

Asin41

)2

CBcos

2

CBcos(

2

Asin21

)2

CBcos

2

Asin(

2

Asin21

2

CBcos

2

Asin2

2

Asin21 2

+

+

edayeKman bc )cp)(bp(2Asin = ab

)bp)(ap(

2

Bsin = p(

2

Csin,

ac

)cp)(a =


127/159


- 120 - PaK1

eKnabc

)cp)(bp)(ap(

2

Csin

2

Bsin

2

Asin

= KTaj

abc

)cp)(bp)(ap(41CcosBcosAcos

+ +

R

r1

R.p

r.p1

R.p

S1

R.SpS1

RR4

abc.p

)cp)(bp)(ap(p1

2

++

=

+

dUcen

R

r1CcosBcosAcos + .

faxRsay r2RtamvismPaB :GMAM .2 eKn )bp)(ap(2)bp()ap( )bp)(ap(2c

)cp)(ap(2bap2

eKTaj )1(21

c)bp)(ap(

cKaEdr )2(2

1

a

)cp)(bp(


128/159


- 121 - PaK1

nig )3(2

1

b

)cp)(ap( TMng Gg wgGg:eKn NTMnak :n)3(,)2(,)1(8

1

abc

)cp)(b eKTaj

p)(ap(

8

1

2

Csin

2

Bsin

2

Asin

eday 2Csin2Bsin2Asinco 41CcosBcosAs + eKn2

3)

8

1(41CcosBcosAcos =

EtR

r1CcosBcosAcos +

ajKT 23Rr1 .

r2R


129/159


130/159


- 123 - PaK1

l atTI6h 6kcUrRsayfa

xcot2

xcot

xcos

11 = KuNKNnapl

)

2

acos

11).....(

2

acos

11)(

2

acos

11)(

acos

11(Pn +

n

+ 2

dMeNaRsaykRsayfaxcot2

xcot

xcos

11 = eyIgtag

xcos

11)x(A +

xcot2

xcot

xtan2

xco =tan

xcos

xsin.

2

xsin

2

xs

2

xsinxcos

xsin2

xcos

2

xsinxcos

2

xsin

2

xcos2

xcos2

xcos2

x

1x22

=

== coscos

dUcenxcot2

xcot

xcos

11 = .

xKNnaplKuN )2

acos

11).....(

22a

cos

11)(

2

acos

11)(

acos

11(P

n

n + acot.

2

atanP =

1nn + .


131/159


- 124 - PaK1

lMhatTI67eK[sItncMnYnBit n kMntelI eday )U( IN

2

2U0 = nig INn,

2

11 UU

2n

1n = aGnuKmn_n .Nna nU C

n

dMeNaRsayKNna GnuKmn_n nU Ca neyIgman4

sin

2

20 = U

8

sin

2

4

sin11

2

U11U1

=2

20 ==

bmafavaBitdltYTIp KW 2pp s 2inU + ayfavaBitdl YTIyIgnwgRs t )1p( + KW3p1p 2

sinU += BitanyIgm

2

U11U

2p

1p

= Ettamkarbma2pp 2

sinU +

eyIgn2

sin11

U 1p+= 2

2p

2 + 3p22 +dUcen

32p sin22cos1 + p2

sin2

2

+ = Bit2nn 2

sinU + .


132/159


- 125 - PaK1

lMhatTI68cUrKNna 97cos94cos9cosS 333 + dMeNaRsayKNna

9

7cos

9

4cos

9

cS = os 333 + tamrUbmn acos3acos4a3cos 3 a3cos

4

1acos

4

3acos3 + GacsresrCa enamEdl[

)3

7cos

3

4s(cos

4

3S = co

3

(cos

4

1)

9

7cos

9

4cos

9

+ tag 97cos94cos9cosM = + yda

9

13cos

9

4cos =

9

13cos

9

7cos

9

cosM + KuNnwg

3

sin2 eKn

0)9

7cos(si

2

3M2 n2

9

16sin

9

2sin

9

10sin9

16sin9

49

10sin)9

2sin(9

4sin3

3

sin

9

13cos2

3

sin

9

7cos2

3

sin

9

cos2

3

sinM2

=

+

eKTajn

sin2

M.2

0M =


133/159


- 126 - PaK1

tag2

3

2

1

2

1

2

1

3

7cos

3

4cos

3

cosN =

eKn 83N41M43S = .dUcen8

3

9

7cos

9

4cos

9cosS

333 =+= .lMhatTI69eK[RtIekaN mYymanmMkgCamMRsYc .ayfa Btan.AtanCtanBtanAtanABCkcUrRs tan. CTajbBaakfa 33CtanBtanAtan .dMeNaRsaykRsayfa tan Ctan.Btan.AtanCtanBtanA =

= yIgman BA + C CBA nK )Ctan()BAtan( CtanBtanAtanCtanBtanAtan

CtanBtan

Btan

Atan1

Atan

+ tan.AtanBtanA

++

dUcen .BCtantan Ctan .


134/159


- 127 - PaK1

xTajbBaakfa 33CtanBtanAtan CamMRsYc ( tamsmtikm )eKTaj PaBkUsIeyIgGacsresr

eday C,B,A0Ctan,0Btan,0Atan >tamvism

3 Ctan.Btan.Atan3CtanAtan eday tan.Btan.AtanCtanBan

Btan + tAtan CeKn 3 CtanBtat nAtan3BtanBtanAan +

2 + (tan27)CtanBtanA(tan 3

27)CtanBtanA(tan

)CtanBtanA + dUcen 33CtanBtanAtan .


135/159


- 128 - PaK1

lMhatTI70KNnaplbUk n

999....999....999999S + manelx cMnYn elx ) .9 n

dMeNaRsayKNnaplbUk 9..

99......999. .999999S

n

)9n( elx

+

9

)10n9(10

n9

110.10

n =)1...111()10...10

)110(....)110()110()110(

1n

n32

n32

+=

+

+

dUcen

1010( +

9

)10n9(10

S

2n

n

+=

+ .


136/159


- 129 - PaK1

lMhatTI71cUrkM tRKbKUtmKtviCman ebIeKdwgfacMnYn

)b,a(

1bab2

a32

2

+ CacMnYnKtviCmanEdr .dMeNaRsayntRKbKUtmKtviCman k

kM )b,a(y k

1bab2

a32

2 =+ Edl *INk 32eKn 2a2 )1(0)1b(kakb =

42

DIsRKImINgnsmIkar )1b(k4 3bk4

22 k4)bkb2( 2b mIkar mancemykg luRtaEt)1( *IN CakaerRkdmannyfa 2222 dbk4)bkb2( =d

Edl CacMnYnKt.-ebI 0bk4 2 = 4bk 2=

eyIgTTYln2

bbb

2kb2a

3

2 = 2

ba =


137/159


- 130 - PaK1

eday a b, CacMnYnKtviCman ehtueneKRtUv[ INp,p2b eKTaj pp8p

4ehIy)p2( 4

2 )p2(2a 2=p

p2 = .2

a

2,p(;)p2,pp4dUcen 8()b, *INp,)pa( -ebI 0b k4 2 > eKn 4)2 bkb2( 22222 *INk,)1bkb2(dbk 2 )1b(k4 2 0)1b( neKTajsmI 1b =kgkrN kar )1( kayCa 0ak2a2 = naM[ k2a =

dUcen )1,k2()b,a( = cMeBaRKb *INk .-ebI 0bk4 2 < )bkb Kn k42( 222222 )1bkb2(db

)bkb2 2222


138/159


- 131 - PaK1

TIhat 72cUrbgHajfa 1

ab8ca 22c

ca8bbc8 2ba +++ RKbcMnYnBit man .eBa viC c,b,a

dMeNaRsayajfagH 1ab8

c cca8bbc8a

ba222 +++

CadMbUgeyIgRtUvRsayfa3

4

3

4

3

4

3

4

aa2

cbabc8a +

+mmUlvismPaBens )bc8a(a)cba( 2342343434 +

smmUl bca8cb2ca2ba2cb 3 3434343434343438 8 + amvismPaB eKman

+t GMAM 34343838 cb2cb eKTaj bca8cb2ba2cb 3333 ca2 34343434344488 Bit

tuenh ( )1cba

a

bc8a

a

3

4

3

4

3

4

3

4

2 ++


139/159


- 132 - PaK1

dUcKaEdreKTaj ( )2cba

b

ac8b

b

3

4

3

4

3

4

3

4

2 +

+

ehIy ( )3cba

c

ab8c2c

3

4

3

4

3

4

3

4

+ edaybUkvismPaB Gg:nigGg:eKn+ )3(,)2(,)1(1

ab8c

c

ca8b

ba

b8a c 222++ .+

lMhatTI73eK[ a( n )CasItncMnYnBitEdl2

1a

1=

bcMnYnKtviCman eyIgmangcMeBaRK n 1aa aa n2n 2n1 =n + cUrRsaybBaakfacMeBaRKbcMnYnKtviCman n eyIgman n3

aa21

1a.........a < .dMeNa

+Rsay

RsaybBaakfa 1a.........aaa n321 < eyIgman1aa

aa

1n=

n

2

n

2

n +


140/159


- 133 - PaK1

tagn

na

1b = enaTMnakTMngEdl[kayeTACa

n

2 + bb1 1bbb n1n+eKTaj 1nb + n2n

nnnn1n1b)b(b1b

nn1)1b(b1

n

b

11

1

11

b +

+ eKTaj

1b1bba

1nnn

n + 111n21

n21.aa +

b

1......

b

1

b

1a... +

11b

111b1b

1n1 +11

1b1b121+ )1b

1

1b

1(...)

1b

1

1b

1()

1n

nn32


141/159


- 134 - PaK1

lMhatTI74cUrkMntRKbKUtmKt 3n,m ebIeKdwgfacMeBaRKbcMnYnKtviCman eKmana

1aa

1aa2n

m

CacMnYnKt .dMeNaRsay

KUtmKtviC ntRKb man ( )n,medIm,I[1aa

an1aa2n

m

CacMnYnKtluRtaEt 2 1a aktarYmn 1aam ehIy .gyk nm >ey *INk,knm gn

k1k2nk

knm

++

,I[ey

)1aa)(a1()1aa(a

1aa1aa

=tamTMnakTMngenedIm 1aa 2n CaktarYmn a 1am luRtaEt 1kn + n g 2k= ..dUcen )3,5()n,m( =


142/159


- 135 - PaK1

lMhatTI75eK[bIcMnYnKt c,b,a Edl baviCman = 10c .cUrrktmFMbMputn cbaP .dMeNaRsayrktmFMbMputn cbaP aM[ cyIgman 10cba = n ba01

)b10(b10(ab 2 yIgn baP aa)b + 2tag )ba(P ab10() 2ba

eKn 2bb10ab2)a('P bI eKTajn0)a('P =2

b5a eday a(''P *INb,0b2) Gtibrma mhtuen )a(P mantm cMeBat

2

b5a .

a RtUvEt YnKUeday *INa en CacMnbeb k2b = ena k5a ehIy k5k2)k5(10c


143/159


- 136 - PaK1

eday ,a ena *INc,b

1k5c

1k2b

1k5a

4k1 = n-cMeBa k 1eK 4c,2b,4a = naM[ 32424P =

n 3a-cMeBa 2k=eK c,4b, 3 PnaM[ = 36343 -cMeBa 3k= eKn 2c,6b,2a = naM[ 242P 26 = -cMeBa 4k=

n aeK 1b,1 c,8 = naM[ 32181P = tmGtibrmandUcen c.b.aP = es g .

n 36


144/159


- 137 - PaK1

lMhatTI76eKmansmPaB

ba

1

b

xcos

a

xin44

s + 0dl b,a 0ba,0 arbgHajf

33

8

3

8

)b

xcos

a

xsin = .eNa

ba(

1+dM RsaygHajfa

33

8

3

8

)ba(

1

b

xcos

a

xsin + eKmanba

1

b

xcos

a

xsin 44 + [a abxcos)ba(axsin)ba(b 44 = 0)xsinbxcosa(

0xsinbxsinxcosab2xcosa

)xcosx(sinabxcosabxcosaxsinbxsinab

222

422242

422442424

==

+

eKTaj ba1

ba

x

a

xsin =cosxsinb

xcos 2222

++= aM[

44

8

4

8

)ba(

1

b

xcos

a

xsin +


145/159


- 138 - PaK1

eKTaj )1()ba(

a

a

xsin43

8

+ nig )2()ba( bb xcos 438 + nig Gg:nigGg: eKn ksmIkar )1( )2(343

8

3

8 cxsin + )ba(

1

)ba(

ba

b

xos

a +++=dUcen33

8

3

8

)ba(

1

b

xcos

a

xsin + .lMhatTI77eK[RtIekaN ABC mYy .bgHajfaebI

3

Ctan,

3

Btan,

3

Atan

kar bxaxx:)E( 23Cassm c 0nnaeK cb3a3 + .dMeNaRsaykarbgHaj eyIgman CBA kgRtIekaN )yIgn( plbUkm ABCe ]

3

C)

3

B

3

Atan[()

3

C

3

B

3

Atan( +


146/159


- 139 - PaK1

)1(

)3

Ctan

3

Btan

3anta

3tan

3(tan1 + Ct

3n

BA3

Ctan

3

Btan

3

Atan

Cta

3

Btan

3

3

Ctan.

3

Btan

3

Atan

3

Btan

3

Atan

3

Ctan

3

Btan

3

A3

Btan

A

3Ctan

31

3

Ctan)

3

B

3

Atan(

++

+++

+=

A3

ntan

3

=A +1

1

tan1

3

tan

3tan =

)3BAtan(

)

3

tan( +CBA +

eday 3Ctan,3Btan,3Atan CasrbssmIkarenatamRTwsbTEvteKmanTMnakTMng )E()2(a

3

Ctan

3

Btan

3

Atan

)4(c2

Ctan.

2

Btan.

2

Atan

)3(b2

Ctan

2

Atan

2

Ctan

2

Btan

2

Btan

2

Atan

=

ykTMnakTMng nig CYskgsmIkar eKn )3(,)2( )4( )1(


147/159


- 140 - PaK1

b1

ca3 += cab33 + dUcen cb3a3 + .

lMhatTI78k cUrKNnatmkdn 10sin nig 10cos x cUrRsayfa

10sin)yx(4)yx(x

22222 + RKbcMnYn IRy,x .dMeNaRsayk KNnatmkdn

10sin

nig10

cos

eKman10

32

210

eKn

10

3cos)

10

3

2sin(

10

2sin

== tamrUbmnRtIekaNmaRt nigacosasin2a2sin = acos3acos4a3cos 3


148/159


- 141 - PaK1

)1010

sin1(43sin2

10cos43

10sin2

10cos4

10cos

10cos

10sin2

10

3cosco

10sin2

2

3

= 310

s

2

=

0sin2sin4 2 =11010 tag 010eKn t4 2 sint > >=0541',01t2 = seKTaj 0

4

51t1 < ( minyk )

4

51t, 2

+= dUcen 4 5110sin += .eday 1

10cossin

22

10=+

naM[4

5210)

4

51(1

10

2 = cosdUcen

410cos = 5210 .

sayfa R10

sin)yx(4)2 yx(x 2222 +

tagGnuKmn_10

sin)yx(4)yx(x)y;x(f22222 +


149/159


- 142 - PaK1

eKn 222222 )

451)(yx(4yxy2xx)y;x(f ++

IRy,x,0y2

15x

2

15

y2

15xy2x

2

15

y5 22

1xy2x2

51

2

53)yx(yxy2x2

16

26)yx(4yxy2x2

22

2

2222

2222

++

+=

++++

5=

2+

+=

dUcen10

sin)yx(4)yx(x22222 + .


150/159


- 143 - PaK1

lMhatTI79eK[sItncMnYnBit kMnteday)U( n 4nsin.2U nn = Edl .k-cUrbgajfa*INn

4

nsin

4

ncos

4

)1n(cos.2

= x-Taj[nfa

4

)1n(cos)2(

4

ncos)2(U

1nn

n

= + K-KNnaplbUk n321n

U.........UUUS + CaGnuKmn_n .dMeNa

n

Rsaya-cUrbgHajf 4nsin4ncos4 )1n(cos.2 = tamrUbmn bsinasinbcosacos)bacos( gn y

)4sin4

nsini4cos4

n(cos2

)44

ncos(2

4

)1n(cos2

=

+= 4

nsin

4

ncos)

4

nsin

2

2

4

ncos

2

2(2

4

)1n(cos2

==


151/159


- 144 - PaK1

dUcen4

nsin

4

ncos

4

)1n(cos.2

= .x-Taj[nfa

4

)1n(cos)2(

4(

1n

n

ncos)2U

n = + anyIgm

4

nsin

4

ncos

4

)1n(osc.2

= naM[

4

)1n(cos2

4

ncos

4

nsin

= NGg:TaMgBIrnwg n)2( eKn

4

)1n(cos)2(

4

ncos)2(

4

1nnnsin)2(

n = + cen

d

4

)1n( cos)2(

4

ncos)2(U

1nn n

= + .-KNnaplbUk

n321nU.........UUUS +

eyIgn4

)1n(cos)2(

4cos2

4

)1 k(cos)2(

4

kcos)2(

1n

n

1k

1kk

= +Sn =

+=

+ dUcen 4 )1n(cos)2(1S 1nn + .


152/159


- 145 - PaK1

lMhatTI80eK[ n cMnYnBit 0a.......;;a;a;a n321 .eKtag *NSn In,n

a.....aaan321 =yfaebIcUrRsa 1n

1na

n3211na.......a.a.aS + + enaeKn n 3n .....S

n21 aaaa .

dMeNaRsayebIsayfa 1n1nn3211n

aa.......a.a.aS + + enaeKn n n321neyIgman a.....aaaS 1n1nn3211n

aa.......a.a.aS + + edayn

a.....aaaS n321

n

+ ena

1n

aa....aaaS 1n

1nn321 + += + eKn +( )1aa.........

aaa 21

aaa1n

aa.....1n

1nn321

1nn3 ++

+ ++ k

+y

n

a.....aaaa n321

1n

+= CYskg ( )1 eKn


153/159


- 146 - PaK1

nn21

n21

n21a.a

n

n2

n21

n21

1n

n21

1n n2

21

n2

1n n21

n21

n21

n

a.........a.an

a.......aa

a......n

a.......a

n

a...aaa.....a.a

n

a.........aa

n

a....a....a.a

n

a.......an

a....aa.a...a.an

a.....aa

++

+++

1a

1

n

1a

.aa

1n ++21

a....aa

...

++

++

+

+

eKTaj n n321n dUcen ebI a.....aaa S 1n1nn3211n

aa.......a.a.aS + enaeKn n n321n a.....aaaS .lMhatTI81cUrbgHajfa

10

1

100

99......

6

5.

4

3.

2

1

15

1 < dMeNaRsaybgHajfa 10110099......65.43.21151 < anyIgm )1n(n)1n( + 2n1n2:*INn + eKTaj

1n

n

2n2

1n2 +++


154/159


- 147 - PaK1

eKn 2515049......32.2110099......65.43 = KuNnwg2

1 eKn210

1

100

99........

6

5.

4

3.

2

1 > eday

15

1

210

1 > eKTaj )1(15110099....65.43.21 > ageTot

)1n2)(1:*INn n2(2)1n2()1n2(n4 + eKTaj1n2

1n2

1n2

n2 +> rW 1n2 1n2n2n2 1 +< eKn 101110199....75.53.3110099.....65.43.21 = eday10

1

100

1

101

1 = eKTaj )2(

10

1

100

99....

6

5.

4

3.

2

1 < ng nwg eKn amTMnakT )1( )2(

10

1

100

99......

6

5.

4

3.

2

1

15

1 < < .


155/159


- 148 - PaK1

lMhatTI82eK[ n2 ....;cMnYnBit ;b;b;a.....;;a;a .a....b

2

n

2

2

2

1

2

n

2

2

2

1

2

n22+

n21n21bcUrRsayfa

)b.....bb)(a....aa()baba(n11

+dMeNaRsay

.....bb)(a....aa()ba....baba(22

2

2

1

2

n

2

2

2

1

2

nn2211+ eyIgeRCIserIsGnuKmn_mYykMnt

Rsayfa )b

n

IRx eday )bx..)bxa()b

2

n

22

1nn2211

22

n

2

2

2

2

n

2

22

2

1

++

eday )b...bb(x)ba..baba(2x)a...aa()x(f 21 +a(...xa()x(f

n1+

IRx RtIFa 0)x(f Canicenaeday KnCanic

>

0'

0af

0a.....aaa2

n

2

2

2

1f>ehtuene 0' .

( ) 0)b....bb)(a....aa(ba. ...baba' 2221n211 +.... 2n22212n2n2 dUcen a()ba....baba( 2n

2

2

2

1

2

n

2

nn2211+ )b.....bb)(a....a 2

2

2

1+


156/159


- 149 - PaK1

lMhatTI83eK[ )( nig y( nxn CasItcMnYnBitkMntelI0 = nigTMnakTMngkMenIn

n1nyx3xx + nig RKbcUrKNna nig CaGnuKmn_n .

dMeNa

) INeday ,5x0 =3

1y

2

nn

3

nn

2

n1nyyx3y + INn

nx ny n

RsayeKman nnn1n + nig kar eK

)yx(y

n

3

nn1n ++

ln{ n

KNna nx nig ny CaGnuKmn_n23

n

)1(yx3xx

)2(yyx3y 3nn2

n1n +bUksm nig 2( n )1( ))yxln(.3)yxln(

x1n+

n1n1n +TMnakTMngenbBaakfa x( n })y CasItFrNImaRtmanersug nigtYdMbUg3q = 6ln)yxln( 00 = .Kn 6ln3)yxln( nnn

=


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- 150 - PaK1

eKTajnig eKn

.3y

y 3n

akfa

)3(6yxn

3

nn=

dkksmIkaryx = )1( )2(

)yx()xln(

)x(

nn1nn

nn11n + ln1TMnakTMngenbBa })yxln({ nn CasItFrNImaRtmanersug nigtYd

003q = bUg ln( 4ln)yx = .

Taj nig eKneKn 4ln3)yxln( nn n =eK )4(4yx n3nn =bUksmIkar 3( ) )4( nn 33n 46x2 +eKTaj

2

46x

nn33= .n +ar nig eKn ksmIk y2)3( )4( nn 33n 46eKTaj

2y

n

46nn

33 = .dUcen

2

46x

nn33 +

n = nig2

46y

nn33

n

= .( anPaKTI2 sUmG )sUmrgcaMG rKuN


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- 151 - PaK1

ksareyag( Referent )

Hu hematical Chal esdward J. Bar beau, M ray S. Klamkin)

plex Numbers Fro A to Zitu Andresscu , Dorin Andrica )

a Oly in China

at ly hallengesitu Andresscu & n Gelca)ematical Oly d Treasures

ndreescu , Bogdan enescu )7. International Mathematical Olympiads 1959-1977

8. The IMO Compendium ( 1959-2004 )

(By Titu Andreescu , Dorin Andrica )

1. 103 Trigonometry Problems(From the training of the USA IMO Team)

2. Five ndred Mat leng(By E ur

3. Com m(By T

4. Mathem tical mpiad

5. Mathem ical O mpiad C(By T Razva

6. Math mpia(By Titu A

( Samuel L. Greitzer )

9. 360 Problems for Mathematical Contest


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