117 Inequality Selected Exercises.pdf

7/25/2019 117 Inequality Selected Exercises.pdf

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cUrbgajfa 1an1

1 n

.!!$> cMeBaHRKb; c,b,a FMCagmYy cUrbgajfa

)blogalogc(log4ablogcalogbclog cabcabcba


21/231

vismPaBeRCIserIs


!!%> eKtag ,, CargVas;mMukgRtIekaN ABC mYyEdlmanbrimaRt p2

nigkaMrgVg;carwkeRkA R .

/a cUrRsaybBaak;fa

1p

R.93cotcotcot

2

2222

/b etIeBlNaeTIbeK)ansmPaB ?

!!^> eKyk c,b,a CabIcMnYnBitviCman . cUrbgajfa

5

3

c)ba(

)cba(

b)ac(

)bac(

a)cb(

)acb(22

2

22

2

22

2

.

!!&> eK[bIcMnYnBitminGviCman c,b,a nigminsUnRBmKaBIr .

cUrRsaybBaak;fa

49

)ac(1

)cb(1

)ba(1)cabcab(

222

www.mathtoday.wordpress.com


22/231

vismPaBeRCIserIs


CMBUkTI2

EpkdMeNaHRsaylMhat;TI1

cUrRsayfa ab2

baba

22

RKb;cMnYnBitviCman a nig b

]bmafa ab2

baba

22

Bit

elIkGgTaMgBIrnvismPaBCakaereK)an 2

222 ab

2

ba)ba(

0ab2

ba

ab2

ba

.ab22

ba

0ab2

ba.ab2

2

babab2a

ab2

ba.ab2

2

babab2a

2222222

222222

222222

dUcenH ab2

baba

22

.


23/231

vismPaBeRCIserIs


lMhat;TI2

cUrRsaybBaak;facMeBaHRKb; 0c,b,a eKmanvismPaB

)cabcab)(cba(9

8)ca)(cb)(ba(

eKmansmPaB

abc2bcacababcaba)ac)(cb)(ba( 222222

abc3bcacababcaba)cabcab)(cba( 222222

eFIVpldksmPaBBIrenHeK)an

)1(abc)ac)(cb)(ba()cabcab)(cba(

tamvismPaB GMAM eKman

ac2ac

bc2cb

ab2ba

eK)an )2(abc8)ac)(cb)(ba(

yk )1( CMnYskg )2( eK)an

)cabcab)(cba(8)ac)(cb)(ba(9

dUcenH )cabcab)(cba(9

8)ca)(cb)(ba( .


24/231

vismPaBeRCIserIs


lMhat;TI3

cUrbgajfa 27

8

)cba(

)ac)(cb)(ba(

3



3

3

)ac)(cb)(ba(3)cba(2

)ca)(cb)(ba(3)ac()cb()ba(

elIkGgTaMgBIrCasVyKuN 3 eK)an )ac)(cb)(ba(27)cba(8 3

dUcenH27

8

)cba(

)ac)(cb)(ba(3

.


25/231

vismPaBeRCIserIs


lMhat;TI4

eK[cMnYnBitviCman z,y,x,c,b,a .

cUrRsayfa 333 abcxyz)cz)(by)(ax(

eKman

abcbcxacyabzcxybxzayzxyz)cz)(by)(ax(


3 222

zyabcx3cxybxzayz

3 222 xyzcba3bcxacyabz

eK)an

abcxyzcba3zyabcx3xyz)xz)(by)(ax( 3 2223 222

333 )abcxyz()cz)(by)(ax(

dUcenH 333 abcxyz)cz)(by)(ax( .


26/231

vismPaBeRCIserIs


lMhat;TI5

cMeBaHRKb;cMnYnBitviCman b,a nig ccUrRsaybBaak;fa

2

3

ba

c

ac

b

cb

a

.

rebobTI tamvismPaB SchwarzCauchy

)bbb)(aaa()bababa( 2

32

22

12

32

22

12

332211

ykba

ca,

ac

ba,

cb

aa 321

nig )ba(cb,)ac(bb,)cb(ab 321

)ca2bc2ab2)(ba

c

ac

b

cb

a()cba( 2

eKTaj )1()cabcab(2

)cba(

ba

c

ac

b

cb

a 2

tamvismPaB GMAM

eKman

acbcab2

ac

2

cb

2

ba 222222

b cabcabcba 222

b )2()cabcab(3)cba( 2

tamvismPaB )1( nig )2( eKTaj)an

2

3

ba

c

ac

b

cb

a

.


27/231

vismPaBeRCIserIs


rebobTI2

eKman 2

3

ba

c

ac

b

cb

a

eK)an 32

3)1

ba

c()1

ac

b()1

cb

a(

b2

9

ba

cba

ac

cba

cb

cba

b2

9)

ac

1

cb

1

ba

1()cba(

tamvismPaB GMAM

eKman

)1()ac)(cb)(ba(3)cba(2

)ca)(cb)(ba(3)ac()cb()ba(

3

3

)2()ac)(cb)(ba(

3

ac

1

cb

1

ba

13

KuNvismPaB )1( nig )2( Ggnig GgeK)an 9)

ac

1

cb

1

ba

1)(cba(2

eKTaj2

9)

ac

1

cb

1

ba

1()cba(

Bit

dUcenH2

3

ba

c

ac

b

cb

a

.


28/231

vismPaBeRCIserIs


lMhat;TI6

eK[ z,y,x CabIcMnYnBitviCman .

cUrbgajfa)zx)(zy)(yx(

)zyx(2

)xz(

1

)zy(

1

)yx(

1222


)3()yx)(xz(

2

)yx(

1

)xz(

1

)2()xz)(zy(

2)xz(

1)zy(

1

)1()zy)(yx(

2

)zy(

1

)yx(

1

22

22

22

bUkvismPaB )2(,)1( nig )3( eK)an

)zx)(zy)(yx()zyx(4

)xz(1

)zy(1

)yx(12 222

dUcenH

)xz)(zy)(yx(

)zyx(2

)xz(

1

)zy(

1

)yx(

1222

.


29/231

vismPaBeRCIserIs


lMhat;TI7

cUrRsaybBaak;vismPaB

2

23)a1(c)c1(b)b1(a 222222

cMeBaHRKb;cMnYnBit c,b,a .

tamvismPaB Minkowsky eKman 2

3212

3212

32

32

22

22

12

1 )yyy()xxx(yxyxyx

yk cx,bx,ax 321

nig a1y,c1y,b1y 321 nig cbas

eK)an2

9)s3(s)yyy()xxx( 222321

2321

eRBaH2

9]9)3s2[(

2

19s6s2)s3(s 2222

dUcenH

2

23)a1(c)c1(b)b1(a 222222 .


30/231

vismPaBeRCIserIs


lMhat;TI8 Junior TST 2002, Romania

ebI )1,0(c,b,a cUrbgajfa

1)c1)(b1)(a1(abc eyIgdwgfacMeBaH )1,0(x eKman 3 xx

ehtuenH )1(abcabc 3

nig )2()c1)(b1)(a1()c1)(b1)(a1( 3

bUkvismPaB )2(&)1( eK)an )3()c1)(b1)(a1(abc)c1)(b1)(a1(abc 33


)5(3

)cba(3

3

c1b1a1)c1)(b1)(a1(

)4(3

cbaabc

3

3

bUkvismPaB )5(&)4( eK)an

)6(1)c1)(b1)(a1(abc

3

)cba(3)cba()c1)(b1)(a1(abc

33

33

tamvismPaB )6(&)3( eK)an 1)c1)(b1)(a1(abc Bit .


31/231

vismPaBeRCIserIs


lMhat;TI9

eK[ c,b,a CabIcMnYnBitviCmanEdl 1abc .

cUrRsayfa 3cbab

ac

a

cb

c

ba

tamvismPaB GMAM eKman ab2ba

eK)an )1(c

ab2

c

ba

RsaydUcKaEdreK)an )2(

a

bc2

a

cb nig )3(

b

ca2

b

ac

bUkvismPaB )3(&)2(),1( eK)an

)4()b

ca

a

bc

c

ab(2

b

ac

a

cb

c

ba

ma:geTot )5(a2b

ca

c

ab

)6(b2c

ab

a

bc nig )7(c2

b

ca

a

bc


)8(3cba)cba(2)b

ca

a

bc

c

ab

(2

eRBaH 3abc3cba 3 1abc

tam )8(&)4( eK)an 3cbab

ac

a

cb

c

ba .


32/231

vismPaBeRCIserIs


lMhat;TI10

eK[ d,c,b,a CabYncMnYnBitEdl 4dcba 2222 .

cUrRsaybBaak;fa 8dcba 3333 .

eKman 4dcba 2222

eKTaj 0a4dcb 2222 b 4a2 b 2a

eday 0a2 enaH )1(a2a 23

RsaydUcKaEdr )2(b2b 23

)4(d2d,)3(c2c, 2323

bUkvismPaB )4(&)3(),2(,)1( Gg nig GgeK)an

8)dcba(2dcba 22223333 Bit

dUcenH 8dcba 3333 .


33/231

vismPaBeRCIserIs


lMhat;TI11

eK[ c,b,a CabIcMnYnBitminGviCman .

cUrRsayfa 3333 )a2

cb(2abc3cba .

tamvismPaB GMAM eKman abc3cba 333

enaH 0abc3cba 333 .

ebI 0a2

cb enaHvismPaBxagelIBitCanic .

eyIg]bmafa 0a2

cb .

tag 3333 )a2

cb(2abc3cbaT

yk x2ab nig y2ac enaH y2x2a2cb

Edl 0y,0x

.eK)an

3333 )yx(2)y2a)(x2a(a3)y2a()x2a(aT

bnab;BIbRgYmrYceK)an 2222 )yx)(yx(6)yx)(yx(6)yxyx(a12T

0)2

cb)(a

2

cb(6T 2 Bit

dUcenH 3333 )a2

cb(2abc3cba .


34/231

vismPaBeRCIserIs


lMhat;TI12


cUrbgajfa 5222

3

cba

3

cba .

]bmafa 5222

3

cba

3

cba

Bit

smmUl3

cba

3

cba 2225

smmUl )cba(abc81)cba(81)cba( 2222225


)zxyzxy(3)zyx(

zxyzxy)zxyzxy(2)zyx(zxyzxyzyx

zxyzxy2

xz

2

zy

2

yx

2

2

222

222222

yk caz,bcy,abx eK)an

)cba(abc3)cabcab( 2

enaH )cba)(cba(abc3)cabcab)(cba( 2222222 edIm,IRsayfa )cba(abc81)cba( 2225 Bit

eyIgRtUvRsayfa 22226 )cabcab)(cba(27)cba(


35/231

vismPaBeRCIserIs


tag 22226 )cabcab)(cba(27)cba(T

yk cbaS nig cabcabP

enaH P2S)cabcab(2)cba(cba 22222

eK)an 226 P)P2S(27ST

0)P6S()P3S(

P54PS27S

2222

4226

dUcenH 5222

3cba

3cba

.


36/231

vismPaBeRCIserIs


lMhat;TI13

eK[ c,b,a CabIcMnYnBitxusKa .

cUrRsayfa 2)ba(

c

)ac(

b

)cb(

a2

2

2

2

2

2

.

tamsmPaB xz2yz2xy2zyx)zyx( 2222

eKTaj )zxyzxy(2)zyx(zyx 2222

eday0)zyx( 2

RKb;cMnYnBitz,y,x

enaHeK)an (*))zxyzxy(2zyx 222

ykba

cz,

ac

by,

cb

ax

enaHeK)an

)cb)(ba(

ac

)ac)(ba(

bc

)ac)(cb(

abzxyzxy

1)ac)(cb)(ba(

)ab)(cb)(ac(

)ac)(cb)(ba(

)ac(ac)acb)(ac(b

)ac)(cb)(ba()ac(ac)cb(bc)ba(ab

tamTMnak;TMng (*) eKTaj)an 2)1(2zyx 222

dUcenH 2)ba(

c

)ac(

b

)cb(

a2

2

2

2

2

2

.


37/231

vismPaBeRCIserIs


lMhat;TI14

eK[ c,b,a CabIcMnYnBitminGviCmanEdl cbacba 222 .

cUrRsayfa cabcabaccbba 222222 .

eKman cbacba 222

eK)an 22222 )cba()cba(

)cabcab(2cba)accbba(2cba 222222222444

222444222222 cbacba)accbbacabcab(2

tamsmPaBenH edIm,Ibgajfa cabcabaccbba 222222

eyIgRtUvbgajfa 222444 cbacba .

tamvismPaB Holder eKman

)ccc)(bbb)(aaa( 3

33

23

13

33

23

13

33

23

13

333222111 )cbacbacba(

yk 32

333

2

223

2

11 cba,bba,aba

nig 34

33

4

23

4

1 cc,bc,ac eK)an32224442222 )cba()cba()cba(

naM[222444

cbacba

BitdUcenH cabcabaccbba 222222 .


38/231

vismPaBeRCIserIs


lMhat;TI15

eK[ c,b,a CabIcMnYnBitminGviCman . cUrRsayfa 3222222 )cabcab()acac)(cbcb)(baba(

tamvismPaB GMAM eKman xy2yx 22

naM[ xy4y3x3xy2xy4y3x3yx 222222

b 222 )yx(3)yxyx(4 RKb;cMnYnBitviCman x nig y .

ehtuenHcMeBaH c,b,a CabIcMnYnBitminGviCman eKTaj)an 2223222222 )ac()cb()ba()

4

3()acac)(cbcb)(baba(

eyIgnwgRsayfa 33222 )cabcab()3

4()ac()cb()ba(

eKmansmPaB

abc2bcacababcaba)ac)(cb)(ba( 222222

abc3bcacababcaba)cabcab)(cba( 222222

eFIVpldksmPaBBIrenHeK)an

)1(abc)ac)(cb)(ba()cabcab)(cba(


ac2ac

bc2cb

ab2ba

eK)an )2(abc8)ac)(cb)(ba(


39/231

vismPaBeRCIserIs


yk )1( CMnYskg )2( eK)an

)cabcab)(cba(8)ac)(cb)(ba(9

b )3()cabcab()cba(81

64)ca()cb()ba( 22222


acbcab2

ac

2

cb

2

ba 222222

b cabcabcba 222

b )4()cabcab(3)cba( 2

tam )3( nig )4( eKTaj)an 33222 )cabcab()

3

4()ac()cb()ba( Bit

dUcenH 3222222 )cabcab()acac)(cbcb)(baba(


40/231

vismPaBeRCIserIs


lMhat;TI16

eK[ c,b,a CabIcMnYnBitminGviCmanEdl 1abc . cUrRsayfa

2

3

ac

acb

cb

cba

ba

bac22

33

22

33

22

33

.

tag22

33

22

33

22

33

ac

acb

cb

cba

ba

bacT

tamvismPaB SchwarzCauchy eKman

)ba)(ba()b.ba.a()ba( 33223

21

23

21

222

eKTaj 3322 ba.baba

b )1(ba

c

ba

bac22

33

. RsaydUcKaEdreK)an

)2(cb

a

cb

cba22

33

nig )3(acb

ac

acb22

33


ba

c

ac

b

cb

aT

tamvismPaB SchwarzCauchy eK)an

)4()ba

c

ac

b

cb

a(3

ba

c

ac

b

cb

a 2222

eday baba

cac.

ac

bcb.

cb

acba


41/231

vismPaBeRCIserIs


eK)an

)ba

c

ac

b

cb

a)](ba()ac()cb[()cba(

2222

)ba

c

ac

b

cb

a)(cba(2)cba(

2222

eKTaj )5(2

3

2

abc3

2

cba

ba

c

ac

b

cb

a 3222

tam )5(&)4( eK)an 29

ba

c

ac

b

cb

a 2

naM[2

3

ba

c

ac

b

cb

a

enaH2

3T

dUcenH2

3

ac

acb

cb

cba

ba

bac22

33

22

33

22

33

.


42/231

vismPaBeRCIserIs


lMhat;TI17

eK[ c,b,a CabIcMnYnBitviCman . cUrRsaybBaak;vismPaB

1)ba(c

c

)ac(b

b

)cb(a

a33

3

33

3

33

3

.

tag33

3

33

3

33

3

)ba(c

c

)ac(b

b

)cb(a

aS

yk cba

z,b

acy,

a

cbx

eK)an

333 z1

1

y1

1

x1

1S

tamvismPaB GMAM RKb; 0u eKman

22

22

23)x

2

11(

2

xx1x1)xx1)(x1(x1

eKTaj)an

2

223

a2

)cb(1

1

x2

11

1

x1

1

eday2

222

2

22

2

2

a

cba

a2

)cb(21

a2

)cb(1

eKTaj)an222

2

3 cba

a

x1

1

dUcenH 1cba

c

cba

b

cba

aS

222

2

222

2

222

2

.


43/231

vismPaBeRCIserIs


lMhat;TI18

cMeBaHRKb;cMnYnBitviCman c,b,a cUrRsayfa

2

3

ac

c

cb

b

ba

a

ykac

c

cb

b

ba

aS

tagc

az,

b

cy,

a

bx ehIy 1xyz

eK)an222 z1

1

y1

1

x1

1S

]bmafa zyx eK)an 1xyzx3 b 1x ehIy 1x

1yz


)z1

1

y1

1(2)

z1

1

y1

1( 22

2

22

eday2222

22

2222

22

22 zyzy1

zy11

yzzy1

zy2

z1

1

y1

1

ehIyyz1

yz1

zyyz21

)yz1)(yz1(

zyzy1

yz1222222

22

eK)anyz1

4)

yz1

yz11(2

z1

1

y1

1 2

22


44/231

vismPaBeRCIserIs


eKTaj)anyz1

2

z1

1

y1

1

22

ehtuenHyz1

2x1

1S2

eyIgnwgRsayfa2

3

yz1

2

x1

1

2

eKman )x1(2)x1( 22 b 2x1.2x1

enaH yz1yz2

yz

11

2

x1

2

x1

1

2

naM[yz1

2

yz1

yz2

yz1

2

x1

1

2

ehIy2)yz1(

yz122yz2)yz1(3

yz1

2

yz1

yz2

2

3

02)yz1(

)yz12(

2)yz1(

yz122yz3

2

eKTaj2

3

yz1

2

yz1

yz2

yz1

2

x1

1

2

Bit

dUcenH2

3

ac

c

cb

b

ba

a

.


45/231

vismPaBeRCIserIs


lMhat;TI19

cMeBaHRKb;cMnYnBitminGviCman z,y,x cUrRsayfa

)xyz1)(z1)(y1)(x1()z1)(y1)(x1(2 222

ebI xzy enaHvismPaBeTACa )x1()x1()x1(2 3332

smmUl 0)x1()x1()x1(2 3332

smmUl 0)xx1()x1( 24 Bit

ehtuenHeK)an

)z1()z1()z1(2

)y1()y1()y1(2

)x1()x1()x1(2

3332

3332

3332

KuNvismPaBenHGgnigGgeK)an

P)]z1)(y1)(x1[()]z1)(y1)(x1(2[ 33222 b (*)P)z1)(y1)(x1()z1)(y1)(x1(2 3222

Edl )z1)(y1)(x1(P 333 333333333333 zyx)xzzyyx()zyx(1

tamvismPaB GMAM eK)an

33333 6663 333 )xyz1(zyxzyx3zyx31P

tamTMnak;TMng (*) eK)an

)xyz1)(z1)(y1)(x1()z1)(y1)(x1(2 222 Bit.


46/231

vismPaBeRCIserIs


lMhat;TI20

cMeBaHRKb;cMnYnBitminGviCman d,c,b,a cUrRsayfa 22222 )abcd1()dd1)(cc1)(bb1)(aa1(4

ebI abcd vismPaBkayCa 2442 )a1()aa1(4

smmUl 422 a1)aa1(2

smmUl 0)a1()a1()aa1(2 4422 Bit

ehtuenH

422

422

b1)bb1(2

a1)aa1(2

eK)an )b1)(a1()bb1()aa1(4 442222

eday 4422444444 baba21ba)ba(1)b1)(a1(

b 22244 )ba1()b1)(a1(

eKTaj 2222222 )ba1()bb1()aa1(4

b )1(ba1)bb1)(aa1(2 2222

dUcKaEdr )2(dc1)dd1)(cc1(2 2222

ehIy 22222 )abcd1()dc1)(ba1(

eFIVplKuNvismPaB )1( nig )2( eK)an 22222 )abcd1()dd1)(cc1)(bb1)(aa1(4


47/231

vismPaBeRCIserIs


lMhat;TI21 ( Tournament of the Towns,1993)

ebIsmIkar 01bxx2axx 234 manbsy:agticmYyCacMnYnBit

enaHbgajfa 8ba 44 .

tag IRx0 Edl 0x0 Cabsrbs;smIkarenaHeK)an

01bxx2axx 02

03

04

0 b 22

003

0 )1x(bxax


)ba)(xx()bxax()1x( 222

06

0203

040

eKTaj2

06

0

42022

xx

)1x(ba

eyIgnwgRsayfa 8xx

)1x(2

06

0

420

eKman 1x4x6x4x)xx(8)1x( 02

03

04

02

06

042

0

b 0)1x()xx(8)1x( 402

06

042

0

eKTaj)an 8xx

)1x(2

06

0

420

Bit

dUcenH 8ba 44 .


48/231

vismPaBeRCIserIs


lMhat;TI22

cUrRsaybBaak;fa

)cba(4

9

)ba(

c

)ac(

b

)cb(

a222


tamvismPaB SchwarzCauchy 2

3322112

32

22

12

32

22

1 )bababa()bbb)(aaa(

ykba

ca,

ac

ba,

cb

aa 321

nig cb,bb,ab 321 eK)an 2

222 ba

c

ac

b

cb

a)cba(

)ba(

c

)ac(

b

)cb(

a

eday 23ba cac bcb a emIllMhat;TI5

eKTaj4

9)cba(

)ba(

c

)ac(

b

)cb(

a222

dUcenH)cba(4

9

)ba(

c

)ac(

b

)cb(

a222

.


49/231

vismPaBeRCIserIs


lMhat;TI23 ( JBMO 2002 Shortlist)

cUrRsaybBaak;fa ac

c

b

b

a

a

c

c

b

b

a 222

2

3

2

3

2

3


tamvismPaB SchwarzCauchy 2

3322112

32

22

12

32

22

1 )bababa()bbb)(aaa(

yk ca,ba,aa 321

niga

ccb,

c

bbb,

b

aab 321 eK)an

)1(a

c

c

b

b

a

a

c

c

b

b

a)cba(

2222

2

3

2

3

2

3

ebIeKyka

ca,

c

ba,

b

aa 321

nig ab,cb,bb 321 eK)an

)2()cba()cba(a

c

c

b

b

a 2222

KuNvismPaB )1( nig )2( Gg nig Gg rYcsRmYlkta 2)cba(

nigac

cb

ba

222

eK)anac

cb

ba

ac

cb

ba

222

2

3

2

3

2

3

Bit .

dUcenHa

c

c

b

b

a

a

c

c

b

b

a 222

2

3

2

3

2

3

.


50/231

vismPaBeRCIserIs


lMhat;TI24 ( Crux Mathematicorum)

eK[ z,y,x CabIcMnYnBitviCman . cUrRsaybBaak;fa

1)yz)(xz(z

z

)xy)(zy(y

y

)zx)(yx(x

x

eKman zxyzx2xyxzyzxyx)zx)(yx( 2

)zyx(xxzxyx)zx)(yx(x

xzxy)zx)(yx(

)xzxy()zx)(yx( 2

eKTaj )1(zyx

x

)zx)(yxx

x

RsaydUcKaEdr )2(zyx

y

)xy)(zy(y

y

nig )3(zyxz

)yz)(xz(z

z

eFIVplbUkvismPaB )3(&)2(),1( eK)an

1)yz)(xz(z

z

)xy)(zy(y

y

)zx)(yx(x

x


51/231

vismPaBeRCIserIs


lMhat;TI25

RKb;cMnYnBitviCman c,b,a cUrRsaybBaak;fa

ba

c

ac

b

cb

a4

c

ba

b

ac

a

cb

tamvismPaB GMAM RKb; 0y,0x eKman

xy2

yx

b xy4)yx( 2

b

y

1

x

1

xy

yx

yx

4

yka

bx nig

a

cy eK)an )1(

c

a

b

a

cb

a4

dUcKaEdr )2(a

b

c

b

ac

b4 nig )3(

b

c

a

c

ba

c4

eFIVplbUkvismPaB )3(&)2(),1( eK)an

b

ac

a

cb

c

ba

ba

c4

ac

b4

cb

a4

dUcenH

ba

c

ac

b

cb

a4

c

ba

b

ac

a

cb .


52/231

vismPaBeRCIserIs


lMhat;TI26 ( MOSP,2001)


cUrRsayfa )1cba(4)ac)(cb)(ba(

man )1(abc)ac)(cb)(ba()cabcab)(cba(


)2(abc8ac2.bc2.ab2)ca)(cb)(ba(

yk )1( CMnYskg )2( eK)an )cabcab)(cba(

9

8)ac)(cb)(ba(

edIm,Ibgajfa )1cba(4)ac)(cb)(ba(

eyIgRtUvRsayfa )1cba(4)cabcab)(cba(9

8 Bit

b 1cba

1)cabcab.(92

Bit


32

)cba(81

)cabcab(3

cba

1

9

cbcab

9

cabcab

b 1)cba(3)cabcab(

cba

1)cabcab(

9

23

2

Bit

eRBaH )cba(3)cba(abc3)cabcab( 2 .

dUcenH )1cba(4)ac)(cb)(ba( .


53/231

vismPaBeRCIserIs


lMhat;TI27 ( APMO,2004)

cUrbgajfa )cabcab(9)2c)(2b)(2a( 222


eKman )cabcab(3)cba( 2

edIm,Ibgajfa )cabcab(9)2c)(2b)(2a( 222

eyIgRtUvbgajfa 2222 )cba(3)2c)(2b)(2a(

eKman3

1u13

2u 22

enaHvismPaBGacsresr

9

)cba()

3

1c1)(

3

1b1)(

3

1a1(

2222

tamsmPaB xyyx1)y1)(x1(

ebI x nig y mansBaadUcKaenaH 0xy

ehtuenH yx1)y1)(x1( RKb;cMnYnBit y,x .

-ebI 0z,0y,0x b 0z,0y,0x eK)an

)1(zyx1)z1)(yx1()z1)(y1)(x1(

-ebI 0z1,0y1,0x1 enaH yx nig z

mansBaadUcKa ehIy zyx1)z1)(yx1(

eK)an zyx1)z1)(y1)(x1(

eyIgsikSavismPaBtambIkrNIdUcxageRkam


54/231

vismPaBeRCIserIs


!-ebI 1c,b,a enaH 03

1c,0

3

1b,0

3

1a 222

tamlkN )1( eK)an

3

3cba1)

3

1c1)(

3

1b1)(

3

1a1(

222222

3

cba)

3

1c1)(

3

1b1)(

3

1a1(

222222

eday 9)cba(

3

cba 2222

SchwarzCauchy

eK)an

9

)cba()

3

1c1)(

3

1b1)(

3

1a1(

2222

Bit

@-ebI 1c,b,a0 enaH 03

1c,0

3

1b,0

3

1a 222

tamlkNH )2( eK)an

3

3cba1)

3

1c1)(

3

1b1)(

3

1a1(

222222

9

)cba(

3

cba)

3

1c1)(

3

1b1)(

3

1a1(

2222222

Bit

#-ebI c,b,a manBIrEdlFMCag 1enaHeyIgsntfa 1c,1b,1a

eK)an 3 ba13 2ba1)3 1b1)(3 1a1(222222

enaH9

)2c)(ba1()

3

1c1)(

3

1b1)(

3

1a1(

222222


55/231

vismPaBeRCIserIs


eday 2222 )cba()11c)(ba1(

eK)an 9)cba(

)3

1c

1)(3

1b

1)(3

1a

1(

2222

Bit .$-cMeBaHkrNI EdlRtIFatu c,b,a manEtmYyKt;EdlFMCag 1sikSadUc

krNITI! .

srubcMeBaHRKb;cMnYnBitviCman c,b,a eK)an

)cabcab(9)2c)(2b)(2a( 222 .


56/231

vismPaBeRCIserIs


lMhat;TI28 ( Japan,1997)

eKyk c,b,a CabIcMnYnBitviCman . cUrbgajfa

5

3

c)ba(

)cba(

b)ac(

)bac(

a)cb(

)acb(22

2

22

2

22

2

tag22

2

22

2

22

2

c)ba(

)cba(

b)ac(

)bac(

a)cb(

)acb(T

ykc

baz,

b

acy,

a

cbx

eK)an 1cba

cba

z1

1

y1

1

x1

1

ehIykenSam T eTACa

1z

)1z(

1y

)1y(

1x

)1x(T

2

2

2

2

2

2

.

ebI 2zyx eK)an5

3T CakrNIEdlvismPaBkayCasmPaB .

eyIgsntfamanBIrcMnYnBit a nig b EdlnaM[ b1x

a

1x

)1x(2

2

Bit

RKb; 0x ehIyeday 11z

1

1y

1

1x

1

enaHedIm,I[5

3T

luHRtaEt5

3)b

1z

a()b

1y

a()b

1x

a(

naM[ (*)5

3b3a

ma:geTotedaysmPaBekIteLIgcMeBaH 2x enaH 2x CabsDb


57/231

vismPaBeRCIserIs


rbs;smIkar b1x

a

1x

)1x(2

2

b 0)bxba)(1x()1x)(1x( 22

tag )bxba)(1x()1x)(1x()x(f 22

2x CabsDbn 0)x(f enaH 0)2('f

eday )1x(b)bxba(x2)1x(2)1x()x('f 222

enaH0b5

5

347b5)b3a(461)2('f

eKTaj)an

25

23b ehIytam (*) eK)an

25

54b3

5

3a

eK)an)1x(25

54

25

23

1x

)1x(2

2

smmUl 0)1x)(1x(25

)7x()2x(22

2

Bit

ehtuenH)1(

)1x(25

54

25

23

1x

)1x(

2

2

)2(

)1y(25

54

25

23

1y

)1y(2

2

nig )3()1z(25

54

25

23

1z

)1z(2

2

bUkvismPaB )3(&)2(,)1( eK)an

5

3

25

15

25

5469)

1z

1

1y

1

1x

1(

25

543

25

23T

Bit

dUcenH5

3

c)ba(

)cba(

b)ac(

)bac(

a)cb(

)acb(22

2

22

2

22

2

.


58/231

vismPaBeRCIserIs


lMhat;TI29

eK[ t,z,y,x CabYncMnYnBitviCman . cUrRsayfa

1xt

t

tz

z

zy

y

yx

x 2222

.

tagz

tc,

y

zb,

x

ya nig

t

xd Edl 1abcd

vismPaBsmmUl 1)d1(

1

)c1(

1

)b1(

1

)a1(

12222


0)ab1()b1()a1(

)ab1()ba(ab

ab1

1

)b1(

1

)a1(

122

22

22

eKTaj )1(ab1

1

)b1(

1

)a1(

122

RsaydUcKaEdr )2(ab1

ab

cd1

1

)d1(

1

)c1(

122

bUkvismPaB )2(&)1( eK)an

1ab1

ab

ab1

1

)d1(

1

)c1(

1

)b1(

1

)a1(

12222

dUcenH 1xt

ttz

zzy

yyx

x 2222

.


59/231

vismPaBeRCIserIs


lMhat;TI30

eK[ c,b,a nig z,y,x CacMnYnBit . cUrRsaybBaak;fa 2222222 )abzcaybcx(3)zc)(yb)(xa(4


])bzcy(cb)[ax()]bzcy(a)bc(x[ 222222

b 222222 )abzacybcx(])bzcy(cb)[xa(

edIm,IRsayfa2222222

)abzcaybcx(3)zc)(yb)(xa(4

eyIgRKan;EtRsayfa ])bzcy(cb[3)zc)(xb(4 2222222

b 0])bzcy(cb[3)zc)(xb(4 2222222

b 0bcyz4bcyz2zy4yczbcb 22222222

b 0)yz2bc()bzcy( 22 Bit

dUcenH 2222222 )abzcaybcx(3)zc)(yb)(xa(4


60/231

vismPaBeRCIserIs


lMhat;TI31

eK[bIcMnYnBit 1c,1b,1a . cUrRsaybBaak;fa

2ba1

c1

ac1

b1

cb1

a12

2

2

2

2

2

.

yk2

2

2

2

2

2

ba1

c1

ac1

b1

cb1

a1T

eKman 0b1 nig 0c2 enaH 0b1cb1 2

ehIy 0)b1( 2

b2b1b

2

naM[2

)c1(2b1c1

2

b1cb1

222

22

eKTaj)an )1()c1(2b1

)a1(2

cb1

a122

2

2

2

RsaydUcKa )2()a1(2c1

)b1(2

ac1

b122

2

2

2

nig )3()b1(2a1

)c1(2

ba1

c122

2

2

2

bUkvismPaB )2(,)1( nig )3( Gg nig GgeK)an

y2xz2

x2zy2

z2yx2T

Edl 222 c1z,b1y,a1x .


61/231

vismPaBeRCIserIs



)zxyzxy(3

)zyx(2

yz2xz

z2

xy2yz

y2

xz2xy

x2

T

2222

eday )zxyzxy(3)zyx( 2 enaH 2T

dUcenH 2ba1

c1

ac1

b1

cb1

a12

2

2

2

2

2

.


62/231

vismPaBeRCIserIs


lMhat;TI32

eK[bIcMnYnBitviCman c,b,a Edl 1abc . cUrRsaybBaak;fa

8

3

)c1(

1

)b1(

1

)a1(

1333

.

tag333 )c1(

1

)b1(

1

)a1(

1T

tamvismPaB Holder

)ccc)(bbb)(aaa( 333231333231333231 3333222111 )cbacbacba(

ykc1

1ba,

b1

1ba,

a1

1ba 332211

ehIy 1ccc 321 eK)an 3

222

2

)c1(

1

)b1(

1

)a1(

1T3

b3

222

2

)c1(

1

)b1(

1

)a1(

1

3

1T

edIm,IRsayfa8

3T eyIgRtUvRsayfa

64

9

8

3

)c1(

1

)b1(

1

)a1(

1

3

1 23

222

b4

3

)c1(

1

)b1(

1

)a1(

1222


63/231

vismPaBeRCIserIs


eKman

0)ab1()b1()a1(

)ab1()ba(ab

ab1

1

)b1(

1

)a1(

1

22

22

22

eKTaj)anab1

1

)b1(

1

)a1(

122

ehIy2222 )c1(

1

ab1

1

)c1(

1

)b1(

1

)a1(

1

eyIgnwgRsayfa 43

)c1(

1

ab1

1

2

eday 1abc

eK)an 0)c1(4

)1c(

4

3

)c1(

1

c1

c

4

3

)c1(

1

ab1

1 2

22

eKTaj4

3

)c1(

1

ab1

12

Bit

dUcenH 83

)c1(

1

)b1(

1

)a1(

1333

.


64/231

vismPaBeRCIserIs


lMhat;TI33 USAMO 2011

eK[bIcMnYnBitviCman c,b,a Edl 4)cba(cba 2222 .

cUrRsaybBaak;fa 3)ac(

1ca

)cb(

1bc

)ba(

1ab222

.

eKman 4)cba(cba 2222

2cabcabcba

4)cabcabcba(2

222

222

eK)an ab22)caab()bcc(baab2 222

)1ab(2)ac)(cb()ba( 2

eKTaj )1()ba(2

)ac)(cb(

2

1

)ba(

1ab22

RsaydUcKaEdreK)an )2()cb(2

)ca)(ba(

2

1

)cb(

1bc22

nig )3()ac(2

)cb)(ba(

2

1

)ac(

1ca22

bUk )3(&)2(),1( eK)an T2

3

)ac(

1ca

)cb(

1bc

)ba(

1ab222

Edl 2

3

)ac(2

)cb)(ba(

)cb(2

)ca)(ba(

)ba(2

)ac)(cb(T

222

dUcenH 3

)ac(

1ca

)cb(

1bc

)ba(

1ab222

.


65/231

vismPaBeRCIserIs



eK[bIcMnYnBitviCman b,a nig c .

cUrRsayfa 3252525 )cba()3cc)(3bb)(3aa(

cMeBaHRKb; 0x eyIgnwgRsayfa (*)2x3xx 325

eK)an 0)1x)(1x(1xxx 32235

Bit

tag )3cc)(3bb)(3aa(P 252525

tamvismPaB (*) eK)an )2c)(2b)(2a(P 333

edIm,IRsayfa 3)cba(P eyIgRKan;EtRsay[eXIjfa 3333 )cba()2c)(2b)(2a(

eday3

1t1

3

2t 33 enaHvismPaBGacsresrCa

27)cba()

31c1)(

31b1)(

31a1(

3333

eKmansmPaB uvvu1)v1)(u1(

ebI u nig v CaBIrcMnYnBitmansBadUcKaenaHplKuN 0uv

ehtuenH vu1)v1)(u1(

cMeBaH 1w

eK)an )w1)(vu1()w1)(v1)(u1(

-ebI 0w,v,u enaH wvu1)w1)(vu1(

eKTaj (**)wvu1)w1)(v1)(u1(


66/231

vismPaBeRCIserIs


-ebI 0w,v,u enaH wvu1)w1)(vu1(

eKTaj *)*(*wvu1)w1)(v1)(u1(

edIm,IRsayfa27

)cba()

3

1c1)(

3

1b1)(

3

1a1(

3333

Bit

eyIgRtUvsikSakrNIdUcxageRkam

-krNITI1 ebI 1c,b,a b 1c,b,a enaHtam (**)nig *)*(*

eKGacsresr

3

3cba1)

3

1c1)(

3

1b1)(

3

1a1(

333333

3

cba)

3

1c1)(

3

1b1)(

3

1a1(

333333

tamvismPaB Hoder eKman 3333 )cba()cba(9

eKTaj)an 27 )cba()3 1c1)(3 1b1)(3 1a1(3333

Bit-krNITI2 ebIbIcMnYn c,b,a manmYytUcCag 1nig BIrcMnYneTotFMCag 1

sntfa 1c nig 1b,a enaHeK)an

3

1ba

3

2ba1)

3

1b1)(

3

1a1(

333333

ehIy3

c11

3

2c

3

1c1

333

enaHtamvismPaB Hoder

27

)cba(

3

111.

3

c11.

3

1ba 3333


67/231

vismPaBeRCIserIs


ehtuenH27

)cba()

3

1c1)(

3

1b1)(

3

1a1(

3333

Bit

srubmk3252525

)cba()3cc)(3bb)(3aa(



68/231

vismPaBeRCIserIs



cUrRsayfacMeBaHRKb;cMnYnBitviCman c,b,a eKmanvismPaB

abc

1

abcac

1

abccb

1

abcba

1333333

.

eKman 0)ba)(ba( 22 RKb;cMnYnBitviCman a nig b

eK)an 2233 abbaba

)cba(ababcba

abcabbaabcba

33

2233

eKTaj )1(cba

c.

abc

1

)cba(ab

1

abcba

133

RsaydUcKaEdr )2(cba

a.

abc

1

abccb

133

nig)3(

cba

b.

abc

1

abcac

1

33

bUkvismPaB )3(&)2(),1( GgnigGgeK)an

abc

1

abcac

1

abccb

1

abcba

1333333

.


69/231

vismPaBeRCIserIs



eKyk nm

1n1m

nm

nma

Edlm

nign

CacMnYnKt;viCman .cUrRsayfa nmnm nmaa .

eyIgeRCIserIs 1m

ax nig 1

n

ay eday

nm

1n1m

nm

nma

enaH nmn

nm

1m

n

.nx

nig nmm

nm

1n

m

.my

tamvismPaB Bernoullie eKman mx1)x1( m

eK)annm

nm

nm

mn.n1)1

m

a1(

naM[ )1(nm

mn.nmma

nm

nmmm

ehIy ny1)y1( n

eK)annm

mn

nm

nm.m1)1

n

a1(

naM[ )2(nm

nm.nmna

nm

nmnn

bUkvismPaB )2(&)1( eK)annmnm

nmaa

BitdUcenH nmnm nmaa .


70/231

vismPaBeRCIserIs


lMhat;TI37

eK[ z;y;x CabIcMnYnBitviCmanEdl 1xyz .

cUrRsaybBaak;fa

2xxzz

xz

zzyy

zy

yyxx

yx6336

99

6336

99

6336

99

.

RsaybBaak;fa

2xxzz

xz

zzyy

zy

yyxx

yx6336

99

6336

99

6336

99

eyIgman ))(( 63363399 yyxxyxyx

)(2))(( 333363363399 yxyxyyxxyxyx

eKTaj6336

333333

6336

99 )(2

yyxx

yxyxyx

yyxx

yx

tamvismPaB GMAM eKman 336336 3 yxyyxx

eK)an )(3

2)(2 336336

3333

yxyyxx

yxyx

eKTaj )(3

2 33336336

99

yxyxyyxx

yx

b )1()(3

1 336336

99

yxyyxx

yx

RsaydUcKaEdreK)an )2()(3

1 336336

99

zyzzyy

zy


71/231

vismPaBeRCIserIs


)3()(3

1 336336

99

xzxxzz

xz

eFIVplbUk )2(;)1( nig )3( Gg nig Gg eK)an )(

3

2 3336336

99

6336

99

6336

99

zyxxxzz

xz

zzyy

zy

yyxx

yx

tamvismPaB GMM

eK)an 33333

xyzzyx eRBaH 1xyz

dUcenH 2633699

6336

99

6336

99

xxzz

xz

zzyy

zy

yyxx

yx

.


72/231

vismPaBeRCIserIs


lMhat;TI38

eK[ c,b,a CacMnYnviCmanEdl 3cabcab .

cUrbgajfaabc

1

)ba(c1

1

)ac(b1

1

)cb(a1

1222

tamvismPaB GMAM eKman 3 2)abc(

3

cabcab1 b 1abc .

eKman )abc1(a3

1

)bc3(a1

1

)acab(a1

1

)cb(a1

12

eday 1abc b 0abc1 enaH )1(a3

1

)cb(a1

12

dUcKaEdr )3(c3

1

)ba(c1

1,)2(

b3

1

)ac(b1

122

bUkvismPaB )2(;)1( nig )3( eK)an

abc3

cabcab

c3

1

b3

1

a3

1

)ba(c1

1

)ac(b1

1

)cb(a1

1222

eday 3cabcab dUcenH

abc

1

)ba(c1

1

)ac(b1

1

)cb(a1

1222

.


73/231

vismPaBeRCIserIs


lMhat;TI39 (Austrian Olympiad 1989)eK[ a nig b CaBIrcMnYnBitminGviCmanEdl 4ba 22 .

cUrRsayfa 122ba

ab

.

eyIgman ab24ab2ba)ba( 222

eKTaj )2ba)(2ba(4)ba(ab2 2

naM[ 22ba

2ba

ab

)1(12

ba

2ba

ab

eKman )ba(2)ba()ba( 2222

eKTaj 8)ba(2)ba()ba(2)ba( 222222

naM[ 22ba b )2(22ba

tam )1( nig )2( eKTaj)an 12

2ba

ab

.


74/231

vismPaBeRCIserIs


lMhat;TI40 (Selection test for JBMO 2007)

cUrRsayfa|)xz)(zy)(yx(|

4

3xyz

3

zyx 333

cMeBaHRKb; 0z;y;x .

|)xz)(zy)(yx(|4

3xyz

3

zyx 333

tag |)xz)(zy)(yx(|p

eKmanklkNHPaB)i()zxyzxyzyx)(zyx(xyz3zyx 222333

cMeBaHRKb; 0z;y;x eKman

|xz|xz,|zy|zy,|yx|yx eK)an |xz||zy||yx|)zyx(2

tamvismPaB GMAM eK)an )1(p3)zyx(2 3

ma:geToteKmansmPaB

222222)xz()zy()yx(

2

1zxyzxyzyx


)2(p2

3zxyzxyzyx 3 2222


75/231

vismPaBeRCIserIs


eFIVviFIKuNvismPaB )1( nig )2( eKTTYl)an

)ii(p4

9)zxyzxyzyx)(zyx(

p2

9)zxyzxyzyx)(zyx(2

222

222

tam )i( nig )ii( eK)an p4

9xyz3zyx 333

eKTaj p4

3xyz

3

zyx 333 eday |)xz)(zy)(yx(|p

dUcenH |)xz)(zy)(yx(|4

3xyz

3

zyx 333 .


76/231

vismPaBeRCIserIs


lMhat;TI41 RMC 2000

eK[ c,b,a CaRbEvgRCugnRtIekaNmYy .

cUrRsayfa 3cba

c

cba

b

cab

a

.

tag

zcba

ycba

xcab

enaH2

yxc;

2

xzb;

2

zya

yk cbac

cba

b

cab

aT

3)222(2

1)

z

x

x

z()

y

z

z

y()

y

x

x

y(

2

1

z

y

z

x

y

x

y

z

x

z

x

y

2

1

z2

yx

y2

xz

x2

zy

eRBaH 2z

x

x

z;2

y

z

z

y;2

y

x

x

y

dUcenH 3cba

c

cba

b

cab

a

.


77/231

vismPaBeRCIserIs


lMhat;TI42 RMC 2004

eK[ c,b,a CabIcMnYnBitEdlepgpat; 3cba 222 .

cUrbgajfa 4abc|c||b||a| .

tamvismPaB Cauchy-Schwarz eK)an

9)cba(3|c||b||a| 2222

eKTaj )1(3|c||b||a|

tamvismPaB GMAM eK)an 3 2222 )abc(3cba

eKTaj 13

cba)abc(

32222

naM[ 1abc1

eKTaj)an )2(1abc

eFIVplbUkvismPaB )1( nig )2( eK)an 4abc|c||b||a| .


78/231

vismPaBeRCIserIs


lMhat;TI43 JBMO 2003


cUrbgajfacabcab

6

cba

31

.

eyIgtagz

1c;

y

1b;

x

1a enaHvismPaBGacsresr

zyx

6

zxyzxy

31

eRBaH 1abc

1xyz

eyIgman 0)xz()zy()yx( 222 eKTaj zxyzxyzyx 222

EfmGgTaMgBIrnwg zx2yz2xy2 eK)an

)zxyzxy(3)zyx( 2

naM[ zyx6

)zyx(

91

zxyzxy

31

2

eRBaH 0zyx

31

)zyx(

9

zyx

61

2

2

dUcenHcabcab

6

cba

31

.


79/231

vismPaBeRCIserIs


lMhat;TI44

eK[ 1a nig 1b .

cUrbgajfa )2

ba(log2blogalog 222

.

cMeBaH 1a nig 1b

eyIgman b.a2ba

naM[)blogalog(

2

1)

2

ba(log

222

)1()2

ba(log2blogalog 222

ma:geToteKman

blog.alog2blogalog 2222

)2()blogalog(2

1blogalog

)blogalog()bloga(log2blogblog.alog2alog)bloga(log2

2

2222

2

2222

222222

tamTMnak;TMng )1( nig )2( eyIgTaj


80/231

vismPaBeRCIserIs


)2

ba(log2blogalog

)2

ba(log4)blogalog(

)2

ba(log2)blogalog(

2

1

222

22

22

2

2

22

dUcenH )2

ba(log2blogalog 222

.


81/231

vismPaBeRCIserIs


lMhat;TI45

eK[ z,y,x CabIcMnYnBitviCmanEdlepgpat; 1xyz .

cUrbgajvismPaB

zyxy

)1xz(

x

)1zy(

z

)1yx( 222 .

rebobTI1 tamvismPaB GMAM eK)an

)3()1xz(2|1xz|2yy

)1xz(

)2()1zy(2|1zy|2xx

)1zy(

)1()1yx(2|1yx|2zz

)1yx(

2

2

2

bUkvismPaB )3(,)2(,)1( GgnwgGgeK)an 6)zyx(3

y

)1xz(

x

)1zy(

z

)1yx( 222

eday 3xyz3zyx 3 eRBaH 1xyz

eK)an 6)zyx(2

b 06)zyx(2

b zyx6)zyx(3

dUcenH zyxy

)1xz(

x

)1zy(

z

)1yx( 222


82/231

vismPaBeRCIserIs


rebobTI2

)zyx(y

)1xz(

x

)1zy(

z

)1yx(T

222

edayeRbIvismPaB

cba

)zyx(

c

z

b

y

a

x 2222

eK)an

zyx

)1zyx)(3zyx(3T

zyx

)zyx()3z2y2x2(T

zyx

)zyx()1xz1zy1yx(T

)zyx(zyx

|1xz||1zy||1yx|T

22

22

2

tamvismPaB GMAM

eKman 3xyz3zyx 3 eRBaH 1xyz

eKTaj)an 03zyx nig 21zyx

ehtuenH 0zyx

)1zyx)(3zyx(3T

Bit

dUcenH zyxy

)1xz(x

)1zy(z

)1yx( 222

.


83/231

vismPaBeRCIserIs


lMhat;TI46 BMO 2010

eK[ c,b,a CacMnYnBitviCman .

cUrbgajfa 0ac

)ba(ac

cb

)ac(cb

ba

)cb(ba 222

.

bgajfa

0ac

)ba(ac

cb

)ac(cb

ba

)cb(ba 222

eKman

abcba

ca.ab

ba

)ba(abc)ca(ab

ba

)cabbca()cabba(

ba

bcaba

ba

)cb(ba

22

222222222

eK)an)1(abc

ba

ca.ab

ba

)cb(ba 22

RsaybMPWdUcKaEdreK)an )2(abc

cb

abbc

cb

)ac(cb 22

nig )3(ac

bcca

ac

)ba(ac 22

bUksmPaB )2(,)1( nig )3( GgnigGgeK)an

abc3acbcca

cbabbc

babcabT 222

Edlac

)ba(ac

cb

)ac(cb

ba

)cb(baT

222


84/231

vismPaBeRCIserIs



abc3

ac

bcca

cb

abbc

ba

bcab 222

b 0abc3ac

bcca

cb

abbc

ba

bcab 222

dUcenH 0ac

)ba(ac

cb

)ac(cb

ba

)cb(ba 222

.


85/231

vismPaBeRCIserIs


lMhat;TI47

eK[ c,b,a CabIcMnYnBitviCman . cUrbgajfa 2

3333

5555

)cba(9

10

)cba(cba

)cba(cba

eKmansmPaB

)cabcabcba)(ac)(cb)(ba(5)cba(cba

)ac)(cb)(ba(3)cba(cba

2225555

3333

eK)an

)cabcabcba(3

5

)cba(cba

)cba(cba 2223333

5555

eyIgnwgRsayfa 2222 )cba(9

10)cabcabcba(

3

5

b

2222 )cba(2)cabcabcba(3

b cabcabcba 222

tamvismPaB GMAM eKman 222

222222

cba2

ac

2

cb

2

bacabcab Bit

dUcenH2

3333

5555

)cba(9

10

)cba(cba

)cba(cba

.


86/231

vismPaBeRCIserIs


lMhat;TI48

cMeBaHRKb;cMnYnBit)

4

,0(x

cUrbgajfa xsinxcos )x(sin)x(cos .

tamvismPaB Bernoulli cMeBaHRKb;cMnYn x nig a Edl 1x nig 1a

eyIgman ax1)x1( a .

ehtuenHcMeBaH4

x0

eK)an

xsin

xcos

xsin

xcos

xsin

xcos

2 )xsin1.()xsin1()x(cos

eday xcos1)xsin1( xsinxcos

nig xcos1)xsin1( xsinxcos

eK)an xsin)xcos1)(xcos1()x(cos 2xsinxcos

2

eKTaj xsin)x(cos xsin

xcos

xsinxcos

xsin

xcos

)xln(sin)xln(cos

)xln(sinxsin)xln(cosxcos

)xln(sin)xln(cosxsin

xcos

)xln(sin)xln(cos

dUcenH xsinxcos )x(sin)x(cos .


87/231

vismPaBeRCIserIs


lMhat;TI49

eK[ d,c,b,a CabIcMnYnBitEdlepgpat;vismPaB 22222 )1bdac()1dc)(1ba(

cUrbgajfa 1ba 22 nig 1dc 22 .

tag 22 ba1x nig 22 dc1y

eyIg]bmafa 0x nig 0y

vismPaB22222

)1bdac()1dc)(1ba(

smmUl 2)1bdac(xy

b 2)2bd2ac2(xy4

eday 2222 dcba2yx

enaH yxbd2ac2dcba2bd2ac2 2222

]yx)db()ca[( 22

eKTaj 2222 )yx()]yx()db()ca([xy4

b 22 yxy2xxy4

b 0)yx( 2 minBit . naM[kar]bmaxagelIpyBIkarBit .

dUcenHeKTaj 0x

nig 0y

naM[ 1ba 22

nig 1dc 22

.


88/231

vismPaBeRCIserIs


lMhat;TI50

eK[ c,b,a CabIcMnYnBitviCmanEdl 1cabcab .

cUrkMnt;tmGb,brmankenSamac

c

cb

b

ba

aE

222

.

kMnt;tmGb,brmankenSam E

eKmanba

aba

ba

ababa

ba

a 22

tamvismPaB GMAM

eKman ab2ba

eK)an

ab2

1

ba

1

b2

ab

ba

ab

eKTaj )1(2

aba

ba

aba

ba

a2

dUcKaEdr )3(2

cac

ac

c;)2(

2

bcb

cb

b 22

bUkvismPaB )3(&)2(;)1( eKTTYl)an

2

1cba

2

cabcabcbaE


1cabcabcba

eKTaj2

1

2

11E . dUcenHtmGb,brman E KW

2

1E

min .


89/231

vismPaBeRCIserIs


lMhat;TI51

eK[ n21 x...,,x,x Edl 2n CacMnYnBitviCmanEdlepgpat;

1998

1

1998x

1....

1998x

1

1998x

1

n21

cUrbgajfa 19981n

x...x.xn n21

.

eKman1998

1

1998x

1....

1998x

1

1998x

1

n21

b 11998x 1998....1998x 19981998x 1998 n21 tag

1x

1998y

i

i

eK)an 1y....yy n21

eKTaj

ijii )y(y1 Edl ni1 nig n1

tamGMAM

eKman

ij 1n ij ii

)y()1n()y(

eK)an 1n

ijii )y()1n(y1

eKTaj

n

1ii

nn

1ii )y()1n()y1( b

n

1i

n

i

i )1n()y

y1(

Et 1998x

y

y1 i

i

i

enaHn

n

n21 )1n(1998

x...xx

dUcenH 1998

1n

x...x.xn n21

.


90/231

vismPaBeRCIserIs


lMhat;TI52

eK[ z,y,x CabIcMnYnBitviCmanEdl zyxxyz .

cUrRsayfaxyz2

27

y1

xz

x1

zy

z1

yx222

.

bgajfaxyz2

27

y1

xz

x1

zy

z1

yx222

tag222 y1

xz

x1

zy

z1

yxT

b)xz(yxz

)xz(

)zy(xzy

)zy(

)yx(zyx

)yx(T

2

2

2

2

2

2


n321

2n321

n

2

n

3

2

3

2

2

2

1

2

1

b...bbb

)a...aaa(

b

a....

b

a

b

a

b

a

eK)an

)xz)(zy)(yx()zyx(4T

xyzyzxyxyzxzxyz2

)zyx(4T

)xz(y)zy(x)yx(z)zyx(2

)xz()zy()yx(T

2

222222

2

222

2


91/231

vismPaBeRCIserIs



)xz)(zy)(yx(27)zyx(8

)xz)(zy)(yx(3)zyx(2

)xz)(zy)(yx(3)xz()zy()yx(

3

3

3

eKTajxyz2

27

)zyx(2

27

)xz)(zy)(yx(

)zyx(4 2

naM[xyz2

27T

dUcenHxyz2

27

y1

xz

x1

zy

z1

yx222

.


92/231

vismPaBeRCIserIs


lMhat;TI53

cUrbgajfaebI 8abc nig 0c,b,a enaHeK)an

3

4

)a1)(c1(

c

)c1)(b1(

b

)b1)(a1(

a33

2

33

2

33

2

tag)a1)(c1(

c

)c1)(b1(

b

)b1)(a1(

aT

33

2

33

2

33

2

tamvismPaB GMAM eKman2222

23

2

a2

2

aa1a1)aa1)(a1(a1

eKTaj2

2aa1

23 .

ehtuenH )1()2b)(2a(

a4

)b1)(a1(

a22

2

33

2

RsaydUcKaEdreK)an

)3()2a)(2c(

c4

)a1)(c1(

c

)2()2c)(2b(

b4

)c1)(b1(

b

22

2

33

2

22

2

33

2


93/231

vismPaBeRCIserIs


bUkvismPaB )3(,)2(,)1( GgnwgGgeK)an

8)c2b2a2accbba(2cba

)c2b2a2accbba(4

)2c)(2b)(2a(

)]2b(c)2a(b)2c(a[4

)2a)(2c(

c4

)2c)(2b(

b4

)2b)(2a(

a4T

222222222222

222222222

222

222222

22

2

22

2

22

2

eday 8abc

enaHeK)an

36t

t2

36)c2b2a2accbba(

)c2b2a2accbba(2T

222222222

222222222

Edl 222222222 c2b2a2accbbat


72cba6cba3t 3 2223 444

eRBaH )8abc

eK)an

2

3

2

11

t

361 b

2

3

t

36t naM[

3

2

36t

t

ehtuenH3

4

36t

t2T

Bit .

dUcenH

34

)a1)(c1(

c

)c1)(b1(

b

)b1)(a1(

a33

2

33

2

33

2

.


94/231

vismPaBeRCIserIs


lMhat;TI54

eK[RtIekaN ABC mYymanRCug cAB,bAC,aBC ehIy

manmMukgCamMuRsYc .

cUrRsayfa )cba(4Bcos

ac

Acos

cb

Ccos

ba .

bgajfa )cba(4Bcos

ac

Acos

cb

Ccos

ba

tag Bcosac

Acos

cb

Ccos

ba

Bcos)ac(

)ac(

Acos)cb(

)cb(

Ccos)ba(

)ba( 222

tamvismPaBn21

2n21

n

2

n

2

2

2

1

2

1

y...yy

)x...xx(

y

x....

y

x

y

x

eK)an Bcos)ac(Acos)cb(Ccos)ba()]ac()cb()ba[( 2

)AcosbBcosa()CcosaAcosc()BcoscCcosb(

)cba(4 2

)cba(4cba

)cba(4 2

dUcenH )cba(4Bcosac

Acos

cb

Ccos

ba

.


95/231

vismPaBeRCIserIs


lMhat;TI55

eK[ 0z,y,x Edl 1zyx .

cUrRsayfa4

1

)z1(

z

)y1(

y

)x1(

x2

3

2

3

2

3

.

Rsayfa4

1

)z1(

z

)y1(

y

)x1(

x2

3

2

3

2

3

eKBinit2

232

2

3

)x1(

)xx2()xx2x(

)x1(

x

2

2

2

22

2

22

2

2

)x1(4

)1x3(

4

1

x)x1(4

)x1()1x3(

x

)x1(4

)xx21()1x6x9(x

)x1(

xx2x

eday 0)x1(4

)1x3(2

2

enaH )1(4

1x

)x1(

x2

3

dUcKaEdr )2(4

1y

)y1(

y2

3

nig )3(4

1z

)z1(

z2

3

bUkvismPaB)3(,)2(,)1(Gg nig GgeK)an

4

1

4

3zyx

)z1(

z

)y1(

y

)x1(

x2

3

2

3

2

3

Bit .


96/231

vismPaBeRCIserIs


lMhat;TI56

eK[ d,c,b,a CacMnYnBitviCmanEdl 1abcd .

ebIeKdwgfaa

d

d

c

c

b

b

adcba enaHcUrbgajfa

d

a

c

d

b

c

a

bdcba .

bgajfad

a

c

d

b

c

a

bdcba

eKmanadd

cdc

bcb

aba

ad

dc

cb

badcba

2222

dacdbcab

)dcba(dcba

2

naM[ )1(cbadacdbcab

ma:geTot dab)ab(

cda

)ad(

bcd

)cd(

abc

)bc(

d

a

c

d

b

c

a

b

2

2

2

2

2

2

2

2

(*)

c

b

b

a

a

d

d

c

)abdacdbc(

c

b

)ab(

b

a

)ad(

a

d

)cd(

d

c

)bc(

2

2222

tam )1( eK)an )2()dcba()abdacdbc( 22

tamsmtikma

d

d

c

c

b

b

adcba


97/231

vismPaBeRCIserIs


eKTaj )3(dcba

1

a

d

d

c

c

b

b

a

1

KuNvismPaB )2( nig )3( GgnigGgeK)an

(**)dcba

a

d

d

c

c

b

b

a

)abdacdbc( 2

tam (*) nig (**)eK)an dcbad

a

c

d

b

c

a

b

dUcenHd

a

c

d

b

c

a

bdcba .


98/231

vismPaBeRCIserIs


lMhat;TI57 Czech and Slovak Republics 2005

eKyk c,b,a CacMnYnviCmanEdlepgpat; 1abc .

cUrbgajfa4

3

)1a)(1c(

c

)1c)(1b(

b

)1b)(1a(

a

.

tag)1a)(1c(

c

)1c)(1b(

b

)1b)(1a(

aT

cabcabcba2

21

cabcabcba2

cabcabcba

abccabcabcba1cabcabcba

)1c)(1b)(1a(

)1b(c)1a(b)1c(a


8cba8cabcabcba11 8 333

eKTaj

8

1

cabcabcba2

1

naM[4

3

8

21T

dUcenH4

3

)1a)(1c(

c

)1c)(1b(

b

)1b)(1a(

a

.


99/231

vismPaBeRCIserIs


lMhat;TI58 APMO 1998

eKyk c,b,a CacMnYnviCman.

cUrbgajfa )abc

cba1(2)

a

c1)(

c

b1)(

b

a1(

3

.

vismPaBenHsmmUl

3

3

abc

)cba(2

a

b

b

c

c

a

a

c

c

b

b

a

)abc

cba1(2

abc

abc)

a

b

b

c

c

a()

a

c

c

b

b

a(1

tag 333 zc,yb,xa eK)an

xyz

)zyx(2

x

y

y

z

z

x

x

z

z

y

y

x 333

3

3

3

3

3

3

3

3

3

3

3

3


)3(xy

z31

y

z

x

z

)2(zx

y31

x

y

z

y

)1(yzx31

zx

yx

2

3

3

3

3

2

3

3

3

3

2

3

3

3

3

6xy.

yz.

zx.

xz.

zy.

yx6

xy

yz

zx

xz

zy

yx 6

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

3

b )4(3)x

y

y

z

z

x

x

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2

13

3

3

3

3

3

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3

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3


100/231

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bUkvismPaB )3(,)2(,)1( nig )4( eK)an

xyz

)zyx(3)

xy

z

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y

yz

x(3)

x

y

y

z

z

x

x

z

z

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2

3 333222

3

3

3

3

3

3

3

3

3

3

3

3

eKTaj

xyz

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x

y

y

z

z

x

x

z

z

y

y

x 333

3

3

3

3

3

3

3

3

3

3

3

3

Bit

dUcenH )abc

cba1(2)

a

c1)(

c

b1)(

b

a1(

3

.


101/231

vismPaBeRCIserIs


lMhat;TI59 Romania 2002

eKyk c,b,a CacMnYnBitncenaH )1,0( .

cUrbgajfa 1)c1)(b1)(a1(abc .

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,0(

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eday )2

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)yxcos(zsinysinxsinzcosycosxcos

eday 1)yxcos( enaH 1zsinysinxsinzcosycosxcos Bit

dUcenH 1)c1)(b1)(a1(abc .


102/231

vismPaBeRCIserIs


lMhat;TI60 Poland 2006

eKyk c,b,a CacMnYnBitviCmanEdl abccabcab

cUrbgajfa 1)ac(ca

ac

)cb(bc

cb

)ba(ab

ba33

44

33

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a

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a

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2

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1zyxxz

xz

zy

zy

yx

yx33

44

33

44

33

44

Bit .


103/231

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lMhat;TI61 Kazakhstan 2008

eKyk z,y,x CacMnYnBitviCmanEdl 1xyz

cUrbgajfa2

3

yxy

1

xzx

1

zyz

1

.

taga

cz;

c

by;

b

ax enaH 1xyz

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ba

c

ac

b

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tag bcacc

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b

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ba

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222

222222

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3T Bit .

dUcenH2

3

yxy

1

xzx

1

zyz

1

.


104/231

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lMhat;TI62 Korea 2000

cMnYnBit z,y,x,c,b,,a epgpat; 0cba nig 0zyx

cUrbgajfa

4

3

)bxay)(byax(

zc

)azcx)(axcz(

yb

)cybz)(czby(

xa 222222

.

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X.

2

1

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xa

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xa2222

2222

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1

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XZ

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2

1

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zc

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222222

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Z

XZ

Y

ZY

X(

2

1T

.

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b )i()XZ)(ZY)(YX(2

3ZYX 3


105/231

vismPaBeRCIserIs


ehIy )ii()XZ)(ZY)(YX(

3

YX

1

XZ

1

ZY

13

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2

33

2

9

YX

Z

XZ

Y

ZY

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2

91

YX

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9

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XZ

ZYX

ZY

ZYX

eKTaj43T Bit

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4

3

)bxay)(byax(

zc

)azcx)(axcz(

yb

)cybz)(czby(

xa 222222

.


106/231

vismPaBeRCIserIs


lMhat;TI63 Ireland 2002

cUrbgajfa 33

xyz1

xyz3

z1

z

y1

y

x1

x

cMeBaHRKb; 1z,y,x0 .

tagGnuKmn_ 1x1

1

x1

x)x(f

Edl 1x0

eK)an2

)x1(

1)x('f

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2)x(''f

3

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eK)an

)3

zyx(f3)z(f)y(f)x(f

eday 0)x1(

1)x('f

2

naM[ )x(f CaGnuKmn_ekIn .

tamvismPaB GMAM eKman 3 xyz3

zyx

eKTaj )xyz(f)3

zyx(f 3 ehtuenH )xyz(f3)z(f)y(f)x(f 3

dUcenH3

3

xyz1

xyz3

z1

z

y1

y

x1

x

.


1

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