eroberogeday lwm pl:ún briBaØabR&tKNitviTüa nig BaNiC¢kmμ€¦ · Mathematical Olympiad in...

eroberogeday lwm pl:ún briBaØabR&tKNitviTüa nig BaNiC¢kmμ

Problems and Solutions

sñaédeá£Bum<pßay

esovePAKNitviTüa ½ 1-dMeNa£RsaylMhatKNitviTüa eá£Bum< qñaM2000 ( sRmabeRtomRbLgcUlsaklviTüal&y nig GaharUbkrN_ ) 2-BiPBsVIútcMnYnBit ( sRmabfñak´TI11 nig sisßBUEkKNitviTüa ) 3- GnuKmn_RtIekaNmaRt ( sRmabfñak´TI11 nig sisßBUEkKNitviTüa )

4- dMeNa£RsayKMrU cMnYnkMupøic lImIt edrIev ( sRmabfñak´TI12) 5- segçbrUbmnþKNitviTüa ( sRmabfñak´TI11-12 ) 6- KMrUsikßaGnuKmn_ ( sRmabfñak´TI11-12 ) 7- kMEnlMhatKNitviTüafñakTI10kmµviFIsikßafµI ( PaK1 qñaM2008 ) 8- 151 KNnalImIt ( sRmabfñak´TI11-12 )

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esovePA KNitviTüaCMuvijBiPBelak PaK2 EdlGñksikßakMBugkan enAkñúgéden£ xMJúáTáneroberogeLIgkñúgeKalbMngTukCaÉksar sRmab CaCMnYydl´GñksikßaykeTAsikßaRsavRCavedayxøÜnÉg nig müageTot kñúgeKalbMngcUlrYmelIksÞÜyvis&yKNitviTüaenARbeTskm<úCaeyIg [kanEtrIkceRmInEfmeTotedIm,Ibeg;InFnFanmnusß[mankanEteRcIn edIm,ICYyGPivDÄn_RbeTsCatirbseyIg . enAkñúgesoven£eyIgxMJúánxitxMRsavRCaveRCIserIsyklMhatya¨g sRmaMgbMputBIÉksarbreTsyageRcInbMputdUcCa 103 Trigonometry Problems ,

Five Hundred Mathematical Challenges,Complex Numbers From A to … Z , Mathematical Olympiad in China,Mathematical Olympiad Challenges, Mathematical Olympiad Treasures, International Mathematical Olympiads 1959-1977, The IMO Compendium ( 1959-2004 ) , 360 Problems for Mathematical contest…

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lwm plþún nig Esn Bisidæ

Problems and Solutions

rkßasiT§iRKb´yag

KNitviTüaCMuvijBiPBelak

- 1 -


- 2 -

1-eK[BIrcMnYnBitviC¢man a nig b . cUrbgHajfa 2)ab1()b1)(a1( +≥++ Gnuvtþn_ rktémøtUcbMputénGnuKmn_ ½

)41)(41()x(f xcosxsin 22++= Edl IRx∈ .

2-eK[bIcMnYnBitviC¢man b,a nig c . cUrbgHajfa cabcabcba 222 ++≥++ 3-eK[bIcMnYnBitviC¢man b,a nig c . cUrbgHajfa abc8)ac)(cb)(ba( ≥+++ 4-eK[ n cMnYnBitviC¢man n321 a;....;a;a;a EdlplKuN 1a....a.a.a n321 = . cUrRsaybBa¢akfa n

n21 2)a1)....(a1)(a1( ≥+++ 5-eK[

2a0 π<< nig

2b0 π<< .

cUrbgHajfa 1bcosacos

bsinasin

2222

=⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

lu¼RtaEt ba = .


- 3 -

6-cMeBa¼RKb´cMnYnBit x cUrRsayfa ½ 2

23)xcos1)(xsin1( +≤++

7-eK[ 632a0 ++= nig )2a(2

5aan

2n

1n +−

=+

cMeBa¼RKb´ 0n ≥ .

cUrRsayfa 23

2cota3n

n −⎟⎟⎠

⎞⎜⎜⎝

⎛ π=

−

cMeBa¼RKb´ INn∈ .

8-eda¼RsayRbB&næsmIkarxageRkam ½

⎪⎪⎩

⎪⎪⎨

⎧

=+++

=+

18)y

11()x

11()y

1x

1(

9y1

x1

3333

9-cUrRsaybBa¢akfa ½ )

n11(n1

n1...

31

211)11n(n

nn −+<++++<−+

cMeBa¼RKb´cMnYnKtviC¢man n . 10-eK[ n cMnYn )1,0(a....,,a,a,a n321 ∈ ehIyeKtag

n321

n321n a...aaa

a....a.a.a.nt++++

= .


- 4 -

cUrRsayfa ( ) n)1n(tlogn

1knak

−≥∑=

. 11-eK[ mnig nCaBIrcMnYnKt´viC¢man .

cUrRsayfa x

1xm

1x nmn −≥

− cMeBa¼RKb´cMnYnBitviC¢man x . 12-eK[ 12x

n2n += cMeBa¼RKb´ ...,3,2,1n = .

cUrRsayfa 31

x2....

x2

x2

x1

n

1n

3

2

21

<++++−

13-eK[ IR),0(:f →∞+ ehIymancMnYnBit 0a > Edl 1)a(f = . cUrbgHajfaRbsinebI )xy(f2)

ya(f)

xa(f)y(f)x(f =+

cMeBa¼RKb´ ),0(y,x ∞+∈ ena¼ )x(f CaGnuKmn_efr . 14-eK[ ABC CaRtIekaNmYyEdlepÞógpÞat´lkç&x&NÐ

AcosCsinBsin1CsinBsin 22 +=+ . bgHajfa ABC CaRtIekaNEkg .


- 5 -

15- cUrRsaybBa¢akfa ½

)1n(3

3

3

3

3

3

32)1n(

1n1n......

1313

1212 −−>

+−

+++−

++−

16-eK[GnuKmn_ )1x(2

8m3mx2xy 2

2

+−++

=

Edl IRx∈ nig mCaá¨ra¨Em¨Rt . etIeKGackMnt´témø m edIm,I[GnuKmn_en¼Gactag[témø kUsIunUsénmMumYYyán¦eT? 17-eK[GnuKmn_ IRx,)x1xln()x(f 2 ∈++= cUrRsaybBa¢ak´fa )

b1b

a1a(f)

ba1ba(f

++

+<

+++

cMeBa¼RKb´ 0b,0a >> . 18-cUrRsaybBa¢akfa ½

38

2x4cos1x4sin4x4cos2 ≤

+++

≤− cMeBa¼RKb´cMnYnBit x . 19-eK[sIVúténcMnYnBit 66....666Un +++++= man n ra¨DIkal´ .


- 6 -

cUrRsayfa 3Un < cMeBa¼RKb´ INn∈ . 20-eK[cMnYnkMupøic ½

)xsin

1x.(sini)xcos

1x(cosZ 22

22 +++=

Edl x CacMnYnBit. cUrkMntrkm¨UDulGb,brmaéncMnYnkMupøicen¼ ?

21-eK[GnuKmn_41x

21x.....xxx)x(f +++++++=

cUrkMntbNþatémø INx∈ edIm,I[GnuKmn_ )x(f mantémøelx CacMnYnKt . 22-eK[ x CacMnYnBitEdl 021x71x60 2 <+− . cUrbgHajfa 0

1x3sin <⎟

⎠⎞

⎜⎝⎛

−π .

23-eK[GnuKmn_ ½

IRy;x,)y1()x1(

)yx1)(yx()y,x(f 2222

2222

∈++−−

=

cUrbgHajfacMeBa¼RKb´ IRy,x ∈ eKán 41|)y;x(f| ≤


- 7 -

24- eK[sIVút )n(

n 32.....222U +++++=

cMeBa¼RKb´ *INn∈ . k-cUrkMnt´ nU CaGnuKmn_én n . x-cUrbgHajfa

n

n321

2.3sin2

3U......UUUπ

=×××× .

K-eKBinitüsIVút )n(

nn 32.....2222V ++++−= .

cUrKNna nV nig lImIt nnVlim

+∞→ .

25-eK[ 1a ≥ nig 1b ≥ .

cUrbgHajfa )2

ba(log2blogalog 222+

≤+

26-eK[ θ CacMnYnBitEdl 2

0 π<θ< .

cUrbgHajfa ( ) ( ) 1cossin sincos >θ+θ θθ 27-eK[RtIekaN ABC mYy .

k-cUrRsaybBa¢akfa 2

33CsinBsinAsin ≤++ .


- 8 -

x-bgHajfa 8

332Ccos

2Bcos

2Acos ≤ .

28-eK[smIkar ½ 08m8m4x)3m2(2x:)E( 22 =+++++

k-cUrkMnt´bNþatémø INm ∈ edIm,I[smIkaren¼man¦s CacMnYnKt´rWLaTIhV . x-rk¦sEdlCacMnYnKt´rWLaTIhVrbs´smIkar . 29-MCama¨RTIskaerlMdab´bIEdlmanFatuTaMgGs´CacMnYKt´ . eKdwgfaplKuNénFatuTaMgbItamCYredknImYy@ plKuNén FatuTaMgbItamCYrQrnImYy@nig plKuNénFatuTaMgbItam Gg;t´RTUgesμInwg mdUcKña . bgHajfa m CaKUbéncMnYnKt´ . 30-eK[bYncMnYnviC¢man d,c,b,a . cUrbgHajfa ½

2bad

dadc

cdcb

bcba

a1 <++

+++

+++

+++

<


- 9 -

31-eKmanbIcMnYnBit 0c;0b;0a >>> . cUrRsayfa abc6)ac(ca)cb(bc)ba(ab ≥+++++

32-eK[BIrcMnYn x nig y xusBIsUnü nig mansBaØadUcKña . cUrbgHajfa 04

xy

yx3

xy

yx

2

2

2

2≥+⎟

⎠

⎞⎜⎝

⎛+−+ .

33-eK[smIkar ca,0a,0cbxax2 ≠≠=++ . tag αtan nig βtan Ca¦srbssmIkarxagelI . cUrKNnatémøénkenßam ½

)(cosc)cos()sin(b)(sinaA 22 β+α+β+αβ+α+β+α= 34-cMnYnKt;viC¢man n Ecknwg * [sMNl; ! . cMnYn n enaHEcknwg % [sMNl; @ . k-ebIcMnYn n enaHEcknwg $0 [sMNl;b:un μan ? x-rkcMnYn n enaHedaydwgfa 4000n3940 << . 35-edaHRsayRbBn½æsmIkar ⎩⎨⎧

=

=+

64yx

26)ylogxlog(5 xy


- 10 -

36-edaHRsayRbBn½æsmIkar ⎪⎪⎩

⎪⎪⎨

⎧

=

=

90016.5

40015.4

yx

yx

37-eda¼RsaysmIkar ½ 1339

222 )1x(x1xx +=+ −−− 38-eK[ c;b;a CabIcMnYnBitviC¢mandac´xat . cUrbgHajfa

2cba

bac

acb

cba 222 ++

≥+

++

++

39-cUrkMnt;rkGnuKmn_ )x(f nig )x(g ebIeKdwgfa ³

2x)1x3(g2)1x2(f =++− nig 1x2x2)2x6(g)3x4(f 2 ++−=−−− cMeBaHRKb; IRx∈ . 40-eK[RtIekaN ABC mYymanRCúg c,b,a . kMnt´RbePTénRtIekaN ABC ebIeKdwgfa ½ )

c1

b1

a1(

21

cCcos

bBcos

aAcos

++=++


- 11 -

41-k¿ cUrRsaybBa¢ak´TMnak´TMng x2cot2xcotxtan −= x¿ cUrKNnaplbUkxageRkam ½ nn22n 2

atan21....

2atan

21

2atan

21atanS ++++=

42-eK[GnuKmn_ 2x)x(f 2 −= Edl IRx∈ k-eKyk )x(fU1 = nig )U(fU n1n =+ cMeBa¼RKb´ *INn∈ . cUrbgHajfa [ ][ ][ ].......)x(ff.......ffU nn = . x-RsayfaebI 2x > eKán 2Un > RKb´ *INn∈ . K-eKtag 4UUV 2

nnn −−= RKb´ *INn∈ nig 2x > . cMeBa¼RKb´ *INn∈ cUrbgHajfa 2

n1n VV2 =+ . X-sn μtfa 2lnVlnW nn −= cMeBa¼RKb´ *INn∈ . cUrrkRbePTénsIVút nW . g-eRbIlTæplxagelIcUrTajrkGnuKmn_ ½

[ ][ ][ ].........)x(ff........ff)x(F nn = . 43-eK[GnuKmn_

1x3x31x3x3x)x(f 2

23

+−+−+

= RKb´ IRx∈ cMeBa¼RKb´cMnYnBitviC¢man a nig b


- 12 -

cUrRsaybBa¢akfa ½ ⎟

⎠⎞

⎜⎝⎛

+++++

≥⎟⎠⎞

⎜⎝⎛ ++

ba2abba1f

2ba1f .

44-eK[Exßekag xm

1m4mx4x)x(fy:)C(22

mm −+−+

== cUrbgHajfamanExßekagBIrénRKYsarExßekag )C( m Edlkat´tamcMnuc )y;x(M 000 cMeBa¼RKb´( ) 2

00 IRy;x ∈ . mCaá¨raËm¨Rt . 45-eK[ )x(P CaBhuFadWeRkTIbI . eKdwgfa 2)x(P + Eckdacńwg 2)1x( + ehIy 2)x(P − Eckdacńwg 2)1x( − . cUrkMnt´rkBhuFa )x(P . 46-eK[ INn,i

31i

31A

nn

∈⎟⎠⎞

⎜⎝⎛ −−⎟

⎠⎞

⎜⎝⎛ += .

cUrbgðajfa 3

nsin.)3(

2.iA n

1n π=

+ cMeBaHRKb; INn∈ .

47-eK[sIVúténcMnYnBit )U( n kMnt;eday ³ 3lnU0 = nig INn,)e1(lnU nU

1n ∈+=+ . cUrKNna nU CaGnuKmn_én n .


- 13 -

48-edaHRsaysmIkar 1y29x47 =+ kñúgsMNMucMnYnKt;rWLaTIhV . 49-cUrbgðajfa 3n22n3

n 73A ++ += Eckdac;nwg !! Canic©RKb;cMnYnKt;FmμCati n. 50-cUrRsaybBa¢ak;facMnYn nnnn

n 124122462447E −−+= Eckdac;nwg 221Canic©cMeBaHRKb;cMnYnKt;Fm μCati n . 51-eK[GnuKmn_ xcos.e)x(f x= k> KNna )x('f rYcbgðajfa )

4xcos(e2)x('f x π+=

x> edayeFIVvicartamkMeNIncUrbgðafaedrIevTI n kMnt;eday )

4nxcos(.e2)x(f xn)n( π

+= 52-eK[RtIekaN ABC mYymanRCug

cAB,bAC,aBC === . ]bmafa M CacMnucmYyenAkñúgRtIekaNenH ehIyeKtag y,x nig z CacMgayerogKñaBIcMnuc M eTARCug AC,BC nig AB énRtIekaN . cUrKNnatémøGb,brmaén 222 zyxT ++= CaGnuKmn_én c,b,a .


- 14 -

53-cUrkMnt;rkGnuKmn_ )x(fy = ebIeKdwgfa ³

1x1x)2x2xx(f 2

22

+−

=+−+ cMeBaHRKb;cMnYnBit x .

54-kñúgbøg;kMupøic )j,i,O(→→ eK[bYncMnuc D,C,B,A

EdlmanGahVikerogKña i45Z,i54Z,i61Z CBA +=+=+= nig i32ZD −−= .

cUrRsayfactuekaN ABCDcarikkñúgrgVg;mYyEdleKnwgbBa¢ak;p©it nig kaMrbs;va . 55-eK[RtIFa cbxax)x(f 2 ++= Edl IRc,b,a,0a ∈≠ k-cUrRsayfaebI 0ac4b2 <−=Δ nig 0a > ena¼eKán IRx,0)x(f ∈∀> x-kñúgkrNIen¼eKsnμtfa 0ac4b2 <−=Δ nig 0a > . ebI ab > cMeBa¼RKb´ IR∈λ cUrbgHajfa ½ k

abc)k(b)k(a 2

>−

++λ+−λ Edl k CacMnYnBitefrmYyEdleK[ . K-Gnuvtþn_ cMeBa¼RKb´RtIFa cbxax)x(f 2 ++=


- 15 -

Edl IRc,b,a,0a ∈≠ ebI 0ac4b2 <−=Δ nig 0a > ena¼bgHajfaeKman ½

3ab

aba>

−++ .

56-eK[ 2

x0 π<< . cUrRsaybBa¢ak´fa ½

21xcos

11xsin

11 +≤⎟⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛ +

57-cUreda¼RsayRbBnæ&smIkar⎪⎩

⎪⎨⎧

=+

=++

+

1723.28

171273.4y21xx

y1yx

58-cUreda¼RsayRbBnæ&smIkar ½

⎪⎩

⎪⎨⎧

=−+−

=+−−

344y28xy33yx21x9

342y26xy21yx3x73223

3223

59-eK[smIkar 0)b1)(a1(x.ba22x:)E( 222 =+++++− Edl a nig b CaBIrcMnYnBit . k-cUrkMnt´témø a nig b edIm,I[smIkaren¼man¦sDub rYcKNna¦sDubena¼ . x-eRkABItémø a nig bxagelIcUrbgHajfasmIkar )E( man¦sBIr Canic©kñúg IR.


- 16 -

60-eK[BhuFa ½ n1n

1n2

21n xxc.......xcxc1)x(P +++++= −−

Edl 0c.......,,c,c,c 1n321 ≥− . snμtfasmIkar 0)x(Pn = man n ¦sCacMnYnBitGviC©man . cUrbgHajfa n

n 3)2(P ≥ . 61-k-RbsinebI 1p −≥ cMeBa¼RKb´ *INn∈ cUrbgHajfa ½ )1(np1)p1( n +≥+ . x-eK[ n321 a......,,a,a,a Can cMnYnminGviC¢man . eKtag

na....aaa

A n321n

++++=

nig nn321n a......a.a.aG = .

bgHajfaRbsinebI kk AG ≤ ehIy 0Ak ≠ eKán )p1(Aa.G 1k

k1kk

k +≤ ++ Edl 1

Aa

pk

1k −= + .

K-edayeRbIlTæplxagelIcUrRsaybBa¢akfa ½ n

n321n321 a....aaana.....aaa ≥++++ .


- 17 -

62-eK[ C,B,A CargVas´mMukñúgrbsRtIekaN ABC mYy . k-cUrbgHajfa

23CcosBcosAcos ≤++

x-cUrbgHajfa 49

2Ccos

2Bcos

2Acos 222 ≤++

K-cUrbgHajfa 43

2Csin

2Bsin

2Asin 222 ≥++

X-cUrbgHajfa 8

332Ccos

2Bcos

2Acos ≤

63-eK[smPaB

3)1n2(n

1n2ncot......

1n23cot

1n22cot

1n2cot 2222 −

=+π

+++π

++π

++π

cUrbgHajfa 6n

1...........31

21

11lim

2

2222n

π=⎟

⎠⎞

⎜⎝⎛ ++++

+∞→.

64-eKtag r nig R erogKñaCakaMénrgVg´carikkñúg nig carikeRkA Rbs´RtIekaNEkg ABC mYy . cUrRsaybBa¢ak´fa r)21(R +≥ ? 65-eK[ c,b,a CaRCugrbs´RtIekaNmYy . cUrRsaybBa¢akvismPaB ½ 0)ac(ac)cb(cb)ba(ba 222 ≥−+−+− ?


- 18 -

66-eK[ c,b,a CaRCugrbs´RtIekaNmYyEdlmanépÞRkTLa esμInwg S . cUrRsayfa S34cba 222 ≥++ ? 67-eKyk I Cap©itrgVgćarwkkñúgRtIekaN ABC mYyEdl

cAB,bAC,aBC === . cMeBa¼RKbćMnuc X cUrRsayfa ½

abcXI)cba(XC.cXB.bXA.a 2222 +++=++ 68-eKtag r nig R erogKñaCakaMrgVgćarikkñúg nig carwkeRkA énRtIekaNmYy . cUrRsaybBa¢akfaebI d Cacm¶ayrvagp©it rgVgćarwkkñúg nig carwkeRkArbsRtIekaNena¼eKán ½

)r2R(Rd2 −= ? 69-eKyk I Cap©itrgVgćarwkkñúgRtIekaN ABC mYyEdl manRCug cAB,bAC,aBC === . cUrRsayfa 1

abIC

caIB

bcIA 222

=++ . 70-cMeBa¼RKbćMnYnBitviC¢man c,b,a cUrbgHajfa ½ )cabcab(9)2c)(2b)(2a( 222 ++≥+++ .


- 19 -


- 20 -

lMhat´TI1 eK[BIrcMnYnBitviC¢man a nig b . cUrbgHajfa 2)ab1()b1)(a1( +≥++ Gnuvtþn_ rktémøtUcbMputénGnuKmn_ ½

)41)(41()x(f xcosxsin 22++= Edl IRx∈ .

dMeNa¼Rsay bgHajfa 2)ab1()b1)(a1( +≥++ eyIgman )1(ab)ba(1)b1)(a1( +++=++ tamvismPaB GMAM − eKman ab2ba ≥+ tam )1( eKTaján abab21)b1)(a1( ++≥++ eday 2)ab1(abab21 +=++ dUcen¼ 2)ab1()b1)(a1( +≥++ edayeRbIvismPaBxagelIen¼eKTaján ½

9)41()41)(41()x(f 2xcosxsinxcosxsin 2222=+≥++= +

dUcen¼témøGb,brmaénGnuKmn_esμInwg 9 .


- 21 -

lMhat´TI2 eK[bIcMnYnBitviC¢man b,a nig c . cUrbgHajfa cabcabcba 222 ++≥++ dMeNa¼Rsay bgHajfa cabcabcba 222 ++≥++ tamvismPaB GMAM − eKman ½

)3(ca2

ac

)2(bc2

cb

)1(ab2

ba

22

22

22

≥+

≥+

≥+

bUkvismPaB )2(,)1( nig )3( Gg:nwgGg:eKán ½

cabcab2

accbba 222222

++≥+++++

dUcen¼ cabcabcba 222 ++≥++ .


- 22 -

lMhat´TI3 eK[bIcMnYnBitviC¢man b,a nig c . cUrbgHajfa abc8)ac)(cb)(ba( ≥+++ dMeNa¼Rsay bgHajfa abc8)ac)(cb)(ba( ≥+++ tamvismPaB GMAM − eKman ½

)3(ca2ac)2(bc2cb)1(ab2ba

≥+

≥+

≥+

eFIVviFIKuN )2(,)1( nig )3( Gg:nwgGg:eKán ½ ca.bc.ab8)ac)(cb)(ba( ≥+++

dUcen¼ abc8)ac)(cb)(ba( ≥+++ .


- 23 -

lMhat´TI4 eK[ n cMnYnBitviC¢man n321 a;....;a;a;a EdlplKuN 1a....a.a.a n321 = . cUrRsaybBa¢akfa n

n21 2)a1)....(a1)(a1( ≥+++ dMeNa¼Rsay RsaybBa¢ak´fa n

n21 2)a1)....(a1)(a1( ≥+++ tamvismPaB GMAM − eKman ½

nn

33

22

11

a2a1

a2a1

a2a1

a2a1

≥+

−−−−−−−−

≥+

≥+

≥+

eKán n21n

n21 a....a.a2)a1)....(a1)(a1( ≥+++ eday 1a....a.a.a n321 = . dUcen¼ n

n21 2)a1)....(a1)(a1( ≥+++ .


- 24 -

lMhat´TI5 eK[

2a0 π<< nig

2b0 π<< .

cUrbgHajfa 1bcosacos

bsinasin

2222

=⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

lu¼RtaEt ba = . dMeNa¼Rsay karbgHaj

eKman 1bcosacos

bsinasin

2222

=⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

smmUl 1bcosacos

bsinasin)bcosb(sin 2

4

2

422 =⎟⎟

⎠

⎞⎜⎜⎝

⎛++

0)acosbcosbsinasin

bsinbcos

1acosbcosbsinasin

bsinbcosacosasin21

1acosbcosbsinasin

bsinbcosacosasin

222

42

24

2

222

42

24

2

244

=⎟⎠⎞

⎜⎝⎛ −

=++−

=+++


- 25 -

eKTaj acosbcosbsinasin

bsinbcos 22 =

smmUl bcosbsin

acosasin

2

2

2

2

= smmUl btanatan 22 = eday

2a0 π<< nig

2b0 π<< ena¼eKTaj ba = .

lMhat´TI6 cMeBa¼RKb´cMnYnBit x cUrRsayfa ½

223)xcos1)(xsin1( +≤++

dMeNa¼Rsay RsaybBa¢ak´fa 2

23)xcos1)(xsin1( +≤++

eyIgman 0)ba( 2 ≥− cMeBa¼RKb´ IRb,a ∈ eKán 0bab2a 22 ≥+−

eKTaj 2

bab.a22 +

≤


- 26 -

edayeRCIserIsyk xsin1a += nig xcos1b +=

eKán 2

)xcos1()xsin1()xcos1)(xsin1(22 +++

≤++

)1(2

)xcosx(sin23)xcos1)(xsin1( ++≤++

eKman 2)4

xsin(2xcosxsin ≤π

+=+

tam )1( eKTaj 2

223)xcos1)(xsin1( +≤++

dUcen¼ 223)xcos1)(xsin1( +≤++ .


- 27 -

lMhat´TI7

eK[ 632a0 ++= nig )2a(2

5aan

2n

1n +−

=+

cMeBa¼RKb´ 0n ≥ .

cUrRsayfa 23

2cota3n

n −⎟⎟⎠

⎞⎜⎜⎝

⎛ π=

−

cMeBa¼RKb´ INn∈ .

dMeNa¼Rsay

RsaybBa¢ak´fa 23

2cota3n

n −⎟⎟⎠

⎞⎜⎜⎝

⎛ π=

−

cMeBa¼ 0n = eKán 224

cota0 −π

=

26264

426

4261

)43

sin(

)43

cos(1

12sin

12cos1

24cos

24sin2

24cos2

24sin

24cos

24cot

2

−++

=−

++

=π

−π

π−

π+

=

π

π+

=ππ

π

=π

π

=π


- 28 -

63224

348)26(44

)26()26(424

cot2

+++=

+++=

+++=

π

eKTaj 632a0 ++=

ehtuen¼ 23

2cota3n

n −⎟⎟⎠

⎞⎜⎜⎝

⎛ π=

−

BitcMeBa¼ 0n = .

snμtfavaBitdl´tYTI k KW 23

2cota3k

k −⎟⎟⎠

⎞⎜⎜⎝

⎛ π=

−

Bit

eyIgnwgRsayfavaBitdl´tYTI 1k + KW ½

23

2cota2k

1k −⎟⎟⎠

⎞⎜⎜⎝

⎛ π=

−

+ Bit .

eyIgman )2a(2

5aak

2k

1k +−

=+ eday 23

2cota3k

k −⎟⎟⎠

⎞⎜⎜⎝

⎛ π=

−

ena¼ ⎟⎟⎠

⎞⎜⎜⎝

⎛ π

−⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛ π

=−

−

+

32cot2

523

2cota 3k

23k

1k


- 29 -

2

22cot2

13

2cot

32cot2

13

2cot43

2cota

3k

3k2

3k

3k3k2

1k

−

⎟⎟⎠

⎞⎜⎜⎝

⎛ π

−⎟⎟⎠

⎞⎜⎜⎝

⎛ π

=

⎟⎟⎠

⎞⎜⎜⎝

⎛ π

−⎟⎟⎠

⎞⎜⎜⎝

⎛ π−⎟⎟

⎠

⎞⎜⎜⎝

⎛ π

=

−

−

−

−−

+

edayeRbIrUbmnþ acot2

1acota2cot2 −

=

eKán 23

2cota2k

1k −⎟⎟⎠

⎞⎜⎜⎝

⎛ π=

−

+ Bit .

dUcen¼ 23

2cota3n

n −⎟⎟⎠

⎞⎜⎜⎝

⎛ π=

−

.


- 30 -

lMhat´TI8 eda¼RsayRbB&næsmIkarxageRkam ½

⎪⎪⎩

⎪⎪⎨

⎧

=+++

=+

18)y

11()x

11()y

1x

1(

9y1

x1

3333

dMeNa¼Rsay eda¼RsayRbB&næsmIkar

⎪⎪⎩

⎪⎪⎨

⎧

=+++

=+

18)y

11()x

11()y

1x

1(

9y1

x1

3333

edayeRbIÉklkçN¼PaB )ac)(cb)(ba(3cba)cba( 3333 ++++++=++ )

y1

x1)(

y11)(

x11(3

y1

x11)

y1

x11(

33333

33++++++=++

64)18(391)y1

x11( 3

33=++=++


- 31 -

eKTaj 4y1

x11

33=++ ¦ 3

y1

x1

33=+

eKán 27)y1

x1( 3

33=+

81xy

3xy11

27)3(xy39

27)y1

x1(

y.x3

y1

x1

3

3

3333

=

=+

=+

=+++

eKánRbB&næ ⎪⎪⎩

⎪⎪⎨

⎧

=

=+

81xy

9y1

x1

¦ ⎪⎩

⎪⎨

⎧

=

=+

81xy

89yx

bnÞab´BIeda¼RsaysmIkarEvüt 081z

89z2 =+−

eKTTYlánKUcemøIy ½ (

81y,1x == ) ¦ ( 1y,

81x == ) .


- 32 -

lMhat´TI9 cUrRsaybBa¢ak´fa ½

)n

11(n1n1...

31

211)11n(n

nn −+<++++<−+

cMeBa¼RKb´cMnYnKtviC¢man n .

dMeNa¼Rsay RsaybBa¢ak´fa ½

)n

11(n1n1...

31

211)11n(n

nn −+<++++<−+

tamvismPaB GMAM − eKman ½

∑ ∏ ⎟⎠⎞

⎜⎝⎛ +

>⎟⎠⎞

⎜⎝⎛ +

= =

n

1kn

n

1k k1k

k1k

n1

¦ ∑ +>⎟⎠⎞

⎜⎝⎛+

=

n

1k

n 1nk1

n11

¦ )1()11n(nn1....

31

211 n −+>++++

dUcKñaEdrtamvismPaB GMAM − eKman ½


- 33 -

n

n1n....

32.

21.1

n1n....

32

211

n1 −

>⎟⎠⎞

⎜⎝⎛ −

++++

)2(1)n

11(nn1...

31

211

n1)

n1....

31

21(

n11

n1)

n11...

311

2111(

n1

n1)

n1n...

313

2121(

n1

n1

n1n....

32

211

n1

n

n

n

n

n

+−<++++

>+++−

>−++−+−+

>−

++−

+−

+

>⎟⎠⎞

⎜⎝⎛ −

++++

tam )1( nig )2( eKTaján ½

)n

11(n1n1...

31

211)11n(n

nn −+<++++<−+


- 34 -

lMhat´TI10 eK[ n cMnYn )1,0(a....,,a,a,a n321 ∈ ehIyeKtag

n321

n321n a...aaa

a....a.a.a.nt++++

= .

cUrRsayfa ( ) n)1n(tlogn

1knak

−≥∑=

. dMeNa¼Rsay Rsayfa ( ) n)1n(tlog

n

1knak

−≥∑=

tamvismPaB GMAM − RKb´ )1,0(a....,,a,a,a n321 ∈ eKman n

n321n321 a...a.a.a

na...aaa

≥++++

eKTaj ( ) n1n

n321n a....a.a.at−

≤ eday 1a0 k << eKTaj )a...a.a(log.

n1n)t(log k21ana kk

−≥

¦ [ ] ( )[ ] ( )*a...a.alog.n

1n)t(logn

1kk21a

n

1kna kk

∑−

≥∑==

tag [ ]∑==

n

1kk21an )a....a.a(logS

k


- 35 -

)aloga(log....

...)aloga(log...)aloga(logn

1nana

na1a1a2a

n1n

1n21

−++

+++++++=

−

tamvismPaB 0t,2t1t >∀≥+ eKTaján ½

2n n]2...)2n(2)1n(2nS =++−+−+≥

tamTMnak´TMng ( )* eKTaján ½ ( ) n)1n(tlog

n

1knak

−≥∑=

.

lMhat´TI11 eK[ mnig nCaBIrcMnYnKt´viC¢man .

cUrRsayfa x

1xm

1x nmn −≥

− cMeBa¼RKb´cMnYnBitviC¢man x . dMeNa¼Rsay Rsayfa

x1x

m1x nmn −≥

− eday 0x > ena¼vismPaBxagelIsmmUl ½


- 36 -

[ ] (*)0mx....)x(x)x(x)1x(0)1x(m)1x(x

2mn1mnn

nmn

≥−+++−

≥−−−−−

- cMeBa¼ 1x = vismPaB (*)køayCasmPaB . - cMeBa¼ 1x > eKman 01xn >− nig 0mx...)x(x)x(x 2mn1mn >−+++ −− naM[vismPaB (*) BitCanic© . - cMeBa¼ 1x < eKman 01xn <− nig 0mx...)x(x)x(x 2mn1mn <−+++ −− naM[vismPaB (*) BitCanic© .

srubmkeKán x

1xm

1x nmn −≥

− .


- 37 -

lMhat´TI12 eK[ 12x

n2n += cMeBa¼RKb´ ...,3,2,1n = .

cUrRsayfa 31

x2....

x2

x2

x1

n

1n

3

2

21

<++++−

dMeNa¼Rsay Rsayfa

31

x2....

x2

x2

x1

n

1n

3

2

21

<++++−

tag 12yn2

n −= RKb´ 1n ≥ eKán )12)(12(12y

nn1n 2221n −+=−=

+

+ ¦ nn1n xyy =+ . eKman

nn

n

nnn1nn yx2x

yx2

y1

y2

y1 −

=−=−+

eday n22

n y122122xnn

=−=−+=− eKTaj

n1nn x1

y2

y1

=−+

naM[ 1n

1n

n

n

n

1n

y2

y2

x2

+

+−

−=


- 38 -

cMeBa¼ ....,3,2,1n = eKán ½

11n

1n

1n

1n

3

2

21 y1

y2

y1

x2....

x2

x2

x1

<−=+++++

+−

eday 312y 21 =−=

dUcen¼ 31

x2....

x2

x2

x1

n

1n

3

2

21

<++++−

.

lMhat´TI13 eK[ IR),0(:f →∞+ ehIymancMnYnBit 0a > Edl

1)a(f = . cUrbgHajfaRbsinebI )xy(f2)

ya(f)

xa(f)y(f)x(f =+

cMeBa¼RKb´ ),0(y,x ∞+∈ ena¼ )x(f CaGnuKmn_efr .

dMeNa¼Rsay bgHajfa )x(f CaGnuKmn_efr eKman )xy(f2)

ya(f)

xa(f)y(f)x(f =+


- 39 -

yk 1yx == eKán )1(f2)a(f)1(f 22 =+ eday 1)a(f = ena¼ ( ) 01)1(f 2 =− ¦ 1)1(f = . yk 1y = eKán )x(f2)a(f)

xa(f)1(f)x(f =+

eKTaj )xa(f)x(f = .

yk xay = eKán )a(f2)x(f)

xa

(f)xa

(f)x(f =+

eKTaj 1)a(f)xa(f)x(f == eday )

xa(f)x(f =

ehtuen¼ 0x,1)x(f 2 >∀= . yk tyx == eKán )t(f2)

ta(f)t(f 22 =+

eKTaj 0t,0)t(f >∀> . dUcen¼ 1)x(f = CaGnuKmn_efrRKb´ 0x > .


- 40 -

lMhat´TI14 eK[ ABC CaRtIekaNmYyEdlepÞógpÞat´lkç&x&NÐ

AcosCsinBsin1CsinBsin 22 +=+ . bgHajfa ABC CaRtIekaNEkg .

dMeNa¼Rsay bgHajfa ABC CaRtIekaNEkg tamRTwsþIbTsIunUseKman R2

Csinc

Bsinb

Asina

=== eKTaj CsinR2c,BsinR2b,AsinR2a === eday Acosbc2cba 222 −+= ( RTwsþIbTkUsIunUs ) eKTaján ½

)AcosCsinBsin2CsinB(sinR4AsinR4 22222 −+= )1(AcosCsinBsin2CsinBsinAsin 222 −+= Et )2(AcosCsinBsin1CsinBsin 22 +=+ yksmIkar )2( CYsenAkñúg )1( eKTaj 1Asin2 = naM[ 090A = . dUcen¼ ABC CaRtIekaNEkg .


- 41 -


)1n(3

3

3

3

3

3

32)1n(

1n1n......

1313

1212 −−>

+−

+++−

++−

dMeNa¼Rsay RsaybBa¢ak´ tamvismPaB GMAM − eKán ½

)1n(3

3

3

3

3

3

3

3

3

3

3

3

1n1n.....

1313.

1212)1n(

1n1n......

1313

1212 −

+−

+−

+−

−>+−

+++−

++−

eKman 1nn1nn.

1n1n

1n1n

2

2

3

3

+−++

+−

=+−

cMeBa¼ n.....;;4;3;2n = eKán ½

1nn1nn.

1n1n

1n1n

713.

42

1313

37.

31

1212

2

2

3

3

3

3

3

3

+−++

+−

=+−

−−−−−−−−

=+−

=+−


- 42 -

KuNsmPaBen¼Gg:nwgGg:eKán ½

32

)1n(n11

32

31nn.

)1n(n2

1n1n......

1313.

1212 2

3

3

3

3

3

3

>⎥⎦

⎤⎢⎣

⎡+

+=

+++

=+−

+−

+−

dUcen¼ )1n(3

3

3

3

3

3

32)1n(

1n1n......

1313

1212 −−>

+−

+++−

++−

lMhat´TI16

eK[GnuKmn_ )1x(2

8m3mx2xy 2

2

+−++

=

Edl IRx∈ nig mCaá¨ra¨Em¨Rt . etIeKGackMnt´témø m edIm,I[GnuKmn_en¼Gactag[témø kUsIunUsénmMumYYyán¦eT? dMeNa¼Rsay kMnt´témørbs´ m edIm,I[GnuKmn_en¼tag[témøkUsIunUsénmMumYYylu¼RtaEtcMeBa¼


- 43 -

RKb´ IRx∈ eKán 1)1x(2

8m3mx2x1 2

2

≤+

−++≤−

edayeKman IRx,0)1x(2 2 ∈∀>+ eKTaj 2x28m3mx2x2x2 222 +≤−++≤−−

¦ ⎩⎨⎧

≥+−−

≥−++

)2(010m3mx2x)1(06m3mx2x3

2

2

cMeBa¼ 06m3mx2x3:)1( 2 ≥−++

smmUl ⎩⎨⎧

≤+−=Δ

>=

018m9m'03a

2

eday )6m)(3m(18m9m2 −−=+− eKán 0)6m)(3m(' ≤−−=Δ naM[ 6m3 ≤≤ ¦ ]6,3[m∈ cMeBa¼ 010m3mx2x:)2( 2 ≥+−−

smmUl ⎩⎨⎧

≤−+=Δ

>=

010m3m'01a

2

eday )5m)(2m(10m3m2 +−=−+ eKán 0)5m)(2m(' ≤+−=Δ naM[ 2m5 ≤≤− ¦ ]2,5[m −∈


- 44 -

edayykcemøIy ]6,3[m∈ RbsBVnwg ]2,5[m −∈ ena¼eKán φ∈m . dUcen¼eKminGackMnttémø m edIm,I[GnuKmn_en¼Gactag[témø kUsIunUsénmMumYYyáneT .

lMhat´TI17 eK[GnuKmn_ IRx,)x1xln()x(f 2 ∈++= cUrRsaybBa¢ak´fa )

b1b

a1a(f)

ba1ba(f

++

+<

+++

cMeBa¼RKb´ 0b,0a >> . dMeNa¼Rsay RsaybBa¢ak´fa )

b1b

a1a(f)

ba1ba(f

++

+<

+++

eyIgman IRx,)x1xln()x(f 2 ∈++=

eyIgán 2

2

2

2

x1xx12

x21

x1x)'x1x()x('f

+++

+=

++

++=


- 45 -

222

2

x11

)x1)(x1x(xx1)x('f

+=

+++

++=

eday IRx,0x1

1)x('f2

∈∀>+

=

dUcen¼GnuKmn_ )x(f CaGnuKmn_ekInCanic©elI IR. müa¨geTotcMeBa¼RKb´ 0b,0a >> eKman ½

a1a

ba1a

+<

++ nig

b1b

ba1b

+<

++

eKTaj b1

ba1

aba1

bba1

a+

++

<++

+++

¦ b1

ba1

aba1

ba+

++

<++

+ edaysarEtGnuKmn_ )x(f CaGnuKmn_ekInCanic©elI IR ehtuen¼tamlkçN¼GnuKmn_ekIneKán½

b1b

a1a

ba1ba

++

+<

+++

naM[ )b1

ba1

a(f)ba1

ba(f+

++

<++

+

dUcen¼ )b1

ba1

a(f)ba1

ba(f+

++

<++

+ cMeBa¼RKb´ 0b,0a >> .


- 46 -


38


+++

≤− cMeBa¼RKb´cMnYnBit x .

dMeNa¼Rsay RsaybBa¢ak´fa

38


+++

≤−

eyIgtag IRx,2x4cos

1x4sin4x4cosy ∈∀+

++=

eyIgán y2x4cosy1x4sin4x4cos +=++ ¦ )1(1y2x4sin4x4cos)y1( −=+− eyIgeRCIserIsviucT&rBIr ½

)4,y1(U −→

nig )x4sin,x4cos(V→

tamkenßamviPaKplKuNs;aEl ½ )2(x4sin4x4cos)y1(V.U +−=

→→


- 47 -

tam )1( nig )2( eKTaj 1y2V.U −=→→

. müa¨geTottamniymn&y θ=

→→→→

cos.||V||.||U||V.U eday IR,1cos1 ∈θ∀≤θ≤− eKTaj ||V||.||U||V.U||V||.||U||

→→→→→→

≤≤−

¦ 222

||V||.||U||V.U→→→→

≤⎟⎠⎞

⎜⎝⎛

eday 1||V||,16)y1(||U|| 22 =+−=→→

nig 1y2V.U −=

→→

eKán 16)y1()1y2( 22 +−≤− ¦ 016y2y3 2 ≤−− eday )8y3)(2y(16y2y3 2 −+=−− ehtuen¼ 016y2y3 2 ≤−− smmUl

38y2 ≤≤− .

dUcen¼ 38


+++

≤− cMeBa¼RKb´cMnYnBit x .


- 48 -

lMhat´TI19 eK[sIVúténcMnYnBit 66....666Un +++++= man n ra¨DIkal´ . cUrRsayfa 3Un < cMeBa¼RKb´ INn∈ . dMeNa¼Rsay Rsayfa 3Un < cMeBa¼RKb´ INn∈ sm μtikm μ 66....666Un +++++= eyIgman 36U1 <= Bit 336U666U 12 <+<+=+= Bit . snμtfavaBitdl´tYTI k KW 36...66Uk <+++= eyIgnwgRsayfavaBitdl´tYTI 1k + KW

36....666U 1k <++++=+ eyIgman k1k U6U +=+ eday 3Uk < naM[ 9U6 k <+ eKTaján 3U6U k1k <+=+ Bit . dUcen¼ 3Un < cMeBa¼RKb´ INn∈ .


- 49 -

lMhat´TI20 eK[cMnYnkMupøic )

xsin1x.(sini)

xcos1x(cosZ 2

22

2 +++= Edl x CacMnYnBit. cUrkMntrkm¨UDulGb,brmaéncMnYnkMupøicen¼ ? dMeNa¼Rsay rkm¨UDulGb,brmaéncMnYnkMupøic

eyIgán 22

222

2 )xsin

1x(sin)xcos

1x(cos|Z| +++=

tag 22

222

2 )xsin

1x(sin)xcos

1x(cos)x(f +++=

)xcosxsinxsinxcos()xsinx(cos4

)xcos

1xsin

1()xsinx(cos4

xsin12xsin

xcos12xcos

44

4444

4444

44

44

++++=

++++=

+++++=


- 50 -

[ ]

)x2sin

161)(x2sin211(4

)x2sin

161(xcosxsin2)xsinx(cos4

)xcosxsin

11)(xsinx(cos4)x(f

42

422222

4444

+−+=

+−++=

+++=

edayeKman 1x2sin 2 ≤ naM[ 21x2sin

211 2 ≥−

nig 17x2sin

161 4 ≥+

eKTaj 225

2174)

x2sin161)(x2sin

211(4 4

2 =+≥+−+

eyIgán 225)

x2sin161)(x2sin

211(4)x(f 4

2 ≥+−+=

eday )x(f|Z| = eKTaján 2

252

5|Z| =≥

dUcen¼m¨UDulGb,brmaén Z KW 2

25|Z| min = .


- 51 -

lMhat´TI21

eK[GnuKmn_ 41x

21x.....xxx)x(f +++++++=

cUrkMntbNþatémø INx∈ edIm,I[GnuKmn_ )x(f mantémøelx CacMnYnKt .

dMeNa¼Rsay kMnt´témø INx∈ ´ eyIgman ½

2)41x

21(

)41x(.

41x

21.2

41

41x

21x

++=

++++=+++

eKTaján 41x

21)x(f ++= .

sm μtfamancMnYnKt´ *INk∈ Edl k)x(f =

eKán k41x

21

=++

¦ 41kk)

21k(

41x 22 +−=−=+


- 52 -

eKTaj )1k(kx −= eday *INk ∈ ena¼ IN)1k(kx ∈−= dUcen¼bNþatémø x EdlRtUrkena¼KW ½ *INk,)1k(kx ∈∀−= . lMhat´TI22 eK[ x CacMnYnBitEdl 021x71x60 2 <+− . cUrbgHajfa 0

1x3sin <⎟

⎠⎞

⎜⎝⎛

−π .

dMeNa¼Rsay bgHajfa 0

1x3sin <⎟

⎠⎞

⎜⎝⎛

−π

tag 21x71x60)x(f 2 +−= ebI 021x71x600)x(f 2 =+−⇔=

150405041)21)(60(4)71( 2 =−=−−=Δ eKTaj¦s

53

1203971x,

127

120171x 21 =

+==

−=


- 53 -

eyIgán 021x71x60)x(f 2 <+−= naM[

53x

127

<< ¦ 59x3

47

<<

¦ 541x3

43

<−< naM[ 34

1x31

54

<−

<

eKTaj 3

41x35

4 π<

−π

<π naM[ 0

1x3sin <⎟

⎠⎞

⎜⎝⎛

−π .

dUcen¼ ebI x CacMnYnBitEdl 021x71x60 2 <+− ena¼eKán 0

1x3sin <⎟

⎠⎞

⎜⎝⎛

−π .

lMhat´TI23 eK[GnuKmn_ ½

IRy;x,)y1()x1(

)yx1)(yx()y,x(f 2222

2222

∈++−−

=

cUrbgHajfacMeBa¼RKb´ IRy,x ∈ eKán 41|)y;x(f| ≤


- 54 -

dMeNa¼Rsay

eyIgman 2222

2222

)y1()x1()yx1)(yx()y,x(f

++−−

=

22

2

22

2

2222

222222

2222

422422

2222

2422242222

2222

422242

)y1(y

)x1(x

)y1()x1()x1(y)y1(x

)y1()x1()xx21(y)yy21(x

)y1()x1(yxyx2yyxyx2x

)y1()x1(yxyyxx

+−

+=

+++−+

=

++++−++

=

++−−−++

=

+++−−

=

eyIgán½

IRy,x,0)y1(

y)x1(4

)x1()y1(

y)x1(4

)x1(x4)y1(

y41

)x1(x

41)y,x(f

22

2

22

22

22

2

22

222

22

2

22

2

∈∀≤+

−+−

−=

+−

++−

=

+−−

+=−


- 55 -

eKTaj 041)y,x(f ≤−

naM[ )1(IRy,x,41)y,x(f ∈∀≤

müa¨geToteyIgman ½

IRy,x,0)y1(4

)y1()x1(

x)y1(4

y4)y1()x1(

x)y1(

y41

)x1(x

41)y,x(f

22

22

22

2

22

222

22

2

22

2

22

2

∈∀≥+−

++

=

+−+

++

=

+−+

+=+

eKTaj 041)y,x(f ≥+

naM[ )2(IRy,x,41)y,x(f ∈∀−≥

tamTMnak´TMng )1( nig )2( eKTaján IRy,x,

41)y,x(f

41

∈∀≤≤− .

dUcen¼ 41|)y;x(f| ≤ cMeBa¼RKb´ IRy,x ∈ .


- 56 -

lMhat´TI24 eK[sIVút

)n(

n 32.....222U +++++=

cMeBa¼RKb´ *INn∈ . k-cUrkMnt´ nU CaGnuKmn_én n . x-cUrbgHajfa

n

n321

2.3sin2

3U......UUUπ

=×××× .

K-eKBinitüsIVút )n(

nn 32.....2222V ++++−= .

cUrKNna nV nig lImIt nnVlim

+∞→ .

dMeNa¼Rsay k-kMnt nU CaGnuKmn_én n eyIgman

)n(

n 32.....222U +++++=

eyIgán 6

cos23U1π

==


- 57 -

24cos2

12cos22U2322U

12cos2

6cos22U232U

23

12

π=

π+=+=++=

π=

π+=+=+=

--------------------------------------------------------- eyIgsnμtfavaBitdltYTI kk 2.3

cos2U π= .

eyIgnwgRsaybBa¢ak´favaBitdl´tYTI )1k( + KW ½

1k1k 2.3cos2U ++

π= Bit

eyIgman ½

k1k U232...2222U +=++++++=+ edaytamkarsn μteKman kk 2.3

cos2U π=

eyIgán ½

1k1k2

k1k 2.3cos2

2.3cos4

2.3cos22U +++

π=

π=

π+= Bit .

dUcen¼ nn 2.3cos2U π

= . x-bgHajfa

n

n321

2.3sin2

3U......UUUπ

=××××


- 58 -

eyIgman n

n

nn

2.3sin

2.32sin

2.3cos2U

π

π

=π

=

( rUbmnþ acosasin2a2sin = ) eyIgán ½

⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

π

π

=

−−−−−−−−−−−−−−

π

π

=

π

π

=

n

n

n

2

1

2.3sin

2.32sin

U

12sin

6sin

U

6sin

3sin

U

dUcen¼ nn

n321

2.3sin2

3

2.3sin

3sin

U......UUUπ

=π

π

=×××× .


- 59 -

K-KNna nV nig lImIt nnVlim

+∞→

eyIgman )n(

nn 32.....2222V ++++−=

1n

nn

)1n(

nn

U22V

32....2222V

−

−

−=

++++−=

eday nn 2.3cos2U π

= ena¼ 1n1n 2.3cos2U −−

π=

eyIgán ½

n1n

n2n

1nn

n

2.3sin2

2.3sin42

2.3cos222V

π=

π=

π−=

+

−

dUcen¼ n1n

n 2.3sin2V π

= + .

ehIy 3

22.3

sin2limVlim n1n

nnn

π=

π= +

+∞→+∞→ .


- 60 -

lMhat´TI25 eK[ 1a ≥ nig 1b ≥ .

cUrbgHajfa )2


≤+

dMeNa¼Rsay

bgHajfa )2


≤+ cMeBa¼ 1a ≥ nig 1b ≥ eyIgman b.a2ba ≥+ ( vismPaB GMAM − ) ¦ b.a

2ba≥

+

naM[ )blogalog(21)

2ba(log 222 +≥

+

¦ )1()2


≤+ müa¨geToteKman ½

blog.alog2blogalog 2222 ≥+


- 61 -

)2()blogalog(21blogalog

)blogalog()bloga(log2

blogblog.alog2alog)bloga(log2

22222

22222

222222

+≥+

+≥+

++≥+

tamTMnak´TMng )1( nig )2( eyIgTaj½

)2

ba(log2blogalog

)2

ba(log4)blogalog(

)2

ba(log2)blogalog(21

222

22

22

22

22

+≤+

+≤+

+≤+

dUcen¼ )2


≤+ .


- 62 -

lMhat´TI26 eK[ θ CacMnYnBitEdl

20 π

<θ< . cUrbgHajfa ( ) ( ) 1cossin sincos >θ+θ θθ

dMeNa¼Rsay bgHajfa ( ) ( ) 1cossin sincos >θ+θ θθ tamvismPaB Bernoulli eKman 0,1x,x1)x1( >α−>∀α+≤+ α eyIgman ½

θθθ−θ+θ

<⎟⎠⎞

⎜⎝⎛

θ

θθ−θ

+<⎟⎠⎞

⎜⎝⎛

θ

⎟⎠⎞

⎜⎝⎛

θθ−

+=⎟⎠⎞

⎜⎝⎛

θ

θ

θ

θθ

sincossincossin

sin1

sin)sin1(cos1

sin1

sinsin11

sin1

cos

cos

coscos

¦ ( ) )1(cossincossin

sinsin cos

θθ−θ+θθ

>θ θ


- 63 -

RsaydUcxagelIen¼EdreyIgán½ ( ) )2(

cossincossincoscos sin

θθ−θ+θθ

>θ θ bUkvismPaB )1( nig )2( xagelIen¼eyIgán ½ ( ) ( )

θθ−θ+θθ+θ

>θ+θ θθ

cossincossincossincossin sincos

edayeKman 1cossincossin

cosson>

θθ−θ+θθ+θ

dUcen¼ ( ) ( ) 1cossin sincos >θ+θ θθ . lMhat´TI27 eK[RtIekaN ABC mYy .

k-cUrRsaybBa¢akfa 2


x-bgHajfa 8

332Ccos

2Bcos

2Acos ≤ .

dMeNa¼Rsay

k-RsaybBa¢akfa 2

33CsinBsinAsin ≤++


- 64 -

eyIgmanplbUkmMukñúgRtIekaN π=++ CBA . eyIgán )1(

2BAcos

2BAsin2BsinAsin −+

=+

edayeKman 12

BAcos ≤− nig 0

2BAsin2 >

+

eKán )2(2

BAsin22

BAcos2

BAsin2 +≤

−+ tam )1( nig )2( eKTaj ½

)3(2

BAsin2BsinAsin +≤+

dUcKñaEdreKman ½

23

CBACsin2)

3CBA(sinCsin

+++

≤++

+

)4()6

C4BA(sin2)3

CBAsin(Csin ++≤

+++

bUkvismIkar )3( nig )4( eyIgán½ ⎥⎦⎤

⎢⎣⎡ ++

++

≤++

+++ )6

C4BAsin(2

BAsin2)3

CBAsin(CsinBsinAsin

)3

CBAsin(2)2

6C4BA

2BA

sin(2)6

C4BAsin(2

BAsin ++=

+++

+

≤++

++


- 65 -

eKTaj ½ )

3CBAsin(4)

3CBAsin(CsinBsinAsin ++

≤++

+++ ¦

233

3sin3)

3CBA(sin3CsinBsinAsin =

π=

++≤++

dUcen¼ 2


x-bgHajfa 8

332Ccos

2Bcos

2Acos ≤

eyIgman ½

2Ccos

2Csin2

2BAcos

2BAsin2CsinBsinAsin +

−+=++

eday 2Ccos)

2C

2sin(

2BAsin =−

π=

+

nig 2Csin

2BAcos =

+ eKán

2Ccos

2Csin2

2BAcos

2Ccos2CsinBsinAsin +

−=++

2Ccos

2Bcos

2Acos4CsinBsinAsin

)2

BAcos2

BA(cos2Ccos2CsinBsinAsin

)2Csin

2BAcos(

2Ccos2CsinBsinAsin

=++

++

−=++

+−

=++


- 66 -

eday 2

33CsinBsinAsin ≤++

eKTaj 2

332Ccos

2Bcos

2Acos4 ≤

naM[ 8

332Ccos

2Bcos

2Acos ≤

dUcen¼ 8

332Ccos

2Bcos

2Acos ≤ .

lMhat´TI28 eK[smIkar 08m8m4x)3m2(2x:)E( 22 =+++++ k-cUrkMnt´bNþatémø INm ∈ edIm,I[smIkaren¼man¦s CacMnYnKt´rWLaTIhV . x-rk¦sEdlCacMnYnKt´rWLaTIhVrbs´smIkar .

dMeNa¼Rsay kMnt´témø m smIkar 08m8m4x)3m2(2x:)E( 22 =+++++


- 67 -

eKman 1m4)8m8m4()3m2(' 22 +=++−+=Δ edIm,I[smIkaren¼man¦sCacMnYnKt´rWLaTIhVluu¼RtaEt½ 1m4' +=Δ CakaerRákd mann&yfa *INk,k1m4 2 ∈∀=+

eKán 4

1km2 −

= EtedaysarEt *INm∈ dUcen¼eKRtUv[ kCacMnYness BIeRBa¼ebI *INp,1p2k ∈∀−=

ena¼ )1p(p4

1)1p2(m2

−=−−

= CacMnYnKt´ . dUcen¼ *INp,)1p(pm ∈∀−= . x-rk¦sEdlCacMnYnKt´rWLaTIhVrbs´smIkar tamsRmayxagelIebI *INp,)1p(pm ∈∀−= eKán 2)1p2(1)1p(p41m4' −=+−=+=Δ eKTaj¦s ⎢

⎣

⎡

−−=−−+−=

−+−=−++−=

2p2)1p2()3m2(x

4p4p2)1p2()3m2(x2

2

21

dUcen¼ *INp,2p2x,4p4p2x 22

21 ∈∀−−=−+−= .


- 68 -

lMhat´TI29 M Cama¨RTIskaerlMdab´bIEdlmanFatuTaMgGs´CacMnYKt´ . eKdwgfaplKuNénFatuTaMgbItamCYredknImYy@ plKuNén FatuTaMgbItamCYrQrnImYy@nig plKuNénFatuTaMgbItam Gg;t´RTUgesμInwg mdUcKña . bgHajfa m CaKUbéncMnYnKt´ .

dMeNa¼Rsay bgHajfa m CaKUbéncMnYnKt´ ½

tag ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

333231

232221

131211

aaaaaaaaa

M Edl 3;2;1j,3;2;1i,INaij ==∈

tambMrab´eyIgán

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

=

=

=

=

=

=

=

=

)8(ma.a.a)7(ma.a.a)6(ma.a.a)5(ma.a.a)4(ma.a.a)3(ma.a.a)2(ma.a.a)1(ma.a.a

132231

332211

332313

322212

312111

333231

232221

131211


- 69 -

KuNsmIkar )1( nig )3( eKán ½ )9(ma.a.a.a.a.a 2

333231131211 = KuNsmIkar )7( nig )8( eKán ½

)10(ma.a.a.a.a.a 2132231332211 =

EcksmIkar )9( nig )10( eKán ½

2

2

132231332211

333231131211

mm

a.a.a.a.a.aa.a.a.a.a.a

=

naM[ 2223212 aa.a = ykeTACYskñúg )5(

eKán m)a( 322 = .

dUcen¼ 322 )a(m = CaKUbéncMnYnKt´ .


- 70 -

lMhat´TI30 eK[bYncMnYnviC¢man d,c,b,a . cUrbgHajfa ½

2bad

dadc

cdcb

bcba

a1 <++

+++

+++

+++

< dMeNa¼Rsay bgHajfa 2

badd

adcc

dcbb

cbaa1 <

+++

+++

+++

++<

cMeBa¼RKb´ 0d,0c,0b,0a >>>>

eyIgman ⎪⎪⎩

⎪⎪⎨

⎧

+>++>++++>++>++++>++>++++>++>+++

dbbaddcbaacadcdcbadbdcbdcbacacbadcba

eKTaj

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

+<

++<

+++

+<

++<

+++

+<

++<

+++

+<

++<

+++

dbd

badd

dcbad

cac

adcc

dcbac

dbb

dcbb

dcbab

caa

cbaa

dcbaa


- 71 -

edaybUkTMnak´TMngTaMgen¼Gg:nwgGg:eKán ½ 2

badd

adcc

dcbb

cbaa1 <

+++

+++

+++

++<

dUcen¼vismPaBRtUvánRsaybBa¢ak´ .

lMhat´TI31 eKmanbIcMnYnBit 0c;0b;0a >>> . cUrRsayfa abc6)ac(ca)cb(bc)ba(ab ≥+++++

dMeNa¼Rsay Rsayfa abc6)ac(ca)cb(bc)ba(ab ≥+++++ tag abc6)ac(ca)cb(bc)ba(abA −+++++=

0c,b,a,0)ba(c)cb(a)ca(bA)ab2ba(c)bc2cb(a)ac2ca(bA

)abc2cbca()abc2acab()abc2bcba(Aabc6caacbccbabbaA

222

222222

222222

222222

>∀≥−+−+−=

−++−++−+=

−++−++−+=

−+++++=

dUcen¼ abc6)ac(ca)cb(bc)ba(ab ≥+++++ .


- 72 -

lMhat´TI32 eK[BIrcMnYn x nig y xusBIsUnü nig mansBaØadUcKña . cUrbgHajfa 04

xy

yx3

xy

yx

2

2

2

2≥+⎟

⎠

⎞⎜⎝

⎛+−+ .

dMeNa¼Rsay bgHajfa 04

xy

yx3

xy

yx

2

2

2

2

≥+⎟⎠

⎞⎜⎝

⎛+−+

tag xy

yxP += eKán 2

xy

yx

xy

yxP 2

2

2

222 ++=⎟

⎠

⎞⎜⎝

⎛+=

yk 4xy

yx3

xy

yxM 2

2

2

2

+⎟⎠

⎞⎜⎝

⎛+−+=

eKán )2P)(1P(2P3P4P32PM 22 −−=+−=+−−= eday 0

xy)yx(2

xy

yx2P

2

≥−

=−+=−

( eRBa¼ x nig y mansBaØadUcKña ) ehtuen¼ 0)1P)(2P(M ≥−−= .

dUcen¼ 04xy

yx3

xy

yx

2

2

2

2

≥+⎟⎠

⎞⎜⎝

⎛+−+ .


- 73 -

lMhat´TI33 eK[smIkar ca,0a,0cbxax2 ≠≠=++ . tag αtan nig βtan Ca¦srbssmIkarxagelI . cUrKNnatémøénkenßam ½

)(cosc)cos()sin(b)(sinaA 22 β+α+β+αβ+α+β+α=

dMeNa¼Rsay KNnatémøénkenßam ½

)(cosc)cos()sin(b)(sinaA 22 β+α+β+αβ+α+β+α= eday αtan nig βtan Ca¦srbssmIkareKán ½

⎪⎪⎩

⎪⎪⎨

⎧

=βα

−=β+α

actan.tan

abtantan

eyIgán ac

bca

b

ac1

ab

tantan1tantan)tan(

−=

−−=

−

−=

βα−β+α

=β+α

eyIgán )(cosc)cos()sin(b)(sinaA 22 β+α+β+αβ+α+β+α=


- 74 -

[ ][ ]

c)ac(

)ac(c)ac(babb)ac(

)ac(

cac

b)ac(

abb)ac(

)ac(

c)tan(b)(tana)(tan1

1c)tan(b)(tana)(cosA

2

222

22

2

2

2

2

22

2

22

22

=−

−+−+×

+−−

=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+

−⎥⎦

⎤⎢⎣

⎡

+−−

=

+β+α+β+αβ+α+

=

+β+α+β+αβ+α=

dUcen¼ c)(cosc)cos()sin(b)(sinaA 22 =β+α+β+αβ+α+β+α=

lMhat´TI34 cMnYnKt;viC¢man n Ecknwg * [sMNl; ! . cMnYn n enaHEcknwg % [sMNl; @ . k-ebIcMnYn n enaHEcknwg $0 [sMNl;b:un μan ? x-rkcMnYn n enaHedaydwgfa 4000n3940 << .

dMeNa¼Rsay k> ebIcMnYn n enaHEcknwg 40 [sMNl;b:un μan ?


- 75 -

]bmafa n Ecknwg * [plEck INq1 ∈ nigsMNl; ! nig cMnYn n enaHEcknwg % [plEck INq2 ∈ nigsMNl; @ . tamGWKøIt eyIg)an

)16()15(

2q5n1q8n

2

1 −

⎩⎨⎧

+=+=

b¤ ⎩⎨⎧

+=−−=−

)2(32q80n16)1(15q120n15

2

1

bUksmIkar ¬!¦ nig ¬@¦ eyIg)an 17q4017q120q80n 12 +=+−= Edl 12 q3q2q −= . tamTMnak;TMng 17q40n += bBa¢ak;faebIcMnYn n enaHEcknwg 40 [sMNl; 17r = x> rkcMnYn n enaHedaydwgfa 4000n3940 << eyIgman 17q40n += eday 4000n3940 << eKTaj 400017q403940 <+< b¤

4017100n

40398 +<<+

eday INq∈ naM[eKTaj)an }100,99{q = ehIy }4017,3977{n = .


- 76 -

lMhat´TI35 edaHRsayRbBn½æsmIkar ⎩⎨⎧

=

=+

64yx

26)ylogxlog(5 xy

dMeNa¼Rsay edaHRsayRbBn½æsmIkar ⎩⎨⎧

=

=+

)2(64yx

)1(26)ylogxlog(5 xy

lkçx½NÐ ⎩⎨⎧

≠≠>>

1y,1x0y,0x

smIkar )1( Gacsresr 526

xlog1xlog

yy =+

b¤ 01xlog.526)x(log y

2y =+− tag xlogt y=

eK)an 25

144125

169',01t526t 2 =−=Δ=+−

naM[ 55

125

13t,51

512

513t 21 =+==−=

- cMeBaH 51t = eK)an

51xlogy = naM[ )3(yx 5

1

=


- 77 -

yk ¬#¦ CYskñúg ¬@¦ eK)an 64y.y 51

= naM[ 32y =

ehIytam ¬#¦ eK)an 2)32(x 51

== . - cMeBaH 5t = eK)an 5xlogy = naM[ )4(yx 5= yk ¬$¦ CYskñúg ¬@¦ eK)an 64y.y 5 = naM[ 2y = ehIytam ¬$¦ eK)an 322x 5 == . dUcenHRbB½næmanKUcemøIy )32y,2x( == b¤ )2y,32x( == lMhat´TI36

edaHRsayRbBn½æsmIkar ⎪⎪⎩

⎪⎪⎨

⎧

=

=

90016.5

40015.4

yx

yx

dMeNa¼Rsay edaHRsayRbBn½æsmIkar ³

⎪⎪⎩

⎪⎪⎨

⎧

=

=

90016.5

40015.4

yx

yx


- 78 -

eyIgman )4001ln()5.4(ln yx =

b¤ )1(5ln24ln25lny4lnx −−=+ ehIy )

9001ln()6.5ln( yx =

b¤ )2(5ln26ln26lny5lnx −−=+ tam )2(&)1( eK)anRbBnæ½ xageRkam

)5ln()6ln(

5ln26ln26lny5lnx5ln24ln25lny4lnx −

⎩⎨⎧

−−=+−−=+

⎩⎨⎧

−−=+

+=−−

)4(5ln25ln.6ln26ln.5lny5lnx

)3(6ln.5ln26ln.4ln26ln.5lny6ln.4lnx22

bUksmIkar )4(&)3( eK)an )5ln.6ln.4(ln2x)6ln.4ln5(ln 22 −=− naM[ 2x −=

tamsmIkar 40015.4 yx = eKTaj

40015.4 y2 =−

naM[eKTaj 2y −= . dUcenHRbB½næsmIkarmanKUcemøIy 2y,2x −=−= .


- 79 -

lMhat´TI37 eda¼RsaysmIkar ½

1339222 )1x(x1xx +=+ −−−

dMeNa¼Rsay eda¼RsaysmIkar ½

1339222 )1x(x1xx +=+ −−− mann&yRKb´ IRx∈

smIkarGacsresr 1333222 )1x(x1)xx(2 +=+ −−−

tag X3 )xx(2 2=− nig Y3

2x1 =− Edl 0Y,0X >> eKán 222 )1x(x1)xx(2 33Y.X −−+− == smIkarkøayCa 1Y.XYX +=+

0)Y1)(1X(

0)Y1()Y1(X0)Y1()XYX(

=−−=−−−=−−−

eKTaj ⎢⎣

⎡==

1Y1X smmUl

⎢⎢⎣

⎡

=

=−

−

13

132

2

x1

)xx(2


- 80 -

smmUl ⎢⎢⎣

⎡

=−

=−

0x1

0)xx(22

2

eKTaj¦s }1,0,1{x −∈ . lMhat´TI38 eK[ c;b;a CabIcMnYnBitviC¢mandac´xat . cUrbgHajfa

2cba

bac

acb

cba 222 ++

≥+

++

++

dMeNa¼Rsay bgHajfa

2cba

bac

acb

cba 222 ++

≥+

++

++

tag ba

cac

bcb

aT222

++

++

+=


- 81 -

( )

( ) )1(4)ba

1ac

1cb

1()cba(cbaT

4)1ba

c()1ac

b()1cb

a(cbaT

)1ba

ccb

bcb

a)(cba(T

)cba(ba

)bac(cac

)acb(bcb

)cba(aT

)cba()cba

c()bac

b()acb

a(T222

⎥⎦⎤

⎢⎣⎡ −

++

++

+++++=

⎥⎦⎤

⎢⎣⎡ −+

+++

+++

+++=

−+

++

++

++=

++−+++

++++

++++

=

++−++

+++

+++

=

tamvismPaB GMAM − eyIgman ½

3

3

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

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

+++≥++

+++≥+++++

eKTaj )2()ac)(cb)(ba(23cba 3 +++≥++

nig )3()ac)(cb)(ba(3ac

1cb

1ba

1 3 +++≥+

++

++

KuNvismPaB )(2 nig )(3 Gg:nwgGg:eKán ½

29)

ac1

cb1

ba1()cba( ≥

++

++

+++

214)

ac1

cb1

ba1()cba(

4294)

ac1

cb1

ba1()cba(

≥−+

++

++

++

−≥−+

++

++

++


- 82 -

KuNGg:TaMgBIrnwg 0cba >++ eKán ½ )5(

2cba4)

ac1

cb1

ba1)(cba()cba( ++

≥⎥⎦⎤

⎢⎣⎡ −

++

++

+++++

tamTMnak´TMng )(4 nig )(5 eKán 2

cbaT ++≥

dUcen¼ 2

cbaba

cac

bcb

a 222 ++≥

++

++

+ .

lMhat´TI39 cUrkMnt;rkGnuKmn_ )x(f nig )x(g ebIeKdwgfa ³

2x)1x3(g2)1x2(f =++− nig 1x2x2)2x6(g)3x4(f 2 ++−=−−− cMeBaHRKb; IRx∈ .

dMeNa¼Rsay kMnt;rkGnuKmn_ )x(f nig )x(g ³ eKman )1(x)1x3(g2)1x2(f 2=++− nig )2(1x2x2)2x6(g)3x4(f 2 ++−=−−− eyIgtag 3t41x2 −=− naM[ 1t2x −=


- 83 -

yk 1t2x −= CYskñúg ¬!¦ eK)an ³ [ ] [ ]

)3(1t4t4)2t6(g2)3t4(f

)1t2(1)1t2(3g21)1t2(2f2

2

+−=−+−

−=+−+−−

ebIeKyk tx = CYskñúg¬@¦ eK)an )4(1t2t2)2t6(g)3t4(f 2 ++−=−−−

tam ¬#¦ nig ¬$¦ eK)anRbB½næ

⎪⎩

⎪⎨⎧

++−=−−−

+−=−+−

)4(1t2t2)2t6(g)3t4(f

)3(1t4t4)2t6(g2)3t4(f2

2

dksmIkar¬#¦nig¬$¦eK)an t6t6)2t6(g3 2 −=− naM[ 2t2)2t6(g 2 −=−

yk 2t6x −= naM[ 6

2xt +=

ehIy 18

)8x)(4x(2)6

2x(2)x(g 2 +−=−

+=

tam ¬$¦naM[ 1t2t2)2t2()3t4(f 22 ++−=−−− naM[ 1t2)3t4(f −=− yk 3t4x −= naM[

43xt +

= eKTaj

21x1)

43x(2)x(f +

=−+

=

dUcenH 18

)8x)(4x()x(g,2

1x)x(f +−=

+= .


- 84 -

lMhat´TI40 eK[RtIekaN ABC mYymanRCúg c,b,a . kMnt´RbePTénRtIekaN ABC ebIeKdwgfa ½ )

c1

b1

a1(

21

cCcos

bBcos

aAcos

++=++

dMeNa¼Rsay RbePTénRtIekaN ABC tamRTwsþIbTkUsIunUskñúgRtIekaN ABC eKman ½

bc2acbAcos

222 −+= naM[ )1(

abc2acb

aAcos 222 −+=

dUcKñaEdreKTaj )2(abc2

bcab

Bcos 222 −+=

nig )3(abc2

cbac

Ccos 222 −+=

bUkTMnak´TMng )3(,)2(,)1( Gg:nwgGg:eKán ½

abc2cba

cCcos

bBcos

aAcos 222 ++

=++ eday )

c1

b1

a1(

21

cCcos

bBcos

aAcos

++=++


- 85 -

eKTaján ½

0)ac()cb()ba(

0)aac2c()cbc2b()bab2a(

ac2bc2ab2c2b2a2

acbcabcbaabc2

abcabcabc2

cba

)c1

b1

a1(

21

abc2cba

222

222222

222

222

222

222

=−+−+−

=+−++−++−

++=++

++=++

++=

++

++=++

eKTajánsmPaB cba == . dUcen¼ ABC CaRtIekaNsmát .


- 86 -

lMhat´TI41 k¿ cUrRsaybBa¢akTMnak´TMng x2cot2xcotxtan −= x¿ cUrKNnaplbUkxageRkam ½ nn22n 2

atan21....

2atan

21

2atan

21atanS ++++=

dMeNa¼Rsay k¿ RsaybBa¢akTMnak´TMng x2cot2xcotxtan −= tag x2cot2xcotA −=

eday ⎪⎪⎩

⎪⎪⎨

⎧

−==

=

xtan2xtan1

x2tan1x2cot

xtan1xcot

2

eKán

xtanxtan

xtan11

)xtan2

xtan1(2xtan

1A

2

2

=+−

=

−−=

dUcen¼ x2cot2xcotxtan −= .


- 87 -

x¿ KNnaplbUkxageRkam ½

a2cot22acot

21

2acot

21

2acot

21

)2

acot22a(cot

21

2atan

21

2atan

21....

2atan

21

2atan

21atanS

nn

n

0k1k1kkk

n

0k1kkk

n

0kkk

nn22n

−=⎟⎠⎞

⎜⎝⎛ −=

⎥⎦⎤

⎢⎣⎡ −=

⎟⎠⎞

⎜⎝⎛=

++++=

∑

∑

∑

=−−

=−

=

dUcen¼ a2cot2

2acot

21

2atan

21....

2atan

21atanS nnnnn −=+++=


- 88 -

lMhat´TI42 eK[GnuKmn_ 2x)x(f 2 −= Edl IRx∈ k-eKyk )x(fU1 = nig )U(fU n1n =+ cMeBa¼RKb´ *INn∈ . cUrbgHajfa [ ][ ][ ].......)x(ff.......ffU nn = . x-RsayfaebI 2x > eKán 2Un > RKb´ *INn∈ . K-eKtag 4UUV 2

nnn −−= RKb´ *INn∈ nig 2x > . cMeBa¼RKb´ *INn∈ cUrbgHajfa 2

n1n VV2 =+ . X-sn μtfa 2lnVlnW nn −= cMeBa¼RKb´ *INn∈ . cUrrkRbePTénsIVút nW . g-eRbIlTæplxagelIcUrTajrkGnuKmn_ ½

[ ][ ][ ].........)x(ff........ff)x(F nn = . dMeNa¼Rsay k-bgHajfa [ ][ ][ ].......)x(ff.......ffU nn = eyIgman )x(fU1 = Bit ( tamsm μtikm μ ) [ ])x(ff)U(fU 12 == Bit ( eRBa¼ )U(fU n1n =+ )


- 89 -

[ ][ ])x(fff)U(fU 23 == Bit eyIgsnμtfavaBitdl´tYTIkKW ½

[ ][ ][ ].......)x(ff.......ffU kk = Bit eyIgnwgRsayfavaBitdl´tYTI 1k + KW ½

[ ][ ][ ].......)x(ff.......ffU 1k1k ++ = Bit eyIgman

( ) [ ][ ][ ][ ]( )[ ][ ][ ].......xff.......ff

......)x(ff........fffUfU

1k

kk1k

+

+

===

dUcen¼ [ ][ ][ ].......)x(ff.......ffU nn = . x-RsayfaebI 2x > eKán 2Un > RKb´ *INn∈ eyIgman 2U)U(fU 2

nn1n −==+ ebI 2x > ena¼ 22x)x(fU 2

1 >−== ¦ 2U1 > Bit eyIgsnμtfavaBitdltYTI k KW 2Uk > Bit eyIgnwgRsayfavaBitdl´tYTI 1k + KW 2U 1k >+ Bit eyIgman 2UU 2

k1k −=+ eday 2Uk > naM[ 4U 2

k > ¦ 2242U 2k =−>−

eKTaj 22UU 2k1k >−=+ Bit .


- 90 -

dUcen¼ ebI 2x > eKán 2Un > RKb´ *INn∈ . K-bgHajfa 2

n1n VV2 =+ eyIgman 4UUV 2

nnn −−= eyIgán 4UUV 2

1n1n1n −−= +++ Et 2UU 2n1n −=+

( ) 2n

22nn1n

22n

2nn

2n1n

2nn

2n1n

2nn

2n1n

2n

4n

2n1n

22n

2n1n

V4UUV2

)4U(4UU2UV2

4UU2U2V2

4UUUV

U4UUV

4)2U(UV

=−−=

−+−−=

−−=

−−=

−−=

−−−=

+

+

+

+

+

+

dUcen¼ 2n1n VV2 =+ .

X-rkRbePTénsIVút nW . eKman 2lnVlnW nn −= cMeBa¼RKb´ *INn∈ eKán 2lnVlnW 1n1n −= ++ eday 2

n1n VV2 =+

nn

2n

1n W22ln2Vln22ln2

VlnW =−=−=+

dUcen¼ )W( n CasIVútFrNImaRtmanersug 2q = .


- 91 -

g-rkGnuKmn_ [ ][ ][ ].........)x(ff........ff)x(F nn = eday [ ][ ][ ].......)x(ff.......ffU nn = eKTaján nn U)x(F = . tamsRmayxagelIeyIgman )W( n CasIVútFrNImaRtman ersug 2q = . tamrUbmnþ 1

1n1n1n W.2qWW −− =×=

eday ⎟⎠⎞

⎜⎝⎛=−=

2V

ln2lnVlnW 111

Et 4)x(f)x(f4UUV 22111 −−=−−=

2222

2421

2221

)4xx(214xx2x

x4x2xV

4)2x(2xV

−−=−−−=

−−−=

−−+−=

eKán 2

222

1 24xxln

4)4xx(lnW ⎟

⎟⎠

⎞⎜⎜⎝

⎛ −−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−=

ehtuen¼ n2

22

21n

n 24xxln

24xxln.2W ⎟

⎟⎠

⎞⎜⎜⎝

⎛ −−=⎟

⎟⎠

⎞⎜⎜⎝

⎛ −−= −

eday ⎟⎠⎞

⎜⎝⎛=−=

2V

ln2lnVlnW nnn


- 92 -

eKTaj n2

2n

24xxln

2V

ln ⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=⎟

⎠⎞

⎜⎝⎛

¦ n2

2

n 24xx2V ⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

müa¨geToteKman 4UUV 2nnn −−=

nn

n

2n

n

2nnn

2n

2nnn

2n

2nnn

V2V

21

V24V

U

4VVU2

4UVVU2U

4UVU

+=+

=

+=

−=+−

−=−

eday n2

2

n 24xx2V ⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

eKán n

n

22

22

n

24xx

12

4xxU

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−+⎟

⎟⎠

⎞⎜⎜⎝

⎛ −−=

KuNnwgkenßam n2

2

24xx⎟⎟⎠

⎞⎜⎜⎝

⎛ −+ eKán ½


- 93 -

nn

n

n

n

nn

n

n

22

22

n

222

22

22

n

22

22

22

22

n

24xx

24xxU

44xx

24xx

24xxU

24xx

24xx

24xx

24xxU

⎟⎟⎠

⎞⎜⎜⎝

⎛ −++⎟

⎟⎠

⎞⎜⎜⎝

⎛ −−=

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

+⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+⎟⎟⎠

⎞⎜⎜⎝

⎛ −−

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

+⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

dUcen¼

nn 22

22

n 24xx

24xx)x(F ⎟

⎟⎠

⎞⎜⎜⎝

⎛ −++⎟

⎟⎠

⎞⎜⎜⎝

⎛ −−= .


- 94 -

lMhat´TI43 eK[GnuKmn_

1x3x31x3x3x)x(f 2

23

+−+−+

= kMntćMeBa¼RKb´ IRx∈ . cMeBa¼RKbćMnYnBitviC¢man a nig b cUrRsaybBa¢ak´fa ½

⎟⎠⎞

⎜⎝⎛

+++++

≥⎟⎠⎞

⎜⎝⎛ ++

ba2abba1f

2ba1f .

dMeNa¼Rsay RsaybBa¢ak´fa ⎟

⎠⎞

⎜⎝⎛

+++++

≥⎟⎠⎞

⎜⎝⎛ ++

ba2abba1f

2ba1f

eyIgman 1x3x3

1x3x3x)x(f 2

23

+−+−+

= kMntćMeBa¼RKb´ IRx∈

IRx,0)1x3x3(

)1x(x3)1x3x3(x3x6x3

)1x3x3()1x3x3x)(3x6()1x3x3)(3x6x3()x('f

22

22

22

234

22

2322

∈∀≥+−

−=

+−+−

=

+−+−+−−+−−+

=

dUcen¼ )x(f CaGnuKmn_ekInelI IR . müa¨geToteyIgsnμtfa

ba2abba1

2ba1

+++++

≥++

eKán abba1

ba2ba1

2+++++

≤++


- 95 -

b1

1a1

1ba1

2)b1)(a1()b1()a1(

ba12

++

+≤

++

+++++

≤++

eday a1

1ba1

1+

≤++

nig b1

1ba1

1+

≤++

RKb´cMnYnBitviC¢man a nig b. eKTaj

b11

a11

ba12

++

+≤

++

naM[karsnμt ba2abba1

2ba1

+++++

≥++ Bit.

dUcen¼tamlkçN¼GnuKmn_ekIneKTaján ½ ⎟⎠⎞

⎜⎝⎛

+++++

≥⎟⎠⎞

⎜⎝⎛ ++

ba2abba1f

2ba1f .


- 96 -

lMhat´TI44 eK[Exßekag

xm1m4mx4x)x(fy:)C(

22

mm −+−+

== cUrbgHajfamanExßekagBIrénRKYsarExßekag )C( m Edlkat´tamcMnuc )y;x(M 000 cMeBa¼RKb´( ) 2

00 IRy;x ∈ . mCaá¨ra¨Em¨Rt . dMeNa¼Rsay karbgHaj ½ ebI )C(M m0 ∈ eKán

0

20

20

0 xm1m4mx4x

y−

+−+=

smmUl 1m4mx4x)xm(y 20

2000 +−+=−

)1(01yxxm)x4y(m4

1m4mx4xyxmy

002

0002

20

20000

=−−−−+

+−+=−

DIsRKImINg´énsmIkar )1( sresr ½

IRy,x;0)1x(16)x4y(

)1x(16)x16yx8y(

16yx16x16x16yx8y

)1yxx(16)x4y(

002

02

00

20

2000

20

002

02

0002

0

002

02

00

∈∀>+++=

++++=

++++−=

−−−−−=Δ


- 97 -

eday 0>Δ naM[smIkar )1( man¦sBIrepßgKñaCanic© . dUcen¼manExßekagBIrénRKYsarExßekag )C( m Edlkat´tam cMnuc )y;x(M 000 cMeBa¼RKb´ ( ) 2

00 IRy;x ∈ . lMhat´TI45 eK[ )x(P CaBhuFadWeRkTIbI . eKdwgfa 2)x(P + Eckdacńwg 2)1x( + ehIy 2)x(P − Eckdacńwg 2)1x( − . cUrkMnt´rkBhuFa )x(P . dMeNa¼Rsay kMnt´rkBhuFa )x(P ½

tambMrabéKGacsresr ⎪⎩

⎪⎨⎧

+−=−

++=+

)2()dcx()1x(2)x(P

)1()bax()1x(2)x(P2

2

eKán ⎩⎨⎧

=−=+−02)1(P

02)1(P naM[ ⎩⎨⎧

=−=−

2)1(P2)1(P

edayeFIVedrIevelI )1( nig )2( eKán ½


- 98 -

⎪⎩

⎪⎨⎧

−++−=

++++=2

2

)1x(c)dcx)(1x(2)x('P

)1x(a)bax)(1x(2)x('P

⎩⎨⎧

−++−=++++=

)4()1x(c)dcx(2)[1x()x('P)3()]1x(a)bax(2)[1x()x('P

tam )3( nwg )4( bBa¢ak´fa )x('P Eckdac´nwg )1x)(1x( −+ eKTaj )1x)(1x(k)x('P −+= ( eRBa¼ )x(P CaBhuFadWeRkTI3 ) eKán ∫ +−=−= r)x

3x(kdx).1x(k)x(P

32

cMeBa¼ 1x ±= eKán ⎪⎪⎩

⎪⎪⎨

⎧

=+−=

−=+=−

2rk32)1(P

2rk32)1(P

eda¼RsayRbB&næen¼eKán 0r,3k == dUcen¼ x3x)x

3x(3)x(P 3

3−=−= .


- 99 -

lMhat´TI46 eK[ INn,i

31i

31A

nn

∈⎟⎠⎞

⎜⎝⎛ −−⎟

⎠⎞

⎜⎝⎛ += .

cUrbgðajfa 3

nsin.)3(

2.iA n

1n π=


dMeNa¼Rsay bgðajfa

3nsin.

)3(2.iA n

1n π=


eyIgman INn,i3

1i3

1Ann

∈⎟⎠⎞

⎜⎝⎛ −−⎟

⎠⎞

⎜⎝⎛ +=

tag ⎟⎠⎞

⎜⎝⎛ π

+π

=⎟⎟⎠

⎞⎜⎜⎝

⎛+=

+=+=

3sin.i

3cos

32

23i

21

32

33.i1i

31Z

tamrUbmnþdWmr½eK)an ⎟⎠⎞

⎜⎝⎛ π

+π

=⎟⎠⎞

⎜⎝⎛ +=

3nsin.i

3ncos

)3(2i

31Z n

nnn

ehIy ⎟⎠⎞

⎜⎝⎛ π

−π

=3

nsin.i3

ncos)3(

2Z n

nn

eKTaj ⎟⎠⎞

⎜⎝⎛ π

−π

−⎟⎠⎞

⎜⎝⎛ π

+π

=3

nsin.i3

ncos)3(

23

nsin.i3

ncos)3(

2A n

n

n

n

3nsin.

)3(2.i

3nsin.i

3ncos

3nsin.i

3ncos

)3(2

n

1n

n

n π=⎟

⎠⎞

⎜⎝⎛ π

+π

−π

+π

=+

dUcenH 3

nsin.)3(

2.iA n

1n π=

+

.


- 100 -

lMhat´TI47 eK[sIVúténcMnYnBit )U( n kMnt;eday ³ 3lnU0 = nig INn,)e1(lnU nU

1n ∈+=+ . cUrKNna nU CaGnuKmn_én n . dMeNa¼Rsay KNna nU CaGnuKmn_én n eKman INn,)e1(lnU nU

1n ∈+=+ eKTaj n1n UU e1e +=+ b¤ 1ee n1n UU =−+ efr naM[ ( )nUe CasIVútnBVnþmanplsgrYm 1d = nigtYTImYy 3ee 3lnU0 == eK)an n3e nU += naM[ )3nln(Un += .


- 101 -

lMhat´TI48 edaHRsaysmIkar 1y29x47 =+ kñúgsMNMucMnYnKt;rWLaTIhV . dMeNa¼Rsay edaHRsaysmIkar 1y29x47 =+ kñúgsMNMucMnYnKt;rWLaTIhV eyIgman 1812947 +×= naM[ 294718 −= 1111829 +×= naM[ 47229)2947(29182911 −×=−−=−= 711118 +×= naM[ )47229()2947(11187 −×−−=−= b¤ 3292477 ×−×= 41711 +×= naM[ )329247()47229(7114 ×−×−−×=−= b¤ 5294734 ×+×−= 1247 −×= naM[ )329247(2).529473(7241 ×−×−×+×−=−×=


- 102 -

b¤ )13(29)8(471 ×+−×= eK)an )13(29)8(47y29x47 ×+−×=+ )13y(29)8x(47 −−=+ naM[

⎩⎨⎧

∈∀=−−=+

Zq,q4713yq298x

dUcenH Zq,13q47y,8q29x ∈∀+=−−= . lMhat´TI49 cUrbgðajfa 3n22n3

n 73A ++ += Eckdac;nwg !! Canic©RKb;cMnYnKt;FmμCati n. dMeNa¼Rsay bgðajfa 3n22n3

n 73A ++ += Eckdac;nwg !! eyIgman )11mod(533 ≡ naM[ )11mod(5.93 n2n3 =+ müa:geTot )11(mod572 ≡ naM[ )11(mod57 nn2 ≡ ehIy )11(mod273 ≡


- 103 -

eKTaj)an )11(mod5.27 n3n2 ≡+ eyIg)an

)11(mod05.115.25.973A nnn3n22n3 ≡=+≡+= ++ . dUcenH 3n22n3

n 73A ++ += Eckdac;nwg !! Canic©RKb;cMnYnKt;FmμCati n . lMhat´TI50 cUrRsaybBa¢ak;facMnYn nnnn

n 124122462447E −−+= Eckdac;nwg 221Canic©cMeBaHRKb;cMnYnKt;Fm μCati n . dMeNa¼Rsay RsaybBa¢ak;facMnYn nE Eckdac;nwg 221 eyIgeXIjfa 1713221 ×= Edl 13 nig 17bzmrvagKña dUcenHedIm,IRsayfa cMnYn nE Eckdac;nwg 221eKRtUvRsay [eXIjfacMnYn nE vaEckdac;nwg !#pg nig Eckdac;nwg !& pg. eKman nnnn

n 124122462447E −−+=


- 104 -

)124462()122447(E nnnnn −+−=

tamrUbmnþ )b....baa)(ba(ba 1n2n1nnn −−− +++−=− eyIg)an 11n q).124462(p).122447(E −+−= )q26p25(13q338p325 1111 +=+= Edl *INq,p 11 ∈ . TMnak;TMngenHbBa¢ak;fa cMnYn nE Eckdac;nwg !# . müa:geToteKman )122462()124447(E nnnn

n −+−=

*INq,p,)q20p19(17

q340p323q).122462(p).124447(

2222

22

22

∈+=+=

−+−=

TMnak;TMngenHbBa¢ak;fa cMnYn nE Eckdac;nwg !& . dUcenHsrubesckþIeTAeyIg)an cMnYn nE Eckdac;nwg 221Canic© cMeBaHRKb;cMnYnKt;FmμCati n .


- 105 -

lMhat´TI51 eK[GnuKmn_ xcos.e)x(f x= k> KNna )x('f rYcbgðajfa )


x> edayeFIVvicartamkMeNIncUrbgðafaedrIevTI n kMnt;eday )

4nxcos(.e2)x(f xn)n( π

+=

dMeNa¼Rsay k> KNna )x('f rYcbgðajfa )


eyIgman xcose)x(f x= eyIg)an xsinexcose)x('f xx −=

)4

xcos(e2)4

sinxsin4

cosx(cose2

)xsin22xcos

22(e2

)xsinx(cose

xx

x

x

π+=

π−

π=

−=

−=

dUcenH )4

xcos(e2)x('f x π+= .

x> edayeFIVvicartamkMeNInfa )4

nxcos(.e2)x(f xn)n( π+=


- 106 -

eyIgman )x(f)4

xcos(.e2)x('f )1(x =π

+= Bit ]bmafavaBitdl;tYTI k KW )

4kxcos(.e2)x(f xk)k( π

+= Bit eyIgnwgRsayfavaBitdl;tYTI 1k + KW

⎟⎠⎞

⎜⎝⎛ π+

+=++

4)1k(xcose2)x(f x1k)1k(

eyIgman )')x(f()x(f )k()1k( =+

⎥⎦⎤

⎢⎣⎡ π

+−π

+=

π+−

π+=

)4

kxsin()4

kxcos(e2

)4

kxsin(e2)4

kxcos(e2

xk

xkxk

eday ⎟⎠⎞

⎜⎝⎛ π+

+=π

+−π

+4

)1k(xcos2)4

kxsin()4

kxcos(

eK)an ⎟⎠⎞

⎜⎝⎛ π+

+=++

4)1k(xcose2)x(f x1k)1k( Bit .

dUcenH )4

nxcos(.e2)x(f xn)n( π+= .


- 107 -

lMhat´TI52 eK[RtIekaN ABC mYymanRCug cAB,bAC,aBC === ]bmafa M CacMnucmYyenAkñúgRtIekaNenH ehIyeKtag y,x nig z CacMgayerogKñaBIcMnuc M eTARCug AC,BC nig AB énRtIekaN . cUrKNnatémøGb,brmaén 222 zyxT ++= CaGnuKmn_én c,b,a . dMeNa¼Rsay KNnatémøGb,brmaén 222 zyxT ++= CaGnuKmn_én c,b,a eyIgtag S CaépÞRkLarbs;RtIekaN ABC eyIg)an AMCBMCAMB SSSS ++=

A

B C

M

N

PP


- 108 -

ax

21by

21cz

21S

AC.MP21BC.MN

21AB.MQ

21S

++=

++=

eKTaj)an S2czbyax =++ tamvismPaBEb‘nUyIeyIg)an ³

T.cbaS2

zyx.cbaczbyax222

222222

++≤

++++≤++

eKTaj)an 222

2

cbaS4T++

≥ dUcenHtémøGb,rmaén 222 zyxT ++= KW 222

2

min cbaS4T++

=

Edl )cp)(bp)(ap(pS −−−= nig 2

cbap ++= .


- 109 -

lMhat´TI53 cUrkMnt;rkGnuKmn_ )x(fy = ebIeKdwgfa ³

1x1x)2x2xx(f 2

22

+−

=+−+ cMeBaHRKb;cMnYnBit x . dMeNa¼Rsay kMnt;rkGnuKmn_ )x(fy = ³ eKman )1(

1x1x)2x2xx(f 2

22

+−

=+−+ ebIeyIgtag t2x2xx 2 =+−+ eyIg)an xt2x2x2 −=+−

1t,)1t(2

2tx

2t)1t(x2

2tx2tx2

xtx2t2x2x

)xt(2x2x

2

2

2

222

22

≠−−

=

−=−

−=−

+−=+−

−=+−

yktémø )1t(2

2tx2

−−

= CMnYskñúg )1( eyIg)an ³


- 110 -

8t8t)8t8t(t)t(f

8t8tt8t8t)t(f

4t8t44t4t4t8t44t4t

1])1t(2

2t[

1])1t(2

2t[)t(f

4

3

4

24

224

224

22

22

+−+−

=

+−+−

=

+−++−−+−+−

=+

−−

−−−

=

dUcenH 8x8x

)8x8x(x)x(f 4

3

+−+−

= .

lMhat´TI54 kñúgbøg;kMupøic )j,i,O(

→→ eK[bYncMnuc D,C,B,A EdlmanGahVikerogKña

i45Z,i54Z,i61Z CBA +=+=+= nig i32ZD −−= . cUrRsayfactuekaN ABCDcarikkñúgrgVg;mYyEdleKnwgbBa¢ak;p©it nig kaMrbs;va .


- 111 -

dMeNa¼Rsay RsayfactuekaN ABCDcarikkñúgrgVg; eyIgtag 0cbyaxyx:)c( 22 =++++ CasmIkarrgVg;carikeRkARtIekaN ABC . eyIg)an )c(A∈ naM[ 0cb6a61 22 =++++ b¤ )1(37cb6a −=++ )c(B∈ naM[ 0cb5a454 22 =++++ b¤ )2(41cb5a4 −=++ )c(C∈ naM[ 0cb3a2)3()2( 22 =+−−−+− b¤ )3(13cb3a2 −=+−−

eyIg)anRbB½næsmIkar ⎪⎩

⎪⎨

⎧

−=+−−−=++−=++

13cb3a241cb5a4

37cb6a

bnÞab;BIedaHRsayRbB½næenHeK)ancemøIuy 23c,2b,2a −=−=−= .

smIkarrgVg;carikeRkARtIekaN ABCGacsresr ³ 023y2x2yx:)c( 22 =−−−+


- 112 -

b¤ 25)1y()1x(:)c( 22 =−+− müa:geTotedayykkUGredaen DCYskñúgsmIkar

25)13()12(:)c( 22 =−−+−− vaepÞógpÞat;enaHnaM[ )c(D∈ . edaybYncMnuc D,C,B,A sßitenAelIrgVg;mansmIkar

25)1y()1x(:)c( 22 =−+− EtmYy enaHnaM[ctuekaNABCDcarikkñúgrgVg; )c( manp©it )1,1(I nig kaM 5R = .


- 113 -

lMhat´TI55 eK[RtIFa cbxax)x(f 2 ++= Edl IRc,b,a,0a ∈≠ k-cUrRsayfaebI 0ac4b2 <−=Δ nig 0a > ena¼eKán IRx,0)x(f ∈∀> x-kñúgkrNIen¼eKsnμtfa 0ac4b2 <−=Δ nig 0a > . ebI ab > cMeBa¼RKb´ IR∈λ cUrbgHajfa ½ k

abc)k(b)k(a 2>

−++λ+−λ

Edl k CacMnYnBitefrmYyEdleK[ . K-Gnuvtþn_ cMeBa¼RKb´RtIFa cbxax)x(f 2 ++= Edl IRc,b,a,0a ∈≠ ebI 0ac4b2 <−=Δ nig 0a > ena¼bgHajfaeKman ½

3ab

aba>

−++ .

dMeNa¼Rsay k-RsayfaebI 0ac4b2 <−=Δ nig 0a > ena¼eKán IRx,0)x(f ∈∀>


- 114 -

eyIgman cbxax)x(f 2 ++=

( )

ac4b,a4a2

bxa)x(f

a4ac4b

a2bxa)x(f

ac)

a2b()

a2b()

a2b)(x(2xa

acx

abxa)x(f

22

2

2

22

2222

−=Δ⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δ−⎟

⎠⎞

⎜⎝⎛ +=

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−⎟

⎠⎞

⎜⎝⎛ +=

⎥⎦⎤

⎢⎣⎡ +−++=⎟

⎠⎞

⎜⎝⎛ ++=

eyIgeXIjfaebI 0<Δ ena¼ 0a4 2 >Δ

−

naM[ 0a4a2

bx 2

2

>Δ

−⎟⎠⎞

⎜⎝⎛ + .

ehtuen¼ ⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δ−⎟

⎠⎞

⎜⎝⎛ += 2

2

a4a2bxa)x(f mansBaØadUc a

ehIyebI 0a > ena¼eKán ½

IRx,0a4a2

bxa)x(f 2

2

∈∀>⎥⎥⎦

⎤

⎢⎢⎣

⎡ Δ−⎟

⎠⎞

⎜⎝⎛ += .

dUcen¼ebI 0ac4b2 <−=Δ nig 0a > ena¼eKán IRx,0)x(f ∈∀> . x-bgHajfa k

abc)k(b)k(a 2>

−++λ+−λ

tamsRmayxagelIebI 0ac4b2 <−=Δ nig 0a >


- 115 -

ena¼ 0)x(f:IRx >∈∀ cMeBa¼RKb´ IR∈λ eKán 0cba)(f 2 >+λ+λ=λ EfmGg:TaMgBIrnwg kakb− eKán kakbkakbcba 2 −>−++λ+λ )1()ab(kc)k(b)k(a 2 −>++λ+−λ eday ab > ¦ 0ab >− ena¼eyIgEckTMnak´TMng )1( nwg ab − viC¢man eyIgán k

abc)k(b)k(a 2

>−

++λ+−λ CavismPaBEdlRtUvbgHaj .

K-Gnuvtþn_ bgHajfa 3ab

aba>

−++

tamsRmayxagelIcMeBa¼RKb´RtIFa cbxax)x(f 2 ++= Edl IRc,b,a,0a ∈≠ ebI 0ac4b2 <−=Δ nig 0a > ena¼eyIgman

kab

c)k(b)k(a 2>

−++λ+−λ

Edl IRk,IR ∈∈λ nig :k efr .


- 116 -

ebIeyIgeRCIserIs 3k = nig 2−=λ eyIgán 3

ab)32(b)34(a>

−+−+−

¦ 3ab

cba>

−++ Bit .

dUcen¼ cMeBa¼RKb´RtIFa cbxax)x(f 2 ++= Edl IRc,b,a,0a ∈≠ ebI 0ac4b2 <−=Δ nig 0a > ena¼bgHajfaeKman 3

ababa>

−++ .

lMhat´TI56 eK[

2x0 π<< . cUrRsaybBa¢ak´fa ½

21xcos

11xsin

11 +≤⎟⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛ +

dMeNa¼Rsay RsaybBa¢ak´fa ½

21xcos

11xsin

11 +≤⎟⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛ +


- 117 -

eyIgman ½

xcosxsin1

xcos1

xsin11

xcos11

xsin11 +++=⎟

⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛ + tamvismPaB GMAM − eKman ½

xcosxsin12

xcos1

xsin1

≥+ eKTaj ½

xcosxsin1

xcosxsin121

xcos11

xsin11 ++≥⎟

⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛ +

eday 21x2sin

21xcosxsin ≤= ¦ 2

xcosxsin1

≥

eKán 2)21(2221xcos

11xsin

11 +=++≥⎟⎠⎞

⎜⎝⎛ +⎟⎠⎞

⎜⎝⎛ +

dUcen¼ 21xcos

11xsin

11 +≤⎟⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛ + .


- 118 -

lMhat´TI57 cUreda¼RsayRbBnæ&smIkar ½

⎪⎩

⎪⎨⎧

=+

=++

+

1723.28

171273.4y21xx

y1yx

dMeNa¼Rsay eda¼RsayRbBnæ&smIkar ½

⎪⎩

⎪⎨⎧

=+

=++

+

)2(1723.28

)1(171273.4y21xx

y1yx

edaybUksmIkar )1( nig )2( Gg:nigGg:eKán ½

)3(732

343)32(

343)3()3)(2(3)3()2(3)2(

343273.23.48

yx

3yx

3y2yxy2x3x

yy21x1yxx

=+

=+

=+++

=+++ ++

müa¨geTotdksmIkar )1( nig )2( Gg:nigGg:eKán ½

)4(132

1)23(

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

183.43.227

yx

3xy

3x2xyx2y3y

x1yxy21xy

=−

−=−

−=−+−

−=−+− ++


- 119 -

tam ( )3 nig ( )4 eyIgánRbBn&æ ½

⎪⎩

⎪⎨⎧

=−

=+

)4(132

)3(732yx

yx

bUksmIkar( )3 nig ( )4 eKán 82.2 x = naM[ 2x = dksmIkar( )3 nig ( )4 eKán 63.2 y = naM[ 1y = . dUcen¼ 1y,2x == .

lMhat´TI58 cUreda¼RsayRbBnæ&smIkar ½

⎪⎩

⎪⎨⎧

=−+−

=+−−

344y28xy33yx21x9

342y26xy21yx3x73223

3223

dMeNa¼Rsay eda¼RsayRbBnæ&smIkar ½

⎪⎩

⎪⎨⎧

=−+−

=+−−

)2(344y28xy33yx21x9

)1(342y26xy21yx3x73223

3223

edaybUksmIkar )1( nig )2( Gg:nigGg:eKán ½


- 120 -

)3(7yx2343)yx2(

686)yxy6yx12x8(2

686y2xy12yx24x16

3

3223

3223

=−=−

=−+−

=−+−

müa¨geTotdksmIkar )1( nig )2( Gg:nigGg:eKán ½

)4(1y3x1)y3x(

2)y27xy27yx9x(2

2y54xy54yx18x2

3

3223

3223

=−=−

−=−+−−

−=+−+−

tam ( )3 nig ( )4 eyIgánRbBn&æ ½ ⎩⎨⎧

=−=−

)4(1y3x)3(7yx2

eyIgman 5163112

D −=+−=−−

= 572

1172

D,203117

D yx −=−==−=−−

=

eKán 155

DD

y,4520

DD

x yx =−−

===−−

== dUcen¼ 1y,4x == .


- 121 -

lMhat´TI59 eK[smIkar 0)b1)(a1(x.ba22x:)E( 222 =+++++− Edl a nig b CaBIrcMnYnBit . k-cUrkMnt´témø a nig b edIm,I[smIkaren¼man¦sDub rYcKNna¦sDubena¼ . x-eRkABItémø a nig bxagelIcUrbgHajfasmIkar )E( man¦sBIr Canic©kñúg IR. dMeNa¼Rsay k-kMnttémø a nig b rYcKNna¦sDub½ eKman 0)b1)(a1(x.ba22x:)E( 222 =+++++− edIm,I[smIkaren¼man¦sDublu¼RtaEt 0'=Δ eyIgman )b1)(a1()ba2(' 22 ++−++=Δ

[ ][ ]222

2222

22

22

)1b()1a()ba(21

)1b2b()1a2a()bab2a(21

abbaba1

abba1ba2

−+−+−=

+−++−++−=

−−−++=

−−−−++=


- 122 -

eKán [ ] 0)1b()1a()ba(21' 222 =−+−+−=Δ

naM[ ⎪⎩

⎪⎨

⎧

=−=−=−

01b01a0ba

eKTaján 1a = nig 1b = . cMeBa¼ 1b,1a == eKTaj¦sDub ½

2112ba2xx 2221 =++=++== .

x-bgHajfasmIkar )E( man¦sBIrCanic©kñúg IR ebI 1b,1a ≠≠ ena¼ [ ] 0)1b()1a()ba(

21' 222 >−+−+−=Δ

dUcen¼smIkar )E( man¦sBIrCanic©kñúg IR .


- 123 -

lMhat´TI60 eK[BhuFa n1n

1n2

21n xxc.......xcxc1)x(P +++++= −−

Edl 0c.......,,c,c,c 1n321 ≥− . snμtfasmIkar 0)x(Pn = man n ¦sCacMnYnBitGviC©man . cUrbgHajfa n

n 3)2(P ≥ .

dMeNa¼Rsay bgHajfa n

n 3)2(P ≥ tag n321 x......,,x,x,x Ca¦ssmIkar 0)x(Pn = Edl n,....,2,1k,0xk =∀< eKán )xx)......(xx)(xx)(xx()x(P n321n −−−−= yk n,......,3,2,1k,0xt kk =∀>−= eKán )tx).......(tx)(tx)(tx()x(P n321n ++++= ebI 2x = ena¼eKán )t2).....(t2)(t2)(t2()2(P n321n ++++= .


- 124 -

tamvismPaB GMAM − eKman ½ n....,3,2,1k,t3t11t2 3

kkk =∀≥++=+ eyIgán 3

n321n

n t.....ttt3)2(P ≥ eday 1)1.()1(x......x.x.x.)1(t.......t.t.t nn

n321n

n321 =−−=−= dUcen¼ n

n 3)2(P ≥ .

lMhat´TI61 k-RbsinebI 1p −≥ cMeBa¼RKb´ *INn∈ cUrbgHajfa ½ )1(np1)p1( n +≥+ . x-eK[ n321 a......,,a,a,a Can cMnYnminGviC¢man . eKtag

na....aaa

A n321n

++++=

nig nn321n a......a.a.aG = .

bgHajfaRbsinebI kk AG ≤ ehIy 0Ak ≠ eKán )p1(Aa.G 1k

k1kk

k +≤ ++ Edl 1

Aa

pk

1k −= + .

K-edayeRbIlTæplxagelIcUrRsaybBa¢akfa ½ n

n321n321 a....aaana.....aaa ≥++++ .


- 125 -

dMeNa¼Rsay cUrbgHajfa ½ )1(np1)p1( n +≥+ tamrUbmnþeTVFajÚtuneKman ½

nnn

22n

1n

0n

n pC.....pCpCC)p1( ++++=+ eday

2n)1n(C,nC,1C 2

n1n

0n

−===

eKán n2n p.....p2

n)1n(np1)p1( ++−

++=+ edaycMeBa¼RKb´ 1p −≥ eKman 0p....p

2n)1n( n2 ≥++

− dUcen¼ )1(np1)p1( n +≥+ . x-bgHajfa )p1(Aa.G 1k

k1kk

k +≤ ++

RbsinebI kk AG ≤ ena¼ 1kkk1k

kk aAaG ++ ≤

)p1(A

)]1A

a(1[A

)]Aa(A[AaA

1kk

k

1k1kk

k1kkkk1k

kk

+=

−+=

−+=

+

++

++

eRBa¼ 1A

ap

k

1k −= + .

dUcen¼ )p1(Aa.G 1kk1k

kk +≤ +

+ .


- 126 -

K-RsaybBa¢akfa ½ n

n321n321 a....aaana.....aaa ≥++++ eyIgman 0aaa2a)aa( 2211

221 ≥+−=−

naM[ 2121 aa2aa ≥+ Bit . snμtfavaBitdl´tYTI k KW ½

kk21k21 a.....aaka.....aa ≥+++ Bit

eyIgnwgRsayfavaBitdl´tYTI 1k + K W½ k

1k211k21 a.....aaka.....aa ++ ≥+++ Bit tamkarsn μteyIgman k

k21k21 a.....aaka.....aa ≥+++ smmUl k

k321k321 a......aaa

ka...aaa

≥++++

¦ kk GA ≥ Edl

ka....aaa

A k321k

++++=

nig kk321k a......a.a.aG = .

tamsRmayxagelIeKman )p1(Aa.G 1kk1k

kk +≤ +

+ eday 1k

1k1kk211kkk Gaa.....a.aaG +

+++ == eKTaj )p1(AG 1k

k1k1k +≤ ++

+


- 127 -

Edl 0A,11A

ap k

k

1k ≠−≥−= + .

tamrUbmnþ )1( eKán 1k)1k

p1()p1( +

++≤+

naM[ 1k1kk

1kk )

1kp1(A)p1(A +++

++≤+

eKTaj 1k1kk

1k1k )

1kp1(AG +++

+ ++≤

¦ )1k

p1(AG k1k ++≤+

eday½

1kAa

A)1A

a(

1k11A)

1kp1(A k1k

kk

1kkk +

−+=⎥

⎦

⎤⎢⎣

⎡−

++=

++ ++

1k

1kk21k1kkk A1k

aa....aa1k

AaAkA+

++ =+

++++=

+−++

= eKTaj 1k

1kk1k A)p1(AG +

++ ≤+≤

¦ k1k211k21 a.....aaka.....aa ++ ≥+++ Bit

dUcen¼ nn321n321 a....aaana.....aaa ≥++++ .


- 128 -

lMhat´TI62 eK[ C,B,A CargVas´mMukñúgrbs´RtIekaN ABC mYy . k-cUrbgHajfa


x-cUrbgHajfa 49

2Ccos

2Bcos

2Acos 222 ≤++

K-cUrbgHajfa 43

2Csin

2Bsin

2Asin 222 ≥++

X-cUrbgHajfa 8

332Ccos

2Bcos

2Acos ≤

dMeNa¼Rsay k-bgHajfa


tag CcosBcosAcosT ++= Ccos

2BAcos

2BAcos2 +

−+=

eday π=++ CBA ena¼ 2Csin)

2C

2cos(

2BAcos =−

π=

+ ehIy

2Csin21

2Csin

2CcosCcos 222 −=−=


- 129 -

eKán 2Csin21

2BAcos

2CsinT 2−+

−=

12Csin.

2BAcos

2Csin2 2 +

−+−=

tag 2Csint =

eKánGnuKmn_ ½ 1t

2BAcost2)t(T 2 +

−+−=

2

BAcos211)

2BAcos

21t(2

2BAcos

211

2BAcos

41t

2BAcos

21t2

222

222

−++

−−−=

−++⎟

⎠⎞

⎜⎝⎛ −

+−

−−=

cMeBa¼RKb´ IRt∈ eKTaj ½

23

211

2BAcos

211)t(T 2 =+≤

−+≤ .

dUcen¼ 23CcosBcosAcos ≤++ .

x-bgHajfa 49

2Ccos

2Bcos

2Acos 222 ≤++

tag 2Ccos

2Bcos

2AcosU 222 ++=

)CcosBcosA(cos

21

23

2Ccos1

2Bcos1

2Acos1

+++=

++

++

+=


- 130 -

eday 23CcosBcosAcos ≤++ ( tamsRmayxagelI )

eKTaj 49

23

21

23

2Ccos

2Bcos

2AcosU 222 =+≤++=

dUcen¼ 49

2Ccos

2Bcos

2Acos 222 ≤++ .

K-bgHajfa 43

2Csin

2Bsin

2Asin 222 ≥++

tag 2Csin

2Bsin

2AsinV 222 ++=

⎟⎠⎞

⎜⎝⎛ ++−=

−+−+−=

2Ccos

2Bcos

2Acos3

2Ccos1

2Bcos1

2Acos1

222

222

eday 49

2Ccos

2Bcos

2Acos 222 ≤++

( tamsRmayxagelI ) eKTaj

43

493

2Csin

2Bsin

2AsinV 222 =−≥++=

dUcen¼ 43

2Csin

2Bsin

2Asin 222 ≥++ .

X-bgHajfa 8

332Ccos

2Bcos

2Acos ≤

tamvismPaB GMAM − eKman ½


- 131 -

3 222222

2Ccos

2Bcos

2Acos3

2Ccos

2Bcos

2Acos ≥++

naM[ 3222 )2Ccos

2Bcos

2A(cos

271

2Ccos

2Bcos

2Acos ++≤

eday 49

2Ccos

2Bcos

2Acos 222 ≤++

naM[ 8

332Ccos

2Bcos

2Acos ≤ .

lMhat´TI63 eK[smPaB

3)1n2(n

1n2ncot......

1n23cot

1n22cot

1n2cot 2222 −

=+π

+++π

++π

++π

cUrbgHajfa 6n

1...........31

21

11lim

2

2222n

π=⎟

⎠⎞

⎜⎝⎛ ++++

+∞→ .

dMeNa¼Rsay bgHajfa

6n1...........

31

21

11lim

2

2222n

π=⎟

⎠⎞

⎜⎝⎛ ++++

+∞→


- 132 -

eyIgman ½ - épÞRklaRtIekaN MOΑ KW θ=θΑ= sin

21sin.OM.O.

21S1

- épÞRklacMerokfas OMA KW θ=θ=21OA.

21S 2

2 - épÞRklaRtIekaN OAB KW θ== tan

21AB.OA

21S3

tamrUbxagelIcMeBa¼ 2

0 π<θ< eKman ½ 321 SSS <<

eKán θ<θ<θ tan21

21sin

21

¦ θ<θ<θ tansin

0 Α

ΒM

θ

y

x


- 133 -

θ+=θ

<θ

<θ

θ<

θ<θ

θ<

θ<

θ

222

2 cot1sin

11cot

sin11cot

sin11

tan1

edayCMnYs 1n2

k+π

=θ eKán ½

∑∑ ∑

∑∑ ∑

== =

== =

⎥⎦⎤

⎢⎣⎡

+π

+<⎟⎠⎞

⎜⎝⎛

π+

<⎥⎦⎤

⎢⎣⎡

+π

⎥⎦⎤

⎢⎣⎡

+π

+<⎥⎦

⎤⎢⎣

⎡

π+

<⎥⎦⎤

⎢⎣⎡

+π

+π

+<π+

<+π

n

1k

2n

1k

n

1k22

22

n

1k

2n

1k

n

1k22

22

222

22

)1n2

k(cotnk1)1n2()

1n2k(cot

)1n2

k(cot1k

)1n2()1n2

k(cot

)1n2

k(cot1k

)1n2()1n2

k(cot

edaytambMrab´eKman ½

3)1n2(n

1n2ncot......

1n23cot

1n22cot

1n2cot

)1n2

k(cotS

2222

n

1k

2n

−=

+π

+++π

++π

++π

=

⎥⎦⎤

⎢⎣⎡

+π

= ∑=

eKán 3

)2n2(n3

)1n2(nnk1)1n2(

3)1n2(n n

1k22

2 +=

−+<⎟

⎠⎞

⎜⎝⎛

π+

<−

∑=

¦ 2

2n

1k22

2

)1n2()2n2(n

3k1

)1n2()1n2(n

3 ++π

<⎟⎠⎞

⎜⎝⎛<

+−π

∑=


- 134 -

6k

1lim6

)1n2()2n2(n

3lim

k1lim

)1n2()1n2(n

3lim

2n

1k2n

2

2

2

n

n

1k2n2

2

n

π<⎟

⎠⎞

⎜⎝⎛<

π

++π

<⎟⎠⎞

⎜⎝⎛<

+−π

∑

∑

=+∞→

+∞→=+∞→+∞→

naM[eKTaj 6k

1lim2n

1k2n

π=⎟

⎠⎞

⎜⎝⎛∑

=+∞→ .

dUcen¼ 6n

1...........31

21

11lim

2

2222n

π=⎟

⎠⎞

⎜⎝⎛ ++++

+∞→ .


- 135 -

lMhat´TI64 eKtag r nig R erogKñaCakaMénrgVg´carikkñúg nig carikeRkA Rbs´RtIekaNEkg ABC mYy . cUrRsaybBa¢ak´fa r)21(R +≥ ?

dMeNa¼Rsay RsaybBa¢ak´fa r)21(R +≥ tag CcosBcosAcosT ++=

2Csin

2Bsin

2Asin41

)2

CBcos2

CB(cos2Asin21

)2

CBcos2Asin(

2Asin21

2CBcos

2Asin2

2Asin21

2CBcos

2CBcos2

2Asin21

2

2

+=

−−

+−=

−−−=

−+−=

−++−=

eKán ( )12Csin

2Bsin

2Asin41CcosBcosAcos +=++


- 136 -

tamRTwsþIbTkUsIunUs Acosbc2cba 222 −+= eday

2Asin21Acos 2−=

eKán 2Asinbc4bc2cba 2222 +−+=

eKTaj bc4

)cba)(cba(bc4

)cb(a2Asin

222 −++−

=−−

=

yk 2

cbap ++= naM[

⎩⎨⎧

−=−+−=+−

)cp(2cba)bp(2cba

eKán bc

)cp)(bp(bc4

)cp)(bp(42Asin2 −−

=−−

=

eKTaj bc

)cp)(bp(2Asin −−=

dUcKñaEdr ab

)bp)(ap(2Csin;

ac)cp)(ap(

2Bsin −−

=−−

=

eKán )2(abc

)cp)(bp)(ap(2Csin

2Bsin

2Asin −−−

= tamrUbmnþRkTLaépÞRtIekaN ½

R4abcpr)cp)(bp)(ap(pS ==−−−=

eKTaján ⎪⎩

⎪⎨

⎧

===−−−

=

r.Sp

r.p.SpS)cp)(bp)(ap(

RS4abc2

tam )2( Gacsresr ½


- 137 -

( )3R4r

2Csin

2Bsin

2Asin =

ykTMnak´TMng )3( CYskñúg )1( eKán ½ ( )4

Rr1CcosBcosAcos +=++

eday ABC CaRtIekaNEkgena¼eKGaceRCIserIsyk 2

A π=

ehIy C2

B −π

= CYskñúgTMnak´TMng )4( eKán ½

( )5Rr1CcosCsin

Rr1Ccos)C

2cos(

2cos

+=+

+=+−π

+π

tamTMnak´TMng 2)C4

sin(2CcosCsin ≤+π

=+ ena¼tam )5( eKán 2

Rr1 ≤+

naM[ r)12(12

rR +=−

≥ dUcen¼ r)21(R +≥ . vismPaBen¼køayCasmPaBkalNa ½

2)C4

sin(2CcosCsin =+π

=+ naM[ 4

C π= nig

4B π=

eBalKWRtIekaN ABC CaRtIekaNEkgsmát .


- 138 -

lMhat´TI65 eK[ c,b,a CaRCugrbs´RtIekaNmYy . cUrRsaybBa¢ak´vismPaB ½

0)ac(ac)cb(cb)ba(ba 222 ≥−+−+− dMeNa¼Rsay RsaybBa¢ak´fa 0)ac(ac)cb(cb)ba(ba 222 ≥−+−+− tag yxc,xzb,zya +=+=+= Edl 0z,y,x > vismPaB 0)ac(ac)cb(cb)ba(ba 222 ≥−+−+− smmUleTAnwgvismPaBxageRkam ½

zyxzz

zy

yx

xyzzxyyzxyzxyzx

0xyzzxyyzxyzxyzx

0)zx)(zy()yx()yz)(yx()xz()xy)(xz()zy(

222

222333

222333

222

++≥++

++≥++

≥−−−++

≥−+++−+++−++

tamvismPaB Cauchy-Schwarz eKman ½ )

zz

zy

yx)(zyx()zyx(

2222 ++++≤++


- 139 -

eKTaj zyxxz

zy

yx 222

++≥++ Bit . dUcen¼ 0)ac(ac)cb(cb)ba(ba 222 ≥−+−+− .

lMhat´TI66 eK[ c,b,a CaRCugrbs´RtIekaNmYyEdlmanépÞRkTLa esμInwg S . cUrRsayfa S34cba 222 ≥++ ? dMeNa¼Rsay Rsayfa S34cba 222 ≥++ eKman Asinbc

21S = naM[

bcS2Asin =

ehIy bc2

acbAcos222 −+

= ( RTwsþIbTkUsIunUs )

eKán S4

acbAsinAcosAcot

222 −+==

dUcKñaEdr S4

cbaCcot;S4

bacBcot222222 −+

=−+

=

eKán S4

cbaCcotBcotAcot222 ++

=++ naM[ )1()CcotBcotA(cotS4cba 222 ++=++


- 140 -

CabnþeTAen¼eyIgnwgRsayfa 3CcotBcotAcot ≥++ . eKman π=++ CBA ena¼ )CB(A +−π= eKán ( ) )CBtan()CB(tanAtan +−=+−π=

CtanBtanCtanBtanAtan

Ctan.Btan1CtanBtanAtan

+=+−−

+−=

eKTaj CtanBtanAtanCtanBtanAtan =++ KuNGg:TaMgBIrnwg CcotBcotAcot eKánsmPaB

1AcotCcotCcotBcotBcotAcot =++ edayeRbIvismPaB )zxyzxy(3)zyx( 2 ++≥++ eKTaján 3)CcotBcotA(cot 2 ≥++ naM[ 3CcotBcotAcot ≥++ ( C,B,A CamMuRsYc ) tamTMnak´TMng )1( eKTaján ½

S34cba 222 ≥++ CavismPaBEdlRtUvRsaybBa¢ak´ .


- 141 -

lMhat´TI67

eKyk I Cap©itrgVg´carwkkñúgRtIekaN ABC mYyEdl cAB,bAC,aBC === .

cMeBa¼RKb´cMnuc X cUrRsayfa ½ abcXI)cba(XC.cXB.bXA.a 2222 +++=++

dMeNa¼Rsay Rsayfa ½

abcXI)cba(XC.cXB.bXA.a 2222 +++=++

I

B

A

C x

y


- 142 -

kñúgtRmuyGrtUnrm¨al´ )Bxy( eKman ½ )Bsinc;Bcosc(A;)0,a(C;)0;0(B .

tag r CakaMrgVg´carwkkñúg nig 2

cbap ++= Caknø¼brimaRt

énRtIekaN ABC ena¼eKán )r;bp(I − . ebI S CaépÞRkTLaénRtIekaN ABC ena¼eKman ½

pr)cp)(bp)(ap(pS =−−−= naM[

p)cp)(bp)(ap(r2 −−−

= . tag )y;x(X 00 CacMnucNak¾edayénbøg´ . eKán ½

2200

20

20

2200

20

20

220

222

02

02

0

2200

20

20

20

20

20

20

20

20

222

accaSy4)bp(px4)yx(p2

accaSy4)b2p2)(p2(x)yx(p2

accaSy4)ac2

bca1(acx2)yx(p2

caac

Bsinacy2)Bcos1(acx2)yx)(cba(

]y)ax[(c

)yx(b])Bsincy()Bcoscx([a

cXCbXBaXAM

++−−−+=

++−−−+=

++−−+

+−+=

++

+−+−+++=

+−+

+++−+−=

++=


- 143 -

ehIy abcXI)cba(N 2 +++=

[ ]

abc)cp)(bp)(ap(2)bp(p2Sy4)bp(px4)yx(p2

abcp

)cp)(bp)(ap(p2

)bp(p2Sy4)bp(px4)yx(p2

abcpr2

)bp(p2pry4)bp(px4)yx(p2

abc)ry())bp(x(p2

200

20

20

200

20

20

2

200

20

20

20

20

+−−−+

+−+−−−+=

+−−−

+

+−+−−−+=

++

+−+−−−+=

+−+−−=

yk abc)cp)(bp)(ap(2)bp(p2T 2 +−−−+−=

22

2

22

acca

)ca(ac)bb2p2(acabc)bp(ac2abc]acp)cba(p2)[bp(2

abc]acp)ca(ppbp)[bp(2

+=

+=+−=+−=++++−−=

+++−+−−=

eKTaj 2200

20

20 accaSy4)bp(px4)yx(p2N ++−−−+=

edaykenßam NM = naM[eKTaján ½ abcXI)cba(XC.cXB.bXA.a 2222 +++=++ .

( TMnak´TMngen¼ehAfarUbmnþGWEl ) .


- 144 -

lMhat´TI68

eKtag r nig R erogKñaCakaMrgVgćarikkñúg nig carwkeRkA énRtIekaNmYy . cUrRsaybBa¢akfaebI d Cacm¶ayrvagp©it rgVgćarwkkñúg nig carwkeRkArbsRtIekaNena¼eKán ½

)r2R(Rd2 −= ? dMeNa¼Rsay Rsayfa )r2R(Rd2 −= tag ABC CaRtIekaNEdl cAB,bAC,aBC === ehIy r nig R erogKñaCakaMrgVgćarikkñúg nig carwkeRkA énRtIekaNen¼ . yk I Cap©itrgVgćarwkkññúg nig O Cap©it rgVgćarwkeRkAénRtIekaN . tamRTwsþIbTGWEleyIgán ½

abcOI)cba(OC.cOB.bOA.a 2222 +++=++ eday ROCOBOA === nig

2cbap ++

= eKán abcd.p2Rp2 22 += naM[

p2abcRd 22 −=

Edl OId = .


- 145 -

tamrUbmnþépÞRkTLa R4

abcprS == eKTaj rR2p2

abc=

dUcen¼ )r2R(RrR2Rd 22 −=−= . lMhat´TI69 eKyk I Cap©itrgVg´carwkkñúgRtIekaN ABC mYyEdl manRCug cAB,bAC,aBC === . cUrRsayfa 1

abIC

caIB

bcIA 222

=++ . dMeNa¼Rsay Rsayfa 1

abIC

caIB

bcIA 222

=++ tamRTwsþIbTGWElcMeBa¼RKb´cMnuc X eKman

abcXI)cba(XC.cXB.bXA.a 2222 +++=++ edayyk IX ≡ eKán abcIC.cIB.bIA.a 222 =++ EckGg:TaMgBIrnwg abc eKTTYlán ½

1abIC

caIB

bcIA 222

=++ .


- 146 -

lMhat´TI70 cMeBa¼RKb´cMnYnBitviC¢man c,b,a cUrbgHajfa ½ )cabcab(9)2c)(2b)(2a( 222 ++≥+++ dMeNa¼Rsay Rsayfa )cabcab(9)2c)(2b)(2a( 222 ++≥+++

eyIgeRCIserIs 2

z,y,x0 π<< Edl

⎪⎩

⎪⎨

⎧

=

=

=

ztan2cytan2bxtan2a

vismPaB )cabcab(9)2c)(2b)(2a( 222 ++≥+++ smmUleTAnwgvismPaBxageRkam ½

)xtanztanztanytanytanx(tan18zcosycosxcos

8222 ++≥

94)ycosxsinzsinxcoszsinysinzcosysinx(sinzcosycosxcos ≤++

edayeRbIrUbmnþ ½

ycosxsinzsinxcoszsinysinzcosysinxsinzcosycosxcos)zyxcos(

−−−=++

ena¼eKGacsresr ½


- 147 -

[ ] (*)94)zyxcos(zcosycosxcoszcosycosxcos ≤++−

tamvismPaB GMAM − nig Jensen eyIgán ½ tcos

3zcosycosxcoszcosycosxcos 3

3

≤⎟⎠⎞

⎜⎝⎛ ++

≤ Edl

3zyxt ++

= .vismPaB (*) smmUleTAnwgvismPaB ½

94)t3cost(costcos 33 ≤− eday tcos3tcos4t3cos 3 −=

ena¼ 94)tcos3tcos3(tcos 33 ≤−

274)tcos1(tcos

274)tcost(costcos

24

33

≤−

≤−

tamvismPaB GMAM − eKán ½

271

3

tcos12

tcos2

tcos

)tcos1.(2

tcos.

2tcos

32

22

222

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛−++

≤−

eKTaj 274)tcos1(tcos 24 ≤− Bit .

dUcen¼vismPaBxagedImRtUvánRsaybBa¢ak´.

sUmrgcaMGanPaK3


- 148 -

Éksareyag ( Referent )

1. 360 Problems for Mathematical Contest (By Titu Andreescu , Dorin Andrica )

2. 103 Trigonometry Problems (From the training of the USA IMO Team)

3. Five Hundred Mathematical Challenges (By Edward J. Bar beau, Murray S. Klamkin)

4. Complex Numbers From A to … Z (By Titu Andresscu , Dorin Andrica )

5. Mathematical Olympiad in China 6. Mathematical Olympiad Challenges (By Titu Andresscu & Razvan Gelca) 7. Mathematical Olympiad Treasures (By Titu Andreescu , Bogdan enescu ) 8. International Mathematical Olympiads 1959-1977 ( Samuel L. Greitzer )

9. The IMO Compendium ( 1959-2004 )

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