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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 )
GñkcUlrYmRtYtBinitübec©keTs
elak lwm qun elak Esn Bisidæ elakRsI Tuy rINa elak Titü em¨g elak RBwm sunitü elak pl b�unqay
GñkRtYtBinitüGkçraviruTæ elak lwm miKþsir
karIkMuBüÚT&r kBaØa lI KuNÑaka
GñkniBnæ nig eroberog elak lwm plþún nig elak Esn Bisidæ
id9850103 pdfMachine by Broadgun Software - a great PDF writer! - a great PDF creator! - http://www.pdfmachine.com http://www.broadgun.com
GarmÖkfa
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…
nigÉksarbreTsepßg@eTottamy¼Internet ykmkeFIVdMeNa£Rsay yagek,a£k,ayEdlGac[elakGñkgayylqabcgcaMGMBIsil,£énkar eda£RsayTaMgGs´en£ . b¨uEnþeTa£Caya¨gNak¾eday kgV£xat nig kMhusqþgedayGectnaRákdCamanTaMgbec©keTs nig GkçraviruTæ . GaRs&yehtuen£ eyIgxMJúCaGñkeroberogrgcaMedayrIkrayCanic©nUv mtiri£Kn´EbbsSabnaBIsMNakGñksikßakñúgRKbmCÄdæanedIm,ICYyEklMG esovePAen£[ánkanEtsuRkiRtPaBEfmeTot .
CaTIbBa©ben£eyIgxMJúGñkeroberogsUmeKarBCUnBrdlGñksikßa TaMgGs[mansuxPaBmaMmYn nig TTYlC&yCMn£RKb´Parkic© .
átdMbgéf¶TI 14 kumÖ¼ 2009 GñkniBnæ lwm plþún Tel : 017 768 246
lwm plþún nig Esn Bisidæ
Problems and Solutions
rkßasiT§iRKb´yag
KNitviTüaCMuvijBiPBelak
- 1 -
KNitviTüaCMuvijBiPBelak
- 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 = .
KNitviTüaCMuvijBiPBelak
- 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++++
= .
KNitviTüaCMuvijBiPBelak
- 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 .
KNitviTüaCMuvijBiPBelak
- 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´ .
KNitviTüaCMuvijBiPBelak
- 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| ≤
KNitviTüaCMuvijBiPBelak
- 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 ≤++ .
KNitviTüaCMuvijBiPBelak
- 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 <++
+++
+++
+++
<
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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
++=++
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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¨ra¨Em¨Rt . 45-eK[ )x(P CaBhuFadWeRkTIbI . eKdwgfa 2)x(P + Eckdac´nwg 2)1x( + ehIy 2)x(P − Eckdac´nwg 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 .
KNitviTüaCMuvijBiPBelak
- 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 .
KNitviTüaCMuvijBiPBelak
- 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 ++=
KNitviTüaCMuvijBiPBelak
- 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.
KNitviTüaCMuvijBiPBelak
- 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 ≥++++ .
KNitviTüaCMuvijBiPBelak
- 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 ≥−+−+− ?
KNitviTüaCMuvijBiPBelak
- 18 -
66-eK[ c,b,a CaRCugrbs´RtIekaNmYyEdlmanépÞRkTLa esμInwg S . cUrRsayfa S34cba 222 ≥++ ? 67-eKyk I Cap©itrgVg´carwkkñúgRtIekaN ABC mYyEdl
cAB,bAC,aBC === . cMeBa¼RKb´cMnuc X cUrRsayfa ½
abcXI)cba(XC.cXB.bXA.a 2222 +++=++ 68-eKtag r nig R erogKñaCakaMrgVg´carikkñúg nig carwkeRkA énRtIekaNmYy . cUrRsaybBa¢akfaebI d Cacm¶ayrvagp©it rgVg´carwkkñúg nig carwkeRkArbsRtIekaNena¼eKán ½
)r2R(Rd2 −= ? 69-eKyk I Cap©itrgVg´carwkkñúgRtIekaN ABC mYyEdl manRCug cAB,bAC,aBC === . cUrRsayfa 1
abIC
caIB
bcIA 222
=++ . 70-cMeBa¼RKb´cMnYnBitviC¢man c,b,a cUrbgHajfa ½ )cabcab(9)2c)(2b)(2a( 222 ++≥+++ .
KNitviTüaCMuvijBiPBelak
- 19 -
KNitviTüaCMuvijBiPBelak
- 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 .
KNitviTüaCMuvijBiPBelak
- 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 ++≥++ .
KNitviTüaCMuvijBiPBelak
- 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( ≥+++ .
KNitviTüaCMuvijBiPBelak
- 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( ≥+++ .
KNitviTüaCMuvijBiPBelak
- 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
=⎟⎠⎞
⎜⎝⎛ −
=++−
=+++
KNitviTüaCMuvijBiPBelak
- 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 +
≤
KNitviTüaCMuvijBiPBelak
- 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( +≤++ .
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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=
−
.
KNitviTüaCMuvijBiPBelak
- 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=++=++
KNitviTüaCMuvijBiPBelak
- 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 == ) .
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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 −+<++++<−+
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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 −≥
− .
KNitviTüaCMuvijBiPBelak
- 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
+
+−
−=
KNitviTüaCMuvijBiPBelak
- 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 =+
KNitviTüaCMuvijBiPBelak
- 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 > .
KNitviTüaCMuvijBiPBelak
- 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 .
KNitviTüaCMuvijBiPBelak
- 41 -
lMhat´TI15 cUrRsaybBa¢ak´fa ½
)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
+−++
+−
=+−
−−−−−−−−
=+−
=+−
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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 −∈
KNitviTüaCMuvijBiPBelak
- 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
+++
+=
++
++=
KNitviTüaCMuvijBiPBelak
- 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 >> .
KNitviTüaCMuvijBiPBelak
- 46 -
lMhat´TI18 cUrRsaybBa¢ak´fa ½
38
2x4cos1x4sin4x4cos2 ≤
+++
≤− cMeBa¼RKb´cMnYnBit x .
dMeNa¼Rsay RsaybBa¢ak´fa
38
2x4cos1x4sin4x4cos2 ≤
+++
≤−
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 +−=
→→
KNitviTüaCMuvijBiPBelak
- 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
2x4cos1x4sin4x4cos2 ≤
+++
≤− cMeBa¼RKb´cMnYnBit x .
KNitviTüaCMuvijBiPBelak
- 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∈ .
KNitviTüaCMuvijBiPBelak
- 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
++++=
++++=
+++++=
KNitviTüaCMuvijBiPBelak
- 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 = .
KNitviTüaCMuvijBiPBelak
- 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 +−=−=+
KNitviTüaCMuvijBiPBelak
- 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 =
+==
−=
KNitviTüaCMuvijBiPBelak
- 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| ≤
KNitviTüaCMuvijBiPBelak
- 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
∈∀≤+
−+−
−=
+−
++−
=
+−−
+=−
KNitviTüaCMuvijBiPBelak
- 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 ∈ .
KNitviTüaCMuvijBiPBelak
- 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
==
KNitviTüaCMuvijBiPBelak
- 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
=××××
KNitviTüaCMuvijBiPBelak
- 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
=×××× .
KNitviTüaCMuvijBiPBelak
- 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= +
+∞→+∞→ .
KNitviTüaCMuvijBiPBelak
- 60 -
lMhat´TI25 eK[ 1a ≥ nig 1b ≥ .
cUrbgHajfa )2
ba(log2blogalog 222+
≤+
dMeNa¼Rsay
bgHajfa )2
ba(log2blogalog 222+
≤+ cMeBa¼ 1a ≥ nig 1b ≥ eyIgman b.a2ba ≥+ ( vismPaB GMAM − ) ¦ b.a
2ba≥
+
naM[ )blogalog(21)
2ba(log 222 +≥
+
¦ )1()2
ba(log2blogalog 222+
≤+ müa¨geToteKman ½
blog.alog2blogalog 2222 ≥+
KNitviTüaCMuvijBiPBelak
- 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
ba(log2blogalog 222+
≤+ .
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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
33CsinBsinAsin ≤++ .
x-bgHajfa 8
332Ccos
2Bcos
2Acos ≤ .
dMeNa¼Rsay
k-RsaybBa¢akfa 2
33CsinBsinAsin ≤++
KNitviTüaCMuvijBiPBelak
- 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 ++=
+++
+
≤++
++
KNitviTüaCMuvijBiPBelak
- 65 -
eKTaj ½ )
3CBAsin(4)
3CBAsin(CsinBsinAsin ++
≤++
+++ ¦
233
3sin3)
3CBA(sin3CsinBsinAsin =
π=
++≤++
dUcen¼ 2
33CsinBsinAsin ≤++ .
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
=++
++
−=++
+−
=++
KNitviTüaCMuvijBiPBelak
- 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 =+++++
KNitviTüaCMuvijBiPBelak
- 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 ∈∀−−=−+−= .
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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´ .
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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 ≥+++++ .
KNitviTüaCMuvijBiPBelak
- 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
≥+⎟⎠
⎞⎜⎝
⎛+−+ .
KNitviTüaCMuvijBiPBelak
- 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=
KNitviTüaCMuvijBiPBelak
- 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 $0 [sMNl;b:un μan ? x-rkcMnYn n enaHedaydwgfa 4000n3940 << .
dMeNa¼Rsay k> ebIcMnYn n enaHEcknwg 40 [sMNl;b:un μan ?
KNitviTüaCMuvijBiPBelak
- 75 -
]bmafa n Ecknwg * [plEck INq1 ∈ nigsMNl; ! nig cMnYn n enaHEcknwg % [plEck INq2 ∈ nigsMNl; @ . tamGWKøIt eyIg)an
)16()15(
2q5n1q8n
2
1 −
⎩⎨⎧
+=+=
b¤ ⎩⎨⎧
+=−−=−
)2(32q80n16)1(15q120n15
2
1
bUksmIkar ¬!¦ nig ¬@¦ eyIg)an 17q4017q120q80n 12 +=+−= Edl 12 q3q2q −= . tamTMnak;TMng 17q40n += bBa¢ak;faebIcMnYn n enaHEcknwg 40 [sMNl; 17r = x> rkcMnYn n enaHedaydwgfa 4000n3940 << eyIgman 17q40n += eday 4000n3940 << eKTaj 400017q403940 <+< b¤
4017100n
40398 +<<+
eday INq∈ naM[eKTaj)an }100,99{q = ehIy }4017,3977{n = .
KNitviTüaCMuvijBiPBelak
- 76 -
lMhat´TI35 edaHRsayRbBn½æsmIkar ⎩⎨⎧
=
=+
64yx
26)ylogxlog(5 xy
dMeNa¼Rsay edaHRsayRbBn½æsmIkar ⎩⎨⎧
=
=+
)2(64yx
)1(26)ylogxlog(5 xy
lkçx½NÐ ⎩⎨⎧
≠≠>>
1y,1x0y,0x
smIkar )1( Gacsresr 526
xlog1xlog
yy =+
b¤ 01xlog.526)x(log y
2y =+− tag xlogt y=
eK)an 25
144125
169',01t526t 2 =−=Δ=+−
naM[ 55
125
13t,51
512
513t 21 =+==−=
- cMeBaH 51t = eK)an
51xlogy = naM[ )3(yx 5
1
=
KNitviTüaCMuvijBiPBelak
- 77 -
yk ¬#¦ CYskñúg ¬@¦ eK)an 64y.y 51
= naM[ 32y =
ehIytam ¬#¦ eK)an 2)32(x 51
== . - cMeBaH 5t = eK)an 5xlogy = naM[ )4(yx 5= yk ¬$¦ 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
KNitviTüaCMuvijBiPBelak
- 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 −=−= .
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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
++
++
+=
KNitviTüaCMuvijBiPBelak
- 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(
≥−+
++
++
++
−≥−+
++
++
++
KNitviTüaCMuvijBiPBelak
- 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 −=
KNitviTüaCMuvijBiPBelak
- 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 +−=
+= .
KNitviTüaCMuvijBiPBelak
- 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
++=++
KNitviTüaCMuvijBiPBelak
- 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 .
KNitviTüaCMuvijBiPBelak
- 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 −= .
KNitviTüaCMuvijBiPBelak
- 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 −=+++=
KNitviTüaCMuvijBiPBelak
- 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 =+ )
KNitviTüaCMuvijBiPBelak
- 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 .
KNitviTüaCMuvijBiPBelak
- 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 = .
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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 ⎟
⎟⎠
⎞⎜⎜⎝
⎛ −++⎟
⎟⎠
⎞⎜⎜⎝
⎛ −−= .
KNitviTüaCMuvijBiPBelak
- 94 -
lMhat´TI43 eK[GnuKmn_
1x3x31x3x3x)x(f 2
23
+−+−+
= kMnt´cMeBa¼RKb´ IRx∈ . cMeBa¼RKb´cMnYnBitviC¢man a nig b cUrRsaybBa¢ak´fa ½
⎟⎠⎞
⎜⎝⎛
+++++
≥⎟⎠⎞
⎜⎝⎛ ++
ba2abba1f
2ba1f .
dMeNa¼Rsay RsaybBa¢ak´fa ⎟
⎠⎞
⎜⎝⎛
+++++
≥⎟⎠⎞
⎜⎝⎛ ++
ba2abba1f
2ba1f
eyIgman 1x3x3
1x3x3x)x(f 2
23
+−+−+
= kMnt´cMeBa¼RKb´ 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+++++
≤++
KNitviTüaCMuvijBiPBelak
- 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 .
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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 + Eckdac´nwg 2)1x( + ehIy 2)x(P − Eckdac´nwg 2)1x( − . cUrkMnt´rkBhuFa )x(P . dMeNa¼Rsay kMnt´rkBhuFa )x(P ½
tambMrab´eKGacsresr ⎪⎩
⎪⎨⎧
+−=−
++=+
)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
KNitviTüaCMuvijBiPBelak
- 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−=−= .
KNitviTüaCMuvijBiPBelak
- 99 -
lMhat´TI46 eK[ INn,i
31i
31A
nn
∈⎟⎠⎞
⎜⎝⎛ −−⎟
⎠⎞
⎜⎝⎛ += .
cUrbgðajfa 3
nsin.)3(
2.iA n
1n π=
+ cMeBaHRKb; INn∈ .
dMeNa¼Rsay bgðajfa
3nsin.
)3(2.iA n
1n π=
+ cMeBaHRKb; INn∈ .
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=
+
.
KNitviTüaCMuvijBiPBelak
- 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 += .
KNitviTüaCMuvijBiPBelak
- 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 ×−×−×+×−=−×=
KNitviTüaCMuvijBiPBelak
- 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 ≡
KNitviTüaCMuvijBiPBelak
- 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 −−+=
KNitviTüaCMuvijBiPBelak
- 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 .
KNitviTüaCMuvijBiPBelak
- 105 -
lMhat´TI51 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
+=
dMeNa¼Rsay k> KNna )x('f rYcbgðajfa )
4xcos(e2)x('f x π+=
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+=
KNitviTüaCMuvijBiPBelak
- 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+= .
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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 ++= .
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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 .
KNitviTüaCMuvijBiPBelak
- 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 =−−−+
KNitviTüaCMuvijBiPBelak
- 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 = .
KNitviTüaCMuvijBiPBelak
- 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 ∈∀>
KNitviTüaCMuvijBiPBelak
- 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 >
KNitviTüaCMuvijBiPBelak
- 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 .
KNitviTüaCMuvijBiPBelak
- 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 +≤⎟⎠⎞
⎜⎝⎛ +⎟
⎠⎞
⎜⎝⎛ +
KNitviTüaCMuvijBiPBelak
- 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 +≤⎟⎠⎞
⎜⎝⎛ +⎟
⎠⎞
⎜⎝⎛ + .
KNitviTüaCMuvijBiPBelak
- 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
=−
−=−
−=−+−
−=−+− ++
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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 == .
KNitviTüaCMuvijBiPBelak
- 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
−+−+−=
+−++−++−=
−−−++=
−−−−++=
KNitviTüaCMuvijBiPBelak
- 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 .
KNitviTüaCMuvijBiPBelak
- 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 ++++= .
KNitviTüaCMuvijBiPBelak
- 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 ≥++++ .
KNitviTüaCMuvijBiPBelak
- 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 +≤ +
+ .
KNitviTüaCMuvijBiPBelak
- 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 +≤ ++
+
KNitviTüaCMuvijBiPBelak
- 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 ≥++++ .
KNitviTüaCMuvijBiPBelak
- 128 -
lMhat´TI62 eK[ C,B,A CargVas´mMukñúgrbs´RtIekaN 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 ≤
dMeNa¼Rsay k-bgHajfa
23CcosBcosAcos ≤++
tag CcosBcosAcosT ++= Ccos
2BAcos
2BAcos2 +
−+=
eday π=++ CBA ena¼ 2Csin)
2C
2cos(
2BAcos =−
π=
+ ehIy
2Csin21
2Csin
2CcosCcos 222 −=−=
KNitviTüaCMuvijBiPBelak
- 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
+++=
++
++
+=
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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=⎟
⎠⎞
⎜⎝⎛ ++++
+∞→
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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
∑=
KNitviTüaCMuvijBiPBelak
- 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=⎟
⎠⎞
⎜⎝⎛ ++++
+∞→ .
KNitviTüaCMuvijBiPBelak
- 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 +=++
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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 .
KNitviTüaCMuvijBiPBelak
- 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 ++++≤++
KNitviTüaCMuvijBiPBelak
- 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 ++=++
KNitviTüaCMuvijBiPBelak
- 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´ .
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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
++−−−+=
++−−−+=
++−−+
+−+=
++
+−+−+++=
+−+
+++−+−=
++=
KNitviTüaCMuvijBiPBelak
- 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 ) .
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- 144 -
lMhat´TI68
eKtag r nig R erogKñaCakaMrgVg´carikkñúg nig carwkeRkA énRtIekaNmYy . cUrRsaybBa¢akfaebI d Cacm¶ayrvagp©it rgVg´carwkkñúg nig carwkeRkArbsRtIekaNena¼eKán ½
)r2R(Rd2 −= ? dMeNa¼Rsay Rsayfa )r2R(Rd2 −= tag ABC CaRtIekaNEdl cAB,bAC,aBC === ehIy r nig R erogKñaCakaMrgVg´carikkñúg nig carwkeRkA énRtIekaNen¼ . yk I Cap©itrgVg´carwkkññúg nig O Cap©it rgVg´carwkeRkAénRtIekaN . tamRTwsþIbTGWEleyIgán ½
abcOI)cba(OC.cOB.bOA.a 2222 +++=++ eday ROCOBOA === nig
2cbap ++
= eKán abcd.p2Rp2 22 += naM[
p2abcRd 22 −=
Edl OId = .
KNitviTüaCMuvijBiPBelak
- 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
=++ .
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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
KNitviTüaCMuvijBiPBelak
- 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 )