HU NSEN 3 STR O N G · Five Hundred Mathematical Challenges,Complex Numbers From A to … Z ,...
159
eroberogeday lwm pl:ún briBaØabR&tKNitviTüa nig BaNiC¢kmμ H U N SE N × S T R O N G 3 Problems and Solutions
Transcript of HU NSEN 3 STR O N G · Five Hundred Mathematical Challenges,Complex Numbers From A to … Z ,...
eroberogeday lwm pl:ún briBaØabR&tKNitviTüa nig BaNiC¢kmμ
H U N S E N×
S T R O N G
3
Problems and Solutions
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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,
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lwm plþún nig Esn Bisidæ
H U N S E N× 3
S T R O N G
Problems and Solutions
rkßasiT§iRKb´yag
KNitviTüaCMuvijBiPBelak
EpñkRbFanlMhat
- 1 - PaK1
KNitviTüaCMuvijBiPBelak
- 2 - PaK1
EpñkRbFanlMhat´ 1-BIcMnYnBitviC¢man nig x y
epÞógpÞat´TMnak´TMng 11y3x4 =+ cUrkMntrtémøGtibrmaénGnuKmn_ ½
6 )3)(7y)(x()y,x(f y2x +++= 2 enøa¼-cUrkMntRKb´témø kñúgc x [
2;0] π edaydwgfa ½
2xcos13
xsin13 ++
− 4=
mYy3-enAkñúgRtIekaN Edl cAB,bAC,aBCABC === cUrbgHajfa
cba
2Asin
+≤ rYcsresrTMnak´TMngBIreTotEdl
tag[ce RsedogKña . 4-eKyk I nøa¼ ]
4;
4[ ππ− . cUrkMnt´GnuKmn_
os(s
f
kMntelI [ − ebIeKdwgfa cxsin)x2in]1,1 xf += 2 cMeBa¼RKb´ kñúgcenøa¼ .
jfa t
rYcsRmYl f )x(tan x I
5-cUrbgHa tanatana2tana3an atana2tana3=−− Edl
2ka π
≠ RKb´cMnYnKtrWTLaTIhV . k
KNitviTüaCMuvijBiPBelak
- 3 - PaK1
6-cUrbgHajfa ( )2
ab21xcos
b1xsin
a1 +≥⎟⎠⎞
⎜⎝⎛ +⎟
⎠⎞
⎜⎝⎛ +
cMeBa¼RKb´ 2
xb,0a 0,0 π<>> <
-e ue mYyEdl 7 K[ct kaNEkg ABCD AB3BC = . aBIrcMnucénRCug Edl QB = P nig Q C PP]BC[ QC= .
cUrbgHajfa DPDBCDQC C∠+∠=∠ . CacMnYnkMupøicBIr . 8-eK[ 1z nig 2z
cUrRsayfa ( )22
21
221
22 2|zz||zz| 1 |z||z| +=−++
9-eK[ 2 Ca nig cMnYnkMupøicBIrEdl 1z z 1|z||z| 21 == . nig z.z 21 −≠ 1
cUrRsayfa 21
21
zz1+ CacMnYnBzz + itmYy .
10-KNnalImItxageRkam ½
4x
4x1212...121212lim
4x −
−++++++ man n ¦skaer .
→
KNitviTüaCMuvijBiPBelak
- 4 - PaK1
11-eK[sIVúténcMnYnBit kMnt´eday ½
=
)u( n
0 nig 1u1u4 n +u6u4
uu 2n
3n
4n
1n ++=+ RKb´
INn∈
cUrbgHajfa4
n1n u1
u ⎟⎟⎠
⎜⎜⎝
+=++
111⎞⎛ RKb´ INn∈
CaGnuKmn_én . KNna
1)(1n2 +
+−
. nmMu kat g´cMnuc .
rYcKNna nu n
12- x1).....(xxxx1(P
n2422n −++−= )x
13-eK[RtIekaN ABC mYymanRCugRbEvg c,ba ,
knø¼bnÞatBu¼é ] RtC AB[ D
cUrRsayfa 2Ccos
baab2CD+
= . KNnaplbUk 14-
1Sn !n.n.....!3.3!2.2!1. ++++ =
Edl n(n!n 1.2.3)....2n)(1 −− . KNnatémø
=
15- )29tan3)...(3tan3)(2tan3)(1tan3(A oooo ++++=
KNitviTüaCMuvijBiPBelak
- 5 - PaK1
16- itEdleK[ CacMnYnB epÞógpÞat´lkçx&NÐ ½ z,y,x
ycosxcos 0zcos =++ cos nig x3cos 0z3cosy3 =++ cx2s cUrbgHajfa 2cosy2osco 0z ≤ .
k ug .
cUrRsayfa ½
17-eK[RtIekaN mYy . ABC
bnÞat´Bu¼mMu A at´RCC,B, ]AB[,]AC[,]BC[
erogKñaRtg´ ,'A 'C,'B
0'2
'AA2 =
CC
)BAsin(
'BB
)2
ACsin()CBsin( −
+
−
+
−
18-eK[BhuFa n)asasinx()x(P co+= n∈Edl
nwg *IN
cUrrksMNlénviFIEckrvag )x(P 1x2 + .
&n19-eda¼RsayRbB æsmIkar ⎩⎨
− yyx4 ⎧
−=
+=+
baxbayx
4
3
CacMnYnBitEdl Edl a nig b 0ba ≠+ . 0-cUrbgHajfa 2 9
)1nn)(1nn(19999
1n44 =⎟
⎠
⎞⎜⎝
⎛++++
∑=
KNitviTüaCMuvijBiPBelak
- 6 - PaK1
21- IeK[GnuKmn_ kMnt´RKb´ f }0;1{Rx −−∈ eday ½ 1x)
x1(f)x(f)1x2(x + +=+ .
cUrKNna )2009(f....)3(f)2(f)1(fS ++++= 22- éncMnYnKtcUrkMnt´RKb´KU ´viC¢manebIeKdwgfa ½ )n;m(
)m(13nm 22 n+=+ . 23-eK[RtIekaN C m g´ A .AB YyEkgRt
k nig eKy F]AC[E∈ ]AB[∈ Edl ABCAEF ∠=∠
nig erogKñaCaeCIgcMeNalEkgén nig .
ayfa
nig ACBAF ∠=∠ . E
eKtag E 'F ' E F
elIRCug BC[ ]
cUrRs BC'FFEFE'E ≤++ ?
24-eK[ CabIcMnYnBitviC¢man . cUrRsaybBa¢ak´fa ½ c,b,a
2)bb)cb(4a 33 33 a(4c
ba)ac(4
accb3 33 33
≤++
++
+++
+++
+
it =
25-eK[sIVútcMnYnB−⎩
⎨⎧
≥∀+
==
−+ 1n,2a2a1aa
1n1n
10 an
KNitviTüaCMuvijBiPBelak
- 7 - PaK1
cUrRsayfa 1nn aaa +1n 2 +×= .
26-kñúgRtIekaN mYycUrRsaybBa¢akfa ½ ABC
r2Csin
2Bsin
2Asin
R4111≥++
ig Edl r n R C IekaN . CabIcMnYnBitviC¢manEdl akaMrgVg´carwkkñúg nig carwkeRkARt
27-eK[ b,a c, 1cbaabc4 +++= cUrbgHajfa ½
)cabcab(2c
bab
ca
cb≥
+++
+ a 222222
+++ 28-eK kMnt´eday ½
101n
[sIVút }a{ n
1a0 = nig 4na....a.aa +=+ a¼RKb . a
cMeB ´ n ≥ 1
cUrbgHajf 2a 1n =+ cMeBa¼RKb´ n ≥an − . CabIcMnYnBitviC¢manEdl cab
1
29-eK[ ,a bc,b 1ca =++ . cUrbgHajfa 16)
a1c()
c1b()
b1a( 222 ≥+++++
30-eK[ 432 222,
2cos222,
2cos22
2cos2 π
=++π
=+π
= IcUrrkrUbmnþTUeTA nig RsaybBa¢ak´
mnþena¼pg . BI«TahrN_xagelrUb
KNitviTüaCMuvijBiPBelak
- 8 - PaK1
31-edayeFIVvicart
dac´nwg Canic©cMeBa¼RKb´ μCati n .
n,7 ∈
amkMeNIncUrRsaybBa¢akfacMnYn ½1n1n6
n 92A ++ += Eck 11
cMnYnKtFm32-eK[ N61009743831E nnnn
n I−+−=
acnwg Canic©RKb´ .
cUrbgHajfa nE Eckd 187 INn∈
33-cUrRsaybBa¢akfa 1n21
1 <+n1.....
31
21
++
+++ viC¢man .
cUrRsaybBa¢akfa ½ 34-eK[ n cMnYnBit n321 a.........;;a;a;a
nn321n321 na..........aaa ≥++++ a......a.a.a
35-eda¼RsayRbB&næsmIkar ½
⎩⎨
tanxtan 33
⎧
=+
=++
34ytanytanx28ytanytanxtan6xtan 4224
Edl 2
x0 π< . y <<
´RtIekaNmYynigtag énRtIekaN .
36-eK[ c,b,a CaRCugrbs p
Caknø¼brimaRt
KNitviTüaCMuvijBiPBelak
- 9 - PaK1
cUrbgHajfa ⎟⎠⎞
⎜⎝⎛ ++≥
−+
−+
− bp1
ap1
c1
b1
a12
cp1
-k¿cUrRsaybBa¢akfa ½ 37
3n41n4
1n41n4
1n43n4
<+−
<+−
+− cMeBa¼RKb´
x¿TajbgHajfa ½ *INn∈
3n43
)1n4(.......139)1n4(.......117
1n21
+×××−×××
+ 53
+<
××
<
CabIcMnYnBitviC¢man . 38-eK[ ;b;a c
cUrbgHajfa 23
bac
acb
cba
≥+
++
++
39-k¿KNna n)
....2.1
1++ Edl 2n >
1n(1
4.31
3.21
−++
x¿edayeRbIvismPaB GMAM − én n( )1− cMnYnxageRkam ½
n)11;
3.21;
2.11
− cUrbgHajfa nn
n(;
4.31 )!n( 2< .
CabIcMnYnBitxusBIsUnü .
fa
40-eK[ b;a c;
cUrbgHajac
cb
ba
ac
cb 222
++≥⎟⎠⎞
⎜⎝⎛+⎟
⎞⎜⎝⎛+⎟
⎞ ba
⎠⎠⎜⎝⎛
CabIcMnYnBitviC¢manminsUcUrbgHajfa
41-eK[ y;x nü . z;
222222 zyx9
z1
y1
x1
++≥++
KNitviTüaCMuvijBiPBelak
- 10 - PaK1
42-eKman xlog11
aay nig −= ylog11
aaz −=
cUrRsayfa zlog11
aax −= .
43-eKBinitüsIVúténcMnYnBit kMnt´eday )u( n⎪⎩
⎪⎨
⎧
−−
=
=
+ 1u24u5
u
4u
n
n1n
1
cUrKNna CaGnuKmn_én .
44-eKBinitüsIVúténcMnYnBit kMnt´eday
nu n
)u( n⎪⎩
⎪⎨
⎧
−−
=
=
+ 1u4u3
u
3u
n
n1n
1
45-eKBinitüsIVúténcMnYnBit n kMnt´eday ½ cUrKNna nu CaGnuKmn_én . n
u( )
⎩⎨
∈=+ *INn,uu 1n
⎧ = 4u
n
1
_én 46-eK[ Bit kMnt´eday ½ cUrKNna nu CaGnuKmn n .
sIVúténcMnYn )u( n
5u0 = nig 5nu21u nn ++ ( INn1 =+ ∈ )
rKNna énsIVút CaGnuKmn_én . cU nu )u( n n
KNitviTüaCMuvijBiPBelak
- 11 - PaK1
47-edayeRbIGnumanrYmKNitviTüacUrKNnalImItxageRkam ½
2x
2x22.....222limL
2xn −−++++
.
8-eK[sIVúténcMnYnBit k t´elI eday ½
++=
→
( man n ra¨DIkal´ ) 4 Mn)U( n IN
⎪⎩⎨
∈+=+ INn,UU n1n
0 ⎪⎧ =
2
2U
GnuKmn_én . ..U
k¿cUrKNna nU Ca n
x¿KNnaplKuN n21n U..UUP 0 ××××= . 49-eK[s MnYnBit kMnt´elI eday ½IVúténc )U( n IN
22U0 = ig n INn,
2U11
U 1n =+
2n ∈∀
−−
0-eK[sIVúténcMnYnBit kMnt´eday½
=
2a,INn,
au0
KNna nU CaGnuKmn_én . n
5 )u( n
⎩ −=+ 2uu 2n1n
⎨⎧
>∈∀
KNitviTüaCMuvijBiPBelak
- 12 - PaK1
cUrRsaybBa¢ak´fa nn
4aau ⎟⎞
⎜⎛ −
+⎜⎛ +
=2
22
2
n 2a
24a
⎟⎠
⎜⎝
−⎟⎟⎠
⎞⎜⎝
−
51-k¿cUrkMntelxénGBaØat éncMnYn d,c,b,a abcd ebIeKdwgfa ½
dcbabcd = a9×
cMnYn x¿cMeBa¼témø EdlánrkeXIjxagelIcUrbBa¢ak´fa d,c,b,a
abcd nig dcba suTæEtCakaer®ákd . 52-snμtfa HUNSEN Cac YnmYymanelxRáMmYyxÞg´Edl Mn
Caelx . KuNcMnYne eKTTYl CacMnYnmanelx
Þg´KW
N,E,S,N,U,H
eBlEdleK n¼nwg Tæpl3
RáMmYyx STRONG Edl G,N,O,R,T,S Caelx cUrrkcMnYn HUNSEN nig STRONG cMeBa¼ 3S,6N == nigcMeBa¼ 8N 9S, == . (GkßrxusKñatagedayelxxusKña ). 53-cUrbMeBjRbmaNviFIbUkxageRkam ½
KNitviTüaCMuvijBiPBelak
- 13 - PaK1
YENOM EROM
E
(GkßrxusKñatagedayelxxusKYn ´Ed
DNS+
ña ) 54-eK[ACacMn mYymanelxRáMxÞg lelxxÞg´vaerobtam lMdab´ x1x;x ;1x;2x; ++ CacMnYnmYymanelx
lelxxÞg+ ehIy B
RáMmYyxÞg´Ed ´vaerobtamlMdab´ x;1x;1x;2x;1x;x −+ .
cUrbgHajfae CakaerRákdena¼ B k¾Cakaer®ákdEdr .++
bI A 55-eK[cMnYn
)elxn2(
111.....111A = nig B) elxn(
444.....444=
++ CakaerRákd . MnYnmYymanelxbYnxÞg´EdlelxxÞg´vaerobtamlMdab´
.
7-eKman
cUrbgHajfa A 1B
56-c b;b;a;a
rkcMnYnena¼ebIeKdwgfavaCakaerRákd . 5 111088893333,110889333,108933 222 ===
KNitviTüaCMuvijBiPBelak
- 14 - PaK1
333332 1111088889= . ahrN_xagelIcUrrkrUbmnþTUeTAnigRsaybBa¢ak´rUbmnþen¼pg. BI«T
58-eKman 999800019999,998001999,980199 222 === . 9999800001999992 =
BI«TahrN_xagelIcUrrkrUbmnþTUeTAnigRsaybBa¢ak´rUbmnþen¼pg. 59-eKman 444355566666,443556666,435666 222 ===
4556,115 22222 =
4444355556666662 = . BI«TahrN_xagelIcUrrkrUbmnþTUeTAnigRsaybBa¢ak´rUbmnþen¼pg 60- eKman
111111445556,111162 −=−=
«TahrN_xagelIcUrrkrUbmnþTUeTA nigRsaybBa¢ak´rUbmnþen¼pg
−
1111111144455556 22 =− . BI
KNitviTüaCMuvijBiPBelak
- 15 - PaK1
61-eK[ 011nn aa..........aaA = − nig 0a2B ×− 11nn a.........aa= −
ac´nwg . Rsayfa A Eckdac´nwg 7 lu¼RtaEt B Eckd 7
62-eK[cMnYn 012nn aaa......aaA − 1=
,
nYn ac´nwg kalNa a)a...
Edl n210 a....,aa,a Caelx . .,
cUrRsayfacM A Eckd 6
0n21 aa(4y ++ Eckdacnwg nig Im,I[cMnYn
++= . 6
63-cUrkMnt´elx b eda abba CaKUbéncMnYnKt 64-eKtag r nig R erogKñaCakaMrgVg´carikkñúg nigcarikeRkA énRtIekaN ABCmYy . k¿cUrRsayfa
Rr1CcosBcosAcos +=++
. x¿cUrRsayfa R ≥ r2
65-k¿cUrRsayfa x2cotxcotx2sin
1−=
¿cUrKNna xn2
n
2a
sin
1....
2a
sin
1
2a
sin
1asin
1S ++++=
KNitviTüaCMuvijBiPBelak
- 16 - PaK1
66-k¿cUrRsayfa xcot2x
cot11+
xcos=
¿KNn uN x aplK )
2a
cos
11).a
cos
11)(a
cos
11)(
ac1(Pn +++= ....(
22os1
n2
+
Bit n
67-eK[sIVúténcMnYn )U( kMnt´elI IN eday ½
2U0 = nig 2
INn,2
U11Un+
2n
1 ∈∀−−
= n_én .
cUrKNna KNna nU CaGnuKm n
68-9π7
cos9π4
cos9π
cosS 333 +−= 69-eK[RtIekaN mYymanmMukñúgCamMuRsYc . ABC
k¿cUrRsayfa Ctan.Btan.AtanBtanAtan Ctan =++ x¿TajbBa¢akfa 33CtanBtanAtan ≥++ .
½
70-KNnaplbUk
)9n(999......999....999999Sn
elx++++=
IeKdwgfacMnYn ½ 71-cUrkMnt´RKb´KUtémøKt´viC¢man ) ebb,a(
1bab2
a32 +
CacMnYnKt´viC¢manE2
−dr .
KNitviTüaCMuvijBiPBelak
- 17 - PaK1
72-cUrbgHajfa ½ 1
ab8cca8bbc8a 222≥
cba+
++
++
cMeBa¼RK b´cMnYnBitviC¢man .
YnBitEdlc,b,a
73-eK[ )a( n CasIVúténcMn 21a1 =
nigcMeBa¼RKb´cMnYnKt´v ¢mann eyIgmiC an 1aa
a2
2n
+−= a
nn1n+
´cMnYnKt n eyIgman ½n
cUrRsaybBa¢ak´facMeBa¼RKb ´viC¢man aaa 321 1a......... <++++ .
øKt´ ebIeKdwgfacMeBa¼RKb´ MnYnKt´viC¢man eKman
74-cUrkMnt´RKb´KUtém 3n,m ≥
c a1aa1aa
2n
m
−+−+ CacMnYnKt´ .
YnKt´viC¢man Edl =+75-eK[bIcMn ac,b,a 10cb+ . n cUrrktémøFMbMputé cbaP ××= .
76-eKmansmPaB ba
1b
xcosa
xsin 44
+=+
Edl ,0a 0ba,0b ≠+≠ ≠
cUrbgHajfa 33
8
3
8
)ba(1+b
xcosa
xsin=+ .
KNitviTüaCMuvijBiPBelak
- 18 - PaK1
77-eK[RtIekaN ABC mYYy . bgHajfaebI
3Ctan,
3Btan tan,
3A
ax2Ca¦ssmIkar cbxx:)E( 3 0=+++ án ena¼eK cb3a =+3 + .
atémø®ákdén 78-k¿ cUrKNn10
sin π nig 10
cos π a x¿ cUrRsayf
10sin)yx(4)yx(x 22222 π
+≤−+ RKb´cMnYn IRy,x ∈ .
79- eK[sIVúténcMnYnBit kMnt´)U( n eday4
nsin.2Un
n π=
ðajfa Edl *In∈ . N
k-cUrbg4
nsin4
ncos4
)1n(cos.2 π−
π=
π+ x-Taj[ánfa
4)1n(c1 os)2(ncos)2(U nn
n 4π+
−π
= bUk ½
U.........U
+
K-KNnapln321n UUS ++++= _éCaGnuKmn n .
n32 ≥ . n
80-eK[ n cMnYnBit a1 0a.......;;a;a;
eKtag *INn,n
a.....aaaS 1
nn32 ∈∀
++++ =
KNitviTüaCMuvijBiPBelak
- 19 - PaK1
cUrRsayfaebI 1n1nn3211n aa.......a.a.aS +
++ ≥ ena¼eKán n
n31 a.....a≥ . 2n aaS
gHajfa 81-cUrb101
10099......
65.
43.
21
< 151<
82-eK[ cMnYnB 1n2 b.;b;b;a.....;;a .
2n
22
21
2n
222nn2211 ++++++≤+++
83-eK[ nig CasIVútcMnYnBitkMnt´elI eday nigTMnak´TMngkMenIn ½
nig RKb´ cUrKNna nig CaGnuKmn_én .
n2 it 21 ...;;a n
cUrRsayfa ½ )b.....bb)(a....aa()ba....baba( 21
)x( n )y( n IN
1y,5x 00 ==2
nn3
n1n yx3xx +=+3
nn2
n1n yyx3y +=+ INn∈
nx ny n
KNitviTüaCMuvijBiPBelak
EpñkdMeNa¼Rsay
- 20 - PaK1
KNitviTüaCMuvijBiPBelak
lMhat´TI1 BIcMnYnBitviC¢man nig epÞógpÞat´TMnak´TMngx y 11y3x4 =+ cUrkMntrtémøGtibrmaénGnuKmn_ ½
)y2x3)(7y)(6x()y,x(f +++= dMeNa¼Rsay kMnt´rtémøGtibrmaénGnuKmn_ ½
)y2x3)(7y)(6x()y,x(f +++= tamvismPaB eyIgán ½ GMAM −
3
3
)y,x(f3
13y3x4
)y2x3)(7y)(6x(3
)y2x3()7y()6x(
≥++
+++≥+++++
eKTaj 3
313y3x4)y,x(f ⎟
⎠⎞
⎜⎝⎛ ++
≤ eday 11y3x4 =+
eKán 5123
1311)y,x(f3
=⎟⎠⎞
⎜⎝⎛ +
≤ . dUcen¼témøGtibrmaénGnuKmn_ )y,x(f es μInwg . 512
- 21 - PaK1
KNitviTüaCMuvijBiPBelak
- 22 - PaK1
lMhat´TI2 cUrkMntRKb´témø kñúgcenøa¼ x [
2;0] π edaydwgfa ½
24xcos13
xsin13
=+
+−
dMeNa¼Rsay kMnt´RKb´témø kñúgcenøa¼ x [
2;0] π
)1(24xcos13
xsin13
=+
+−
eKBinitü ⎟⎠⎞
⎜⎝⎛ π
−π
=π
43sin
12sin
4
2621.
22
22.
23
3cos
4sin
4cos
3sin
−=−=
ππ−
ππ=
ehIy ⎟⎠⎞
⎜⎝⎛ π
−π
=π
43cos
12cos
4
2622.
23
22.
21
2sin
3sin
4cos
3cos
+=+=
ππ+
ππ=
KNitviTüaCMuvijBiPBelak
smIkar GacsresreTACa ½ )1(
x2sinx12
sin
xcosxsin212
cosxsinxcos12
sin
2xcos
12cos
xsin12
sin
2xcos
426
xsin4
26
2xcos22
13
xsin22
13
=⎟⎠⎞
⎜⎝⎛ +π
=π
+π
=
π
+
π
=
+
+
−
=
+
+
−
eday 2
x0 π<< ena¼eKTaján
⎢⎢⎢⎢
⎣
⎡
−π=+π
=+π
x2x12
x2x12
eKTaj 12
x π= ¦
3611x π
= ( epÞógpÞat´nwglkçx&NÐEdl[ ) dUcen¼ }
3611;
12{x ππ
∈ .
- 23 - PaK1
KNitviTüaCMuvijBiPBelak
- 24 - PaK1
lMhat´TI3 enAkñúgRtIekaN mYyEdl ABC cAB,bAC,aBC === cUrbgHajfa
cba
2Asin
+≤ rYcsresrTMnak´TMngBIreTotEdl
RsedogKña .
dMeNa¼Rsay bgHajfa
cba
2Asin
+≤
tamRTwsþIbTsIunUs R2Csin
cBsin
bAsin
a===
Edl R CakaMrgVg´carwkeRkARtIekaN . ABC
eKTaj CsinR2c,BsinR2b,AsinR2a === eKman
CsinR2BsinR2AsinR2
cba
+=
+
2CBcos)
2A
2sin(
2Acos
2Asin
cba
2CBcos
2CBsin2
2Acos
2Asin2
CsinBsinAsin
cba
−−
π=
+
−+=
+=
+
KNitviTüaCMuvijBiPBelak
( )1
2CBcos
2Asin
2CBcos
2Acos
2Acos
2Asin
cba
−=
−=
+
eday π<−≤ |CB|0 ena¼ 12
CBcos0 ≤−
<
tam eKTaj )1(2Asin
2CBcos
2Asin
cba
≥−
=+
.
dUcen¼ cb
a2Asin
+≤ .
TMnak´TMngBIreTotEdlRsedogKñaen¼KW ½
acb
2Bsin
+≤ nig
bac
2Csin
+≤ .
- 25 - PaK1
KNitviTüaCMuvijBiPBelak
- 26 - PaK1
lMhat´TI4 eKyk tag[cenøa¼ I ]
4;
4[ ππ−
−
. cUrkMnt´GnuKmn_ f kMnt´elI [ ebIeKdwgfa ]1,1 xcosxsin)x2(sinf += rYcsRmYl f cMeBa¼RKb´ x kñúgcenøa¼ I . )x(tan2
dMeNa¼Rsay kMnt´GnuKmn_ f eKman xcosxsin)x2(sinf += elICakaer f 22 )xcosx(sin)x2(sin +=
x2sin1)x2(sinfxcosxcosxsin2xsin)x2(sinf
2
222
+=
++=
eday )x4
sin(2xcosxsin +π
=+ cMeBa¼RKb´ Ix∈
eKán 0)x2(sinf > . ehtuen¼ x2sin1)x2(sinf += tag x2sinz = dUcen¼ z1)z(f += cMeBa¼ 1z1 <<− . müa¨geTotRKb´ ]
4,
4[x ππ−∈ eKman 1xtan1 ≤≤−
KNitviTüaCMuvijBiPBelak
naM[ . 1xtan0 2 ≤≤
dUcen¼ xcos
1xcos
1xtan1)x(tanf 222 ==+= .
lMhat´TI5 cUrbgHajfa atana2tana3tanatana2tana3tan =−− Edl
2ka π
≠ RKb´cMnYnKtrWTLaTIhV k .
dMeNa¼Rsay bgHajfa atana2tana3tanatana2tana3tan =−− eKman )aa2tan(a3tan +=
atana2tanatana2tana3tana3tan
atana2tan)atana2tan1(a3tanatana2tan1atana2tana3tan
+=−+=−
−+
=
dYcen¼ atana2tana3tanatana2tana3tan =−− .
- 27 - PaK1
KNitviTüaCMuvijBiPBelak
- 28 - PaK1
lMhat´TI6 cUrbgHajfa ( )2
ab21xcos
b1xsin
a1 +≥⎟⎠⎞
⎜⎝⎛ +⎟
⎠⎞
⎜⎝⎛ +
cMeBa¼RKb´ 2
x0,0b,0a π<<>>
dMeNa¼Rsay bgHajfa ( )2
ab21xcos
b1xsin
a1 +≥⎟⎠⎞
⎜⎝⎛ +⎟
⎠⎞
⎜⎝⎛ +
eKman xcosxsin
abxcos
bxsin
a1)xcos
b1()xsin
a1( +++=++ tamvismPaB eKman ½ GMAM −
xcosxsinab2
xcosb
xsina
≥+
eKán xcosxsin
abxcosxsin
ab21)xcos
b1)(xsin
a1( ++≥++
22
2
)ab21()x2sin
ab21()xcos
b1()xsin
a1(
)xcosxsin
ab1()xcos
b1)(xsin
a1(
+≥+≥++
+≥++
BIeRBa¼RKb´ 2
x0 π<< eKman 1x2sin ≤ .
KNitviTüaCMuvijBiPBelak
- 29 - PaK1
lMhat´TI7 eK[ctuekaNEkg mYyEdl BCABCD AB3= . P nig Q CaBIrcMnucénRCug Edl]BC[ QCPQBP == . cUrbgHajfa DPCDBCDQC ∠+∠=∠ .
dMeNa¼Rsay bgHajfa DPCDBCDQC ∠+∠=∠ tag γ=∠β=∠α=∠ DQC;DPC;DBC
eyIgman 1tan,21tan,
31tan =γ=β=α
eday γ==−
+=
βα−β+α
=β+α tan1
61131
21
tantan1tantan)tan(
dUcen¼ α ¦ γ=β+ DPCDBCDQC ∠+∠=∠ .
A
B C
D
P Q
KNitviTüaCMuvijBiPBelak
- 30 - PaK1
lMhat´TI8 eK[ nig CacMnYnkMupøicBIr . 1z 2z
cUrRsayfa ( )22
21
221
221 |z||z|2|zz||zz| +=−++
dMeNa¼Rsay Rsayfa ( )2
22
12
212
21 |z||z|2|zz||zz| +=−++ eKman )zz)(zz(|zz| 2121
221 ++=+
( )1|z|zzzz|z||zz|
| zzzzzzzz|zz2
221212
12
21
222121112
21
+++=+
+++=+
ehIy )zz)(zz(|zz| 21212
21 −−=−
( )2|z|zzzz|z||zz|zzzzzzzz|zz|
222121
21
221
222121112
21
+−−=−
+−−=−
bUkTMnak´TMng ( nig )1 ( )2 eKán ½ ( )2
22
12
212
21 |z||z|2|zz||zz| +=−++ .
KNitviTüaCMuvijBiPBelak
- 31 - PaK1
lMhat´TI9 eK[ nig CacMnYnkMupøicBIrEdl 1z 2z 1|z||z| 21 == nig z . 1z. 21 −≠
cUrRsayfa 21
21
zz1zz
++ CacMnYnBitmYy .
dMeNa¼Rsay Rsayfa
21
21
zz1zz
++ CacMnYnBitmYy
tag 21
21
zz1zzZ ena¼
++
=21
21
z.z1zzZ
++
=
eday 1|z|z.z 2111 == ena¼
11 z
1z = ehIydUcKña 2
2 z1z =
eKán Z1zz
zz
z1.
z11
z1
z1
Z12
12
21
21 =+
+=
+
+=
eday ZZ = ena¼ 21
21
zz1zzZ
++
= CacMnYnBit .
KNitviTüaCMuvijBiPBelak
- 32 - PaK1
lMhat´TI10 KNnalImItxageRkam ½
4x4x1212...121212
lim4x −
−++++++→
man n ¦skaer .
dMeNa¼Rsay KNnalImIt
tag 4x
4x1212...121212limL
4xn −
−++++++=
→
( man ¦skaer ) n
eKán
4x4x1212...1212limL
4x1n −−+++++
=→−
( man ¦skaer ) 1n −
snμtfa 4x1212...121212)x(fn −++++++= nig 4x1212...121212)x(f n +++++++=
KNitviTüaCMuvijBiPBelak
eKman ½
)x(f4x1212.....1212)x(f).x(f
16x1212...121212)x(f).x(f
1nnn
nn
−=−+++++=
−++++++=
eKTaj )x(f)x(f)x(f
n
1nn
−=
eKán )x(f
1lim.4x
)x(flim)x(f)4x(
)x(flimLn
4x
1n
4xn
1n
4xn →
−
→
−
→ −=
−=
)4(f
1.LLn
1nn −= eday 8)4(fn =
eKán 1nn L81L −= naM[ CasIVútFrNImaRtmanersug )L( n
81q = nigtY
4x4x12limL
4x1 −−+
=→
. tamrUbmnþtYTI n eKán 1n
1n qLL −×= eday
81
4x121lim
4x4x12limL
4x4x1 =++
=−
−+=
→→
nig 81q = ena¼ nn 8
1L = . dUcen¼
n4x 81
4x4x1212...121212
lim =−
−++++++→
.
- 33 - PaK1
KNitviTüaCMuvijBiPBelak
- 34 - PaK1
lMhat´TI11 eK[sIVúténcMnYnBit kMnt´eday ½ )u( n
1u0 = nig 1u4u6u4
uu RKb´ n
2n
3n
4n
1n +++=+ INn∈
cUrbgHajfa 4
n1n u11
u11 RKb´ ⎟⎟
⎠
⎞⎜⎜⎝
⎛+=+
+
INn∈
rYcKNna u CaGnuKmn_én . n n
dMeNa¼Rsay
bgHajfa 4
n1n u11
u11 ⎟⎟
⎠
⎞⎜⎜⎝
⎛+=+
+
eyIgman 4n
n2
n3
n
1n u1u4u6u41
u11 +++
+=++
4
n4
n
4n
4n
n2
n3
n4
n
u11
u)1u(
u1u4u6u4u
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
+=
++++=
dUcen¼ 4
n1n u11
u11 ⎟⎟
⎠
⎞⎜⎜⎝
⎛+=+
+
.
KNitviTüaCMuvijBiPBelak
KNna CaGnuKmn_én nu n
eyIgman 4
n1n u11
u11 ⎟⎟
⎠
⎞⎜⎜⎝
⎛+=+
+
eKán ⎟⎟⎠
⎞⎜⎜⎝
⎛+=⎟⎟
⎠
⎞⎜⎜⎝
⎛+
+ n1n u11ln4
u11ln
tag ⎟⎟⎠
⎞⎜⎜⎝
⎛+=
nn u
11lnv naM[ ⎟⎟⎠
⎞⎜⎜⎝
⎛+=
++
1n1n u
11lnv
eKán . n1n v4v =+
TMnak´TMngen¼bBa¢ak´fa CasIVútFrNImaRtmanersug )v( n 4q = nigtY 2ln
u11lnv
00 =⎟⎟
⎠
⎞⎜⎜⎝
⎛+= ( eRBa¼ 1u0 = ) .
tamrUbmnþ 2ln4q.vv nn0n == .
eday ⎟⎟⎠
⎞⎜⎜⎝
⎛+=
nn u
11lnv
eKTaj 2ln4u11ln n
n
=⎟⎟⎠
⎞⎜⎜⎝
⎛+ naM[ n4
n
2u11 =+
dUcen¼ 12
1u n4n−
= .
- 35 - PaK1
KNitviTüaCMuvijBiPBelak
lMhat´TI12 KNna
)xx1).....(xx1)(xx1(P1nn 22422
n
++−+−+−=
dMeNa¼Rsay KNna
)xx1).....(xx1)(xx1(P1nn 22422
n
++−+−+−=
eKman )aa1)(aa1(a)a1(aa1 2222242 +++−=−+=++ eKTaj 2
422
aa1aa1aa1
++++
=+− yk k2xa =
eKán 1kk
2k1k1kk
22
2222
xx1xx1xx1 +
+++
++
++=+−
2
22n
0k 22
22
n xx1xx1
xx1xx1P
2n1n
1kk
2k1k
++++
=⎟⎟⎠
⎞⎜⎜⎝
⎛
++
++=
++
+
++
∏=
dUcen¼ 2
22
n xx1xx1P
2n1n
++++
=++
.
- 36 - PaK1
KNitviTüaCMuvijBiPBelak
- 37 - PaK1
lMhat´TI13 eK[RtIekaN mYymanRCugRbEvg . ABC c,b,a
knø¼bnÞat´Bu¼énmMu C kat´ [ Rtg´cMnuc D . ]AB
cUrRsayfa 2Ccos
baab2CD+
= .
dMeNa¼Rsay Rsayfa
2Ccos
baab2CD +
=
eyIgman S CDBACDABC SS += eday Csinab
21SABC =
2CsinCDb
21SACD = nig
2CsinCDa
21SCDB =
eKán 2CsinCD.a
21
2CsinCD.b
21sibCab
21
+= eKTaj
2Ccos
baab2
2Csin)ba(
CsinabCD+
=+
= .
dUcen¼ 2Ccos
baab2CD+
= .
C
A D B
KNitviTüaCMuvijBiPBelak
- 38 - PaK1
lMhat´TI14 KNnaplbUk
!n.n.....!3.3!2.2!1.1Sn ++++= Edl 1.2.3)....2n)(1n(n!n −−= .
dMeNa¼Rsay KNnaplbUk
!n.n.....!3.3!2.2!1.1Sn ++++= eKman !k.]1)1k[(!k.k −+=
!k)!1k(!k.k!k!k).1k(!k.k
−+=−+=
eKán !n)!1n(...)!3!4()!2!3()!1!2(Sn −+++−+−+−= dUcen¼ 1)!1n(Sn −+= .
KNitviTüaCMuvijBiPBelak
lMhat´TI15 KNnatémø
)29tan3)...(3tan3)(2tan3)(1tan3(A oooo ++++=
dMeNa¼Rsay KNnatémø A
)29tan3)...(3tan3)(2tan3)(1tan3(A oooo ++++= eKman oo
ooo0
1cos60cos61sin1tan60tan1tan3 =+=+
oo
ooo0
oo
oooo
oo
oooo
29cos60cos89sin29tan60tan29tan3
3cos60cos63sin3tan60tan3tan3
2cos60cos62sin2tan60tan2tan3
=+=+
−−−−−−−−−−−−−−−−−−−−−−−−
=+=+
=+=+
eKán 29ooooo29o
oooo
)60(cos1
29cos...3cos2cos1cos)60(cos89sin....63sin.62sin.61sinA ==
eday 2160cos o = dUcen¼ 5368709122A 29 == .
- 39 - PaK1
KNitviTüaCMuvijBiPBelak
- 40 - PaK1
lMhat´TI16 eK[ CacMnYnBitEdlepÞógpÞat´lkçx&NÐ ½ z,y,x
0zcosycosxcos =++ nig cos 0z3cosy3cosx3 =++ cUrbgHajfa cos 0z2cosy2cosx2 ≤ . dMeNa¼Rsay bgHajfa cos 0z2cosy2cosx2 ≤ tamrUbmnþ cos acos3acos4a3 3 −= eKTaj )1(x3cosxcos3xcos4 3 +=
)3(z3coszcos3zcos4)2(y3cosycos3ycos4
3
3
+=
+=
bUkTMnak´TMng eKán ½ )3(,)2(,)1(
0zcos4ycos4xcos4 333 =++
¦ 0zcosycosxcos 333 =++ (eRBa¼ 0zcosycosxcos =++ nig 0z3cosy3cosx3cos =++ ) eKman 0zcosycosxcos =++ eKán zcosycosxcos −=+
KNitviTüaCMuvijBiPBelak
elIkCaKUbeKán ½
zcosycosxcos3xcosycosxcoszcosycoszcosycosxcos3xcos
zcosycos)ycosx(cosycosxcos3xcoszcos)ycosx(cos
333
333
333
33
=++
−=+−
−=+++
−=+
eday 0zcosycosxcos 333 =++ eKTaj 0zcosycosxcos3 = naM[ ¦ 0xcos = 0ycos = ¦ 0zcos = . edaysn μtyk 0xcos = ena¼ zcosycos −= eKán ½
0)1zcos2(z2cosy2cosx2cos)1zcos2)(1ycos2)(1xcos2(z2cosy2cosx2cos
22
222
≤−−=
−−−=
dUcen¼bBaHaRtUvánRsaybBa¢ak´ .
- 41 - PaK1
KNitviTüaCMuvijBiPBelak
- 42 - PaK1
lMhat´TI17 eK[RtIekaN mYy . bnÞatBu¼mMu kat´RCug ABC C,B,A
]AB[,]AC[,]BC[ erogKñaRtg´ . 'C,'B,'A
cUrRsayfa ½
0'CC
)2
BAsin(
'BB
)2
ACsin(
'AA
)2
CBsin(=
−
+
−
+
−
dMeNa¼Rsay
eyIgman S B'CC'ACCABC SS += eday Csinab
21SABC =
2Csin'CCb
21S 'ACC = nig
2Csin'CCa
21S B'CC =
eKán 2Csin'CC.a
21
2Csin'CC.b
21sibCab
21
+=
C
A 'C B
KNitviTüaCMuvijBiPBelak
eKTaj )1(2Ccos
baab2
2Csin)ba(
Csinab'CC+
=+
=
eKman 2Ccos
2Csin2
2BAcos
2BAsin2
CsinBsinAsin
cba
+−
=−
=−
eday 2Csin
2C
2cos
2BAcos =⎟
⎠⎞
⎜⎝⎛ −π
=+
eKán 2Ccos2
BAsin
cba
−
=− naM[ )
2BAsin(.
bac
2Ccos −
−=
tam eKán )1( )2
BAsin(ba
abc2'CC 22
−−
=
eKTaj abc2
ba'CC
)2
BAsin( 22 −=
−
.
dUcKñaEdr abc2
ac'BB
)2
ACsin(;
abc2cb
'AA
)2
CBsin( 2222 −=
−−
=
−
dUcen¼ 0'CC
)2
BAsin(
'BB
)2
ACsin(
'AA
)2
CBsin(=
−
+
−
+
−
.
- 43 - PaK1
KNitviTüaCMuvijBiPBelak
- 44 - PaK1
lMhat´TI18 eK[BhuFa P n)acosasinx()x( += Edl *INn∈ cUrrksMNlénviFIEckrvag nwg x)x(P 12 + .
dMeNa¼Rsay sMNl´énviFIEck tag )x(R CasMNl´énviFIEckrvag nwg )x(P 1x2 +
-ebI ena¼ 1n = acosasinx)x(P += dUcen¼ acosasinx)x(R += CasMnl´énviFIEck . -ebI eKán 2n ≥ )x(R)x(Q)1x()x(P 2 ++= Edl CaplEck nwg )x(Q BAx)x(R += eKán BAx)x(Q)1x()acosasinx( 2n +++=+ ebI ena¼ ix = BAi)acosasini( n +=+ ¦ A.iB)nasin(.i)nacos( +=+ eKTaj nig )nasin(A = )nacos(B = dUcen¼ )nacos()nasin(x)x(R += .
KNitviTüaCMuvijBiPBelak
- 45 - PaK1
lMhat´TI19
eda¼RsayRbB&næsmIkar ⎩⎨⎧
−=−
+=+
byaxyxbayx
44
3
Edl nig b CacMnYnBitEdl aa 0b ≠+ .
dMeNa¼Rsay eda¼RsayRbB&næsmIkar
⎩⎨⎧
−=−
+=+
byaxyxbayx
44
3
smmUl 1y
yxbyax)yx(ba
44
3
⎩⎨⎧
−=−
+=+
443
44
3
yx)yx(y)yx(a
yxbyax)yx(ybyay
−++=+
+⎩⎨⎧
−=−
+=+
eday ena¼ 0ba ≠+ 0yx ≠+ eKTaj 23
443
xy3xyx
yx)yx(ya +=+
−++=
ehIy 323 yyx3a)yx(b +=−+=
eKánRbB&næ naM[ ⎩⎨⎧
=+
=+
byyx3axy3x
32
23
⎩⎨⎧
−=−
+=+3
3
bayxbayx
dUcen¼ 2
babay;2
babax . 3333 −−+
=−++
=
KNitviTüaCMuvijBiPBelak
- 46 - PaK1
lMhat´TI20 cUrbgHajfa 9
)1nn)(1nn(19999
1n44 =⎟
⎠
⎞⎜⎝
⎛++++
∑=
dMeNa¼Rsay bgHajfa 9
)1nn)(1nn(19999
1n44 =⎟
⎠
⎞⎜⎝
⎛++++
∑=
eKman n1n)n1n)(n1n( 4444 −+=−+++
n1n1)n1n)(n1n( 4444
++=−+++
eKTaj 4444 n1n
)1nn)(1nn(1
−+=++++
9110000
)n1n()1nn)(1nn(
1
4
9999
1n
449999
1n44
=−=
−+=⎟⎠
⎞⎜⎝
⎛++++
∑∑==
dUcen¼ 9)1nn)(1nn(
19999
1n44 =⎟
⎠
⎞⎜⎝
⎛++++
∑=
.
KNitviTüaCMuvijBiPBelak
- 47 - PaK1
lMhat´TI21 eK[GnuKmn_ kMnt´RKb´ f }0;1{IRx −−∈ eday ½
1x)x1(f)x(f)1x2(x +=++ .
cUrKNna )2009(f....)3(f)2(f)1(fS ++++=
dMeNa¼Rsay KNna )2009(f....)3(f)2(f)1(fS ++++= eKman )1(1x)
x1(f)x(f)1x2(x +=++
CMnYs eday xx1 kñúgsmIkar ( eKán ½ )1
)2(2xxx)x(f
2xx)
x1(f
1x1)x(f)
x1(f)1
x2(
x1
22
++
=+
+
+=++
dksmIkar nig Gg:nwgGg:eKán ½ )1( )2(
2x2x2)x(f
2x)1x(x2
2xxx1x)x(f
2xx)1x2(x
2
22
++
=++
++
−+=⎥⎦
⎤⎢⎣
⎡+
−+
eKTaján 1x
1x1
)1x(x1)x(f
+−=
+=
KNitviTüaCMuvijBiPBelak
- 48 - PaK1
eKán [ ] ∑∑==
=−=⎟⎠⎞
⎜⎝⎛
+−==
2009
1k
2009
1k 20092008
200911
1k1
k1)k(fS
dUcen¼ 20092008)2009(f....)3(f)2(f)1(fS =++++= .
lMhat´TI22 cUrkMntRKb´KU éncMnYnKt´viC¢manebIeKdwgfa ½ )n;m(
)nm(13nm 22 +=+ .
dMeNa¼Rsay kMnt´RKb´KU ½ )n;m(
eKman )1()nm(13nm 22 +=+ -krNITI1 nm =
eKán naM[ n26n2 2 = 13n = dUcen¼ . 13nm ==
-krNITII2 nm ≠
eyIgBinitüeXIjfaebI ( CaKUcemøIyrbs ena¼eKán )n;m )1(
)m;n( k¾CaKUcemøIyrbs Edr . )1(
snμtfa m . tamvismPaB Cauchy Schwarz n<
KNitviTüaCMuvijBiPBelak
eyIgman )nm(2)nm( 222 +<+
26nm)nm(26)nm( 2
<++<+
eday ena¼eKán nm < 26nmm2 <+< ¦ 13m <
eday ena¼eKTaj *INm∈ 12m1 ≤≤ . müa¨geTotsmIkar Gacsresr ½ )1(
)2(0m13mn13n 22 =−+− DIsRKImINg´ )m13m(4169 2 −−=Δ smIkar mancemøIykñúg kalNa )2( *IN Δ CakaerRákd éncMnYnKtviC©maness . eKyk INk)1k2()m13m(4169 22 ∈∀+=−− eKán )1k(k41)1k2()m13m(4168 22 +=−+=−− eKTaj )m13(m42)1k(k −+=+ eday 12m1 ≤≤ ena¼témøEdlGacrbs´plKuN )1k(k + KW ½
}84,82,78,72,64,54{)1k(k =+ . kñúgtémøTaMgRáMmYyen¼témøEdlCaplKuNcMnYnKt´tKñamanEt témø mYyKt´EdlRtUvnwg 9872 ×= }10;3{m = .
- 49 - PaK1
KNitviTüaCMuvijBiPBelak
-cMeBa¼ eKán 3m = 0)2n)(15n(30n13n2 =+−=−− naM[ . 15n =
-cMeBa¼ eKán 10m = 0)2n)(15n(30n13n2 =+−=−− naM[ . 15n =
srubmkeKTTYlánKUcemøIy®áMKUKW ½ })10;15(;)3;15(;)13;13(;)15;10();15;3({)n;m( = .
- 50 - PaK1
KNitviTüaCMuvijBiPBelak
- 51 - PaK1
lMhat´TI23 eK[RtIekaN mYyEkgRtg . eKyk ABC A ]AC[E∈
nig F Edl ]AB[∈ ABCAEF ∠=∠ nig ACBAFE ∠=∠ . eKtag nig F erogKñaCaeCIgcMeNalEkgén E nig F 'E '
elIRCug [ . ]BC
cUrRsayfa E BC'FFEFE' ≤++ ?
dMeNa¼Rsay Rsayfa E BC'FFEFE' ≤++ tag BC cAB,bAC,a === . RtIekaNEkg nig manmMu ABC AFE ABCAEF ∠=∠ vaCaRtIekaNdUcKña .
A
B C
F
E
'F 'E
KNitviTüaCMuvijBiPBelak
eKánpleFobdMnUc kBCEF
ACAF
ABAE
=== Edl 0k > tag[pleFobdMnUc . eKTaj akEF,bkAF,ckAE === . RtIekaNEkg nig manmMu ABC C'EE ACB∠ rYmvaCaRtIekaN dUcKña . eKán
ECBC
'EEAB
= naM[ BC
EC.AB'EE = eday ckbAEACEC −=−= eKán
a)ckb(c'EE −
= . dUcKñaEdreKTaj
a)bkc(b'FF −
= eKán
a)bkc(bak
a)ckb(c'FFEFE'E −
++−
=++
abc2
a)cba(kbc2
akbbckakcbc
222
222
=−−+
=
−++−=
BIeRBa¼kñúgRtIekaNEkg eKman ABC 222 cba += (BItaK&r) tamvismPaB eKman GMAM − bc2cba 222 ≥+= eKTaj BCa
abc2
=≤ .dUcen¼ BC'FFEFE'E ≤++ .
- 52 - PaK1
KNitviTüaCMuvijBiPBelak
- 53 - PaK1
lMhat´TI24 eK[ c,b,a CabIcMnYnBitviC¢man . cUrRsaybBa¢akfa ½
2)ba(4c
ba)ac(4b
ac)cb(4a
cb3 333 333 33
≤++
++
+++
+++
+
dMeNa¼Rsay Rsayfa ½ Rsayfa ½
2)ba(4c
ba)ac(4b
ac)cb(4a 3 +
cb3 333 3333
≤+
++
+++
++
+
)cb( 3
+
eKman b(bc3cb 33 )c+−+= PaB AM − an +
tamvism M eKmG bc2cb ≥+ eKTaj
2
2cbbc ⎟⎠⎞
⎜⎝⎛ +
≤ naM[ 3)cb(43)cb(bc3 +−≥+−
eKán 33333 )cb(41)cb(b
43)cb(c +=+−+≥+
aj eKT 3 33 )cb(4cb +≤+ ¦ 3 33 )cb(4acba + ++≤+ naM[ ( )1
cbacb
)cb(4a 3
cb33 ++
+≤
++
+
KNitviTüaCMuvijBiPBelak
- 54 - PaK1
dUcKñaEdr ( )2cba
ac)ac(4b
ac3 33 ++
+≤
+++
nig ( )3c)ba(c
ba3 33 ba
ba4 ++
+≤
+
edaybUkTMnak´TMng eKán ½ +
+ .
)3(,)2(,)1(
2)ba(4c
ba)ac(4b
ac)cb(4a 3
cb3 333 3333
≤++
++
+++
++ .
++
lMhat´TI25 eK[sIVútcMnYn
−Bit
=⎩⎨⎧
≥∀+
==
−+ 1n,2a2aaa
1nn1n
10
2 ++
a1
cUrRsayfa a 1nn1naa ×= .
dMeNa¼Rsay a aa 2 +
Rsayf 1nn a +1n×=
a2a + −=
aa −−= RK ≥ −=
2aa2(c nn1nn1n
eKman a 1−
tag c b´ n
nn1n
1nnn
eKán c
1
n1n1n ++ aa
a) 2c +=−+−= −+
KNitviTüaCMuvijBiPBelak
- 55 - PaK1
TMnak´TMngen¼bBa¢ak´fa CasIVútnBVnþmanplsgrYm )c( n 2d = 01nigtY 0c1 11aa =−=−= .
eKán 2n2)1n(2d)1n(cc 1n −=−=−+= 1nneKTaj a 2n2a −=− −
)
( ) (
)1n(n=1a
)12aa
1k2aa
n
0n
n
1k
n
1k1kk
−−
=−
−=− ∑∑==
−
2n(n −
eKTaj 1nna 2n +−= nig 1)1n()1n(a 2
1n ++−+=+ 2 ++= nig ¦ a 1n+ 1n21nn 1nna 24 ++=
+
nn(a −+eday )(1a 222221nn n)1n()1nn =+++−=+
¦ 1nnaa 241nn ++=+ .
dUcen¼
1nn1naaa 2 ++
×= .
KNitviTüaCMuvijBiPBelak
- 56 - PaK1
lMhat´TI26 kñúgRtIekaN ABC mYycUrRsaybBa¢akfa ½
rR4111
≥ 2Csin
22Edl r nig
BsinA ++sin
R CakaMrgVg´carwkkñúg nig carwkeRkARtIekaN .
dMeNa¼
Rsay fa Rsay ( )1
r2Csin
2Bsin
2Asin
R4111++
tag ,bAC,aBCtag A,bAC,aBC
≥
cB cBA ===
wsþIbTkUsIunUskñúgRtIekaN eKman ½ 222 − eday
tamRT ABC
Acos.bc2cba +=2Asin21Acos 2−=
ena¼ )2
sin21(bc2 − Acba 2222 −+=
eKTaj bc4
)cba)(cba()cb( 22 −−+−−bc4
a2Asin2 += =
tag 2
cbap =++ ( knø¼brimaRténRtIekaN )
eKán )cp(2cba −=−+ nig )bp(2cba −=+−
KNitviTüaCMuvijBiPBelak
- 57 - PaK1
eKán bc
)cp)(bp(2Asin2 −−=
naM[ bc
)cp)(bp(2Asin −−= . dUcKñaEdreKTaj ½
ab)bp)(ap(
2Csin;
ac)cp)(ap(
2Bsin =
−−=
−−
eKán abc
)cp)(bp)(ap(2Cs sin
2Bsin
2Ain −−−
= eKman
R4abcpr =)cp)(bp)(ap(pS =−−−=
eKTaj S.R4ab nig c = S.rpS)cp)(bp)(ap( −
2
==−−
eKán R4r
S.R4S.r
2Csin
2Bsin
2Asin == .
vismPaB smmUleTAnwg ½ ( )1
( )22
2Csin
2Bsin
2Asin
2Bsin
2Asin
2Csin
2Asin
2Csin
2Bsin
2Csin
2Bsin
2Asin4
14
2Csin
1
2Bsin
1
2Asin
1
≥++
≥++
eday 2Asin
aap
bca)cp)(bp()si 2
−=
−−ap(2Csin
2Bn
2−=
KNitviTüaCMuvijBiPBelak
- 58 - PaK1
eKTaj a
p −=
a
2Asin
2Csin
2Bsin
. dUcKñaEdreKTaján ½
bbp2
Asin2Csin −
2Bsin
= nig c
cp
2Csin
2Bs
2Asin in −
=
smmUleTAvismPaB ( )2 nwg ½ 2
ccp
bbp
aap
≥−
+−
+−
aB eKán ½ tamvismP GMAM −
a)ap(2a)ap(p −= −≥+ naM[ p
)ap(2a
ap −≥
−
dUcKñaEdr p
)bp(2b
bp −≥
− nig p
)cp(2c
cp −≥
− eKán
Bit2c
cpb
bpa
ap
)cp()bp()ap(2c
cpb
bpap
≥−
+−
+−
−+−+−≥
−+
−+
−
pa
dUcen¼r
4CsinBsin≥++
R
2
1
2
1
2Asin
. 1
KNitviTüaCMuvijBiPBelak
- 59 - PaK1
lMhat´TI27 eK[ CabIcMnYnBitviC¢manEdl abc4c,b,a 1cba +++= UrbgHajfa ½ rbgHajfa ½ c
)cabcab(2c
bab
accb 222 ++
+a
222
++≥+
+
dMeNa¼Rsay Rsayfa ½
)ab(2 + cabcc
bab
aa
22
+≥+
++
aB eKman ½
ccb 2222 ++
tamvismP GMAM −
4 abcbaabc4 ++= naM[ 41c ≥+ eKTaj
1abc ≥
)1(abc31abc4cba ≥−=++ ( eRBa¼ ) 1abc ≥
tamvismPaB GMAM − eKán ½ ( )2
cab2ca2cccb 222 ++
bab2
cba
ba
a
222
++≥+
++ mPaB Cauchy - Schwarz eKman ½ tamvis
[ ]2222 )ca()bc()ab(3)cabcab( ++≤+ +
eday ])ab()ca()bc([abcca
=++2ab2
bca2 222 ++ bc2
KNitviTüaCMuvijBiPBelak
- 60 - PaK1
eKTaj ( )3)cabcab(abc32
cab2
bca2
abc2 2++≥++
tam nig eKTaján½ )2( )3(
( )4)cabcab(abc32
cba
bac
acb 22
++ 2
2222
++≥+
++
22 +
eyIgman⎪⎩
⎪⎨
⎧
≥+
≥+
≥+
bca2)ab()ca(abc2)ca()bc(
cab2)bc()ab(
222
222
222
eKán )ab([2 2 )cba(abc2])ca()bc( +≥++ )cba(abc)ca()bc()ab( 222 ++≥++
aMgBIrnwg )()(ab(2EfmGg:T )caab(2)bc )ca)(bc(2++ a(abc3)cabc 2eKán bab( )c++≥++
≥++ ( tam eKman a abc3cb ( )1 )
naM[
eKTaj 2222 cba9)cabcab( ≥++
¦ abc3cabcab ≥++ ( )5 )cabcab(2)cabcab(
abc32 2 ++≥++
eKTaján ½ tam )4( nig )5(
)cabcab(2c
baaa
cb 22222
≥+
+++
bc2
+++ .
KNitviTüaCMuvijBiPBelak
- 61 - PaK1
lMhat´TI28 eK[sIVút a{ ½ }n kMnteday
= nig 1a0 4a....a.aa n101n +=+ cMeBa¼RKb´ . 1n ≥
cUrbgHajfa 2aa 1nn =− + cMeBa¼RKb´ . 1n ≥
dMeNa¼Rsay bgHajfa 2aa 1nn =− +
g;teXIjfaRKb´ eyIgse INk∈ eKman . eKman
n 0ak > . eKman
0ak >
4a....a.aa n101n 4a....a.aa n101n +=+ 1n102neKán a 4a.....a.a += ++
2n102n
n0n102n
)2a.....aa(
4)a....a.4)a.....aa(
4)4a......a)(a.....a.a(
+=
++= 1
a(a
aa
2n102n
n10
a
++=
+
+
+
eKTaj 2a... n..aaa 102n +=+ 2)4a(a 1n2n +−= ++ ¦ 2aa 2n1n =− ++
dUcen¼ 2aan − 1n =+ .
KNitviTüaCMuvijBiPBelak
- 62 - PaK1
lMhat´TI29 eK[ CabIcMnYnBitviC¢manEdl c,b,a 1cabcab =++ . cUrbgHajfa 16)
a1c()
c1b()
b1a( 222 ≥+++++
¼
dMeNa Rsay a bgHajf 16)
a1c()
c1b()
b1a( 222 ≥+++++
tag 222 )a1c()
c1b()
b1a(A = +++++
⎟⎠⎞
⎜⎝⎛ ++= a ++++++
ac
cb
ba2
c1
b1
a1cb 222
222
eyIgBinitü 22 acabcab
a1 ++= 2a
bcac
ab
++= 22
cab1
bbcab ++
= 2bac
bc
ba+= +
22 ccabcab
c1 ++= 2c
abca
cb
++= PaB eKman ½ tamvism GMAM −
3cab
bca
abc,
ab≥+ 2
ac
ca,2
bc
cb;2ba
222 ≥++≥+≥+ eKán 93222
c1
b1
a1
2 + 22 =+++≥+
KNitviTüaCMuvijBiPBelak
- 63 - PaK1
ehIy 3a
müa¨geT c2
c.cb.
ba3
ac
cb
ba
3 =≥++ ot b,ab2ba 22222 ≥ ca2ac;bc2 +≥+≥+
222naM[ a2 ca2bc2ab2c2b2 ++≥++ aM[ n 1cabcabcba 222 =++≥++
eKán 1A 16)3(29 =+ .
+≥
dUcen¼ 16)acb
1c()1b()1a( 222 ≥+++++ .
lMhat´TI30 eK[
432 2cos2222,
2cos222,
2c22 os π
=++π
=+π IcUrrkrUbmnþTUeTA nig RsaybBa¢ak´
a¼pg .
=
BI«TahrN_xagelrUbmnþen
dMeNa¼Rsay
eKman rkrUbmnþTUeTA ½
,2
cos222,2
cos22 32
π=+
π=
42cos2222 π
=++
KNitviTüaCMuvijBiPBelak
- 64 - PaK1
tamlMnaM«TahrN UeTAdUcxa ½_eyIgGacTajrkrUbmnþT geRkam
1n
)n(2
co222 s2.........2 +
π=+ +++ .
eyIgtagRsaybBaØak´rUbmnþen¼ ½
)n(
n 2......222A ++++= RK b´ n∈ *IN
eyIgman 21 2cos22A π
== Bit favaBitdl´tYTI W ½ eyIg«bma Kp
1p
)p(
p 2222A2
cos2...... +
π=++ Bit
ayfavaBitdl´tYTI
++=
eyIgnwgRs 1p + KW 2p1p 2cos2A ++
π= Bit
eyIgman p1p A2A +=+ aytamkar«bma ed 1pp 2
cos2A +
π=
eyIgán 2p2p
21p1pA
2cos2
2cos4
2cos22 ++++
π=
π=
π+= Bit
dUcen¼ 1ncos22.........222 +
π=++++
)n(2
.
KNitviTüaCMuvijBiPBelak
- 65 - PaK1
lMhat´TI31 edayeFIVvicartamkMeN acMnYn ½ IncUrRsaybBa¢akf
2 + ckdacnwg Ct´Fm
1n1n 9A ++= E anic©cMeBa¼RKb´ n6 11
cMnYnK μCati n .
dMeNa¼Rsay RsaybBa¢ak´facMnYn 1n6A 1n
n 92 ++ += Eckdacn wg11
-ebI 0n = eKán 1192A0 =+= Bit ( Eckdac´nwg11 ) -ebI eKán 091n = 1911292A 27
1 ×==+= Bit wg ).
favaBitdltYTI KW ( Eckdacn 11
eyIg«bma p 1p1p6p 92A ++ +=
eyIgnwgRsayfavaBitdl YTI KW Ec
p 9
Eckdac´nwg ´t11 1P +
2p7p61p 92A ++
+ += kdacnwg11 2peyIgman 1 2A 7p6 ++
+ += 11p1p66
1p+ )9.29(A p62p)92(2 +++
+ −++=
9.55A64
9A64
1p
p
A
)649(A+
p1p+
1pp1
++
−=
−+=
KNitviTüaCMuvijBiPBelak
- 66 - PaK1
edaytamkar«bma Eckdac wg ehIy .
án p 9.55A64A
pA ´n 11 1p9.55 +
Eckdac´nwg11
eKTaj 1pp1
++ −= Eckdacnwg it .
nA
11 BdUcen¼cMnYn n1n6 92 1++ += kdacnwg11 MeBa¼ Ec Canic©cb´cMnYnKt´Fm μCatin .
RK
lMhat´TI32 97831 nn
neK[ INn,617043E nn 0 ∈−+−=
ac ic©R
cUrbgHajfa nE Eckd ´nwg 187 Can Kb´ INn∈ .
dMeNa¼Rsay a Eckdac´nwg Canic b´ bgHajf ©RKnE 187 INn∈ ½
× Edl nig CacMnYnbfmrvagyfa Eckdac´nwg eyIgEtUvRsay
[eXIjfa Eckdac´ nig . ab 1nn −=
eday 187 = Kña . 1711 11 17
dUcen¼edIm,IRsa nE 187
nE 11 17
eKman ba)(ba(a 2n1nn N.)ba()b....+++−=− −−−
KNitviTüaCMuvijBiPBelak
- 67 - PaK1
Edl 1n aaN 1n2n b.......b −−− +++= án 610709743831E nnnn
n ∈eK , INn−+−=
( ) ( )
*INq,p;)q9p8(11q99p88q)610709(p)743831(
610709743831 nnnn
∈+=+=−+−=−+−=
709743831E nnnnn
eKTaján Eckdacnwg 11 . nE
müa¨geToteKman½ INn,610 ∈−+−=
*IN;;)213(1734221
)743709()610831()743709()610831( nnnn
∈βαβ−α=β−α=β−+α−=
−+−=
Taján Eckdacnwg . ´nwg Canic©RKb´
eK nE 17
dUcen¼ n EckdacE 187 INn∈ .
KNitviTüaCMuvijBiPBelak
- 68 - PaK1
lMhat´TI33 cUrRsaybBa¢ak´fa
1n21n
1.....3
112
1 +<+
++++
a¼
dMeN Rsay ybBa¢ak´fa ½ Rsa
1n21n
1.....3
12
11 < ++
+++ man
+
eyIg
1k21
1k1kk1kk1k 11
+=
+++>
++=−+
¦ k1k1k2
1−+<
+
ebI 1222
1:1k = −<
ebI 2332
1:2k = −<
ebI 3442
1:3k −<= ------------------------
ebI ---------
n1n1n2
1:nk −+<+
=
KNitviTüaCMuvijBiPBelak
- 69 - PaK1
eKán 1121
−n)1n
1.........3
12
1( +<+
+++
Taj
eK 21n21n
1........3
12
1−+<
++++
1n2
11n21n
1.......32
+
1
<
−+<+
+++
dUcen¼
11+
1n21n
1.....3
112
1 +<+
++++ . lMhat´TI34 eK[ n cMnYnBitviC¢man . cUrRsaybBa¢akfa ½
n321 a.........;;a;a;a
nn321 ............aa ++ 21n3 a....a.a.anaa ≥++
dMeNa¼Rsay RsaybBa¢ak´fa ½
nn321n321 a.a.ana..........aaa ≥++++
=
a...... -cMeBa¼ eKman 2n 2121 aa2aa ≥+
Ul smm 0)aa( 221 ≥− Bit
KNitviTüaCMuvijBiPBelak
- 70 - PaK1
-«bmafavaBitdltYTI KW ½ k
kk321k321 aaa ++ a......a.a.aka.......... ≥++ Bit
eyIgnwgRsayfavaBitdl´tYTI 1k + KW ½ 1k
1kk3211kk32 a.a......a.a.a)1k(aa..........aaa +++ +≥+++++ 1
tag *INk,0a,0x;
x)a......aaax(
)x(f k
1kk321 ∈≥>∀
+++++=
+ eyIgán ½
2
kk
xax(x)a(
)x('f1k
k2121 )a.....a....aax)(1k +++++−+++++
=
[ ]
( )2
k21k
k21
2k21
kk21
xa....aakx)a....aax(
x)a....aax(x)1k()a....aax(
−−−−++++=
++++−+++++=
ebI naM[ 0)x('f =k
a.....aax k21 +++=
cMeBa¼ k
a..... kaax 21 ++ eKán ½
+=
kk211kk21
k2
ka +1
1kk
k21
k21
ka.....aa
)1k()k
a......aa(f
a....a
)....k
a....aa(
)k
aa(f
⎟⎠⎞
⎜⎝⎛ +++
+=+++
++
+21 aaaa.... ++++++
=++
+
+
+
KNitviTüaCMuvijBiPBelak
- 71 - PaK1
taragGefrPaBén x
)a.....aax()x(f
kk21 ++++
= x o
k..aa 21 a. k+++ ∞+
)x('f −.
+.
)x(f
m
PaBxagelIcMeBa¼ tamtaragGefr 0x ≥∀ eyIgTaján ½
kk3211k
1kk21
ka.....aaa)1k(
x)a......aa
⎟
x()x(f
⎠⎞
⎜⎝⎛ ++++
+
++++
+
+
ma ≥
=
tamkar«b kk321k321 a......a.a.aka..........aaa ≥++++
smmUl k21
ka.....++ k21 a.........a.ak
aa≥⎟
⎠⎞
⎜⎝⎛ +
eKTaj k211k
1kk21 a......aa)1k(
x)a.......aax( +
+
+≥++++
naM[ (*)xa.......a.a)1k(a.....aax 1kk21k21
++≥++++ yk CYskñúg 0ax 1k >= + ( )*
KNitviTüaCMuvijBiPBelak
- 72 - PaK1
eKán 1k
1kk3211kk321 aaa +++ a.a......a.a.a)1k(aa.......... +++ +≥++ Bit
dUcen¼ nn321n321 .....aaa +++ a......a.a.ana..... ≥+ .
lMhat´TI35 eda¼RsayRbB&næsmIkar ½
⎩⎨
ytanxtan3
⎧
=+
=++
34ytanxtan28ytanytanxtan6xtan
3
4224
Edl 2
yx0 π<<< .
dMeNa¼Rsay eda¼RsayRbB&næsmIkar ½
⎩⎨
+ tanxtantanxt 33
⎧
=
=++
34yyan28ytanytanxtan6xtan 4224
smmUl⎩⎨⎧
=+
=++
)tanxtan4y
2(316ytanxtan4
)1(28ytanytanxtan6xtan33
4224
KNitviTüaCMuvijBiPBelak
- 73 - PaK1
bUksmIkar nig eKán ½ )1( )2(
424 4(3162)tax(tan ==+ )31()328yn +=++ eday
2yx0 π<<< ena¼eKán ytanxtan0 <<
eKTaj )3(31ytanxtan +=+ eFIVplsg 1( nig 2() eKán ½ )
424 )31()324(1628x(tan −−− 3)ytan −=== eKTaj )4(1tanxtan −=− eRB3y a¼ ytanxtan0 <<
ar nig án bUksmIk )4( eK tan2)3( 2x = eKTaj 1xtan = naM[
4x π= .
IVplsgsmIkar nig eKán eF )3( )4( 32ytan2 = Taj eK 3ytan = naM[
3y π= .
cen¼eKánKUcemøIy dU )3
y,4
x( π=
π= .
KNitviTüaCMuvijBiPBelak
- 74 - PaK1
lMhat´TI36 eK[ CaRCugrbs´RtIekaNmYynigtag
nRtIekaN . a
c,b,a p
Caknø¼brimaRtécUrbgHajf ⎟
⎞
⎠⎜⎛ ++≥
−+
−+
− c1
b1
a12
cp1
bp1
ap1
dMeNa¼⎝
Rsay ajfa bgH ⎟
⎠⎞
⎜⎝⎛ ++≥
−+
−+
− c1
b1
a12
cp1
bp1
ap1 MnYnBit > eKman ½
2
cMeBa¼RKb´c > 0B;0A
0A)BA( 22 ≥BB.A2 +−2 ≥++
n
=− ¦ A2 B.A4BB.A2
B.A4)BA( 2 ≥+
eKTajáBA
4B.ABA
+≥
+ ¦ )1(BA
4B11
A +≥+
yk B,apA bp −=−= án eK cbap2BA =−−=+ CYskñúg )1(
eKán )2(c4
bp11
≥−
+
dUcKñaEdreKTaj½ ap −
.
KNitviTüaCMuvijBiPBelak
- 75 - PaK1
)3(a4
cp1
bp1
≥−
+−
nig )4(b4
ap1
cp1
≥−
+−
aB Gg:nwgGg:eKán ½ B Gg:nwgGg:eKán ½ bUkvismP )4(,)3(,)2( )4(,)3(,)2(
c4
b4
a4
cp2
bp2
ap2
++≥−
+−
+−
dUcen¼ ⎟⎠⎞
⎜⎝⎛ ++≥
−+
−+
−p1
c1
b1
a12
cp1
bp1
a .
lMhat´TI37 k¿cUrRsaybBa¢ak´fa ½
3n41n4
1n41n4
1n43n4
+−
<+−
<+− cMeBa¼RKb´
¿TajbgHajfa ½ *INn∈
x
3951n2 ×+
n43
)1n4(.......13)1n4(.......1173
+<
+×××−×××× . 1
<
dMeNa¼Rsay k¿RsaybBa¢akfa
3n41n4
1n41n4
1n43n4
+−
<+−
<+−
tamvismPaB eKGacsresr ½ GMAM −
KNitviTüaCMuvijBiPBelak
)3n4)(1n4(1
1n41
)3n4)(1n4(1n4
)3n4)(1n4(22n8
)3n4)(1n4(2)3n4()1n4(
+−<
+
+−>+
+−>+
+−>++−
KuNGg:TaMgBIrnwg )1n4( − eyIgán ½ )1(
3n41n4
1n41n4
+−
<+−
müa¨geTot )1n4)(3n4(2)1n4()3n4( +−>++−
)1n4)(3n4(1n4
)1n4)(3n4(22n8
+−>−
+−>−
EckTaMgBIrnwg eyIgán ½ )1n4( +
)2(1n43n4
1n41n4
+−
>+−
tamTMnak´TMng nig eKTaján ½ )1( )2(
3n41n4
1n41n4
1n43n4
+−
<+−
<+− .
x¿TajbgHajfa½
3n43
)1n4(.......1395)1n4(.......1173
1n21
+<
+××××−××××
<+
- 76 - PaK1
KNitviTüaCMuvijBiPBelak
tamsRmayxagelIeyIgman 3n41n4
1n41n4
1n43n4
+−
<+−
<+−
cMeBa¼ eyIgán ½ n;.....;3;2;1n =
3n41n4
1n41n4
1n43n4
1511
1311
139
117
97
95
73
53
51
+−
<+−
<+−
−−−−−−−−−−−−−−−−−−
<<
<<
<<
KuNvismPaBTaMgen¼Gg:nwgGg:eKán ½
3n43
1n41n4.........
1311.
97.
53
1n41
+<
+−
<+
edayeKman 1n2
14n4
11n4
1+
=+
>+
. dUcen¼
3n43
)1n4(.......1395)1n4(.......1173
1n21
+<
+××××−××××
<+
.
- 77 - PaK1
KNitviTüaCMuvijBiPBelak
lMhat´TI38 eK[ a CabIcMnYnBitviC¢man . c;b;
cUrbgHajfa 23
bac
acb
cba
≥+
++
++
dMeNa¼Rsay bgHajfa
23
bac
acb
cba
≥+
++
++
tag ⎪⎩
⎪⎨
⎧
=+=+=+
pbanacmcb
eKán ( ) ( ) ( ) pnmbaaccb ++=+++++ naM[
2pnmcba ++
=++
eKTaj
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
−+=
+−=
−+=
2pnmc
2pnmb
2mpna
eKán
- 78 - PaK1
KNitviTüaCMuvijBiPBelak
- 79 - PaK1
p2pnm
n2pnm
m2mpn
bac
acb
cba −+
++−
+−+
=+
++
++
⎥⎦
⎤⎢⎣
⎡−⎟
⎠
⎞⎜⎝
⎛++⎟
⎠
⎞⎜⎝
⎛+⎟
⎞⎜⎝⎛ ++
++
+ nm
mn
acb
cba
+⎠
=+
3np
pn
pm
mp
21
bac
amvismPaB eKán ½ t GMAM −
2np
pn;2
pm
mp;2
nm
mn+ ≥+≥+≥
eKán ( )23322
21
bac
acb
ca
=−+≥ 2b
++
++
++
dUcen¼ 23
bac
acb
cba
≥+
++
+ . +
lMhat´TI39 k¿KNna
n)1n(3.21
2.11
−++ Edl 2n > 1....
4.31
++ x¿edayeRbIvismPaB GMAM − én n( )1− cMnYnxageRkam ½
n)1n(1;
4.33.22.11;1;1
− cUrbgHajfa 2n )!n(n < .
dMeNa¼Rsay k¿KNna ∑
=⎥⎦
⎤⎢⎣
⎡−
=−
++++n
2k k)1k(1
n)1n(1....
4.31
3.21
2.11
KNitviTüaCMuvijBiPBelak
- 80 - PaK1
eKman k1
1k1
k)1k()1k(k
k)1k(1
−−
=−
−−=
−
eKán n11
k1
1k1
k)1k(1 n
2k
n
k∑=2
−=⎟⎠⎞
⎜⎝⎛ −
−=⎥
⎦
⎤⎢⎣
⎡−
∑=
dUcen¼ n11
n)1n(1...
2.11+ .
.31
3.21
−=−
+++4
. n(n <
tamvismPaB ½ x¿ bgHajfa 2)!n
GMAM −
nn321n321 a......a.a.ana.....aaa ++ ≥++
eKán ½ 0a; k ≥∀
)1n(
n)1n(1......
3.21.
2.11)1n(
n)1n(1....
3.21
2.11
−
−−>
−+++
2n2
1n
)1n(2
)1n(
)1n(
)!n(n1
>⎟⎠⎞
⎜⎝⎛ n
)!n(n
)!n(n
n1
!)1n(!n1)1n(
n1n
!)1n(!n1)1n(
n11
<⇒
>
−−>
−
−−>−
−
−
−
−
dUcen¼ 2n )!n(n < .
KNitviTüaCMuvijBiPBelak
- 81 - PaK1
lMhat´TI40 eK[ ;b;a c CabIcMnYnBitxusBIsUnü .
UrbgHajfa cac
cb
ba
ac
cb
ba 222
++≥⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝
⎛
dMeNa¼Rsay
bgHajfa ac
cb
ba
ac
ba
+≥⎟⎠⎞
⎜⎝⎛+⎜
⎝⎛
cb 222
+⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
tamvismPaB eKman ½ GMAM −
3b
⎜⎝
+⎟⎠
⎜⎝
ac.
cb.
ba3
ac
cba
32
2
2
2
2
2222
=≥⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞⎛⎞⎛
eKTaj )1(3ac
cb
ba 222
≥⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛
müa¨geToteKman
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
≥+⎟⎠⎞
⎜⎝⎛
≥+⎟⎠⎞
⎜⎝⎛
≥+⎟⎠⎞
⎜⎝⎛
ac21
ac
cb21
cb
ba21
ba
2
2
2
eKTaján )2(ac
cb
ba23
ac
cb
⎜⎝⎛ +≥+⎟
⎠⎞
⎜⎝⎛+⎜
⎝⎛+⎜
⎝⎛
ba 222
⎟⎠⎞+⎟
⎠⎞
⎟⎠⎞
KNitviTüaCMuvijBiPBelak
- 82 - PaK1
bUkvismPaB nig Gg: nig Gg: eKán ½ )1( )2(
⎟⎠⎞
⎜⎝⎛ +++≥⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛+
ac
cb
ba23
ac
cb
ba23
222
⎟⎠⎣ ⎝ b⎞
⎜⎝⎛ ++≥⎥
⎦
⎤⎢⎡
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎛
ac
cb
ba2
ac
cba2
222
Ucen¼ dac
cb
ba
ac
cb
b⎝a 222
++≥⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎛ .
lMhat´TI41 eK[ ;y;x z CabIcMnYnBitviC¢manminsUnü . UrbgHajfa c 222222 zyx
9z1
y1
x
1++
≥++
dMeNa¼Rsay bgHajfa 222222 yxzyx +
≥++z
9111+
vismPaBen¼GacsresreTACa ½
9z
zyxy
zyxx
zyx2
222
2
222
2
222
≥++
+++
+++
6zy
yz
zx
xz
yx
xy
9zy
zx1
yz
yx1
xz
xy1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
≥+++++
≥++++++++
KNitviTüaCMuvijBiPBelak
eday 2yx.
xy2
yx
xy
2
2
2
2
2
2
2
2
=≥+
2zx.
xz2
zx
xz
2
2
2
2
2
2
2
2
=≥+ nig 2yz.
zy2
yz
zy
2
2
2
2
2
2
2
2
=≥+
eKán 6222zy
yz
zx
xz
yx
xy
2
2
2
2
2
2
2
2
2
2
2
2
=++≥+++++ Bit
dUcen¼ 222222 zyx9
z1
y1
x1
++≥++ .
lMhat´TI42 eKman xlog1
1
aay −= nig ylog11
aaz −=
cUrRsayfa zlog11
aax −= .
dMeNa¼Rsay Rsayfa zlog1
1
aax −=
eKman xlog11
aay −= naM[ xlog1
1yloga
a −=
eKTaj )1(ylog
11xloga
a −=
- 83 - PaK1
KNitviTüaCMuvijBiPBelak
dUcKñaEdr ylog11
aaz −= naM[ ylog1
1zloga
a −=
eKTaj )2(zlog
11yloga
a −=
ykTMnak´TMng eTACYskñúg eKán ½ )2( )1(
zlog11
11xlog
a
a
−−=
1)z(logzlog
1xloga
aa −
−=
zlog11xlog
1)z(logzlog1zlog
xlog
aa
a
aaa
−=
−−−
=
eKTaján zlog11
aax −= .
- 84 - PaK1
KNitviTüaCMuvijBiPBelak
- 85 - PaK1
lMhat´TI43
eKBinitüsIVúténcMnYnBit ( kMnt´eday )un⎪
⎪⎨
⎩
⎧
−−
=
=
+ 1u24u5
u
4u
n
n1n
1
cUrKNna u CaGnuKmn_én . n n
dMeNa¼Rsay KNna u CaGnuKmn_én ½ n n
smIkarsMKal´énsIVút 1u24u5
un
n1n −
−=+ KW
1r24r5r
−−
=
¦ man¦s 04r6r2 2 =+− 2acr;1r 21 ===
tagsIVút 2u1uv
n
nn −
−=
eKán nn
n
n
n
n
n
1n
1n1n v3)
2u1u
(32
1u24u5
11u24u5
2u1u
v =−−
=−
−−
−−−
=−−
=+
++
TMnak´TMngen¼bBa¢ak´fa CasIVútFrNImaRtmanersug )v( n
nigtY 3q =23
2u1u
v1
11 =
−−
= .
KNitviTüaCMuvijBiPBelak
tamrUbmnþ 233
23qvv
n1n1n
1n =×=×= −−
eday 2u1uv
n
nn −
−= eKTaj
123
131v1v2u n
n
n
nn
−
−=
−−
=
dUcen¼ 23
)13(2v n
n
n −−
= . lMhat´TI44
eKBinitüsIVúténcMnYnBit kMnt´eday )u( n⎪⎩
⎪⎨
⎧
−−
=
=
+ 1u4u3
u
3u
n
n1n
1
cUrKNna CaGnuKmn_én . nu n
dMeNa¼Rsay KNna CaGnuKmn_én ½ nu n
smIkarsMKal´énsIVút 1u4u3
un
n1n −
−=+ KW
1r4r3r
−−
=
¦ 0)2r(4r4r 22 =−=+− eKTaj¦s 2rr 21 == . tagsIVút
2u1v
nn −=
- 86 - PaK1
KNitviTüaCMuvijBiPBelak
- 87 - PaK1
eKán 2u
1v1n
1n −=
++ eday
1u4u3
un
n1n −
−=+
eKán 2u1u
2u24u31u
vn
n
nn
n1n −
−=
+−−−
=+
eKman 12u
12u1u
vvnn
nn1n =
−−
−−
=−+ efr
naM[ CasIVútnBVnþmanplsgrYm )v( n 1d = nigtY 1
231
2u1v
11 =
−=
−= .
tamrUbmnþ n1n1d)1n(vv 1n =−+=−+= eday
2u1v
nn −= eKTaj
n1n2
n12
v12u
nn
+=+=+=
dUcen¼ n
1n2un+
= .
KNitviTüaCMuvijBiPBelak
- 88 - PaK1
lMhat´TI45 eKBinitüsIVúténcMnYnBit kMnt´eday ½ )u( n
⎩⎨⎧ = 4u1
∈=+ *INn,uu n1n
cUrKNna CaGnuKmn_é .
MeNa¼
nu n n
d Rsay aGnuKmn_én KNna nu C n
eKman n1 uu = n+
eKán n1n ulnln21u =+
gen¼bB¢ak´fa (l n CasIVútFrNImaRtmanersug TMnak´TMn un )21q =
tamrUbmnþ ( ) 1n2
1
1n1
n 4ln4ln1ulnuln −=== − n1 2
q× −
dUcen¼ ( ) 1n2
1
n 4u −= .
KNitviTüaCMuvijBiPBelak
- 89 - PaK1
lMhat´TI46 eK[sIVúténcMnYnBit kMnt´eday ½ )u( n
5u0 = nig 5nu21u n ++ 1n =+ ( INn∈ )
cUrKN CaGnuKmn_én
MeNa¼
na nu énsIVút n . )u( n
d Rsay sIVút CaGnuKmn_én
u
KNna nu én )u( n n
eyIgtag bvnn an ++= Edl IRb,a ∈ eyIgán baanvu 1n1n +++= ++ eday 5nu
21u n1n+ ++=
eKán 5n)banvn ++v(21baan n1 ++=++++
(*))b21a5(n)1a
21(v
21v n1n ⎥⎦
⎤⎢⎣⎡ −−++−+=+
ebI ⎪⎪⎩
⎪⎪⎨
⎧ −+− 01a1
=−− 0b21a5
2 naM[ 3b;2a ==
TMnak´TMng eTACa ( )* n1n v21v =+ naM[ CasIVútFrNImaRt. )v( n
manersug 21
= . q
KNitviTüaCMuvijBiPBelak
- 90 - PaK1
tamrUbmnþ 0 qv × nn v= eday 6n2vbanvu nnn ++=++= eKán 0cMeBa¼ n = 6v00 u +=
u0naM[ 56v 0 16 −=−=−= eKán nnn 2
1211v −=×−= .
dUcen¼ 6n221u nn ++−= .
lMhat´TI47 edayeRbIGnumanrYmKNitviTüacUrKNnalImItxageRkam ½
2x2x22.....222
limLn 2x −−++++++
= →
( man n ra¨DIkal´ ) .
MeNa¼
d Rsay ageRkam ½ KNnalImItx
eyIgman 2x
2x22.....222limLn 2x −
−+++++→
+=
eyIgán
KNitviTüaCMuvijBiPBelak
- 91 - PaK1
41
2x21lim
)2x2)(2x(4)x2(lim
2x2x2limL
2x1 = → 2x2x=
++=
++−−+
=−
−+→→
212x2x2
2x2x2
41L
41
2x2x2lim
2x221limL
)2x22)(2x(2x2lim
2x2x22limL
==−−+
×+++
=
+++−
−+=
−−++
=
→→
→→
eyIgsnμtfavaBitdllMdab´TI KW k kk 41L =
´TeyIgnwgRsayfavaBitdl´lMdab I )1k( + KW 1k1k 41L ++ =
eyIgman 2x
2x2.....222 2limL
2x1k −−++
→+ knigPaKEbgnwg
++++=
KuNPaKy2x2..2222)x(M ++++= . +++
eKán )x(M).2x(
4x2....2222limL2x1k −
−++++++=
→+
Bit1kkkk1k
2x2x1k
41
41
41L
41L
)2(M1L
2x2x2....222lim
)x(M1limL
++
→→+
=×==×=
−−+++++
×=
dUcen¼ n2xn 41
2x2x22.....222
limL =−
−++++++=
→
KNitviTüaCMuvijBiPBelak
- 92 - PaK1
lMhat´TI48 eK[sIVúténcMnYnBit kMnt´elI eday ½ )U( n IN
⎪⎩
⎪⎨⎧ = 2U0
∈+=+ INn,U2U n1n
¿cUrKNna CaGnuKmn_én . KuN 1 U....
k nnU
x¿KNnapl 20n UUUP n××××= .
eNa¼
dM Rsay aGnuKmn_én ½
KNna nU C n
eyIgman4
cos220U π==
8
cos24
cos22U2U 01π
=π
+=+=
itdl´tYTI KW «bmafavaB p 2pp 2cos2U +
π=
I eyIgnwgRsayfavaBitdl´tYT )1p( + KW 3p1p+ 2cos2U +
π= Bit
eyIgman p1p U2U +=+ Et r«bmtamka a 2pp 2cos2U +
π=
eyIgán 3p3p
22p cos4U ++
π=
π 1p 2cos2
22cos22 ++
π=+= Bit
KNitviTüaCMuvijBiPBelak
- 93 - PaK1
dUcen¼ 2nn 2cos2U +
π= .
aplKuN 0n U....UUPx¿ KNn 1U n2 ××××= naM[ tamrUbmnþ 2a2sin = acosasin
ana2n
sisiacos2 =
2n2n
n 1k2sin +
0k
n
0k2k
n
0k2kkn
2sin
1
2sin
2sin
)
2sin
()2
cos2()U(P
++
= =+
=+
π=
π
π
=
π
π
=π
==∏ ∏∏
dUcen¼ 2n
n
2sin
1P+
π= .
Mhat´TI49 l
MnYnBit kMnt´elI eday ½ eK[sIVúténc )U( n IN
22U0 = nig INn,
211
U2
n1n ∈∀
−−+
U=
KNna nuKmn_én .
n CaGU n
KNitviTüaCMuvijBiPBelak
- 94 - PaK1
dMeNa¼Rsay KNna CaGnuKmn_én ½ n
U n
eyIgman4
sin20 ==2 π U
8
sin2
4sin11
2U11
U1 =
22
0 π=
π−−
=−−
favaBitdl´tYTI KW «bma p 2pp 2sinU +
π=
I eyIgnwgRsayfavaBitdl´tYT )1p( + KW 3p1pU + 2sin += Bit π
eyIgman 2
U11U 1p
−−=+
2p kar«bma 2+pp 2
sinU π=
eyIgán 2
2U2p
2+
sin111p+
π−−
=
3p
3+=
p2
2p
2sin
22
sin2
22
cos1+
+ π=
π
=
π−
Bit
dUcen¼ 2nn 2sinU +
π= .
KNitviTüaCMuvijBiPBelak
- 95 - PaK1
lMhat´TI50 eK[sIVúténcMnYnBit )u( n kMnt´eday½
⎧ = au0
⎩⎨
>∈−=+ 2a,IN,2uu 2n1n ∀n
rRsaybBa¢ak´fa cUnn 2
22
2
n 24aa
24aau ⎟
⎟⎠
⎞⎜⎜⎝
⎛ −−+⎟
⎟⎠
⎞⎜⎜⎝
⎛ −+=
dMeNa¼Rsay
RsaybBa¢ak´fa nn 2
22
2
n 24aa
24aau ⎟
⎟⎠
⎞⎜⎜⎝
⎛ −−+⎟
⎟⎠
⎞⎜⎜⎝
⎛ −+=
ebI 0n = eKán a2
4aa2
4aau22
0 =−−
+−+
= Bit
avaBit KW
«bm dltYTI kkk 2
22
2
k 24⎟⎟⎠
aa2
4aau⎞
⎜⎜⎝
⎛ −−+⎟
⎟⎠
⎞⎜⎜⎝
⎛ −+=
itdl´tYTeyIgnwgRsayfavaB I 1k + KW 1k1k 2
22
2
1k 22u +⎟
⎟⎠
⎜⎜⎝
=+
4aa4aa++
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−⎞⎛ −+
eyIgman 2uu 2k1k −=+
KNitviTüaCMuvijBiPBelak
- 96 - PaK1
Ettamkar«bma kk 2
22
2
k 24aa
24aau ⎟
⎟⎠
⎞⎜⎜⎝
⎛ −−+⎟
⎟⎠
⎞⎜⎜⎝
⎛ −+=
eyIgán 22
4aa2
4aau
22
22
2
1k
kk
−⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−+⎟
⎟⎠
⎞⎜⎜⎝
⎛ −+=+
24
4aa22
4aa2
4aau22
22
22
1k
1k1k
−+−
×+⎟⎟⎠
⎞⎜⎜⎝
⎛ −−+⎟
⎟⎠
⎞⎜⎜⎝
⎛ −+=
++
+
1k1k 2
22
2
1k 24aa
24aau
++
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−+⎟
⎟⎠
⎞⎜⎜⎝
⎛ −+=+ Bit
dUcen¼ nn 2
22
2
n 24aa
24aau ⎟
⎟⎠
⎞⎜⎜⎝
⎛ −−+⎟
⎟⎠
⎞⎜⎜⎝
⎛ −+= .
KNitviTüaCMuvijBiPBelak
- 97 - PaK1
lMhat´TI51 k¿cUrkMnt´elxénGBaØat éncMnYn d,c,b,a abcd ebIeKdwgfa ½
dcba9abcd =× x¿cMeBa¼témø ,b,a EdlánrkeXIjxagelIcUrbBa¢ak´fa d,c
cMnYn abcd nig dcba suTæEtCakaer®ákd .
dMeNa¼Rsay k¿kMntelxénGBaØat ½ d,c,b,a
eKman )1(dcba9abcd =× k´TMng eKTajánté EtmYyKtKW . tamTMna mø )1( a 1a =
cMeBa¼ 1a = eKán )2(1dcb9bcd1 =× tamTMna g )2( eKk´TMn Taján 9d = eRBa¼ =×
es d 819
manelxxagcug μI 1 . cMeBa¼ 9d = eKán )3(1cb999bc1 × =
k´TMn Taján tamTMna g )3( eK 0b = inG T ) ( eRBa¼ 9b× m acmanRtaTuke
KNitviTüaCMuvijBiPBelak
- 98 - PaK1
cMeBa¼ b eKán 0= )3(01c999c10 =× k´TMn Taján tamTMna g )3( eK 8c = eRBa¼ 72989c =×=×
Efm 8 [elxxagcugesμI 0 . cMeBa¼ 8c = eKán 91089 9801=× .
adUcen¼ 0b,1 9d,8c, === .= ´facMnYn x¿bBa¢ak abcd nig dcba suT aer®ákd ½
0
æEtCakcMeBa¼ 98c,b,1a d, ==== eKán ½
2331089abcd = = nig 2991980dcba == suTæEtCakaer®ákd
.
KNitviTüaCMuvijBiPBelak
- 99 - PaK1
lMhat´TI52 sn μtfa HUNSEN CacMnYnmYymanelxRáMmYyxÞg´Edl
eBlEdleKKuNcMnYn YlTæplCacMnYnmanelx N,E,S,N,U,H Caelx . en¼nwg 3 eKTT
RáMmYyxÞg´KW STRONG Edl G,N,O,R,T,S Caelx cUrrkcMnYn HU ig NSEN n STRONG
= 8NcMeBa¼ N = nigcMeBa¼ S,3S,6 9== . usKñatageday ). (Gkßrx elxxusKña
STRONG3
HUNSEN×
dMeNa¼Rsay rkcMnYn HUNSEN nig STRONG
KNitviTüaCMuvijBiPBelak
- 100 - PaK1
eKman ½
STRONG
3HU×
NSEN
k¿cMeBa¼ 3S,6N ==
G6TRO33
6E63HU×
tamtaragRbmaNviFIKuNen¼bBa¢ak´fa 1H =
cMeBa¼ 1H = taragviFIKuNeTACa ½
G6TRO3
36E36U1
×
tamtaragviFIKuNen¼eKTaján 8G = eRBa¼
eTACa ½
36×
manelxagcugesμI 8 . cMeBa¼ 8G = taragviFIKuN
68TRO3
36E63U1
×
tamtaragviFIKuNen¼eKTaján 5E = eRBa¼ Efm 3E× 1
KNitviTüaCMuvijBiPBelak
- 101 - PaK1
manelxcugeRkayesμInwg 6 . cMeBa¼ 5E = taragviFIKuN TACa e ½
68TRO3
363U1
×56
tamtaragviFIKuNen¼eKTaján 0O = eRBa¼ Efm
½
33× 1
manelxcugeRkayesμInwg 0 . cMeBa¼ 0O = taragviFIKuNeTACa
068TR33
6U1 356×
tamtaragviFIKuNen¼eKTaján 9R = eRBa¼ Efm 36× 1
manelxcugeRkay esμInwg 9 . cMeBa¼ 9R = taragviFIKuNeTACa ½
9068T33
6U1 356×
KNitviTüaCMuvijBiPBelak
- 102 - PaK1
tamtaragen¼bBa¢ak´fa minGacFMCag eTehIyedayGkßr xusKñatagedayelxxusKñaena¼eKTaján
U 2
2U = EtmYyKt BIeRBa¼ ehIy minGacyktémø ¦ 1H,0O == U 0 1
dEdleTotáneT . cMeBa¼ 2U = taragviFIKuNeTACa ½
90T3 68
56
tamtaragviFIKuNen¼eKTaján
3× 1263
7T = BIeRBa¼fa ½
379068 3
126356×
edayeRbobeFobtaragviFIKuN ½
STRONG nwg 3
HUNSEN×
379068
3126356×
KNitviTüaCMuvijBiPBelak
- 103 - PaK1
eKTaján ½ 126356 nigHUNSEN = 379068STRONG =
CacMnYnEdlRtUvrk . ¿cMeBa¼ x 9S,8N ==
tamrebobdUcxagelIeKán ½ eyIgeda¼Rsay
9567843×
318928
edayeRbobeFobtaragviFIKuN ½
STRONG3
HUNSEN× nwg
9567843
318928×
eKTaján ½ 318928HUNSEN = nig 956784STRONG =
CacMnYnEdlRtUvr
k .
KNitviTüaCMuvijBiPBelak
- 104 - PaK1
lMhat´TI53 cUrbMeBjRbmaNviFIbUkxageRkam ½
YENOM
EROM
dMeNa¼
DNES+
(GkßrxusKñatagedayelxxusKña )
Rsay bMeBjRbmaN UkxageRkam ½ viFIb
YENOMEROM
+DES
N
cemøIyrbs´taragviFIbUken¼KW ½
256015801
7659+
.
( sisßeda¼RsayedayxøÜnÉg )
KNitviTüaCMuvijBiPBelak
- 105 - PaK1
lMhat´TI54 eK[ACacMnYnmYymanelxRáMxÞg´EdlelxxÞg´vaerobtamlMdab´
x;1x;x +x;2x;1 ++ ehIy CacMnYnmYymanelx ´
B
RáMmYyxÞg´EdlelxxÞg´vaerobtamlMdabx;1x;1x;2x;1x;x −++ .
Ha
dMeNa¼
+
cUrbg jfaebI A CakaerRákdena¼ B k¾Cakaer®ákdEdr .
Rsay
eyIgmanelxTaMgRáMxÞg´én A erobtamlMdab´ karbgHaj ½
x;1x2x;1x;x +++ naM[ 92x1x0 ≤+<; x +<< ¦ 7x0 ≤< .
a¼ -ebI 1x = en 211112321A == kd .
-ebI ena¼ A = mnCakaer®ákd . I e minEmnCakaer®ákd .
= minEmnCakaer®ákd .
-ebI 2x = ena¼ 23432A = minEmnCakaer®á3x = 34543 minE
-eb na¼ 4x = 45654A =
-ebI x ena¼ 5 56765A =
KNitviTüaCMuvijBiPBelak
- 106 - PaK1
-ebI ena¼ minEmnCakaer®ákd . 6x = 67876A =
-ebI ena¼ minEmnCakaer®ákd . 7x = 78987A =
srubmkeKán x øEdl[ A Cakaer®ákd 1= Catém . ´vaerobtaml
;1
eday elxxÞg Mdab´ ½ B
x;1;2xx;x x;1x −++
MeBa¼ eKán +
ena¼c 1x = 2351123201B == Cakaer®ákd . ¾Cakaer®ákdEdr . dUcen¼ ebI A CakaerRákdena¼ B k
lMhat´TI55 eK[cMnYn
)elxn2(
11.....11A 11= nig )elxn(
444.....444B =
cUrbgHajfa 1B ++ CakaerRákd . A
dMeNa¼Rsay bgHajfa A 1B ++ CakaerRákd ½ eyIgman
9110111A
n2 −=.....111
n2(
=)elx
KNitviTüaCMuvijBiPBelak
- 107 - PaK1
9
)110(4444.....444n
n(
−=B =
)e
lx
eyIgán 19
)19
1BAn
+−
+=++
10(4110 n2 −
2n
nn2
nn2
32101BA
9410.4101BA
99410.41101BA
⎟⎟⎠
⎞⎜⎜⎝
⎛ +=++
++=++
+−+−=++
dUcen¼ CakaerRákd . 1BA ++
lMhat´TI56 cMnYnmYymanelxbYnxÞg´EdlelxxÞg´vaerobtamlMdab´
b;b;a;a . IeKdwgfavaCakaerRákd . rkcMnYnena¼eb
dMeNa¼Rsay rkcMnYnEdlCakaerRákd ½ tag N CacMnYnEdlRtUvrk
KNitviTüaCMuvijBiPBelak
- 108 - PaK1
eyIgán bb10a100a1000aabbN +++==
++=
)ba100(11Nb11a1100N
+= [ ] )1()ba(a9911+=
eday 9a0 ≤< nig b0 9≤≤ ena¼ 18ba0 ≤+< Mng edIm,I[ GacCakaerRákdlu¼RtaEt huKuNén ehIy
tamTMnak´T )1( N
ba + CaB 11 18ba0 ≤+< ena¼eKRtUv[ Etm .
Ga11ba =+ YyKt
TMnak´TMng )1( csresr ( ) 1a9(1111a9911 2 )N +=+= ehtue k a n¼ N Cakaer®á dkalN 1a9 + Cakeday ≤
aerRákd . 9a ≤ naM[ 101 811a9 ≤+≤ eKTaj ½
)
= (K ñúg )
161a9 =+ (Kμan¦skñúg IN
1a9 =+ (K μan¦skñúg IN ) 25
1a9 + μan¦sk36 IN
(K μan¦skñúg ) 491a9 =+ IN
naM[ 7a641a9 =+ = e Iy h 4711b =−=
cen¼cMnYnEd KW 4
(K μan¦skñúg I ) . 811a9 =+ N
dU lRtUvkMnt´ena¼ 774 288= .
KNitviTüaCMuvijBiPBelak
- 109 - PaK1
lMhat´TI57 eK 0889333,1089 2man 33 22 = 111088893333,11==
. agelIcUrrkrUbmnþTUeTA nigRsaybBa¢ak´rUbmnþen¼pg
1111088889333332 =
BI«TahrN_x
dMeNa¼Rsay eKman 111088893333,110889333,108933 222 === .
IgGacbeg;ItrUbmnþTUeTAdUcxageRkam ½ ...
)1)1n()
2
−−
=
1111088889333332 =
tamlMnaMen¼ey9888.....8880111.....111333..333
n(n(
.
eyIgtag karRsaybBa¢ak´rUbmnþ ½
9888.........8880111..........111A)1n()1n( −−
=
KNitviTüaCMuvijBiPBelak
)n(
2
2n2nnn2
n1nn2
1n1n1n
)1n(
1n
)1n(
333............333
3110
9)110(
9110.210
9818010.81010
910)110(9810)110(
91
910.888.........88810111.........111
=
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
−=
+−=
+−+−=
+−+−=
++×=
+
−+−
−
+
−
dUcen¼eKánrUbmnþ ½ 9888.....8880111.....111333.....333
)1n()1n()n(
2
−−
= .
- 110 - PaK1
KNitviTüaCMuvijBiPBelak
lMhat´TI58 eKman 999800019999,998001999,980199 222 === . 9999800001999992 =
BI«TahrN_xagelIcUrrkrUbmnþTUeTAnigRsaybBa¢ak´rUbmnþen¼pg
dMeNa¼Rsay tambMrab´eKman ½
999800019999998001999
980199
2
2
2
=
=
=
9999800001999992 = tamlMnaMen¼eyIgGacbeg;ItrUbmnþTUeTAdUcxageRkam ½
1000.....0008999.....999999.....999)1n()1n()n(
2
−−
= .
karRsaybBa¢ak´rUbmnþ ½ eyIgtag 1000.........0008999..........999A
)1n()1n( −−
=
- 111 - PaK1
KNitviTüaCMuvijBiPBelak
)n(
22n
nn2
n1nn2
n1n1n
n1n
)1n(
999............999)110(110.210
110.81010110.810)110(
110.810999.........999
=−=
+−=
++−=
++−=
++×=
+
+−
+
−
dUcen¼eKánrUbmnþ½
1000.....0008999.....999999.....999)1n()1n()n(
2
−−
= .
lMhat´TI59 eKman 444355566666,443556666,435666 222 === . 4444355556666662 =
BI«TahrN_xagelIcUrrkrUbmnþTUeTAnigRsaybBa¢ak´rUbmnþen¼pg
dMeNa¼Rsay tambMrab´eKman
444355566666,443556666,435666 222 === 4444355556666662 = .
- 112 - PaK1
KNitviTüaCMuvijBiPBelak
- 113 - PaK1
tamlMnaMen¼eyIgGacbeg;ItrUbmnþTUeTAdUcxageRkam ½ 6555.....5553444.....444666.....666
)1n()1n()n(
2
−−
= .
karRsaybBa¢ak´rUbmnþ ½ eyIgtag ½
6555.........5553444..........444A)1n()1n( −−
=
)n(
2
2
)n(
2
)n(
2nnn2
nn2
nn1nn2
1nn1n1n
)1n(
n1n
)1n(
666............666)999.........999(32
)999.........999(94)110(
94
9410.810.4
94)52740(1010.4
9545010.510.2710.410.4
610)110(9510.310)110(
94
610555.......55510.310444........444
=⎥⎥⎦
⎤
⎢⎢⎣
⎡=
=−=+−
=
+−−−=
+−++−=
+−++−=
+×++×=
+
−+−
−
+
−
dUcen¼eKánrUbmnþ½ 6555.....5553444.....444666.....666
)1n()1n()n(
2
−−
=
KNitviTüaCMuvijBiPBelak
- 114 - PaK1
lMhat´TI60 eKman
111111445556,11114556,1156 22 2222 =−=− .
BI«TahrN_xagelIcUrrkrUbm nigRsaybBa¢ak´rUbmnþen¼pg
=−
1111111144455556 22 =−
nþTUeTA
dMeNa¼Rsay rkrUbmnþTUeTA nigRsaybBa¢ak´ ½
22
2
=−
=−
1111111144455556 =− tamlMnaMEdleK[eyIgGacsresrrUbmnþTUeTAdUcxageRkam ½
)n(
nigRsaybBa¢ak´ ½
11111144555611114556
11
22
2
=−
=−
1111111144455556 22 =− tamlMnaMEdleK[eyIgGacsresrrUbmnþTUeTAdUcxageRkam ½
)n2()n()n(
eKman eKman
11111144555611114556
11562
22 =− 562
22 =−
22
22 111.......111445.....444556....555 =− 22 111.......111445.....444556....555 =−)n2()n(
sRmayepÞógpÞat´rUbmnþen¼ ½
)n(
2445...tag )n(
2n ..444556....555T −=
KNitviTüaCMuvijBiPBelak
- 115 - PaK1
( )
( )9
999.........999
9110)
9110(110
1)110(941)110(
951)110(
941)110(
95
1)110(941)110(
95
)1444......444(1555.....555=
)n2(n2n
n
nnnn
2n
2n
22
=−
=−
+=
⎥⎦⎤
⎢⎣⎡ −−−+−⎥⎦
⎤⎢⎣⎡ +−++−=
⎟⎠⎞
⎜⎝⎛ +−−⎟
⎠⎞
⎜⎝⎛ +−=
+−+
Bit .
dUcen¼ )n2()n(
2
)n(
111.......111445.....5....555 =
)n2(n 111..........111T =
2 44456 − .
lMhat´TI61 eK[ 011nn aa..........aaA = − nig 011nn a2a.........aaB ×−= −
fa Eckdac´nwg t Eckdacnwg . Rsay lu¼RtaE
A 7 B 7
dMeNa¼Rsay ´nwg ena¼naM[man ebI B Eckdac 7 INq∈ Edl q7B =
eKán q7a2a........aa 011nn =×−− .. naM[ )1(a a2q7a..........a 011nn ×+=−
KNitviTüaCMuvijBiPBelak
- 116 - PaK1
eyIgman )2(a10a...........aaaa.........aaA 1nn= − 011nn01 +×= −
yk CUskñúg eKán q7
)1( )2(
00 a)a2(A 10+×+= )a3q10(7a21q07aa20q70A 0000 +=+=++=
naM[ Eckdacnwg . ¼RtaEt Eckdac´nwg .
A 7
dUcen¼ Eckdacnwg luA 7 B 7
lMhat´TI62 eK[cMnYn 0121nn aaa......aaA −=
a.....,,a, Caelx . yfacMnYn Eck alNa
Edl 0 a,a n21
cUrRsa dac´nwg 6 kA
0n21 a)a..aa(4y . ++
dMeNa¼
++= Eckdacnwg 6 .
Rsay eyIgman 0121nn aaa......aaA = −
n2
10 a10.....a10a10aA 2n++++=
KNitviTüaCMuvijBiPBelak
- 117 - PaK1
eyIgman )6(mod410 ≡
eyIgán )6(mod410
)6(mod410)6(mod410
n
3
2
≡
−−−−−−−−−−≡
≡
)6(mod)a.....aa(4aA n210 ++++≡ Eckdac´nwg lu¼RtaEt
n1
edIm,I[ A 6
(mod0)a....aa(4a 20 )6≡++++ cen¼cMnYn Eckdacnwg kalNa
2
dU A 6
n1 a)a...aa(4y 0++++= Eckdacnwg 6 . lMhat´TI63 cUrkMntelx a nig edIm,I[cMnYn b abba CaKUbéncMnYnKt´ .
dMeNa¼
Rsay nig kMnt´elx a b
tag ab10b100a1000abbaN +++== b110a1001
+=+=
NN
)b10a91(11
KNitviTüaCMuvijBiPBelak
- 118 - PaK1
edIm,I[ N CaKUbéncMnYnKt´lu¼RtaEt *INk,k121k1110a91 332b ∈==+
eday 9b0,9a0 ≤≤≤< ena¼ 909b10a910 ≤+< eKán 909k1210 3 ≤<
smmUl121627 +
121909k0 3 =≤< naM[ 1k =
eKTaján 121b10a91 =+ naM[ 10
a9112b 1−=
ena¼ eday 0b ≥ 0≥10
a91121− ¦
91301a +≤
91121
= naM[ 1a = ehIy 310
91121b =−
= . en¼ dUc nig 3b,1a == 3111331abba == .
KNitviTüaCMuvijBiPBelak
- 119 - PaK1
lMhat´TI64 eKtag r nig R erogKñaCakaMrgVg´carikkñúg nigcarikeRkA énRtIekaN ABCmYy . k¿cUrRsayfa
Rr1CcosBcosAs +co +=+
.
x¿cUrRsayfa r2R ≥
dMeNa¼Rsay k¿Rsayfa
Rr1CcosBcosAcos +=++
2BBcosAcos Ccos
2CBcos2
2Asin21Ccos 2 −++−=
++
2Csin
2Bsin
2Asin41
)2
CBcos2
CBcos(2Asin21
)2
CBcos2Asin(
2Asin21
2CBcos
2Asin2
2Asin21 2
+=
−−
+−=
−−−=
−+−=
edayeKmanbc
)cp)(bp(2Asin −−=
ab)bp)(ap(
2Bsin =
p(2Csin,
ac)cp)(a −−
=−−
KNitviTüaCMuvijBiPBelak
- 120 - PaK1
eKán abc
)cp)(bp)(ap(2Csin
2Bsin
2Asin −−−
= eKTaj½
abc)cp)(bp)(ap(41CcosBcosAcos −−−
+=+
+
Rr1
R.pr.p1
R.pS1
R.SpS1
RR4
abc.p
)cp)(bp)(ap(p1
2
+=+=+
+=
×
=
−−−+=
dUcen¼ Rr
1CcosBcosAcos +=++ . fa
x¿Rsay r2R ≥
tamvismPaB :GMAM − βα≥β+α .2 eKán )bp)(ap(2)bp()ap( ≥− −−−+
)bp)(ap(2c
)cp)(ap(2bap2
−−≥
−−≥−−
eKTaj )1(21
c)bp)(ap( −≤
−
dUcKñaEdr )2(21
a)cp)(bp(≤
−−
KNitviTüaCMuvijBiPBelak
- 121 - PaK1
nig )3(21
b)cp)(ap(≤
−− ´TMng Gg wgGg:eKán ½ KuNTMnak :n)3(,)2(,)1(
81
abc)cp)(b≤
−−
eKTaj
p)(ap( −
81
2Csin
2Bsin
2Asin ≤
eday 2C
sin2B
sin2A
sinco 41CcosBcosAs +=++ eKán
23)
81(41CcosBcosAcos =+≤++
Et Rr1CcosBcosAcos +=++
aj eKT 23
Rr1 ≤+ ¦ .
r2R ≥
KNitviTüaCMuvijBiPBelak
- 122 - PaK1
lMhat´TI65 k¿cUrRsayfa x2cotxcot
x2sin1
−= x¿cUrKNna
n2
n
2aS
2a
sin
1....
sin
1
2a
sin
1asin
1++++=
dMeNa¼Rsay k¿Rsayfa x2cotxcot
x2si
tag n1
−= x2cotxcot)x(f − =
x2sin1
xcosxsin21xcos2xcos2
xcosxsin21xcos2
xsinxcos
x2sinx2cos
xs
=
in
22
2
=+−
=
−−
−
¼
xcos=
dUcen x2cotxcotx2sin
1−= .
x¿KNna n22
asin
Sn
2a
sin
1....
1
2a
sin
1asin
1++++=
eyIgán ∑=
=n
0kk
n )
2a
sin
1(S eday x2cotxcot
x2sin1
−=
acot
2a
cot
2a
Sn
kkn ⎟⎠⎞=
=∑ cot
2a
cot
1n
01k
−=
⎜⎝⎛ −
+
+
dUcen¼ acot2a
cotS 1nn −= + .
KNitviTüaCMuvijBiPBelak
- 123 - PaK1
l atTI6Mh 6
k¿cUrRsayfa xcot2x
cot
xcos1
1 =+ KuN x¿KNnapl
)
2a
cos
11).....(
2a
cos
11)(
2a
cos
11)(
acos1
1(Pn +=
n
+++ 2
dMeNa¼Rsay
k¿Rsayfa xcot2x
cot
xcos1
1 =+ eyIgtag
xcos1
1)x(A +=
xcot2x
cotxtan
2xco
== tanxcosxsin
.
2x
sin
2x
s
2x
sinxcos
xsin2x
cos
2x
sinxcos
2x
sin2x
cos2
xcos2x
cos2
x1x
22
=
===+
=
coscos
dUcen¼ xcot2x
cot
xcos1
1 =+ . x¿KNnaplKuN )
2a
cos
11).....(
22
acos
11)(
2a
cos
11)(
acos1
1(P
n
n ++++=
acot.2
atanP = 1nn + .
KNitviTüaCMuvijBiPBelak
- 124 - PaK1
lMhat´TI67 eK[sIVúténcMnYnBit n kMnt´elI eday ½ )U( IN
22
U0 = nig INn,211− U
U2
n1n ∈∀
−=+
aGnuKmn_én . KNna nU C
n
dMeNa¼Rsay KNna GnuKmn_én ½
nU Ca n
eyIgman4π
sin22
0 == U
8π
sin2
4π
sin11
2U11
U1
−=
22
0 =−−
=−
«bmafavaBitdltYTI p KW 2pp s=2π
inU + ayfavaBitdl YTI eyIgnwgRs ´t )1p( + KW 3p1p 2
πsinU ++ = Bit
an eyIgm2
U11U
2p
1p
−−=+ Ettamkar«bma 2pp 2
πsinU +=
eyIgán 2
sin11U 1p+
−−= 2
π2p
2+
3p22 +
dUcen¼
32p sin22π
cos1 +− p2
πsin2
π
2
+=== Bit
2nn 2π
sinU += .
KNitviTüaCMuvijBiPBelak
- 125 - PaK1
lMhat´TI68 cUrKNna
9π7
cos9π4
cos9π
cosS 333 +−=
dMeNa¼Rsay KNna
9π7
cos9π4
cos9π
cS = os 333 +− tamrUbmnþ acos3acos4a3cos 3 −= ¦ a3cos
41
acos43
acos3 += GacsresrCa ½ kenßamEdl[
)3π7
cos3π4
s(cos43
S = co3π
(cos41
)9π7
cos9π4
cos9π
+−++− tag
9π7
cos9π4
cos9π
cosM = +− y eda
9π13
cos9π4
cos =−
9π13
cos9π7
cos9π
cosM ++= KuNnwg 3π
sin2 eKán
0)9π7
cos(πsi23
M2 n29π16
sin9π2
sin
9π10
sin9π16
sin9π4
9π10
sin)9π2
sin(9π4
sin3
3π
sin9π13
cos23π
sin9π7
cos23π
sin9π
cos23π
sinM2
=−=+=
−+−+−−=
++=
eKTaján
sin2
M.2
0M =
KNitviTüaCMuvijBiPBelak
- 126 - PaK1
tag 23
21
21
21
3π7
cos3π4
cos3π
cosN =++=+−= eKán
83
N41
M43
S =+= .
dUcen¼ 83
97cos
94cos
9cosS 333 =
π+
π−
π= .
lMhat´TI69 eK[RtIekaN mYymanmMukñúgCamMuRsYc .
ayfa Btan.AtanCtanBtanAtan
ABC
k¿cUrRs tan. C=++ ¿TajbBa¢ak´fa x 33CtanBtanAtan ≥++ .
dMeNa¼Rsay k¿Rsayfa tan Ctan.Btan.AtanCtanBtanA =++
= ¦ eyIgman BA ++ πC CπBA −=+ án eK )Cπtan()BAtan( −=+
CtanBtanAtanCtanBtanAtan
CtanBtanBtan
Atan1Atan
+−=
−=
tan.AtanBtanA
+−
+
dUcen¼ .BCtantan Ctan=++ .
KNitviTüaCMuvijBiPBelak
- 127 - PaK1
x¿TajbBa¢ak´fa 33CtanBtanAtan ≥++ CamMuRsYc ( tamsm μtikm μ )
eKTaj PaBkUsIueyIgGacsresr ½
eday C,B,A
0Ctan,0Btan,0Atan >>>
tamvism3 Ctan.Btan.Atan3CtanAtan ≥+
eday tan.Btan.AtanCtanBan
Btan + tAtan C=++
eKán 3 CtanBtat nAtan3BtanBtanAan ++≥++
2 ≥++
+ (tan27)CtanBtanA(tan 3
27)CtanBtanA(tan
)CtanBtanA +≥++ dUcen¼ 33CtanBtanAtan ≥++ .
KNitviTüaCMuvijBiPBelak
- 128 - PaK1
lMhat´TI70 KNnaplbUk ½ n 999....999....999999S ++++= manelx cMnYn elx ) . ( 9 n
dMeNa¼Rsay KNnaplbUk ½
9.. 99......999. .999999Sn
)9n( elx++++=
9)10n9(10
n9
110.10n
−−
=
)1...111()10...10)110(....)110()110()110(
1n
n32
n32
+−=
++++−+++
−++−+−+−=
+
dUcen¼
1010( +=
9)10n9(10S
2n
n+−
=+
.
KNitviTüaCMuvijBiPBelak
- 129 - PaK1
lMhat´TI71 cUrkM ntRKb´KUtémøKt´viC¢man ebIeKdwgfacMnYn ½
)b,a(
1bab2a
32
2
+− CacMnYnKt´viC¢manEdr .
dMeNa¼Rsay nt´RKb´KUtémøKt´viC¢man ½ k
kM )b,a(
y k1bab2
a32
2
=+−
Edl *INk∈ 32eKán 2a2 )1(0)1b(kakb =−+
42
−
DIsRKImINg´énsmIkar )1b(k4 3bk4 −−=Δ 22 k4)bkb2( 2b−+−=Δ mIkar mancemøIykñúg lu¼RtaEt s )1( *IN Δ CakaerRákd
mann&yfa 2222 dbk4)bkb2( =−+−=Δ
d
Edl CacMnYnKt´. -ebI 0bk4 2 =− ¦
4 bk
2
=
eyIgTTYlán 2
bbb2
kb2a3
2 −=−= ¦
2ba =
KNitviTüaCMuvijBiPBelak
- 130 - PaK1
eday a b, CacMnYnKt´viC¢man ehtuen¼eKRtUv[ ½ INp,p2b ∈∀=
eKTaj pp8p4
ehIy
)p2( 42
−=− )p2(2a 2=
pp2== .
2a
2,p(;)p2,pp4dUcen¼ 8()b, *INp,)pa( ∈∀−=
-ebI 0b
k4 2 >−
eKán 4)2 bkb2( 22222 *INk,)1bkb2(dbk ∈∀+−≥=−+− 2¦ )1b(k4 2 0)1b( ≤−+− án eKTaj
IsmI1b =
kñúgkrN kar )1( køayCa 0ak2a2 =− naM[ k2a = dUcen¼ )1,k2()b,a( = cMeBa¼RKb´ *INk∈ . -ebI 0bk4 − 2 <
)bkb −eKán k42( 222222 )1bkb2(db −<=−+− )bkb2 2222 <smmUl 4( 2 0)1bkb2(bk −−−−+−
¦ k4(b2 0)1k4()1b(b2)3 <−+−+− ( minBitkñúg ) srubmkeKánKUcemøIybImanragdUcxageRkam ½
*IN
)k2,kk8(;)k2,k(;)1,k2()b,a( 4 −= *INk∈
KNitviTüaCMuvijBiPBelak
- 131 - PaK1
´TIlMhat 72 cUrbgHajfa ½ 1
ab8ca 22
cca8bbc8 2
≥ba
++
++
+
RKb´cMnYnBit ¢man . cMeBa¼ viC c,b,a
dMeNa¼Rsay
ajfa bgH 1ab8
c≥
cca8bbc8aba
222 ++
++
+
CadMbUgeyIgRtUvRsayfa
34
34
34
34
aa2
cbabc8a ++≥
+
mmUl
vismPaBen¼s )bc8a(a)cba( 234
234
34
34
+≥++ smmUl bca8cb2ca2ba2cb 3 3
434
34
34
34
34
34
38
≥8
+++ amvismPaB eKman
+
t GMAM − 34
34
38
38
cb2cb ≥+ eKTaj bca8cb2ba2cb 3333 ca2 3
434
34
34
344488
≥++++ Bit
tuen¼ eh ( )1cba
abc8a
a
34
34
34
34
2
++≥
+
KNitviTüaCMuvijBiPBelak
- 132 - PaK1
dUcKñaEdreKTaj ( )2cba
bac8b
b
34
34
34
34
2
++≥
+
ehIy ( )3cba
cab8c2
c
34
34
34
34
++≥
edaybUkvismPaB Gg:nigGg:eKán +
)3(,)2(,)1(
1ab8c
cca8b
bab8a c 222
≥+
++
. ++
lMhat´TI73 eK[ a( n )CasIVúténcMnYnBitEdl
21a1 =
b´cMnYnKt´viC¢man eyIgman nigcMeBa¼RK n1aa
aan
2n
2n
1 =+n +−
cUrRsaybBa¢ak´facMeBa¼RKb´cMnYnKtviC¢man n eyIgman ½ n3 aa 21 1a.........a <+++ .
dMeNa¼+
Rsay RsaybBa¢ak´fa 1a.........aaa n321 <++++ eyIgman
1aaaa 1n =+
n2
n
2n
+−
KNitviTüaCMuvijBiPBelak
- 133 - PaK1
tag n
n a1b = ena¼TMnak´TMngEdl[køayeTACa ½
n2 +−=
bb1 −=−
1bbb n1n+
eKTaj 1nb + n2
n
nnnn1n 1b)b(b1b−
−=
−=
−
nn1 )1b(b1n
b11
111
b −=−
+
+
eKTaj 1b1bb
a1nnn
n −−
−==
+
111
n21n21 .aa ++
b1......
b1
b1a... +++=+
11b
111b1b 1n1 − +
111b1b 121 −−− +
)1b
11b
1(...)1b
11b
1()
1n
nn32
<−
−=−
−=
−−
++−
−−
+
+
BIeRBa¼
11( −=
2a1b
1
dUcen¼ 1a.........aa
1 == .
a n321 <++++ .
KNitviTüaCMuvijBiPBelak
- 134 - PaK1
lMhat´TI74 cUrkMntRKb´KUtémøKt´ 3n,m ≥ ebIeKdwgfacMeBa¼RKb´ cMnYnKtviC¢man eKman a
1aa1aa
2n
m
−+−+ CacMnYnKt´ .
dMeNa¼Rsay
´KUtémøKt´viC ½ kMnt´RKb ¢man ( )n,m
edIm,I[ 1aa
an1aa2n
m
−+−+ CacMnYnKt´lu¼RtaEt 2 −1a+
aktþarYmén C 1aam −+ ehIy . Igyk
nm >
ey *INk,knm ∈+= Igán ½
k1k2nk
knm
−+−+−+
−+=−++
+
,I[
ey
)1aa)(a1()1aa(a1aa1aa
=
tamTMnak´TMngen¼edIm 1aa 2n −+ CaktþarYmén a 1am −+
lu¼RtaEt 1kn += n
ig 2k = . .
dUcen¼ )3,5()n,m( =
KNitviTüaCMuvijBiPBelak
- 135 - PaK1
lMhat´TI75 eK[bIcMnYnKt c,b,a Edl ba´viC¢man = 10c++ . cUrrktémøFMbMputén cbaP ××= . dMeNa¼Rsay rktémøFMbMputén cbaP ××=
aM[ ceyIgman 10cba =++ n ba01 −−= )b10(b10(ab 2 −eyIgán baP aa)b +−=−−=
2tag )ba(P ab10() 2ba −+−= eKán 2bb10ab2)a('P −+−= bI eKTaján e 0)a('P =
2b5a −=
eday a(''P *INb,0b2) ∈∀<−= øGtibrma mø ehtuen¼ )a(P mantém cMeBa¼té
2b5a −= .
a¼ RtUvEt YnKU I
eday *INa∈ en CacMnb
eb k2b = ena¼ k5a −= ehIy k5k2)k5(10c −=−−− =
KNitviTüaCMuvijBiPBelak
- 136 - PaK1
eday ,a ena¼ ⎪⎩
⎨*INc,b ∈ ⎪⎧
≥−=≥=≥−=
1k5c1k2b
1k5a ¦ 4k1 ≤≤
= n
-cMeBa¼ k 1
eKá 4c,2b,4a === naM[ 32424P =××=
n 3a
-cMeBa¼ 2k =
eKá c,4b, 3== P= naM[ =× 36343×= -cMeBa¼ 3k = eKán 2c,6b,2a == naM[ 242P= 26 =××= -cMeBa¼ 4k =
n a eKá 1b,1 c,8 === naM[ 32181P =××= témøGtibrmaén dUcen¼ c.b.aP = es wg .
μIn 36
KNitviTüaCMuvijBiPBelak
- 137 - PaK1
lMhat´TI76 eKmansmPaB
ba1
bxcos
axin 44s
+=+
0Edl b,a 0ba,0 ≠+≠≠ a cUrbgHajf 33
8
3
8
)bxcos
axsin
=+ .
eNa¼
ba(1+
dM Rsay gHajfa b 33
8
3
8
)ba(1
bxcos
axsin
+=+
eKman ba
1b
xcosa
xsin 44
+=+
M[ na abxcos)ba(axsin)ba(b 44 =+++
0)xsinbxcosa(
0xsinbxsinxcosab2xcosa)xcosx(sinabxcosabxcosaxsinbxsinab
222
422242
422442424
=−
=+−
+=+++
eKTaj ba
1ba
xa
xsin=
cosxsinb
xcos 2222
+++
==
naM[ 44
8
4
8
)ba(1
bxcos
axsin
+==
KNitviTüaCMuvijBiPBelak
- 138 - PaK1
eKTaj )1()ba(
aa
xsin43
8
+= nig )2(
)ba(b
bxcos
43
8
+=
nig Gg:nigGg: eKán ½ bUksmIkar )1( )2(
343
8
3
8 cxsin+
)ba(1
)ba(ba
bxos
a +=
++
=
dUcen¼ 33
8
3
8
)ba(1
bxcos
axsin
+=+ .
lMhat´TI77 eK[RtIekaN ABC mYYy . bgHajfaebI
3Ctan,
3Btan,
3Atan
Ikar bxaxx:)E( 23Ca¦ssm c 0=+++ n ena¼eKá cb3a3 +=+ .
dMeNa¼Rsay karbgHaj ½ eyIgman π=++ CBA
MukñúgRtIekaN ) yIgán ( plbUkm ABC
e ]3C)
3B
3Atan[()
3C
3B
3Atan( ++=++
KNitviTüaCMuvijBiPBelak
- 139 - PaK1
)1()
3Ctan
3Btan
3anta
3tan
3(tan1 +−
Ct3
nBA3Ctan
3Btan
3AtanCta
3Btan
3
3Ctan.
3Btan
3Atan
3Btan
3Atan
3Ctan
3Btan
3A
3BtanA
3Ctan
31
3Ctan)
3B
3Atan(
+
+
+
++
−
++=
A3
ntan3
−=
A+
11
−−
tan1
3
−
tan
3tan =
π
)3BAtan(
)3
tan(+
CBA ++
eday 3Ctan,
3Btan,
3Atan Ca¦srbs´smIkar
ena¼tamRTwsþIbTEvüteKmanTMnak´TMng ½ )E(
)2(a3Ctan
3Btan
3Atan −=++
)4(c2Ctan.
2Btan.
2Atan
)3(b2Ctan
2Atan
2Ctan
2Btan
2Btan
2Atan
−=
=++
ykTMnak´TMng nig CYskñúgsmIkar eKán ½ )3(,)2( )4( )1(
KNitviTüaCMuvijBiPBelak
- 140 - PaK1
b1ca3
−+−
= ¦ cab33 +−=− dUcen¼ cb3a3 +=+ .
lMhat´TI78 k¿ cUrKNnatémø®ákdén
10sin π nig
10cos π
x¿ cUrRsayfa 10
sin)yx(4)yx(x 22222 π+≤−+
RKb´cMnYn IRy,x ∈ .
dMeNa¼Rsay k¿ KNnatémø®ákdén
10sin π nig
10cos π
eKman 1032
210π
−= eKán
ππ
103cos)
103
2sin(
102sin π
=π
−π
=π
tamrUbmnþRtIekaNmaRt ½ nig acosasin2a2sin = acos3acos4a3cos 3 −=
KNitviTüaCMuvijBiPBelak
- 141 - PaK1
)1010
sin1(43sin2
10cos43
10sin2
10cos4
10cos
10cos
10sin2
103cosco
10sin2
2
3
π−−=
π
π−=
π
π−
π=
ππ 3
10s
2
π=
ππ
¦ 0sin2sin4 2 =11010
−π
−π tag 0
10 eKán t4 2
sint >π
>
=
0541',01t2 =+=Δ=−− ¦s eKTaj 0
451−t1 <= ( minyk )
451t, 2
+=
dUcen¼ 4
5110
sin +=
π . eday 1
10cossin 22
10=
π+
π
naM[ 4
5210)4
51(110
2 −=
+−=
π cos
dUcen¼ 410
cos =5210 −π .
sayfa x¿ R10
sin)yx(4)2 ≤yx(x 2222 π+−+
tagGnuKmn_ 10
sin)yx(4)yx(x)y;x(f 22222 π+−−+=
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eKán ½ 222222 )
451)(yx(4yxy2xx)y;x(f +
+−+−+=
IRy,x,0y2
15x2
15
y2
15xy2x2
15
y5 2
21xy2x
251
253)yx(yxy2x2
1626)yx(4yxy2x2
22
2
2222
2222
∈∀≤⎟⎟⎠
⎜⎜⎝
−−
⎟⎟⎠
⎞⎜⎜⎛ +
++−
−
+−−
−=
++−+−=
++−+−
5=
2⎞⎛ +
⎝
+=
=
dUcen¼ 10
sin)yx(4)yx(x 22222 π+≤−+ .
KNitviTüaCMuvijBiPBelak
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lMhat´TI79 eK[sIVúténcMnYnBit kMnt´eday)U( n 4
nsin.2Un
nπ
= Edl . k-cUrbgðajfa
*INn∈
4nsin
4ncos
4)1n(cos.2 π
−π
=π+
x-Taj[ánfa 4
)1n(cos)2(4
ncos)2(U 1nnn
π+−
π= +
K-KNnaplbUk ½ n321n U.........UUUS ++++= CaGnuKmn_én .
dMeNa¼
n
Rsay a k-cUrbgHajf
4nsin
4ncos
4)1n(cos.2 π
−π
=π+
tamrUbmnþ bsinasinbcosacos)bacos( −=+ Igán ½ ey
)4
sin4
nsini4
cos4
n(cos2
)44
ncos(24
)1n(cos2
ππ−
ππ=
π+
π=
π+
4
nsin4
ncos)4
nsin22
4ncos
22(2
4)1n(cos2 π
−π
=π
−π
=π+
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dUcen¼ 4
nsin4
ncos4
)1n(cos.2 π−
π=
π+ .
x-Taj[ánfa 4
)1n(cos)2(4
( 1nn
ncos)2U n π+−
π= +
an eyIgm4
nsin4
ncos4
)1n(osc.2 π−
π=
π+ naM[
4)1n(cos2
4ncos
4nsin π+
−π
=π
KuNGg:TaMgBIrnwg n)2( eKán
4)1n(cos)2(
4ncos)2(
41nnnsin)2( n π+
−π
= +
Ucen¼
π
d4
)1n( π+cos)2(4
ncos)2(U 1nn −nπ
= + . K-KNnaplbUk n321n U.........UUUS ++++=
eyIgán
4
)1n(cos)2(4
cos2
4)1 πk(cos)2(
4kcos)2(
1n
n
1k
1kk
π+−
π=
⎥⎦⎤
⎢⎣⎡ +
−Snπ
=
+
=
+∑
dUcen¼ 4
)1n(cos)2(1S 1nn
π+−= + .
KNitviTüaCMuvijBiPBelak
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lMhat´TI80 eK[ n cMnYnBit 0a.......;;a;a;a n321 ≥ . eKtag *NSn In,
na.....aaa n321 ∈∀
++++=
yfaebI
cUrRsa 1n1nan3211n a.......a.a.aS +
++ ≥ ena¼eKán n
3n .....S ≥
n21 aaaa .
dMeNa¼Rsay ebI Rsayfa 1n
1nn3211n aa.......a.a.aS +++ ≥
ena¼eKán nn321n
eyIgman a.....aaaS ≥
1n1nn3211n aa.......a.a.aS ≥ +
++ eday
na.....aaa
S n321n
++++ =
ena¼ 1n
aa....aaaS 1n
1nn321
++++++
= + eKán ½
+
( )1aa.........aaa 21 aaa
1naa..... 1n
1nn3211nn3 +
++ ≥
++++
k
++
yn
a.....aaaa n321
1n
++++=+ CYskñúg ( )1 eKán ½
KNitviTüaCMuvijBiPBelak
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nn21
n21
n21 a.a≥n
n2
n21n21
1nn21
1n n221
n2
1n n21n21
n21n
a.........a.an
a.......aa
a......n
a.......a
na...aa
a.....a.an
a.........aan
a....a....a.a
na.......a
na....aa
.a...a.ana.....aa
≥+++
⎟⎠⎞
⎜⎝
+++
++≥⎟
⎠⎞
⎜⎛ +++
+++≥
++
+++≥
1a⎛
⎝
1n
1 a.a
a1n+
+
21 a....aa
...+
++++
+
+
+
+++
eKTaj nn321n
dUcen¼ ebI a.....aaa S ≥
1n1nn3211n aa.......a.a.aS +
++ ≥ ena¼eKán n
n321n a.....aaaS ≥ .
lMhat´TI81 cUrbgHajfa
101
10099......
65.
43.
21
151
<<
dMeNa¼Rsay bgHajfa
101
10099......
65.
43.
21
151
<< an eyIgm )1n(n)1n( +>+ 2n1n2:*INn +=+∈∀
eKTaj 1n
n2n21n2
+>
++
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eKán
251
5049......
32.
21
10099......
65.
43
=>
KuNnwg 21 eKán
2101
10099........
65.
43.
21
>
eday 151
2101
>
eKTaj )1(151
10099....
65.
43.
21
> müa¨geTot
)1n2)(1:*INn −∈∀ n2(2)1n2()1n2(n4 +>++−= eKTaj
1n21n2
1n2n2
−+
>−
rW 1n21n2
n2n2 − 1
+−
<
eKán 1011
10199....
75.
53.
31
10099.....
65.
43.
21
=<
eday 101
1001
1011
=<
eKTaj )2(101
10099....
65.
43.
21
< Mng nwg eKán ½tamTMnak´T )1( )2(
101
10099......
65.
43.
21
151< < .
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lMhat´TI82 eK[ n2 ....; cMnYnBit ;b;b;a.....;;a;a .
a....b 2n
22
21
2n
22
21
2n22 ++++++≤++
n21n21 b
cUrRsayfa ½ )b.....bb)(a....aa()baba( n11 +
dMeNa¼Rsay
.....bb)(a....aa()ba....baba( 222
21
2n
22
21
2nn2211 +++++≤+++
eyIgeRCIserIsGnuKmn_mYykMnt´
Rsayfa ½ )bn+
IRx∈∀ eday ½ )bx..)bxa()b
2n
221nn2211
22n
22
2
2n
222
21
+++++++++
+++++
eday )b...bb(x)ba..baba(2x)a...aa()x(f 21 +=
a(...xa()x(f n1 +=
IRx∈∀ RtIFa 0)x(f ≥ Canic©ena¼
eday KánCanic©
⎩⎨⎧
≤Δ>
0'0af
0a.....aaa 2n
22
21f >+++=
ehtuen¼e 0'≤Δ . ( ) 0)b....bb)(a....aa(ba. ...baba' 2
22
1n211 ++++++=Δ .... 2n
22
21
2n
2n2 ≤+++−
dUcen¼ a()ba....baba( 2n
22
21
2n
2nn2211 ++++≤+++ )b.....bb)(a....a 2
22
1 ++
KNitviTüaCMuvijBiPBelak
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lMhat´TI83 eK[ )( nig y( nxn CasIVútcMnYnBitkMnt´elI
0 = nigTMnak´TMngkMenIn ½ n1n yx3xx +=+ nig RKb´
cUrKNna nig CaGnuKmn_én .
dMeNa¼
) IN
eday ,5x0 =3
1y2
nn3
nn2
n1n yyx3y +=+ INn∈
n
x ny n
Rsay
eKman nnn1n +=+ nig
Ikar eKá)yx(y
n
3nn1n
+=+
+=+
ln{ n
KNna nx nig ny CaGnuKmn_én23
n
)1(yx3xx
)2(yyx3y 3nn
2n1n +=+
bUksm nig 2( n ½ )1( )
)yxln(.3)yxln(x 1n ++
n1n1n ++
TMnak´TMngen¼bBa¢ak´fa x( n })y+ CasIVútFrNImaRtman ersug nigtYdMbUg 3q = 6ln)yxln( 00 =+ .
eKán 6ln3)yxln( nnn =+
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eKTaj nig eKán ½
.3yy 3
n
−
−+
¢ak´fa
)3(6yxn3
nn =+
dkksmIkar yx =−
)1( )2(
)yx()xln()x(
nn1nn
nn11n
−=++
+ ln1
TMnak´TMngen¼bBa })yxln({ nn − CasIVútFrNImaRtman ersug nigtYd 003q = MbUg ln( 4ln)yx =− .
Taj nig eKán
eKán 4ln3)yxln( nn− n =
eK )4(4yxn3
nn =−
bUksmIkar 3( ) )4(nn 33
n 46x2 +=
eKTaj 2
46xnn 33
= . n+
ar nig eKán −= dksmIk y2)3( )4(nn 33
n 46
eKTaj 2
yn46
nn 33 −= .
dUcen¼ 2
46xnn 33 +
n = nig 2
46ynn 33
n−
= .
( anPaKTI2 sUmG )
sUmrgcaMG rKuN
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- 151 - PaK1
Éksareyag ( Referent )
Hu hematical Chal es dward J. Bar beau, M ray S. Klamkin)
plex Numbers Fro A to … Z itu Andresscu , Dorin Andrica )
a Oly in China at ly hallenges
itu Andresscu & n Gelca) ematical Oly d Treasures
ndreescu , Bogdan enescu ) 7. International Mathematical Olympiads 1959-1977
8. The IMO Compendium ( 1959-2004 )
(By Titu Andreescu , Dorin Andrica )
1. 103 Trigonometry Problems (From the training of the USA IMO Team)
2. Five ndred Mat leng (By E ur
3. Com m (By T
4. Mathem tical mpiad 5. Mathem ical O mpiad C (By T Razva
6. Math mpia (By Titu A
( Samuel L. Greitzer )
9. 360 Problems for Mathematical Contest
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- 152 - PaK1
