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# $% & ')( !" * +(,-$. 0/12 ,342 5768!9;:= LHMON&PQMSRUTW...

Author: Canadian Mathematical Society

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n2} ^ l Y!j g MbcYHjcNrM jUT]XTW[wNr[ cjM @ dj2[M VXNj ggUh PzcTrMiM g Nr[`M T qt MJ NWY k M V=Y j q MHV?V?jUV q jV h ^ K s N_MJ^f^eM cj j g+h M j2x=^fYDT k j%M q M»]XMONWN_M»bcYHjcNrM jUT @ h jYHj q M VVXjUV K dY]rYDV?VmM T s PQY [eV djMiVX[ hcg ^eM!^eYHT k jM q j g M|bcY jcN_MJjUTu2^wjUVzcYDV?VmM @ djM ^ dj M g V h [wNy^eM gUh PzcTrM q Mix h [eV K g Ma]rYDV?VmM TWYuUYHV n g h jUVuUT h ]_jUTrM q MJj h MJj2x`VMJN hcg h jUV q [\N djMa]XMONWN_M bcYHjcNrM jUT=M VXNM g N_TrM MJN MJ NWY k M V @ [ g ]_^fjUVX[\cM PM g N K n§h jUV h j2^`M q MO cMJ^ h u2uM Tj g u2^fY g dj2[tuM T_PQMJN q 1M|^ [ q 37 M g Nr[fRMHT q M|^eYPQY g [M TrM^fYauc^fjUVvM ]rY]_M n D d[ q MHPQM g N sUhcg u h jUTrTWYH[\N] h P&PM g ]XM TaY jÀuUTrM P&[eMHT MJ NWY k MMONaP hcg N_M Tj g MJ NWY k M Y ^fYx h [eV n MONWN_M VXN_TWYJN M k [eM'u h jUT_TWY [wNu2TrM gUq TrM'j g ] h jUu»V?[^fY&bcYHjcNrM jUTv]XTr[\Nr[ djM&M VXNj g MO NrY k M s P&YH[fV ]XMONWN_M!NrMD]?b g [ cjMiuUTrM gUq TWY [wN¥`jUV dj Y ] h jUuUVVX[]XMJ^fj2[e]_[MHV_N^fYbcY jcN_MJjUT=]XTW[wNr[ cjM n L YVXN_TWYJN M k [eM h udNr[eP&YH^eMdY g MH ]MH V?V?[\N_M T 37 ^`M gUh PzcTrMP'[ g [fPQY ^ q M]?bdjcNrMHV nc Y g Vo^`M=u2[fTrM Vm] M g YDTW[ h s ^`M gUh PzcTrM q M]?bdjcN_M g MD ]_MHV?VY [eTrMQu h jUT![ q M g Nr[wR M T!^eYbcYHjcNrM jUT|]XTW[wNr[ cjM YJcMH]^eY|VXN_TWYJN M k [eM h ucNr[fPQY ^`M|M VXN *

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5H

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n Y g V^eM q [fY k TWYHPQPQM s ABCD M VXNj g ]rYDTrTM qUhcg N o `^ M Vi] h N M V!PM V?jUTrM g N MON cjM^eMHV djUY N_TrMaN_TW[eY gUk ^`M V 17MON V hcg N q MHVN_TW[eY gUk ^`M V s s ABF CDH TrMD]rNWY gUk DAE ^eMHV»] hcgUBCG k TrjUV n g u2^wjUV s F B = 8 n UT h jdcMHT ^ YH[fTrM q j djUY q Tr[w^eYJN MH TrM h PQz2TWY k M n

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n g»k T h jUuM'MHV_NYH^w^ M q [ g MHTYHj»TrMHV_NrY jUTWY g N n ¦ ^eYaR g»q jTrM uUYHV s [f^fV q MD ]_[ q M g N q M uUYHTWNWY k M T^eY gUh NrM M k Y ^`M PM g NM g N_TrMM j n=c [f^fV&] hcg N_TW[`zdjM g N]?bcY]_j g s [w^eVVmM $16 TrMJN_T h jdcM g N Y» ] h jUTWN q M Y ^ W T V c j M V w [ e ^ | V P O M W N _ N M N w [ e ^ | V u J M d j c M i N U u Y s h g $19 s g l MHT!^eY $4[eTrMQMJNv[w^yTrM VXNrM x`Y]rNrjUTrM s Q M u U j X T z Q P c z e [ M M M V O M r N Y ` [ M X N X w [ e ^ i V MJN^eY q h h n h gq k g g 15% $2 _ N i M V M e ^ m M d H Y \ [ N Y ] Q P c z e [ M gUh h g

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n

n M ^w^f[eMaM VXN 5 ZP Y jV?j qq j g ] h jUTWV q M YHj dj2[dYicM TWV^ MHV_N n ^f^eMaM VXN 8 ZP Y ^ h jMHV_N=MJN ZDP% Y j gUh T Q q q M|VY]rYzd[ g M n ^f^eMvcM jcNYzcTrM jdcM T=V hcg ]?b2MmdYH^YHj] h jUTWV 6TrMON jUT MHT Yv H M Y j c u 2 j e [ V V Yv]_YDzc[ g M n jM ^w^`MvM VXN^fYvuc^fjUV] h jUTWNrM q [eV_NrY g ]XM djM M ^w^f[eM qt h g \ [ y N Y H M T qUh hHl k

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n= Y g V|^eM q [fY k TWYHPQPQM s ^eM'N_Tr[fY gUk ^eM ABC M VXNvj g N_Tr[fY gUk ^eM 30◦ E 60◦ E 90◦ q2hcg N ^ Y U g k ^eM q T h [wNM VXN Y C s ^eM 30◦ VMN_T h jdcM'M g B s MON AB PQMHVXjUTrM 20 n LHMVmM k PM g N MHV_NuUMHTruM gUq [e]_j2^eY [eTrMY MON MHV_Ny^`MP'[f^w[`MJj q M n LHMVM k PQM g N EC MHV_N ED AC DMHV_NiuUMHTruM AC uUYHTWYH^w^ MJ ^`M Y H L M V M Q P M N e [ _ ] 2 j e ^ Y e [ r T M Y MON M VXNV?jUT!^`M n k g EF gUq ED F uUT h ^ hcgUk MHPQM AB N M g q AB n E C @ Y K UT h jdcMHT^eM^ hcgUk jMJjUT q M ....................................................... n . .. ... ............... .. ............... ........ ... ED ........ ........ .... .. . ........ ......... . . ........ . . . . . . @ z K UT h jdcMHT^fYi^ hcgUk jMJjUT q M ........ ........ .. n ... ........ ............ .... D . ............ ... . DF

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A

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XM HN{M k [ cM'NXb2MV h ^fjcNr[ hcg V!N h NXb2M hcg ] h jUTWV hcg N_P h rT M g ] l DGDGd E HGG * AOE n

KNMdhd^ M.n_o6UX` dM h cC e] MPoiRVh_^tvjk.kdjdjk.k y= { y C#y =

j ]rN h xtY gl N{ h'h2q2q [ g _N M k MHTWV=[eV=YH^ {|Y l V 2 n @ K b h {4NXbcYJNNXb2MiuUT 2 h q h q2qon @ ½»YHT g [ gUk * ∗M?HYHPQuc^`M VYDTrM gUh N=M gUh j k b n K

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2

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q MHuUM gUq

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[ g ]_M 5 | bcd s NXb2M g d = 5 n¦ ^eV h V?[ g ]XM 4 | abc s NXb2M g c = 2 h T c = 4 n [ g ]_M 3 | cde s {M|bcYOcM 3 | (c + d + e) n } x = 2 s NXb2M g 3 | (2 + 5 + e) s {bd[e]?b[fPQuc^f[eMHVNXbcY N e ≡ 2 (mod 3) n tbd[fV TrMHu2TrMHVM g cNWV Y&] hcg N_TWY q [e]rNr[ hcgts V?[ g ]XM P&jUV_NvzUM q [eV_Nr[ g ]rNx`T h P Y gUq n d = 5 bd[eV[ecP&= |M g ]XM s [eVx`Y ^eVM s Y gUq NXbdjUV s e t 2 b M u2^w[`2 M V n g n c =2 c =4 3 | (4 + 5 + e) NXbcY N s x`T h P¯{bd[`]?b{M|bcYJcM e = 3 n e ≡ 0 (mod 3) h { s s nS [ g ]XM 214 [eV gUh NYP&j2^\Nr[euc^`M h x 4 s {MP&jUV_N|bcYOcM {1 2} Y gUq a b ∈ n a=1 b=2 tb2M TrM x h TrM s NXb2MiuUT h zd^`M P bcYDVNXb2Mij g [ cjM|V h ^fjcNr[ hcg* n abcde = 12453 2004

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gUq ÁyYDPQM ÁyYDPQM A NWYDTN

A

B

C

27

27

27

27 ÷ 3 = 9

27 ÷ 3 = 9

27 + 18 + 18 = 63

57 ÷ 3 = 19

21 ÷ 3 = 7

9÷3=3

3 + 38 + 14 = 55

63 ÷ 3 = 21

9 + 42 + 6 = 57

tbdjUV s¦ TNXbdjUTVXNWYDTN_M q {i[\NXb sp MHT g [`Mv{i[\NXb $19 s Y gUq bcYDTW^f[eM!{i[\NXb $7 n $55

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+ B+D " B"

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b

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a 6= 1

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(a2 − 1) + 2 a2 + 1 a2 − 1 2 2 = = + = a+1+ a−1 a−1 a−1 a−1 a−1

n

½ b l [eVNXbd[eVb2MJ^eucxej2^ ½ M{|Y g NNXbd[eVM?DuUTrM VVX[ hcg N h z2M!Y g [ g N_M k MHT n [ g ]XM{M rT M dj2[eTrM N h z2M'Y g [ g NrM k M T s NXb2M'M?DuUTrM VVX[ hcg {i[f^w^z2M'Y g [ g NrM k M T 2 a a+1+ a − 1 {b2M g MmcM T [eVY g [ g N_M k MHT n tbdjUV s {M g MDM q N h&q [ c[ q M|MmcM g ^ l [ g N h n 2 a−1 2 a − 1 ½M&]_Y ggUh {R g [eVbNXbd[ gUk V cj2[e]?ZD^ lon [ g ]_M [fVY&u h VX[wNr[ cM[ g N_M k MHT s [eVaY N^`M YDV_N s Y gUq NXb2M hcg ^ lgUhcg g M k YJNr[\cM[ g N_M k aMHT q [ c[fV h TWV h x YDTrM Y agUq − 1n 0 2 1 @ NXbcY N[eV s @ NXbcYJN[eV s K h T K n 2 tb2M TrM x h TrM s MJ[wNXb2M T a − 1 = 1 a = 2 a − 1 = 2 a = 3 jz2V_Nr[wNrjcNr[ gUk z2YD]?Z[ g N h M djUY Nr[ hcg @ OK l [`MJ^ q V b = 5 [ g z h NXb]rYDVMHV n b2MHTrMH x h TrM s NXb2M|V h ^fjcNr[ hcg VN h NXb2MM cjUYJNr[ hcg YHTrM Y gUq n (a, b) = (2, 5) (a, b) = (3, 5) } N V g MmcM T!YQz2Y q [ q M YQYJNNXbd[fViV_NrY k MN h ]?b2MH]?Z h jUTvY g Vr{MHTWVvN h PQYDZDM&VXjUTrM NXbcY NNXb2M l { h T_Z&[ g NXb2M h Tr[ k [ g Y ^yM djUY Nr[ hcgon& jzcVXNr[\NrjcNr[ gUk NXb2MN{ h u h V?V?[ezc[w^f[\Nr[`M V s {M k MON 1 2 Y gUq 1 3 s {bd[e]?b'YDTrM|z h NXbaN_TWjM n 1 1 + 5 + 10 =1 + 5 + 15 =1 2 3 tbd[fVPMONXb h2qh xtV h ^wjcNr[ hcg [eVY k h2h2qQhcg Mz2MD]rYHjUVM|[\N[fVcMHT l u h TNrYDzc^eM NXbcYJN [eV s [wN{ h T_ZV|x h T!Y^ h N h x q [f¬MHTrM g NiuUT h zd^`M PQV n iMHTrMQ[fVYVX[eP'[f^fYDTzcjcN q [w¬oM TrM g N YDu2uUT h YD]?bN h V h ^ c[ gUk NXbd[fVVYHPMiuUT h zd^`M P n n'¦=k Y [ gts {MQ]_^`M YDTNXb2M&x`TWYD]rNr[ hcg ViY gUqh zdNWYH[ g @~A K n tbd[fVNr[fPM s {M'N_T l N h V h ^\cM!x h T [ g NrMHTrPQV h x z l TrMF{|Tr[\Nr[ gUk M djUY Nr[ hcg @~A K YHV=Y djUY q TWY Nr[e]M djUY Nr[ hcg b [ g scg YDPQM ^ l a a n a2 + (−b)a + (b + 1) = 0 ½M]rY g&gUh {¿V h ^\cMx h T [ g NrMHTrPQV h x z l jUV?[ gUk NXb2M djUY q TWY Nr[e]=x h TrP&j2^fY * a b

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U j PQzUM TWVbcYJcMzUMHM g ]_M g N_TWY ^[ g NXb2M q MmcMJ^ h u2PM g N h xPQYJNXb2MHP&Y Nr[e]_V h cMHT NXb2M']_M g NrjUTW[`M V n b2M g MHM q N hq M Vm]XTW[`z2M cjUY g Nr[wN l ^eM q N h NXb2M q MmcMJ^ h u2PM g N h xNXb2M g Y NrjUTWY ^ g jUPQzUM TWV s 2 s 3 s . . . } n ¦ VbcYDT q YDVo[\N [eVtN h zUMJ^f[eMmcMN h2q Y l s NXb2M=[ q MHY =MHYH{1 x ? M T [ N ] Q P M ? V w [ ^ h hiq qgUh h l {b2M g [\N q [ q ] h PM s NXb2M g Y NrjUTWY ^ g jUPz2MHTV l VXNrMHP{|YHV M g ^fYDT k M qts Y gUq NXb2M{b h ^eM g jUPz2MHTWV ]rYDPQM[ g N h z2M [ gUkcn b2M s s s s TrMHu2TrMHVM g Nr[ hcg&h x g jUPQzUM TWVV l PQz h ^w[`]rYH^w^ l ={|{0 YHVNX1b2M 2g&3k TrMH.YJ.Nr.^ } l VXN_TrM YDP'^f[ g M q {i[\NXbNXb2M [ g N_T h2q j]rNr[ hcg»h xuc^eYD]XMidYH^wjMV l VXNrMHP&V^f[eZDM|NXb2MQ ![ gUq j ¦ TWYDzc[e] g jUPQzUM TWVNXbcY N{M jUVmM!N h2q Y ln ¦ NdYHTr[ h jUV h NXb2MHTNr[fPM V sg MHM q Y gUq [ g NrM ^w^`MH]rNrjUYH^V h ubd[fVXNr[e]_YJNr[ hcg ^`M q N h NXb2M [ g NrM k M TWV s s s s s s s s s NXb2MTWY Nr[ hcg YH^ g jUPQzUM TWV = {. . . −3 o−2 −1 0 1 2 3 . . . } n a s s Y gUq NXb2M|TrM YH^ g jUPz2MHTWV n s

=

b

H

|a b∈

b 6= 0

+H

tb2MQM cjUYJNr[ hcg bcY qgUh V h ^fjcNr[ hcg VizUMJx h TrMNXb2M q MmcM ^ h uUPQM g N h x x+1 = 0 M J Y r N \ [ c M U j Q P U z M W T V f [ & P w [ e ^ H Y r T ^ X N 2 b g k g n lts M|M djUY Nr[ hcg 2 1 = 0 bcYDV gUh V h ^wjcNr[ hcg VNXbcYJN YDTrMTrMHY ^ g jUPz2MHTWV nt¦ ^fYDT k M T g jUPQzUM TV l V_N_M P4[eV xg MD+ M q M q x h TNXbd[eVyM cjUYJNr[ hcg N h bcYOcM V h ^wjcNr[ hcg V n }~g NXb2MV?[ HNrMDM g NXb»]_M g NrjUT l s YDT q Y gq MmcMJ^ h uUM q PMONXb h2q Vvx h TV h ^\d[ gUk ]_jzd[`] M djUY Nr[ hcg V s [ g {bd[e]?bb2Ma{ h j2^ q PQY g [eucj2^eYJN_MNXbd[ gUk V^f[eZDM √ [ g bd[fVx h TrP&j2^fYDV n tb2M VmM ! j g ?TrM YH^ #4g jUPz2MHTWV k YJcM¯bd[eP dYH^w[ q TrM YH^»TrMHVXj2^wNW−1 V NXb2M ! [fPQY k [ g YHT l # g jUPz2MHTWV{MHTrM|z h T g'& } x{M q M R g M'NXb2M'j g [wNv[eP&Y k [ g YDT lg jUPz2MHT i N h VYJNr[eVXx l i2 = −1 @ V h NXbcYJN h jUT djUY q TWY Nr[e]M djUY Nr[ hcg x2 + 1 = 0 bcYDVvN{ h V h ^wjcNr[ hcg V stg YDPQM ^ l ±iK s {M&]rY g {b2M TrM s q MJR g M&Y " %$¹¶ g jUPQzUM T!YHVY g jUPz2MHT h xoNXb2M'x h TrP n tb2M] h ^f^eMD]rNr[ hcgh xoY ^f^tVXj]?b g jUPQzUM TWV[eVNXb2MaVMJN h x " %a$~¶+ bi 2¶ s a{bdb[`]?∈ b{M q M gUh N_Maz l vn Á[\cM g Ya] h P&u2^eM? g jUPQzUM T a + bi s {Ma]rYH^w^ a NXb2M X¶ %$ % h xtNXb2M g jUPz2MHTY gUq b NXb2M¸ (J¸ % n ½M{i[f^w^DY qUh udNtNXb2M gUh NrYJNr[ hcg Re(a+bi) = a x h TNXb2MTrM YH^ou2YDTN s Y gUq x h T=NXb2Ma[eP&Y k [ g YDT l uUYHTWN n ½M|{i[w^f^YH^fV h + bi) = b Y q2h ucNiNXb2M] hcg cM g Nr[ hcg+Im(a h xjUV?[ gUk z x h T|Y] h PQuc^`MX»dYHTr[fYzd^`M @ YDV h uUu h VmM q N h x s {bd[`]?b{M!N l u2[e]_Y ^f^ l TrM VmM T~cMix h TYTrM YH^cdYHTr[fYzd^`M K n h { qUh {M q MJR g M h uM TWY Nr[ hcg V={i[wNXbQ] h P&u2^eM? g jUPQzUM TWV UT l N h ]rYH^e]_j2^eYJN_M MHYD]?b h xUNXb2Mix h ^w^ h {i[ gUk z2M x h TrM k h [ gUkihcg N h VmMHM|b h {4NXb2M l YHTrM q2hcg M n

+H

H

.H

+H

A n n (1 + 2i) + (−3 + 4i) n (1 + 2i) × (−3 + 4i) n

n (1 + 2i) − (−3 + 4i) n

I n n (1 + 2i) ÷ (−3 + 4i)

ª uM TWY Nr[ hcg V={i[wNXbQ] h P&u2^eM? g jUPQzUM TWV!]_Y g 2 z M q M R g M q [ g Nrj2[\Nr[\cMJ^ l jUVX[ gUk XN b2M ^eYO{|V h xYHTr[\NXbcPMONr[`]»Y gUq Y ^ k MHz2TWY n ® h TQ[ g VXNWY g _] M s w[ NVmMHMHP&V»^ h2k [`]rYH^!N h q M R g M Y q2q [\Nr[ hcg z l (a + bi) + (c + di) = a + bi + c + di = (a + c) + (b + d)i

NXbcY N[eV s ¥`jUVXNY q2q XN b2M'TrM YH^Y gUq e[ P&Y k [ g YDT l 2 u YDTNrVN h»k MJNNXb2M'TrM YH^oY gUq [fPQY k [ g YHT l uUYHTWNWV h xNXb2MVXjUP n jzdN_TWYD]rNr[ hcg f[ VvV?[fP&[w^eYHT n

bdjUV s NXb2MY g V_{M TWVN h NXb2MaRUTWVXN=N{ h M?DM Tr]_[eVMHVYHTrM * A n n n (1 + 2i) − (−3 + 4i) = 4 − 2i n (1 + 2i) + (−3 + 4i) = −2 + 6i hq MJR g MP'j2^wNr[fu2^w[`]rY Nr[ hcgts {M g MHM q N h jUVmMNXb2Max`YD]rN=NXbcY N 2 n} x{M = −1 YDV?V?jUPQMNXbcY N{MPQY l N_TrMHYJNNXb2MV l PQz h ^ YHV{M|{ h j2^ q Y gl»h NXib2M T V l PQz h ^oY gUq jUVmM h jUTy{MJ^f^eZ gUh { g TWj2^`M V h xtY ^ k MHz2TWY s {iM k MJN (a + bi) × (c + di) = ac + adi + bci + bdi2 n = ac + adi + bci + bd(−1) = (ac − bd) + (ad + bc)i

tbdjUV s XN b2MiY g Vr{MHTyN h NXb2MNXbd[eT q M?DM Tr]_[eVMi[fV n (1 + 2i) × (−3 + 4i) = −11 − 2i n h NrYD]?ZD^eM q [\d[eVX[ hcgts {MbcYOcM=N h TrM PM Pz2MHTob h { q [ c[fV?[ hcg [eVTrMJ^eYJN_M q N h P'j2^` Nr[euc^f[e]_YJNr[ hcgon bdjUV s N h R gUq Ya] h PQuc^`MX g jUPQzUM T ?V j]?baNXbcY N x + yi

x + yi =

1 + 2i s −3 + 4i

{M g MHM q N h R U g q x Y gUq y V h NXbcY N (x + yi) × (−3 + 4i) = 1 + 2i n YH^e]_j2^eYJNr[ gUk NXb2Mi^`MJxfNVX[ q M s { M k OM N tbd[fVvPM ] h P&u2^eM? M djUYH^ s Y

+H

H

(−3x − 4y) + (4x − 3y)i = 1 + 2i

n

Y g V NXbcY NNXb2M^`MJxfNY gUq TW[ k bNVX[ q MHV h xNXb2MM cjUYJNr[ hcg rT M uUTrM VmM g N=NXb2MaVYHPM U j P zUM T NXbcYJN![fV s NXb2MTrM YH^yuUYHTWNWV h xNXb2M&^eM xfN!Y gUq TW[ k bN!V?[ q M V|P'jUVXNzUM g Q X V gUq e[ P'[f^fYDTW^ l x h TNXb2M[eP&Y k [ g YDT l u2YDTNrV n tbdjUV s {M k MONXN b2MM cjUYJNr[ hcg V * −3x − 4y 4x − 3y

s

= 1 n = 2

H

Dh \^ d[ gUks {M k MJN x = 1 s y = − 2 n tbdjUV s NXb2MiY g Vr{MHTyN h NXb2Mix h jUTNXbQMXDMHTr]_[fVmM[eV * 5 5 I n n (1 + 2i) ÷ (−3 + 4i) = 51 − 52 i h { s N_T l N h R gUq YVX[eP&u2^eMHTPMONXb h2q x h T qUh [ gUk NXb2M]rYH^e]_j2^eYJNr[ hcg [ g I n @ v¸ Y ^`]_j2^fY NrM n h {7[fVoNXb2MTrMD]_[fuUT h ]rYH^ h xdY] h P&u2^eM? g jUPz2MHTTrM ^fY NrM q N h NXb2M h TW[ k [ g 1÷(−3+4i) YH^ h {)]_Y g!lch j|jUVMNXbd[fVoN h b2MJ^eu!{i[wNXb q [ c[fV?[ hc g K tb2M g VmMHM[fx lch ja]rY g m M c J M ^ u Y M M W T H Y

^ x _ T ' P 2 j e ^ Y x T \ [ d e [ X V [ w ^ ` [ D Z M { M [ x = T q h k g h h q hcg q q h Y qUq [wNr[ hcgts V?jzN_TWY]rNr[ hcgts Y gUq P'j2^wNr[fu2^w[`]rY Nr[ hcgon MXHNaP hcg NXbS{M{i[w^f^] hcg V?[ q M T|NXb2M] h P&u2^eM? g jUPz2MHT YDV'Y z [\NX= u h [g N [ g NXb2M @ ] h PQuc^`MX K u2^fY g M n ® h Tb h PQMO{ h T_Z s YH^ hcgUk {i b lcah jU+ TMXbi DMHTr]_[fVmM b) Yz h cM (a, s ^ h2h ZY NNXb2MMHYD]?b h xNXb2M|x h jUT=MXHYDP&u2^eMHVY gUq VmMHM!{bcY NNXb2M|Y g Vr{MHTWV^ h2h Z ^f[eZDM!{b2M g u2^ h NWN_M q YHVu h [ g NWV n (a, b)

d

U

¢

#

q * O 8

½MVXNWYDTNNXbd[fV g jUPz2MHT={i[\NXbuUT h zd^`M PQV h xNXb2MRUTWVXNY gUq VmMH] hcgUq T h j gUq V h x NXb2M HGG A Gd p Tr[\Nr[eVb Y NXb2M PQYJNr[`]rYH^ ^ l P&u2[fY qn tbcY g ZV k h N h»p [f^w^ Y gUq V s bcYH[fT h xNXb2M }~g N_M T g YJNr[ hcg Y ^ ^ l P&u2[fY q h P&P&[\NWN_MHM h xNXb2M Y g Y q [eY g¿ Y NXb2M PQJY Nr[`]rYH^ Dh ]_[`MON l s x h T h zNrY [ g [ gUk NXb2MHP.x h T h jUTjUVmM n

jk.k djk.kdj J u #\5W` !dW` #K ` p { z" #%$t9&'& )()(

® [U n g q Y ^f^u h V?[\Nr[\cM[ g _N M k MHTWV m s n s {b2MHTrM n [eV h2qUq s NXbcY NV?Y Nr[fV?x l 1

j

m

qH

4

+

1 n 12

n

=

n tb2M djUY q Tr[w^eYJN_M TWYH^ ABCD [eV[ g V]XTr[ezUM q [ g Y]_[eTr]_^eM n tb2M q [fY k hcg Y ^eV AC Y gUq PQMDMON'YJN n b2MSVX[ q MHV DA s MXHNrM gUq M q z2M lchcgUq A s Y gUq CB s M? N_M gUq M q BD Q zUM lchcgUq Q P D M O M N Y N s n P Áy[ cBM g NXbcYJN s u2T h cMNXbcYJN ∠CAD = 60◦ n CD = CP = DQ

+H

® [U n g q Y ^f^u h V?[\Nr[\cMiTrM YH^V h ^wjcNr[ hcg VN h NXb2M|M djUY rN [ c h g x

x +

{b2MHTrM

btc

6

=

x 2

+

2x

H

3

s

^ N h NXb2MvTrM HY ^ g jUPz2MHT n q M gUh NrMHVNXb2M^fYDT k M VXNo[ g _N M k MHT^`M VVNXbcY gh TM cjUY t

P Y { Y l V]rY g n d{M ^ cMiuUM h uc^`MiYHTrMiVmM Y NrM q YHT h j gUq Y|]_[eTr]_j2^fYDTyNWYzd^`M n }~g b h { Q g l | V?[ u2YH[fTWV h xouUM h uc^`MaM gUk Y k M|[ g bcY gUq VbcYZDM VvV h NXbcY N gUh HY T_P&Vv]XT h V V @ h z 2 h qDl [fV K YH^w^ h {M q N h VbcYZDM|bcY gUq V{i[\NXb'P h TrM!NXbcY gQhcg MiuM TWV hcg Y N c X ] M h g n

zUM|Y|xej g ]rNr[ hcg x`T h P n LHMJN 5 [ g NrM k M TWV fs {bd[`]?b&bcYHV=NXb2Mx h ^f^ h @Y K

{ b2MHTrM + N h +s {i[ gUk 2 u Th u M TWrN [eMHV *

xh T MHYD]?b

s

f (n + 1) > f (n) n∈ + @z K x h TYH^w^ f n + f (m) = f (n) + m + 1 m, n ∈

®[ gUq YH^w^u h V?V?[ezc^eMvdY ^fjM V h x n f (2001)

[fV=NXb2M|VmMON h x gUhcg g M k Y Nr[ cM

+

+n

! { z 1$&1D $D' & )()(

x NXb2MacMHTNr[`]_MHV h xY g YD]_jcN_MH?Y gUk `^ M q N_Tr[fY gUk ^eM n b2MQYH^\Nr[wNrj q M'x`T h P hcg M h y ABC PMHMJNWVaNXb2M h uUu h V?[\N_MVX[ q MYJN n ®T h P D s uUMHTruM gUq [e]_j2^eYHTWV DE Y gUq DF YDTrM D q TWYO{ g N h NXb2M h NXb2M TN{ h V?[ q M V n £UT h cMNXbcYJNaNXb2M^`M gUk NXb h x EF e[ VNXb2MVYHPM {bd[`]?b2MmcM TycMHTN_MX[eV]?b h VM gon

j

]?bcYH[fTWV n b2MHTrMYHTrM ] hcg xBMHTrM g ]XMbcY ^f^bcYHVYT h j gUq NrYDzc^eM{i[wNXb q MJ^`M k Y NrMHVN h n[fV Tb2MHTVmM Y NYDT_zc[\N_n B x H M r T M X ] M t 2 b M R W T _ V N M e ^ M J Y _ N | M ? ] b m V M V d b W T D Y TW[f^ lon b2MHTrMH hcg g n q k h2h h X N 2 b M M e ^ M J Y _ N | M X V w [ W N V c u e ^ D Y X ] M = V N X N 2 b | M r T [ b N x X N 2 b M s h k h q M ^eM k YJN_M s x h T q k (k + 1)@ uUYHTWNr[e]_j2^eYHT NXb2kMiVMD] kN NXb2M!RTWV_N K| J M ` ^ M Y r N i M X V w [ W N V ? M y N n }~g s hcgUq&q k g h n h 1 ]?bc≤ YH[fT=k]r≤ Y g nzUM −h 1]X]_jUuc[`M q z l P h TrMNXbcY ghcg M q M ^eM k YJN_M n x h T{bd[`]?baNXbd[fV[fVu h V?V?[ezc^eM n ®[ gUq NXb2M|VMJN h xUdY ^fjM V n n ¦ NXb2M] YHxfNrMHT

H

n £UT h cM!NXbcY NNXb2MiVmM cjM g ]XM q M R g M q z l y = 1 Y gUq 0 yn+1 =

1 2

] hcg VX[eV_NrV hcg ^ lQh x[ g NrM k M TWV n

3yn +

q

2 −4 5yn

@xh T K n≥0

YDTrM V?ub2M TrMHV h xUj g [\NTWY q [wjUV=YHT_TWY gUk M q [ g V?uUYD]XM s s n jUu2u h VMvNXbcYJN 1 N . .j.]?b2B NMXHY]rNr^N N{ V h NXbcY NMHYD]?bQVuUb2MHBTrMi M ! V h l hh NXb2M TWVvM? N_M T g Y ^f^ lon LHMJN P zUMYau h [ g N c j r N X V [ & M H Y w ^ ^ X N 2 b H M V & M V U u 2 b H M r T M V Y e ^ J M N X N 2 b M u h [ g NrV h x] hcg NrYD]rN!zUM h q s gUq s s n CK 1 [eV . . .M CN_N @ tb2M^`M gUk NXb h xtNXb2MNWY gUk M g Nx`T h P N h NXb2N MaV?ub2M TrM M q q U g h P B 1≤i≤N z l n £UT h cMNXb2Miu2T h2q j]rN h xNXb2M cjUY g Nr[wNr[eMHV t [eV gUh NiP h TrM!NXbcY g NXb2MiuUT h2q j]rN h x t i i NXb2M q [fVXNWY g ]_MHV n PC

_H

i

MXHN){M k [ cMNXb2M u2T h zc^eMHP&V¿x`T h P NXb2M h TrMHY g Y NXb2M ^ l PQuc[eY q {|TW[wNWNrM gi¦ u2Tr[w^ A EdA I s HGG nt¦=k YH[ g P l NXbc15 Y g ZV k h N h&p [f^w^ Y U g q b M h xNXb2M }~g N_M T g YJNr[ hcg Y ^ ^ l P&u2[fY q h P&P&[\NWN_MHM h xNXb2M Y g Y q [eY g¿ Y NX2 _ ] ` [ O M N x T z r N Y [ [ X N 2 b H M V | M 2 u T c z e ^ H M & P V x T U j T U j V M Dh l s h h g gUk h h h n

15

Zu ` d W" ` " !>= # K ` p k$ $'

PQYJNr[`]rYH^ V s bcYH[fT PQYJNr[`]rYH^

n ® h TY|uUTW[ePQM p h xNXb2Mx h T_P p

12k + 1

s

= {(a, b) | a b ∈

Y gUq p

s

p

s

s

s

s

= {0 1 2 . . . p − 1}

+H

p - (4a3 + 27b2 )}

s ^`MON

n

® h T s (a0 , b0 ) ∈ s {MvV?Y l XN bcYJN (a, b) Y gUq (a0 , b0 ) YHTrM!M dj2[\dY ^`M g N[wxcNXb2M TrM b) [eVY (a, ? VX j ]?bNXbcYJN n gUhcg M T h M ^eMHPQM g N pc ∈ 0 4 Y gUq p | (b0 − bc6 ) ®[ gUq NXb2M|P&YOH[eP&YH^ g jUPQzUM T h x p[ g M cj2[ dYH^eM g N=pMJ^`|M (a PM g −NrV=ac [ g ) n

H

p

C

j

n ®[ gUq Y ^f^xej g ]rNr[ hcg V f : → V?Y Nr[fV?x l [ gUk f (x − y) = f (x) + xy + f (y) x h T=MmcM T l Y gUq MmcMHT l s {b2M TrM e[ VNXb2MiVmMON h x YH^w^TrMHY ^ x∈ y ∈ {f (x) | x ∈ } g jUPz2MHTWV n

n tb2Mix h w^ ^ h {i[ gUk `x Y]rNWVYHTrMZ gUh { g [ g Y|PQYJNXb2MHP&Y Nr[e]_V] hcg N_M VXN * @ Y K b2M g jUPQU z M T h

x uUT h zd^`M PQVN_M VXNrM q {|YDV n n≥4 @ z K dY]?b'u2T h zc^eMHP¯{|YDVV h ^\cM q z l M?HYD]rNr^ l x h jUT=] hcg N_M VXNWY g NWV n @]K ® h T MHYD]?bu2YH[fT h x u2T h zc^eMHP&V s NXb2MHTrM[eV'M?HYD]rNr^ lhcg M] hcg rN MHV_NrY g Ni{b h z h NX' b uUT h zd^`M PQV n j Y ^N h ¦ VVXjUP&[ gUk NXb2M g jUPz2MHT h xo] hcg N_M VXNWY g NWV[eV k TrM Y NrMHTNXbcY gh T=M cU s R x h T={bd[`]?bNXb2M TrM&Y ^\{|Y l VM?H[eV_NrV!YQ] hcg NrHM V_NrY g N4n {b h P&[ g [eP'jUPdY ^fjM h x YH^w^2NXb2M|uUT h zd^`M PQV n n

H

$k $'

V h ^ cM q gUq XN b2M V h ^ cM q

Y gUq x h Tu h VX[wNr[ cMTrM YH^ n ® h T s ^eMJN a1 +a2 +· · ·+an {b2MHTTrM =NXb2bM 1 b2jU·PQ · zU · bM nTWV j Pz2MHTWnV ≥ s3 s S = YDTrM|uUY [eT¹{i[eVM g U s s s s s s g a a . . . an b1 b2 . . . b n bi q [fVXNr[ g ]rN n 1 2 @ Y K ®D[ gUq NXb2M g jUPQzUM T h x q [fVXNr[ g ]rNTrMHY ^ ?MHT h M V h xNXb2M|u h ^ lcgUh P&[fYH^ f (x) = (x − b1 )(x − b2 ) · · · (x − bn )

H

@z K U £ T h cMNXb2M[ g M cjUY ^f[\N l

l

1 n−1

n X

j=1

1−

aj S

bj

n X

j=1

aj x − bj

n

 1 n−1 n X T aj  n >  

S

j=1

bj

z2M»Y g YD]_jcN_MN_TW[eY gUk ^`M s Y gUq ^`MON zUM[wNWV']_[fTr]_jUP]_[fTr]_^`M n LHMONiNXb2M n LHMJN u M T_uUM gUq ABC [`]_j2^fYDT^w[ g Mix`T h P Nh PQMDMON YJN O n LHMJN zUM|Yu h [ g N hcg s Y gUq P O£UT cM ^`MON z2M&NXb2MQx h2h N h xNXb2MAuM T_uUBC M gUq [`]_j2^fYDT!O ^f[ g MQx`D Th P N h NXb2MQ^w[ g M n h Q[fx P NXbcY N [eV hcg NXb2M h jcNrVX[ q M h x Y gUq NXb2M g AB YHTrM s s s O 2∠QP B = ∠P BC D P Q ] h ^w^f[ g MHYHQT n LHMON zUM!NXb2M 2 [ g [ g ]XTrMHYHV?[ gUk u Tr[fPM|] h j g Nr[ gUk x`T h PNXb2M|V?PQY f^ ^eMHV_Nu2Tr[fPM 2 M T n ®ph n T M?HYHPQucn ^`M s s s s n p1 = 2 p 2 = 3 p 3 = 5 . . . @ Y K ® h TY k [ cM g T V?Y Nr[fV?x l [ gUk s ^eMJN r z2MNXb2M|V?PQY ^f^eMHV_N[ g NrM k M n ≥ 10

n 6 h Tq

2 ≤ r ≤ n−2

s

n − 4 + 1 < pr

s

Y gUq»q M R g M xh T £ T h cMNXbcYJN s s s n U ) YJ− NXb2M TrMMXD[fVXNWV Nss = (sp1 p2 · s · V?· jp]?r−1 baNXbc N gU1hcg M h sx = 1s 2s . . . s pr q [\d[ q HM V n j 1 ≤ j ≤ pr p1 p2 . . . p n Nj @ z K VX[ gUk NXb2MiTrMHVXj2^wN h x @ Y K s R gUq YH^w^u h VX[wNr[ cM[ g NrM k M TWV x h T{d b [e]?b p2m+1

< p 1 p2 · · · p m

n

m

C G ½MabcYOcMTrMH]XMJ[\cM q a Y V h ^fjcNr[ hcg N h £UT h d z ^`M P h x NXb2M DGDGG t ¨ jUV?V?[fY g JY NXb2MD PQYJNr[`]rYH^ ^ l PQuc[eY q DGDGd * I {bd[e]?bYO h [ q VNXb2M!]_Y ^`]_j2^wjUVyjUVmM q [ g NXb2MxBM Y NrU j TrM q V h ^wjcNr[ hcg HGG * C EDCª n

l

£ T h cM!NXbcY N n U 1

° $ e¸

p

1 + x2

+p

1 1+

y2

≤ p

2 1 + xy

xh T s n 0<x y≤1

!² !²$ ° O¶ND² o°O² D ¸ ¶

;

t¶¶ D¸ ½M uUT h cM7NXb2M7[ g M djUYH^w[wN l j U V NXbcYJN g q MHTNXb2M)VX^f[ k bNr^ l {MHYDZDMHT] hcgUq [\Nr[ hcg Y gUq s n x y ≥® 0T xy ≤ 1 h x s y ≥ 0 s ^eMJN

+H

F (x, y) =

tb2M g F (x, y) = = =

{b2MHTrM G(x, y)

tbdjUV s

F (x, y) =

p

2

1 + xy

p

1 + xy

!

−p

1 1 + x2

−p

1 1 + y2

!

n

p

p p p 1 + x2 − 1 + xy 1 + y 2 − 1 + xy p p + 1 + y2 1 + x2 y(y − x) x(x − y) p p p + p 2 2 2 1 + x + 1 + xy 1 + x 1 + y + 1 + xy 1 + y 2 (x − y)G(x, y) s p p p p 2 2 2 2 1 + x + 1 + xy 1 + x 1 + y + 1 + xy 1 + y

p p = x 1 + y 2 + 1 + xy 1 + y 2 p p − y 1 + x2 + 1 + xy 1 + x2 p p p = x − y − xy(x − y) + 1 + xy x 1 + y 2 − y 1 + x2 p 1 + xy(x2 − y 2 ) p = x − y − xy(x − y) + p x 1 + y 2 + y 1 + x2 ! p 1 + xy(x + y) n p = (x − y) 1 − xy + p 2 2 x 1+y +y 1+x (x − y)

1+

x2

+

p

2

! p 1 + xy(x + y) p 1 − xy + p x 1 + y 2 + y 1 + x2

p p p 1 + xy 1 + x2 1 + y 2 + 1 + xy 1 + y 2

} N[fV gUh {-Mmd[ q M g NNXbcY N F (x, y) ≥ 0 w[ x xy ≤ 1 n b2M q M V?[fTrM q TrMHVXj2^wNx h ^w^ h {|V n

n

CA MXHN{MTrMONrjUT ^f[fVXNrM q x h TNXb2M HGGDG k [\cM g [ g HGG * A N h ÁM h T k MaDdY k M ^ h u

g N h TrMHY q M TWV V h ^wjcNr[ hcg V=N }~g _N M T g YJNr[ hcg Y ^ ^ l PQuc[eY q E HA ª Y gUq ] hcg Nr[ g jM q [ g h j2^ h V sH¦ NXb2M g V s ÁyTrMDMH]XMix h

u T h zd^`M PQVuUT h u h VmM q Y gUq Vb h TWNX h U [ g h TrM Y n b2MHVM&u2T h zc^eMHP&Vv{MHTrM HGG * d CJE DG nvl NXbcY g ZV k h TuUT h c[ q [ gUk jUV{i[wNXbV h ^wjcNr[ hcg V n

n @ % K MON_M T_P'[ g M&Y ^f^[ g NrM k M TWV n ≥ 2 VXj]?bNXbcY Nx h TYH^w^o[ g N_M k MHTWV a Y gUq b TrM f^ Y Nr[ cM ^ l u2Tr[fPM!N h s [wxtY gUqQhcg ^ l [fx n n a ≡ b (mod n) ab ≡ 1 (mod n) &½b2M g NXbd[fVuUT h zd^`M P{|YDV k [\cM g [ g HGG * AHª s NXb2M] hcgUk TrjM g ]XM {|YHVVXNWY NrM q [ g ] h T_TrMH]rNr^ l YDV n

$

ab ≡ 1 (mod n)

° $ e¸

%'%$w¸¹¶

ab ≡ −1 (mod n)

y¶

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K

?¶H¶ "_¶

® h TY gl [ g NrM k M T s NXb2M k [ cM g ] hcgUq [\Nr[ hcg [fVM dj2[\dY ^`M g NN h n≥2 @ } K ® h TMmcMHT l [ g N_M k MHT TrMJ^eYJNr[\cMJ^ l u2Tr[fPMN h s 2 n a n a ≡ 1 (mod n) ~} gUq MHM q s NXb2M k [\cM g ] hcgUq [wNr[ hcg [eP&u2^w[`M V @ } K s Y gUq ] hcg cM TWVmMJ^ l s [fx @ } K b h ^ q V s NXb2M g x h T Y gl [ g N_M k MHTWV Y gUq TrM ^fY Nr[ cM ^ l uUTW[ePQMN h s {M|bcYJcM a b n a ≡ b (mod n)

Y gUq LHMJN

ab ≡ 1 (mod n)

8?Dj $ E hcgUq [\Nr[ hcg

H

@

==⇒ ==⇒ ==⇒

ab ≡ a2 ≡ 1 (mod n)

s

a2 ≡ ab (mod n) n a ≡ b (mod n)

K z2MNXb2Mix`YD]rN h WT [ XYJNr[ hcg&h x [ g N h u2Tr[fPM V n n

n = pe11 pe22 . . . pe` ` ei ≥ 1

@ } K [eVM cj2[ dYH^eM g NN h @ }W} K ® h ToMHYD]?b s x h TMmcMHT l [ g NrM k M T TrMJ^eYJNr[\cMJ^ l uUTW[ePQMN h s n i a pi a2 ≡ 1 (mod pei i ) t¶ $ J¸ *| jUu2u h VM&NXbcYJN @ } K b h ^ q V n ½MQuUT h cM @ }W} K n} N![eVM gUh j k bN h ] hcg VX[ q MHT n jUuUu h VmMQNXbcY N a [fVaTrM ^fY Nr[ cM ^ l u2Tr[fPMQN h p nÀ} xV h PQM h xNXb2M uUTW[ePQMHV i s = 1s s q [ c[ q M a s V?Y l p s p s . . . s 1p q [\d[ q M a Y gUq p2 p3 . . . p ` N [ c [ M s s q2h|gUh q q a s NXb2M g a + pe1 p 2 p 3 · · · p [fVkTrM ^fY Nr[ cM ^ l u2Tr[fPM p k+1 k+2 ` 1 K s `{M|bcYJcM N h k+1npc. .l . @ } p n

A

p jcN{MÀYH^fV h

(a + pe11 pk+1 pk+2 · · · p` )2 ≡ 1 (mod n)

bcYOcM

n

n

(a + pe11 pk+1 pk+2 · · · p` )2 ≡ a2 (mod pe11 )

e1 a2 ≡ 1 h (mod {¯VXjUuUpu 1h Vm)MiNXbcY N @ }W} K b h ^ q V s Y gUq ^eMJN

n

iM g _] M s

zUMaTrMJ^eYJNr[\cMJ^ l uUTW[ePQM|N h n b2M g aK {MbcYOcM n @ e[ VTrM f^ Y Nr[ cM ^ l u2Tr[fPMN h M Y]?b

d b U j V z xh T n s l }W} s a pi a2 ≡ 1 (mod pei i ) Y gl s e[ P&u2^ l [ gUk NXbcY N 2 n i a ≡ 1 (mod pe11 · · · pe` ` ) j UuUu h VmM gUh { NXbcY N ? V Y r N f [ V?RMHVNXb2M k [ cM g ] hcgUq [\Nr[ hcgon 1 e2 n @ =K pe1{ p · M · pe` ` 2bcYO· c X N 2 b M z Q M NXbcYJNv[eV s x} s gts l }W} s n 2 22[ ≡ 1M T (mod pei i ) ei ≤ 3 hcg cM TWVmpMJ^il = x T Y r N M s a2 ≡ 1 (mod 8) h gl&h2qUq g k n a } x p > 2 s NXb2M gts z l @ }W} K s {MibcYJcM 22 ≡ 1 (mod pei ) NXbcY N[eV s p = 3 Y gUq i n hcg cMHTWVM ^ lts a2 ≡ 1 (mod 3) x h TY gl [ g N_M k MHT ai TrMJ^eYJNr[\cMJ^ l u2Tri[fPMvN h 3 n ei = 1tbdjUV s n V?Y Nr[fV?RMHVNXb2M k [ cM g ] hcgUq [\Nr[ hcg [fx Y gUqQhcg ^ l [wx n q [ c[ q M V 23 · 3 n b2M c j _ N ] P M V H Y TrM h h s s s s s s n n ∈ {2 3 4 6 8 12 24}

C

H

n @ ´ d¸ K £UT h cM7NXbcY NNXb2MVmMON h x&u h VX[wNr[ cM)[ g N_M k MHTWV{bd[e]?b-]_Y g2gUh NzUM TrMHu2TrMHVM g NrM q YDVY|V?jUP h x q [eV_Nr[ g ]rNuUMHTWxBMD]rNV cjUYHTrMHV=[eV=R g [\N_M n ° $ e¸ %'%$w¸¹¶ y¶ (c¶U· (c¶ $ % %$ U² t¶ ?¶H¶ "_¶

K

u T h uUMHTNr[`M V jUuUu h VmM!{M|bcYOcM|Y|u h VX[wNr[ cM[ g rN M k M T N {i[wNXbaNXb2Mx h f^ ^ h {i[ gUk U Y gUq 2 2 2 2 2 2 s N = a 1 + a2 + · · · + a m

2N = b1 + b2 + · · · + bn

{b2MHTrM YHTrMu h VX[wNr[ cMQ[ g N_M s s s s s x`TWY]rNr[ hcg aV 1 . . . as m b1 s . . . bns [fVYau h {MHT β [\NXa δ M&VbcYH^w^oNXb2M YH^w^ s s aαs /a{i b α /bδ bγs /aβ bγ /b ½ n α β γ δ α 6= β γ 6= δ P >

4N P−2

(2kN +1)2

]rY g zUMTrM uUTrM VmM g N_M q YDVoYVXjUP

k HM TWVVXj]?bNXbcYJN gUhcg M h xNXb2M h x 2 @ [ g ]_^fj q [ gUk 20 = 1K x h T g u2T h cM'NXbcYJNMmcMHT l [ g NrM k M T

H

h x q [fVXNr[ g ]rNtuM TrxBMH]rNV cjUYHTrMHV n

Á k=0 y [ cM g Y gl VXj]?b|[ g NrM k M T s {M{|TW[wNrM [ g XN b2Mx h TrP s {b2M TrM P P P = 4N q + r r−1 n [ g ]XM r ≡ P (2kN + 1)2 (mod 4N ) Y gUq XN b2M|^fY NWNrMHTV?jUP 0 ≤ r ≤ 4N − 1 [eV=^eMHV?VNXbcY g s {MiPQY l {|TW[wNrM k=0 P

P =

r−1 X

(2kN + 1)2 + 4N t

s

k=0

{ M¥`jUVXNyNrYDZDM x h TV h Q P Miu h ?V [\Nr[\cM![ g N_M k MHT s w[ x n} x r = 0 s s {b2M TrM t r≥1 P = 4N t n t=q P 2ui P LHMJN z2MNXb2M|zc[ g YDT l XM u2Y g VX[ hcgQh x n b2M g t= 2 + 22vj +1 t i

j

 r−1 P P P    (2kN + 1)2 + (2ui +1 aα )2 + (2vj +1 bγ )2 k=0 j,γ P = P ui +1 Pi,α vj +1  2 2  (2 a ) + (2 b ) α γ  i,α

H

j,γ

[wx [wx

r≥1 r=0

s n

tbdjUV s ]rY g zUMiTrMHu2TrMHVM g NrM q YDVY|VXjUP h x q [eV_Nr[ g ]rNuM rT xBMH]rN=V cjUYHTrMHV n P } NyrT M PQY [ g V hcg ^ l N h Vb h { NXbcY NNXb2MHTrM!M?H[eV_NrVY!u h V?\[ Nr[\cM[ g NrM k M T N YDVYz h cM n [fVV?j]?b'Y g jUPz2MHT s V?[ g ]XM p jcN 2 2 Y gUq 2 2n

k

29

29 = 2 + 5

58 = 3 + 7

n @ ´ ?¸ K }~g NXb2Muc^eY g M{MvbcYOcM n TrMH]rNrY gUk ^`M V{i[wNXbiu2YDTWY ^f^eM ^VX[ q MHV n tb2MV?[ q M V h x q [fVXNr[ g ]rNTrMH]rNrY gUk ^`M V!^w[`M hcg»q [fVXNr[ g ]rN^w[ g M V n tb2M'z h j gUq YDTW[`M V h xoNXb2MTrMD]rNWY gUk ^eMHV ]_jcNNXb2Muc^eY g Ma[ g N h ] hcg2g MD]rNrM q TrM k [ hcg V n¦ TrM k [ hcg [eVvVY [ q N h zUM d¸ "_¶i[fxo[\NvbcYHV!YJN ^`M YDV_N hcg M h xNXb2McMHTNr[`]_MHV h xoNXb2M TrMH]rNrY gUk ^`M V hcg [\NrViz h j gUq YHT lon tb2M TrMQ]rY g zUM n gUhcg ] hcg cM?TrM k [ hcg V s YHV={M ^w^tYHVTrM k [ hcg V={i[\NXbQP h TrMiNXbcY ghcg Maz h j gUq YHT l ]_jUT~cM n £UT h cMNXbcYJNNXb2MVXjUP h xoNXb2M g jUPQzUM TWV h xNXb2M|cMHTNr[`]_MHV h xyY ^f^ g [`]_MTrM k [ hcg Vv[fVv^eMHV?V NXbcY g n 40n

° $ e ¸ ¶ D¸

zUMJ^ h {

%'%$w¸¹¶ y¶ (d¶

2·

5

(d¶&$ % $ D² t¶

;

X¶D¶ "X¶ D¸!¶

¶

cYD]?b!cMHTN_MX h xcY g [e]XMTrM k [ hcg [eVMJ[wNXb2M T] hcg c MX h T] hcg r] YJcM s DY Vy[ g XN b2MR k jUTrM V @ {b2MHTrM!NXb2M|VmbcY q M q u2YDTN[eVNXb2Mi[ g N_M Tr[ h T h x NXb2M|TrM k [ c h g ] hcg ?V [ q M rT M q K n

C

. .. . .. .. .. .. .. . .... ... .... ... ... ... ... ... .... .... .... .... .... .... .... .... .... .... .

. . .... .... .... .... .

.................................................. .................................. ............................... ..................................... . .................................................................................................... ........................................................................................................................................... .................................

.................................................................................................................................... ....................... ...................... .......................... ....................... .........................

c hcg ]_YOcM h g cMX ® h TY'VX[eP&u2^eM]_^ h VM q ]_jUT~cM ] hcg NrY [ g M q [ g NXb2M'z h j gUq YHT lh xYTrM k [ hcg s {i[w^f^z2M]_Y ^f^eM q Y C % J¸ n ¦ z h j gUq YDT l uUY [eT [eVVY R[ q NXb2Mu2YH[fT @ TrM VuUMD]rNr[ cM ^ l'h jc(R, K n LHMON N h zUM (R, ¹¶ @ C) TrM VuUMD]rNr[ cM ^ l ¸ t¶ K [fx [eV[ g VX[ q M NrVX[ q C) M R C z2M!NXb2MVMJN h x z h j gUq YDT l u2YH[fTWV h x g [`]_MTrM k [ hcg V n LHMON z2M!NXb2MVMJN h x z h C j gUq YDT l B uUY [eTWV[ g {bd[e]?bYHTrM h jcNrMHT s Y gUq ^eMJN zUM|NXb2MVMJN{bdO [e]?bYDTrM[ g2g MHT n b2M g [fV B I NXb2M q [eVr¥ h [ g Nj g [ hcgh x Y gUq n h N_MNXbcY NNXb2MQVMJN [eV gUh N!MHP&ucN lts VX[ g ]_M'BNXb2M O I I hcg ^ l TrM k [ hcg {i[wNXbQ[ g R g [\N_MYHTrMHYa[eVvY g [e]XMTrM k [ hcg Y gUq Y ^f^ h xo[wNWVz h j gUq YHT l ]_jUT~cM V YDTrM[ g2g M T=z h j gUq YDTW[`M V n ® h T=Yz h j gUq YHT l u2YH[fT Y gUq z2M|NXb2M g jUPQzUM TWV h x s ^eMJN ] hcg cMX|cMHTNr[`]_MHVY gUq ] hcg ]_YOcbM= cMHT(R, Nr[`]_MHC) V h x ∈ Bs TrMHV?uMHv]rbNr[\cMJ^ l cs b{bd[`]?bQzUMJ^ hcgUk N h NXb2M R z h j gUq YDT l ]_jUT~cM n cYD]?bcMHTN_MX h xyNXb2M k [ cM g TrMH]rNrY gUk ^`M Va]XTrM Y NrMHVM?HYD]rNr^ lhcg M C ] hcg cMXcMHTN_MX'Y gUqhcg M] hcg ]_YOcMicMHTN_MX n hcg cM TWVmMJ^ l s M Y]?b] hcg ]_YOcMicMHTN_MX h xoY TrM k [ hcg ] h PM V=x`T h P%Y!cMHTN_MX h x hcg M h xNXb2M h Tr[ k [ g Y ^TrMD]rNWY gUk ^eMHV n b2MHTrMJx h TrM s

X

cb = 4n

n

+H

b∈B

® h TY&z h j U x NXb2Max h ^f^ h {i[ gUk M djUY Nr[ hcg g q YHT l u2YH[fT s z h NXbV?[ q M V h M?DuUTrM VVNXb2MV?jUP h xoNXb2MY gUk b^eMH= V!YJ(R, NNXb2C) M|cMHTNr[`]_MHV h x s { b2MHTrMNXb2M'Y gUk ^`MJY NvMHYD]?b C cMHTN_MX[eV]?b h VM g N h z2MNXb2M|Y gUk ^`M[ g NrMHTW[ h TyN h * R ◦

vb · 90 + cb · 270

◦

=

} Nx h ^w^ h {|VNXbcYJN

(

(vb + cb − 2) · 180◦ (vb + cb ) · 360◦ − (vb + cb − 2) · 180◦

vb − c b =

[ g ]_M I [eV gUhcg ?MHP&ucN lts {M|bcYJcM

{b2MHTrM

X

b∈B

(

4 −4

[wx s b∈O w[ x n b∈I

[fx s b∈O f[ x n b∈I

(vb − cb ) ≤ 4(k − 1) − 4 = 4k − 8

s

e[ VNXb2M g jUPz2MHT h xz h j gUq YHT l u2YH[fTWV[ g n ® kh TvM Y]?bz h j gUq YHT l uUY [eT xyY g [`B ]_M'TrM k [ hcg h s YJN^eMHYHVXN hcg M h xNXb2M C) cMHTNr[`]_MHV h x [fV|Y'cM TWNrM? h x hcg (R, M h xy NXb2M k [ cM g TrMH]rNrY gUk ^`M RV n» [ g ]XMMmcMHT l cMHTN_MX C h xNXb2M k [ cM g TrMH]rNrY gUk ^`M V|YDu2uM YDTWVi[ g MXHY]rNr^ l N{ h z h j gUq YHT l uUY [eTWV s {MQVMDM'NXbcYJN n k ≤ 8n ®[ g YH^w^ l s NXb2MiV?jUP h xNXb2M g jUPQzUM TWV h xNXb2M!cM TWNr[e]XM V h x YH^w^ g [`]_M|TrM k [ hcg V=[eV X

b∈B

(vb + cb ) =

X

b∈B

2cb + (vb − cb )

≤ 2 · 4n + 4k − 8 ≤ 40n − 8

n

C I

j

n o¶ $ ®D[ gUq HY ^w^DuUY [eTWV h xDxej g ]rNr[ hcg V f Y gUq g x`T h P4NXb2MVmMON h xTrMHY ^ g jUPz2MHTWV N h w[ NWVm(MJ^fxVXj]?b)NXbcY N x h TYH^w^UTrM YH^ g U j PQzUM TWV f x + g(y) = xf (y) − yf (x) + g(x) Y gUq n

x

y

° $ e ¸ ¶ D¸

%'%$w¸¹¶ y¶ (d¶

2·

5

(d¶&$ % $ D² t¶

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¶

+H

} N[fVMHYHV l N h VmMHMNXbcY N[wx f [fVi] hcg XV NWY g NNXb2M g f (x) = g(x) ≡ 0 n¦ V?V?jUPQM [eV gUhcg ?] hcg V_NrY g N n ¨tM u2^fY]_[ gUk z l [ g NXb2M k [\cM g M djUY rN [ c h gts {M|bcYOcM f x g(x) n f g(x) + g(y) = g(x)f (y) − yf g(x) + g g(x)

NXbcY N

[ g _] MNXb2M|^`MJxfNVX[ q M[fVV l P&PMON_Tr[e][ g x Y gUq y s NXb2MaTW[ k bNV?[ q MP&jUV_Nz2MaYH^fV h s Y U g q {M h zNrY [ g n @~A K

H

g(x)f (y) − yf g(x) + g g(x)

2YDZD[ gUk

= g(y)f (x) − xf g(y) + g g(y)

[ g NXb2M h TW[ k [ g YH^ M cjUYJNr[ hcg {M k MON x=0 f g(y)

*H

@ FK

= ay + b

{b2MHTrM Y gUq n MJNWNr[ gUk y = 0 [ g @ FK k [ cMHV f (b) = b n b { = g(0) @BA K −f djUY Nr[ hcag = {i[\NX(0) b [ c H M V gUh k y=0 @ OK n = ag(x) + bf (x) − bx + g(b)

g g(x)

}~g VmM TWNr[ gUk @ OK Y gUq @ FK [ g N h @BA K s {M h zdNWYH[ g

H

g(x)f (y) + ag(x) + bf (x) = g(y)f (x) + ag(y) + bf (y)

s

h T=M cj2[ dYH^eM g rN ^ t l s

g(x) − b

f (y) + a

=

g(y) − b

f (x) + a

@ IHK s

x h TY f^ ^ Y gUq n ¨txMDr] YH^w^f[ gUyk NXbcY N [fV gUhcg ] hcg XV NWY g N s {M]?b h2h VM VXj]?b!NXbcY N n y0 f (y0 )+a 6= 0 tb2M g {MiVmMHMx`T h P @ IDfK i { [\NXb XN bcY N y = y0

{b2MHTrM

A=

xh T Y f^ ^ x∈

g(y0 ) − b n f (y0 ) + a

g(x) − b = A f (x) + a

LHMJNWNr[ gUk

B = Aa + b

s

s {M|bcYJcM

H

g(x) = Af (x) + B

@ FK s

n}~g VmM TWNr[ gUk NXbd[eV=[ g N h NXb2M h WT [ k [ g YH^ M cjUYJNr[ hcgts {M h zdNWYH[ g n

= xf (y) − yf (x) + Af (x) + B £2jcNWNr[ gUk k [\cM V f x + g(A) = xf (A) + B s {bd[`]?b»Vb h {|VvNXbcYJN f [eVY y = A ^f[ g MHYHTxej g ]rNr[ hcgonpcl @ FK s [eVY ^eV h ^w[ g M YDT n h { s Y q [eTrMH]rN=] h P&u2jcNWY Nr[ hcgal [eM ^ q V g f x + g(y)

f (x) =

c (x − c)) c+1

xh T Y gl ] hcg VXNWY g N n c 6= −1

Y gUq

g(y) = c(y − c)

s

C

$

n ± D¸ ) LHMON O zUMaNXb2M&]_[fTr]_jUP]_M g N_TrM'Y gUq H NXb2M h TNXb h ]XM g N_TrM h xyY g Y]_jcNrM N_Tr[fY gUk( ^eM n £UT h cMNXbcY NNXb2M TrMiMXD[fVXNyu h [ g NrV D s E s Y gUq F hcg VX[ q MHV BC s CA s ABC Y gUq Y gUq TrM VuUMD]rNr[ cM ^ lts VXj]?baNXbcYJN s + DH = OE + EH = OF + F H NXb2Mi^fAB [ g MHV YDTrM|] OD hcg ]_jUTrTrM g N n s s Y gUq

AD BE

CF

; ) <N 0

5

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¶ ?¶H¶ "_¶ c¸ '°d¶F¸ '¸ c º D¸ % ¶ ( ¸~·H ¶ ¸'$)$~¶ ·H¶ X¸

$

........................................................ ................ .

............ .. ......... K ........... A . ......... ........................................................................................................................................

... .... .................. ... ... ........... ...... .............. ..... .. .. ..... .. ...... ... ..... ..... .. ..... ..... .... .... .... .... ........ . . . . . .... . . . . .. ... .... .... .... .. ... .. .. .... ... .. ... .... .. .... .... ..... ........ . . . . . ... ..... . ... ... .. . ... .. . . . . . . . .... .... ... . .... ... .... .. . ... . . . . . ... . . ... .... ... ... ... . .. .. . . . . ... . . . . . ..... ... ... ... ... . .. .. . . ... . . . . . . . .... E . ... ... .. .. . .. . .. . . . . . . . . . . . . ... ...... .. ... ... ... . .. . .. . . . . . . . . ... .. . ... .. ... ... .... .......... . . .. .. . . . . . . ... . . .. ..... ... ....... ... ... . .... .. . . . . . . . ... . . . . .. F ................... ..... ...... ................... .... .......... ..... .. ... ............... . .. . . . . . . . . . . . . .. . . . .... ... ... .. ............. ... ........ ... .. . . . . . ... . . . . . . . . . .... ... ...... .................. ... ..... .. . . . .... . . . . . . . . . . H . . . . ..... ... .............. .... .................. ... ...O .. ... .. . . . . . ............ ... ... ...... ............. .. ... ... . .. . . . . . . . . . . . . . . . . . ............ .. ...... ... . ....... .... ... . . . . . . . . . . . . . . . . . . . . . . . . . ..... .. ... ... ... ............... . ... . . ..... ... .... ............ .... .. ....... .... ... ... ... ... ..... ............. .. .. .. ....... ... .. .... ... .... .... . .. .. ............ .. ...... ... ... .. .. ....... ..... ....................... ........ ... .... 0 ... ...... ... . .. .. ...... . . . . . . . . . . . . . . ..... . . . D ........ ........................................................................................................................................................................................................................................................ . . . ... ... .. .. C B ........ A0 ....... ... D .... .... ... . .. ... ... ... . . . . ... ... .... . .. ... . . . . . . . .... . ... ... . .... .... .. ... ... ..... ... ..... ... ... .... .... .... ..... ... ... ..... ...... ..... .... ... ... ........... ....... ..... ......................................................................................................................................... ........ ... ..... ........ H 0 ............................................................................................. K 0

q

q

q

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x X N s h g gUq N hcghcg q h b2Mi^f[ g M AB s Y gUq M Ni BNi+1 M1 1 4 Y gUq X N 2 b M 2 u u X V w [ r N i M ? V [ M hcg h h q n N4 tb2MY gUk ^`M V Y gUq YHTrMM Y]?b k TrMHYJN_M TÀNXbcY g n Mi AM NYHiTrBN 60◦ tb2M TrM x h TrM s Y gUq i+1 MM Yi+1 ]?b^eMHV?VNXbcY g ◦ n |M g ]XM s ∠M AM ∠N BN 180 1

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b h2h VMu h [ g rN V s V s X s Y DY T_zc[\N_TWYDTW[f^ lhcg NXb2MTWY l V M A s N B s AM s BN u T h2q j]XM q zUM lchcgUq NXb2U U M djU Y q Tr[w^eJY N_M TWHY ^ n ¨tMD]rYH^w^ NXbcYJN n ∠M AB + ∠N BA < 180◦ tbd[fV[fPQuc^f[eMHVNXbcYJN ∠U AB + ∠ABV > 180◦

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P

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} N[eVM gUh j k bN h ] hcg V?[ q M TyNXb2M!^f[ g MHV CA Y gUq CB n LHMON CA PQMDMONyNXb2M|]_[fTr]_^`M Y k Y [ g Y N s Y gUq ^`MON PMHMJNNXb2M|]_[fTr]_^`M Y k YH[ g Y N ny [ g ]_M (AXY CB (BXY ) z NXb2Ma£ Q{MHT x2NXb2CM ^f[eMHV hcg ) s {M!bcP YJcM s l h h · CP = CX · CY = CB · CQ £ h [ g N!tXY b2M h TrMHP n tbdjUVCA s NXb2M&N_Tr[fY gUk ^eMHV CAB Y gUq CQP YHTrMVX[eP'[f^fYDT s [eP&u2^ l [ gUk NXbcY N n ∠CAB = ∠CQP TWYJ{¿NXb2M^f[ g M CR NrY gUk M g N=N h Γ s {i[wNXb R ^ l [ gUkhcg NXb2MaV?YDPQMaV?[ q M h xo^w[ g M YHV n b2M g s [fPQuc^ l [ gUk NXbcY N CR k P Q n XY = ∠CAB =YJN ∠CQP hcg VX[ q MHB TNXb2MN{ h b ∠BCR P X N 2 b J M r N e [ H M V _ ] M _ N r T M Y h h s g q gUq s TrM VuUMD]rNr[ cM ^ lts NXbcYJNyP&YDu Γ ANXb2MJNr[eMHB N X N 2 b = M N { \ [ c M _ ] f [ r T _ ] ` ^ M V M D x X N 2 b M b P V _ N TWY g V?x h TrPQVNXb2M^f[ g M Nh hcg h hk g n g h h h NXb2Mi^f[ g MNrY gUk M g NY N N h&hcg M h xNXb2MHVM]_[fTr]_^`M V s Y gUq NXb2M h NXb2MHT=b h P h NXb2MJN l CR NrYDZDMHV N h NXb2M^f[ g M|NrY gUk PM g NY N N h NXb2M h NXb2M T]_[eTr]_^eM np h NXb&NXb2M VmM[fPQY k Ma^w[ g M VvYDTrM CR uUYHTWYH^w^`MJ^cN h b2M g ]XM s NXb2M l Q] h [ g ]_[ q M{i[wNXb|NXb2M^w[ g M s {bd[`]?b[eV s NXb2M TrM x h TrM s Y PQ ] h P&P hcg NWY CR M N N X N b m V | M _ ] f [ r T _ ] ` ^ M V gUk g h h n

CDC

n (´

X¸ )

LHMON zUMY] hcg cMXu h ^ lck hcgts £ Th c MNXbcYJN n U . An^ [fxvM Y]?bcMHTN_MX [fV&] l ]_^wA [`]a1 A [wx2Y .gU.qÀ ]_Y g n 2 z ≥ MYH4VVX[ kJg M q Y uUY [eT c h g l A1 A2 . . .x A Ai h TrMHnY ^ g jUPQzUM TWVV h NXbcY N

(bi , ci )

@~A K

Ai Aj = b j c i − b i c j

xh T Y f^ ^ Y gUq {i[\NXb n i j 1≤i<j≤n ° $ e¸ %'%$w¸¹¶ y¶ (c¶U· (c¶ $ % %$ U² t¶

K

?¶H¶ "_¶

jUuUu h VmMNXbcY N=NXb2MHTrMYHTrMuUY [eTWV (b , c ) h xTrMHY ^ g jUPz2MHTWV!VYJNr[eVXx l [ gUk @~A K na}~g T M ! T xh T N h z2M] l ]_^w[`] s [\iNVXj i ]_MHVNXbcYJN s x h TM Y]?b h q s s ns A A . . . A =] 4 Vm.MH.]_jc . Nr[\c 1 2 n NXb2M'u h [ g NWV ^f[eM hcg Y&]_[eTr]_^eM n b2MHVMu h [ g NrV!YDTrMNXib2M& M s s s c h g cMHTNr[`]_MHV h xA Ya1] hcg Ac2M? A cjU3 Y q ATWi[f^fY NrMHTWY ^ X N 2 b a M ] c H M W T V M x c £ N e ^ H M P npcl hcg h h lt V tb2M h TrMHP s Y|V?j ]_[`M g N=] hcgUq [wNr[ hcg x h A T 1 A2 A3 Ai N h zUM|] l ]_^w[`][eVNXbcYJN A A A A

H

1

}~g d[`MF{

2

3

i

n

A1 A2 · A 3 Ai + A 2 A3 · A 1 Ai = A 1 A3 · A 2 Ai

h x @BA K s NXbd[fVN_TWY g VX^eYJN_M V[ g N h

(b2 c1 − b1 c2 )(bi c3 − b3 ci ) + (b3 c2 − b2 c3 )(bi c1 − b1 ci )

H

= (b3 c1 − b1 c3 )(bi c2 − b2 ci )

n

M Tr[wx l NXbcYJNNXb2Mi^fYDV_N=M cjUY ^f[\N l [eVY g [ q M g Nr[\N ln } N[fVMHYHV l N h c hcg cM TWmV MJ^ l s ^eMJN zUMÀ] l ]_^w[`] n ® h T s s s s VmMON A1 A2 . . . A n i = 1 2 ... n @ Y M R w [ r N [ ? V j M X V r N M z c £ N e ^ H M P V t 2 b M h TrMHP K n Y gUq A A q g hcg k k q l h lt b i = A 2 Ai ci = 1 i 2 zUM TWV ½M gUh {-]?b2MH]?Z|NXbcYJNNXb2M Ag 1 A jUPQ V?Y Nr[fV?x l @BA K n s bi ci x} X N 2 b M s g A s A s A s A YDTrMiNXb2Ma] hcg VMD]_jcNr[ cMcMHTNr[`]_MHV h x 1 NXb2M] l ]_^w[`3] ≤ cjUYi q
H

1

2

i

j

A1 A2 · A i Aj = A 1 Ai · A 2 Aj − A 2 Ai · A 1 Aj

n

[\d[ q [ U g k z h NXb&VX[ q MHVz l A A l [eM ^ q V 1 2 Ai Aj = A 2 Aj ·

A1 Ai A1 A2

− A 2 Ai ·

A1 Aj A1 A2

= b j ci − b i cj

s

YDV q M V?[fTrM qn ½MYHTrM|^`MJxfN{i[\NXbNXb2Ma]_YHVmM V Y gUq n ® h T i = 1 Y gUq 2 ≤ j ≤ n s i=1 i=2 NXb2M q MJR g [\Nr[ hcg V h x s k [\cM bj cj

bj c1 − b1 cj = bj · 0 − (A2 A1 )cj = A1 A2 ·

YDV q M V?[fTrM qn [eP'[f^fYDTW^ l s w[ x Y gUq s NXb2M g i=2 3≤j≤n

A1 Aj s = A 1 Aj A1 A2

b j c 2 − b 2 c j = b j · 1 − 0 · c j = A 2 Aj

tb2M|u2T h2h x [eV] h PQuc^`MON_M n

n

DGDG

n ( c¸ ¹¶

¸ (K

LHMJN s s Y gUq zUM)u h V?[\Nr[\cM7[ g N_M k MHTWVVXj]?b4NXbcYJN V?j]?bNXbcYJNNXb2MNXbcTrMHM £UT h cMaNXbc) Y NNXb2M TraMQMXDb[fVXNWVY&TrcMHY ^ g jUPQzUM T n c jU> 2b > 4a λ g Pz2MHTWV λa s λb s Y gUq λc Y ^f^ bcYOcMNXb2MJ[eTx`TWY]rNr[ hcg YH^u2YDTNrV=[ g NXb2Mi[ g N_M T~dYH^ 1 , 2 n

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YHVx h w^ ^ h {|V * ® h TMHYD]?b[ g rN M k M T s {M q MJR g Mi[ g N_M T~dYH^fV s s n A n B n Cn An =

1 3

n+ a

,

2 3

n+ a

Cn =

s

Bn = 1 3

n+ c

,

n+ c

2 3

1 3

n+ b

,

n+ b

s

2 3

s

® h TY gl TrMHY ^ g jUPQzUM T s NXb2Max`TWY]rNr[ hcg YH^uUYHTWN h x ^w[`M Vv[ g 1 2 [fxY gUqhcg ^ l [wx λ λa , 3 3 NXb2MHTrM[eVY g [ g NrM k M T VXj]?biNXbcYJN x h TV h PQM n 1 2 NXbcY No[eV s i i + 3 < λa ≤ i + 3 λ ∈ Ai i tb2M|NXbcTrMHM g jUPQzUM TWV Y f ^ ^ c b O Y c | M X N 2 b J M e [ T ` x W T Y r ] r N [ H Y t ^ 2 u D Y T r N V [ g 1 2 [wx Y hcg s λb s gUq λc λa , Y gUqhcg ^ l [wx x h TV h PQM s s n tbdjUV[wN{i[f^w^tV?j ]XM|N h 3 uUT 3h cM i j k j ]?∩ NXbcY NNXb2M TrMM?λ H[eV_∈N As i s ∩ BVXj bC NXbckYJN n i j k A ∩ B i j ®[fTWVXN{M h z2VMHT~cM!NXbcYJN s x h TMHYD]?b[ g N_M k ∩MHT Cks NX6=b2MH∅TrM|M?H[eV_NrV=Y g [ g N_M k MHT V?j]?b j k NXbcY N n h VMDMNXbd[fV s {M¥`jUV_NbcYJcM»N hgUh NrM»NXbcYJN|NXb2M^eM gUk NXb h x Bj ∩ Ck 6= ∅ NXb2M»[ g NrMHT~dY ^ [eV n + 32 n + 31 1 s {bd[f^eMNXb2M q [eV_NrY g ]XMz2MJN{MHM g Y gl Bj

−

b

=

b

3b

N{ h ] hcg VMD]_jcNr[ cMa[ g NrMHT~dY ^eV Y gUq [eV (k + 1) + 31 k + 23 2 1 Ck Ck+1 − = < c c 3c 3b @ VX[ g ]_M K n ch >] h 2b P&u2^eMJNrM+NXb2MÀuUT h2h x s [\NQ{i[f^w^zUM7M gUh j k bN h Vmb h {NXbcY NNXb2MHTrM MXD[fVXN V?j]?baNXbcY N ] hcg NrY [ g V [ g NrM k M TWV Y gUq n dj2\[ dY ^`M g Nr^ l s NXb2MHTrM|YDTrMi[ g NrM k M TWV i Ai Bj Y gUq VXj]?ibNXbcYJNj

*H

j

i+ a

+H

1 3

j+ b

≤

1 3

H

Y gUq

j+ b

2 3

® h T s NXb2MHVMi[ g M djUYH^w[wNr[eMHVYHTrMM cj2[ dYH^eM g NN h j≥0

≤

3i + 1 a 3i + 2 ≤ ≤ 3j + 1 b 3j + 2

h { s [eV[ g NXb2Mv[ g N_M T~dYH^ a/b VYJNr[eVXx l [ gUk @ OK n tbd[fVi]_Y g z2M q2hcg M j g [ hcgah x2YN{ h rNrY [f^eM q VmM cjM g ]XM h

H

0,

1 2

= ... ∪

1 , 2 25 26

∪

i+ a

2 3

n

@ FK n

s 1 s Y gUq {M{|Y g NoN h R gUq [ g NrM k M TWV i j≥0 2 [ V_NrY x T X ] M z T r M U u r T M V m M r N [ s h g g s l g gUk 0, 1 YHV!NXb2M 2 x h ^f^ h {|V * xU] hcg VMD]_jcNr[ cM ^ l Y q ¥B Y ]_M g No[ g _N M T~dYH^fV YHVy 0,

1 , 2 13 14

∪

1

7

,

2 8

∪

1 4

,

2 5

∪

4 , 5 10 11

∪ ...

@ FK

tb2M&M gUq Xu h [ g NrV h xM Y]?b[ g N_M T~dYH^oYHTrMx`TWYD]rNr[ hcg V!YHVv[ g @ OK s x h TVXj2[wNWYzd^ l ]?b h VM g Y gUq n @ ® h TNXb2Mi[ g NrMHT~dY ^ 1 2 Y gUq NXb2Mi[ g NrMHT~dY ^eVVX[wNrjUYJN_M q N h NXb2M|^eM xfN h x 1 2 is [4, 5] [ , ] NrYDZDM j Y gUq s 1 s 2 s . . . x h TNXb2M[ g NrMHT~dY ^eV!^ l [ gUk N h4 NX5b2M m x h T i = 0 j = 2 m = 0 Tr[ k bN h x 1 2 s NWYZDM Y gUq xh T s s ... n K m m+1 tb2M g jUPQ[zU4M ,T 5 ] ^f[eMHVi[ g =V h 2PM|− [ g N_1M T~dYH^ ^fj[fVXNr= M q 2[ g @ FK s − Y gU1q b2M g m ]_M[w= N=VYJ1Nr[eVX2R M V NXb2M OK x h TV h PM Y gUq n tbd[fV] h P&u2^eMJNrMHVNXb2M|u2T h2h x n [ g M cjUY ^f[\Nr[`M V @ a/b i j

H

DG A

jk

D

n d¸ ~¶ ¸ ( [ g NrM k (M WT VµN h w[ NWVmMJ^fx

[eV q MJR g M q x`T h P-NXb2M!VMJN h x gUhcg g M k Y Nr[ cM ¦ ex j g ]rNr[ hcg XV )j]?b XN bcYJN s Fx h T MmcM T l¼gUhcg g M k YJNr[\cM [ g N_M k MHT s Y gUnq s s F (4n) = F (2n) + F (n) F (4n + 2) = F (4n) + 1 n £UT h cMNXbcY N s x h T&M Y]?bÀu h V?[\Nr[\cM[ g NrM k M T m s NXb2M FjU(2n + 1) = F (2n) + 1 g Pz2MHT h x [ g NrM k M TWV n {i[\NXb 0 ≤ n < 2m Y gUq F (4n) = F (3n) [fV F (2m+1 ) n

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½MabcYJcM x`T h P.NXb2M|RTWV_N k [ cM g M cjUYJNr[ hcgts Y U { MVMDMNXbcY N [fV g q F (0) FNXb2M j g [ djM ^ lq MON_M T_P' [ g M q =z l 0 NXb2M k [\cM g M djUY Nr[ c V t 2 b M i x f ^ ^ i { [ N W Y d z ` ^ | M V b | { = V h g n h h g k U h dYH^wjMHV h x xh T * F (n) 0 ≤ n ≤ 16 n F (n)

0 0

1 1

2 1

3 2

4 2

H

5 3

6 3

7 4

8 3

9 4

10 11 12 13 14 15 16 4 5 5 6 6 7 5

H

®dT h P.NXb2M|RTWV_N k [ cM g M cjUYJNr[ hcgts z l [ gUq j]rNr[ hcgts {M h zdNWYH[ g s k [eVNXb2M®D[`z hcg Y]_]_[VmM cjM g ]XMFsU(2 M R )g = M q fz k+1 xh T {2 b M TrM q l s s s s ∞ k = 0 1 2 ... {fn }n=1 xh T s n f1 = f 2 = 1 fn = fn−1 + fn−2 n≥3 }~g x`Y]rN s Y k M g MHTWY ^yx h TrP&j2^fY&x h T F (n) P&Y l z2M k [\cM g [ g N_M T_P&V h xNXb2M®[ez hcg YD]X]_[ g jUPz2MHTWV * [fx n bcYHVNXb2Mzc[ g YDT l TrMHu2TrMHVM g NWY Nr[ hcg {b2MHTrM

s

n = εk 2k + εk−1 2k−1 + · · · + ε1 2 + ε0

εi ∈ {0 1}

s

x h T=M Y]?b s NXb2M g i

@~A K n tbd[fV'x h T_P'j2^eYP&Y l z2McMHTW[fRM q VX[eP&u2^ l z l ]?b2MD]?ZH[ gUk NXbcYJNa[\NaVYJNr[eVXRVaNXb2M k [ cM g M djUY Nr[ hcg V q M R g [ gUk n ½M gUh { uUT h cM=FNX(n) bcY N x h TY ^f^ gUhcg g M k Y Nr[ cM[ g NrM k M TWV n ½M F zU (4n) NXb2M g ≤jUPQ M T h x q [ k [\NrV[ g NXb2Mzc[ g YDT l TrM uUTrM VmM g n NrYJNr[ hcg uUT h ]_MDM q z l [ gUq j]rNr[ hcghcg F (3n) s m h x n n ½b2M g m = 1 s {MbcYJcM n = 0 h T n = 1 s [ g z h NXb h x={bd[`]?b+]_YHVmM V s YHV]rY g z2M|VMDM g x`T h PNXb2M!NrYDzc^eM|YDz h cM n F (3n) ={ F^`(4n) MONvY g [ g NrM k M T zUMRDM qn|¦ V h jUT[ gUq j]rNr[ hcg b l u h NXb2MHVX[eV s {M h m ≥ YDV?V?jUPQMNXbcYJN {b22M g MmcMHT bcYDVtxBMO{M TUNXbcY g q [ k [wNWV[ g [wNWVozc[ g YDT l ≤ F (4n) n m TrMHu2TrMHVM g NWY Nr[ hcFgo(3n) n hcg VX[ q MHTY gl u h V?[\Nr[\cMa[ g NrM k M T n {bd[e]?bbcYHViMXHY]rNr^ l m q [ k [\NrV [ g [\NrV=zd[ g YHT l TrM uUTrM VmM g NrYJNr[ hcgon bdjUV s s {b2M TrM 0 ≤ p < 2m−1 npcl n = 2m−1 + p @BA K NXb2Mi[ gUq j]rNr[ hcg b l u h NXb2M V?[fV s VX[ gUk n s {M h zNrY [ g F (3p) ≤ F (4p) @ FK n m+1 F (n) = εk fk+1 + εk−1 fk + · · · + ε1 f2 + ε0 f1

+H

F (4n) = F (2

g NXb2M h NXb2MHTbcY gUq

+ 4p) = fm+2 + F (4p) ≥ fm+2 + F (3p)

@ FK n h 2 u T h ]XMHM q xejUTWNX2 b M T{i[wNXbNXb2M]_Y ^`]_j2^fY Nr[ hcgh x s {M] hcg XV [ q MHTNXbcTrMHM F (3n) ]_YHVmM V * F (3n) = F (3 · 2m−1 + 3p) = F (2m + 2m−1 + 3p)

8 3

n m−1 n b23p t M g NX< b2M 2g jUPQzUM T h x q [ k [wNWV[ g NXb2Mzc[ g YDT l rT MHu2TrMHVM g WN Y Nr[ hcgh x [eVo^`M VVNXbcY g 3p Y X N 2 b M d z [ H Y T r T H M 2 u r T H M V M W N Y r N [ x M V = N _ ] H Y _ T T [ N X N c b Y N s gUq g l g hcgQh 3p qUh g h U l g h h x 3 · 2m−1 n m

DG NrYHTWNr[ gUk x`T h P

8 3

@ FK s {M]rY g YHuUuc^ l B@ A K Y gUq NXb2M g @ OK N h&k MON

F (3n) = fm+1 + fm + F (3p) = fm+2 + F (3p) ≤ F (4n)

n

@ IHK

n m−1 mn b22M g NXb2M
= F 2m + 2m−1 + 2m−1 + (3p − 2m−1 ) = F 2m+1 + (3p − 2m−1 ) n = fm+2 + F (3p) − fm = fm+1 + F (3p)

@ V?[ g ]XM K Y gUq U h Nr[ gUk NXbcY N j V?[ gUk @ FK s {MR gUq NXbcYJN fm+1 < fm+2 m ≥ 2 n F (3n) < F (4n) n 2m < 3p < 3 · 2m−1 @ = 2m + 2m−1K n tb2M g NXb2M g jUPQzUM T h x q [ k [wNWV[ g NXb2Mazd[ g YHT l TrM uUTrM VmM g rN YJNr[ hcgh x [eVM?HYD]rNr^ l 3px Y X N 2 b M c z [ D Y T r T H M 2 u r T H M V M W N Y r N [ x _ ] H Y _ T W T ` [ M V [ N X N c b J Y N s gUq g l g hcgh 3p g h h 3 · 2m−1 n m+1 10 NrYHTWNr[ gUk x`T h P @ FK Y gUq jUVX[ gUk @BA K s {M|]_Y ^`]_j2^fY NrM

8 3

F (3n)

= F 2m + 2m−1 + 2m + (3p − 2m ) = F 2m+1 + 2m−1 + (3p − 2m ) n = fm+2 + fm + F (3p) − fm+1 = fm+2 − fm−1 + F (3p)

h rN [ U Y gUq jUV?[ gUk @ OK s {MR gUq NXbcY N n g k NXbcYJN f F (3n) < F (4n) m−1 > 0 H M D Y ? ] ' b _ ] H Y m V M { i M c b J Y c M m V b { X N c b Y N

d b e [ V ] & P 2 u ^eMJNrMHVNXb2M }~g s h g n h F (3n) ≤ F (4n) [ gUq j]rNr[ hcg uUT h2h x n LHMJN q M gUh NrMiNXb2M|VmMON=] hcg VX[eV_Nr[ gUkih x N h2k MONXb2MHTy{i[wNXb'YH^w^ u h V?[\Nr[\cM[ g NrM k M TWV [ g {b h VmMaE zc[ g YDT l TrM uUTrM VmM g NrYJNr[ hcg NXb2M VvYHTr0M[fV h ^fY NrM q @ PM Y g [ gUk NXbcY N gUh N{ h V 1 YDu2uM YDT[ g Y q ¥BYD]XM g Nu h V?[\Nr[ hcg V K n ® h TMX1HYDP&u2^eM s s 2 ∈ E s zcjcN 3 ∈ n ½M 1 ∈ E / E {i[f^w^ gUh {¿uUT h cMNXbcYJN [wxtY gUq&hcg ^ l [fx n = F (4n) ®[fTWVXNvVXjUuUu h VmMNXFbc(3n) YJN x NXb2M g n ∈ E n } s n n ∈ P' Ej2^wNr[fu2n 0 z @F h {¿YDV?V?jUPQMNXbcY N ^l [= NXbc(3n) YJN[eV s = F[ g (0) zd[ g = YHT l FK s (4n) MHYD]?b n g U g k l n 3 11 @ Y gUq NXb2M^`M Y q [ gUnk >K 0[fVTrMHuc^eYD]XM q z l n hcg VM djM g Nr^ lts F (3n) [fV h zNrY [ g M q 01 1 11 x`T h P z l TrM u2^fY]_[ gUk MHYD]?b [ g @BA K z l n g NXb2M h NXb2MHT fi+2 = fi+3 @~A K bcY gUq s F (n) [eV h zNrY [ g M q x`T h fP i+1 z l TrMHfuci+1 ^eYD]_[ + H M D Y ? ] b [ z l n gUk g F (n) fi+1 fi+3 tbdjUV s F (4n) n = F (4n) Fhc(3n) g cM TWVmMJ^ l s VXjUuUu h VmM.NXbcYJN n [fV-Y gUhcg g M k Y Nr[ cM%[ g NrM k M T VXj]?b NXbcYJN n }x h T s NXb2M g n ∈ E n h {¡YDV?V?jUPQMNXbcY N F (3n) =LHMJFN (4n)z2MQNXb2M n jU =Pz2 0MHT n = 1 x [ \ [ r N | V [ g NXb2M»zd[ g YHT l TrMHu2TrMHVM g NWY Nr[ hcgh x n n g h q k n 2MXHYDP'[ m n u ≥ [ X N 2 b M [ j r ] r N [ U u T x D Y z hcg g gUk gUq hcg h2h h cM s {M h z2VMHT~cMNXbcYJNiNXb2MM djUY Nr[ hcg h ]_]_jUTWV hcg ^ l [ g YDVM A Y gUq»hcg ^ l {b2M g F (3p) = F (4p) n ½M F (3n) = NXFbc(4n) _ N a M H Y f ^ V Y N [ gUh h s g YDVM A s {M'bcYOcM m−1 m−1 m−2 s {bd[e]?b [eP&u2^w[`M V=NXbcY NNXb2M|VmMH] hcgUq zc[ g YDT lQq [ k [wN p h < x 2 [eV n /3 tb2< M TrM 2x h TrM s /2 [ g =YHVm2M A s {MbcYOcM n 0 [wxY gUqhcg ^ l [wx n7pcl TrM R g [ gUk NXb2M[ gUq j]rNr[ hcg uUT h2h x=V?^w[ k bNr^ lts {M n ∈ E p ∈ E ] hcg ]_^wj q MNXbcYJNx h TYH^w^ gUhcg g M k Y Nr[ cMa[ g NrM k M TWV s NXb2M'M djUYH^w[wN l n F (3n) = F (4n) [eP&u2^w[`M VNXbcY N n n∈E

+H

+H

+H

DG ½M'bcYOcM'u2T h cM qh jUT]_^eY [eP%NXbcY N [wxyY gUq»hcg ^ l [fx n n z2∈ ®[ g YH^w^ l s {M!{i[f^w^ uUT h cMiNXbcY N s x h TM Y]?b gUFhcg (3n) g M k = Y Nr[ cFM|(4n) [ g NrM k M T NXb2M g jUP MHT Eh x s m VXj]?baNXbcY N [ g NrM k M TWV n [ g ]_M f m [fV m+1 s {M n ∈ E 0 ≤ n < 2 f ) m+2 m+2 = F (2 {i[f^w^ bcYJcMiuUT h cM q NXb2M q MHVX[eTrM q TrM V?j2^\N n ½Mu2T h ]XMHM q z l [ gUq j]rNr[ hcg&hcg n ® h T m = 0 s NXb2M g jUPQzUM T h x n ∈ E V?j]?b NXbcY N xh T m s NXb2M g jUPz2MHT h x n ∈ E VXj]?b»NXbcYJN m [eV 0 ≤ n < [e2 1 = f 2 {%Rd» mY =[ 1NrM M T n h g g k s Y gUq YDV?V?jUPQM s YDViNXb2M m V 0 ≤ n < 2 m ≥ 2 [ gUq j]rNr[ hcg b l u h NXb2M2V?[f= V s NXfbc3YJNNXb2M g jUPQzUM T h x VXj]?bNXbcY N [fV n P ∈ E NXb2M'[ N_M MH0TWV ≤ n < 2V?kj]?b x h TvY ^f^o[ g NrM k M TWV {i[\NXb n ¦ c h U g k g k f k 0 ≤ k < m n ∈ E NXbck+2 Y N m YDTrM|NXb h VMV?j]?b'NXbcY N m−1 s NXb2M g jUPQzUM T h x {bd[e]?b 0 ≤ n < 2 0 ≤ n < 2 @ jUV?[ gUk NXb2M'[ gUq j]rNr[ hcg b l u h NXb2MHVX[eV K n tb2M&TrMHP&YH[ g [ gUk VXj]?b»NXbcYJN [eV fm+1 nRU∈ EN{ H Y r T Q M X N b V M ? V j ? ] b X N c b J Y N

2 b M W T X V v N h n h zd[ g YHT l 0[ ≤ n x< 2nV?j]?b jUPz2MHT 2M m−1 Y ≤ nNXb2< 2MHm ' P U j X V N 2 z w [ W N V Y | M r T & P H Y [ [ [ \ [ r N V r T H M u2TrMHVM g N q k h gl g s gUq g gUkq k 10 tbdjUV NXb2M jUPQzUM T xVXj]?b Y g jUPz2MHT[ g {bd[e]?b[fV!^`M VnVNXbcY g m−2 [fV n s g h @ jUVX[ gUk NXb2Mv[ gUq E j]rNr[ hcg b l u h NXb2MHVX[eVY k YH2[ g K n tb2MN h NWYH^ g jUPz2MHT h x V?jn]?b|NXbcfYJnN n∈E [fV n tbd[fV] h PQuc^`MON_M VNXb2M|u2T h2h x n 0 ≤ n < 2n f +f =f

j

m+1

m

K

(

m+2

¸ ¹¶ ¸ ( c tb2MNWY gUk M g NrViY N Y gUq n Y]_jcNrM'N_Tr[fY gUk ^eM PQ)MDMONvNXb2MNrY gUk M g NYJN B Y N AY C T ^f[ g MHV Y gUq ABC PQMDMONYJN s Y gUq Q [eVNXb2MP&[ q Xu h AT BC P Y gUq PMHMJNYJN s Y gUq [eVNXb2MiP&[ q Xu h [ g N h x n CA R S BR @ Y K £UT h cMNXbcYJN n

N h XN b2MQ]_[fTr]_jUP]_[fTr]_^`M h x Y g r T H M ? V u H M r ] r N \ [ c J M ^ b2M gUq s ln

[g N U x X N 2 b a M f ^ [ H M V g h AP BU

∠ABQ = ∠BAS

@ z K OM N_M T_P'[ g M s [ g NrMHTrPQV h x TWY Nr[ h V h x VX[ q M!^`M gUk NXbcV s NXb2M!N_TW[eY gUk ^`M Vx h T{bd[e]?bNXbd[eV Y gUk ^`M[eVY|P&YOH[eP'jUP n

° $ e¸

%'%$w¸¹¶

y¶

(c¶U· (c¶ $ % %$ U² t¶

}~g 4ABC s ^`MON α = ∠A s β = ∠B s npcl NXb2MaWN Y gUk M g NX]?b h T q NXb2M h TrMHP s γ M = MJ∠C { k N ∠T BC = ∠T CB = ∠BAC = α n LHMJN q M gUh NrMNXb2M!]_[fTr]_jUPQTWY q [wjUV h x 4ABC n tb2MiR [fV h V]XMJ^`M VN_Tr[fY gUk ^eM k [\cM V BCT

BT =

BC 2 cos α

K

?¶H¶ "_¶

........................................... ....... .............. .............. ......... ........... ........ ....... ....... ..... . . . .... . .............. . ................ ....... ... . . . . . . ......... .... .... ............... .. . . . . . . ... ..... .. .. .. ... .... ..... ... .. ... .. ..... .. .. ..... .... .... .... .... ........ ... .. .. ..... ..... ... . ... .. .. ..... .... ..... ...... . . .. ... . .... . .. ....... .. .. .... . . . . . . . . . . . . . . . . . .... . . . . . . . ... . ....... .. . . ..... .. .................... ... ......... .... .. .......... ..... ..... .... .......... ..... .......................... .. .... ..... ... ..... ... ......... ..... .... ... .. .. ... ... ........... .......................... ..... . . . . . . .. .. . . . . . ... .. . . . ... .. ..... .......... ................... ... . . .... . . . ... ... . . . . . . . . . . ... .. .. . ... . .... ...... .......... ............ ...... ... ... ...... ..... .... .......... ............. .. ...... ... ...... ..... ...................... .. . .. .... .............................................. .................................................................................................................................................................... ... .. . . . . ..... . .. . . . . . . . ...... . ...... ........ .. ........ . .. ......... . . . . .... . ... ......... ... ....... .. ... .......... .. .......... .... ........... ... . ............................................................ ... ... . ... .. . . .. ... .. ... ... .. . . . . ... . . ... .. ... ... ... ... ... .. .. ... .. ..... . . ... ... ... ... ... ... ... ... .. .. ... .. .... . . ... .. ... ... ..... ......... .... .. .. .. ...

A

U

Q

R

= R tan α

pcl NXb2ML YJ{

n

h x [ g HM V|[ g 4ABT s {M bcYJcM AB sin ∠AT B n LHMJN n = ϕ = ∠BAT BT h Nr[ gUk NXbcYJN sin ∠BAT {s M k MJN ∠ABT = α + β sin(α + β + ϕ) AB = BT sin ϕ

n

S

B

P

C

T

V?[ gUk NXb2M|TrMJ^eYJNr[ hcg V s s Y gUq α + β + γ = 180◦ s AB = 2R sin γ BT = R tan α {M h zNrY [ g @BA K n cot ϕ = cot γ + 2 cot α

DGJI MXHN s {M'VMJN Y gUq YHuUuc^ l NXb2M»L YJ{ bd[eV k [\cM V ∠ABQ = ψ n 4BP Q BQ QA

=

Y gUq

sin ϕ

BQ

sin ψ

PQ

h x [ g MHVvN h 4BAQ Y gUq

sin ∠AP B

=

sin(β + ϕ) n sin(β − ψ)

=

sin(β − ψ)

[ g _] M Q f[ V'NXb2MSP'[ q ?u h [ g N h x AP s {MSbcYOcM P Q = QA n tbdjUV s NXb2MTWY Nr[ h V [ g NX2 b MN{ h M cjUYJNr[ hcg VaYz h c M YDTrMM cjUY ^ n tbd[fV k [ cMHV sin ϕ sin(β + ϕ) s=h T = sin ψ sin(β − ψ) sin(β + ϕ) sin(β − ψ) n iM g ]_M s s = sin β cot ϕ + cos β = sin β cot ψ − cos β sin ϕ sin ψ ^`M Y q [ gUk N h n}~g NXbd[eV=^fYDV_N=M cjUYJNr[ hcgts {MjUVmM @~A K N h'k MJN cot ψ = 2 cot β + cot ϕ @ FK n

H

H

H

cot ∠ABQ = cot ψ = 2(cot α + cot β) + cot γ

} N7x h f^ ^ h {|V s z l V l P&PMON_T lts NXbcYJN cot ∠BAS = cot ∠ABQ n @ V?[ g ]XMaNXb2M&] h NWY gUk M g Nxej g ]rNr[ hcg [fVV_N_Tr[e]rNr^ l»q MH]XTrMHYHV?[ gUk ∠ABQ = ∠BAS @Y K n u M [ _ N M ~ T d H Y ^ h g g ◦ ◦ K n ½M|bcYJcM|u2T h cM q (0 , 180 )M dj2[\dY ^`M NN P&[ [eP'[ _[ O Y H e [ ' P [ _ [ e [ & V gUk g h g gUk cot ψ s {bd[`]?b+]_Y g z2M jUV?[ gUk x h TrP&j2^fY @ OK nψ [ g ]XM

+H

cot α + cot β

= = ≥

{MbcYOcM cot ψ

≥

sin β cos α + sin α cos β

cos(α − β) − cos(α + β) 2 sin γ s 1 + cos γ 4 sin γ 1 + cos γ

= 4 tan

H

γ 2

+

=

sin α sin β 2 sin γ cos(α − β) + cos γ

+ cot γ 1 − tan2 2 tan

γ 2

γ 2

=

7 2

tan

γ 2

+

¦ uUuc^ l [ gUk NXb2M ¦ E Á ;}~g M cjUY ^f[\N l s {M h zdNWYH[ g

H

® h TM cjUY ^f[\N l s [wN[fV b2M VmM|] hcgUq [\Nr[ hcg V t NXb2ML YO{ h x [ g M V s h x 4ABC YHTrM

cot ψ ≥

q2hcg M

sin γ

=

sin α sin β 2 sin γ

tb2M g [ g XN b2M

s

7 tan

γ 2

·

1 tan

γ 2

=

1 2 tan

γ 2

n

√ n 7

g HM ]XM VV?YDT l XN bcYJN α = β = 90◦ − γ/2 Y gUq 7 tan γ = 1/ tan γ n 2 k [\cM sin γ = √7/4 Y gUq sin α = sin β = √14/4 n iM g ]_M s z 2l NXb2MY gUk ^eM f[ VYaPQYFD[fP&jUP{b2M g NXb2MaTWYJNr[ h V h xoV?[ q M|^eM gUk NXbcV ψ

a:b:c =

P

√ √ n 2: 2:1

tbcYJN=] h PQuc^`MON_M V NXb2M ¶ y x h TNXbd[eV=[fVVXjM ny M gUq PQM lch jUT g [`]_MiV h ^wjcNr[ hcg V s k M g M TWYH^w[ XYJNr[ hcg V s Y gUq ] hcg NrMHV_NP&Y rN MHTW[eY ^eV n

DG /

F

K6 P Mv6 : ° ¸ ¸ 2¸~¶ $

>

*

Kd

?<6

(

2º t¶ $ (

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%¶e¸ e¸ u jzc^w[eVb2M q z l XN b2MS h j k b N hcg) [ &[ g h P&uUY glts MF{ h T_Z s pcl+ N_MmcM ^eV hcgts c DGDG I M T s OII uUY k MHV s OI n }¹2p G ªdA A I A s bcYHT q ] h c ´y¶ ·F¸¹¶Hº!¶ %$ X ¸¹¶ d¸¹·H¶ X¸ $w¸ ° n

2!#"$&%

; }~g 2º s NXb2M&Y jcNXb h Tvu2T h RU^eMHVvNXb2M'V?[ QVXNrj q M g NWVv{b h TrMHu2TrMHVM g NrM q

aH

NXb2Ma g [wNrM qQ NrYJN_M VYJNNXb2M HGG A }~g NrMHT g Y Nr[ hcg YH^ Y NXb2M PQYJNr[`]rYH^ ^ l PQuc[eY qon}~g NXb2M uUT h ]_MHV?V s b2MSVb2M q VV h PM g MO{[ g V?[ k bN hcg [eP&u h TNrY g N djMHV_Nr[ hcg VQVXj]?b7YHV'{b l ¦ PM Tr[e]_Y g V&YDTrMV h Y x`TWYH[ qÀh xvP&Y NXb2M PQYJNr[`]rV s {b l V h xBMO{ k [eTW^eV'YDTrM[ g h ^\cM q [ g bd[ k b2M TX_^`MmcMJ^ PQYJNXb2MHP&Y Nr[e]_V s Y gUq {b2MJNXb2M T h T gUh NNrY ^`M g N[eV=[ g2g Y NrM n ^eV hcg u h TN_TWY l VNXb2MV?[ VXNrj q M g NWV{b h YHTrM»NXb2MPM Pz2MHTWV h xUM YDP O¦ YDVaMXD]XM ucNr[ hcg YH^w^ l {M ^w^`XT h j gUq M qlch j gUk PQM g {b h VbcYDTrMY g [ g ]XTrM q [`zd^`MuUYHVVX[ hcg x h T^eMHYHT g [ gUkcn tb2M l YHTrM']_jUTW[ h jUVN_MHM g Y k MHTWV{i[wNXbY|{i[ q MTWY gUk M h xyb h zUzd[`M Vvx`T h P {|Y NrMHTu h ^ h Y gUq j2^wNr[fPQYJN_Mx`TW[eVzUMHM|N h uc^eY l [ gUk P'jUV?[e]Y gUq ]?b2MHV?V n tb2Mz h2h Z q2h MHV Y g MXD]XMJ^f^eM g N¥ h z h xiu2TrMHVM g Nr[ gUk Y gUqq MmcM ^ h u2[ gUk NXb2M+]?bcYHTWY]rNrMHTWV h xvNXb2MHVMV?[ [ gUq [\d[ q jUYH^fV nt¦ VY!PQMHPQzUM T h xdNXb2M Y g Y q [fY gq M ^eM k YJNr[ hcg N h NXb2M HGG A }r sd} bcY q NXb2MuUTW[\d[f^eM k M h xPQMDMONr[ gUk Y ^f^HVX[ h xHNXb2MHVM }r V_Nrj q M g NrV s Y gUq} d[\d[ q ^ l TrMHPQMHPQzUM T NXb2M [fT{M ^w^`XT h j gUq M q2g M VVY gUqq [\cM TWV?[\N lh xt[ g N_M TrMHV_NrV n @ } YHPMHPQz2YHT_TWYHVVM q N h V?Y l NXbcY Nv[ g NXb2MQY g2g jUYH^ Y g Y q Y'cV| O¦4k YDPQM h xyj2^wNr[fPQYJN_M'x`TW[eVzUMHM s NXb2M Y g Y q [eY g V ^ h V_N=] hcg c[ g ]_[ gUk ^ lon K ¦ N=NXb2M }r s NXb2MaVXNrj q M g NWV!YHTrM k [ cM g NXbcTrMDMu2T h zc^eMHP&V hcg NXb2MaRTWV_N q Y lh x ] h P&uMONr[wNr[ hcgts Y gUq NXbcTrMDMuUT h zd^`M PQV hcg NXb2Mx h ^w^ h {i[ gUk!q Y lon¦ u2uUT h u2Tr[fY NrM ^ lts NXb2M TrM YDTrMVX[ PQY [ g ]?bcYDudN_M TWVo[ g NXb2Mz h2h Z s {i[wNXb!YzcTr[eM xd]?bcYDudN_M ToM g Nr[wNr^eM q"!J}~g NrMHTW^fj q M *c¦g ¦ xfN_M T gUh2hcg N h ¨tM VXN # [ g NXb2MP'[ q2q ^eM n xo] h jUTWVmM s NXbd[fV[fV gUh ] h [ g ]_[ q M g ]XM n}~g MHYD]?b h x NXb2M|V?[ ] h TrM]?bcYDudN_M TWV s NXb2M TrM|[fVYu2T h RU^eM h x hcg M h xNXb2M }r N_M YDP%PQMHPQzUM TWV s Y q MJNWYH[w^`M qQq MHV]XTr[fucNr[ hcg Y gUq Y g Y ^ l V?[fV h x hcg M h xNXb2M|VX[ u2T h zc^eMHP&V=NXbcYJNYDu2uM YDTrM q hcg NXb2M }r s Y gUq YV h ^fjcNr[ hcg N h NXb2M|u2T h zc^eMHP%z l NXbcY NV_Nrj q M g N n tb2M!VX[ i]?bcYDudN_M TWVYDTrMNr[\Nr^`M q&}~g VX[ k bN s h PQuUMJNr[\Nr[\cM g MHV?V s 2Y ^`M g N s TrM Y Nr[ c[\N l s r T H M Y p q NXb s Y gUq Y M g VmM h x½ hcgUq MHT n}~g NXb2MHVM]?bcYHucNrMHTWV s ^fV hcg YH^fV h [ g cMHV_Nr[ k YJN_M V MHYD]?b h xNXb2MHVM]?bcYDTWYD]rN_M Tr[fVXNr[e]_V Y gUq uUT h2q j]XM V4Yx`TrMHVb¤Y g YH^ l VX[eV h xSY k MD h ^ q cjM VXNr[ hcg V!VXj]?bYDV !J} V!P&Y NXb2M PQYJNr[`]rYH^YDzc[w^f[\N l [ g2g YJN_M #Hs%!J} V k M g [fjUV[ g b2M Tr[\N_M q #Hs !X¦ TrM ¦ VX[eY g VXNrj q M g NWV[ g b2MHTrM g Nr^ l P h TrMNrY ^`M g N_M q [ g P&Y NXb #Hs Y gUq !X¦ TrMÀz h l V zUMONWN_M TNXbcY gak [eTW^eVy[ g PQYJNXb # |M qUh M VY{ hcgUq MHTWxej2^D¥ h z h x2u2TrMHVM g Nr[ gUk NXb2M[eV?V?jM V n [fVTrMHVMHYHTr]?b'[fVMXHNrM g VX[\cM s Y gUq b2M]_[\N_M V{M ^w^`?Z gUh { g uM q Y k h2k jMHVV?j]?b'YHVvd^w^`M g ½[ g2g M T s h {|YDT q ÁyYDT qcg MHT s Y gUqi¦ ^fRM| h b gts Y gUq jUVMHVyNXb2Mvx`YDP h jUV }rD VXNrj qDl N h ] h P&uUYHTrMP h2q MJ^eV h xÀP&Y NXb2M PQYJNr[`]rV N_M Y]?bd[ gUk [ g¤¦ PM Tr[e]_Y{i[wNXbdYDTW[ h jUV ] h j g N_TW[`M V|[ g»¦ VX[eY n iMY ^eV h u2T h2q j]_MHVY g M?D]_M ^w^`M g NuUT h R2^`M h x M ^fY g [eMQ½ h2h2qts Y|x h T_PQMHTN{ h rNr[ePQMiPM Pz2MHT h xNXb2M ¦ PM Tr[e]_Y g NrMHYHP s Y gUq NXb2M hcg ^ l xBMHP&YH^eMMmcM T N h TrM uUTrM VmM g NNXb2M| O¦ Y NNXb2M }r n

H

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j uUu h VmM lch j»YHTrM k [\cM g»g [ g M'] h [ g V h x{bd[e]?bM [ k bNYHTrM k M g j2[ g MY gUq U hcg M] h j g N_M TrxBMJ[wN n h jZ gUh {.NXbcY NY ^f^NXb2M k M g j2[ g M] h [ g V{MJ[ k b»NXb2M VYHPM YHV hcg M Y gUh NXb2M TY gUq NXbcY N»NXb2M] h j g N_M TrxBMJ[wN] h [ g [fVVX^f[ k bNr^ l b2MHYOc[eMHT nnJn n ½bcY N[fVNXb2MV?PQY ^f^eMHV_N g jUPQzUM T h x{MJ[ k bd[ gUk V {i[\NXbY z2Y ^eY g ]XMiVm]rYH^eM g MHM q M q N h [ q M g Nr[fx l NXb2M] h j g NrMHTWxBM [\N=] h [ g

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√ s F 3 ≤ 3 3(abc)2 /64 √ s {bd[e]?b k M h PMON_Tr[e]_Y ^f^ l ] h _T TrM V u hcgUq V=N h Y gUq V?[ g ]XM 2 3) NXb2M=x`YDabc ]rNNXbc= YJNNX4F b2MR N_TW[eY gUk ^`R M h xH≥ ^eYHT 4F/(3 M g ]_[fTr]_^`M k MHV_NtYHTrMHY{bd[`]?b!]_Y g zUM[ g V]XTr[ezUM q [ g Y k [ c

H H

[eVNXb2MM cj2[w^eYJN_M WT YH^ c ½ M U j VmM¤NXb2M x h ^f^ h {i[ gUk h g M n g h { U b [e]?b&b h ^ q x h T Y ^f^u h V?[\Nr[\cM s s * ¦ E Á }~g M cjUY ^f\[ N l {d a b c n

(abc) 3 ≤

q

dYHTr[fY g NWV

h x NXb2M

(bc)n + (ca)n + (ab)n /3 ≤ (an + bn + cn )/3

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H

V n−1 ≤ n!

s

n+1 Y

(n + 1)n−1 n3n

k=1

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n !n+1

@~A K s

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n+1 2

≤ c(n)

{b2MHTrM

N Y

ek

@ FK s

k=1

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+H

½»Tr[\N_M7[ g M djUYH^w[wN l

@~A K [ g NXb2M x h TrP

V

n−1

≤ λ(n)

n+1 Q k=1

Fk

n n+1

Y gUq

s

YDV?V?jUPQMNXb2M=dY ^f[ q [\N l|h x @ FK x h T q [fPM g V?[ hcg V n b2M g F n2 ≤ c(n − 1) Q e s (n − 1) ij i

!

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j

DG C {b2MHTrM s xh T s s s s YHTrMNXb2M!M qUk M^`M gUk NXbcV h xcNXb2M i x`YD]XMON n tbd[fV[fPQeucij^f[eMHVNXbcjYJN= 1 2 . . . n(n − 1)/2 n+1 Y

Fi

i=1

!n2

≤ c(n − 1)

N Y

n+1

ek

k=1

!n−1

s

V?[ g ]XM|MHYD]?b&M q2k M h xNXb2M k [\cM g VX[eP&u2^eM?az2M ^ hcgUk V=N h x`YD]XMONrV n tb2M TrM x h TrM s n−1 V n−1 ≤ λ(n)c(n − 1)2

|M g ]XM s V

n+1 2

≤ λ(n)

n+1 2(n−1)

c(n − 1)

n+1 n−1

N Y

ek

k=1

N Y

ek

k=1

!2(n−1) n+1

!

n

N Y

= c(n)

ek

n

k=1

QN h Nr[ gUk NXbcY N n+1 [eV[ g dYDTW[eY g Nj gUq MHTaVX[eP'[f^fYDTW[wN l+k [ cMHVajUVNXb2M»x`YD]rN 2 ek NXbcY N [eVNXV b2M|] hcg /V_NrY k=1 N H M V X ] r T [ezUM q YDz h cM n bdjUV s NXb2MvdY ^f[ q [\N l x h TyNXb2M g q (n − 1) V?[fPQucc(n) ^`MXDMHV=[fPQuc^f[eMHVNXb2MvdYH^w[ q [wN l x h T q [fPM g V?[ hcg n n ½M gUh {4jUVmM!NXb2M YD]_^eY jUTr[ g»}~g M cjUY ^f[\Nr[`M V A n LHMONWNr[ gUk

H

m (x + em 1 ) · · · (x + en+1 )

= xn+1 +

H

n+1 n+1 s n p x + p2 xn−1 + · · · + pn+1 1 1 2

H

1 1 1 {b2MHTrM s {MbcYJcM p n+1 s {i[wNXbM cjUY ^f[\N l [fxY gUq 2 m > 0 VYHTrMM cjU ≤ p[ennP&u2≤^w[`M ·V · ·NXbc≤ p[f2xo ≤ p1 M xtNXb2M n+1 ^ f [ x H Y w ^ ^ X N 2 b M Y ^

d b e [ V J Y = N Y V[eV k [ cM gts hcg l n glhcg h ej NXb2M g YH^w^NXb2M V{b2M TrM Y gUq Y]?bd[eMmcMNXb2MJ[eTP&YOH[eP&Yv[fxUY gUqhcpg i^ l [fxUY ^f^NXb2M j>i V VYHTrMM cjUY p^ n j e

H f!$3 n i

A

Á n n iYHT ql s n n LJ[wNWNr^eMO{ 2 h h2q s Á n £ h ^ l Y s ±t¶ V?[\N l £UTrM VV s L hcgUq2hcgts AHC OI s U u u n AFE D n

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1LHMJNSNXb2M uM T_uUM gUq [`]_j2^fYDT x`T h P PQMDMONNXb2M hcg M 0 N h x`T h P B0 N h CAYJN n ½M|¥`jUVXN bcYJcMiN Ch Vb h {AB NXbcYJN M0 [eVuUMHT_ AM uM gUq [e]_j2^eYHTN h n h { s BC

B

C

− → − − → − − → − → → − → − → → − → − → → − → − n ( A0 − M ) · ( C − B ) = ( A0 − M ) · ( C − A ) − ( A0 − M ) · ( B − A )

hcg VX[ q MHTW[ gUk NXb2MiRUTWVXNNrMHTrP

hcg NXb2MiTr[ k bNV?[ q M s {M|bcYJcM

− → − − → − → − → − → − → − → s → → − → − → − → − ( A0 − M ) · ( C − A ) = ( A0 − B 0 ) · ( C − A ) = ( A0 − B 0 ) · ( C 0 − A0 )

{b2MHTrMaNXb2MRUTWVXNV_N_M u»x h ^w^ h | { Vvx`T h P NXb2Mx`Y]rNNXbcYJN Y gUq NXb2M'VmMH] hcgUq M B 0 YH⊥^N CA x`T h P1NXb2Mx`YD]rNiNXbcYJNaz h NXb Y D Y r T M T X N b g q CC 0 U h h2k hcg h A0 B 0 n7 [fP&[w^eYHTr^ lts AA0 →0 − →0 − →0 − →0 − →0 n b2MHTrMJx h TrM s {MR gUq NXbcYJN − − → → − → − (A − M ) · ( B − A ) = ( A − C ) · ( B − A ) − → − → → − → − n ( A0 − M ) · ( C − B ) = 0

[eV[ g A mK YH^fV h h TWNXb

@ {i[\NXb'TrM VuUMD]rNN h K n} N tb2Miu h [ g N [fV]_Y ^f^eM q NXb2M % $¹¶ h x 4ABC NrMHTrM VXNr[ gUk N h M ] h P&uUYHTrM'NXb2MQVb h TWN!u2T h2h xYDz m h cM{i[\NXbV l2g NXb2MONr[`]uUT h2h x`V @ s h TY|uUT h2h x jUVX[ gUk ] h PQuc^`MX g jUPQzUM TWV @ K n tb2MiV?[fPQuc^f[e]_[wN l&h xUcMH]rN h TuUT h2h x`V V_NrY gUq V h jcN|[ g NXb2M»x h ^f^ h {i[ gUk N{ h ^eMHV?VmM TX?Z gUh { g NXb2M h TrMHP&VaNXbcY N^w[ g ZNXb2M h u h ^`M!N h NXb2M gUh Nr[ hcgQh x [fV h N h P&[e]N_TWY g VWcM TWVY ^ n

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m

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t tb2M|PQMJNXb h2q [eVY g HY ^ 2h k h jUVN h b2M h TrM P

C

A n ®D[eTWV_Ny{Mv{|Tr[\N_M

− → − → − → → − → → − → → − − → → − − − → → − − n (M − H ) · ( F 0 − D 0 ) = ( M − H ) · ( F 0 − B ) − ( M − H ) · ( D 0 − B )

hcg VX[ q MHTW[ gUk NXb2MiRUTWVXNNrMHTrP − → → − → → − − (M − H ) · ( F 0 − B )

hcg NXb2MiTr[ k bNV?[ q M s {M|bcYJcM

− → → − → → − → − → − − → − → − = (M − H ) · ( A − F ) = (C 0 − C ) · ( A − F ) − → → − → s − → − = ( C 0 − C ) · ( A − A0 )

{b2MHTrM=NXb2M=RTWV_NVXNrMHux h ^w^ h {|Vx`T h P4NXb2M q M R g [wNr[ hcg|h x 0 s NXb2MVMD] hcgUq x`T h P NXb2Mx`YD]rN F NXbcY N 0 Y gUq YHTrM h TWNXb h2k hcg Y ^ N h s Y gUq NXb2MNXbd[fT q zUMH]_Y jUVmM F A0 ⊥ CC 0 n CM CH AF − → → − → → − → n bdjUV s → → − − − → − [eP'[f^fYDTW^ l s {MiVMDMvNXbcY N (− M − H ) · ( D 0 − B ) = ( A0 − A ) · ( C − C 0 ) − → − → − → → − n (M − H ) · ( F 0 − D 0 ) = 0

1" B"

z MiNXb2MauM T_uUM gUq `[ ]_j2^fYDTN h n b h2h VmMY gl u h [ g N K hcg m Y gUq `^ MON n U NXbcT h j k b b2M h TNXb h u h e^ M

x ^w[`M V c NX2 b M|uM _T uUM gUq [`]_j2^fYDTx`T h P N h NXb2mM n h h g K M m K [eV h N h P'`[ ]N_TWY g WV cMHTWV?YH^ 0 h x n n n .. ... . .. .. .. .. .. .. .... .. .. ....... ...... .. . ... ..... ......... ... ..... ... ...... ..... ..... ... ... ............. ......... ... ... .. . .......... .. ... .. .... ........ . . .. .. ..... ...... .. . .. ... ... ........ ..... . ... .. . ...... .. ............. .... .. . ....... . . . .............. ..... . ............. ..... .... ... . ... .............. . . . . . ..... ... .... . ... ... .... . . . . . . . . . ..... ........ ... ... ... ...... ... ....................... ... .. .. ..... ................... ... ... .... .... ..... .............. ... .. ........................... . . . . ...... . . . . . .... ............. .... .. .. ..... ... 0 .............. .................. ... ..... .... .............. ..... . ..... 0 . ....................................................................... .... ..... ..... . . . . . .... . . . . . . . . . . . . . . . . . . . . . . ................. ..... ........ ... . . . . . . . . . . . . . . . . . . . . . . . . . ... ................... .... . ......... . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . ................................................................................................................................................................................................................................ ........... ................. .... . ... . . . . . ..... . ... . . 0 .................. ... 0 .... . . . . . . . .... . . ..... .. . 0 ............................................ ....

Y

n

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− → − → − → → − → → − → → − − → → − − − → → − − (M − K ) · ( Y 0 − X 0 ) = ( M − K ) · ( Y 0 − C ) − ( M − K ) · ( X 0 − C )

{i[wNXb

− → → − → → − → → − − − → → − → − → − − → − → − (M − K ) · ( Y 0 − C ) = ( M − K ) · ( A − Y ) = ( B 0 − K ) · ( A − Y ) − → → − → → − − n = ( B 0 − K ) · ( A0 − K ) − → → − → → − → → → → − − − − n [eP'[f^fYDTW^ l s {M|bcYJcM (− M − K ) · ( X 0 − C ) = ( A0 − K ) · ( B 0 − K ) }~g ] hcg ]_^wjUV?[ hcgts {MuUT h u h VmMNXb2M=x h ^w^ h {i[ gUk uUT h zd^`M P s MHYHV?[w^ l V h ^ cM q {i[wNXbvNXb2M b2M ^fu h xNXb2Mu2TrMmc[ h jUVNXb2M h TrM PQV * LHMJN q M gUh N_MNXb2M [eP&V hcg ^w[ g M h xY'u h [ g N P sP hcg NXb2M]_[eTr]_jUPQ]_[eTr]_^eM h x n 4ABC

@ Y K Vb h {4NXbcYJNNXb2M h TNXb h u h ^eM h x Y|^w[ g MiuUMHTruM gUq [e]_j2^eYHTN h ^f[eMHV hcg sP sP @ z K Vmb h {NXbcYJNiNXb2M h TWNXb h u h ^`M h x @ NXb2M»^w[ g M ^w[`M V hcg NXb2M N_MJ[ g M T|^f[ g M h x K n P½bcYJN[fV|[wNWV NXbcT h j k bNXb2MTrMJ« MH]rNr[ hcg V h x sP[ g NXb2MVX[ q MHV s s AB M?HYD]rNu h VX[wNr[ hcg»hcg NXbd[eVv^w[ g M P v¸ ½bcYJN[eBC VNXb2MaCA [eV h N h P' [`]=N_TWY g VWcM TWVY ^ h x sP

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® h TTrM Y q MHTWV{i[eVbd[ gUk N h ucjUTWV?jM|NXbd[eVxejUTNXb2MHT s NXb2M TrMYHTrMaPQY g l [ g N_M rT MHV_Nr[ gUk TrMHVXj2^wNWVi] hcg2g MD]rNr[ gUk NXb2M h TNXb h u h ^eMaN h» [eP&V hcg ^w[ g M Vv[ g Y gUq s Y gUq N h NXb2M ] h P&u2^eMJNrM djUY q Tr[w^eYJN_M TWYH^[ g A n

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¨ n Á 2 P M TNXb h u h ^eMÀ b2M h TrM PQV s h h TrPQY k b Nr[ k b i s ¦g Y ^ l Nr[`]UTrMHYJN_PM g N h xaV h Q ª @BADC C K s u2u n ª E ªC n ²»¶ t³ ³ $ I a

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f (x)+f (y)+f (z)+3f

x+y+z 3

≥ 2f

x+y y+z z+x +2f +2f 2 2 2

s

[eVY] hcg cMXaxej g r] Nr[ hcghcg Y g [ g N_M T~dYH^ Y U {b2MHTrM g q s s z ∈ I n¦!n JL jUu2YDV f u h VX[wNr[ cM g U j Pz2MHTWV K b MHTrM s I s Y gUq x yYDTrMi k M g M TWYH^w[ ?M q NXbd[eV=[ g ADC d HY V=x h ^f^ h {|V @ {2 p q r

px + qy + rz p+q+r

pf (x) + qf (y) + rf (z) + (p + q + r)f px + qy qy + rz rz + px n ≥ (p + q)f + (q + r)f + (r + p)f p+q

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}~g NXbd[eVu2YDuUMHT{Mu2TrMHVM g NN{ hQk M g MHTWY ^f[*XY Nr[ hcg V h x£ h u h c[e]_[fj V }~g M djUYH^w[wN l Nh dYDTW[eYDzc^eMHVo{i[\NXb|V h PMYHuUuc^f[e]_YJNr[ hcg V n tb2MRTWV_N k M g M TWYH^w[ XYJNr[ hcg {|YHVu2jzd^f[fVmb2M q [ g nA {i[wNXbY'P h TrM q [ ]_j2^\NuUT h2h xoNXbcY g NXb2M hcg M'zUMJ^ h { n tb2M'VMD] hcgUqk M g MHTWY ^` [ XYJNr[ hcg @ {i[wNXb h jcNiV h ^wjcNr[ hcg K {|YDVu h V_N_M qhcg NXb2M Y NXbd^w[ g ZDV [wNrM E }~g M djUYH^w[wNr[eMHV ® h TWjUP[ g HGGJI nS} N![eV|u h VVX[`zd^`M'NXbcYJN hcg M h Tz h NXb k M g M TWYH^w[ XYJNr[ hcg V|P'[ k bNibcYOcM zUMHM g uUTrMmd[ h jUVX^ l u2jzd^f[fVmb2M q M ^fVmMF{b2MHTrM s zdjcN gUh NYHV=x`YDTYDV{M|Z gUh { n jUTuUT h2h xTrM ^w[`M V hcg tYHTWYDP&Y NWY V }~g M cjUY ^f[\N l x h T] hcg cM?ixej g ]rNr[ hcg V s VmMHM I Y gUq − s {bd[`]?b{M gUh { TrMD]rYH^w^ n ½MQV?Y l NXbcYJNY'cMH]rN h T → A = [a1 , a2 , . . . , an ] {i[wNXb PQYO¥ h Tr[*?MHVaYcMD]rN h T → {i[\NXb − a1 ≥ a 2 ≥ . . . ≥ a n B = [b1 , b2 , . . . , bn ] Y gUq {|Tr[\N_M!NXbd[fVYHV → − → − s [fx A ≥ B b ≥ b ≥ ... ≥ b

+H

H

1

2

n

a1 a1 + a 2

≥ b1 ≥ b1 + b2

n n n

a1 + a2 + · · · + an−1 a1 + a 2 + · · · + a n

H

≥ b1 + b2 + · · · + bn−1 n = b1 + b2 + · · · + b n

tb2M|tYDTWYHPQYJNrY }~g M cjUY f^ [\N l VXNWY NrMHVNXbcYJNy[wx [fVY gl ] hcg cM?xej g ]rNr[ hcg Y gUq → − → −s f A ≥ B NXb2M gQhcg MbcYHV=NXb2Mx h ^f^ h i { [ gUk f (a1 ) + f (a2 ) + · · · + f (an ) ≥ f (b1 ) + f (b2 ) + · · · + f (bn )

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f (a1 ) + f (a2 ) + · · · + f (an ) + n(n − 2)f

a1 + a 2 + · · · + a n n

{b2MHTrM

≥ (n − 1) f (b1 ) + f (b2 ) + · · · + f (bn )

bi =

1 X aj n−1

x h TYH^w^ n i

s

j6=i

¡ ½[wNXb h jcN+^ h VV h x k M g MHTWY ^f[\N l s {M¿PQY l YHVVXjUPM4NXbcYJN n ≥ 3 Y gUq n tb2M g NXb2M TrMQ[eV|Y g [ g NrM k M T m {i[\NXb 1 ≤ m ≤ n − 1 a Y gU1 q ≤ a2 ≤ · · · ≤ an s {b2MHTrM a = (a + · · · + a )/n n 1 ½MYHa^f1V h ≤bc·YJ·c·M ≤ am ≤ a ≤ am+1 ≤ · · · ≤ an n } N![eV]_^`M YDnTvNXbcYJN b ≥ · · · ≥ b ≥ a ≥ b ≥ · · · ≥ b 1 m m+1 n NXb2M[ g M djUYH^w[wN l NXbcY Nv{MQYDTrM&N_T l [ gUk N h uUT h cMQ[eViNXb2MVXjUP h xNXb2MQx h ^f^ h {i[ gUk N{ h [ g M cjUY ^f[\Nr[`M V *

H +H

f (a1 ) + f (a2 ) + · · · + f (am ) + n(n − m − 1)f (a) s ≥ (n − 1) f (bm+1 ) + f (bm+2 ) + · · · + f (bn ) f (am+1 ) + f (am+2 ) + · · · + f (an ) + n(m − 1)f (a) n ≥ (n − 1) f (b1 ) + f (b2 ) + · · · + f (bm )

@~A K @ FK

+H

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f (a1 ) + f (a2 ) + · · · + f (am ) + (n − m − 1)f (a) ≥ (n − 1)f (b)

{b2MHTrM

b=

a1 + a2 + · · · + am + (n − m − 1)a n n−1

s

bdjUV s {MiVXNr[w^f^bcYJcM!N h Vb h {

(n − m − 1)f (a) + f (b) ≥ f (bm+1 ) + f (bm+2 ) + · · · + f (bn )

Y gUq [ g ]_M a ≥ b m+1 ≥ bm+2 ≥ · · · ≥ bn

n

s

(n − m − 1)a + b = bm+1 + bm+2 + · · · + bn {MVmMHMNXbcY N → P&YJ¥ h TW[ ?M V → − − n tb2Mi[ M djUYH^wA [wN n−m x ^w^ = {|V[a, z . .t.YH,TWa, YDP&b]Y NWY V M djUYHB ^w[wN n−m x T== ] [bcm+1 M?|xej , bm+2 ]rNr[ , V . . . , bn ]

g

+H H

+H

f (am+1 ) + f (am+2 ) + · · · + f (an ) + (m − 1)f (a)

H

n−1

N h NXb2M[ g M cjUY ^f[\N l

H

+H

l h h l }~g l h h g c g h g n c tb2M[ g M cjUY f^ [\N l @ OK ]_Y g z2MuUT h cM q XV [eP'[f^fYDTW^ l z l Y qUq [ U g k M g mV M t g V ~} g M djUYH^w[wN l ≥ f (c)

f (c) + (m − 1)f (a) ≥ f (b1 ) + f (b2 ) + · · · + f (bm )

H

s

{b2MHTrM n b2M^fYDV_N am+2 + · · · + an + (m − 1)a)/(n − 1) [ g M cjUY ^fc[\N l =x h (a ^w^ h m+1 {|Vx`T + P t D Y W T H Y Q P J Y r N Y V M c U j Y f ^ \ [ N U z H M _ ] Y U j m V M h }~g l s b1 ≥ · · · ≥ b m ≥ a Y gUq Y gUq NXb2M TrM x h TrM → − s c + (m − 1)a = b1 + b2 + · · · + bm C m = [c, a, . . . , a] PQYO¥ h Tr[*?MHV → − n D = [b , b , . . . , b ] m

1

2

m

A

1"N+$ B"% #D B" n 1" B" n} x [fVY] hcg cMXxej g ]rNr[ hcg'hcg NXb2M f

g

Y g [ g NrMHT~dY ^ Y gUq s s s s I a 1 a2 . . . a n ∈ I

a1 + a 2 + · · · + a n (n − 2) f (a1 ) + f (a2 ) + · · · + f (an ) + nf n X ai + a j n ≥ 2 f

H

2

1≤i<j≤n

'½MQ{i[f^w^=u2T h c MNXbd[eVajUVX[ gUk [ gUq j]rNr[ hcgon ® h T shcg MbcYDV'M cjUY ^f[\N ln j uUu h VmM gUh {NXbcY N U Y gUq NXbcY NiNXb2M»[ g M cjUY ^f[\N nl [f= Vd2Y ^f[ q x h T n ½M ≥ 3T n − Y 1 ^`MON {i[f^w^Vmb h {¤NXbcYJN[wNab n ^ ' V x H L O M N h q h nn gUq a = NX(a + j a]r2Nr[ + · b · · u + NXa b2M1 [ gU b2nMHVX)/n [eV s {MibcYOcM n c p l q c h g l h x = (a + a + · · · + a )/(n − 1)

H

1

2

n−1

(n − 3) f (a1 ) + f (a2 ) + · · · + f (an−1 ) + (n − 1)f (x) X ai + a j n ≥ 2 f 2

1≤i<j≤n−1

tbdjUV s [\NVXj ]_MHVN h Vb h {4NXbcYJN

f (a1 ) + f (a2 ) + · · · + f (an−1 ) + (n − 2)f (an ) + nf (a)

+H

≥ (n − 1)f (x) + 2

®dT h P¯M g VM gt V }~g M djUYH^w[wN lts {M|bcYJcM

n−1 X

f

i=1

f (a1 ) + f (a2 ) + · · · + f (an−1 ) ≥ (n − 1)f (x)

ai + a n 2

n

n

|M g ]XM s {Mv¥`jUVXN=bcYOcMN h Vmb h { NXbcY N

(n − 2)f (an ) + nf (a) ≥ 2

n−1 X i=1

f

ai + a n 2

n

P ai + an s {M+{i[w^f^|Y k Y [ g jUVM tYHTWYDP&Y WN Y V [ g ]_M (n − 2)a + na = 2 n−1 n 2 i=1 }~g M djUYH^w[wN l x h TN{ h ]_YHmV M V n

+H 8 3

s s s s a s . . . s a }n n 2NXbcY N n ½[w2a NXb h ≥ jcN=min{a ^ h VV h x 1k M ag 2M TW.YH.^w[w.N ltas n{}M+ PQmax{a Y l YHVVXjU1PMi n a ≥ · · j ≥ b' an 1 tb2M g ]_] h T q [ gUk N h tYHTWYDP&Y NWY V }~g M djUYH^w[wN lta [w2N≥ [eVM · gU Nh n ¦ s h k Vmb h {¿aNXbc≥ Y N (a1 + an )/2 Y gUq s s s an ≤ min{(a1 + an )/2 (a2 + an )/2 . . . (an−1 + an )/2} t 2 b = M R W T _ V o N ] w [ r N s s s n hcgUq [ hcg a +a 1 +a n )/2[\Nr.[ . . Tr(a n−1 n )/2} [eV≥ ]_^eMHmax{(a YHTr^ l N_TWjM| Y gUq nNX)/2 b2M|V(a MD] 2hc+a ] M j _ ] H M = V N gUq hcgUq hcg q h a ≥ (a + a )/2 n

8 3

+H

1

n

n s s s s a s . . . s a }n 2NXbcY N n ½[w2a NXb h < jcN=min{a ^ h VV h x 1k M ag 2M TW.YH.^w[w.N ltas n{}M+ PQmax{a Y l YHVVXjU1PMi n a1 ≤ a 2 ≤ · · · ≤ a n tb2M g ]_] h T q [ gUk N h tYHTWYDP&Y NWY V }~g M djUYH^w[wN lts [wN[eVM gUh j k b'N h n ¦ Y gUq Vmb h {aNXbc≤ Y N (a1 + an )/2 s s s a ≤ min{(a1 + an )/2 (a2 + an )/2 . . . (an−1 + a nb2)/2} Q M m V H M ] s s s n hcgUq a + a )/2 (a + a )/2 . . . (a j]XM + ] hcngUq ≥[wNr[ max{(a VN ah n )/2} hcg [eV=]_^eMH1YHTr^ l N_TrnjM!Y gUq NX2b2M!RUTWVXnN] hcgUq [\Nr[ hcg TrM n−1 q n a ≤ (a + a )/2

+H

1

n

AHª

+ B" . / D*j, B"3 B 1B 3, B"

n

n LHMON a s a s . . . s a zUMvu h V?[\Nr[\cM g jUPQzUM TWV{i[wNXb a a · · · a = 1 n 1 2 n 1 2 n

tb2M g

an−1 1

+ an−1 2

+ · · · + an−1 n

H

+ n(n − 2) ≥ (n − 1)

1 1 1 + + ··· + a1 a2 an

n

»tbd[fV&[ g M cjUY ^f[\N l x h ^w^ h {|V'x`T h P; b2M h TrM P A [fx{MS] hcg V?[ q M TNXb2MS] hcg cMX xej g ]rrN [ hcg Y gUq TrM u2^fY]_M {i[wNXb s s s s f (x) = ex (n − 1) ln a1 TrM VuUaMD1]rNr[ caM 2^ lon . . . an s s s (n − 1) ln a . . . (n − 1) ln a

f $

2

H

n

n ® h T n = 3 Y gUq a = x2 s a = y 2 s a = z 2 s NXbd[fV[ g M cjUY f^ [\N l rT M q j]XM V 1 2 3

Nh

yz

+H B B13, B"

zx

xy

n

tbd[fV[ g M djUYH^w[wN l {|YHVuUT h u h VmM q z l» jUTrTWY l t ^eYHPZH[ g [ g n } x a s a s . . . s a HY TrMu h VX[wrN [ cM 1 NXb22 n s M g a + a + ··· + a = n 1

n

x6 + y 6 + z 6 + 3(xyz)2 ≥ 2(y 3 z 3 + z 3 x3 + x3 y 3 )

2

g jUPz2MHTWV

V?Y Nr[fV?x l [ gUk

n

1

(n − a1 )(n − a2 ) · · · (n − an ) ≥ (n − 1)n (a1 a2 · · · an ) n−1

n

t ½MiYHuUuc^ l b2M h TrM P A N h NXb2M|] hcg cMX|xej g ]rNr[ hcg f (x) = − ln x x h T x > 0 n @ NXb2M nQ [ g ]_M [fPQuc^f[eMHVvNXbcY N a^f[\1N + a +YDz · · c · Mi +[ aM ndjU=YH^w[wn aY 1 aNX2b2·M· [ · aM ncjU≤ 1N 2 M| E K Á M c U j Y X N 2 b N e [ V V c b D Y r T u M T X N c b Y f ^ \ [ ¦ }~g l s h g l g g l

f $

H

+H

H

+H

(n − a1 )(n − a2 ) · · · (n − an ) ≥ (n − 1)n a1 a2 · · · an

s

{bd[`]?b&MHYHV?[w^ l x h ^f^ h {|Vz l P&j2^\Nr[euc^ l [ gUk NXb2Mi[ g M djUYH^w[wNr[eMHV

1

s

1 n−1

s

n − a1

= a2 + a3 + · · · + an ≥ (n − 1)(a2 a3 · · · an ) n−1

n − a2

= a1 + a3 + · · · + an ≥ (n − 1)(a1 a3 · · · an )

n n n

B

B13, B"

n

1

n − an

xh T Y f^ ^ s NXb2M g i

= a1 + a2 + · · · + an−1 ≥ (n − 1)(a1 a2 · · · an−1 ) n−1 n=} x a s a s . . . s a YDTrMiu h VX[wNr[ cM g jUPz2MHTWVY gUq b = 1 X a 1 2 n i j n−1

b1 b2 bn a1 a2 an n + + ··· + ≥ + + ··· + a1 a2 an b1 b2 bn

H

t ½Mi]rY g u2T h cMvNXbd[eV{MJ^f^eZ gUh { g [ g M cjUY ^f[\N l z l YDu2u2^ l [ gUk b2M h TrM P n V?[ gUk NXb2M|TrMJ^eYJNr[ hcg V

a = (a1 + a2 + · · · + an )/n (n − 1)bi ai

=

na ai

−1

Y gUq

H

ai bi

=

na bi

j6=i

@ FK A n LHMON

+H

−n+1

xh T s s s s {M|VMDM!NXbcYJN @ FK [eVM cj2[ dYH^eM g NN h NXb2Mi[ g M djUYH^w[wN l i = 1 2 ... n 1 1 1 n(n − 2) + + ··· + + ≥ (n − 1) a1 a2 an a

1 1 1 + + ··· + b1 b2 bn

n

A p jcN7NXbd[eV4M YDVX[f^ l xh T

B

x h ^w^ h {|V)x`T h P n x>0

B13, B"G n jUuUu h VmMNXbcYJN

f (x) = 1/x

A

b2M h TrM P s

s

s

z l

x1 x2 . . . x n

x1 + x 2 + · · · + x n =

tb2M g

1 x1

+

jUVX[ gUk

NXb2M] hcg cM?xej g ]rNr[ hcg

YHTrM|u h V?[\Nr[\cM g jUPQzUM TWVV?j]?baNXbcYJN 1

x2

+ ··· +

@ IHK

1 n xn

1 1 1 n + + ··· + ≥ 1 1 + (n − 1)x1 1 + (n − 1)x2 1 + (n − 1)xn

H

bd[eVo[ g M cjUY f^ [\N l PQY l z2M q M Tr[ cM q `x T h P NXbcY N 1 1 + (n − 1)x1

+

1 1 + (n − 1)x2

@ FK

@ FK [ g NXb2Mx h w^ ^ h {i[ gUk {|Y lon jUu2u h VM

+ ··· +

1 1 + (n − 1)xn

< 1

@Bª K

[ gQh NXb2M Ty{ h T q V s {MiV?jUu2u h VMNXbcYJN @ F K e[ Vx`YH^fVmM n t b2M g {M!{i[w^f^UVmb h { NXbcY N @ IDK YH^fV h xh T 1 q h M V U U g h Nb h ^ qon h XN bd[fVM gUq s ^`MON a = s s s n h N_M i = 1 2 ... n i 1 + (n − 1)xi NXbcY N Y gUq NXbcYJN 1 − ai x h TYH^w^ s s s n ½MY ^eV h bcYOcM ai > 0 xi = i = 1 2 ... n (n − 1)ai P z l @`ª K n iM g ]_M s ai < 1

1 − ai >

x h TY f^ ^ s s s n tbdjUV s i = 1 2 ... n x1 + · · · + x n =

n X

i=1

X j6=i

1 − ai (n − 1)ai

@ mK

aj = (n − 1)bi

> ≥ >

z l @ OK

n X ai

i=1 n X i=1

=

z l @ FK

n X bi a i=1 i

bi

z l @ FK

(n − 1)ai 1 − ai

1 1 1 n + + ··· + x1 x2 xn

|M g X] M s { MbcYOcMuUT h cM q XN bcY N|[fx @ OK qUh M V U g h Nab h ^ q s XN b2M g @ IDK U q h M V gUh Nab h ^ qn tb2M TrM x h TrM s @ IDK [fPQuc^f[eMHV @ OK s z l NXb2M|] hcg N_TWYHu h XV [wNr[ cM n nQ jzcVXNr[\NrjcNr[ gUk xh T [ g @ OK Y U V V_Nr[f^w^VYJNr[eVXR M q g qgUh Nr[ gUk XN bcY N @ HI K [fi 1/xi xi k [\cM VjUV

f $

x1 n − 1 + x1

+

x2 n − 1 + x2

+ ··· +

xn n − 1 + xn

≥ 1

n

A [ g ]_M xi

+H

n−1+x

= 1−

n−1

n−1+x

i i x h T=M Y]?b s s s s NXb2Mi[ g M djUYH^w[wN l ]_Y g zUMiTrMO{|TW[wNWNrM g [ g NXb2Mix h T_P i = 1 2 ... n

+H

1

n − 1 + x1

B13, B" #

+

1

+ ··· +

n − 1 + x2

1 n − 1 + xn

≤ 1

n

tbd[fV[fVY g [ g M djUYH^w[wN l [ g NXbcYJNy{|YDV q HM TW[\cM q U j V?[ gUk L Y k TWY gUk M j2^wNr[fu2^w[`M WT V n n} x x s x s . . . s x YHTrM|u h V?[\Nr[\cM g jUPQzUM TWV s NXb2M g 1 2 n

B

H

n

1

(n − 1)(x21 + x22 + · · · + x2n ) + n(x21 x22 · · · x2n ) n ≥ (x1 + x2 + · · · + xn )2

vtbd[fV[eViYZ gUh { g [ g M cjUY ^f[\N l ª {bd[`]?b»x h ^w^ h {|V!x`T h P tb2M h TrMHP jUVX[ gUk Y r T H M c u e ^ D Y _ ] [ i { w [ X N b NXb2M] hcg cMXxej g ]rNr[ hcg U g q U g k s s s s x f a1 a2 . . . a n 2 ln x1 TrM (x) V uUMD]r= Nr[ ceM ^ lon s s s 2 ln x . . . 2 ln x

/HTrM xB"M *TrMDB M VDjx %1T)b2 M ^fu2xe j2n^ VXtj b2MMHYHV_Nrjc[ NXb h V Ti{ h k k hcg n f!$3E 2

A I ª

n

j ^ q ^w[`ZDM»N h MXu2TrMHV?VQbd[eV k TWY Nr[\Nrj q MN h NXb2M h 2

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§!n TWN h JY ¥~M s g £ h u h d[`]_[wj V }~g M djUYH^w[wN l x h T c h g cMX4®j g ]rNr[ hcg V s F¶ ³ ¹¶! e ¸#"! ² s I @ HGGOK s JIU E n I d E I2 n $ A G @~HA CCDª K s 2 §!n TWN h YJ¥~M s £UT h zd^`M P A GdD s ²»¶ ³ ³ Un ÁyjM T hcgtsc jz2V_Nr[wNrjcNr[ hcg V sU}~g M cjUY ^f[\Nr[`M V s Y gUq ![eV_N h T l s ³ t¶! e¸#" @ HGGOK s EC G n ¤º=¸ »³

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 a2  b2  s c2

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HG {b2MHTrM Y gUq x h TM Y]?b n cYD]?b&uM T_P'jcNrYJNr[ hcg s s 0≤a b c ≤1 a +b +c = 1 i h x=NXb2MHVM]_YHT q iVTriMHVXj2i^wNWV[ g YV?jUP i h x=NXibcTrMDM i g jUPQzUM TWV s Y gUq {MQ{i[f^w^ q M gUh N_MNXb2M uM T_P'jcNrYJNr[ hcg z l NXbd[eVV?jUP n ® h T[ g V_NrY g ]XM s [fx2NXb2Mi]_YHT q V=YHTrM h T q M TrM q YDV k [\cM gts NXb2M uM T_P'jcNrYJNr[ hcg [fV q M gUh N_M q z l NXb2MV?jUP s {bd[w^`M![wxU{MV_{i[\N_]?b|NXb2MRTWV_N N{ h ]rYDT q VNXb2M|uM T_P'jcNrYJNr[ hcg z2MD] h PM V a1 + b2 + c3 n ½M!{i[eVbN h u2T h cMNXbcY N=YJN c3 VXjUP¯z2MJN{MHM ^`M YDV_N hcg M h xNXb2MV?[ vu h V?V?[ezc^eMuUMHTrP&jcNWY Nrb[ hc1 g + V l a[`2MJ^ + Y gUq q VyYv g s {i[wNXb'MJ[wNXb2M TM? N_TrMHPQMvdY ^fjMiz2M [ gUk YH^w^ h {M qon tbd[fVx h ^f^ h {|V[eP&PM q [fY Nr1/2 M ^ l x`T h P3/2 NXb2M x h ^f^ h {i[ gUk N{ h ]_^fYH[fPQV * A n¦ NP h V_NUN{ hvh xJNXb2M=VX[ =uM T_P'jcNrYJNr[ hcg VobcYOcM=Y=VXjUP4V_N_Tr[e]rNr^ lvk TrMHYJN_M TUNXbcY g

3/2

N P h V_NNXbcTrMHM h xdNXb2MvVX[ !uUMHTrP&jcNWY Nr[ hcg VbcYJcMvY!VXjUP¯VXN_TW[`]rNr^ l ^`M VVNXbcY g n¦ n 1/2 A * b h2h VmMaY gl NXbcTrMDMuUMHTrP&jcNWY Nr[ hcg V n h NrMNXbcY NYH^w^oVX[ 'uUMHTrP&jcNWY Nr[ hcg V _] Y g z2M q [\d[ q M q [ g N h N{ h VMJNWV h xNXbcTrMDMiYDV=x h ^f^ h {|V * s

s

a1 + b 2 + c 3 b 1 + c 2 + a 3 c 1 + a 2 + b 3 s s n a1 + c 2 + b 3 b 1 + a 2 + c 3 c 1 + b 2 + a 3

tb2MuUMHTrP&jcNWY Nr[ hcg V|[ g M Y]?bVMJN ] l ]_^w[`]Vmbd[wxfNrV h xoMHYD]?b h NXb2M T K n d{ ^ hcgUk N h NXb2M'VYHPMVmMON h xoNXbcTrMHM s NXb2M [fTVX[ M g N_Tr[eMHVo[fVN h2k MJNXb2M TtY NtP tb2M TrM x h TrM sUhcg M h xNXb2MHVMN{ h uM

x NXbcTrMDMYHTrMu2YH[fT{i[fVmM q [eVr¥ h [ g N @ V?[ g ]XM'NXb2M l YDTrM h hQh x h jUTNXbcTrMHM]?b h VM g uM T_P'jcNrYJNr[ hcg VP&jUV_Nz2MD Y gUq YHTrMaNXb2MHTrMJx h TrM q [fV_¥ h [ g N n bdjUV s NXb2M'VXjUP h x h V_NNXb2MV?jUP h xHNXb2MNXbcTrMDM]_YHT q V s {bd[`]?bv[eV ¥`jUVXN 3 n T_P'jcNrYJNr[ hcg VP&jUV_N=bcYJcMiV?jUP.Y NP h VXN n 3/2

* b h2h VMY gl x h jUTuM T_P'jcNrYJNr[ hcg V n ½M]_^eY [eP NXbcY NtNXb2M VmMuUMHTrP&jcNWY Nr[ hcg V P&jUV_NijUVMYH^w^yNXb2M»M g N_TW[`M V h x=Y N^eMHYHVXN!N{ hSh xNXb2M»]rYDT q V n [ g ]_MQNXb2MHTrMYHTrM hcg ^ l NXbcTrMDM'M g N_TW[`M V hcg MHYD]?b»]_YHT qts V h PQM'M g N_T lhcg NXb2MaRTWV_N]_YHT q P'jUVXNYDu2uM YDT=N{i[`]_M n pcl V l PQPQMJN_T l Y gUq TrMJ^eYDzUMJ^f^w[ gUks {MP&Y l V?jUu2u h VM&NXbcY N[\N[eV a s Y gUq NXbdjUV|N{ h h xoNXb2M]?b h VmM g uUMHTrP&jcNWY Nr[ hcg V!YDTrM a + b + c Y gUq a + c +1 b n} x h jUT=x h jUT uM T_P'jcNrYJNr[ hcg V qUhgUh N[ g ]_^fj q M&Y ^f^M 1g N_TW[`M V 2 h xoNXb23 M'RUTWVXNv1]_YHT qts 2NXb2M g 3z l V l P&PMON_T l {M'PQY l V?jUu2u h VMaNXbcY NNXb2M h NXb2MHTN{ h ]?b h VM g uM T_P'jcNrYJNr[ hcg VYDTrM b +a +c Y gUq s Y gUq YH^w^M g N_TW[`M V h x ]_YHT q V 2 Y gUq 3 YHTrM[ g ]_^fj q M qn 1g NXb2M 2 h NXb2MH3T b + c + a bcY gUq s 1[wx{M 2 q2h [ 3g ]_^wj q M|YH^w^tM g N_Tr[eMHV h xNXb2M|RUTWVXN=]_YHT qts NXb2M g {MiPQY l YHVVXjUPMNXbcYJN hcg M h x2NXb2M h NXb2MHTN{ h ]?b h VmM g uM T_P'jcNrYJNr[ hcg V[eV b + a + c s Y gUq'gUh {{MibcYOcM 1 2 3 YH^w^ M g N_TW[`M V h x]rYDT q V Y gUq n 1 2 [ g ]_MNXb2MVXjUP h xNXb2Max h jUT]?b h VM g uUMHTrP&jcNWY Nr[ hcg V!P&jUV_N] hcg NWYH[ g YH^w^tNXb2M M g N_TW[`M Vyx`T h P-YJN^eMHYHVXN S N{ h|h xNXb2Mv]rYDT q V s Y gUq NXb2MVXjUP h x2M Y]?b|]_YHT q [eV s [wNx h ^w^ h {|V 1 NXbcY N n tbdjUV gUh NYH^w^ h x=NXb2M»x h jUT]?b h VmM g uM T_P'jcNrYJNr[ hcg V&]rY g bcYOcMV?jUP S ≥ 2 VXN_TW[`]rNr^ l ^`M VVNXbcY g n 1/2 tb2M'[ g N_M T~dYH^ [fVvNXb2MQzUM VXNvu h VVX[`zd^`M s YDVVXNWY NrM q [ g DGDGd * A n [1/2, 3/2] tb2MYDz h cM=N{ h ]_^fYH[fPQ VYHTrM= NXb2Mz2MHV_Nu h V?V?[ezc^eMN h2h s [ g NXb2MVM g VM=NXbcYJNt[wNt[fVu h V?V?[ezc^eM x h TtN{ h uM T_P'jcNrYJNr[ hcg V?jUP&VoN h z2M k TrM Y NrMHTtNXbcY g sDh Tx h T NXbcTrMDMV?jUP&VN h z2M^eMHV?V NXbcY g ® h TMXHYDP&u2^eM s NXb2M=RTWV_NVmMON h xd]_YHT q VyzU3/2 MJ^ h { bcYDVN{ h uM T_P'jcNrYJNr[ hcg VXjUPQV n M djUYH^U1/2 N h s Y gUq NXb2MiVmMH] hcgUq VmMON=bcYDVNXbcTrMHM|V?jUP&VM djUYH^UN h * 2 1/3

+H

+H



 1  0 s 0



 0  1/2  s 1/2



 0  1/2  1/2



 1  0 s 0



 0  1 s 0



 1/3  1/3  n 1/3

A ½M U g h { k M g M TWYH^w[ ?M!NXb2M|u2T h zc^eMHP n

¶ " % $~¶% n LHMJN z2MY g [ g NrM k M T s Y gUq VXjUuUu h VmM lch jSYDTrM n n ≥ 4 k [\cM g n ]_YHT q V s MHYD]?bÀ] hcg NrY [ g [ gUk n gUhcg g M k Y Nr[ cMTrM YH^ g jUPQzUM TWV s {|Tr[\NWN_M g7hcg M zUMJ^ h { NXb2M h NXb2MHT s V h NXbcY NNXb2MV?jUP h x NXb2M g jUPQzUM TWV hcg MHYD]?b]rYDT q [eV n h jYDTrM YH^w^ h {M q N hh T q M ToNXb2M]rYDT q VYHTXzd[wN_TWYHTr[w^ ln tb2M g|lch j|{|TW[wNrM q2h { g NXb2MvRUTW1VXN g jUPz2MHT x`T h P¿NXb2MRUTWVXN]_YHT qts NXb2MvVMD] hcgUqg jUPQzUM Tx`T h P NXb2MvVMD] hcgUq ]rYDT q s s Y gUq NXb2M n ... g jUPz2MHTx`T h P%NXb2M n ]_YHT qts Y gUq&lch jY q2q NXb2MHVM n g jUPQzUM TWVN h2k MONXb2MHT n ½bcYJN[fV NXb2M'V?PQY ^f^eMHV_N[ g N_M T~dYH^ ] hcg NrY [ g [ gUk VXj]?bQNXbcY N=NXb2MV?jUP%{i[f^w^oY ^\{|Y l V^w[`Ma[ g

"

[a, b]

[a, b]

1

tb2MiVMD] hcgUq YHjcNXb h TbcY q VXj k k MHV_N_M q @ DGDGd * A K NXbcYJNNXb2MiY g V_{M TyN h NXb2M ]rYDT q 2 u T h zc^eMHP%P'[ k bN=z2MNXb2M[ g rN MHT~dY ^eV n h i x T xh T MmcM g|s 2 2 h n h2q2q|s n 1− ,1 + n+1 n+1 {bd[`]?b{ h j2^ q [eP&u2^ l [ g u2YDTNr[`]_j2^fYDTNXbcYJNNXb2Mi^eM gUk NXb h xNXb2Mi[ g N_M T~dYH^YDu2uUT h YD]?b2MHV G YDV n} NyNrjUT g V h jcNyNXbd[fV] hcg ¥~MH]rNrjUTrMi[fVzcY q ^ l'h jcN h x2{bcYD]?Z & tb2M!x h ^f^ h {i[ gUk M?HYHnPQ→ uc^`M ∞ V s x h j gUq z l NXb2MRTWV_NiYHjcNXb h T s Vmb h {.NXbd[eVa] hcg d[ g ]_[ gUk ^ ln¦gUq {M hcg ^ l g MDM q N{ h&q [f¬MHTrM g N]rYDT q V=N h&q2h [wN n n ® h T h2q2qts ] hcg V?[ q M TNXb2M]rYDT q V h

1−

2 2 ,1 + n n

4 $ Dj        |

1 0 0 .. . 0

i

n





    s  s  ...     

1 0 0 .. . 0

{z

    s   

     

} |

(n−1)/2

0 1/(n − 1) 1/(n − 1) .. . 1/(n − 1)





    s  s  ...      {z

0 1/(n − 1) 1/(n − 1) .. . 1/(n − 1)

(n+1)/2



  s    }

{b2MHTrMMmcMHT l ]_YHT q c b YDV M g N_TW[`M V ni} N[fV!M YDV l N h ]?b2MH]?ZNXbcYJNY gl uUMHTrP&jcNWY Nr[ hcg»h x NXb2MHVM]rYDT q V{i[w^f^TrM ?V j2^\Nn[ g | Y VXjUP h xMJ[wNXb2M T

n+1 −1 2

1 n−1

=

n−1 −1 2

1 n−1

=

1 2

h T

1+

n+1 2

1 n−1

n−1 2

=

n

3 1 + 2 n−1

} x{MTrM u2^fY]_M hcg M h xNXb2MTW[ k bNX?bcY gUq ]_YHT q Vaz l Y gUh NXb2MHT!^eM xfNX?bcY gUq ]rYDT q s NXb2M VmM V?jUP&Vz2MD] h PM

4 $ Dj

1 1 − 2 n−1

h T

1+

1 n−1

=

3 2

n

MmcM t { M|¥`jUV_NjUVmM x MHYD]?b»ZH[ gUq»h x]rYDT q [ g HHYHPQuc^`M A n n ® h T g s h y tb2M g NXb2M|VXjUP.x h n TY g l u M T_' P jcNrYJNr[ hcgQh xn/2 NXb2M mV M ]_YHT q V{i[f^w^ zUM|M [\XN b2MHT n n 1 1 1 1 3 1 h T 1+ n n −1 = − = + 2

n−1

2

2n − 2

2

n−1

2

2n − 2

® h T=xejcNrjUTrMTrM xBM TrM g ]_M s {MaYH^fV h PM g Nr[ hcg N { h dYDTW[eY g NrV h x HYDP&u2^eM n ® h T NXb2MRUTWVXN s jUVM h xUNXb2M!^`MJxfNXbcY gUq ]rYDT q V h xH HYHPQuc^`M A Y U g q (n/2) + 1 h x −1 NXb2M|Tr[ k bNXbcY gU(n/2) _ ] H Y T V

2 b M X N 2 b | M X V U j Q P V H Y r T M q q n g n 2

1 n−1

=

1 1 + 2 2n − 2

h T

1+

n +1 2

1 n−1

=

3 3 + 2 2n − 2

n

D ® h T NXb2M'VmMH] c j VM h gUq s U h xoNXb2M'^eM xfNX?bcY gUq ]_YHT q VY gUq (n/2) − 1 h xNXb2M +1 Tr[ k b NXbcY gUq r] YDT q V XN b2M (n/2) g NXb2M|V?jUP&VYDTrM

n −2 2

1 n−1

=

1 3 − 2 2n − 2

h T

1+

n −1 2

1 n−1

=

3 1 − 2 2n − 2

n

HHYHPQuc^`M V A Y gUq Vmb h {¯NXbcYJNx h TY gl sgUh [ g NrMHT~dY ^VYJNr[eVXx l [ gUk NXb2M NXbdjUV s NXb2M ]rYDT q uUT h zd^`M P%]rY g zUMiV_N_Tr[e]rNr^ l ] hcg NrY [ g M q [ g nNX2 b ≥M[ 3g N_M T~dYH^ n @ x h T[1/2, ] hcg ¥~MH]rNrjUTrM hcg DGDGd * A ]XM TWNWYH[ g ^ l x`YH[w^eV=x h T^eHY T k M Y ^f^ 3/2]s [ g x`Y]rN K n n n≥4 |M TrMi[fVNXb2M|TrMHY ^f[\N l x h T n

8 $%1 B

n=4

n tb2M[ g NrMHT~dY ^ [1/3, 4/3] V?Y Nr[fV?RMHVNXb2M 4 ]rYDT q u2T h zc^eMHP n 4 M gUh N_MiNXb2Ma]_YHT q Vvz l s x h T i = 1 s 2 s 3 s 4 n tb2M|[ q HM Y C = (a , b , c , d ) h xNXb2Mu2T h2h x=[eViNXb2MVYHPMYHVx h i T!NXbcTrMHiM]riYDT q iV n i YDPQM ^ lts NXb2MNXb2M h TrMHPx h ^w^ h {|V [eP&PM q [fY NrM ^ l x`T h PNXb2Mix h ^w^ h {i[ gUk N{ h ]_^eY [eP&V * A n¦ NP h VXN x NXb2M uM _T P'jcNrYJNr[ c h U h g ?V jUP&VYDTrMiVXN_TW[`]rNr^ l&k TrMHYJN_M TNXbcY g 4/3 n 12 24 x NXb2M u M _T P'jcNrYJNr[ c n¦ NP h VXN 11 h U h g ?V jUP&VYDTrMiVXN_TW[`]rNr^ l ^`M VVNXbcY g 1/3 n 24

^eY [eP A *o} x{M k T h jUu|N h2k MONXb2MHTuUMHTrP&jcNWY Nr[ hcg V{bd[e]?bYHTrM!] l ]_^f[e]yVmbd[wxfNrV h x MHYD]?b h NXb2MHT s YHVy{M q [ q [ g NXb2M ]rYDT q ]_YHVmM s {M k MJNVX[ VmMONrV h x2x h jUTuUMHTrP&jcNWY Nr[ hcg V MHYD]?b n |M g ]XM s k [\cM g Y gl uU3MHTrP&jcNWY Nr[ hcg V s NXbcTrMHM h xNXb2M PP&jUV_Nz2M ^ hcgUk N h NXb2M 13 VYHPM|] l ]_^f[e]=VmMON s Y gUq NXbdjUV{i[f^w^ z2M|u2YH[fT{i[fVmM q [eVr¥ h [ g N n tb2M|VXjUP h xNXb2MM g N_Tr[eMHV=[ g NXb2MHVM!NXbcTrMDMiuM T_P'jcNrYJNr[ hcg V{i[f^w^2NXb2M TrM x h TrM|zUMiYJNP h VXNyNXb2MiV?jUP h xUNXb2MM g N_Tr[eMHV=[ g NXb2M'x h jUT!]rYDT q V s {bd[e]?b[eV n bdjUV sgUh NvY ^f^NXbcTrMHM h xNXb2M q [fV_¥ h [ g NvuUMHTrP&jcNWY Nr[ hcg V ]_Y g bcYOcM|VXjUP k TrM Y NrMHTNXbcY 4g n 4/3 ^fYH[fP [eVbcYHT q M TN h uUT h cM s Y gUq {MRUTWVXN g MHM q V h PQM g MO{ [ q MHYHV n ½M+^`MON s s . . . s n} s Y gUq4q M gUh N_MNXb2M7VMJN h xaYH^w^ixej g ]rNr[ hcg V [n]z = {1 2 b2M VmMON h x=Y ^f^ hcg MDrN h hcg M hcg N h xej g ]rNr[ hcg V[ g l n f F [n] [eV sd:h x [n] ] h jU→ TWVmM s[n] NXb2MVMJF N h [n] xUYH^w^dNXb2M!uM T_P'jcNrYJNr[ hcg V h x s Y gUq [fV q M gUh N_M q S n Áy[ cM g [n] n N{ h xej g ]rNr[ hcg V s s {MQVY l NXbcYJN f d· H¸ g [wx f (x) 6= g(x) x h TY ^f^ ® h T=Y fg YDgT_zc∈ [\N_TWYDFT l [n] e¸ h x n s ^eMJN A(f ) z2M|NXb2MaVmMON h x %¶ x ∈{[n] bd[e]?b'YJ h [ q NXbcY N[fV s f ∈ F [n] [n] f

x h TYH^w^ n x ∈ [n] } } x f ∈ S s NXb2M g |A(f )| [fV¥`jUVXN d(n) s NXb2M g jUPz2MHT h x U¶ (c¶!»¶ h x [n] n @ HM M n nK tb2M^`M YDV_Nu h V?V?[ezc^eMidY ^fjM h x h cMHTY ^f^txej g ]rNr[ hcg V f [eV s h x] h jUTWVM s G s |A(f )| {bd[`]?b[fVaY]?bd[eMmcM q {b2M g [fVaY] hcg V_NrY g Nixej g ]rNr[ hcgon @ }~g x`Y]rN shcg ^ l {b2M g [fV Y] hcg V_NrY g Nxej g ]rNr[ hcgts YDV'TrfM Y q MHTWV'P&Y l ^w[`ZDMN h u2T h cM n K p jcNi{bcY N|xej g ]rNr[ hcg fV f PQYFD[fP&[*?M |M TrMi[fVNXb2M gUh NXrN h2h XV?jUTruUTW[eVX[ gUk Y g Vr{MHT n |A(f )| [fVPQYFD[fP&[*?M q [wxtY gUq&hcg ^ l [fx n ® h T s n

:

A(f ) = {ϕ ∈ Sn | ϕ(x) 6= f (x)

n ≥ 4 |A(f )|

f ∈ Sn

iLHMJN z MRDM q Y gUq U b2M g xh T n

[n] fM 6∈ [eS f (a) = w[ fN(b) =N c N n N V h PQM s s f : [n] → { 2 b M r T M [ X ] V H M _ N M f [ V s na g gUh c h g h c h g s gUh hcg h a b c ∈ [n] a 6= b f

M [\NXb2MHT {b2M g ]XM s NXb2MHTrM[fV=V h PQM {bd[`]?ba[eV gUh N[ g s NXb2M[fPQY k M h x f n Im(f ) hcg V_N_Trj]rN YHV=x h ^f^ h d{|∈V * [n] g : [n] → [n]

g(x) =

(

f (x) d

[fx s x 6= b f[ x n

@~A K

x=b

tb2M g NXbdjUV s [fV !mg M YDTrM T # N h zUMJ[ gUk YiuUMHTrP&jcNWY Nr[ hcg NXbcY g g [fV n Im(g) = Im(f ) ∪ {d} f ½M&]_^eY [eP NXbcY N n h uUT h cMNXbd[fV s N h MHYD]?b ϕ ∈ A(f ) {M {i[f^w^tYHVV h ]_[eYJN_M|Yxej g ]rNr|A(g)| [ hcg 0 ≥ |A(f s)|YDVx h ^f^ h {|V n LHMON n!} x ϕ NXb2∈MHT¹A(g) YH^fV h s NXb2M g ucjcN 0 {i[eVM s z l @BA K s [\N!P'jUVXNϕ zUM∈N_TrA(f jM'NX)bcYJN ϕ ∈ A(g)n n ϕ V?= ϕNXbcYJN ϕ(b) = d tb2M g R gUq j]?ba s Y gUq ] hcg VXN_TWj]rN ϕ0 z l e ∈ [n] ϕ(e) = c   ϕ(x) 0 ϕ (x) = c   d

[fx Y gUq s x 6= b x 6= e f[ x s x=b f[ x n x=e

bdjUV s Y gUq NXbcYJN 0 } N[fViM YDV l N h V MDMNXbcY N 0 n t ≥ |A(f )| {i[f^w^UzUMu2T h cM q uUT h c[ q M ϕq NX∈ bcYJS NnNXb2Mu2YH[fTr[ gUk ϕ ∈ A(g) M _N h hc|A(g)| g M n h XN bcY NM gUqts 0 [eV hcg H ϕ → ϕ Y gUq V?jUu2u h VMNXbcYJN s Y gUq NXbcY N ϕ1 → ϕ01 ϕ2 → ϕ02 @ FK 0 0 n ϕ1 = ϕ 2

½M!{|Y g N N h mV b h { NXbcY N n ½M|bcYJcM!NXbcTrMDM]rYDVMHV n ϕ1 = ϕ 2 n s n tb2ϕ M g 1 ϕ2 ∈ A(g) s Y gUq [wNx h ^f^ h {|V=x`T h P @ FK NXbcY N ϕ = ϕ n 0 Y gUq ϕ1 = ϕ 1 ϕ2 = ϕ02 1 2 n n ϕ s ϕ ∈ tb2M g 1 2 / A(g) Y gUq ϕ (b) = d = ϕ (b)

8 3 8 3

1

Y gUq

2

  ϕ1 (x) 0 ϕ1 (x) = c   d   ϕ2 (x) ϕ02 (x) = c   d

[wx Y gUq s x 6= b x 6= e1 w[ x s x=b w[ x s x = e1

[wx Y gUq s x 6= b x 6= e2 w[ x s x=b w[ x s x = e2

+H

@ FK

{b2MHTrM Y gUq n [ g ]_M ϕ Y gUq ϕ YHTrMuM T_P'jcNrYJNr[ hcg V s ϕ2 (e^e2V ) = c@ FK Y 1 YDz @ FK [fPQucϕ ^f[e1MH(e V|NX1bc)Y N = c z nS¦ h s l gUq NXb2M» h cM2M djUY Nr[ hcg V s {MbcYOcM e = e 1 2 x h TY ^f^ s n} Nx h ^f^ h {|VNXbcY N ϕ = ϕ n ϕ1 (x) = ϕ2 (x) x 6= e1 e2 1 2 n ½[wNXb h jcN^ h V?V h x k M g MHTWY ^f[\N l s ϕ ∈ A(g) Y gUq ϕ ∈/ A(g) n @ FK b h ^ q VvYDV1{M ^w^ npcl @ OK s {2MbcYOcM tb2M g 0 Y gUq 0 0 s ϕ = ϕ 1 Y gUq [ g uUYHTWNr[e]_j2^eYHT s z 1l @ FK s {MbcYJcM bd[eV[fPQuc^f[eMHVNXϕ bcY 1N = ϕ1 = ϕ2 s n =c / A(f ) Y gUq x`T h PNXbd[eV] hcg N_TWY q [e]rNr[ hcg {M|] hcg ϕ]_^f1j (b) [fV gUh NTrMHY ^f[*Xϕ Yzd1^`M ∈ q MNXbcY N YDVM n

8 3

OI {i[wNXb&{bd[e]?b pcl TrMHuUMHYJN_M q YDu2u2^ l [ gUk NXb2MYz h cM'] hcg V_N_Trj]rNr[ hcgts Y gl ∈ /{SM nbcYOcMVb {MV_NrYHTWN{i[f^w^z2M&N_TWY g V?x h TrPM q [ g N h YuUMHTrP&jcNWY Nr[ hcgts Y gUq NXbdfjUV h { g NXbcY Nx h TyY gl NXb2M TrM![fVY V?j]?biNXbcY N NXb2M s f ∈ /S g∈S |A(f )| ≤ |A(g)| = d(n) g jUPz2MHT h x q M TWY gUk M nPM g NrV h x [n] n b2Mnu2T h2h xD{i[w^f^zUM] h PQuc^`MON_M q z l Vmb h {i[ gUk NXbcYJN [fx s NXb2M g |A(g)| > |A(f )| x h TNXb2M&uM T_P'jcNrYJNr[ hcg g h zdNWYH[ g M q )| = n − 1 x`T h |Im(f P n f jUuUu VmMNXbcYJN x h TV h PQM h n bd[eVPM Y g VvNXbcY N ) = [n] − {d} x h Im(f TV h PQ M Y gUq s Y gUq TrMHdV_N_∈Tr[e]r[n] N_M q N h [eV!Y f f (b) = c a 6= b c f [n] − {b} zc[ (a) ¥~MD]rNr[= hcg»hcg N h [n] − {d} n tb2M gts ] hcg VXN_TWj]rNr[ gUk g : [n] → [n] YDV[ g @BA K s {M k MON Y gUq x h TvY ^f^ s Y gUq NXbcYJN g ∈ S [ g NXbd[eViVuUMD]_[wR ] g(b) f (x) x 6= b ]_YHVmM n =½M d g MDM q g(x) N h R gU= V Q P | M e x j r ] r N [ q h g hcg ϕ00 ∈ A(g) {bd[`]?b&[eV gUnh NPQYJN_]?b2M q {i[\NXb Y gl n ϕLH∈ MJN A(f ) Y gUq

s

s

[n] − {a b} s [n] − {c d}

s

s

= {v1 v2 . . . vn−2 } s s s = {w1 w2 . . . wn−2 }

V h XN bcY N x h TYH^w^ n= MJR g M 00 xh T g(vi ) = f (vi ) =Y wi YH^fV i ϕY (vi ) = wi+1 1 ≤ i ≤ n−3 Y gUq s U g q h U g q n b2M g ϕ00 [eVY 00 00 00 ϕ (v ) = w ϕ (a) = d ϕ (b) = c n−2 1 uM T_P'jcNrYJNr[ hcg Y gUq n ½MVmMHM&NXbcY N ϕ0 (e) = d x h TY ^f^ ϕ0 V?j]?bNXbcYJN ϕ00 ∈ A(g) [ ]_M x y T V Q P M Y gUq 00 h h n g s [\NyP&jUV_Nz2MNXb2M]_YHVmM ϕ → ϕ0 ϕ ∈ A(f ) a 6= e ϕ (a) = d NXbcY N x T H Y w ^ ^

c b Y N f [ V h n s n ϕ 6→ ϕ00 ϕ ∈ A(f ) |A(g)| > |A(f )| h {¡{M]_Y g bcY gUq ^`M ^fYH[fP Y gUq ] h PQuc^`MON_M h jUT&uUT h2h x h x!NXb2M YHT q 4 tb2M h TrMHP n * ½M|zUM k [ g z l YHVVXjUP&[ gUk NXbcY N uM T_P'jcNrYJNr[ hcg VXjUPQVYHTrMiVXN_TW[`]rNr^ l ^eMHV?V 12 K ]?b @ NXbcY g ® v T M Y ? ] » b r ] D Y T n h q C i = 1s 2s 3s 4 s h2h VmMY g M g N_T l e h ]X]_jUTrTr[ gUk Y i VM ^`M YDV_N 1/3 U j P 2 z H M T x r N f [ P M y V [ X N 2 b H M ? V U j & P V ½ v M r ] H Y w ^ ^ NXb2M '¸ c¸ $U¶ i x h To]_YHT q g h g n 12 e i n bdjUV s s s c s d } x h TyM Y]?b i s Y gUq e h ]_]_jUTWV[ g NXb2M 12 uM T_P'jcNrYJNr[ hcg C ∈NX{a i M i jUPQ V?jUiP&ViYJNvP eh iV_N b2M&iV?YDbPQ g i zUM T h xNr[fPM VYDViMHYDi ]?b h xNXb2M h NXb2MHTNXbcTrMDMQM g N_Tr[eMHV hcg ]rYDT q C n ½M{i[eVbN h ] h PQu2YDTrM!NXb2MiVXjUP e + e + e + e {i[wNXb'MHYD]?b h xNXb2M uM T_P'jcNri YJNr[ hcg VXjUPQV n h NrMNXbcY NY'uUMHTrP&jcNW1Y Nr[ hcg 2 V?jUP%3{i[w^f^o4bcYOcM gUh MJ^`M PM g NrV 12 [ g ] h P&P hcg {i[wNXb u2TrMD]_[fVmMJ^ l [fx NXb2MuM T_P'jcNrYJNr[ hcg @ ] hcg VX[ q MHTrM q eN1 +[\Nre e 3 + e4NXb2Mxej ]rNr[ 2^wx+ K YDVY|PQYHu&x`T h P V M J Y [ V h h q g hcg α : [4] → [4] q M R g M q z l [4]

 1    2 α(i) =  3    4

bcYDu2uM g V N h z2M&YQuUMHTrP&jcNWY Nr[ hcgh }x {bdα[`]?b'YJ h [ q [eVV?[fPQuc^ l n αYO [ [ d(4) =P 9 V_N uM T_P'jcNrYJNr[ hcg V f [ V Y N h q gUk α h

8 3

[wx s ei = a i w[ x s ei = b i w[ x s ei = c i w[ x n e =d i

i

x

s XN b2M g NXb2M g jUPQzUM T h xuUMHTrP&jcNWY Nr[ hcg V NXb2[4] MHT¹{i[eVM s z l NXb2Mi^eMHP&PQY s NXb2M g jUPz2MHT h x n 8

n n ∈ /YOS tb2α MiP& H[e4P'jUP g jUPQzUM T h xuM T_P'jcNrYJNr[ hcg V?jUP&V{bd[e]?bYJ h [ q [ g NXbd[fV]_YHVmM [eV [\N'x h ^f^ h {|V&NXbcY N'[ g Y gl ?] b h [e]XM h x uUMHTrP&jcNWY Nr[ hcg V?jUP&V s αYJN&e^ MHYHVXNx h jUT 8 12

D P&jUV_NjUVMY N|^`M YDV_N hcg MP'[ g [fPQY ^M g N_T l n [ g ]XM e [eVNXb2MM g N_T l+hcg ]rYDT q C iV=NXbcY NMmcMHT {bd[`]?bQYHuUuUMHYHTWVNXb2M^eMHYHVXN g jUPz2MHT h x Nr[ePQeMHiV s [\Nx h ^f^ h {| l M g N_T lh xtNXbcYJNi TrMHV?uMH]rNr[\cM]rYDT q [eVY]_] h j g NrM q x h TY N^`M YDV_NYHVPQY gl Nr[ePQMHVYDVNXb2MP'[ g [fPQY ^tM g N_T l @ {i[\NXbd[ g NXb2M ]?b h VM g uM T_P'jcNrYJNr[ hcg V?jUP&V K n tbcYJN[fV s x h TvM Y]?b»P'[ g [fPQY ^yM g N_T lts {MQYHTrM k jUYDTW12 Y g N_MHM qhcg MQV?jUP ] hcg NWYH[ g M q [ g NXb2MQV?jUP h xNXb2M + ] ciT_+ ]?b h VM g uM T_P'jcNrYJNr[ hcg V s Y gUq aNXibd+ [fVV?bjUi P TrM Vdu ihcgUq VN h NXb2M]rYDT q N h {bd[`]?b'NXb2M h 12 P&[ g [eP&YH^tM g N_T l z2M ^ hcgUk V n [ g ]XMNXb2MHTrM|YHTrMYJN^`M YDV_N I P'[ g [fPQY ^tM g N_Tr[eMHV s [\Nx h ^w^ h {|V NXbcY NyNXb2M TrMiYHTrM!x h jUTVXjUPQV [ g NXb2MV?jUP h xYH^w^ uM T_P'jcNrYJNr[ hcg V n tb2M TrM x h TrM s NXb2M'VXjUP h xyY ^f^ ai +ubM iT_+ P'jccNri YJ+ Nr[ hcdg i V!P'jUVXNzUMY N^eMHYHVX12 N s {bd[`]?bPM Y g V uUMHTrP&jcNWY Nr[ hcg V12 ]_Y g2gUh NYH^w^ bcYJcMiV?jUP.VXN_TW[`]rNr^ l ^`M VVNXbc4Y g NXbcY NNXb2M n 12 1/3

8 3

n α∈S n 4 jUuUu h VmM s {i[\NXb h jcN^ h V?V h x k M g MHTWY ^f[\N l s NXbcYJN α(i) = i x h T i = 1 s 2 s 3 s 4 n tb2M TrMYDTrM uM T_P'jcNrYJNr[ hcg V?jUP&V={bd[e]?b&YDTrM] h P&u h VM q M g Nr[fTrM ^ lh x gUhcg XP&[ g [eP&YH^ M g N_TW[`M V sg YD9PQM ^ l NXb h VmM»] h TrTrMHV?u hcgUq [ gUk N h NXb2M q M TWY gUk M PM g NrV h x s s s 4} n tbdjUV s [ g Y gl VMJN h x uM T_P'jcNrYJNr[ hcg V?jUP&V s Y Nt^`M YDV_N h xNXb2M P-P'jUVXN{1 jUVM2P&[ 3g [eP& YH^ M g N_TW[`M V n ½M gUh N_M=NXbc12 Y N[fxdx h jUTouM T_P'jcNrYJNr[ hcg V?jUP&VyjUVm3MP'[ g [fPQY ^cM g N_Tr[eMHV s NXb2M g NXb2M uM T_P'jcNrYJNr[ hcg Yz h cM] hcg N_TWY q [e]rNr[ hcg b h ^ q V n tbdjUV s {M|YDV?V?jUPQM!{MbcYJcMM?HYD]rNr^ l V?jUP&Vv{bd[`]?b»jUVM&P&[ g [eP&YH^M g N_Tr[eMHV s Y gUq NXbcY N!Y ^f^ q MHTWY gUk MHPQM g NW3V|YDTrM'[ g ]_^wj q M q YDP hcgUk NXb2M uM T_P'jcNrYJNr[ hcg VXjUPQV ny [ g ]_M q M TWY gUk M 9PM g NrV]rY g2gUh Ny[ g ]_^fj q M s s 12 a b s2h T s z l V l P&PMON_T l NXb2MiV?jUP h xNXb2M q M TWY gUk M PM g NrV[eVM YDVX[f^ l VMDM g N h 1 z2M 2

c3

d4

9

3(b1 + c1 + d1 + a2 + c2 + d2 + a3 + b3 + d3 + a4 + b4 + c4 )

n

}~g NXb2MTrMHP&YH[ g [ gUk NXbcTrMHMauUMHTrP&jcNWY Nr[ hcg VXjUPQV s {MP&Y l YHVVXjUPMNXbcY N a YDu2uM YDTWV n p jcNyV?[ g ]XM[\NyYHuUuUMHYHTWV[ g YJNyP h VXNN{ h VXjUPQV gUh N] hcg NrY [ g [ gUk b s c sh T d1 @ g YDPQM ^ l K s {MP&Y l YHVVXjUPMNXbcYJN 2 3YHuUuUMHYHTW4VYHV {M ^w^ n Y gUq a +c +d +b a +d +b +c 1 2 3 4 1 2 3 4 h { VX[ g ]_M YDu2uM YDTWV[ g V h PMuM T_P'jcNrYJNr[ hcg V?jUP-^eMHV?VyNXbcY g b2 s {M!Z gUh {)NXbcYJN a 1/3 1 {bd[`]?b'PM Y g V n [eP'[f^fYDTW^ l s a + c + d > 2/3 n a1 < 1/3 b1 + c1 + d1 > 2/3 2 2 p jcN gUh {4NXb2M|VXjUP h xNXb2M g [ g M q M TWY gUk M PM g NrVP&jUV_N=zUMiY N^eMHYHVXN 2 3(b1 + c1 + d1 + a2 + c2 + d2 ) > 3

2 2 + 3 3

= 4

s

{bd[`]?b&] c h g N_TWY q [e]rNrVNXb2M|YHVVXjUPQudNr[ hcg NXbcY NNXb2M l M Y]?b&bcYOcM|VXjUP.^eMHV?VNXbcY g 1/3 n h NrMQNXbcY N!NXb2M»]rYDVM&{b2M TrM uM T_P'jcNrYJNr[ hcg V?jUP&VaYDTrMVXN_TW[`]rNr^ l ^eMHV?ViNXbcY g [eVTrM YH^w[ XYDzc^eM{i[\NXbiNXb2Mx h ^f^ h {i[ gU11 ]rYDT q ?VMJN s {bd[`]?bbcYDV uM T_P'jcNrYJNr[ hcg VXjUPQV k 1/3 11 M djUYH^UN h n 1/4

+H



+H

 1/4 1  1/4   0    1/4   0 1/4 0

+H



 0 0  1  0    0  1 0 0

   

xh T bcYHV V?jUP&V k TrM Y NrMHTNXbcY g ¦ e^ V

h s HHYHPQuc^`M nQ}~g x`YD]rN[\NvbcYDV 4/3dYDTW[eY N xtHHYHPQuc12 V?jUP&V=M djUYH^dN h Y gUnq = 4V?jUP&V=M 12 djUYH^dN h t 2 b ! M V D M ] ^`M n hcgUq g h 1/3 xh T bcYD5/3 VV?jUP&VM 12 cjUY ^UN h T t 2 b M m V | M ? M H H Y Q P c u ` ^ M = V N J M X N 2 b M T m V b 4 { X N c b J Y N h n h2k h NXb2Mi[ g nNrMH= T~dY 4^ [ gQh jUTNXb20M h TrMH4/3 P[eVz2MHV_Nu h V?V?[ezc^eM n [1/3, 4/3]

H

ª

H

h {MmcM T s P&Y gl cjM VXNr[ hcg VTrM Q P Y [ gon Y V h ^wjcNr[ hcg N h NXb2M ?]_YHT q uUT h zd^`M Px h T YH^w^ n} VNXb2M[ g N_M T~dHY ^ [1/2, 3/2] i n h2qUq n ≥ 3 HHYHPQuc^`M A Vb h {|VNXbcY N s [fxN_TWjM s NXbd[fVvTrMHVXj2^wN{ h j2^ q z2M'z2MHV_Nu h V?V?[ezc^eM n ® h T mM cM g s HHYHPQuc^`M Y gUq [wNWVN{ h dYHTr[fY g NWVVb h {4NXbcYJNNXb2Mi[ g N_M T~dYH^fV n

B1&D h

1 1 3 1 − , − 2 2n − 2 2 2n − 2

Y gUq

i

h

1 1 3 1 + , + 2 2n − 2 2 2n − 2

i

@ IHK

{ h 2 j ^ q z h NXb&z2MzUM VXNu h V?V?[ezc^eM[fxNXb2M l {MHTrMiV h ^wjcNr[ hcg V h xNXb2M ?]_YHT q uUT h zd^`M P n n n¦ TrMNXb2MN{ h [ g N_M T~dYH^fVv[ g @ IDK V h ^fjcNr[ hcg V h xNXb2M n ]rYDT q u2T h zc^eMHPx h T YH^w^MmcM g Y|V h ^fjcNr[ hcg'h xUNXb2M ]rYDT q uUT h zd^`M P &}~g u2YDTNr[`]_j2^fYDT s [eV 5/3] 4 V ^wjcNr[ £UT h nzc^e≥ MHP 4 VXj k k MHV_NrV!NXbcY NNXb2[2/3, M TrMQP& Y l zUM&P h TrMNXbcY ghcg M& h hcg N h NXb2M ]rYDT q uUT h zd^`M Px h TV h PM n}~g x`Y]rN s [\N[eV!M YDV l N h VmMHMNXbcYJN=NXb2Ma[ g NrMHT~dY ^ n n [0, 1] [eV|YV h ^fjcNr[ hcg x h T!Y gl n b2MTrMHYHV hcg [fV|V?[fPQuc^ l NXbcY NvNXb2MYOcMHTWY k M'dYH^wjM h xYH^w^ n uM T_P'jcNrYJNr[ hcg VXjUPQV[eV b2M g ]_M s NXb2MHTrM![fV=Y ^\{|Y l VV h PM!uM T_P'jcNrYJNr[ hcg {b h VmM!V?jUP [eVz2MHV_Nu h V?V?[ezc^eM n [eVYJNP h V_N n g ]_M|Y k 1YH[ gts2hcg M]rY g Vb h {4NXbcYJN ® h TvNX1b2M]_YHVmM h x [wN![eV gUh NibcYDT q [0, N h 1] Vb h {%NXbcYJN!NXb2M hcg ^ l P&[ g [eP&YH^ s V h ^wjcNr[ hcg VYDTrM|NXb2Ma[ g N_M T~ndYH= ^fV 3 s s Y gUq n tb2Mu2T h2h xNXbcYJN [1, 2] 2]N NXb2MiTrM Y MHT {i[\NXba { h TXZDV=[fV=V?[fP&[w^eYHTN h NXb2Miu2T h2h [0, x x h 1]T [1/2, 3/2]Y gUq [fV[1, ^`MJxfNy NXb2M h q s 3/2] bd[ g N * Vmb h { NXbcY NNXb2M TrMMXD[fVXNWVYJNP [1/2, _ V N | M U u H M r T & P c j W N Y r N [ i { w [ X N ' b X V U j P r T M Y r N H M y T X N c b Y g h hcg hcg k Y Y N P X V N U u H M r T & P c j W N Y r N [ V i { \ [ X N ' b ? V U j P ` ^ M V V X N c b Y s gUq h hcg g 1n 2 4 n)} V [1, 2] YH^fV h YV h ^wjcNr[ hcg x h T n = 4 ½bcYJNi[eVNXb2M»VX[wNrjUYJNr[ hcg x h T ^eYHT k MHT

B1&D

B1&D B1&D

n

G

n ®[ g YH^w^ l s [wNyVmMHMHP&VNXbcYJN s x h T s P&[ g [eP&YH^c[ g N_M ~T dYH^fVVYJNr[eVXx l [ gUk XN b2M T [fV ]rYDT q uUT h zd^`M P Y ^\{|Y l VbcYJcMi^eM gUk NXbQMXHnY]r>Nr^ l 2 n} V=NXbd[eV=N_TWjM } xV h s {b l n 1 [wNy¥`jUV_Ny{i[eVbdxej2^UNXbd[ g ZD[ gUk

f!$3 A

OO

O

O

KO

bNWN_u * O{!{!{ n PQYJNXb n j]_Y ^ k YHT lon ]rY DM q j]_YJNr[ hcg O?T YJNXb MXHYDP DGDGG ¥BTrPQYJNXb DGDGG n bN_P&^

OO

DO

bNWN_u * HPQYJNXbD{ h Tr^ qon { h ^wx`TWYDP n ] h P

cTW[`]!LHM g XY MHu2YDTN_PM g N h x JY NXb2MHP&Y Nr[e]_V Y gUqQ NrYJNr[eV_Nr[`]rV g [\cM TWV?[\N lQh x Y ^ k HY T l Y ^ k YHT ltsD¦ ^`z2MHTNrYi c A I Y g Y q Y eric [email protected]

O

M TWY gUk M PM g N n bN_P&^ p [w^f^ Y gUq V M uUYHTWN_PQM g N h x YJNXb2MHP&Y Nr[e]_V Y gUqQ NrYJNr[eV_Nr[`]rV g [ cMHTWVX[wN lQh x Y ^ k YHT l YH^ k YDT l s¦ ^ezUM TWNWYi d A I Y g Y q Y [email protected]

L

aZ*][hga5[i\e\]hm¥l$h\[l1]5a`9[Uhgu \cc,=Z hm\l1c},[v0z Z*ij]`^ hj,ij5`n5eo1cqpZ\[0[qqF`9c5^B]e^bNl$hg][ce0i*[=^¨B¦ ],dZ eeg]5p"b ][5\[ hg` [et1im]Z*a]5[q``1d5¦hjqi ij[`-p", [a e_ c,p5ij`Zgb d$a5hji\i\hm\c5l]`[Vhg^t[ ^Fn^Deol1hmclc"n]d,chrZmas^]F[]5egn^Ue|eoei\chg,pc [#Z\[`1p"q c[ ]^rneo¥cl1n[vc^[,[,a5aNi\hgc5reri\hmhmllc"]5d$`h¦5^-](§,^[c,]5Zg`1d$hji\¨/c5`-]5eo¥[e|ei\[e-]5`a¨/]5ehjij`s©fc,Zma^h\[ ij`chml[Lª-`i\[eg^oi\hjbLc ¨Vc5`hmeo[ ],Z1\c5e_hmeg]5`^Z*]hji\c5`^Hchml1[_neocpZ\[q^, /

7¢#

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1−x 1−y

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x y z

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T q f[ Vu2T h c MNXb2Mvx h ^f^ h {i[ gUkk M g M TWYH^w[ XYJNr[ hcg [ g h ^\d[ gUk u h V?[\Nr[\cMvTrM YH^ n {bd[e]?b'V?jUPN h * s s s x1 x2 . . . x n 1 n Y

xi

i=1

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a3 + b 3 + c 3 a2 + b 2 + c 2 − abc ab + bc + ca

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% O¶ n LHMJN q M gUh N_MSNXb2M7®D[`z hcg fn xh T f TrMHnP&= YH[ gUqfn−1 M Ty{b2+M g fn−2 [fV nq [ c≥[ q fkn−r @Y K @z K r=1

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|M TrM P&Y l z2MaY g lQgUhcg g M k Y Nr[ cM[ g N_M k HM T s Y gUq NXb2MMXu2Y g VX[ hcg [fV=dYH^w[ q x h T=YH^w^ ] h P&u2^em M? g jUPz2MHTWV ?V j]?baNXbcY N n z |z| < 1 z MRdDM qts Y gUq ^eMJN LHMJNY g [ g N_M k MHT 2 1 n tb2M gts x h T s {M|bcYJcM 0 < |z| < 1

r

f (z) =

f (z) =

∞ X n+r+k k

k=0

z r+1 (1 − z)n+r+1

z k−r−1

~} g NXbd[fVL HY jUTrM g NVmM Tr[eMHVx h T f (z) s XN b2Mv] h M ]_[eM g N h x z −1 [fV TrMHVX[ q jM h x M ¨tM ?V [ q jMi b2M h TrM P s YJN npcl NXb2& f (z) 0

{b2MHTrM

n + 2r r

1 = 2πi

I

f (z) dz

n

n + 2r n tbd[fVy[eVNXb2M r

s

C

q M U g h N_M V=NXb2M|]_[eTr]_^eM |z| = 1 _N TWYOcMHTWVM qQhcg ]_M] h j g N_M rT ]_^ h ]?ZJ{i[eVM n C z2M'RdDM qon ® 2h v LHMJN T Y ^f^ hcg s {MQbcYOcM |1 − z| ≥ x ∈ − 41 , 14 z C b2M g _] M s x z(1 − z) ≤

tb2M gts x h TYH^w^ hcg s z C ∞ X

r=0

x z(1 − z)

r

=

1 2

s Y gUq

|x| 1 = 4|x| < 1 n

1 2

2

1

−z(1 − z) s

= 2 x z −z+x 1− z(1 − z)

Y U { [wNXb TrMHV?uMH]rNN h g q NXb2M] hcg cM T k M g ]_M[fVSj g f[ x h rT P i h g c n tb2M j g [fx h TrP z { [\NXC ] hcg cM T k M g ]_MY ^f^ h {|VjUVaN h [ g _N M Tr?] bcY U bSNXb2MVXjUPQP&Y Nr[ hcg g k M»NXb2M[ g N_M k TWY Nr[ hcg i [ g NXb2Mix h ^w^ h {i[ gUk ]_Y ^`]_j2^fY Nr[ hcg* ∞ X n + 2r

r=0

r

x

r

I ∞ X 1 = f (z) dz xr 2πi C r=0 r I ∞ X 1 1 x = dz 2πi C z(1 − z)n+1 r=0 z(1 − z) I 1 −1 n = dz n 2 2πi C (1 − z) (z − z + x)

JIcG √

tb2M'u h ^ lcgUh P&[fYH^ bcYDVT h2h NWV 1 + 1 − 4x Y gUq p(z) = z 2 − z + x ζ+ = 2 √ √ Y gUq 1 − 1 − 4x n h NrM|NXbcY N n b2M ζ− = ζ+ + ζ − = 1 ζ+ − ζ− = 1 − 4x [ g NrM k TWY gUq YDz 2h cMabcYHVV?[fPQuc^`M|u h ^`M VY N Y gUq d z c j N ^ e [ = V [ X V [ M s hcg l ζ g q C npcl ζ+ ζ− − NXb2M&¨tMHVX[ q jMi b2M h TrM P s ∞ X n + 2r r

r=0

xr

= Res

−1

=

(1 −

z)n p(z)

, ζ−

=

−(z − ζ− )

lim

z→ζ− (1 − z)n p(z) 1 −1 = n n (1 − z) (z − ζ+ ) z=ζ− ζ+ (ζ+ − ζ− ) n @~A K 1 2 n √ √ 1 − 4x 1 + 1 − 4x

=

H

tbd[fV[fVNXb2M|TrM cj2[fTrM q ]_^ h VM q x h TrP.x h TyNXb2M]rYDVM n n ≥ 0Y h {-V?jUu2u h VM|NXbcYJN LHMONWNr[ gUk n gUq s = r − m s {MabcYJcM s n<0 m = −n x`T h P @~A K s X n + 2r r

r≥−n

xr = x m

∞ X −m + 2r

r=m

= xm

∞ X

s=0

lim

m + 2s s

5

¶% % O¶ D¸!¶

z2VMHT~cM!NXbcY N

r→∞

n + 2r + 2 r+1 n + 2r r

= lim

r→∞

∞ X m + 2s

s=0

xs =

P

$w¸ ( $

m+s

xs

(2x)m m √ √ 1 + 1 − 4x 1 − 4x

(2x)−n n −n √ √ 1 + 1 − 4x 1 − 4x

=

7 7 n ° $ e¸

r

xr−m = xm

;

¶¶ D¸

(n + 2r + 2)(n + 2r + 1) = 4 (r + 1)(n + r + 1)

{b2M g ]_M=NXb2MTWY q [fjUV h xc] hcg cMHT k M g ]XM h xDNXb2M k [ cM g VMHTW[`M Vo[eV s z l NXb2M¨ Y Nr[ h 2MHV_N n 1/4 LHMJN M _ N M X N 2 b ! M ? V U j P x 2 X N 2 b ! M m V M r T e [ H M V ® f [ W T X V N { M ] X V [ q gUh h n s hcg q MHToNXb2Mi]rYDVM n ≥ 0 n yn{MbcYOcM ® h T s n=0 y0 =

® h T s n=1 y1

=

r≥0

r

X 2r + 1

r≥0

=

X 2r

X

r≥0

r

2−

xr =

2

r=0

xr =

1 r+1

∞ 1 X −

X 2r + 1 2r

r≥0

r

1 n (−4x)r = √ 1 − 4x

r+1

r

xr

X 2r X 1 2r n 2r r x = 2 xr − xr r r r + 1 r r≥0 r≥0

JI A

H

H

tb2MS^eYHVXN&VXjUPQP&Y Nr[ hcg YDz h cM] h TrTrMHV?u hcgUq VN h XN b2M k M g HM TWYJNr[ gUk xej g ]rNr[ hcgh x!NXb2M √ VmM cjM g ]XM h x YJNrY ^eY gg jUPz2MHTWV|Y gUq [fVZ gUh { g N h M cU j Y ^ n 1 − 1 − 4x /(2x) |M g ]XM s y1

= =

2

− √ 1 − 4x

1−

√ 1 − 4x

= 2x ! √ 1 − 1 − 4x n 2x

1 √ 1 − 4x

®jUTNXb2MHTrP h TrM s x h T s {MbcYOcM n≥1 yn − yn−1 = =

X n + 2r r

r≥0

−

X n − 1 + 2r r−1

r≥1

4x −

n − 1 + 2r r

xr =

√ 1 − 4x + (1 − 4x) √ 2x 1 − 4x

xr

X n + 1 + 2r r

r≥0

xr+1 = xyn+1

n

bcY gUk [ gUk Nh s {M h zNrY [ gts x h T n ≥ 2 s NXb2M|TrMD]_jUTrTrM g ]_M|TrM ^fY Nr[ hcg n n−1 n xy − y +y = 0

H

n

n−1

n−2

Y gUq NXb2M]?bcYDTWYD]rN_M Tr[fVXNr[e]T h2h NWV tb2M]?bcYDTWYD]rN_M rT [fVXNr[e]vM cjUYJNr[ hcg e[ V 2 xλ − λ+1 = 0 √ YDTrM | M ] X M n g s λ = 1 ± 1 − 4x /(2x) yn = A

{b2MHTrM

1+

!n √ 1 − 4x 2x

+B

1−

!n √ 1 − 4x s 2x

Y gUq YHTrMixej g r] Nr[ hcg V h x l MJNN h zUM q MON_M T_P'[ g M qon A B x JM NWNr[ gUk n = 0 s {M h zNrY [ g 1 s A + B = y0 = √ 1 − 4x

Y gUq VMJNWNr[ gUk s {M k MJN n=1 A

1+

√

1 − 4x +B 2x

1−

√ √ 1 − 4x 1 1 − 1 − 4x = y1 = √ 2x 2x 1 − 4x

n Dh ^\d[ gUk @ h T!VX[eP&u2^ l z l [ g VuUMD]rNr[ hcg K s {M'R gUq NXbcY N A = 0 Y gUq B = √ 1 1 − 4x |M g ]XM s x h T YH^w^ s n≥0 1−

1

yn = √ 1 − 4x

!n √ 1 − 4x n 2x

h P&YZDMNXb2MiTr[ k bNV?[ q MiPM Y g [ gUk xej2^ Y N s {MiTrMO{|TW[wNrM y YHV x=0 n yn = √

$ ttb2M]rYDVMv{b2MHTrM

1 1 − 4x

n<0

2 √ 1 + 1 − 4x

n

n

{|YHVbcY gUq `^ M q YDV=[ gQDh ^fjcNr[ hcg} Yz h cM n

n

JI2

=@B- 8 - 6 ='1 !<- ' L+4 J %": 4 6 !56, @ ' C 6P! % @!8 "M1 3 ! C & 6 ') 3 '(6 ';C !; !, ' , ;5F ?H 6 %- , !>"'5 , & & % & ! <: !?4 C % , ') N? J 6D<6 B* !6 @B 3 J3 6! ?- % % %, & % &%('*)$+,).-/013@FHGI )>D<.' I$NvJVIBC"<'w/ -$GPNV8V) 2<>!FM<" di d2q"b!cwZtiZts!dKZ\[idn$df=c k Zti*j > 1 ^["] aPd@i a b c d5q(b!c*ZriZ s!d fbs!di*j!^$i fd^$a?[ "g md$f=cncA^ iZMc*e !Z\["n am+1 + bm+1 + cnm+1 =1 a b c (m + n)1+ 1 − man

+

1 − mbn

+

1 − mcn

≥

m n

n

?4 ) />;GM-1.32G9C?/`/.0.FQGQ)' jDd`n@Zts!d$[5Z\["d "^arZri k jZM_wjmZ\c^[b sZPb "c^i=id$g2qi ib n$d$["df=^$a\Ztuwd !#"%$'&)(+*-,.0/1* fb aMdg .23 =..$ 045@ 76V.. 3 04988$# i*jDd cw^.g2d qDfb!q(b!cAdf Z\cm_aMd^fa eP^$aMcAd ^cKci^id] : jZ\c k ^cKq"bZM[id.] b !i cL dLs!d$f=^aVfd^.]"d$f=c Yb!fb!["dKi*jZM["n cAb!gd2beVi*jDd2]"d$["b!g5Z\[D^$ib!f=c`beVi*jDdmidfg2cRb! [ i*jDdaMd$e\iVcwZM]"d_ b DaP]Kd^cwZr;a d 0 = @BAC@ED Y DGFHAD@ c*Ii (]Dd[i J dcid$f [ K ZP_wjZMn^.[ D [!Zts!d$f=cwZtEi L ^a\^.g5^$uwbDb K @7DCMON ["bid]5i*j!^ilZre √ =P b UI= 1/< DC2M! n =JN 2 aQS=RTFV NCWX ^.["] c = (1 − 1/√2)1/3 i*jDd[ 1 − man m D fYc Da\Z\["d["n !g2[D^cwZ "g Z@ [D[Dc [f (]_ \ PN "cifZ\^ = L_@B?`Ua<XQbWXM
JI

7 7 n

G

% X¸ ~¶ ?¸ '¸)$ $ e¸ d¸ °c¶m¸ '¸ dº D¸ % ¸ $ d¸ $)$~¶!(d¶ ¸ ~¶ d·H¶ o°O² t¶ % % O¶ n LHMJN zUMNXb2M]_M g N_TrM h x n [ g ]XM Y gUq M YDTrMvNWY gUk M g ONyN h Y N Y gUq Y N Γs TrMHV?uMH]rNrM [\cMJB ^ l s {Mbc YJcCM A................................................... E Γ B C ........ ............ ....... ....... .. .... . .... s ..... .... ...... ◦ n b2MHTrMJx h TrM s NXb2Mau h [ g NrV . . Γ ... .... . .... ∠OBM = ∠OCM = 90 O ... .. .. ...... . . . . .... . .. . ... Y f ^ e [ M a Y _ ] f [ r T _ ] ` ^ v M i { \ [ X N b f [ D Y Q P J M r N H M T . . . . . . . gUq hcg q s n .... . . .. .... ... .. .. .. B M OM .. .. ... .... .. .. .. .. . .. f [ V ½MbcC YOcM U z H M _ ] Y U j m V M . . . . . . . . s .... . ... O .. ... . . . . .. . ∠M BC = ∠BAC M B . ... ..... .. ............. . . . . . . . . . ... .. .. .. ..... F . ... NrY gUk M g N'N h Y N ^fV h s z2MD . . . . . . . n ¦ s ... .. .. ... .. .... ... . . . . . . . . . . . . . . ... .. . Γ B ∠BAC = ∠M F C ..... ..... .. . . . . . . . . . ... .. .. . . ]_Y jUVmM .. ... ......... ... n tbdjUV s ∠M BC = ∠M F C s {bd[e]?b ... ... .... ........................................................................................................................ ... .. .. .. ..... [eP&u2^w[`M M V!NXF bcY N k AB Y gUq YHTrM] l ]_^f[e] np jcN{M&YH^e .... .. ... .......... s s s ....... C B .................. . ... .......... B M C F .... ................. ... .............................. ................. TrMHY ql Z gUh {NXbcY N .... ... s s Y gUq C ^w[`M hcg NXb2MQ]_[fTr]_^`M{i[\NXb . .... ... ... D...... .... B M ... ... .... .... .... .. .... ...... q [fYDPQMJNrMHT OM n b2MHTrMJx h TrM s F ^w[`M V hcg NXbd[eVVYHPMa]_[eTr]_^eM n .... ... .. ... .... ......... ............ [fVNXb2MQP&[ q Xu h [ g N h x |M g ]XM s .. ◦ n b2M g ∠OF M = 90 F M NXb2M]?b h T q n DE ¹¶F¸ U¶ &³S¸)$'$ $'$¹¶&( ¸ ~¶± e¸ ~¶ n ° $ e¸ n M lcq MF{i[w)N Vtb2M h TrMHPV_NrYJN_M V *o} xYvN_TW[eY gUk ^`M[eV[ g V]XTr[ezUM q [ g Yi] hcg [`]VmMH]rNr[ hcgts NXb2M g Y gl ^f[ g M&] hcg ¥`j k YJN_MaN h»hcg M'V?[ q MPMHMJNWVvNXb2M h NXb2MHTN{ h VX[ q MHVv[ g ] hcg ¥`j k YJN_M u h [ g NrV @ VMDM A K n |M TrM={MbcYOcM ] hcg ¥`j k YJN_MN h {i[wNXbTrMHV?uMH]rNtN h s V?[ g ]XM DM BC Γ t d b U j V NXb2Miu h ^`M h x [eV [ _ N M W T m V H M r ] r N V Y n s DM g gUq AC YJNu h [ g NWV{bd[`]?baYDTrM BC M AB bcYDTrP hcg [e]] hcg ¥`j k YJN_M Vy{i[\NXb|TrMHV?uMH]rNoN h NXb2MvVM k PQM g N ny [ g ]XM DM PMHMJNWV AB DE Y N & P U j _ V = N U z M X N 2 b i M & P [ X u [ N x s q h g h DE n ∞ F * A nwUn`)n h DMJNrMHT s X¶ " e¸~·H¶ y¶ »¶ s M q [\Nr[ hcgts2 u2Tr[ gUk MHT s AHC d n

2

77 7

K

$

f !

' '4 L @<J 6 @ " 8; 6 F 6 * ! E -$1 1J - - -L J- 6 C CM @# 6M <%"3!: ?4 %? , ')?C% , ,0&<, 6 &",0& & !- %:4" 6<- &"1 ,0C&# -D- % -4";6 @ "L - '( !5C & J # &L & ! & : J # %, 7 %% ,08@ & 5 47- ! && J 1K,11 - & JL+6 @ C C M & @<M J % * ; J : @ & !FE/@'=-*FHGMFQBH<50"/ 4D"3$<NvJ?<'wB/ X )>"<$NUT+;<FP>7 di dK^5q"b!cwZtiZts!d`ZM[id.n$d$f Y!b!f d$^_wjq"b!cwZtiZts!d`ZM[id.n$d$f k aPd@i f d i*jDd n Y.Z b![D^__Z?[ "g df 6 i*j!^i:Z\c f = 1 f = 1 ^.["] f = f + fk e9b!f:^ara k"! 1 2 k+2 k+1 k fbs!di*j!^$i k≥1

n X

k=1

2 fk+1

!

n X 1

k=1

f2k

!

≥ n2

JIDI

° $ e¸

o¶ $¹¶<

W¸

< ~¶ W¸

; $w¸ ° °F²

$'$¹¶&(c¶

?¶

tb2M'TrMHVXj2^wNx h w^ ^ h {|Vv[fPQPQM q e[ YJN_MJ^ l x`T h P¤NXb2M h z2VMHT~dYJNr[ hcg VNXbcY N fn+1 ≥ n n x h T=MmcM T l u h VX[wNr[ cM[ g NrM k M T Y gUq P 1 1 n ≥ =1 n f2k f tb2Mix h ^w^ h {i[ gUk ]_^eY [eP k [ cMHVYk=1 VbcYDTru M T[ g M 2djUYH^w[wN l x h T * n 6= 3 n ® h TMmcMHT l u h V?[\Nr[\cM[ g N_M k MHT n MXD]XM ucN n = 3 s

+H

8?Dj $

n X

fk+1

k=1

!

n X 1

f2k

k=1

!

≥ n2

n

} N[fVVXN_TWY [ k bNrx h T¹{|YDT q N h ]?b2MH]?ZaNXbcYJNNXb2M]_^fYH[fP¤b h ^ q Vx h T s s n=1 n=2 @ zcjcN s YH^fYDV scgUh N K n Y gUq n=4 n=3 h YHuUuc^ l [ gUq j]rNr[ hcgts {M&YDV?V?jUPQM'NXbcYJNNXb2M]_^fYH[fPb h ^ q Vix h TvV h PQM'[ g NrM k M T n K s Y gUq 2 Y gUq P b M g ]XM s 1 4 @ V?[ g ]XM n ½M|bcYJcM f n≥4 ≥ 2n n≥4 ≥ n+2

k=1

n+1 X k=1

=

fk+1

!

n X

k=1

n+1 X

1

fk+1

3

!

f2k ! ! n X 1

k=1

f2k

k=1

f2k

n X 1

+ fn+2

k=1

n

≥ n2 + 2n(4/3) ≥ (n + 1)2

f2k

!

n+1 X

+

fk+1

k=1

!

1 f2n+2

@ - ! 1 - ' 1 'M !4 C %"C : L< !L ?4- C 1 6 E')- ?-! 8 6 % H- 5( &6 H % & p!*;<*!?4"! ' %% ,' 6 ') ?@ J - 66 & ! b- - !5C8 % ?@ 6 66 ' - - - !C % M C 6 1
R :. 5$# %('*)$+,).-/0 1.3 < @F 0 )D/ /@'NR-$GI0"/>;GPN'=FP>FHGP3 4?)X\X /@W/@Nv4D< 1'=F 0W!/ N E di di*jDdR^["n@aPd$c be?^if=ZM^["n@aPd fbs!di*j!^$i A B C tan2

A 2

tan

A 2

Z\e;^["]2b![!a;KZre

+ tan2

+ tan

B 2

B 2

+ tan2

C 2

C 2

+ tan

< 2

< 2

JI

° $ e¸

+ U¶

U ± ²

>

;

»³¸'$)$

$'$¹¶&( ¸ ~¶'± e¸ ¹¶

[ g ]_M A + B + C = π s {M|bcYJcM tan

Y gUq NXbdjUV s tan

A 2

= cot

A 2

tan

B 2

+ tan

B+C 2

B 2

B C 1 − tan tan 2 2 s = C B + tan tan

2

tan

C 2

2

+ tan

C 2

+H

tan

A 2

= 1

n

|M g ]XM s [fVM dj2[\dY ^`M g NN h A B C tan2 + tan2 + tan2 < 2 2

2

tan

A 2

+ tan

A 2

+ tan

{bd[`]?b&]_Y g zUM|M?DuUTrM VVM q YDV tan

2

+H

B 2

B 2

+ tan

+ tan

C 2

2

C 2

< 4

< 2

s

s

V?[ g ]XM!NXb2Mi^eM xfNVX[ q M h xNXb2Mi^fYDV_N[ g M djUYH^w[wN l [fV]_^`M YDTW^ l u h V?[\Nr[\cM n

9 *

;

§

w

9

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
2

2

0

2

T!)XHIDGMF )> 1.3 6 ?Fr+"+)CD>D-@)>DNVJU/@</@'LGQ)>!Nl87N UT jDd c*i=^$idg2d[i,bei*jDd q"fb aPd$g&Z\cv["biUcY _ZMd[ia Rq"fd_ZMcAd ?J dcLj!^$a\aq"fb$s!d i*j!^$i"i*jDdln@Z s!d[ i^["n$d[i;_*b!["]ZriZMb! [ZMc?if[ (d:Zre.^.["]b![!a Z\e$i*jDdEif=ZM^["n@aPd$c7^.fdEZMcAb!cL_d$aMdc k Zri*jmfd$cAq"d._iZ s!dR^q(Yd e ^["n@aPd$c^i A ^["] A0

JI ª

H

pcl NXb2MUTr[fY gUk ^eM }~g M cjUY ^f[\N l [ g 4A0 B 0 C 0 s {MiP&jUV_N=bcYJcM

H

(s − a) + (s − b) > s − c

s

NH H

{i[wNXbiVX[eP'[f^fYDT [ g M cjUY ^f[\Nr[`M V h zdNWYH[ g M q z l uM T_P'jcNr[ gUk s s n b2MYz h cM[ g M cjUY ^f[\N l a b c V?[fPQuc^f[wR M VN h n bdjUV s MHYD]?bV?[ q M h x 4ABC P'jUVXNvM?D]_MDM qhcg M djUYDTN_M T c > s/2 h x2NXb2MuM Tr[fPMON_M T n dj2[\dY ^`M g Nr^ l s NXbcTrMHMvNr[ePQMHVNXb2M!Vmb h TN_M VXNVX[ q M!P&jUV_NM?D]_MDM q NXb2M|V?jUP h xNXb2M h NX2s b2MHTyN{ h VX[ q MHV n h NrMNXbcYJNNXb2M|VMHP'[euUMHTW[ePQMJNrMHT h x * 0 0 0 [fVbcYH^wxNXbcY N h xUNXb2M

*H

4A B C

4ABC

(s − a) + (s − b) + (s − c) 3s − (a + b + c) s = = 2 2 2

s0 =

½MQ{i[f^w^ g MHM q NXb2M»x h ^w^ h {i[ gUk Z gUh { g [ q M g Nr[\N l* B tan =

p TW[`MJ« l s

2

{M)bcYJcM

tan

B 2

V?[ gUk NXbd[eV=[ q M g Nr[\N l s {M k MJN tan

A0 2

= = =

Y gUq

tan r r

B0 2

=

r s−b

(s0 − (s − b))(s0 − (s − c)) · s0 (s0 − (s − a))

(s0 − s + c)2 = (s0 )2 2c s −1 s

tan2

tan

B 2

= =

r

=

r

[ABC] s(s − b)

=

r

r

n

(s − a)(s − c) n s(s − b)

(s − a)(s − c) n s(s − b)

(s0 − (s − a))(s0 − (s − c)) s0 (s0 − (s − b))

(c − s/2)2 c − s/2 = (s0 )2 s0

VX[ g ]_M

c > s/2

(s − a)(s − c) s2 − (a + b + c)s + bs + ac = s(s − b) s(s − b)

−s2 + bs + ac ac = −1 s(s − b) s(s − b)

n

tb2M»] hcgUq \[ Nr[ c h g NXbcY N tan B [fV|NXb2M k M h PMON_Tr[e]|PM Y gh x tan A0 B0 2

2

H

2

]rY g NXb2M TrM x h TrMz2M!{|TW[wNWNrM g [ g NXb2Mix h w^ ^ h {i[ gUk M cj2[ dYH^eM g N x h TrPQV * ac 2c s −1 = −1 s(s − b) s c(b − c) s = 0 s(s − b) s b = c n b0 = c 0

Y gUq

JI

»¶ n¤} N'[fV]_^`M YDTaNXbcYJNa{b2M g { M[ g NrMHTr]?bcY gUk M Y gUq { M{i[f^w^!bcYOcM s A B 0 0 @ M dj2[\dY ^`M g rN ^ l s 0 K n [wx2Y gUqhcg ^ l [fx A B A = tan tan tan2 a=c a = c0 2 2 2 z l u2T h2q j]_MHViYV h PMF{bcY N q [f¬MHTrM g N] hcg ]_^wjUV?[ hcgon }~g N_M TrMHV_Nr[ gUk ^ lts TrM u2^fY]_[ gUk

+H

B

C

}~g NXbd[eVy]_YHVmM s YVX[eP'[f^fYDTtYHT k jUPM g NoVmb h {|VNXbcYJN tan2 C = tan A0 2 @ h TM dj2[\dY ^`M g Nr^ l s 0 2 0 0 K n [fxtY gUqQhcg ^ l [wx a=c b=c a =c b = c0

f

. v

7

+H

tan

B0 2

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
R 8!2$# %('w)$+,).-@/.0&13 I.<> wJU).-B)&8V) /@'w) < '?I/ *N U>FE/@'=-*FQ0<00"/ !<$XrXP<0")X\F 0N !<$XrXP<0")X\F 0NUT@+;<$FM>U di ^.["] d`i*jDd aMd["n i*j!cbe7i*jDdKgd]!Z\^.[ i*jDd ^.["n@aMd Z\cLd_ib!f ^.["] i*jDd ^matiZraIi (w ]Dd a fd$cAq"d.h_aiZ s! d$a ePfb!g i*jDd fZMn$j i w^["n@aPd] s!dfHidYe A be if=ZM^["n@aPd ib`i*jDdRjDq"bid$[ "cLd H M "qDq(b!cAdi*j!^ivi*jDdc*ZP]Ddc a ^["] c ^fdq e.d]KZ\[maMd["n i*j ABC j \ Z M a d * i jDdaPd$["n i*j2be?cwZM]"d b s^.f=ZPd$cl]c d._ivib a > b ≥ c k R s^$a "^id m −h lim b→c w − h T!)XHIDGMF )> 1.3 F C?)(INV%()X 4?) I.>FHGP324?)XrXQ/ W!/ N FM>7G / ' < />!N $N UT YZ\f=c*i k d["bid2i*j!^ii*jDdqDfb aMdg ^c ci^id] Z\c`Z\["_*b!ffd._i @ elc*ZP]Ddc a ^["] jDd ^.f d qe.d.] ZM[ aMd["n i*j i*jDd[ cLb ZMcRcwZM]"d b i*jDd i*j!^.n$b!fd^[ jDd.b!fd$g o c h J q"fb aPd$g k Zra\a d5_*b!ffd._iZre k dK^.cwc] "g2d i*j!^ib![!a c Z\cqe.d.] d k Zra\a,]!fb!q i*jDd _*b!["]ZriZMb![ ∠A = 90◦ ^.["]5cAbats!di*jDdq"fb aPd$ghe9b!f:^[ KifZ\^.["n@aMd k Zti*j a > b > c ^.cw]c "gmZM["ni*j!^i c Z\%c q e.d] jDd$fd k Zra\a d ["b _b![!ie "cwZMb![ Z\e k d A ]Dfb!qKi*jDdYc cA_*fZ\q! i=cp^.["] k f=Zrid m w ^.["] Z\[Dc*id^.] be ^.["] diRi*jDd h gd]!Z\^.[ i*jDd^m["n@aaPd w aZMcAd._ib!f h^.a["] i*jDd^a iZrIi (]Dd ePfb!g s!dfHiYd e A Z\[id$f=cLd_i`i*jDd c*ZP]Dd ^i?q"bZM[i=c M E ^["] D fd$cAq"d._iZ s!d$a BC c i*jDd i*j!^.n$b!fd^[ jDdb!fdg k d j!^s!d j "c m2 − h2 = M D 2 a

a

a

a

M

2

ZMgmZ\a\^.f=a;

m−h =

MD m+h

w 2 − h2 = ED 2

.. ............. .... ......... ... ........ .... .... . . . ... ... .. .. ... ....... ..... ..... .... .... .. . .... .... .. .. .... .... ... ... . . ..... .... .... ... . . . . . . .. .. .. .. ..... .... .. . ..... ... .... .. ... . ... ... .. ... . . ..... . .. . . .... ... ... .. .. .... .. ... . ... . . . . . .. .. ... ... . . . . . . . . .. . .. ... . . . . . . . . . .. ... .. ... . . . . . . . ... . ... .. ... . . . . . . ... . ... .. .. . . . . . . . ... . .. . ... . . . . . ... . . .. ... .. ... . .. . . . . . ... ... .. .. . . . . . . . ... . .. . ... . . . . . . . ... . ... ................ ... . . . . ... . ... ... .. ... . . . . . ............................................................................................................................................................................................

m

B

k jZP_wjKZ\g2q!a\ZMdc:i*j!^i

w

ME 2

w−h=

ED w+h

h

D

C

JI ½M|Y ^eV h bcYJcM MB =

Y gUq

a 2

s

ac b+c

EB =

a2 + c 2 − b2 2a

BD = c cos B =

{b2MHTrMNXb2M^eYHVXN&M?DuUTrM VVX[ hcg x h f^ ^ h {|VQx`T h P;NXb2M L YO{ TrMHVXj2^wNWV s {M h zdNWYH[ g MD

Y gUq

ED

tb2M TrM x h TrM s m−h w−h

=

s

h x h ?V [ g HM V n

VX[ gUk NXb2M VmM

(b − c)(b + c)

= M B − BD =

2a (b − c) (b + c)2 − a2 n = EB − BD = 2a(b + c)

M D 2 (w + h) ED 2 (m + h)

(b + c)4 (w + h) n 2 (b + c)2 − a2 (m + h)

=

h { s n |M g X] M s lim m = lim w = h b→c

s

b→c

lim

b→c

m−h = w−h

(2c)4 (2c)2

−

2 a2

n

}~g NXb2M|]_YHVmMv{b2M g ∠A = 90◦ s {M|bcYJcM lim a = c√2 {b2M g X] M s b→c

lim

b→c

m−h w−h

= 4

n

'>'4 E L @# 6
%, 7% 85- 54 ! & C%"@ : - -$ C -$ 16 1 ;- % !- 6? C ,0& !6& E : - M 1 L C L B6;5?H %& ' '('; , 5 & % !54 %, ') 6 @ & * % B ,0&,(:;%& 6 J6 - @ C @ 6 '('; ;5?H !
a

∠A = 90◦

JI C

j

HGG O¶ K K s $f¸( $'$ K $f¸(o° ¸ n

n G GJI * C d¸¹·H¶ X ¸ U¶ $'$

"

%" -´ »¶ ³ " d¶

%

%

LHMJN Y gUq 0 0 zUMN{ h TW[ k bN|N_TW[eY gUk ^`M V{i[wNXbTr[ k bNaY gUk ^eMHV'YJN ABC A0 B CNXb2M[ NrMHTW[ TY A Y gUq x Y H Y r T M Q n } U g q g h gUk ^eMQzd[eVMD]rN h TWV h xY gUk ^eMHV A Y gUq A0 s 0 0 A w w a a TrMHV?uMH]rNr[\cMJ^ l s u2T h cMvNXbcYJN 0 0 0 s {i[wNXbM cjUY ^f[\N l [fxY gUq'hcg ^ l [fx z h NXb aw w aM a a0 ≥ bcb c Y gUq D Y r T M e [ V m V _ ] e ^ H M V h n ABC A0 B 0 C 0

H

O¸ · " ?¶ ´ d¸ n zUMaNXb2M'YH^\Nr[wNrj q Mx`T h P LHMJN np MD]rYHjUVMaNXb2M'Vmb h TN_M VXN q [eV_NrY g ]XMx`T h P Nh [eVYh^ ahcgUk NXb2MiuM T_uUM gUq [`]_j2^fYDT s A {M|bcYJcM BC ° $ e¸

*H

awa ≥ aha = 2[ABC] = bc

A

n

djUYH^w[wN l b h ^ q V&[fx!Y gUq hcg ^ l [fx NXbcY N[fV s [wx!Y gUq hcg ^ l [wx [fV 4ABC a^eMHVYDV{Mv{i[fVmb {M|]rY [eV h Vm]_M ^eMHV n tbd[fV[fVN_TrjMix h TYHVP&h Y gal = N_Tr[fY w c b J Y c M M gUk g hcg h T PQY gl N_TW[eY gUk ^`M V[ g V_N_M Y q¤h xN{ hn [ g ]XM-NXb2M cjUY g Nr[wNr[eMHV[ g h ^\cM q YDTrMYH^w^ u h VX[wNr[ cM shcg M]_Y g P'j2^wNr[fu2^ l NXb2MTrM VuUMD]rNr[ cMVX[ q MHV h xMHYD]?b[ g M djUYH^w[wN l N h2k MJNXb2M T {bd[f^eM|P&YH[ g NrY [ g [ gUk NXb2Mi[ g M cjUY ^f[\N ln »¶ n b2Ma] hcg ]_^wjUV?[ hcg ] hcg Nr[ g jM VN h b h ^ q [fx NXb2Mazd[eVMD]rN h TWVYHTrMTrMHuc^eYD]XM q z l Y glQh NXb2M T=]_Mmc[fY g V n

?

U

H

H

+H

'>'4 E L @# 6 6 ';& E , : '); 56 ?H %H , % ! @ 5H5 & *;% *
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0

0

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j d

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c /d 3s2 − r2 − 4Rr

≤

8sRr

X ma s2 − r 2 − 4Rr R ≤ 7sRr e f7eLg h e awa

X m2 c d 3 4R + r a . ≤ ≤ 2 2 4 4R e f7eLg h e b + c

i kjCl ) [lm3n Pl'oLPprqs>tA u s>tv"lw i w'xtyw i t z)@E?$F9-C{|,A1 V)-7,v-9B /--9BC,A?$@:B)-}@E1 ,~ 5 /;@-m2^@
e f7eLg h e awa

=

3 a

6

>

7a

=

s2 − r 2 − 4Rr 7sRr

,>{3@;E;C@<1CF6-6,/)=J ?7VK ,-9B /_

X ma 3s2 − r 2 − 4Rr s2 − r 2 − 4Rr ≤ ≤ 8sRr 2sRr e f7eLg h e awa

_

,35 F],>-9BC,3XV ;E;
= 2(b2 + c2 ) − a2

wa2

=

abc a2 + b 2 + c 2 ab + bc + ca

= = =

z ?7V H-9BC,3a ?$F9-'-k{|V@=C,1 -7@L-7@,FD/

V{{|,`V

.

bc (b + c)2 − a2 (b + c)2 4sRr 2s2 − 2r 2 − 8Rr s2 + r 2 + 4Rr

VK , {|,`:,-

.

m2a (b + c)2 2(b2 + c2 ) − a2 = · 2 wa 4bc (b + c)2 − a2

cm d

FP,?mK ,-9B /-

2bc ≤ (b + c)2 − a2 ≤ 4bc

c d

{|B @<4B C B ,I;,XE->@E1 ,~ 5 /;@-m2[@-6?75, ,46/5 FP,I@L->@EF|,~ 5C@ K /;<,1 -^-7V 2 b @@EF @
,~ 5C@LK /;,1 -v-7V

|b − c| ≤ a

(b − c)2 4bc

{|B @4BZ@EFy-6?75,3XV ?v/1)2-6?7@E/1 :;<,

+1 ≤

-9B /-v@EF

(b − c)2

(b + c)2 − a2

z)?7V H

(b − c)2

+1 ≤

2bc

c d {|,`:,+1

R

(b + c)2 2(b2 + c2 ) − a2 b2 + c 2 ≤ ≤ 4bc (b + c)2 − a2 2bc

,47/;;E@E1 : c d {|,`:,-

(b + c)4 16b2 c2

m2a

≤

≤

wa2

(b + c)2 (b2 + c2 )

V{ 5 F9@<1 :^-9BC,`,/F$2 87-6VC89J ?7VK ,3@E1 ,~ 5 /;@-m2

8b2 c2

(b + c)2 ≤ 2(b2 + c2 )

{|,`V .

-7/@<1

(b + c)4 m2a (b2 + c2 )2 ≤ ≤ 16b2 c2 wa2 4b2 c2

C/@E1 :3F]~ 5 /?7,`?7VCV)-7Fy-9B ?7V 5:BCV 5 -y/1 =*= @LK @= @<1 : (b + c)2

ma

≤

4abc

awa

bwb

e f7eLg h e awa /1 =

≤

X ma e f7eLg h e awa

X b2 + c 2

e f7eLg h e

≥

:@LK ,F

mc {|,|V -7/@<1 . cwc

a2 + b 2 + c 2 abc

s2 − r 2 − 4Rr

=

2sRr

1 (2s2 − 2r 2 − 8Rr + s2 + r 2 + 4Rr) 8sRr 3s2 − r 2 − 4Rr 8sRr

=

a

  X (b + c)2 X X 1  = a2 + bc 2abc e f7eLg h e e f7eLg h e 4abc e f7eLg h e

=

/F046;
2abc

=

2

b2 + c 2 2abc

≤

{|BC,?7,|,~ 5 /;E@L-"2IVC49465 ?$FA@EX /1 =*V 1 ; 2@X c 0F@E1 :F@EHI@;
.

~ 5 /;@-m2*VC46465 ?$FD@E1

.

V)-9BZ@E1 ,~ 5 /;@-7@<,FA@X /1 =*V 1 ;L2Z@EX

i kjCl )[lm3n Pl'oq#prq)s Y P}Cl< Emw & j w 7 6 ,`JC?7VK ,>-9B /_ 3 4

<

X

m2a

e f7eLg h e b2 + c2

≤

4R + r 4R

a = b = c

@<1 46, 2 2 2 2 @-r@
X

a2

< 3 e f7eLg h e b2 + c2 @<1 46,Z-9BC,-6?7@E/1 :;<,@EF/465 -6, -9BC,*46V F@E1 ,F3V XG/;;\/1 :;,F/?7,JV F9@-7@ K , 0 F9@<1 :|-9BC, V F9@<1 ,+/{ {|,|V -7/@<1 . 2−

X

a2

=

R

≤

X b2 + c2 − 2bc cos A

= 3−

X 2bc cos A

b2 + c 2 e f7eLg h e b2 + c2 e f7eLg h e e f7eLg h e 1Z-9BC,|V)-9BC,?AB /1 = 5 F@E1 :>-9BC,^{|,;;8)1 V{`1Z@=C,1 -7@L-"2 cos A + cos B + cos C = 1 +

/1 =Z-9BC,`,/F$2@E1 ,~ 5 /;@-m2 X

e f7eLg h e

2

b2

a + c2

2

2bc ≤ b + c

2

{|,`V .

-7/@<1

b2 + c 2

< 3

r R

X 2bc cos A X ≥ 3− cos A 2 2 e f7eLg h e b + c e f7eLg h e r r = 3− 1+ = 2− R R = 3−

~ 5 /;@-m2*BCV ;<=CFA@EX /1 =*V 1 ; 2@X

a = b = c

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