Canadian Mathematical Society | CMS-SMC · a _ oe`AED./*?C/12-l6

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A/4>=81-z3

120 _D./-zBz68/F.1433Bz68RAED<?C;>=l2DAED>Bz3</2?AC/ËK1687MFl/2?|F8/687>B b =.-z14?AC;M;<68/:;>e>AED./174¤o1 b /6>A3BJ78/23*;>eAED./MA|?CBz1468-J/O7>B>B07./3AED./(A|?CBz1468-J/Bz6>A|;IAyM;OA|?CBz1468-J/3 ;<68/O;>eryMD>B b DBz3;.G>AC=835/14687MD<13L146*BJ6.3 b ?CBJG8/7 b BJ? b -J/Ry*BAED?147<B@=83/4>=81-4AC;1_RÌ¡/A|/2?|HBJ68/¡AED./1?C/1;>e 4ABC

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2

2 · 5+

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2

2000 · 2003

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a b c 351YACBJ3Ee@9a

b + c+

b

c + a+

c

a + b= 1

F8?C;2</KAED<12Aa2

b + c+

b2

c + a+

c2

a + b= 0

_

/EËAryM/F8?C/23+/26>AL1:35;>-z=<ACBJ;<6Me0?C;<H AED./ 44> ?U9</?L;<6>A|/23EABz6AED./K4/2F<AC/HI~G8/2? 44> Bz353E=l/ 44> <4 _È _U6IAED./7<Bz1Y?1H ABCDBz3135<=81?C/(14687OAED./ b ;.;<?C7>BJ6.1YA|/23:;>e

A14687

D14?C/143¡3+D.;4y6_

01 ∗ `D./F8;>BJ6>A PD<143 b ;.;<?C7<Bz6.12AC/3

(10, 0)_"4D.;4y AED<1YAAED./ 14?C/21&;>eA|?CBz1468-J/

PCBBz3

10_

oG ∗ ,<;>Bz6>A E(a, 0)Bz3;<6cAED./

x1^Ë4Bz3¦3E= b DcAED<1YAA|?BJ168-0/

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ABCD_oeLAED./1?C/1;>eLAED./A|?CBz1468-J/IBz3M/4>=81-A|;dAED./14?C/21(;>elAED./*3+>=814?C/ yMD<1YABz3RAED./>1-@=l/M;>e a É b ∗ 4D.;4yAED<1YA¡AED./2?C/Bz3:68;Fl;>Bz6>A

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x 1ËBJ3eo;<? yMD>B b DOAED./1?C/1;>elA|?CBz1468-J/

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ABCD_

8SWTSUT r ∗ GBJ3 1*F8;>BJ6>A;<6MAED./:-zBz68/KF.1433Bz68AED<?C;>=l2DMAED./KFl;>Bz6>AC3

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_2BJ6 b /yM/:468;4ycAED./ b ;.;<?C7>BJ6.1YA|/23 ;>eA = (1, 4)

14687D = (1, 8) 3BJ78/

ADHO=83|ArG8/

4_2BJ6 b /

ABCDBz31¡3+>=814?C/ CB = AD = AB = CD = 4

_0`/ b 1=835/

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A yM/HO=83|AMD<1YeA|?CBz1468-J/

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b ;.;<?C7<Bz6.12AC/(;>eEBJ3Y?C/1YA|/2?AED<16

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2

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= a3 + b3 − a3 − 3a2b − 3ab2 − b3

= 3ab(−a − b)

= 3abc_

È _L ∗ a&b /2?A1Bz6?C/ b AC168-0/*D<143R14?C/21 61687O7>BJ142;<6.1-l;>e8-0/268ÂED

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√5 B>BJ68

a2 + b2 = 20_D>=83

(a + b)2 = a2 + b2 + 2ab = 32 ==⇒ a + b = 4√

2 (a − b)2 = a2 + b2 − 2ab = 8 ==⇒ a − b = 2

√2_

4;>->BJ68:AED./39.3|A|/2H yM/M2/YA a = 3√

214687

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y = 2x2 + 4x − 2 y = −2x2 + 4x + 2

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(a, b)3E= b D AED<1YA AED./ /4>=812ACBJ;<6

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(x, y)_

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È

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arctan 34

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y*BAEDx0 y0 > 0

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2x0

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arctan x − arctan y = arctan

(

x − y

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3y0x0

2(x20 + y2

0)

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3

4

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T.S UTV"3[WW*Ã ETW%ÃWVU UT8W < YWB P/(_`-J/1?C- 9 Bze N = 2 3 4 B@ABz3Fl;<33BJG<-J/ 13LAED./l?3|A[F.-z1Y9</2? b 146*¤0=83EA[?C/2H;2</

1 2 ;<? HI1?EG>-0/23(1687eo;<? b /IAED./3+/ b ;<687F.-z1Y9</2?:AC;AC144/IAED./I-J13EAKHO14?|G<-J/8_ oeN = 5 6 AED./(8?3EA¡14687OAED>BJ?C7F<-J19</?3 b 16.68;A¡eo;<? b /AED./35/ b ;<687F<-J19</? A|;O-0;<35/G8/ b 12=83+/MD./ b 16I1- y1Y9<3¡F.-z1Y9O35;(AED<12ALAED./M3E=8H ;>erD>BJ31687AED./*8?3EA F.-z1Y9</2?+ 3F.-z1Y9BJ3

4@yMD>B b D(eo;<? b /23 AED./AED>Bz?C7F.-z1Y9</2?[AC;(-J;<3+/Bze

N = 5 ;<?AED./:l?3|AF<-J19</?A|;M-0;<35/BzeN = 6 _

%

;<?N > 6

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N − 4 ≤ k ≤ N − 2_D>=83 3 ≤ k

_ ;4y AED./AED>BJ?C71687AED./Ol?3|AKF.-z1Y9</2?3 b 146 b ;<6.35F<BJ?C/OAC;-J/1</

k′ HO14?|G<-J/3 k′ ≡ 1 (mod 5)

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1 (mod 5)14687My*Bz-@-/+</6>AC=812-z- 9IG8/K-0/YezA[y*B@AED

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0! = 9_

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SUTX D./M?C/217./?RHB02DA -@B04/KA|;3+//(D.;4y½3BzHOB@-J1?B07./1314?C/(D<1687<-J/47OBz6AED./I198D./2H 14?ACB b -J/ 08BJ68;<HBJ12- U6</2?3BJ;<6 >yM;,8?C;.;>e0314687I146 a F.F.-@B b 12ACBJ;<6OAC;Ì¡/?1682/HI/6>A3 G9¥/G.1¥*1YAED.;>=<A 44> <7È 7È* _

® ¾Á _ $lXT*¿TW"V" IÀ YSoX& -, " TXEWB C ,._

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_4D.;4yAED<1YAa ± 2c

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;>eAED./¡<=8147.?1YACB b /<=81YACB0;<6ax2+bx+c = 0

1?C/ −b ±√

b2 − 4ac

2a

_4/AACBJ68LAED./23+/AyM;(?C;.;AC3LAC;

cos θ1687

sin θ AED./:BJ78/26>ACB@AU9 sin2 θ + cos2 θ = 1Bz3R/4>=.B>12-0/26>A[AC;

b2 − 2ac = a2 ;<? b2 = a(a + 2c)_D./26

b2 − 4ac = a(a − 2c)_ 08=<AG.;AED3BJ78/23:;>erAED>BJ3:/<=81YACB0;<6HO=83|AG8/Fl/2?Ceo/ b AL35<=81?C/3 3Bz6 b /AED./:?C;.;A3 cos θ1687

sin θ14?C/:?1YACB0;<6.12-_2BJ6 b /

(a, c) = 1 B@Aeo;>-z-J;4y3:AED<1YAa1687

a − 2cD<1Y</I68; b ;<HOH;<6e01 b AC;<?3 1687/1 b D¦H=83EAKG8/I1Fl/2?Ceo/ b A35<=81?C/¦3+/2F81?12AC/2- 9_ a -J35; e0?C;<H AED.//4>=812ACBJ;<6

b2 = a(a + 2c) AED./b ;<687>B@ACBJ;<6(a, a +2c) = 1

eo;<? b /23a +2c

AC;G./K1KF8/?eo/ b AL3+>=814?C/ 3EBJ6 b / a1687

b214?C/*Fl/2?Ceo/ b AR35<=81?C/32_ K 066)% L&%C%?6 J;.%&; <1;K,MM6K')-6 A GH(6K.+6K; $N:? OP?Q8 SR

GE*8"3 : (/= !" #&%(') *',:, *')-: #& 02$3P4(6 7)86B9-1;,%(&:; <1;< 6 : >& 6 & & @6 A- *')-: 25D;-( >& ?6 & 2F GH*I3 :"%; ( N:.6 O.GH(6K.+6K(99I!I%""6 +)A(:; 6 C

® ¾*Æ _ $lXT*¿TW"V"OSW < ""`WÅ 2S%ÆL_U6¦6<=8HG./?AED./4;<?U9AED./eJ=86 b ACBJ;<6

ω(n)BJ3AED./O6<=8HIG8/2?K;>e YS `Q5SLF8?BJHI/37AC/42/2?n

ln n ≥ ω(n) ln 2_

T.S UTV 4 2SÃT rX PÅ3 <WYXOÂ DÃWX @!6@ - <WT JXW RX <WrS ._Ê/YA

n = pα11 pα2

2 · · · pαω(n)

ω(n) yMD./2?C/ p1 p2 . . . pω(n)14?C/*7<Bz3EACBz6 b AF8?BJHI/314687

α1 α2 . . . αω(n)1?C/Fl;<3EB@ACB </(Bz6>A|/2/?32_M2Bz6 b /

αi > 01687

pi ≥ 2 yM/D<1Y</

ln n = ln

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i=1

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i

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ω(n)∏

i=1

pi

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ω(n)∏

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2

= ln 2ω(n) = ω(n) ln 2_

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/10123054768059;:=< *> 230@?BAÁ _ Bz687O12-z-l35;>-z=<ACBJ;<6.3;>e

x1 + x2 + · · · + x2000 = 2000 x4

1 + x42 + · · · + x4

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2 + · · · + x32000

_Æ _ WB</26

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x1 x2 . . . x20013E= b DAED<12A

0 ≤ xn ≤ 1eo;<?/1 b D

n = 1 2 . . . 2001 l687AED./*HI1^Ë4BzHO=8H>1-@=l/M;>e(

1

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y*B@AEDx;.787 3= b D(AED<12A

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y

(x + y)(f(x) − f(y)) = f(x2) − f(y2)_

Bz687 01 ;<68/*3= b DeJ=86 b ACB0;<6<0G 12-z-l3E= b DeJ=86 b ACB0;<6.32_Ä _ a -@BJ68/7e.1¡A|?CBz1468-J/Bz6>A|;:AyM;/4>=81-F81?A3_,8?C;</KAED<1YAAED>BJ3R-zBz68/F813535/3 AED<?C;>=l2DAED./*BJ6 b /6>A|?C/M;>e`AED./KA|?CBz1468-J/8_Ì¡;./23AED./ b ;<6</2?3+/*3EA12AC/HI/6>A 12-y19.3¡D.;>-07.É¾ _D./M/<=81YACB0;<6

x2 + y2 + z2 + u2 = xyzu + 6BJ3B </6_ Bz687

J1 12AL-0/2143|AR;<68/*3+;>-@=<ACB0;<6Bz6OFl;<3EB@ACB </KBJ6>AC/42/2?3oG 12AL-0/2143|A 333= b D35;>-z=<ACBJ;<6.3

b 12AL-0/2143|A 1003E= b DO35;>-z=<ACBJ;<6.3_

;4y yM/ AC=8?|6 AC;?C/217./?3:3E=lG.HBJ33BJ;<6.3eo;<?dF8?C;.G>-0/2HI3 ;>eAED./ ]`=8353EBJ1612AED./2HI1YACB b 1- R-9<HIF<BJ147 4 ;<?CH B </6 44>*884 _

Á _ 5P $lT" 0¿ Ì¡;IAED./2?C/O/EË4Bz3EA19

7>B!\r/?C/26>AFl;<3EB@ACB </(Bz6>A|/2/?3AED<12A3=8H AC;1999

14687O3E= b D(AED<12ALAED./*3=8H";>elAED./M7./ b BzHI12-7>B0BAC3;>er/21 b DBz3 AED./314HI/4É TKÃW"V" $WX|XW -TX SUW < T(U¦SSW Ä4X Q|W @.X YST*¿`WYX -`X"WB -`X YST>±Â .,¤ÁMT U> YS8WrS JX 2SUT@ XQ >?Á! Q`T.T @ > "X& - W KÃW;-`X"WB !X SW ¿L_

Ê/YAS(n)

G./KAED./M7>B0BAC12-l3=8H";>en_D>Bz3¡eJ=86 b ACBJ;<6OBz368;4y6AC;351YACBJ3Ee@9

S(m + n) ≡ S(m) + S(n) (mod 9)_

oesBJ3KAED./ b ;<HOH;<6 7<BJB@A1-L3=8H$;>eLAED./

196<=8HG./?3*AED<12AyM/y*Bz3+DdAC;¦3=8H¶AC;

1999 B@ALeo;>-z-J;4y3 AED<1YA19s ≡ 1 + 9 + 9 + 9 (mod 9)

_¥/26 b / s ≡ 1 (mod 9)

_

`-J/1?C- 9 sb 16.68;A(/4>=81-

1 AED>=83 AED./d8?3EAFl;<33BJG<B@-zBA9 Bz3 s = 10

_ `D./35HO1-@-0/23EA19

6<=8HG./?3 y*B@AEDO7<BJB@A1-l3=8H10 6.1H/Y-9 19 28 37 46 55 64 73

82 91 109 118 127 136 145 154 163 172 181 14687 190 3=8H AC;1990

_2BJ6 b /AED./M68/Ë2ARBJ6>AC/42/2?y*BAED7>B0BAC12-3=8H /<=812-`AC;10

Bz3208 AED./(68/EËA¡3HI12-z-J/3|AFl;<33BJG<-J/*3=8HBJ3

1990 + 208 − 190 = 2008_R¥*/6 b / s b 16.68;AR/4>=81- 10 _ ;<?

s ≥ 19AED./35HO1-@-0/23EAKFl;<33BJG<-J/

196<=8HG./?3(17.7 =8FAC; H= b D H;<?C/AED<146

19994yMD./26 b /*BABz3RBJHOFl;<33BJG<-J/:A|;;.G>A1Bz6AED./?C/<=.Bz?C/47O3=8H¦_

Æ _ -WX9T7Ã a eJ=86 b ACBJ;<6f : → BJ3 b ;<6.3BJ78/2?C/47r_O,8?C;</MAED<12ARAED./?C/O/EË4Bz3EAAyM;?1YACB0;<6.12-`6<=8HIG8/2?3

a1687

b3= b D(AED<1YA

f(a) + f(b)

2≤ f

(

a + b

2

) _@`TÅV .S UTMT"7T9<S T V"$WX|XW#-TX SW < T(U¦SSW Ä4X Q|W ,-`X <Q|W @.XT"7T.T8SÂ DÃWX S3@`T?W <WMT"7[S`W @X V.T.T" ,Å UT.T*¿ - @r_

a 353E=8H/KAED./ b ;<6>A|?14?U9@y*BAEDAED./M12BJH ;>er?C/1 b D>BJ681 b ;<6>A|?17>B b ACBJ;<6 _R`D./6 eo;<? 12-z-`7>BJ3|ACBJ6 b A ?1YACB0;<6.12-l6<=8HG./?3a14687

b f

(

a + b

2

)

<f(a) + f(b)

2

_ D>Bz3 b ;<687<BACB0;<6 Bz368;A&1*\/ b A|/7 G914787>BJ68 1 b ;<6.3|AC16>A A|; AED./ eJ=86 b ACB0;<6

f_D./2?C/2eo;<?C/ yM/HI19 1353E=8H/AED<1YA f(−1) ≤ 0

14687f(1) ≤ 0 y*B@AED.;>=<AM-J;<353;>er2/68/2?1-@B@AU9_ / b -z1BzH"AED<1YA¡eo;<?¡12-z-

n ∈ ∪ 0 1687 x ∈ −2−n 0 2−n yM/D<1</f(x) ≤ −n

_d.;F.?C;</OAED>Bz3 yM/I=835/IBz687<= b ACBJ;<6;<6 n_ ;<?

n = 0 yM/IHO=83|Ab ;<6.3EB07./?x ∈ −1 0 1 _ /D<1</ f(±1) ≤ 0 1687ID./26 b / =83EBJ68* E

f(0) = f

(

1 − 1

2

)

<f(1) + f(−1)

2≤ 0

_D>=83 AED./ b -J12BJHBz3 A|?C=l/*eo;<?

n = 0_

;4y b ;<6.3BJ78/2?[1469(Ë4/47neo;<?ryMD>B b DAED./ b -z1BzHBJ3LA|?=l/8_½[3Bz68: 14687MAED./BJ687>= b ACB0;<6ID49.F8;AED./3EBJ3 yM/D<1Y</

f(±2−n−1) = f

(

0 ± 2−n

2

)

<f(0) + f(±2−n)

2≤ −n − n

2= −n

_2BJ6 b /

fAC144/3*;<6<-9BJ6>AC/42/2?¡>1-@=l/3 yM/HO=83|A:D<1Y</ f(±2−n−1) ≤ −(n + 1)

_D./26 =83EBJ68* 1Y12BJ6 f(0) = f

(

2−n−1 − 2−n−1

2

)

<f(2−n−1) + f(−2−n−1)

2≤ −(n+1)

_D>Bz3 b ;<HIF<-0/A|/23RAED./KBJ687>= b ACB0;<6O1687OF8?C;2</3 AED./ b -z1BzH_ /D<1Y</35D.;4y6IAED<1YA

f(0) ≤ −neo;<?¡1-@-r68;<68~68/Y12ACB </(Bz6>A|/2/?3

n_`D>BJ3BJ3BJHOFl;<33BJG<-J/8_*D./2?C/2eo;<?C/ AED./?C/Bz3K68;eJ=86 b ACB0;<6 f : → AED<12A351YACBJ3EBzl/3: eo;<? 12-z-`7>BJ3|ACBJ6 b A ?1YACB0;<6.12-l6<=8HG./?3

a14687

b_

È _&< T `D./Bz6 b BJ? b -J/M;>e<=8147.?Bz-z12AC/?12-

ABCDAC;>= b D./3RAED./M3EB07./3

DA AB BC CD

BJ6K L M N ?C/235F8/ b ACB </2- 9_Ê/YA

S1 S2 S3 S4G./AED./BJ6 b Bz? b -0/23*;>eA|?CBz1468-J/3

AKL BLM CMN DKN ?C/235F8/ b ACB </2- 9_IÊ/YA l1 l2 l3 l4 G8/AED./ b ;<HOH;<6/EËAC/?C6.1->A14682/26>AC3LAC;*AED./:F.1Bz?3 S1

14687S2 S2

1687S3 S314687

S4 S414687

S1 7>B!\r/?C/26>A¡e0?C;<H"AED./(3BJ78/23:;>e>=817<?CB@-J1YA|/2?1-ABCD

_,8?C;</AED<12Al1 l2 l3 l4 BJ6>AC/?35/ b ABz6AED./:</2?ACB b /23;>e`1M?|D.;<HG>=83_

T.S UTV"[email protected] YST*¿`WYX +-`X9"UWB -`X 2SUTÂ ¨ÅdTrWV"OS`WMWSUTX^_

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.

A

B

C

D

I

I1 I2

I3

I4

K

L

M

N

`1

`2

`3

`4

Ê/YArG8/:AED./?147<B@=83;>elAED./KBJ6 b Bz? b -0/;>er>=817<?CB@-J1YA|/2?1-

ABCD_¡Ê/A

α = 12∠A β = 1

2∠B γ = 1

2∠C δ = 1

2∠D

_ ;AC/KAED<1YA

α + β + γ + δ = 180 _8?CBz1468-J/3AKI

1687ALI

1?C/ b ;<68Y?C=l/26>A 3EBJ6 b /AED./+9 1?C/?CBJ2DAE~168-0/7 y*B@AED1 b ;<HIHI;<6 D49.F8;A|/26<=83+/14687KI = LI = r

_ D./2?C/2eo;<?C/ AK = AL14687∠IAK = ∠IAL = α

_2Bz6 b /AI

BJ3AED./OG<Bz3+/ b A|;<?;>e∠A

BJ6AED./Bz3+;<3 b /Y-0/23A|?CBz1468-J/KAL yM/D<1Y</ AI ⊥ KL

_*D./26 3Bz6 b / ∠ALI = 90 yM/35/4/MAED<1YA∠KLI = ∠IAL = α

01687O1-z3+;∠LKI = α _ ¥/26 b / KL = 2r cos α

_Ê/YAI1G8/MAED./Fl;>Bz6>A1YARyMD>B b DIAED./Bz6 b BJ? b -J/;>e

ABCDBJ6>AC/?35/ b A3AED./(-@BJ68/3+/YH/26>A

AI 14687-J/YA r1G8/AED./¦7>BJ3|AC16 b /de0?C;<H

I1A|;

ALyMD>B b D&/4>=81-z3(AED./7<Bz3EA146 b /e0?C;<H

I1A|;

AK _ / b -z1BzHAED<12AI11687

r11?C/KAED./ b /6>A|?C/M1687I?17>Bz=83;>e

S1_.;F8?C;2</MAED>Bz3 l?3|A68;AC/MAED<1YA AI = AI1 + r y*B@AED AI = r csc α

14687AI1 = r1 csc α

_D>=83 yM/D<1</ r csc α = r1 csc α + r4AED<1YABJ3

r1 = r(1 − sin α)_

2

2BJ6 b /*AED./(7>BJ3|AC16 b /*e0?C;<HIAC;

KLBz3

r sin ∠KLI = r sin α AED./(7>BJ3|AC16 b /*e0?C;<HI1AC;

KLBz3

r − r sin α = r1_ / b ;<6 b -@=l78/OAED<1YAAED./Fl;>Bz6>A

I1Bz3*AED./I351H/7<Bz3EA146 b /

r1e0?C;<H"/1 b D3BJ78/;>e 4AKL yMD>B b DF8?C;2</3RAED./ b -J12BJH¦_2BJHBz-z14? b ;<6.3BJ78/2?12ACBJ;<6.314F.F.- 9*A|;KAED./Bz6 b BJ? b -J/3

S2 S3 1687 S4_ /:7./68;AC/AED./2Bz? b /6>A|?C/23RG9

I2 I3 14687 I4 14687AED./2Bz?L?147<B@B8G9 r2 r3 1687 r4 ?C/3Fl/ b ACB</Y-9r_ ;4y b ;<6.3EB07./? 4II1I2_ 0AMBz3BJ35;<3 b /2-J/3 y*BAED II1 = II2 = r

_c2BJ6 b /∠I1II2 = ∠AIB = 180 − α − β yM/D<1Y</

∠II1I2 = ∠II2I1 = 12(α + β)

14687ID./26 b / I1I2 = 2r cos

[

12(α + β)

] _ Ê/YAθG8/¡AED./1468-J/:G./YAyM//6KAED./-@BJ68/23

I1I21687

`2AED<1YA b ;<6>AC12BJ6.3AED./Fl;>Bz6>A

IBJ6BAC3RBz6>A|/2?CBJ;<?_ /y*Bz-@-lF.?C;</KAED<1YA

θ = γ + 12(α + β)

_nF%9`_

β < γo;<? /4>=.B>12-0/26>AC-9 r2 > r3 13¡Bz6AED./*l=8?C/ _D./26(AED./:-@BJ68/23

BC1687

`2H//YA;<6MAED./*/EËAC/6.3EB0;<6;>e

BCG8/+9<;<687

C_RÊ/AAED./KF8;>BJ6>AryMD./?C/AED./+9HI/4/AG8/K7./68;AC/47G9

S 14687M-0/A TG8/AED./KF8;>BJ6>AryMD./?C/

`2BJ6>AC/?35/ b A3I1I2

_D>=83 ∠I1TS = θ_

?C;<H$ E G9 39.HOH/A|?U9 yM/D<1Y</ ∠II2I3 = 12(β + γ)

_¦D>Bz3(168-0/Bz3/Ë2A|/2?CBJ;<?:A|; 4I2BS;<F8F8;<3BA|/IAED./Bz6>A|/2?CBJ;<?168-0/23

∠I2BS = β14687

∠I2SB_D./2?C/2eo;<?C/ ∠I2SB = 1

2(β + γ) − β = 1

2(γ − β)

_:2Bz6 b /SI2

G<Bz3+/ b AC3∠TSB yM/12-J35;D<1Y</

∠I2ST = 12(γ − β)

_ ;4y

θBz3¦16 /EËAC/?B0;<?1468-J/ ;>e 4TI2S

;<F.Fl;<3EB@AC/&AED./ BJ6>AC/?B0;<?1468-J/3∠I2ST = 1

2(γ − β)

1687∠TI2S = 1

2(α + β) + 1

2(β + γ) = β + 1

2(α + γ)

_D./2?C/2eo;<?C/ θ = β + 1

2(α + γ) + 1

2(γ − β) = γ + 1

2(α + β)

_nF%98_

β = γo;<? /4>=.B>12-0/26>AC-9 r2 = r3 _D./26AED./:AED<?C/4/*-@BJ68/23

BC I2I3 14687 `214?C/*F81?1-@-0/Y-_ ¥*/6 b /

θ = ∠I1I2I3 = 12(α+β)+ 1

2(β +γ) = β + 1

2(α+γ) = γ + 1

2(α+β)

_nF%9Mx8_

β > γo;<? /4>=.B>12-0/26>AC-9 r2 < r3 _D./26AED./-zBz68/3

BC14687

`2HI/4/AL;<6*AED./:/Ë2A|/26.3BJ;<6;>e

BCG./+9<;<687

B_`D./1468-J/*12AyMD>B b DMAED./+9HI/4/ALBJ3

β − γJe0?C;<H"`1435/ y*B@AED β

1687γBJ6>AC/? b D<1682/47 _a 6O14?C=8HI/6>AR3EBJHBz-z14?A|;(AED<1YAR;>er`1435/ -0/217<3 A|; θ = γ + 1

2(α + β)

_U6I12-z-.AED<?C// b 1435/3

θ = γ + 12(α + β) = γ + 1

2(180 − γ − δ) = 90 + 1

2(γ − δ)

_0<9 39<HIHI/YA|?U9 AED./1468-J/dG./YAyM//6 I1I2

1687`4yMD>B b D b ;<6>AC12BJ6.3

IBJ6B@A3BJ6>AC/?B0;<?Bz3

90 + 12(δ −γ)

_[2Bz6 b /AED./3E=8H ;>e>AED>BJ3L1468-J/14687θBJ3

180 AED./-@BJ68/23`21687

`414?C/F.14?12-z-J/2-_2BzHOB@-J1?C- 9 `1

14687`31?C/F81?1-@-0/Y-_:D>=83 `1 `2 `3 `4BJ6>AC/?35/ b ABz6AED./:</2?ACB b /23;>e1F.14?12-z-J/2-J;.Y?14H¦_

3È

2BJ6 b /θBz3:;<68/;>erAED./AyM;I1468-J/3:G./YAyM//6

I1I21687

`2@AED./;AED./2?G8/YBJ68

180 − θ E AED./Fl/2?|F8/687>B b =.-z14?:7<Bz3EA146 b /IG8/AyM/4/26AED./-zBz68/3KF.14?12-z-J/2-rA|; `2AED<1YAF81353AED<?C;>=l2D

I11687

I2Bz3

I1I2 sin θ = I1I2 cos[12(γ −δ)] yMD./2?C/ I1I2

BJ3B </6G9I _D./2?C/2eo;<?C/ AED./F8/?CFl/2687<B b =.-J1?R7<Bz3EA146 b /MG./YAyM//6 `214687

`4BJ3

2r cos[

12(α + β)

]

cos[

12(γ − δ)

]

− r1 − r2

= r cos[

12(α + β + γ − δ)

]

+ r cos[

12(α + β − γ + δ)

]

− r1 − r2

= r cos[

12(180 − 2δ)

]

+ r cos[

12(180 − 2γ)

]

− r1 − r2

= r sin δ + r sin γ − r1 − r2

= (r − r4) + (r − r3) − r1 − r2 = 2r − (r1 + r2 + r3 + r4)_

D>Bz3 G9M39.HOH/A|?U9 BJ3/<=812-AC;KAED./Fl/2?|F8/687>B b =.-z14?[7>BJ3|AC16 b /G8/AyM/4/26 `114687

`3_¥/26 b / `1 `2 `3 `4

Bz6>A|/2?3+/ b ABz6AED./:</?ACB b /3;>e1?|D.;<HG>=83_Ä _ : *)WXW `D./F<-J168/

αF.1433Bz68AED<?C;>=l2DIAED./M</2?AC/Ë

A;>eAC/YA|?14D./47<?C;<6

ABCDBJ3:AC1682/6>AAC;AED./ b Bz? b =8HI3FlD./2?C/;>e[AED./A|/A|?1D./7.?C;<6_ ,8?C;</AED<1YAAED./1468-J/3KG./YAyM//6AED./(-@BJ68/23K;>eBJ6>AC/?35/ b ACBJ;<6;>e

αy*BAEDIAED./F<-J168/3

ABC ACD 14687ABD

14?C//4>=81-8Bze1687I;<6<-9B@eAB · CD = AC · BD = AD · BC

_ T.S UTV"[email protected] YST*¿`WYX +-`X9"UWB -`X 2SUTÂ _

Ê/YAK

G8/AED./ b /6>A|?C/;>eLAED./ b Bz? b =8HI3FlD./2?C/;>eABCD

_ B@AED.;>=<A-0;<33;>e2/68/2?1-@B@AU9 1353E=8H/AED<12AAED./:?17>Bz=83 ;>e<AED./:35F8D./?C/Bz3 1_<14/

AA|;(G./AED./K;<?B0Bz6;>e</ b AC;<?3 168778/268;A|/ −→

AB −→AC −−→AD 1687 −−→

AKG9

b c d 14687 k ?C/3Fl/ b ACB</Y-9r_2BJ6 b /KB = 1

1687 |k| = 1 yM/D<1Y</1 = KB2 = |k − b|2 = 1 + |b|2 − 2(k · b)

_D./2?C/2eo;<?C/ |b|2 = 2(k · b)

_2BzHOB@-J1?C- 9 |c|2 = 2(k · c)14687 |d|2 = 2(k · d)

_a e8BJ6>AC/?35/ b ACBJ;<6;>e α y*BAEDAED./F.-z1468/ABC

Bz3k × (c × b) = (k · b)c − (k · c)b = 1

2

(

|b|2c − |c|2b) _

2BJHBz-z14?-9 1eLBz6>A|/2?3+/ b ACB0;<6¦;>e αy*BAEDAED./OF.-z1468/

ACDBJ3 12

(

|c|2d − |d|2c) 14687O1Ke`BJ6>AC/?35/ b ACBJ;<6;>e α

y*BAEDAED./F.-z1468/ADB

BJ3 12

(

|d|2b − |b|2d) _Ê/YA

h i j G8/*3 b 1-J/47M</2?3BJ;<6.3;>e8AED./35/:-@-0;4y3 h = |d|2

(

|b|2c − |c|2b)

i = |b|2(

|c|2d − |d|2c)

j = |c|2(

|d|2b − |b|2d) _

D./26h i j 14?C/ b ;<F<-J16.14? 3Bz6 b /AED./[email protected]/:BJ6 α

_ a -z3+; yM/*D<1Y</ h + i + j = 0_

/ b 1- b =.-J1YA|/|h|2 = |d|4

∣

∣

∣|b|2c − |c|2b∣

∣

∣

2

= |d|4(

|b|4|c|2 + |c|4|b|2 − 2|b|2|c|2b · c)

= |d|4|b|2|c|2(

|b|2 + |c|2 − 2b · c)

= |d|4|b|2|c|2|b − c|2 = |b|2|c|2|d|2(AD2BC2)

AED<12ABJ3 |h| = |b||c||d|(AD · BC)_M2BzHOB@-J1?C- 9 |i| = |b||c||d|(AB · CD)

1687|j| = |b||c||d|(AC · BD)

_ ;4yyM/K1F8F<-9(AED./*68;4y6?C/23=.-A[AED<12A[AED<?C// b ;<F<-J16.14?e120 y*BAED;<68/¦168;AED./?MB@e1687;<6<-9&B@e

|h| = |i| = |j| _ ¡68/MBz6>A|/2?|F.?C/YA12ACBJ;<6;>eAED>Bz3Bz3¡AED<12AAED<?C/4/ b ;<F<-J16.14? eo;<? b /3Ry*BAED?C/3E=.-@A146>Al/?C;14?C/¡/4>=81-@-9BJ6 b -@BJ68/7AC;K/21 b DK;AED./2?`B@e14687K;<6<-9BzeAED./+9:14?C/¡;>e>/4>=81-HI14Y6<B@AC=l7./8_ / b ;<6 b -z=l7./AED<12A:;>=8?F.14?ACB b =.-z14?Re120 y*B@AED¦;<68/O1468;AED./2?Bze1687 ;<6<- 9B@e AD · BC = AB · CD = AC · BD yMD>B b DBJ3 AED./*?C/4>=.BJ?C/7 b ;<687>B@ACBJ;<6_

/EËAlyM/AC=8?|6:A|;AED./KÌ¡/ b /HIG8/2? 6<=8HG./?;>eAED./ @`TX `WYX`1687*35;>-z=<ACBJ;<6.3G9;>=8?¡?C/217./?3¡A|;OF8?C;.G>-0/2HI3B</26OAED./?C/._ /(G8/BJ6y*B@AEDAED./;<HIF8;<3BACB0;<678/12AED /2HI1YACB0>=l/3 4 `-z1433+/K./?CHOBz6.1-` %* _

Æ _.]/23+;>=l7<?C/M7.16.3 -U /<=81YACB0;<6I/6n

(n + 3)n =

n+2∑

k=3

kn _ T.S UTV"!#4X ) IÁ >>7oX U4X UWYX¡Â KÃWYX S <SWX9T.T"LP/(_

D./M;<6<- 93+;>-@=<ACB0;<6.3R14?C/n = 2

1687n = 3

_Ê/YA

f(n) =n+2∑

k=3

kn 1687g(n) = (n + 3)n _D./26

f(1) = 3 6= 4 = g(1) f(2) = 32 + 42 = 52 = g(2) f(3) = 33 + 43 + 53 = 216 = 63 = g(3)

_D>=83 n = 2

14687n = 3

14?C/(35;>-z=<ACBJ;<6.3_.;F8?C;2</MAED<1YARAED./2?C/1?C/(68;35;>-z=<ACBJ;<6.3n ≥ 4 yM/y*B@-z-l35D.;4yAED<1YAeo;<? 12-z-

n ≥ 4 f(n) < g(n)

_

<

U68/4>=81-@B@AU9I BJ3/2143EBz- 9 b D./ b 4/7eo;<? n = 4

f(4) = 34 + 44 + 54 + 64 = 81 + 256 + 625 + 1296

= 2258 < 2401 = 74 = g(4)_

a 3¡146BJ687>= b ACB0;<6ID49<Fl;AED./23Bz3 3E=8F8F8;<3+/:AED<1YAR D.;>-07<3¡eo;<?3+;<HI/ n ≥ 4_D./26

f(n + 1) =

n+3∑

k=3

kn+1 =

n+2∑

k=3

k · kn + (n + 3)n+1

< (n + 2)

n+2∑

k=3

kn + (n + 3)n+1

= (n + 2)f(n) + (n + 3)g(n)

< (n + 2)g(n) + (n + 3)g(n) = (n + 3)n(2n + 5)_

2BJ6 b /g(n + 1) = (n + 4)n+1 B@A 3E= b /3 A|;3+D.;4y AED<12A

(n + 3)n(2n + 5) < (n + 4)n+1 _ 2BJ6 b /

(n + 3)n(2n + 5) < (n + 3)n(2n + 6) = 2(n + 3)n+1 BJ68/<=812-zBA9I yM;>=.-07OG8/:A|?C=l/*B@e(

n + 4

n + 3

)n+1

> 2_

08=<A(

n + 4

n + 3

)n+1

− 2 =

(

1 +1

n + 3

)n+1

− 2

> 1 +n + 1

n + 3+

(

n + 1

2

)

1

(n + 3)2− 2

=n + 1

n + 3+

n(n + 1)

2(n + 3)2− 1 =

n2 − 3n − 12

2(n + 3)2

>n2 − 3n − 18

2(n + 3)2=

n − 6

2(n + 3)

_D./2?C/2eo;<?C/ Bz68/4>=81-@B@AU9I E 14687OD./6 b /M E D.;>-07<3¡eo;<?12-z- n ≥ 6

_ D./6n = 4 BJ68/<=812-zBA9 G./ b ;<H/23 74 × 13 < 85 ;<? 31213 < 32768

yMD./6n = 5 Bz68/4>=81-@B@AU9 G./ b ;<H/23 85 × 15 < 96 ;<? 491520 < 531441

_D>=83 D.;>-J7.3Reo;<?1-@- n ≥ 4 14687I;>=8?LBJ687>= b ACB0;<6BJ3 b ;<HIF<-0/A|/._

/EËA.yM/AC=8?|6AC;:35;>-z=<ACBJ;<6.3eo;<?`F8?C;.G>-0/2HI3;>e2AED./ "2 U?16<BJ16(12AED./2HI1YACB b 1-R-9<HIF<BJ147

44 BJ?3|A];>=8687 B </6 %

2 _

Á _L2=8F8F8;<3+/KAED<1YAa1 < a2 < · · · < an

14?C/?C/21-l6<=8HIG8/2?3_,8?C;2</KAED<1YA a1a4

2+a2a43+ · · ·+an−1a4

n +ana41 ≥ a2a4

1+a3a42+ · · ·+ana4

n−1+a1a4n

_ T.S UTV" < T%Å5ÅdW"/ ? SoX9 YV.T64X ½Ä4X Q|W>_

/F.?C;</MAED./?C/23=.-AG9BJ687>= b ACB0;<6d;<6n_ ;<?

n = 2 yM/D<1Y<//<=812-zBA9r_D./ b 1435/n = 3

y*B@-z-lG8/*68/4/78/7IG8/Y-0;4yI_ ;<?n = 3 yM/D<1</KA|;3+D.;4y AED<12A

a1a42 + a2a4

3 + a3a41 ≥ a2a4

1 + a3a42 + a1a4

3

_D>Bz3¡Bz3 A|?C=l/ 3EBJ6 b /a1a4

2 + a2a43 + a3a4

1 − a2a41 − a3a4

2 − a1a43

= 12(a2 − a1)(a3 − a2)(a3 − a1)

·[

(a1 + a2)2 + (a2 + a3)

2 + (a3 + a1)2]

≥ 0_

a 353E=8H/dAED<12A(AED./ b -z1BzH Bz3A|?C=l/¦eo;<?n − 1 1687-J/YA(=83IF.?C;</¦B@A(eo;<? n

_0<914F.F.- 9<Bz68AED./IBz687<= b ACBJ;<6D49.F8;AED./3EBJ3 yM/I8687¦AED<1YA:B@A:BJ33E= b BJ/6>A:AC;¦F8?C;2</AED<12A

an−1a4n + ana4

1 − an−1a41 ≥ ana4

n−1 + a1a4n − a1a4

n−1 yMD>B b DBJ3 AED./ b 13+/n = 3

_Æ _ 2=8F8F8;<3+/dAED<12A

nBz3O16.12AC=8?12-6<=8HG./?^_D./

nAC=8F<-0/

(a1, a2, . . . , an)Bz33512B07A|;G./; <T.T Bze a1 + a2 + · · · + an = 2n

1687IeJ=8?AED./2?|HI;<?C/ 68;I3=lG<3+/A;>ea1 . . . an D<1313E=8H /<=812-8A|;

n_ BJ6871-@-`2;.;.7

n AC=8F.-J/32_

T.S UTV" $UWYX|XEW;-TX SW (< T(U¦SSW ;ÄX9`QEW_ Bz?3EA[68;AC/AED<12AAED./:F.?C;.G<-J/HBJ368;A b -0/214?^_[2BJ6 b /AED./

ai 3 1?C/:68;A[1433=8HI/47A|;&G./¦0F8;<3BACB</ Bz6>A|/2/?3 AED./2?C/¦/ËBJ3|AM16&Bz6<l6<BA|/d6<=8HG./?(;>e2;.;.7 n

AC=8F.-J/32_&;<?C/4;2</? AED./:68;A12ACBJ;<6 a1 . . . an BJ314HIG<BJ=l;>=83RG8/ b 12=83+/BA[3=8F.Fl;<35/3LAED<1YAAED./ai 314?C/(F.1Bz?y*Bz3+/(7<Bz3EACBz6 b A_oerAED>BJ3Bz3AED./ b 1435/ 1687B@erAED./ ai

3:1?C/(Fl;<3EB@ACB </BJ6>AC/42/2?3 AED./?C/y*Bz-@-`G8/*68;O2;.;.7 n AC=8F.-J/3Reo;<?

n > 1_

;>-@-0;4y*Bz68 yM/y*Bz-@-[1433=8HI/AED<12AAED./ ai 31?C/IFl;<3EB@ACB </OBz6>A|/2/?31687AED<12A5eo;<? /+</?U9

k ∈ 1 . . . n 68; k;>elAED./

nBJ6>AC/42/2?317.7=8FAC;

n_ `-J/1?C- 9 AED./M;<6<- 9O2;.;.7 1

AC=8F.-J/*Bz3(2)

_D./2;.;.72 AC=8F.-J/3¡14?C/

(1, 3)1687

(3, 1) 1687AED./2;.;.7 3AC=8F<-0/231?C/

(2, 2, 2)14687O12-z-lF8/?CHO=<A12ACBJ;<6.3;>e

(1, 1, 4)_ /y*Bz-@-F8?C;2</&G9BJ687>= b ACB0;<6c;<6

nAED<1YA B@e a = (a1, a2, . . . , an)

BJ312;.;.7nAC=8F<-0/y*B@AED

a1 ≤ a2 ≤ · · · ≤ an AED./6 /YB@AED./2? a = (2, 2, . . . , 2);<?

a = (1, 1, . . . , 1, n + 1)_

4

D./(?C/3E=.-@ARBJ3¡A|?C=l/Meo;<?n ∈ 1 2 3 _K2=8F.Fl;<35/MAED<1YARAED./(?C/23=.-AD.;>-J7.3eo;<?

n − 1 eo;<?:35;<H/ n ≥ 4_¦Ê/A

a = (a1, a2, . . . , an)G8/O12;.;.7

n AC=8F.-J/y*BAED

a1 ≤ a2 ≤ · · · ≤ an_*2Bz6 b /

aBz3:2;.;.7 yM/D<1</ n

∑

i=1

ai = 2n yMD>B b DIBJHOF.-@B0/23AED<12ARAED./(1</?142/;>erAED./ai 3BJ3

2_oe[/YB@AED./2?

a1 ≥ 2;<?

an ≤ 2 AED./26OyM/(HO=83|AD<1Y</a1 = a2 = · · · = an = 2

_D./2?C/2eo;<?C/ yM/HI19I1353E=8H/KAED<12A a1 = 11687

an ≥ 3_Ê/YA

j = maxi | ai = 1 _D./26 1 ≤ j < n_MÌ¡/Yl68/AED./

(n−1) AC=8F.-J/

a′ 13Reo;>-z-J;4y3 a′

i = ai+1eo;<?

i 6= j a′

j = aj+1 − 1_

;AC/KAED<1YA1 ≤ a′

1 ≤ a′2 ≤ · · · ≤ a′

n−1

1687 n−1∑

i=1

a′i = 2(n − 1)

_ ;4y 3E=8F8F8;<3+/ AED<1YA

a′ Bz3 68;A 2;.;.7r_ `D./6 AED./?C/"/EË4Bz3EA3 1 3=lG<3+/AE ⊆ 1 2 . . . n − 1 y*BAED E 6= ∅ 3= b D(AED<1YA ∑

i∈E

a′i = n − 1

_ / b ;<6.3EB07./?AyM; b 1435/3 nF%9`_

j /∈ E_Ì¡/2868/

E = 1 ∪ i + 1 | i ∈ E _D./26∑

i∈E

ai = a1 +∑

i∈E

ai+1 = 1 +∑

i∈E

a′i = 1 + n − 1 = n

_nF%98_

j ∈ E_Ì¡/2868/

E = i + 1 | i ∈ E _D./26∑

i∈E

ai =∑

i∈E

ai+1 = (a′j + 1) +

∑

i∈E\ja′

i = 1 +∑

i∈E

a′i = n

_U6/YB@AED./2? b 13+/RyM/D<1A|?17>B b AC/47KAED./D49.F8;AED./3EBJ3AED<1YA

aBJ32;.;.7r_`D./?C/Yeo;<?C/

a′ Bz32;.;.7_0<9 AED./&Bz687<= b ACBJ;<6D49<Fl;AED./23Bz3 yM/ D<1Y</ /2BAED./? a′ = (1, 1, . . . , n)

;<?a′ = (2, 2, . . . , 2)

_A08=<Aa′ = (2, 2, . . . , 2)

Bz3*68;A:F8;<353EB0G>-0/ 3Bz6 b /AED>Bz3yM;>=.-07BJHOF.- 9AED<1YAa = (1, 3, 2, . . . , 2) yMD>B b D7.;./368;AKD<1Y</ a2 ≤ a3

_`D./?C/Yeo;<?C/ a′ = (1, 1, . . . , 1, n)

_`D./6a = (1, 1, . . . , 2, n)

;<?a = (1, 1, . . . , 1, n + 1)

_G<BJ;>=83- 9

(1, 1, . . . , 2, n)BJ3¡68;AR2;.;.7 <D./26 b /

a = (1, 1, . . . , 1, n + 1)_D>=83 a = (1, 1, . . . , 1, n + 1)

;<?a = (2, 2, . . . , 2) 14687IAED./(Bz687<= b ACBJ;<6BJ3 b ;<HIF<-0/A|/._

0Aeo;>-z-J;4y3/2143EBz- 9(AED<12ALAED./2;.;.7nAC=8F<-0/2314?C/

J1 AED.;<3+/;.GAC12BJ68/7IG9O1F8/?CHO=<A12ACBJ;<6;>e (1, 1, . . . , 1, n + 1)

oG (2, 2, . . . , 2) B@e nBJ3;.7.7_

)Q&'# < <SWÅ3.S UQ9 @`TrSUWYS NN dP(%Åb¿ $lXET8VWÅ , T.S UT

7oXETÅ XT6 S`W TX9 /7<BA|/7IG9r_ a 687<?C/4/23 b =I14687£<_ /68 a[a _

4È _:Ê/YA

IG./*AED./Bz6 b /26>A|?C/;>e`AED./*A|?BJ168-0/

ABC14687

AIH//YAAED./ b Bz? b =8H b Bz? b -0/;>e

ABC12AFl;>Bz6>A

D_Ì¡/68;AC/MAED./eo;.;A;>eFl/2?|F8/687>B b =.-z14?3*7<?C;<F8F8/47e0?C;<H

I;<6

IE + IF = 12AD 8687AED./:>12-z=l/;>e ∠BAC

_ TKÃW"V @8X 2SUT*¿WX /-`X9"UWB-`X 2SUTbÂ OÀWÅ3X& V" )2T T W PÅ !# " > ¿ _

D>Bz3¡Bz3¡F8?C;.G>-0/2H 44 7È %> 4"2 _ _Ê/YA

ABCG./(1MA|?BJ168-0/Ky*B@AED

BC > CA > AB_4/2-J/ b A¡F8;>BJ6>A3

D;<6

BC14687E;<6AED./K/EËAC/6.3EB0;<6(;>e

AB3E= b D*AED<12A

BD = BE = AC_rD./ b Bz? b =8H b Bz? b -0/;>e

BEDBJ6>AC/?35/ b A3

AC1YAFl;>Bz6>A

P1687

BPH//YA3AED./ b Bz? b =8H b Bz? b -0/;>e

ABC12A F8;>BJ6>AQ_L4D.;4yAED<1YA

AQ + CQ = BP_

TKÃW"V @8X 2SUT*¿WX /-`X9"UWB-`X 2SUTbÂ OÀWÅ3X& V" )2T T W PÅ !# " > ¿

D>Bz3¡Bz3¡F8?C;.G>-0/2H % <2* < _5_ 2=8F8F8;<3+/AED<1YA

nBz31&F8;<3BACB</Bz6>A|/2/?O1687

d1 < d2 < d3 < d414?C/AED./(eo;>=8?3HI12-z-J/3|A¡Fl;<3EB@ACB </MBJ6>AC/42/2?3K7>B>B07>BJ68

n_ Bz68712-z-Bz6>A|/2/?3

n312ACBz3e@9>BJ68

n = d21 + d2

2 + d23 + d2

4

_ TKÃW" V" < T%Å5ÅdW" / ? SoX 2V.T6X9 zÄX9`QEW , $UWYX|XEWA-TX SUW < T(U¦SSW ;ÄX9`QEW W DÃW / > !X SUW ¿L_

oenBJ3*;.7.7 AED./26 d2

1 + d22 + d2

3 + d24 ≡ 1 + 1 + 1 + 1 ≡ 0 (mod 4) 14687yM/ b 146.68;A:D<1Y</

n = d21 + d2

2 + d23 + d2

4

_D>=83 yM/ b 1461433=8HI/AED<1YA 27=83 4 - n

_¥/26 b / (d1, d2, d3, d4) = (1, 2, p, q);<?

(d1, d2, d3, d4) = (1, 2, p, 2p)eo;<?35;<H/*;.7.7F.?CBzH/23p q _U6(AED./K8?3EA b 1435/ n ≡ 3 (mod 4) 1 b ;<6>A|?17>B b ACBJ;<6_D>=83 n = 5(1 + p2)14687

5 | n_D./2?C/2eo;<?C/ p = d3 = 5

14687n = 130

_ _2=8F8F8;<3+/AED<12A

A = a1 a2 . . . an 14687 B = b1 b2 . . . bn 14?C/AyM;0/1

35/4>=l/6 b /32_M`D./I7>BJ3|AC16 b /;>eAe0?C;<H

BBJ3K7./2868/47A|;dG./(AED./6<=8HG./?;>e

ieo;<?[yMD>B b Dai 6= bi (1 ≤ i ≤ n)

14687Bz378/268;A|/7IG9d(A, B)

_2=8F8F8;<3+/KAED<1YA

A B C 1?C/KAED<?C//0/1

3+/<=l/26 b /231687d(A, B) = d(A, C) = d(B, C) = δ

_01 ,8?C;</:AED<12A δ

BJ3¡16I/+</6O6<=8HG./?^_oG ,8?C;2</:AED<12ALAED./?C//ËBJ3|AC3¡1 0/1

3+/<=l/26 b /D3E= b DAED<1YA

d(D, A) = d(D, B) = d(D, C) =1

2δ_

4

TKÃW"V3< UQW *-,<S>?W .ÀT6<WvÄX9`QEW ,A$WX|XW -TX SW *< YT U ¦SSUWB;Ä4X Q|W W DÃW -,.S ?>UW T9<S Tr_

/M7./68;AC/MG9N(α)

AED./6<=8HIG8/2?¡;>e1 3¡BJ61469

0/135/4>=l/6 b /

α_

oeX = x1 x2 . . . xn 1687 Y = y1 y2 . . . yn 1?C/ 0/1

3+/<=l/26 b /23 -0/AX + Y

G8/AED./0/1

35/4>=l/6 b / x1 + y1 x2 + y2 . . . xn + yn yMD./2?C/17.7<BACB0;<6IBJ3Fl/2?Ceo;<?CH/7HI;.7<=.-J;2_:l-0/214?-9 d(X, Y ) = N(X + Y )

_MÊ/AkG./AED./I6<=8HG./?K;>eLBz687B b D

xi = yi = 1_OD./26

xi = 1 yi = 0eo;<?

N(X) − kBJ687>B b /3 14687 xi = 0 yi = 1

eo;<?N(Y ) − k

BJ687>B b /32_O`D./?C/Yeo;<?C/ d(X, Y ) = N(X + Y ) = N(X) + N(Y ) − 2k

_2=8F8F8;<3+/68;4y AED<12AA B C 14?C/

0/135/4>=l/6 b /3y*B@AED

d(A, B) = d(A, C) = d(B, C) = δ_

Ê/YAmG8/MAED./6<=8HG./?;>eBz687<B b /23

i3= b DOAED<12A

ai + bi = bi + ci = 1_¡6AED./;<68/MD<1687

N(

(A + B) + (B + C))

= N(A + C) = d(A, C) = δ_

¡6AED./;AED./? D<14687 N(

(A + B) + (B + C))

= N(A + B) + N(B + C) − 2m

= d(A, B) + d(B, C) − 2m = 2δ − 2m_

D>=83 δ = 2(δ − m) 14687IJ1 BJ3¡F.?C;</7_ U6O17.7<BACB0;<6 yM/35/4/:AED<1YA 12δ = m

_ ;4y b ;<6.3EB07./? D = d1 d2 . . . dn 78/Yl68/7 G9AED./Ieo;>-@-0;4y*Bz68O?=.-0/

di = 1Bze

N(ai, bi, ci) ≥ 2 1687 di = 0;AED./2?y*Bz3+/zeo;<?

1 ≤ i ≤ n _U6;<?C7./?[A|;O;.GAC12BJ6d(D, B) yM/M;.G<3+/2?U</KAED<1YA

•

BzeiBz3¡3= b D(AED<12A

bi = 0 AED./6 di = 013yM/2-@- =86<-0/2353 ai = ci = 1

•

BzeiBz3¡3= b D(AED<12A

bi = 1 AED./6 di = 113yM/2-@- =86<-0/2353 ai = ci = 0

_ /D<1Y</

di 6= bi/EË1 b AC- 9yMD./26

ai + bi = bi + ci = 1_ID>=83 AED./I6<=8HG./?;>eRBJ687>B b /3(3= b D¦AED<1YA

di 6= biBz3

m = 12δ_D>Bz3*9<BJ/2-J7.3

d(D, B) = 12δ_=0<939.HOH/A|?U9 d(D, C) = 1

2δ = d(D, A) 14687OoG Bz3¡F8?C;2</47r_

/EËA[14?C/:35;>-z=<ACBJ;<6.3[A|;MF.?C;.G<-J/HO3R;>e.AED./K4/ b ;<687¶];>=8687;>e<AED./ 2 U?146<Bz14612AED./2HI1YACB b 1- R-9<HIF<BJ147 44 B</26 44>*#"2 _Á _¡Ì¡/2868/KAED./*3+/<=l/26 b / xi∞

i=0

G9x0 = 0

1687 xn = xn−1 +

3r − 1

2 Bze

n = 3r−1(3k + 1) xn = xn−1 − 3r + 1

2 Bze

n = 3r−1(3k + 2) yMD./?C/

k14687

r14?C/(Bz6>A|/2/?32_,8?C;</MAED<1YA/+</2?U9Bz6>A|/2/?; bEb =8?3K/EË1 b AC- 9d;<6 b /(Bz6AED>BJ3¡3+/<=l/26 b /._

TKÃW" V" < T%Å5ÅdW" / ? SoX 2V.T6X9 zÄX9`QEW , $UWYX|XEWA-TX SUW < T(U¦SSW ;ÄX9`QEW W DÃW;-TX SUW YT.S UT pl_ oe

n =p∑

i=0

αi3i BJ3AED./¡A|/2?|6.1?U9M/Ë4F816.3BJ;<6M;>e<168;<68~E68/4Y1YACB</BJ6>AC/42/2? n AED./6

xn =∑

i∈Un

3i −∑

i∈Tn

3i yMD./?C/

Un14687

Tn14?C/*AED./(35/YA3;>e`>1-@=l/3;>eAED./MBz6878/EË

i3= b DAED<12A

αi = 11687

αi = 2 ?C/235F8/ b ACB </2- 9_$lXT.T"7_ /R=83+/¡Bz687<= b ACBJ;<6M;<6

n_;U.>=812ACBJ;<6M BJ3 b -0/214?-9:A|?=l/¡eo;<? n ∈ 0 1 _RÊ/A

n ≥ 1G./:>Ë4/7 14687(-J/YA n =

p∑

i=0

αi3i G8/AED./A|/2?|6.1?U9O/EËF.146.3EB0;<6;>e n _L2=8F.Fl;<35/AED<12AR D.;>-J7.3Reo;<? n

_nF%9 l_

α0 = 0_

D./26n + 1 = 1 +

p∑

i=1

αi3i = 30(3k + 1) eo;<?R3+;<HI/BJ6>AC/42/2? k

_¥*/6 b / xn+1 = xn +

31 − 1

2= xn + 1

_ /MD<1</

xn+1 =

(

∑

i∈Un

3i −∑

i∈Tn

3i

)

+ 1 =

(

∑

i∈Un

3i + 30

)

−∑

i∈Tn

3i

=∑

i∈Un+1

3i −∑

i∈Tn+1

3i yMD>B b DF8?C;2</3 AED./?C/23=.-ABJ6AED>Bz3 b 13+/._nF%98_

α0 = 1_

D./26n + 1 = 2 +

p∑

i=1

αi3i = 30(3k + 2) eo;<?R3+;<HI/BJ6>AC/42/2? k

_¥*/6 b / xn+1 = xn − 31 + 1

2= xn − 2

_ /MD<1</

xn+1 =

(

∑

i∈Un

3i −∑

i∈Tn

3i

)

− 2 =

(

∑

i∈Un

3i − 30

)

−(

∑

i∈Tn

3i + 30

)

=∑

i∈Un+1

3i −∑

i∈Tn+1

3i yMD>B b DF8?C;2</3 AED./?C/23=.-ABJ6AED>Bz3 b 13+/._nF%9Mx8_

α0 = 2_Ì¡/2868/

αp+1 = 0_¦Ê/A

t = mini | αi 6= 2 _D./26 1 ≤ t ≤ p + 1_ /MD<1</

n = 2

t−1∑

i=0

3i +∑

i≥t

αi3i = 3t − 1 +

∑

i≥t

αi3i

14687xn =

∑

i∈Un

3i −∑

i∈Tn

3i =∑

i∈Un

3i −∑

i∈Tni>t

3i −t−1∑

i=0

3i

=∑

i∈Un

3i −∑

i∈Tni>t

3i − 3t − 1

2

_D./26

n+1 = 3t +∑

i≥t

αi3i = 3t(3k+αt +1) eo;<?35;<H/BJ6>AC/42/2? k

_D>Bz3L-J/147.3A|;(AyM;3=lG b 13+/23 <V.Q9 YW 01 _ αt = 0

_D./26

n + 1 = 3t(3k + 1) 14687OD./6 b / xn+1 = xn +3t+1 − 1

2

_D>=83 xn+1 =

(

∑

i∈Un

3i −∑

i∈Tni>t

3i − 3t − 1

2

)

+3t+1 − 1

2

=

(

∑

i∈Un

3i + 3t

)

−∑

i∈Tni>t

3i =∑

i∈Un+1

3i −∑

i∈Tn+1

3i 14687AED./*?C/3E=.-@ARD.;>-J7.3RBJ6AED>Bz3 b 13+/._ <V.Q9 YW oG _ αt = 1

_D./26

n + 1 = 3t(3k + 2) 14687OD./6 b / xn+1 = xn − 3t+1 + 1

2

_D>=83 xn+1 =

(

∑

i∈Un

3i −∑

i∈Tni>t

3i − 3t − 1

2

)

− 3t+1 + 1

2

=

(

∑

i∈Un

3i − 3t

)

−(

∑

i∈Tni>t

3i + 3t

)

=∑

i∈Un+1

3i −∑

i∈Tn+1

3i 14687AED./*?C/3E=.-@ARD.;>-J7.3RBJ6AED>Bz3 b 13+/._

D>=83 Bz6*/+</?U9 b 1435/LyM/D<1</ AED./¡78/23Bz?C/47*/EËF.?C/33BJ;<6:eo;<? xn+1 yMD>B b DK/687<3AED./*BJ687>= b ACB0;<6O1687OF8?C;2</3 AED./*-J/HOHI18_Ê/YA

pG8/1>Ë4/7MF8;<3BACB</¡BJ6>AC/42/2? 1687-J/YA Ep = 0 1 . . . 3p+1 −1 1687

Fp =

− 3p+1 − 1

2 . . . −1 0 1 . . . 3p+1 − 1

2

_ 0ABJ3yM/2-@->468;4y6KAED<1YA/+</2?U9BJ6>AC/42/2?

n ∈ Epb 146G8/y?B@AAC/6BJ6¦1I=86<BJ<=l/y1Y9Bz6dAED./Oeo;<?|H

n =p∑

i=0

αi3i yMD./?C/

αi ∈ 0 1 2 eo;<?/21 b Di_D>Bz3MBz3*¤0=83|AKAED./OA|/2?|6.1?U9&/EËF.146.3EB0;<6;>e

n_

=8?AED./?CH;<?C/ /+</?U9dBJ6>AC/42/2? m ∈ Fpb 146G./(y?CBAA|/26BJ61O=86<B0>=l/y19dBJ6dAED./

eo;<?|Hm =

p∑

j=0

βj3j yMD./2?C/ βj ∈ −1 0 1 eo;<?/1 b D j

035/4/ 42<><> _

2

Ê/YAfp : Ep → Fp

G8/:AED./*eJ=86 b ACB0;<6I7./2868/47IG9fp

(

p∑

i=0

αi3i

)

=

p∑

j=0

βj3j

yMD./?C/βi = αi

B@eαi 6= 2 14687 βi = −1

Bzeαi = 2

_R`D./MeJ=86 b ACBJ;<6fp

Bz3 b -J/1?C- 9BJ6¤U/ b ACB</ 14687O3Bz6 b / |Ep| = 3p+1 = |Fp| yM/M78/7<= b /*AED<1YA fpBJ3GAC;

Fp_D./*-J/HOHI1*BzHIF<-zBJ/3 AED<1YA

fp(n) = xneo;<? /1 b D

n ∈ Ep_Ê/YA

kG8/*16Bz6>A|/2/?^_[`D./?C//ËBJ3|AC3R1F8;<3BACB</:BJ6>AC/42/2?

p3E= b D(AED<1YA

k ∈ Fp_D./26 e0?C;<H 14G8;2</ AED./?C/K/EË4Bz3EA316MBJ6>AC/42/2? n ∈ Ep

3E= b D*AED<12Ak = fp(n) = xn

_0A eo;>-@-0;4y3AED<1YAR/+</?U9Bz6>A|/2/?R; b|b =8?3¡Bz6AED./35/4>=l/6 b / xn _ ;4y3E=8F8F8;<3+/MAED<12A¡AED./?C/1?C/(AyM;68;<68~E68/4Y1YACB</Bz6>A|/2/?3

m1687

n3= b DAED<12A

xm = xn_[D./2?C/(/EË4Bz3EA31Fl;<3EB@ACB </*Bz6>A|/2/?

p3= b DAED<1YA

m1687

n1?C/(G.;AEDBJ6

Ep_2Bz6 b /

fpBJ3Bz6¤U/ b ACB </ yM/D<1</ m = n

_D>=83 146IBz6>A|/2/? b 146.68;A; b|b =8?H;<?C/:AED<146I;<6 b /KBJ6AED./*3+/<=l/26 b / 1687(yM/*14?C/78;<68/._Æ _[2=8F8F8;<3+/AED<12An(r)

7./68;AC/3[AED./6<=8HIG8/2?L;>e.Fl;>Bz6>AC3y*BAED*BJ6>AC/42/2? b ;.;<?C7<Bz6.12AC/3;<6O1 b Bz? b -0/;>e?147<B@=83r > 1

_,8?C;2</KAED<1YA n(r) < 6

3√

πr2_

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oen ≤ 8 AED./26 3Bz6 b / r > 1

146876 3√

π > 8 yM/D<1</ n < 63√

πr2 143M?C/<=.Bz?C/47r_ ;4y 3E=8F8F8;<3+/OAED<1YAKeo;<?K3+;<HI/r > 1

yM/D<1</n > 8

_ Ê/YA:AED./Fl;>Bz6>AC3y*BAEDBJ6>AC/42/2? b ;.;<?C7<Bz6.12AC/3AED<12A-zBJ/O;<6AED./ b BJ? b -J/OG./P1 P2 . . . Pn Bz6b ;>=86>AC/? b -J; b 2y*Bz3+/;<?C78/2?_2BJ6 b /

P1P3

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

+ P2P4

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

+ · · · + PnP2

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

= 4π ;<68/;>e`AED./14? b 3PiPi+2

............................................................................................................... BJ3¡12A HI;<3EA

4π/n_LD./KA|?BJ168-0/

PiPi+1Pi+2Bz3¡Bz6.3 b ?B0G./47Bz6I16I1? b;>e1468-J/1YA H;<3|A

4π/n_8;3BzHIF<-zB@e@9 AED./68;A12ACBJ;<6 y?B@AC/ A B C

Bz6 F<-J1 b /;>ePi Pi+1 Pi+2 ?C/3Fl/ b ACB</Y-9r_¡Ê/YA

θ = AC......................................................... 14687

t = AB........................................................... _D./26

0 < t < θ ≤ 4π/n 1687

[ABC] =abc

4r=

(

2r sin t2

) (

2r sin θ2

)

(

2r sin θ−t2

)

4r

≤ 2r2(

t

2

) (

θ

2

) (

θ − t

2

)

=r2θt(θ − t)

4

≤ r2θ(

θ2

)2

4=

r2θ3

16≤ r2

(

4πn

)3

16=

4r2π3

n3

_D<1683RAC;d,.B b . 3R`D./4;<?C/2H yM/68;4yAED<1YA

[ABC] ≥ 12

_`D./?C/Yeo;<?C/ 1

2≤ 4r2π3

n3

n ≤ 3√

8r2π3 = 2π3√

r2 < 63√

πr2_

ÈÈ _ 2=8F8F8;<3+/ AED<12A

ABCDEFBJ3 1 b ;<6</EË¶D./EË12;<6 y*BAED

AB = BC CD = DE 14687 EF = FA

_,8?C;2</KAED<1YABC

BE+

DE

DA+

FA

FC≥ 3

2

_ TKÃW"¦V" $UWYX|XEW -TX SW /< YT U ¦SSUWB.ÄX9`QEW 4 %)T UT W PÅ !# " > ¿ W KÃW W Å YT.S UT_

..

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

A

B

C

D

E F

a

b

c

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

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...........

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c

c

/F.=<AAC = a CE = b 14687 AE = c

_ 0<9cAED./yM/2-@-0~68;4y62/68/2?1-@B@E1YACB0;<6I;>e[,<A|;>-J/H(9 3RD./;<?C/H eo;<? <=8147.?Bz-z12AC/?12-ABCE yM/D<1Y</

AC · BE ≤ AB · CE + BC · AE = BC(CE + AE)

AED<12ABz3 a · BE ≤ BC(b + c)_ ¥/26 b / BC

BE≥ a

b + c

_2BJHBz-z14?-9

DE

DA≥ b

c + a

14687 FA

FC≥ c

a + b

_D>=83

BC

BE+

DE

DA+

FA

FC≥ a

b + c+

b

c + a+

c

a + b

_ 0<9AED./ a

WR U68/4>=81-@B@AU9 yM/MD<1</

(

1

b + c+

1

c + a+

1

a + b

)

≥ 3 3

√

1

b + c· 1

c + a· 1

a + b

14687

(b + c) + (c + a) + (a + b) ≥ 3 3√

(b + c)(c + a)(a + b)

AED<12ABz3 a + b + c ≥ 3

23√

(b + c)(c + a)(a + b)

%

[email protected] 9<Bz68 G9O E yM/M2/Aa + b + c

b + c+

a + b + c

c + a+

a + b + c

a + b≥ 9

2

yMD>B b D3BzHIF<-zB@`/23 AC;a

b + c+

b

c + a+

c

a + b≥ 3

2

_ Bz6.1-@-9 e0?C;<H" 14687I |

BC

BE+

DE

DA+

FA

FC≥ 3

2

_ _ Bz687O12-z-8eJ=86 b ACBJ;<6.3

f : → 312ACBz3e@9>BJ68 f(

f(x) + y)

= f(x2 − y) + 4f(x)y eo;<? 12-z-l?C/21-l6<=8HG./?3x y ∈ _

TKÃW"V"3<hTÅ5ÅdW / ? SoX9 YV.T64X vÄ4X Q|W *< UQW 4-,.S ?>UWBOÀT6<WÄX9`QEW 4,$UWYX|XEW -TX SW /< YT U ¦SSUWB.Ä4X Q|W W KÃW3-,<S>?W YT.S UT_

D./eJ=86 b ACB0;<6.3x 7→ 0

14687x 7→ x2 1?C/ b -J/1?C- 9I35;>-z=<ACBJ;<6.3_ /68;4y3+D.;4yAED<12ALAED./2?C/*BJ3¡68;O;AED./2? 35;>-z=<ACBJ;<6_2=8F8F8;<3+/

f : → 351YACBJ3E`/23f(

f(x) + y)

= f(x2 − y) + 4f(x)y eo;<? 12-z-

x y ∈ _¡Ê/YA a = f(0)_.144Bz68

x = 0BJ6I B </3

f(a + y) = f(−y) + 4ay eo;<?12-z-

y_ U6 yM/(8?3EARAC144/ y = 0

A|;2/YAf(a) = a AED./26 y = −a

AC;d2/Aa = a − 4a2 _0A¡eo;>-@-0;4y3¡AED<12A a = 0

_D./26 e0?C;<H | fBz3146d/+</26eJ=86 b ACB0;<6_r;<HOF81?CBz68OAED./?C/23=.-AC3(;>eAED./3=lG<3EACBAC=<ACB0;<6.3

y = −f(x)14687

y = x2 Bz6 /13B@-9-J/147.3 AC;4(

f(x))2

= 4x2f(x)_[`D>=83 f(x) = 0

;<?f(x) = x2 _a 353E=8H/68;4yAED<1YAAED./2?C/I/ËBJ3|AC3

x03E= b DAED<1YA

f(x0) 6= 0_`D./6

x0 6= 014687f(x0) = x2

0

_L2Bz6 b /fBz3R/+</6 yM/:HO1Y9(3E=8F8F8;<3+/AED<12A x0 > 0

_¡Ê/YAxG8/:16968;<68~|/2?C;?C/12-`6<=8HIG8/2?_#0<9O y*BAED y = −x0 yM/;.G>A1Bz6

f(

f(x) − x0

)

= f(x2 + x0) − 4f(x)x0_

oef(x) = 0 AED./26

f(x2 + x0) = f(−x0) = f(x0) = x20 6= 0

_

D>Bz3MBzHIF<-zBJ/3*AED<1YAf(x2 + x0) = (x2 + x0)

2 14687¦AED>=83 (x2 + x0)2 = x2

0

_D>Bz3¡Bz3¡68;A Fl;<33BJG<-J/ 3Bz6 b / x0 > 014687

x 6= 0_D./2?C/2eo;<?C/ f(x) = x2 _D>=83 f(x) = 0

eo;<?1-@-x;<?

f(x) = x2 eo;<?1-@- x _ _ U6¦A|?BJ168-0/

ABC AED./168-0/G<Bz3+/ b A|;<?*;>e ∠BACHI/4/AC3

BC1YA*Fl;>Bz6>A

D_2=8F8F8;<3+/AED<12A

ΓBJ3LAED./ b BJ? b -J/yMD>B b DMBJ3LAC1682/6>A[AC;

BC1YA

D14687F.1433+/23LAED<?C;>=l2DAED./F8;>BJ6>A

A_Ê/A

MG./AED./35/ b ;<687¦Fl;>Bz6>A*;>eLBJ6>AC/?35/ b ACBJ;<6 ;>e

Γ14687

AC1687

BMHI/4/AC3 AED./ b BJ? b -J/1YA

P_R,8?C;</:AED<12A

APBJ3¡1H/7<Bz146I;>elA|?BJ168-0/

ABD_

TKÃW" V" < > >W OÅdW " @`T7ÃB @ Ä > >WYX9 < >UTXQ ¿ @8X 2SUT*¿WX /-`X9"UWB -`X 2SUTÂ 4 )2T T W PÅ ,!# ¿ W KÃW OÅdW YT.S UT_

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

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A

B CD

MN

P

Q

Γ

Ê/YAN

G./OAED./35/ b ;<687Fl;>Bz6>A*;>e Bz6>A|/2?3+/ b ACB0;<6 ;>eΓ14687

AB 14687¦-J/YA APH//YABC

12AQ_n]/ b 12-z-@BJ68MAED<12AAED./I168-0/IG./YAyM//61AC1682/6>A:16871 b D.;<?C7Bz3/4>=81-A|;AED./O168-0/O3E=lG>AC/687./47¦G9dAED./ b D.;<?C712A1IF8;>BJ6>A:;<6AED./ b BJ? b =8Heo/?C/26 b /;<6AED./M;<F8F8;<3BA|/*3EB07./;>elAED./ b D.;<?C7 yM/M2/A ∠MDC = ∠CAD = 1

2∠A

_D./26∠ADM = ∠ADC − ∠MDC

= (180 − ∠CAD − ∠DCA) − ∠MDC

=(

180 − 12∠A − ∠C

)

− 12∠A

= 180 − ∠A − ∠C = ∠B_

a -J35; ∠ADM = ∠ANM 3EBJ6 b /AED./23+/¦1468-J/3O14?C/¦3=lGA|/26878/7hG9&AED./351H/1? b ;>eΓ_c`D>=83 ∠ANM = ∠B

yMD>B b D&35D.;4y3(AED<1YANM

Bz3F.14?12-z-J/2-A|;BC _(¥/26 b / ∠QPB = ∠APM = ∠ANM = ∠B

_2BJ6 b /MyM/12-J35;D<1Y</∠BQP = ∠BQA AED./A|?BJ168-0/23 BPQ

1687ABQ

14?C/:3EBJHBz-z14?^_D./26 e0?C;<HAED./F8?C;<F8;<?ACBJ;<6.1-l3BJ78/23 yM/2/YABQ

QA=

QP

BQ

B>BJ68BQ2 = QP · QA

_ /M1-z3+;ID<1</QP · QA = QD2 @AED./(F8;4yM/?¡;>eAED./Fl;>Bz6>A

Qy*BAED?C/3Fl/ b ALAC;

Γ _ ¥*/6 b / BQ = QD /23EA1G>-zBz3+D>Bz68 AP13¡1H/7<Bz146;>e 4ABD

_

"24 _[2=8F.Fl;<35/AED<12A

ABCBJ3L1A|?CBz1468-J/8_ oeyM/F.1Bz6>AAED./Fl;>Bz6>AC3;>e>AED./F.-z1468/BJ6M?C/714687Y?C/4/26 F.?C;</AED<12AK/2BAED./?AED./2?C//EË4Bz3EAAyM;?C/7¦Fl;>Bz6>AC3:yMD>B b D¦1?C/;<68/O=86<[email protected]?A¡;<?[AED<?C//MY?C/4/26IF8;>BJ6>A3Reo;<?|HBJ68K1KA|?CBz1468-J/M/<=812-8A|;

ABC_

T.S UTV" $UWYX|XEW;-TX SW (< T(U¦SSW ;ÄX9`QEW B@AED:68;:-J;<353;>e>2/68/2?1-@B@AU9 yM/¡HO1Y9K3E=8F8F8;<3+/ AED<1YA`AED./2?C//EË4Bz3EA31YA-J/13EAr;<68/?C/47dFl;>Bz6>A:zBze[68;A AED./ b ;<6 b -z=83EB0;<6D.;>-J7.3A|?B>BJ12-z- 9 14687AED<1YAAED./3EB07./3K;>e ABCD<1Y</*-J/68^AED<3

a b c y*B@AED a ≤ b1687

a ≤ c_a 353E=8H/ eo;<?AED./(F.=8?CFl;<35/;>e b ;<6>A|?147Bz6>AC3yMD>B b DM14?C/:;<68/=86<B@A[14F.14?AL68;<?rAED<?C/4/KY?C/4/26F8;>BJ6>A3eo;<?|HBJ681A|?BJ168-0//4>=81-8AC;

ABC_2=8F8F8;<3+/AED<12A AED./?C/(1?C/AyM;I?C/47F8;>BJ6>A3

M1687

N3= b DAED<12A

MN = a_Ê/YA

PG8/3E= b DIAED<12ARA|?CBz1468-J/

PMNBz3K/<=812-A|;OA|?CBz1468-J/

ABC_Ê/A

ΓP ΓM 14687ΓN

G8/AED./ b BJ? b -J/3;>e?147<B@=831b /6>A|?C/7 1YA

P M 1687 N ?C/3Fl/ b ACB</Y-9r_D./26ΓM

1687ΓN

1?C//26>ACBJ?C/Y-9Y?C//6_oeΓP

Bz3M/26>ACBJ?C/Y-9¦?C/47 AED./26 3Bz6 b / ΓPD<143(?147<B@=831 AED./2?C/14?C/IAyM;¦?C/47 F8;>BJ6>A3;<6 ΓP

yMD>B b D 1?C/d;<68/=86<BA1F81?A1b ;<6>A|?147BJ6>A;<6 ΓP 3519 X_ ?C;<H

M1687

N =83EBJ68:AED./KA|?146.3E-J1YACB0;<6(y*B@AED(Bz6>AC3

Y;<6

ΓM1687

Z;<6

ΓN3= b DIAED<12A

XY ZBz3:1IY?C//6A|?BJ168-0/O/<=812-rAC;

ABC B>BJ68K1 b ;<6>A|?147=83 AED./2?C/M7.;68;AR/ËBJ3|A ?C/7OF8;>BJ6>A3 M1687

N3E= b D(AED<12A

MN = a_

;4y -J/YA ΩG./1?C/47Fl;>Bz6>A 1687-0/A CΩ

G./MAED./ b Bz? b -0/*y*B@AED?147<B@=83a1687b /26>A|?C/

Ω_ ?C;<H 1G.;</ CΩ

Bz3:/6>ACBz?C/2- 9dY?C//6_(Ê/YAU ∈ CΩ

_(Ê/AV ∈ CΩ

G8/1(Fl;>Bz6>A¡3E= b DAED<1YAUV = a

_zD>=83 ∠UΩV = π3

(mod 2π)_ 2BJ6 b / U

1687VG8;AEDO-@B0/M;<6

CΩ AED./+9I14?C/(G.;AEDY?C//6_2BJ6 b / a = mina b c AED./?C/(/EË4Bz3EA31Fl;>Bz6>AT;>=<AC3EB07./

CΩ3= b DOAED<12ARAED./MA|?BJ168-0/

TUVBJ3:/4>=81-AC;

ABC_*l-0/214?-9

THO=83|ARG8/?C/7IzBze68;A yM/yM;>=.-07OD<1</1MY?C/4/26A|?BJ168-0//4>=81-8AC; ABC _ D./6OyM/(?C;AC1YA|/

U;<6

CΩAED./35/YA;>e b ;<?|?C/235F8;<687<Bz68(F8;>BJ6>A3

TBJ31 b Bz? b -0/

Γy*BAED b /26>A|?C/

Ω14687?147<B@=83

r > a_2BJ6 b /

ΓBz3/6>ACBz?C/2- 9?C/47 yM/(HO1Y9Ol687OAyM;?C/47OF8;>BJ6>A3;<6BA 31Y9 M

14687N 3E= b D(AED<12A MN = a 186.1- b ;<6>A|?17>B b ACBJ;<6_

*&`_ &;<?C/:2/268/?12-z- 9 ;<68/ b 146MF8?C;2</: WB</26 A B C D 16914?|G<BA|?14?U9 b ;<6<`=8?1YACB0;<6¦;>e[eo;>=8?Fl;>Bz6>AC3:BJ6AED./F.-z1468/ 14687B</26¦169 b ;>-J;>=8?CBz68;>e AED./dF.-z1468/Iy*BAED¦AyM; b ;>-J;>=8?3 31Y9 ?C/471687&Y?C/4/26 y*B@AED 68;¦AyM; ?C/7Fl;>Bz6>AC312A7<Bz3EA146 b /1e0?C;<H /21 b D ;AED./? AED./?C//EË4Bz3EA3O1Y?C/4/26 b ;<6<l=8?12ACBJ;<6yMD>B b D&Bz3b ;<68Y?=l/6>AAC; A B C D _

)Q&'# ] _ 1E=lD 13E ]`1HI35/+9>9.F8/R`D./4;<?C/2HI3rBJ6AED./K,.-z1468/ YT6X .T"76@`TÅV <STX )`W4TX 4/?B0/23 a M 47È E F_ 4 "24 _

"2

/EËAyM/*AC=8?C6OA|;O3+;>-@=<ACB0;<6.3¡eo;<?¡F.?C;.G<-J/HO3;>eAED./D>Bz?C7f];>=8687;>eAED./ "2 U?146<Bz146d12AED./2HI1YACB b 1- R-9<F.Bz17 44

4 B </6 %È _

È _2=8F8F8;<3+/AED<1YAC1 . . . Cn

14?C/ b BJ? b -J/3;>e?147<B@=83:;<68/Bz6OAED./MF.-z1468/M3= b DAED<1YA68;:AyM;;>eAED./2H 1?C/¡A14682/26>A 1687KAED./3E=lG.35/YA;>eAED./F<-J168/ eo;<?CH/7MG9KAED./¡=86<B0;<6;>elAED./35/ b Bz? b -0/23 BJ3 b ;<6.68/ b A|/7_ oe S = Ci ∩ Cj | 1 ≤ i < j ≤ n F.?C;</KAED<1YA|S| ≥ n

_ T.S UTV" $UWYX|XEW;-TX SW (< T(U¦SSW ;ÄX9`QEW_

;<?L/21 b Di 7./68;AC/*G9 n(Ci)

AED./:6<=8HG./?L;>e`/Y-0/2H/26>AC3R;>eSyMD>B b DG./2-J;<68A|;

Ci_*2Bz6 b /MAED./(=86<BJ;<6;>erAED./ b Bz? b -0/23BJ3 b ;<6.68/ b A|/7 yM/D<1</ n(Ci) > 0

_ ;<?/1 b DM ∈ S 78/268;A|/OG9 n(M)

AED./O6<=8HIG8/2?K;>e b BJ? b -J/3Ci

yMD>B b D b ;<6>A1Bz6M_D>=83 n(M) ≥ 2

_Ê/YAM ∈ S 1687-J/YA Ci

G8/ 169K;>eYAED./¡B </6=86<BA b BJ? b -J/33= b DAED<1YAM ∈ Ci

_2BJ6 b /AED./?C/BJ368;AC1682/6 b 9 /1 b D;>e.AED./ n(M) − 1;AED./2? b BJ? b -J/3[yMD>B b D b ;<6>A1Bz6

MH=83EABJ6>AC/?35/ b A

CiBz6168;AED./?:Fl;>Bz6>A_D./23+/IFl;>Bz6>AC31?C/F.1Bz?y*Bz3+/I7<Bz3EACBz6 b A G8/ b 12=83+/ eo;<?169AyM;¦B </6Fl;>Bz6>AC3;>eAED./IF<-J168/ AED./2?C/I1?C/O12AH;<3|AAyM;=86<B@Ab BJ? b -J/3 yMD>B b D b ;<6>AC12BJ6AED./KAyM;O;>elAED./H¦_`D>=83 Bz6I14787>B@ACBJ;<6A|; M AED./ b BJ? b -J/ Cib ;<6>A1Bz6.3M1YA:-0/2143|A

n(M) − 1;AED./?K/2-J/HI/6>A3(;>e

S_0AKeo;>-@-0;4y3:AED<12A eo;<?K/1 b D

M ∈ S14687I/21 b D b Bz? b -0/

Ci3= b D(AED<1YA

M ∈ Ci yM/MD<1</ n(M) ≤ n(Ci)_

Ê/YAN =

∑

(M,Ci)

1

n(Ci) yMD./2?C/RAED./3E=8H BJ3eo;<?`AED./F.1Bz?3

(M, Ci)3= b D:AED<1YA

CiBz3;<68/;>elAED./MB </6=86<B@A b Bz? b -0/23¡14687

M ∈ S ∩ Ci_ /D<1Y</

N =∑

Ci

(

∑

M∈Ci∩S

1

n(Ci)

)

=∑

Ci

n(Ci)1

n(Ci)= n

_¡6AED./;AED./? D<14687 3EBJ6 b / n(M) ≤ n(Ci) yM/D<1Y</

N ≤∑

(M,Ci)

1

n(M)=

∑

M∈S

(

∑

Ci:M∈Ci

1

n(M)

)

=∑

M∈S

n(M)1

n(M)=

∑

M∈S

1 = |S| _D>=83 |S| ≥ n

_ _2=8F8F8;<3+/ AED<12A

ABCDEFBJ31 b ;<6</ËD./Ë142;<6y*B@AED

∠B+∠D+∠F = 36014687AB

BC· CD

DE· EF

FA= 1

_,8?C;</:AED<12A

BC

CA· AE

EF· FD

DB= 1

_

"2

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A

BC

D

E

F

T

2BJ6 b /∠B + ∠D + ∠F = 360 yM/2/YA

∠A + ∠C + ∠E = 360 _ / b ;<6.3|A|?C= b A 4BAT7>BJ?C/ b AC-93EBJHBz-z14?<A|; 4BCD 13r3+D.;4y6BJ6AED./ l=8?C/8_r`D./6

∠CBA = ∠DBT14687

BC/AB = DB/BT_D./2?C/2eo;<?C/ 4BCA ∼ 4BDT

_¥/26 b / BC

CA=

BD

DT

_ 2BJ6 b / 4BCD ∼ 4BAT yM/MD<1</

AB

BC=

AT

CD

_ 2BJ6 b / AB

BC· CD

DE· EF

F A= 1 e0?C;<H yM/(3+//AED<1YA AT

CD· CD

DE· EF

F A= 1

AED<12ABz3 FA

AT=

EF

DE

_ ?C;<H | yM/D<1Y</

∠BCD + ∠BAF + ∠FED = 360 _ 2BJ6 b /∠BAT + ∠BAF + ∠FAT = 360 yM/Ol687AED<1YA ∠FAT = ∠FED

_`D>BJ3A|;.2/AED./?Ly*BAEDI BzHIF<-zBJ/3 AED<12A 4FAT ∼ 4FED_`D./6 4FAE ∼ 4FTD yMD>B b DBJHOF.-@B0/23 AED<1YA

AE

EF=

TD

DF

_ ?C;<H" 14687I yM/;.GAC12BJ6

BC

CA· AE

EF=

BD

DT· TD

DF=

BD

DF

14687ID./26 b / BC

CA· AE

EF· FD

DB= 1

_

"2 _¶2=8F.Fl;<35/ AED<12A

r1 . . . rn14?C/&?C/21-:6<=8HG./?32_,8?C;2</ AED<12AAED./?C/ /EË4Bz3EA3

I ⊆ 1 2 . . . n 3= b DAED<12A IHI/4/AC3 i i + 1 i + 2 Bz61YA -J/13EAR;<68/M1687I12AH;<3|ALAyM;/2-J/HI/6>A3 eo;<? 1 ≤ i ≤ n − 2

1687∣

∣

∣

∣

∣

∑

i∈I

ri

∣

∣

∣

∣

∣

≥ 1

6

n∑

i=1

|ri|_

T.S UTV" < T%Å5ÅdW"/ ? SoX9 YV.T64X ½Ä4X Q|W>_Ê/YA

r =n∑

i=1

|ri|_ ;<?

i = 0 1 2 78/Yl68/

si =∑

rj ≥0

j≡i (mod 3)

rj14687

ti =∑

rj <0

j≡i (mod 3)

rj_

D./26yM/D<1Y</r = s0 + s1 + s2 − t0 − t1 − t2 16872r = (s0 + s1) + (s1 + s2) + (s2 + s0)

− (t0 + t1) − (t1 + t2) − (t2 + t0)_

D./2?C/2eo;<?C/ AED./?C//ËBJ3|A i11687

i2y*BAED

i1 6= i23= b DdAED<12A

si1 + si2 ≥ 13r;<?

ti1 + ti2 ≤ −13r_ a 33=8HI/ y*B@AED.;>=<A-J;<353¡;>er2/268/?12-zBA9 AED<1YA

si1 + si2 ≥ 13r

14687si1 + si2 ≥ −(ti1 + ti2)

_D./26

si1 + si2 + ti1 + ti2 ≥ 0 1687(yM/MD<1</(si1 + si2 + ti1) + (si1 + si2 + ti2) ≥ si1 + si2 ≥ 1

3r_

¥/26 b / ;<68/;>e si1 + si2 + ti1

1687si1 + si2 + ti2

H=83EARG./12AL-0/2143|A 16r_

8; b ;<HOF.-J/YAC/MAED>BJ3:6<=8HIG8/2?;>erAED./A@`TX `WYXyM/(-J;.;.O1YA35;>-z=<ACBJ;<6.3e0?C;<H¶;>=8??C/1478/2?3 AC; F8?C;.G>-0/2HI3&;>e(AED./ 44 rD>Bz68/35/½1YAED./HO12ACB b 12- R- 9.HOF.Bz17 B </6 %#

#4 _

Á _ U61 b =<A|/A|?CBz1468-J/ 4ABC ∠ACB > ∠ABC_R,<;>BJ6>A

DBz3 ;<6

BC3= b D*AED<1YA

∠ADBBz3;.GAC=83+/._¡Ê/YA

HG./:AED./M;<?AED.; b /26>A|?C/M;>e 4ABD

_L2=8F.Fl;<35/F8;>BJ6>AFBz3BJ6.3EB07./ 4ABC

14687;<6(AED./ b Bz? b =8H b Bz? b -0/*;>e 4ABD_R,8?C;</AED<12AF8;>BJ6>A

FBJ3AED./;<?AED.; b /6>A|?C/;>e 4ABC

Bze1687;<6<- 9IB@eHD

BJ3F.14?12-z-J/2-lAC;CF

14687HBz3:;<6OAED./b BJ? b =8H b BJ? b -J/M;>e 4ABC

_

"2

TKÃW"[email protected] YST*¿`WYX -`X"WB ,-`X YST>;Â4 ", )2T T W Å ,!# ¿ W DÃW W PÅ !X SW ¿L_

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A

B C

D

F

H

BJ=8?C/ Bz?3EA-0/A=83 3=8F.Fl;<35/AED<12A

FBJ3LAED./K;<?AED.; b /26>A|?C/*;>e 4ABC

035/4/ B0=8?C/ _D./26CF ⊥ AB

_ a 3HBJ3RAED./M;<?AED.; b /6>A|?C/M;>e 4ABD yM/MD<1Y</ HD ⊥ AB

_D>=83 HD ‖ CF_2BJ6 b /

A B D 14687 F14?C/ b ;<6 b 9 b -zB b ∠AFB = ∠ADB

_ a 3F1687

H14?C/;<?AED.; b /26>A|?C/3;>e 4ABC

14687 4ABD ?C/235F8/ b ACB </2- 9 yM/D<1</AF ⊥ BC BF ⊥ AC AD ⊥ BH 1687 BD ⊥ AH

_ 0AMeo;>-z-J;4y3MAED<12A∠AFB + ∠ACB = 180 1687

∠ADB + ∠AHB = 180 _MD./26 =83Bz68 E∠ACB = ∠AHB

_D./2?C/2eo;<?C/A B C 1687 H

1?C/ b ;<6 b 9 b -@B b _ ;4yyM/ b ;<6.3EB07./?LAED./ b ;<6</2?3+/(J3+// B0=8?C/ _ a 33=8HI/*AED<1YA HD ‖ CF14687AED<12A

A B C 14687 H14?C/ b ;<6 b 9 b -@B b _2BJ6 b /

A B D 14687 F1?C/ b ;<6 b 9 b -zB b

∠AFB = ∠ADB_2Bz6 b /

A B C 14687 H1?C/ b ;<6 b 9 b -zB b ∠ACB = ∠AHB

_a 3H

BJ3MAED./;<?AED.; b /6>A|?C/;>e 4ABD yM/D<1Y</ ∠ADB + ∠AHB = 180 _D>=83 ∠AFB + ∠ACB = 180 _Ê/YAGG./*AED./?C/l/ b ACBJ;<6;>e

FAED<?C;>=l2D

AB_LD./26

∠AGB = ∠AFB1687

∠ABG = ∠ABF_D>=83

∠AGB + ∠ACB = ∠AFB + ∠ACB = 180 _¥/26 b / A G B 14687 C

1?C/ b ;<6 b 9 b -zB b _2BJ6 b /CF ‖ HD

14687HD ⊥ AB yM/ID<1Y</ CF ⊥ AB

_I`D./6 3Bz6 b /FG ⊥ AB AED./*Fl;>Bz6>AC3 C F 14687 G

H=83EARG./ b ;>-@-zBz68/1?_¥/26 b / ∠ACF = ∠ACG = ∠ABG = ∠ABF

_2BJ6 b /

CF ⊥ AB yM/D<1</∠ACF + ∠BAC = 90 yMD>B b D B </3 =83

∠ABF + ∠BAC = 90 _ D./26 BF ⊥ AC_ `D>=83 F

BJ3MAED./¦;<?AED.; b /26>A|?C/;>e 4ABC_

"2

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A

B C

D

F

G

H

BJ=8?C/ _¡Ê/YA

mG./*1MB </6BJ6>AC/42/2?_¡,8?C;</AED<12A[AED./?C//EË4Bz3EA[BJ6>AC/42/2?3

a b 1687 k3= b DAED<12ARG.;AED

a b 14?C/*68;AR7_Ê/YA

nG8/1OF8;<3BACB</Bz6>A|/2/? 14687-0/A r

G./O16¦;.7.7F8;<3BACB</(BJ6>AC/42/2?_ ;<?1469I;.7.7OF8;<3BACB</KBz6>A|/2/?3x1687

y

xr ≡ yr (mod 2n) ⇐=⇒ x ≡ y (mod 2n) G8/ b 12=83+/xr − yr = (x − y)

(

xr−1 + xr−2y + · · · + yr−1)

14687xr−1 + xr−2y + · · · + yr−1 Bz3¡;.7.7_0ALeo;>-@-0;4y3LAED<1YA[AED./K3+/A;>e b ;<68Y?C=l/26 b /b -J13535/3;>e

1r 3r 5r . . . (2n − 1)r HI;.7<=.-J; 2n BJ3MAED./d314HI/d143MAED./d35/YAM;>eb ;<68Y?=l/6 b / b -J13535/3r;>e1 3 5 . . . 2n−1 yMD>B b DBz3`AED./R3+/A;>e41-@-;.787 b ;<68Y?C=l/26 b /b -J13535/3RH;.7>=.-0;

2n _.144Bz68r = 19

1687n = 1999 yM/M3+//*AED<1YA AED./2?C//EË4Bz3EA316;.7.7I6<=8HG./?

a03E= b D*AED<12A

2m−1 ≡ a190

(

mod 21999) _[rD.;.;<35/

a ≡ a0

(

mod 21999) AC;(G8/3= b B0/26>AC-9O68/Y12ACB </35;(AED<1YA

2m − 1 − a19 > 0_[`D./6O1M35;>-z=<ACBJ;<6BJ3

(a, b, k) =

(

a, 1,2m − 1 − a19

21999

) _

D<1YA b ;<HIF<-0/A|/23AED>BJ3[6<=8HG./?;>e>AED./ @`TX `WYX^_ /1?C/BJ6 R- 9.HOF.Bz1735/13+;<6_4/687OH/9<;>=8? 6-@=<ACB0;<6.3R14687I2/268/?12-zBE12ACBJ;<6.3¡143yM/Y-z-l13 R-9<HIF<BJ147.32_

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xi = 1 AED./26 p(i + 1) = p(βi+1|βci ) · p(βc

i ) =1 − p(i)

n − i − 1 1687B@e xi = 0 AED./6

p(i + 1) = p(βi+1) = p(βi) = p(i)_OU<<=81YACB0;<6I eo;>-@-0;4y32_

0 > 0@A86 =9 < A76 A76 < 6 0$30@A

U6AED./M;<?B0Bz6.1-lY14HI/ AED./G8/23EA3EA|?1YA|/+9My143 AC; b D<1682/KAED./78;.;<?1687AED./yM;<?3EA.y13AC;:4/4/2F:B@A_ D<12A`1?C/AED./ b ;<?C?C/3Fl;<687>BJ683EA|?1YA|/B0/23rBz6AED./¡2/268/?12-zB/47Y14HI/4É8;*eo;<?CHO=.-z12AC/146M146.3CyM/?`yM/:D<1</A|;3|AC=l79*AED./35/4>=l/6 b / px,n(i)n−1i=1

_&u.u792 zu' ;<?1469OB</26

i = 1 2 . . . n − 1 yM/D<1</1

n≤ px,n(i) ≤ i

n

_ $lXT.T"7 `D./ b -J12BJH eo;>-@-0;4y3G9IBz687<= b ACBJ;<6d;<6

iy*B@AED

nË4/47r_oe

i = 1 AED./26IAED>BJ3BJ3 A|?B>BJ12-_L2=8F.Fl;<35/*BABJ3 A|?=l/*eo;<?3+;<HI/i ∈ 1 2 . . . n − 2 _D./26

p(i + 1) ≥ xi

n

n − i

n − i − 1+

1 − xi

n≥ 1

n14687

p(i + 1) ≤ xi

n − i − 1

n − 1

n+ (1 − xi)

i

n

≤ xi

n(i + 1) + (1 − xi)

i + 1

n=

i + 1

n

_ /D<1</R=83+/7AED./Re01 b AlAED<1YA eo;<? i = 1 2 . . . n−2 yM/D<1</ n − 1

n − i − 1≤ i+1

_U6F81?ACB b =.-J1? AED./KF8?C;.G<1G>Bz-@B@AU9;>e.y*Bz6.6<BJ68AED./KF8?B@/*@AED<12A[BJ3 px,n(n − 1)1- y1Y9<3-zBJ/3LG./YAyM//6

1/n14687

1−1/n_ / y*Bz-@-35D.;4y AED<1YAG8;AED;>eAED./23+/ >1-@=l/3b 16G./1YAAC12BJ68/7IG9 b D.;.;<3EBJ68:14F.F8?C;<F.?CBz12AC/MJ=86<BJ<=l/ 3|A|?12AC/4BJ/32_L4/4/*1-z3+; _

&u.u792 zu'x ;<?1469n ≥ 3

AED./Keo;>-z-J;4y*BJ68K/4>=.B>12-0/26 b /23D.;>-J7 px,n(n − 1) =

1

n⇐=⇒ xi = 0 ∀ i = 1 2 . . . n − 2

px,n(n − 1) =n − 1

n⇐=⇒ xi =

0 ∀ i ≤ n − 3 1 i = n − 2

_

7È2

$lXT.T"7 [;<6.3BJ78/2?`AED./¡8?3EAr3|AC1YA|/2H/26>A_ oexi = 0

eo;<?r12-z-i = 1 2 . . . n−2 AED./6

px,n(n−1) = 1/neo;>-@-0;4y3e0?C;<H _ /F.?C;</¡AED./ b ;<6</?35/:G9 b ;<6>A|?17>B b ACBJ;<6_Ê/YA

i0G./AED./3HI12-z-J/3|AR>12-z=l/My*BAED

xi0 = 1_M`D./6 G9/4>=812ACBJ;<6¦ | yM/OD<1</

p(i0) > 1/n_[3Bz68O,8?C;<Fl;<3EB@ACBJ;<6 yM/(D<1Y</ p(i) > 1/n

eo;<?¡169i ≥ i0 1687AED>BJ3 b ;<6>A|?147ACB0;<6AED<1YA

p(n − 1) = 1/n_

;4y b ;<6.3EB07./?KAED./d35/ b ;<6873EA12AC/HI/6>A_ oe xi312ACBz3l/3AED./ b ;<687<BACB0;<6.3;<6¦AED./?B02DAK3BJ78/ AED./6 px,n(n − 1) = 1 − 1/n

_;<6</?35/2- 9 1353E=8H/OAED<1YAp(n − 1) = (n − 1)/n

_oexn−2 = 0 AED./6 p(n − 2) = p(n − 1) = (n − 1)/nG9O | 14687MAED>BJ3 b ;<6>A|?147BJ3RBJ3;<6<- 9Fl;<33BJG<-J/KBze

xn−3 = 0 = · · · = x2 = x1_

D>=83 B@e<;<68/Reo;>-@-0;4y3AED./G8/23EA3EA|?1YA|/+9Keo;<?lAED>Bz3[Y14HI/ AED./26:AED./F.?C;.G.14G<B@-zBA9;>e1Ky*Bz6.6<BJ68*/2687<Bz68*1F8F.?C;<1 b D./23 13 AED./6<=8HIG8/2?¡;>erG.;Ë4/3Y?C;4y32_ 6 0@< @0 *0@< @2!!" 6 68A > 0@A76 A76 < 6 0$ =9

e /¡68/Ë2A b ;<6.3EB07./?8yMD<1YAD<1F8F8/6.3B@eyM/ b D<14682/RAED./ b D.;>B b /;>e>G8;ËRyMD./68/+</2?Fl;<33BJG<-J/ AED<12ABJ3 xi = 1eo;<?

i = 1 2 . . . n−1_D./F.?C;.G.14G<B@-zBA9M;>e.1y*Bz6.6<BJ68/687>BJ68AC=8?C6.3M;>=<AA|;¦G./I?C/Y-J1YA|/7dA|; 14F<B0/2?+ 36<=8HG./?

e 13*3EA12AC/47Bz6dAED./I68/Ë2AF8?C;<F8;<3BACB0;<6_&u.u792 zu' Ê/A

n ≥ 314687

xi = 1 eo;<? i = 1 2 . . . n − 1_[`D./6

px,n(n − 1) = 1 −n∑

j=0

(−1)j

j!

14687lim

n→∞px,n(n − 1) = 1 − 1/e

_$lXT.T"7 MÌ¡/Yl68/M1(eJ=86 b ACB0;<6

fr(x) = (1 − x)/r eo;<? x ∈ 1687r > 0

_Ì¡/68;AC/G9qn

AED./*F8?C;.G<1G>Bz-@B@AU9px,n(n − 1)

b ;<?|?C/235F8;<687<Bz68:AC;(AED./ b D.;<3+/26O3|A|?12AC/4+9r_4/AFn = f1 f2 · · · fn−2

_d,8?C;<Fl;<3EB@ACBJ;<6 BJHOF.-@B0/23AED<12Aqn = Fn(1/n)

_0<9BJ687>= b ACB0;<6I;<6n B@ALBJ3/21439(A|;3+D.;4yAED<1YAFn(x) − Fn(y) = (−1)n x − y

(n − 2)! ∀ x y ∈ _

¥/26 b / qn+1 = Fn+1

(

1

n + 1

)

= Fn

(

fn−1

(

1

n + 1

))

− Fn

(

1

n

)

+ Fn

(

1

n

)

=(−1)n

(n − 2)!

(

fn−1

(

1

n + 1

)

− 1

n

)

+ qn =(−1)n

(n + 1)!+ qn

_2BJ6 b /

q3 = 2/3 B@Aeo;>-@-0;4y3AED<1YAqn+1 =

n∑

j=3

(−1)j

(j + 1)!+

2

3= 1 −

n+1∑

j=0

(−1)j

j!−→ 1 − 1

e

0ARBz3/139AC;O3+//KAED<12AAED>BJ3¡BJ3RAED./(G./3|AR3EA|?1YA|/+9;>e`AED./(Y1H/*BzelyM/H;.7>Bze@9YA|/2F AC;OAED./Meo;>-z-J;4y*BJ68 B@erAED./ b ;<6>AC/3|AC16>A78/ b B07./368;ARAC; b D<14682/G8;Ë4/23 AED./6

7È

AED./Y1H/I/687<31687dAED./ b D.;<3+/26G8;ËIBz3;<Fl/268/47r_ aRbEb ;<?C7<Bz68(A|;AED>BJ3*14787>B@ACBJ;<6.1-?C=.-J/ AED./M;<6<-93EA|?1YA|/B0/23RAED<1YAR1?C/147.HB@AAC/47O1?C/KAED.;<3+/Keo;<?[yMD>B b D xi = 1eo;<? 1-@-

i < i01687

xi = 0eo;<?1-@-

i ≥ i0 eo;<? 35;<H/ i0 ∈ 1 2 . . . n − 2 _&u.u792 zu' `D./G./3|A3EA|?1YA|/+9BJ3

xi = 1eo;<?L1-@-

i = 1 2 . . . n − 214687(AED./

F8?C;.G<1G>Bz-@B@AU9I;>e1:y*BJ6.6<Bz68*/687>BJ68KBz31 −

n∑

j=0

(−1)j/j!_

$lXT.T"7 G<3+/2?U</KAED<1YAeo;<? 169i yM/MD<1</ p(i) > 1/(n − i)

_R¥*/6 b / B@e xi = 1 p(i + 1) =

1 − p(i)

n − i − 1= p(i) +

1 − p(i)(n − i)

n − i − 1> p(i)

_2BJ6 b /*AED./(Y14HI/Bz3;</2?Ly*BAEDOF8?C;.G<1G>Bz-@B@AU9

p(i)B@er1469

xi = 0 AED./(G8/23EAR3|A|?12AC/4+9BJ3x = (1, 1, . . . , 1)

_¡,8?C;<Fl;<3EB@ACBJ;<6 68;4y 9>B0/Y-07<3 AED./ b ;<6 b -z=83EB0;<6_*& oerAED./ b ;<6>A|/23EA146>A¡Bz368;A12-z-J;4yM/47OAC; b D.;.;<35/(1F.?C/+<BJ;>=83- 9 b D.;<3+/26dG8;ËJBzHIF<-9>BJ68OAED<12A

nBz3;.787 E AED./26AED./dF.?C;.G.14G<B@-zBA9&;>eR1y*Bz6.6<BJ68/687>BJ68Ieo;<?KAED./G8/23EA*3|A|?12AC/4+9¦BJ3 +3E=8?|F.?CBz3Bz68-9 ?C/2-z12AC/47A|;AED./6<=8HG./?

π_ 4//IAED./F.14F8/?*G9£2;<?|B _

Ck+ +u& 9&b*&D./?C/217./? b 16¤0=83|ACBze@9HO1469I;>e8AED./1?C=8H/26>AC3B </6O1G.;</G9I4//F<BJ68:Bz6OHBJ687(AED./Keo;>-z-J;4y*BJ68:F.?CBz6 b BJF<-0/._ D./26AED./D.;<3EA /Y-zBzHOBz6.12AC/3R1MG8;Ë AED./d?C/2HI12BJ6<Bz68=86 b D.;<35/6&G8;Ë4/23 5BJ68D./2?CBA AED./2Bz?F.?C;.G.14G<B@-zBACB0/23 F8?C;<F8;<?ACBJ;<6.1-@-97A|/23EA146>A b D.;.;<35/3M1G8;Ë14687&78;./2368;A b D<14682/=86>ACBz-AED./-z143|A3EAC/F AED./6 B@A3F.?C;.G.14G<B@-zBA9&;>e b ;<6>A1Bz6<BJ68AED./F.?CB/dBz31/n yMD>B@-0/AED./¦;AED./2??C/HO1Bz6<BJ68dG8;Ë¦1YA*AED./d-z143|A3EAC/Fy*Bz-@-¡D<1</F8?C;.G<1G>Bz-@B@AU9(n − 1)/n 3Bz6 b /BA:D<13:BJ68D./2?CBA|/71-@-rAED./ 1/n

F8?C;.G<1G>Bz-@B@ACBJ/3:e0?C;<HAED./n − 2

G.;Ë4/3AED<12AMAED./ D.;<3EAD<143I/[email protected]|/7&A|;.2/AED./?y*BAED BAC3I;4y61/nF8?C;.G<1G>Bz-@B@AU9_

0 # 0@<0 =4 0@A _ _ 0l14F8/3Cy14?1±]`14; 14687 c_ 0`D<143514?1±]`1; a AED<?C/4/~57.;.;<?Y1H/3+D.;4y14687I35;<H/M;>e`B@A3 >14?BJ16>AC3 )`W;< <SWÅ3.S UQ9 Q WrS 2S 4 44> E ;r_

_

a _4£2;<?|B a =<AC;<H;.G>Bz-@B b 1F8?C/K/ π RXEQ ÅdW"8W p^s 44> E ;r_ È4 È% _

1G>B0;(£= bEb 1Ì BJF.14?ACBJHI/6>AC;O7<B12AC/HO12ACB b 1,<;>-@B@AC/ b 6B B@-J168;,.BJ1YCE1OÊ/;<6.14?C7.;O7.1 BJ6 b B 4>4 B@-J168; [email protected]

7È

2rI|%~Pw*u¡~P¡%rPsuwn~*0u0uu|%su7|2rxZtQIPswr0t%~Ps~*tw¶6*222#wtu~IsK0u¸

(?)tI~PsFt.w|¡s#w7xPPt%~Psu;~%t%~t %rPs tuFus2xz6P~7|%~

tu2rI|*~Pw3£t%K%rPs¤0u¥2P2sww¡£*w¥2r 0uztwx¡¦sw7D§¨~7s©I0t2rr0tw¥|t2¥%s(u.ª%tw*t*xt

«w0uu|%s(u¬(§v®§ ¯§;tw7xn°§.£w*¥2r 0u±6 r rsQsxs¡¦swK*§;twxbw0uu2|*su²§;³§.´(§;twx¶µ§¦swK6 r r7sQs2xs£w*¥2r 0u3«w~7s.u2rI|%~Pw*u us~Pw§~7sv%rPs S r rSsOu~0t%~Is2xzw~s½r0tw*¥|2t2¥%sF~7s½IztKy.Is(t%~D|s2xu2rI|*~Pw3

·7sOs2xP~IS~%tw%¸u3¹2stw7¢º5t·sDKPs3twxzº5t~w»2rxu~Ps(wz#~s¼3wPsIuP~yºnw~s(t2r7P½~Itw%uQr0t%~Pw%uvF~7s½%rPszu2

Æ *Æ _ $lXET*¿TYWV : ÅdW4W 68 SV.TÅ5ÅdW [S`W W4SWX9 2_U6 b BJ? b -J/

Γy*B@AED b /26>A|?C/

O1687?147<B@=83

R yM/OD<1Y</(AED<?C//F81?1-@-0/Y- b D.;<?C7<3A1A2 B1B2 14687 C1C2

_4D.;4y AED<12ARAED./;<?AED.; b /6>A|?C/23K;>eAED.//YB02DARA|?CBz1468-J/3D<1Y>BJ68</?ACB b /3Ai Bj 14687 Ck

i j k ∈ 1 2 14?C/ b ;>-@-zBz68/1?_Æ *Æ:Ä ? _ $lXT*¿TW"V" WB7 .W4SRX Q9Â KÃWYX SIT"7 XW*ÃT" 4X9 EW*ÃT"

-T ¶ÁMWX ^WB >TÃ 8_2=8F8F8;<3+/KAED<1YA

a b 1687 c1?C/F8;<3BACB</*?C/21-l6<=8HIG8/2?3_,8?C;2</KAED<1YA

a3

b2 − bc + c2+

b3

c2 − ca + a2+

c3

a2 − ab + b2≥ 3(ab + bc + ca)

a + b + c

_Æ *Æ¾ _ $lXET*¿TYWV @8X 2SUT*¿WX +-`X"WB -`X 2SUTÂ _

2=8F8F8;<3+/AED<12AABC

Bz3*16/<=.B@-J1YA|/2?1-A|?BJ168-0/O1687AED<12APBz3*1OFl;>Bz6>ABz6AED./KF.-z1468/*;>e 4ABC

_D./KF8/?CFl/2687B b =.-z14?Le0?C;<HPA|;

CAHI/4/AC3

BC1YA

Y 14687MAED./KF8/?CFl/2687AC/?B0;<? ;>e 4ABC F8?C;2</KAED<1YA [XY Z] ≤ [ABC]

_ _oe

P-zBJ/3;<6 AED./ b BJ? b =8H b BJ? b -J/;>e

ABC F.?C;</¦AED<1YA X Y 1687 Z14?C/b ;>-@-zBz68/1?_

Æ *Æ _ $lXT*¿TW" V -TQET ÀTÅdWYXET < X ,>W VÂ DÃWX 8W ?8T >8T ¿ r_

2=8F8F8;<3+/OAED<12A 4ABCD<13

∠A = 90 14687∠B > ∠C

_&Ê/YAH

G./IAED./eo;.;A*;>eLAED./IFl/2?|F8/687>B b =.-z14?:e0?C;<HAAC;

BC_D./IFl;>Bz6>A

B′ -@B0/23M;<6 BC14687¦Bz3AED./HBJ?C?C;<?:BzHI142/;>e

BBz6¦AED./I-@BJ68/

AH_2=8F.Fl;<35/OAED<12A

DBJ3KAED./Ieo;.;A*;>eAED./Fl/2?|F8/687>B b =.-z14?e0?C;<H

B′ A|; AC AED<1YA EBz3AED./eo;.;A;>e>AED./Fl/2?|F8/687>B b =.-z14?e0?C;<H

DA|;BC AED<12A F

BJ3LAED./:eo;.;A;>e.AED./KFl/2?|F8/687>B b =.-z14?Le0?C;<HBAC;

AB′ 1687(AED<1YA GBz3AED./*eo;.;AR;>e8AED./F8/?CFl/2687<B b =.-J1? e0?C;<H

FAC;

BC_¡,8?C;2</KAED<1YA

AH = DE + FG_

7È Æ *ÈÇ _ $lXT*¿TW" V YTW U : -,4X|XEWYXET" Â KÃWYX S<S $lT S"TW4Q Q [email protected] -,XQ|W UT ¿ r_

2=8F8F8;<3+/KAED<1YAa b 1687 c

1?C/F8;<3BACB</*?C/21-l6<=8HIG8/2?3_,8?C;2</KAED<1YA1

a2+

1

b2+

1

c2− 27

(

ab

c+

bc

a+

ca

b

)−2

≥ 1

3

[

(

1

a− 1

b

)2

+

(

1

b− 1

c

)2

+

(

1

c− 1

a

)2] _

Æ *È¡Á _ $lXET*¿TYWV : ÅdW4W 68 SV.TÅ5ÅdW [S`W W4SWX9 2_W[B </6&>=817<?CB@-J1YA|/2?1-

ABCD -0/A P Q R S M1687

NG./AED./HOBJ78~Fl;>Bz6>AC3;>e

AB BC CD DA AC14687

BD ?C/235F8/ b ACB </2- 9_2=8F.Fl;<35/AED<1YAAED./ 7>BJ142;<6.1-z3AC

1687BD

Bz6>A|/2?3+/ b A1YAE_ Ê/A

OG8/dAED./¦F8;>BJ6>A(3= b DAED<1YA<=8147.?Bz-z12AC/?12-

NEMOBz3¡1MF.14?12-z-J/2-J;.Y?14H¦_,8?C;</AED<12A

[OPAS] = [OQBP ] = [ORCQ] = [OSDR]yMD./?C/

[WXY Z]?C/2F8?C/23+/26>AC3RAED./*14?C/21(;>e<=8147.?Bz-z12AC/?12-

WXY Z_ Æ *È:Æ _ $lXET*¿TYWV )SD87ÃT4X - <Q4XEWYS;ÀTÅ3 ._

U6 4ABC 3=8F.Fl;<35/AED<1YAAED./:F8;>BJ6>A3 M N -zBJ/K;<6AED./-@BJ68/3+/YH/26>ABC AED./IFl;>Bz6>A

P-zBJ/3;<6dAED./-zBz68/I35/4YHI/6>A

CA 1687dAED./IF8;>BJ6>A Q-@B0/23;<6dAED./O-@BJ68/3+/YH/26>A

AB 3E= b D(AED<12A MNPQBz3¡1M35<=81?C/8_L2=8F8F8;<3+/KeJ=8?AED./2?AED<1YA

AM

AN=

AC +√

2AB

AB +√

2AC

_rD<1?1 b AC/?B@/ 4ABC

_Æ *È:È _ $lXET*¿TYWV )SD87ÃT4X - <Q4XEWYS;ÀTÅ3 ._

,8?C;</ y*B@AED.;>=<ALAED./K=83+/;>e1 b 12- b =.-z12AC;<? AED<12A sin(40) <√

37

_Æ *È _ $lXET*¿TYWV : ÅdW4W 68 SV.TÅ5ÅdW [S`W W4SWX9 2_

U6 4ABCy*B@AED b Bz? b =8HI?147<B@=83

R -0/A AD BE1687

CFG8/AED./:12-@ACBAC=l78/23_Ê/YA

PG8/O1469dBz6>A|/2?CBJ;<?:F8;>BJ6>A:;>e[AED./A|?BJ168-0/._O`D./O-@BJ68/AED<?C;>=l2D

PF81?1-@-0/Y-AC;

EFBz6>A|/2?3+/ b AC3:AED./-@BJ68/

AC12A

E114687AED./-@BJ68/

AB1YA

F1_D./-@BJ68/(AED<?C;>=l2D

PF81?1-@-0/Y-`A|;

FDBJ6>AC/?35/ b A3AED./M-zBz68/

AB12A

F214687OAED./M-zBz68/

BC12A

D2_`D./-zBz68/AED<?C;>=l2D

PF.14?12-z-J/2-<AC;

DEBz6>A|/2?3+/ b AC3AED./:-zBz68/

BC1YA

D31687(AED./:-zBz68/

AC12AE3_4D.;4yAED<12A

E1F1 cot A + F2D2 cot B + D3E3 cot C = 2R_

7ÈÆ *È _ $lXET*¿TYWV )SD87ÃT4X - <Q4XEWYS;ÀTÅ3 ._

2=8F8F8;<3+/hAED<12Aa b 14687 c

1?C/Fl;<3EB@ACB </?C/12-6<=8HG./?3 yMD>B b D 351YACBJ3Ee@9a2 + b2 + c2 = 1 14687AED<1YA n > 1

BJ3¡1Fl;<3EB@ACB </KBJ6>AC/42/2?_¡,8?C;</:AED<12Aa

1 − an+

b

1 − bn+

c

1 − cn≥ (n + 1)1+

1n

n

_Æ *È _ $lXET*¿TYWV )T UT W Å ,!# ¿ _

r;<6.3EB07./? 4ABCy*B@AED

∠ABC = 2∠ACB14687

∠BAC > 90 _W[B </6AED<12ALAED./F8/?CFl/2687=l2DCHI/4/AC3

AB1YA

D F.?C;</KAED<1YA1

AB− 1

BD=

2

BC

_Æ *È:Ä _ $lXET*¿TYWV )T8TX#< SW*ÃBÂ KÃWYX ST"7½ÀT6 YW ;ÀT6 YW - > "X ._

2=8F8F8;<3+/AED<1YAx1 . . . xn

n ≥ 2 1?C/KF8;<3BACB</:?C/21-.6<=8HG./?32_,8?C;2</AED<1YA

(

x21 + · · · + x2

n

)

(

1

x21 + x1x2

+ · · · +1

x2n + xnx1

)

≥ n2

2

_

Æ *È¾ _ $lXET*¿TYWV )T8TX#< SW*ÃBÂ KÃWYX ST"7½ÀT6 YW ;ÀT6 YW - > "X ._2=8F8F8;<3+/KAED<1YA

x1 . . . xn α 14?C/*Fl;<3EB@ACB </*?C/12-`6<=8HIG8/2?3_,8?C;2</KAED<1YAJ1 n

√

(x1 − α) · · · (xn − α) ≥ α + n√

x1 · · · xn

oG n√

(x1 − α) · · · (xn − α) ≤ α +x1 + · · · + xn

n

__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _Æ *Æ _$lXT*¿TW¿X.: ÅdW4W 68 SV.TÅ5ÅdW > $" -, _

ÌR146.3L=86 b /? b -J/Γ7./ b /26>A|?C/

O/YA[78/?1Y9<;<6

R ;<6M1A|?C;>BJ3 b ;<?C78/23F81?1-@-/Y-0/23A1A2 B1B2

/AC1C2

_.&;<6>A|?C/?[>=l/-0/23;<?AED.; b /6>A|?C/23 7./3D>=.B@ArA|?CBz1468-J/3L19>146>AFl;>=8? 35;<HIHI/YA3Ai Bj

/YACk

i j k ∈ 1 2 3+;<6>A b ;>-@BJ6/21Bz?C/32_Æ *Æ:Ä ? _ $lXT*¿TW¿X W 7 <WS RX Q9zÂ KÃWYX S W 8W 4X9 EW*ÃT" 4X9 EW*ÃT"

-T WMW4S;ÁMWYX W <T7Ã W_2=8F8F8;<3+;<6.3<=l/

a b /YA c3+;<6>A 78/2368;<HIG.?C/23? //2-z3¡Fl;<3EB@ACB@e03_`&;<6>A|?C/2?R<=l/

a3

b2 − bc + c2+

b3

c2 − ca + a2+

c3

a2 − ab + b2≥ 3(ab + bc + ca)

a + b + c

_

7È Æ *Æ¾ _$lXT*¿TW¿X @8X 2SUT*¿WX +-`X"WB -`X 2SUT -[_

4;>B@AP

=86"F8;>BJ6>A7<146.3 -J/ F.-z146"7 =86A|?CBz1468-J/ /4>=.Bz-z12A /2?1-ABC

_Ê21Fl/2?|F8/687>B b =.-z1Bz?C/d1G<1Bz353/4/78/P3E=8?

BCb ;>=8F8/

AB/6

X b /Y-z-J/1G<1Bz353/4/78/P3E=8?

CAb ;>=8F8/

BC/26

Y /YA b /2-@-0/*14G.12BJ33//78/ P3=8?

ABb ;>=8Fl/

CA/6

Z_ _M&;<6>A|?C/? <=l/

[XY Z] ≤ [ABC]3EB

P/23EA 1-U Bz6>A/?B0/Y=8?R7>=A|?CBz1468-J/

ABC_ _M&;<6>A|?C/?(<=l/

X Y/A

Z35;<6>A b ;>-zBz6 /12BJ?C/233B

P/23EAM3BAC= /3=8?-0/ b /2? b -0/F81353146>AF81?-J/3¡35;<HIHI/YA378/

ABC_

Æ *Æ _6$lXET*¿TWO¿X -TQETbÀ[TÅdWXT;< 4X ,<W OÂ DÃWX SW 8W >8T ?8T (¿ `W>_

4;>B@AABC

=86MA|?BJ168-0/:?C/ b AC168-0/K7` D49<Fl;A /26<=83+/BC

/YAL7 1468-J/K/6BF<-z=83Y?14687<=l/ b /2-@=.Bl/6

C_<4;>BA

H-J/KF<B0/778/-J1KFl/2?|F8/687>B b =.-z1Bz?C/*14G.12BJ33//K78/

A3=8?

BC/YA

B′ -0/39.H /YA|?B0>=l/:78/ BF.14??1F8F8;<?A1 b /YAAC/Fl/2?|F8/687>B b =.-z1Bz?C/8_8Ì /3EB0Y68;<6.3?C/3Fl/ b ACB</2H/26>AKF81?

D E F/A

G-0/23F<B0/7.3M78/23Fl/2?|F8/687>B b =.-z1Bz?C/3M14G.12BJ33//378/

B′ 3E=8? AC 7./ D3E=8?

BC 7./ B3=8?

AB′ /A78/ F3=8?

BC_&;<6>A|?C/?>=l/

AH = DE + FG_

Æ *ÈÇ _$lXT*¿TW ¿4X YTW U : -,4X|XEWYXET"¶Â DÃWX SW $lT.SUW4Q ,<W [email protected] UT" `WB -,XQ|W UTW (¿ `W>_

2=8F8F8;<3+;<6.3<=l/a b /A c

35;<6>AR7./3¡68;<HIG.?C/23? //2-z3¡Fl;<3EB@ACB@e03_l&;<6>A|?C/?R>=l/1

a2+

1

b2+

1

c2− 27

(

ab

c+

bc

a+

ca

b

)−2

≥ 1

3

[

(

1

a− 1

b

)2

+

(

1

b− 1

c

)2

+

(

1

c− 1

a

)2] _

Æ *È¡Á _$lXT*¿TW¿X.: ÅdW4W 68 SV.TÅ5ÅdW > $" -, _ÌR146.3=86>=817<?CB@-J1YA/?C/7.;<6.6 /

ABCD 3+;>BAR?C/3Fl/ b ACB</2H/26>A P Q R S M

/AN

-0/23*Fl;>Bz6>AC3*HOB@-zBJ/2=Ë78/2335/4YHI/6>A3AB BC CD DA AC

/ABD

_2=8F8F8;<3+;<6.3*>=l/-0/23*7>BJ142;<6.1-J/3AC

/ABD

35/ b ;>=8Fl/26>A:/26E_r4;>BA

O-0/Fl;>Bz6>AA|/Y->=l/*-0/>=817<?CB@-J1YA/?C/

NEMO3+;>BAL=86IF.14?12-z-/2-J;.Y?14HOH/._&;<6>A|?C/?>=l/

[OPAS] = [OQBP ] = [ORCQ] = [OSDR]0; =

[WXY Z]?C/2F8? /35/6>AC/*- 1Bz?C/M7>=<=8147.?Bz-z12A /2?C/

WXY Z_ Æ *È:Æ _$lXT*¿TW¿X )SD8ÃTX - <Q4XEWYS;ÀT6Å3 UW>_

ÌR146.3*=86dA|?CBz1468-J/ABC ;<63E=8F8F8;<3+/>=l/O-0/23F8;>BJ6>A3 M N

3+;<6>A:3E=8?-0/b;A/BC <=l/*-J/*Fl;>Bz6>A P

/23EA3E=8?-J/ b;A/CA

/YAR>=l/*-J/*Fl;>Bz6>AQ/3|A 3E=8?-J/ b;A /

AB 78/KAC/2-@-0/*35;<?AC/<=l/ MNPQ35;>B@AL=86 b 1?|? /8_ ¡6O3E=8F8F8;<3+/78/*F.-@=83>=l/

AM

AN=

AC +√

2AB

AB +√

2AC

_R=l/F8/2=<AE~;<67>BJ?C/7<=A|?CBz1468-J/

ABCÉ

7È 2Æ *È:È _$lXT*¿TW¿X )SD8ÃTX - <Q4XEWYS;ÀT6Å3 UW>_

146.3 b 12- b =.-z12A|?B b / HI;<6>A|?C/? <=l/ sin(40) <√

37

_Æ *È _$lXT*¿TW¿X.: ÅdW4W 68 SV.TÅ5ÅdW > $" -, _

ÌR146.3*=86dA|?CBz1468-J/ABC

78;<6>A-0/O?1Y9<;<67<= b /? b -J/ b BJ? b ;<6.3 b ?B@AK/23EAR 35;>B@A

AD BE/YA

CF-J/3MD<1=<AC/2=8?32_L4;>BA

P=86Fl;>Bz6>A*<=l/Y- b ;<68<=l/ 1-U Bz6>A/?B0/Y=8?*7<=A|?CBz1468-J/8_rÊ21F.14?12-z-/2-J/ 1

EFF.14?

Pb ;>=8Fl/K-J1M7.?C;>BA|/

AC/26

E1/YA-z1M7.?C;>BA|/

AB/6F1_LÊ21OF81?1-@-/Y-0/ 1

FDF.14?

Pb ;>=8Fl/-J1I7<?C;>B@AC/

AB/26

F2/A-z1I7.?C;>BA|/

BC/6D2

_Ê21F.14?12-z-/2-J/ 1DE

F.14?Pb ;>=8Fl/M-z17<?C;>B@AC/

BC/26

D3/AR-J17.?C;>BA|/

AC/6E3_&;<6>A|?C/?R>=l/

E1F1 cot A + F2D2 cot B + D3E3 cot C = 2R_

Æ *È _$lXT*¿TW¿X )SD8ÃTX - <Q4XEWYS;ÀT6Å3 UW>_2=8F8F8;<3+;<6.3O<=l/

a b/A

c35;<6>A7./368;<HIG.?C/23? //2-z3F8;<3BACBze03351YACBJ3Ee01Bz3516>A

a2 + b2 + c2 = 1 /YAR>=l/ n > 1/3|A=86I/6>ACBJ/?Fl;<3EB@ACB@eC_&;<6>A|?C/2?R<=l/

a

1 − an+

b

1 − bn+

c

1 − cn≥ (n + 1)1+

1n

n

_Æ *È _$lXT*¿TW¿X )T UT W Å ,!# ¿Tr_

¡67.;<6.68/R=86:A|?BJ168-0/ABC

A|/Y->=l/R- 1468-J/ABC

/3|A`-J/7.;>=lG<-J/78/R-U 168-0/ACB

/A>=l/(-U 168-0/BAC

3+;>BA¡3E=8F /2?CBJ/2=8?190 _[Ì¡/(F<-z=83 -z1F8/?CFl/2687=8Fl/

AB/6

D_`&;<6>A|?C/?R>=l/1

AB− 1

BD=

2

BC

_Æ *È:Ä _$lXT*¿TW¿X )T8TX#< SW*ÃBÂ KÃWYX S W 8WÀT6 YW ½À[T6 WB - > "X W_

2=8F8F8;<3+;<6.3>=l/x1 . . . xn

n ≥ 2 35;<6>AM78/2368;<HIG.?C/23? //2-z3F8;<3BACBze032_&;<6>A|?C/?R>=l/

(

x21 + · · · + x2

n

)

(

1

x21 + x1x2

+ · · · +1

x2n + xnx1

)

≥ n2

2

_Æ *È¾ _$lXT*¿TW¿X )T8TX#< SW*ÃBÂ KÃWYX S W 8WÀT6 YW ½À[T6 WB - > "X W_

2=8F8F8;<3+;<6.3¡>=l/x1 . . . xn α 3+;<6>AL78/23R68;<HIG.?C/23R? //2-z3 Fl;<3EB@ACB@e03_8&;<6>A|?C/2?<=l/J1 n

√

(x1 − α) · · · (xn − α) ≥ α + n√

x1 · · · xn

oG n√

(x1 − α) · · · (xn − α) ≤ α +x1 + · · · + xn

n

_

7ÈÈ

Tn¿XET8VWÅ W*ÃWX¿WYXÅ3 WrS&QTW" )`WWSUTX !# b¿6WB W" SUTQET 8WYX47TX¿ .V Q.S UT `W"! YT.S UT:TX`W"! rS KT¿ 2SF¿XT8VUWÅ

Æ:Ä:Ä¾ _ 7È 7.. $lXT*¿TW"¦V" < -WQ ^W -`X YT7Ã

À[TÅ3 8_2=8F8F8;<3+/AED<12A

z 6= 1Bz3*1 b ;<HOF.-J/Ë6<=8HIG8/2?:3E= b DAED<1YA

zn = 1n ≥ 1 _,8?C;</:AED<12A

∣

∣nz − (n + z)∣

∣ ≤ (n + 1)(2n + 1)

6|z − 1|2 _

STX ÀWÅ3X& `_ a A AED./(/2687;>e`AED./eo/1YAC=8?C/47I3+;>-@=<ACB0;<6AC;AED>BJ3F8?C;.G>-0/2H AED./2?C/y143K1 b ;<6¤U/ b AC=8?C/G9 1-AED./? 1168;>=83:AED<1YAAED./IG8/[email protected]@B0/2?:;>e |z − 1|2 ;<6AED./?CBJ2DA 3BJ78/;>elAED./KBJ68/<=812-zBA9MyM;>=.-07OG8/√

4n(n − 1) sin2(

πn

)

+ 1

4 sin2(

πn

)

_ /MD<1</?C/ b /YB</7O1MF.?C;.;>er;>elAED>Bz3 b ;<6¤U/ b AC=8?C/._ T.S UTV" U8`T6 $lT @`TÅ5Å S3@`T?W <W rSUWYXÁ 7ÃW;ÄJU Â :_

/1?C/MB </6AED<1YAz = e2kπi/n eo;<?3+;<HI/ k ∈ 1 2 . . . n − 1 _D>=83

|z − 1|2 =(

cos2kπ

n− 1

)2

+(

sin2kπ

n

)2

= 2(

1 − cos2kπ

n

)

= 4 sin2 kπ

n14687|nz − (n + z)| =

√

(

(n − 1) cos2kπ

n− n

)2

+(

(n − 1) sin2kπ

n

)2

=

√

(n − 1)2 + n2 − 2n(n − 1) cos2kπ

n

=

√

1 + 2n(n − 1)(

1 − cos2kπ

n

)

=

√

1 + 4n(n − 1) sin2 kπ

n

_Ê/YA

aG8/1dF8;<3BACB</?C/12- 6<=8HIG8/2?_ D./26 |nz − (n + z)| ≤ a|z − 1|2Bze1687I;<6<-9B@e

16a2 sin4 kπ

n− 4n(n − 1) sin2 kπ

n− 1 ≥ 0

_

7È*

0<9AED./<=8147.?1YACB b eo;<?|H=.-J1 AED>BJ3RBz3 A|?C=l/*B@e1687I;<6<-9Bzesin2 kπ

n≥ n(n − 1) +

√

n2(n − 1)2 + 4a2

8a2

=1

2(

√

n2(n − 1)2 + 4a2 − n(n − 1))

_2BJ6 b /

sin2 kπ

n≥ sin2 π

n

eo;<?12-z-k ∈ 1 2 . . . n − 1 AED./1G.;</BJ68/<=812-zBA9OBJ3RA|?C=l/eo;<?R12-z-

k ∈ 1 2 . . . n − 1 B@e1687;<6<- 9OB@erBARBz3¡A|?=l/KyMD./6k = 1 BJ6(yMD>B b D b 13+/

√

n2(n − 1)2 + 4a2 − n(n − 1) ≥ 1

2 sin2 πn

a ≥

√

4n(n − 1) sin2 πn

+ 1

4 sin2 πn

_

"!#%$'&()!"+*-,.#-0/1!.2,.354,*76.,*98: ;<()=>-8?,.;@ .A:'3<$.B?,-

CDFEGIHJD>KHML9NOE>PQMR>Q.PTSVUWFXZY+[\Q]H_^`Sa^?K.P_bQc[d]H+Q.bQ.PnL9[eHf[?Uhg-E>UiUJ[\j^\QkH+EVUgF^e[eH

HJDFQc[d]H+Q.bQ.PU1L2L. . .Ln[d]H+ElHmGkE<n[?U_oTE[?]HUWj>UMQ"H_UhUJWYDZHJD>KpHfHJDFQcUW-qrENHJDFQ

Q^?Q.qlQ.]HUs[d]lE>]-QUiQpHtQuW-K.^dUvHJDFQg-P_EFnWY_HsENHJDFQQ^?Q.qlQ.]HUs[d]wHJDFQkE'HJDFQP

x p!"*y8?B?,AeAO&z8O|! i,3V!~ p.AmF8?BT'*$'&FB:!"+!()'+*F +"8:!MB\*>;~,pB? "'*F pi.,88:!; ,*|2J!M @ FB? p8:9|!p()F!p*>;fh,A?,*|#B?;yJ!!.2J!M @ '*v3V'!l-$.AT!p3y"|; |!f,.*>Bd M)*B:.!" JBm8\& x #|>2m,F8?BT'*F; @ !p*F-,=>!"*F;w!"*-3l,.+ @ >B` @ .M8?B? ;Bd i "B6F8O|!p+*56F8?,F8T! @ .AdAT!"=!";hpAeB\*>;l51;V)6"!MBm8O x $.A?,>s; ,pAdA?, ,AeA\,;) t;)6¡,F8?,AeB:hw¢'>!" "!p*'",Bd=.;1y*>BT.!pm BT#,># @ x @ x ;>(1>!"*9 B\J!" ;v=!p*98?B\*F,p MpB\*F*F; p8_>#-!"*y8\;.A\ @ '3l3*Bm8\& @ .AdAT!"=!";l hB\*98:!"Zh,>.!"*F; ;6yM,3V! k.Am8T'*F;t p8_>#-!"*y8\;.A\ @ '3l3.*>B:8O& @ .AeA:!=>!; hB\*98:!"h,>.!"*F; ;t6£ ,pAm8O|!p|M,*|. ;v)_ AeB\*|!p*>='&.3l*-, Bm3;y¤T*F*F p$'>2_';' 8OB?,y¢|B`-p'-*> p'*>;()!",>.!"M8:'*>;9s/y;))6"F. p8\,.hyB\3l>!p_;y*>BT.!pm BT#,># @ x @ x ;F(1>!"*9 B\J!" ;¥vm=>!"*y8?B?*-,pr)!!p ,pB¥.,F.;Ik'*>=r)'*F=; @ B\*F,y*9#B?*9!~!..A:#'; 8_#-!p*98\;kF#-vB?= i=&3l*F,. JB.3;k)F!'8O|!p*>;1!"+3l,*-&f,"BT#¦F!"§<!"_; p8_>#-!"*y8\; B\*Bm8\& @ .AdAT!"=!"; @ ,.3<$'B:#'=!";r|1-88\OBT!#!"mJ;¨!" p8\,pA:_+Bd='&.3l*-, Bm3;y_,"J; 8OB?,V/1-$F!"M8c|¢6!p,pAO&;l©*98AdAeB? "'*ªy*>BT.!pm B:8O&;c6,2_-"BeAdAT!";¡k(-6>>8O9! 8kBd i "B6>8\,>8:!5y*>BT.!pm B:8O&¦-$.AT!p3«6.A:B\*>=¬yJp|;k6"9B\*>=.9!AT#';51;6£B?F!6pBT.!"&.;16,.3m'J#®)*B:.!" JBm8\&.;t(-B?+3cB\*>=,.3;|;6pB9¯9;.A\ @ '3l3.*>B:8O& @ .AeA:!=>!;v hB\*98:!"h,>.!"*F; ;f69,.*9#Z8O|!9J p!"

° QY_K.^e^-HJD>KpHn∑

i=1

i =n(n + 1)

2

± E>PtEFn-nn ≥ 5

Ln(n + 1)

2− 1 − n − 1

2− (n − 1) = (1)

(

n − 1

2

)

(n − 1)L

7È2

14687eo;<? /+</6n ≥ 6 n(n + 1)

2− 1 − n − 2

2− n = (1)

(

n − 2

2

)

(n)_

*..E6 :;K&%;; -&(;ED; 6K; 6K 6 ; 7)n = 1 (

%+6 (AB OE;C%'+;)

n = 3

3-:(=.:D; 6K; 6KB6 E+6 K$7)n = 2

n = 4

@/;) (6 7;-D; 6K; 6K 6 /%(6C%(( -E( 6 7J; -:D; 6; 6 6 B%6C% 7)

6 C(6K;K8 OP&n

hf_.'' L p' E>P_P_QY_H_Qn vJ p!.#k$&£p 94!FpB? t4 ,""i(|,.+!",*|#.,.*kp 94! x =2M>!;y*>BT.!pm B:8\,>8.AeBm8!.2_*>BT2,c#-! @ ,>8\,pA.*&,;£(|,J2J!MA:'*F,.;y6p9,pB\*y QpH

njFQK]-E>]]-Q.bpKH_[ R>Q[d]H+Q.bQ.P Q"H+QP+qc[?]-Q

n∑

k=0

tanh(2k)

1 + 2 sinh2(2k)

x #Bm8T'9 DFQcgFP_EFj>^?Q.q[dUhU+H_KpH+Q.nVK'jFE.R>QcK.U[`HvGKU[?]H_Q.]-nFQnj.SZHJDFQcgFP_E>gE>UiQ.PU tU-PUJHtb"[`R>Q]c[?] _.'' ' p L[`HtYJE>]HK.[d]-Qn 8m QP+P_E>PU"LFE>]>^ SE>]-QENFGkD[\YDkGK.UYJE>P_P_QY_H_Qn<^dKH_Q.P 9 '"! DFQlQn[eH_E>PUG[?UiDaH_EQ#".H_Q.]-nK.]K.gE^?EFbMSaH+EaHJDFQg-P_E>g-E>UMQPUfK.]-nwHJDFQP_QK'nFQ.PUvNOE>PtUgE[e^d[d]-bhKR>Q.PTS]>[?YJQg-P_EFj^\Qq 6>.AmF8?BT'*<$&8O|!9 "!pm .

$<QG[e^d^-g-P_ER>QhHJD>KHNOE>PK]'ScP_Q.K^|]>W-qlj-QPxL

n∑

k=0

tanh(2kx)

1 + 2 sinh2(2kx)= tanh(2n+1x) − tanh(x)

%T'&

( HsG[d^e^yHJDFQ.]VNOE^e^\EGcLyj.S<UiQpHH_[d]-bx = 1

L|HJD>KpHfHJDFQUJW-q b"[ R>Q.]V[?]ZHJDFQgFP_EFj>^?Q.q [dUtanh(2n+1) − tanh(1)

± E>PtQ.KYD

k = 0L1L2L p LGkQD>KpR>Q

tanh(2kx)

1 + 2 sinh2(2kx)=

sinh(2kx)

cosh(2k+1x) cosh(2kx)

=sinh(2k+1x − 2kx)


=sinh(2k+1x) cosh(2kx) − cosh(2k+1x) sinh(2kx)


=sinh(2k+1x)

cosh(2k+1x)− sinh(2kx)

cosh(2kx)

= tanh(2k+1x) − tanh(2kx)

E>]FUiQuWQ.]H_^ S9LpHJDFQUW-q E>]HJDFQs^\QpNdH)U[?n-QEN %T'& H_Q^?Q.UiYJE>g-Q.U)H_Eb"[`R>QvHJDFQP_[?bD'HUJ[\nFQ

4

> 4I$<P0J0I!C6K GH(6K.+6K; # M6 .? 9/GE*8"3:#R@/0 G> 9OPO %(6K; P9 K8A(8<=6 ;)J@=((IL&#?$GE*8= - $ :6 :',,';:)%(; 6E

R 7; -: C(6K;K?%OPOP; 6K=6 E C ('+ $$.6K"J6;-kA6 A$7 O

0;) ∞ (

E6 1; -:$.+67"; -: KO ; -&(;I$ .6 A?6 I%( 6 -:

)I;- ;- ?%O 6

1 − tanh(1):=/'KP5

K;); 6 An → ∞ 6 ; -: ?%?K; &((

_.'' ' p J p!.#©$'&Z2_>BeAdAT!p, <>,"AT <'_T&.B\,#B? ; p8_>#-!"*y8\;v3V!pBT2,.* @ .AeA:!=>! |!" M,pA:'*B\BTv*F,F8T.AdB?,; 9! i i,AT'*>B?B?;1yJ!!.2J!

(,>#,"18:!.#

$'&8O|! x #Bm8T' )

CW-g-g-E>UMQhHJD>KpHfUiKpH_[?U#|QUtHJDFQhN?W-]-Y_H_[\E>]FK^9Q.u>W-KpH_[\E>]

f(x) + 2f(

x + 2000

x − 1

)

= 4011 − x

± [d]-nwHJDFQRK^dWQENf(2002)

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QpHg(x) =

x + 2000

x − 1

LNOE>Px 6= 1

DFQkb"[`R>Q]lQuW-KH_[?E>]lj-Q.YJE>qlQ.U

f(x) + 2f(

g(x))

= 4011 − x %T'&

( H[dUQ.K.US<H+E YDFQ.YaHJD>KpHg(x) 6= 1

K.]-ng(

g(x))

= x ° QgF^dK'Y+[d]-b

xj.S

g(x)[?] %O & L'GkQbQpHf(

g(x))

+ 2f(x) = 4011 − g(x) %T'&

ê[?qc[?]FKpH_[?]-bf(

g(x)) N\P_E>q %O & K.]-n %T'& LGkQD>KpR>Q

3f(x) = 4011 − 2g(x) + x GkDFQ.]-Y+Q>L

f(x) =4011 − 2g(x) + x

3=

x2 + 4008x − 8011

3(x − 1)

( ]lgFKPmH_[\Y+WF^dKPiLf(2002) = 2003

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F8:&%JK1;- O $A. KOIL6 :5 &7.%'; 6Kf : \ 1 → (; 6 +76 A;-7.%'; 6KE#C%; 6K

f(x) + af

(

x + b

x − 1

)

= c − x $-: |a| 6= 1

:b 6= −1

GH+6 A ;-BO B '')- =6 M;- ;)K%; 6K(J-B--:($ ; -&(;"; -: K%; 6KB6

f(x) =x2 +

(

a(c − 1) − c − 1)

x − a(b + c) + c

(a2 − 1)(x − 1) -5.6 A?6 E#C%; 6KB6 &; ?6 1$-:

a = 2b = 2000

:c = 4011

0 1' p % E>P_P_QY_H_Qn 1 p & "!#®$'&¢ B?*y8d BdO, i; |!" M,pA:'*B\Bd;yJ!!.2J!'

1[ R>Q.] 4ABCG[eHJDcUJ[\nFQ.U

aLbLcL>gFP_E.R>QhHJD>KpH

3(

a4 + b4 + c4)

(a2 + b2 + c2)2 +

ab + bc + ca

a2 + b2 + c2≥ 2

6.A>8?B:'*V$'&cpB¯9;1.A\ @ '3l3.*>B:8O& @ .AdAT!"=!";v hB\*98:!"h,>.!"*F; ;f6$a[eHJDFEW>Hc^\E>UUVENbQ.]-QPK.^e[eHTS9LKUUW-qlQ HJD>KH

a ≤ b ≤ c DFQn-QU[dP_Qn

[?]-Q.u>W-K^d[`H:Sc[dUfQuWF[`RK^\Q]H£H+E

3(a4 + b4 + c4) + (ab + bc + ca)(a2 + b2 + c2) − 2(a2 + b2 + c2)2 ≥ 0L

GkD[\YDw[?UfQ.u>WF[ RK.^?Q.]H£H_E

14

(

a(2a − b − c)2(a + 2b + 2c)

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≥ 0

DFQw^dKU+Hs[?]-Q.u>W-K^d[`H:Sa[dUhY+^?Q.K.P_^ SZH+PWQ u>W-K^d[`H:SVDFE^\n>U[dN1K.]-n<E>]>^ Sa[eNa = b = c

E'H_Q<HJD>KpHHJD[dUg-P_EFENnFEFQ.U]-E'HwP_QuWF[?P_Q<HJD>KpH

aL

bLfK]-n

cKP_QVHJDFQ UJ[\nFQ.UlENK

H+P_[dK]-b"^?Q>L>K.Us^?E>]-bhK.UvHJDFQMScKP_Q]-E>]]-Q.bpKH_[ R>QK]-na2 + b2 + c2 > 0

mm|_ T.S UTV" S`WYX `T6 ÂrX W "Å 9%Å C 2VX >Q 2SoX 8_D./M7./3EBJ?C/7Bz68/4>=81-@B@AU9Bz3/4>=.B>12-0/26>AAC;

3(a4 + b4 + c4) + (ab + bc + ca)(a2 + b2 + c2) − 2(a2 + b2 + c2)2 ≥ 0_

2BJ6 b /dAED>BJ3BJ68/<=812-zBA9BJ339.HOH/A|?CB b yM/ b 161353E=8H/AED<1YA a ≤ b ≤ c_Ê/A

b = a + t1687

c = a + t + s yMD./2?C/ s t ≥ 0_[`D./*Bz68/4>=81-@B@AU9IG./ b ;<H/23

3(

a4 + (a + t)4 + (a + t + s)4)

+(

a(a + t) + (a + t)(a + t + s) + (a + t + s)a)

·(

a2 + (a + t)2 + (a + t + s)2)

− 2(

a2 + (a + t)2 + (a + t + s)2)2 ≥ 0 yMD>B b D3BzHIF<-zB@`/23 AC;

5a2(s2 + st + t2) + 2a(3s3 + 7s2t + 3st2 + 2t3) + s4 + 5s3t + 5s2t2 ≥ 0_

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a(n) =

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∣

∣

∣

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j=0

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j + 1

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(1 + axk)2 !

b G 0 < a < 1p\dUjPei\`kaXw\cDaaXwUè\dUd3e\fXgaP`kgSm`\nc7lRghaPg`7fwUelq !

" wU`BdjPe\de\fJ`7jw\cfdjPeUefGe\jaXwU`BvcfJ`7f n = 3c'mq

n = 4 ! . 4CLUESQ4,K<'>=A\ZQ;@4U:ZRC:/2PMH97EPL;:DKEMD2%QSK\QEG>WH4CC:ZT\:ZMUA'V<,2%Q;,T\:ZMN! c/uG

upv > 1

p'`Bw\cDi\`(u − 1)(v − 1) > 0

pUe\j1 + uv > u + v ! +

¥/26 b / n +

n∑

k=1

ayk =n∑

k=1

(

1 + aλkxkaxk+1λk+1

)

>n∑

k=1

aλkxk +n∑

k=1

axkλk

_&/16yMD>Bz-J/ G9AED./ a

WR U68/4>=81-@B@AU9 aλkxk + a

xkλk ≥ 2a

12xk

(

λk+ 1λk

)

≥ 2axk_

r;<HIG<Bz6<BJ68AED./35/?C/23=.-AC3 yM/;.G>A1Bz6n +

n∑

k=1

ayk > 2n∑

k=1

axk_

2=lG.3|ACB@AC=<ACBz68:AED./KBJ68/<=812-zBA9I Bz6>A|;(B@A3+/Y-ze.9<BJ/2-J7.32 + uvw > 1 + uv + w > u + v + w

_ a -J35; yk + yk+1 = λkxk +

(

λk+1 +1

λk+1

)

xk+1 +xk+2

λk+2

≥ λkxk + 2xk+1 +xk+2

λk+2

_Ì¡/68;ACBz68AED./35/R-J13EAlAED<?C//RAC/?CHI3[G9

u v 1687 w ?C/3Fl/ b ACB</Y-9 Bz6 E yM/;.G>A1Bz63n +

n∑

k=1

ayk+yk+1 ≥ n +

n∑

k=1

aλkxk +

n∑

k=1

a2xk +

n∑

k=1

axkλk

14687=83EBJ68* | yM/;.G>A1Bz63n +

n∑

k=1

ayk+yk+1 ≥ n + 2

n∑

k=1

axk +

n∑

k=1

a2xk =

n∑

k=1

(1 + axk)2_

oG `D./ b -z1BzH¶Bz368;AA|?C=l/._ Ê/YAxk = λk = 1

eo;<?1-@-k_D./26AED./Ol?3|ABJ68/<=812-zBA9IG./ b ;<H/23

n + na2 ≥ 2na ⇐=⇒ n(1 − a)2 ≥ 0 yMD>B b DBJ3 b -J/1?C- 9(A|?C=l/Keo;<? 12-z-

a_L2BJHBz-z14?-9 AED./35/ b ;<687Bz68/4>=81-@B@AU9IG./ b ;<H/23

3n + na4 ≥ n(1 + a)2 ⇐=⇒ (a − 1)2(a2 + 2a + 2) ≥ 0 yMD>B b DBJ312-J35;dA|?=l/Ieo;<?:1-@-

a_D>=83 AED./?C/+</?35/IBJ68/<=812-zBACB0/23M78;¦68;A*D.;>-J7¦eo;<?

0 < a < 1_

"2

0A¡Bz368;A b -J/1?yMD./YAED./2? AED./MBJ68/<=812-zBACB0/23:D.;>-J7IBz6IAED./YBJ?;<?B0Bz6.1-èo;<?|HO3eo;<?0 < a < 1

[email protected]?AC3;>e8AED./*F.?C;.;>e032;MAED<?C;>=l2D=86 b D<1682/47I/EË b /2F<A E 1YA[yMD>B b DFl;>Bz6>A[yM/:yM;>=.-J7O68//47(A|;35D.;4yAED<1YAf(t) + f(u) ≥ 2f

(

t + u

2

) yMD./?C/

f(x) = e−ex 12At = loge

(

λkxk(− loge a)) u = loge

(

xk

λk(− loge a)

) _0<9 b D./ b BJ68AED./*3+/ b ;<687O7./?B>1YACB</;>e

f yM/*35/4/AED<12A[AED>BJ3¡78;./23RBz6878//47OD.;>-0713-0;<68:143t14687

u14?C/*12-y19.3RF8;<3BACB</ AED<12ALBJ3 Bze`[email protected]./>12-z=l/23 λkxk

14687 xk

λk

14?C/12A-J/13EA13K-J1?C2/I13 1− loge a

_ AED./?y*BJ35/ AED>Bz3K-zBz68/I;>e14?C=8HI/6>Ae01B@-J3 G>=<AAED./BJ68/<=812-zBA9OHO1Y93EACB@-z-lG8/:A|?C=l/._ 5KP B#Rd@B0=G:I> 9OPO %6;19 8A8< 6 :;K$@ IL#JGE*8=

c c>p "!#Z$&cBT2_9!MA1(|,>8\,pBdAeA:!;£/1!p*>; ,*|2J!' QpH

f1 = f2 = 1K]-n

fn = fn−1 + fn−2NOE>P[d]H+Q.bQ.PU

n > 2 DFQ.]

n-Q ]-Qgn = fn+6 + 3fn+2 + 3fn−2 + fn−6NOE>P[d]H+Q.bQ.PU

n > 6 ± [?]-n

gcdgf6666

Lgf666

6>.AmF8?BT'*<$& @ >B` @ .M8?B? ; B? p.B96>>8O9!"+*l6F8?,F8T! @ .AdAT!"=!";>pAeB\*>;51;f)6",*|#Z B\>!6"B:.!"&;)6.,3:'# y*>BT.!pm B:8O&;£(-B?+3cB\*>=,.3;F;f)6"° Q.g-Q.KpH+Q.nK.g-g>^d[?Y+KpH_[\E>]EN<HJDFQIP_Q.Y+W-P+P_Q]-YJQ P_Qp^?KpH_[\E>] NOE>PHJDFQ ± [?j-E>]FKYJY+[

UMQ.u>WQ]-YJQ fib"[ R>Q.U

fn−2 = 3fn−5 + 2fn−6L

fn = 8fn−5 + 5fn−6L

fn+2 = 21fn−5 + 13fn−6L

fn+6 = 144fn−5 + 89fn−6L

UMEkHJD>KH

gn = 216fn−5 + 135fn−6 = 27(8fn−5 + 5fn−6) = 27fn

DFQNOEê^\EG[d]-bhgFP_E>gQPHTSlENHJDFQ ± [?j-E>]FKYJY+[UMQ.u>WQ]-YJQ[dUGkQê^ ']-EG]

gcdfmLfn = fgcdm

Ln

% CQ.Q>LNOE>P¥Q#".KqgF^?Q>LyQ.]F]-Q"HJD c ° E>UUMQ]9L®Qn[eH_E>PML h,*|#$FF' vBd p2_J!'8T!,*|# @ '3<$.B\*F,F8T'B\,pA®,F8O|!p3l,>8?B:2: L ° P_QUiU"L .' Ltg tpF & tgFgF^ S>[d]-bHJD[?Ug-P_E>g-Q.PmH:SK.]-ncW-U[d]-b

gcd6666L666 = 6

Lf6 = 8

L>K]-nf8 = 21

LGkQEFjHK.[d]

gcdgf6666

Lgf666

= gcd27ff6666

L27ff666

= 27 gcdff6666

Lff666

= 27fgcdf6666

Lf666 = 27ffgcd6666

L666

= 27ff6= 27f8 = 27(21) = 567

7È

M 9@ "R*0>H@ !I0< 25@=? :.6 :'+D39@$"#(* RKSR)2 25R (A8S*(; (;) G:6.+6;?1* (A8 GH*I3B2J $#R.0 @1 $G E* &2) R G:6.+69 C9 !I% 6 +P)A; 6 &

(VK%; 6K

)3 <# (@ F+J2/0 GH*MG:,?% 6 AOP+6K%O1

R& 8%(' (%; .6 3 I R #&0 CLL&# :;,%(&:; .6 (6K; 59 K8A(8H9O .6KA(8G"NC3#(R#@B0=G:> 9OPO %(6K; 9 8A8< 6 :;K@ "L#GH*I3: ; -: E ?-: = P;)=56 :',,';)%(; 6&J?% O16;.;K

h c>7. "!#Z$&l¢ B?*y8d BdO, i; |!" M,pA:'*B\Bd;£y!.!2+!± E>P)]-E>]J]-QbpKpH_[`R>QP_Q.K^]>W-qZjFQ.PU

xK]-n

yL]-E'H1jFE'HJDkQ.u>W-K^'H_E

0Lg-P_ER>QfHJD>KpH

x4 + y4

(x + y)4+

√xy

x + y≥ 5

8

@ '3w JBm8T!z !" p!"*y8?B?,AeAO&BT#-!p*98?B:2m,A% p.AmF8?BT'*F $& 6!"\F!80v JA\,.*-,=BM42;)*B:.!" JBm8\&F6,._,!.;6.,,J!.;y() i*>B?,,.*9#k!"m!"=>B\*F,yFBT!p+J!()'+*> J"8T!B?*F;,pB? "'*F pi.,88:!; ,*|2J!M @ >B` @ .M8?B? ; Bd i "B6F8O|!p+*6F8?,F8T! @ .AdAT!"=!";pAdB?*F;51;V)6" @ F,AT!" tB\3cB?*-*B:!; x Ad BT! @ ,3w1$F!MAdAd;w,.*9#5vBT'*-*|!(|,BeA:!"&;v*F=!AT 6F8?,F8T!©)*B:.!" JBm8\&.;h6.,*v*F=!AT; ;c6h)!!p ,pBh.,-;kk'*>=®)'*F=;@ >B?*-,p9,*|#ZpB¯9;1.A\ @ '3l3.*>B:8O& @ .AdAT!"=!";v hB?*y8T!ph,>.!"*F; ;f6

( Ny = 0

L-HJDFQ]GkQwD>KpR>Q1 ≥ 5

8

CW-g-g-E>UMQHJD>KHy 6= 0

K]-nZ^?QpHu =

√

x

y

DFQ]cHJDFQkb"[ R>Q.]c[d]-QuW-K.^e[eHTSljFQY+E>qZQU

u8 + 1

(u2 + 1)4+

u

u2 + 1≥ 5

8

D[dUY_K]lj-QP_Q"GP[eHH_Q.]K.U

8(u8 + 1) + 8u(u2 + 1)3 − 5(u2 + 1)4 ≥ 0L

E>P(u − 1)2(3u6 + 14u5 + 5u4 + 20u3 + 5u2 + 14u + 3) ≥ 0

LGkD[\YDc[dUY+^?Q.K.P_^ SwH+P_WQ uW-K.^e[eHTSlDFE^\n>Us[dN9K]-nlE>]>^`Sw[dN

u = 1 HJD>KH[?U"L[dN9K]-nlE>]>^`S[dNx = y

@ '3w JBm8T! ".A>8?B:'*> $&a B:2_|!At(|,F8?,BeAdAT!";s/>!"*F; _,.*92+!,.*9#1!9!"&¢y9,.*9#,pAdAd;£h,.3V#-!p*>; @ ;f6

(3VF#Bdy!.#w JAdBd=98?AO&c$&8O9!!#B:8:'

)

SlDFE>qlEFbQ.]-Qp[eHTS9LGkQqKpScUW-gFgE>UiQx + y = 1

DFQ]

x2 + y2 = (x + y)2 − 2xy = 1 − 2xyL

K]-nx4 + y4 = (x2 + y2)2 − 2x2y2 = 1 − 4xy + 2x2y2

Q]-YJQL>[`HtUJWFXaY+Q.UtH+EcUiDFEG0HJD>KpH

1 − 4xy + 2x2y2 +√

xy ≥ 58

Ê/YAt =

√xy_`D./6

0 ≤ t ≤ 12 1687AED./KBJ68/<=812-zBA9O14G8;2</MG./ b ;<H/23

2t4 − 4t2 + t + 38

≥ 0 16t4 − 32t2 + 8t + 3 ≥ 0 ;<?

(4t2 − 3)(4t2 − 1) + 8t(1 − 2t) ≥ 0 yMD>B b D*BJ3A|?=l/8_ U<<=812-zBA9MD.;>-07<3[Bze.1687(;<6<-9*B@et = 1

2

YAED<1YABJ3 Bze.1687(;<6<-9*B@e x = y_

mmmE_[WWX .S UT(V" S`WYX T6 ÂX `W %Å Å C 2VX >Q SoX ._2BJ6 b / √

xy ≥ 2xy

x + y AED./¦B </6&Bz68/4>=81-@B@AU9&eo;>-@-0;4y3(e0?C;<H$AED./3EA|?C;<682/2?

BJ68/<=812-zBA9x4 + y4

(x + y)4+

2xy

(x + y)2≥ 5

8

_ B@AED.;>=<A-J;<353¡;>er2/68/2?1-@B@AU9 1353E=8H/ x 6= 0

_Ê/Ay = tx

_[`D./6I G8/ b ;<HI/3t4 + 1

(t + 1)4+

2t

(t + 1)2≥ 5

8

_ YA|?1BJ2DACeo;<?y1?C7 b ;<HIF<=<AC1YACB0;<6.3¡3+D.;4y AED<12AR Bz3/4>=.B>12-0/26>ALA|;

3t4 − 4t3 + 2t2 − 4t + 3 ≥ 0 ;<?(t − 1)2(3t2 + 2t + 3) ≥ 0 yMD>B b DIBz3 b -J/1?C- 9OA|?=l/ y*B@AED/<=812-zBA9IBze1687d;<6<- 9IBze t = 1

_`D./?C/Yeo;<?C/ /<=812-zBA9D.;>-07<3:BJ6 B@e[14687;<6<-9Bze x = y_ 0AAED./26deo;>-z-J;4y3AED<1YA:/<=812-zBA9D.;>-J7.3:Bz6AED./;<?CBJBJ6.12-lBz68/4>=81-@B@AU9B@e1687I;<6<-9Bzex = y

_ 1K "N 4$#"*BF8J9"=GE*8"3" $2G C#! &25R 0I 0"*&9S9R G "R

SR.#(R,0 L I2 $2 #A:=*?6 3B> G"#! C 9N &21G:6.+6; 7+ C6 : C%)A GH*I3 9@ "R*0>H@ FP! C #C4& !.6 ;K SG"NC3 > G"# &R I $2 2*&%(;-; M6 &%.6E*(; (;) G:6.+6;?9E 6 .&&%?V0E=GH*I3"2J $#R.0V@1 $G E* &2) R =G:6.+69 C9J!I%"6 +H)A(:; 6

(PK%; 6K

)3 "R,9@J R+I@ (**? :'.- >?K &+$9"

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

; '++6K& K .?"8 (;) KOPMR D; 6K'% .J-: (; -&(;7)n ∈

n ≥ 2

I;-6 :#C% 6K;

xn + yn

(x + y)n+

√xy

x + y≥ 1

2+

1

2n−1

-:/6 7J:M 6 72 ≤ n ≤ 7

(25;K ; -&(;=;-PA?6K6 #C%? 6;6 P;-P E')6 'K $-:

n = 2)F8:&%$? 1&;K:&1; -: ?%?K;E;)5O 6 :#C% 6K; 6K6 :K86 (A1;-5>(=SH

4Æ¾*È _ 44> <7È* $lXT*¿TW"V"3 ) rS >7 *)`W T > rXWW4Q|W>_

2=8F8F8;<3+/¡AED<12A 4ABCBJ3L/4>=.Bz-z12AC/?12-<1687*AED<1YA

PBz3L146Bz6>A|/2?CBJ;<?Fl;>Bz6>A_`D./-zBz68/3

AP BP CPBz6>A|/2?3+/ b A:AED./d;<F8F8;<3BA|/3BJ78/23M12A

D E F ?C/3Fl/ b ACB</Y-9r_2=8F8F8;<3+/KAED<1YAPD = PE = PF

_¡Ì¡/A|/2?|HBJ68/:AED./*-J; b =83;>eP_

T.S UTV" : Ã U>T.WWYX 2S8WS )YX S @`T?W <W @ÅVX <W YÂ Å PÅ3 >¡ÅdTrWV"OS`WMWSUTX^_

/3+D<12-z-35;>-</MAED./(eo;>-z-J;4y*BJ68MH;<?C/2/268/?12-F8?C;.G>-0/2H !#.S S`W UT.Q T"7P

<QdS<SPD = PE

É2=8F8F8;<3+/ eo;<? b ;<6</26<B0/26 b / AED<1YA AB = BC = CA = 1_&Ê/A

x y zG8/*AED./-0/268^AED<3;>eAED./MF8/?CFl/2687Bz6>AR;>eBC B@ALeo;>-z-J;4y3 AED<1YA

MD =∣

∣

∣

1

2− BD

∣

∣

∣ =

∣

∣

∣

∣

1

2− y

y + z

∣

∣

∣

∣

=1

2

∣

∣

∣

∣

y − z

y + z

∣

∣

∣

∣

_2BJ6 b /KAED./12-@ACBAC=l78/KBJ6 4ABC

BJ3 √3

2 yM/D<1</

AD2 =

(√3

2

)2

+ MD2 =3

4+

1

4

(

y − z

y + z

)2

=y2 + yz + z2

(y + z)2_

D./26 3EBJ6 b / P D

AD=

x√3/2

yM/D<1Y</

PD2 =4x2

3AD2 =

4x2(y2 + yz + z2)

3(y + z)2_

a F8F<-9>BJ68IAED./d314HI/d14?C=8HI/6>A*AC;PE yM/d35/4/IAED<12A PD = PE

B@e1687;<6<-9Bzex2(

y2 + yz + z2)

(y + z)2=

y2(

x2 + xz + z2)

(z + x)2_

D>Bz3¡Bz3/4>=.B>12-0/26>AAC;x2(x + z)2(y2 + yz + z2) − y2(y + z)2(x2 + xz + z2) = 0

_ 1 b A|;<?BJ68*B </3(x−y)(x+y+z)(x2y2+zx2y+z2x2+zxy2+z2xy+z3x+z2y2+z3y) = 0

_2BJ6 b /MAED./(3+/ b ;<68714687OAED>BJ?C7IAC/?CHI3:;>eAED./(e01 b A|;<?C/7F8;>-9<68;<HOBz1-r1?C/F8;<3BACB</ BAeo;>-z-J;4y3¡AED<1YA

PD = PEB@e14687;<6<-9OB@e

x = y_¡`D>=83 AED./M-J; b =83Bz3¡AED./(12-@ACBAC=l78/AED<?C;>=l2D

C_

0Areo;>-z-J;4y3rAED<12AAED./35;>-z=<ACBJ;<6:A|;AED./;<?B0Bz6.1-F.?C;.G<-J/H BJ3rAED./Bz6>A|/2?3+/ b ACB0;<6;>eAED./*AED<?C//M12-@ACBAC=l78/234AED<12A Bz3 AED./M;<?AED.; b /6>A|?C/M;>e`AED./M/4>=.Bz-z12AC/?12-8A|?CBz1468-J/ ABC LBJ6 AED>BJ3 b 13+/ 1-z3+;¦AED./ b /26>A|?C;>B07 AED./ b Bz? b =8H b /6>A|?C/ 1687 AED./dBz6 b /26>A|?C/¦133EA12AC/47OG9(>14?B0;>=83¡35;>-</2?3 _

44

- .6 A6 &? KO ? K SR,9@ C# !R)## I&%(L.'+D3 0J! !ER.#(R)2=*(N"R 0$%(;O ; J939@ "R*0>H@ F:! C #C4&"!.6 ;) GIN3"9@/R)>M9"G (R*IM6 &%.6*&%(;-,*; ;K 9 8A8EFC 6 ($V0I5GE*8"39@J "# * R)SRK2 25R I(A8 *; ;KG:6.+6;?*BA(8K GH*I3 "R.9@$ R,H@ ** ::')-:>?K &+"9"GE*8"3<# (@ F+J2/0 GH*GH,?%? 6 :(AOP&+6)%O1R & 8%' C %;.6 ?3 0*(@/R.0* CRKSR,4?""N:$ 6 CFD(3"#R`@/0 G"> 9OPO %(6K; 19 K8A(8<=6 ;)@=((EL&#?GE*8"3H> M4I:<P0J0IH!C6K GH(6K.+6K; H#/M6 .?I9"GE*8"3; -:E

# $ 17)&% ;-5K'%/7" E6 ;P?%')-M; -&(;

AD = BE <B5P; -6 /;) ;-

&.$;)B6 '+(@/? B =6 :;K;C6 ; -:=')57=')?K ;.6 (A8E C%(;:; ;K1; -&(;-: -/:;I &1 5;)5OP &&A ; -&(;"')(

F8:&%$?%?AA(; ; -:$7) (J6 (A1&;K&+6(7)J$-(6K')-P- -&J:5)%(; 6P(;"; -: O O ;

) ABC

! "#

P $&%' ( )

AP BP CP !)*+,&%-""#)! .)! /)0D E F )&"#+ ( 1 4DEF /23 #4&%4&%, +5 6

P)!5+!%7&%

8 P

)9&%7+ ! /5:; P

)9&%7<&%+! := P

)9&%, #+ !

h c>7. "!#Z$& @ -Bd 8T99!").¢y(|,>#AT!"&.;£(|Bd 8T.A?;f£CW-g-g-E>UMQHJD>KpH

Γ[dUKY+[dP_Y+^\QK]-nHJD>KpH

ILJLK]-n

KK.P_QHJD>P_QQn>[dUJH_[d]-Y_H1gE[d]H_U

[?] HJDFQag>^?K.]-Q<ENΓLj>W>H]-E'HE>]

Γ QpH

AjFQVK.]'S gE[d]HkE>]

Γ E[?]HU

BL

CL

DL

EL

FLvK]-n

GE>]

ΓK.P_Qn-Q ]-Q.n¦j.S HJDFQYJE>]-n[eH_[?E>]FUHJD>KpHwYDFE>P_n>U

ABK.]-n

DE[d]H+QPUMQ.Y_HKpH

IL)YDFE>P_nFU

BCK]-n

EF[?]H_Q.PUiQY_HKpH

JL)K.]-nYDFE>P_nFU

CDK.]-n

FG[?]H_Q.PUiQY_HvKpH

K% H_K.]-bQ.]Hv[dUvH+Ej-QkP_Q.bpKP_nFQnlK.UKYDFE>P_nwG[`HJDc[eHUfgE[d]HsEN

YJE>]HK'Y_HvnFQ -]-QnwH_Ej-QKkgFK.[dPvEN9YJE[d]-Y+[\nFQ.]Htg-E[?]HU &( Uv[eHg-E>UiUJ[\j^\QH+EwUiQ^?QY_H1HJDFQg-E>U[`H_[\E>]FUsEN

ILJL>K.]-n

KUMEkHJD>KpH

GYJE[d]-Y+[\nFQ.U

G[eHJDANOE>PK.ê^gE[d]H_U

A^`S[?]-bE>]

Γ >%<? W-U+H_[ Y+KpH_[\E>]cP_QuWF[?P_Q.n0@ &

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% E>PiL)qlE>P_QlbQ]-Q.PK^d^ S9L)EN£KZb"[ R>Q.]Y+E>]>[\YΓ& L)j>W>H

]-E'H£E>]Γ ± E>PK.]'Sg-E[?]H

AE>]

ΓLHJDFQ |b"W-P_Q

ABCDEF[dUKhDFQ ".KbE>][d]FUMYJP[\jFQn

[?]VKY+E>]>[\Y Q.]-Y+Q>L-GkQwqKpSaKgFgF^ S K.UMY_K.^&C U DFQ.E>P_Q.q~H_EZ[`H SVnFQ -]>[eH_[?E>]ABqZQ.QpHU

DEKH

IGkD[d^?Q

BCqZQ.QpHU

EFKH

JD E>P_QER>Q.PiL

KqcW-UJHfjFQHJDFQgE[d]H

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[?]H_Q.PUiQY_HUIJL)U[d]-YJQwGkQlD>KpR>QcHK1Q]

KH+EVê[\QlE>]VHJDFQ^d[d]-Q

IJK.]-n

n-Q ]-Q.n[`H-H+Eê[\QvE>]CD

DFQ.P_QpNOE>P_Q>Lj.S K.UMY_K.^&C U DFQ.E>P_Q.q % GkD[\YDUJHKH_Q.U|HJD>KpHHJDFQ[?]H_Q.PUiQY_H_[?E>]FUfEN-HJDFQE>g-g-E>U[`H+QhU[?n-QUsEN

ABCDEFKP_QY+E^dê[?]-QKP & L.GkQY+E>]-Y+^dWnFQ

HJD>KHAF

qcW-UJHK.^dUME gFKUUHJD>P_EWbDK C[?]-Y+Q

GGK.UwnFQ -]-QnH+E j-QlHJDFQagE[d]H

GkDFQ.P_QFK

qZQ.QpHUΓK'bpK[?]9LGkQfYJE>]-Y+êWn-QtHJD>KH

GY+E[?]-Y+[?n-QU9G[eHJD

ANOE>P9K.]'SYDFE[?YJQ

ENA

6>.AmF8?BT'*<$&c B:2_|!A1(|,>8\,pBdAeA:!;£/>!"*F; ,*|2J!'x #B:8:'5E v2J'3l3V!"*y8_ K.UMY_K.^&C U DFQ.E>P_Q.q7NOE>P_Y+Q.U

ILJL.K]-n

KH+EjFQYJEê^d[d]-Q.K.P

DFQHJDFQ.E>P_Q.qrY+E>]H_[?]>WQUH+EaDFE^\naNOE>PfgE>UJ[eH_[?E>]FUhENAHJD>KHY+E[?]-Y+[?n-QkG[`HJDaK]'SVEN

HJDFQE'HJDFQ.PR>QsR>Q.PmH_[\Y+Q.U NOE>P1Q ".K.qlg>^\QLUMDFEWF^?n AIjFQfH_K.]-bQ.]H)H_E

ΓLHJDFQ]

B = A

4

U6&e01 b A AED./AED./4;<?C/2H?C/HO1Bz6.3M>12-zBJ7eo;<?143HI169&143MAED<?C/4/F.1Bz?3O;>e¡BJ78/26>ACB b 1-</?ACB b /3O1H;<68AED./¦3EB Ë 35;-0;<68d13AED./dAED<?C//F8;>BJ6>A3 I J 1687 K1?C/yM/2-@-78/Yl68/7_ 0l12A1B@-z-J/ ;<6 AED./d;AED./?MD<1687 Bz6>A|/2?|F.?C/YAC/47AED./dF.?C;.G<-J/H7>B!\r/?C/26>AC-9 y*B@AED

Γ I J 14687 KË4/47 AED./26Meo;<?169MF8;>BJ6>A A

;<6Γ Fl;>Bz6>A B

Bz378/Yl68/7*AC;MG8/AED./*F8;>BJ6>AyMD./?C/AI

H//YA3Γ1Y12BJ6 F8;>BJ6>A C

Bz3LyMD./2?C/BJ

H//YA3Γ1Y12BJ6 16873+;eo;<?AEDr_U6.3|A|/217;>e

AB1687

DEBJ6>AC/?35/ b A12A

I ?C/147 AB

14687DE

F.1433AED<?C;>=l2DI_ `D./35/KyM/?C/Bz6878//470l?17>-0/+9` 3RyM;<?C7<3BJ6D>BJ3;<?B0Bz6.1-`3EA12AC/HI/6>A¡;>eAED./:F.?C;.G<-J/H yMD>B b DKAED./K/7<BA|;<? b D<14682/7 | 3+; 0l1YAC12Bz-@-0/> 3[Bz6>A|/2?|F.?C/YA12ACBJ;<6 b 1F<AC=8?C/23AED./A|?=l/O3F.Bz?CBA:;>eAED./OF.?C;<Fl;<31- /+</26AED.;>=l2DAED./F.?C;.G<-J/Hd 3 b =8?|?C/26>AyM;<?C7>BJ683=l22/23EA3RAED<1YAR;<F.Fl;<3EB@AC/*3BJ78/23;>elAED./*?C/3E=.-@ACBz68*D./Ë142;<6OHO=83|ABJ6>AC/?35/ b ABz6AED<?C//yM/2-@-0~78/Yl68/7OF8;>BJ6>A3_ ¥*/?C/ AED./6 BJ3 0l1YAC12Bz-@-0/> 3R3+;>-@=<ACB0;<6_

G! ?>8QET Q 8W ! S

A7TX+ ?

AT

Γ7 MT 7lW S`WYX

IJKXWQET? WB4XTX 4IJK

* W >7 K¿TXRSoX UW ! SXW (¿WQ5SSUTΓ@AED<12ABz3 /1 b DK</?A|/EËKBz3LAED./Fl;>-J/;>eAED./;<F.Fl;<3EB@AC/(3BJ78/ _U6AED./M-J1YAA|/2? b 13+/*yM/D<1Y</ A = D B = E 1687

C = F_8;:F.?C;</AED>BJ3 yM/ BJ78/26>ACBze@9:Fl;>Bz6>AC3`y*BAED b ;<HIF<-0/EË6<=8HG./?314687 y*BAED.;>=<A-0;<33;>e82/268/?12-zBA9 3E=8F8F8;<3+/¡AED<12A Γ Bz3AED./=86<BA b BJ? b -J/8_ ;<? A

;<6Γ14687

M68;A[;<6

Γ yM/7./68;AC/MG9 M(A)AED./*Fl;>Bz6>AR;>e

ΓyMD./2?C/KAED./*-@BJ68/

MABJ6>AC/?35/ b A3

Γ14Y1Bz601687

M(A) = AB@e`B@ALBJ3 A14682/26>AAC;

Γ1YA

A _ a 6I/21439 b 12- b =.-z12ACBJ;<6(9>B0/Y-07<3M(A) =

A − M

AM − 1

yMD./?C/M

Bz3 AED./ b ;<6¤0=lY12AC/(;>e8AED./ b ;<HIF<-0/EË(6<=8HIG8/2?M_ /?C/217>Bz- 9O;.G>A1Bz6

D = K J I(A) =AU − V

AV − U

yMD./?C/U = 1 − IJ − JK + KI

14687V = I − J + K − IJK 1687

G = (K J I)2(A) =A(U2 − V V ) + V U − UV

A(UV − UV ) + U2 − V V

_0A eo;>-@-0;4y3AED<1YA

G = Aeo;<? 12-z-

A;<6

ΓBze`14687I;<6<- 9B@e

U2 − V V = U2 − V V

14687V U − UV = UV − UV = 0

_D>Bz3 ?C/47>= b /23LA|;

(U = U);<?

(U 6= U14687

V = 01687

U +U = 0)_L;<687<BACB0;<6

U = UBzHIF<-zBJ/3

I(J − K) − I(J − K) + JK − JK = 0 yMD>B b DH/2146.3RAED<1YAI-zBJ/3;<6AED./K-zBz68/:AED<?C;>=l2D

J1687

K_

¡6AED./M;AED./2?RD<1687 b ;<687>B@ACBJ;<6.3 V = 0 = U + U9>B0/Y-07

I − J + K = IJK 14687

a − b + c = 0 yMD./?C/yM/7./68;AC/MG9a b c AED./?C/21-86<=8HG./?3 JK + KJ − 2 KI + IK − 2 14687

IJ + JI − 2 ?C/3Fl/ b ACB</Y-9r_

4

2BJ6 b /K HO1Y9(12-J35;(G./y?CBAA|/2613 aI − bJ + cK = 0 yM/:3+//AED<1YAL 1687 1?C//4>=.B>12-0/26>ArAC; a+c = b1687

a(I −J)+c(K −J) = 0_rD./-z12AAC/?-0/217<3A|;

a = c = 0 G./ b 1=835/ I J K1?C/O68;A b ;>-@-zBz68/1?_ Bz6.1-@-9 AED./O3+/ b ;<687 b 13+/; bEb =8?3:Bze[14687¦;<6<- 9Bze

a = b = c = 0_(`D>BJ3KHI/16.3AED<12A 4IJK

Bz3K3+/Y-zeo~EFl;>-z14?3Bz6 b /ZBJ3 ;<6MAED./:F8;>-J1?L;>e

My*B@AEDM?C/235F8/ b AA|;

ΓB@el1687;<6<- 9MB@e

MZ +MZ = 2 143RB@ALBJ3¡?C/217>Bz- 9 b D./ b 4/47r_T8SW _ ;<?¡13+/Y-zeo~EFl;>-z14? A|?BJ168-0/

IJK yM/1 b AC=81-@-9D<1</ K J I(A) = Aeo;<?1-@-

A;<6

ΓAED>=83 D = A = G E = B F = C

_ _D./*?C/3E=.-@A b 146IG./M/EËAC/687./47A|;1 b ;<6=l2DO1MF.?C;4¤U/ b ACB <BA9 _

K I R #&0 CLL&# : ;.%&; ?.6 6;9 K8A(819O .6A8 GIN3 ;-E

!.5:P# $ &;-1A/ C 6K'K6K; E+6K; 6KJ7)IJH:

K; -&(;H5+6 5 ;K

K=;- O 6 '; ?:J6;-&%;&6 6 (A/>')? ?-:O J< 6;-P;-E6 ; " :;K/ '+OCPH%O &.6 K%; 6K R.R !.(K')-:1; -: E6 ;

0rI

1/r 6 ;-1&

IJ

KEJ6;-

r?C

0 6= r 6= 1 E$-(6 5#( $')-: r0:: −r

r'+OC/:

0 6= r 6= 1

? f ' "!#$&¬'F,3l3V!.#tv,. BeA\,;>6>8O_, $F.=; _,.*92+! QpH

PjFQwKcP_Q.K^)gE^ SF]-E>qc[?K^|G[`HJDl[?]H_QbQPY+EFQXZY+[\Q]H_UUJWYDlHJD>KpHsHJDFQP_Qw[dU

K]<[d]]>[`H+QUJWjFUiQuWQ.]-Y+QZENHJDFQUMQ.u>WQ]-YJQ P (k)∞k=1

G[`HJDZHJDFQlgFP_E>gQPHTSVHJD>KpHHJDFQUWj>UMQ.u>WQ]-YJQkD>K.UE>]>^ S ]>[`H+Qp^`ScqlK.]'SgFP_[dqZQkn[`R[?UiE>PU

P_E.R>QHJD>KHP[dUEN-HJDFQNOE>P_q

P (x) = (ax + b)n

x #B:8:'M DFQ.P_QZGkQP_QV]-EUME^eW>H_[\E>]FUUWj>q[`HH+Q.nNOE>PhHJD[?UwgFP_EFj>^?Q.q tUKP_QUWF^`HMLg-P_EFj^\Qq P_QqlK[?]FUlK.]¬E>g-Q.]¦g-P_EFj^\QqGkD[\YDHJDFQ P_Q.Kn-QPUaEN KP_QQ.]-Y+EW-PK'bQ.nwH+EcP_QMR[?UJ[eH

! #"%$ &' ("*),+-&# .0/#&12$ &3"*),+46587 9:8;<46=?>!;@7 9BADC<EGF<H>!56IKJ@9L>MAD;NO>!=?>!;@7 9BADCQPSRT;BADJL>VUDW F<W8XZY6IM[]\8>!;

^`_?ab $ &' ("*),+-&# .0/'c _?ade6:KATfD587 fDgh4T587 9L:8;@C<EGF<H>!56IKJ@9L>MAD;@CiNjk:8fT56IQ9>QAT;BCQPlUDH>!:8mD:Kn 5GX!IQA6opH>rqse8;B>Z5T>Q;B7 JOtvuwW RlWDx(I!CitK>Mn n46587 9:8;BCy>Q=z>Q;B7 97DEGF<H>!56IKJ@9L>MAD;NO>!=?>!;@7 9B7@P0uwW 'W8X!IZfT56CQ|F<W 4~WK:6:656;:Z[V|RT;BADJL>GUTW F<W6XZY8IQ[w\6>Q;

$ &' ("*) d &# .0/#&1$ &3"*) de6:KATfD587 fDgh4T587 9L:8;@C<EGFyH>Z56IZJB9>QAT;BCiNj :8fD56IM9L>MAD;@CQPl8IQ9;B7 JOtrXQAD;B;\?qFIMo876pIKt!7 n4T587 9L:8;@Cy>!=?>!;@7 9B7EVFyH>Z56IZJB9>QAT;BCiNO>Q=z>Q;B7 97PSDYK7 n 7 mvO:8fTg8|!O>Q.]7 gQY6I!=|

!W 0W6ul;B:8COCO=?I!f~|!SfD56;B>r0Y8IZfTg8|TwIZ:6tZ7X!IM9L:|86\6;@ADCCL7 I

Canadian Mathematical Society | CMS-SMC · a _ oe`AED./*?C/12-l6

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Transcript of Canadian Mathematical Society | CMS-SMC · a _ oe`AED./*?C/12-l6