No title

Ae-E(I01M14575'3§?905CB…3TFQ5*1T:=KT<=L814L8K01TB…5>Ae E(I01Œ"@0/214B ’ > c FO1§E{< ` >?Bg41 3T36LJ0/O1?˜¹~I6=57>?L815*>A@;B c 1T5 c BG1T:?F„ER¨436LMK;14B…E ~F„E(I E(I01N‹;B…5(EuJ;BG>0>Ae¹E(I?3§E e FQ5‡FQBHB…3§EGF2>?9034/0~IAFQ/O1>"E(I01TB…5 c BG1T:?F„EŒA@0/O1TBHÔ ‰ I0136905G~1TB~N3T5 9;>"E1T3T5'FD/„<eq>A@;9;:—gE(I014BG183TBG1 6™"™ FDEG1TL5FO9E(I01834@?E(I0>?BGF„EG3§EGF„ƒ?1ŒA9;1T5HEHB$>?# L /DFO5HEC>Ae ŒA@0/O1TB - 5J?@nK?/DF c 34EGFO>?905§˜uL369"<©>Aeu~IAF c IS36BG1B…34E(I014BN/O3TBG¢41PK0>0> “ 5X˜u369;:ªE(I014BG1=36BG1 3T/Q5*> "6 >Ae—IAFQ5‡/O1§E…EG1TB…541 3TE…369AE `}Ì EŠEG@;BH9;1T:8>A@?EŠE(I?34E ŒA@0/O1TBuBG143T/D/„<:?FO:J;BG>4ƒ?1E(I?3§E FO5‡FOBGB…34EGFO>?903T/{˜;3T/„E(I0>A@n¢4IE(I01R:;1XEG34FQ/Q536BG15*>?LM1X~I?3§E‡K?@;B…F21T:MFO9M3/2>?9;¢NJ036J;1TB‡>?9 e c >?9AEGFQ9?@n16:e2B…3 c EGF2>?905}e2BG>?L *‘ "‘0` 






















6£ a 9;>"E(I01TBJ;34FOBu>Aet1T5'573X<05uFO905'J0FQBG16:KT<E(I01RŒA@0/O1TB ’ > c F21XE†<FQ9"ƒ?>A/„ƒ?1gE(I01§ŒA@0/O1TB eq>?BHL@0/O3  V − E + F = 2 `’ >?LM18I?34/Qeq‚GEHBG@?E(I?5I?3Xƒ?183TBGFQ5*149 c >?9 c 1TBG9?FO9;¢R~I?3§E ŒA@0/O1TB :AF2:M369;:~I?3§E I01RJ0BG>Tƒ?1T: ` ‰ >@;9 c >Tƒ?14BuE(I01~I0>A/21EHB…@?E(I¹˜ ’ 369;:AFQeq14B I?3":EG> 5(EG@n:6<ˆEð~>J036J;1TB…5N>Ae}Œ"@0/214B EG>0¢41§E(I014B‡~FDE(I3J;3TJn14Bg>Ae ‡1T5 c 3TB…EG1T5ge2BG>?L ™  ˜-3 c 149AEG@;B{<=K014eq>?BG1ŒA@0/O1TB - 5‡~>?B “ ˜0~IAF c Iˆ1T5HEG36K?/DFO57I016:83Req>?BHL@0/O3E(I?3§E‡FO5g1TÏ?@0F ƒA3T/O1T9AE EH>8Œ"@0/214B - 5 ` ’ >?L81>Ae ’ 3T9;:?FDeq1TB - 5g5HEH>?B…F2145m3TBG1K?365716:ˆ>?98FDEG1TL5gI0 1 2@;5HE 5(EG@;LMK?/O16:83 c BG>?5'5 ~IAFQ/O1ˆ/2>0> “ FO9;¢Meq>?B5*>?LM1§E(IAFQ9;¢1§/O571 `ǚ 9;1=41 3T36LJ0/O1=>AeuE(IAFO5RFQ53 ‘6‘T J;3TJn14BNFQ9 ~IAF c IPŒ"@0/214B3T9AEGF c FQJ;3§EH1T:=>?BðE(I0>0¢4>?903T/-L34EHB…F c 1T5g57>?L81 6 ?BG1RE(I01 c >?9;‚ c 14J?E c 36LM1NFQ9AEH>LM3§E(I01TL34EGF c 5 ` Œ"@0/214Bue234FQ/O16:EG>5'3§?BHL@0/O3§EH1T:E(I01J;BG>0KA/214LE(I?34E/O16:IAFOL EH>E(I01T571NBG145'@0/„EG5 ` Ì /O1T3TBH9;1T:57>?L81XE(IAFO9;¢e2BG>?LÎ1*ƒ?1TB{<1T5'573X?5*1 >?9mEH>?J?F c 536K;>A@?En~IAF c I Ì 34/OBG143":6< “ 9;1X~·5*>?LM1§E(IAFQ9;¢ ` Ì /O1T3TBH9;1T:Ç3"K0>A@?EE(I01PFOLJn>?BðEG3T9 c 1[>Ae/21XE…EH14B…5eq>?B E(I0141 3 c I?369;¢41>Ae—F2:01T3T5FQ9ME(I01 c 149AEG@;B{< `Ì /O1T3TBH9;1T:ME(I?3§E 9?@;LMK;14B}E(I01T>?B{<834E(‚ EHB…3 c EG16:=/QF„E…EG/21FO9AEG1TBG145(Eg34EE(I01EGFOLM1?˜57>E(I?3§EmE(I01FQLMJ03 c EC>Ae Œ"@0/214B - 5N5'@nK?5(E…369AEGFQ3T/ ~>?B “ FO9NE(I01g36BG143~N3T5u:;1§/O3X?B3C¢41T9;14B…34EGFO>?9>?B—Eð~> ` G46.7(8)?$ #"$ BG1T36:;14B…5ªLFO¢4I"E=BG1 c 3T/D/  ` – `N’ I014JnI?3TBG2: - 5S3TB…EGF c /21i>?9œŒ"@0/214B - 5PEHB…FO3T9;¢X/21©E(I016>?BG14L *  
       §¤ *£   /«    ! V£  ! ! * g£  !  « *  !  O ABC


£ *ª

!      !

      !         " * *  ! D

CO

E

 ! 

F

g« «

3ª

 *  " *    !



AO BO

 ! 

˜ ~I01TBG1ˆE(I01PÏA@n>"EGF2149AEG53TBG1=Jn>?5(FDEGF ƒ?1M~I01T9 O FQ5K01§Eð~1T1T9ªE(I018ƒ?1TBðEH143[3T9;:€eq>0>"E*˜ 369;:Ç9;16¢§3§EGF„ƒ?1[>"E(I01TB†~FO571 ` ŒA@0/O1TBJ0BG>TƒAF2:016:dIAFQ5‹;B…5(EJ;BG>0>Aeg>Ae E(I01=E(I01T>?BG1TL FQ9 ‘3" ˜ 2@;5(E‡E(I?BG161?BG1I01:AF21T: `RÑ{Ì 9PIAFO5 ƒ?1TB…5(F2>?9=>Ae-E(I015HEH>?B{<¹˜ ’ 369;:AFQeq14B B…3TFQ5*145gE(I01MÏ?@n145(EGFO>?9[>AeI0>6~ù3MJ;1TB…57>?9ˆ~I0>=I?365NK016149[K?/DFO9;:eq>?Bm5*>?LM1 £ A@0/O:8J;>?575(F2KA/„<M:?FQ5 c >Tƒ?14B}5'@ c IM3NE(I01T>?BG1TL gÓN1 c 369ˆ143"J?/O34FO98>?9?/„<E(I?3§EmŒA@0/O1TB‡I?36: 365'5'FQ5(E…369AE…5¹~I0> ~NBG>"EG1 :0>6~N9IAFO5-J0BG>0>Ae25 ` Ò ¹>"E(I ’ I014JnI?3TBG:3T9;: ’ 369;:AFQeq14B c >?LJ0/Q3TFQ9 >Ae-E(I013X~ “ ~N3TBG:09;14575C>Ae—E(I?34E‡‹nB…5HEJ0BG>0>Ae `N’ 3T9;:?FDeq1TB7˜—I0>6~1*ƒ?1TB7˜tJ0BG>TƒAF2:01T5NŒA@0/O1TB - 5 K;143T@?EGFDeO@0/34/DEG1TBG9034EGF ƒ?1J0BG>0>Ae-E(I?3§ECŒA@0/O1TB 571614LM5EG>I?3Xƒ?13TJ;J;1T9;:016:8EG>IAFQ5gJ;3TJn14B 34EN3/Q34EG1TBN:03§EH1 ` Ó15'Jn1 c @0/O3§EH145E(I?3§EŒ"@0/214B/Q3 c'“ 1T:S5'@  c F2149AEEGFQL81K;1§eq>?BG1=IAFO5 :;1434E(IEG>BG1*ƒ?FQ5*1m3T9;:RJ;>A/QFQ5*ICE(IAFO5J036J;1TB " E(IA@;5§˜TFDE¢XF„ƒ?145Š@;53C¢X/QFQLMJ05*1g>AenI0>6~$ŒA@0/O1TB :?FQ5 c >Tƒ?14BG16:CE(IAFQ9;¢§5Š365ŠI01u~NBG>"EH1 3mJ036J;1TB-3T9;:I0>6~bI01 c 36LM1mK?3 c'“ /Q34EG1TB¹EH>gFOLJ;BG>4ƒ?1 IAFO5‡5*>A/D@?EGF2>?905 ` a /Q/Neq>?BðE†< c >A/D@;LM905[36BG1dJ0/O1T3T573T9AE=369;:ŽFO9?eq>?BGLM3§EGF„ƒ?1 ` ‰ I01”K0>0> “Žc 369ÎK;1 1T9 {>4A@MBG1T36:FDEe2BG>?L c >4ƒ?1TBŠEH> c >4ƒ?1TB}>?B‡K?BG>6~N5*1CE(I?BG>A@n¢4IB…369;:0>?L 1T5'573X<05 ` ;@?E 57I0>A@0/2:A@K?@A<E(I01K0>0> “ Ô a eQEH14B‡3T/D/†˜?E(I01~I0>A/21E(IAFO9;¢ Ñ 369;:8L8>?BG1 BG1 c 1T9AEL34EG1TB…FO34/„Ò FQ5BG143":AFQ/ <[3§ƒA34FQ/Q3"KA/218>?9;‚(/DFO9;1 `[š 9=E(I01ˆ>?9;1ˆI?369;:¹˜FDEgFO5N5'@;BG1§/„< 3€¢4>0>0:ÇF2:01T3PEH>Ç5'@;J0Jn>?BðEE(I01 •Paa ц• 34E(I014LM3§EGF c 3T/ a 5'5*> c FQ34EGFO>?9d>Ae a LM1TB…F c 3§ÒH˜ ~IAF c IRJ0@nKA/QFQ5*I01T:E(I01CK;>0> “n` ¹145'FO:;145§˜4E(I01K0>0> “ ~>A@0/O:K;1C3T9'1 3 c 14/D/2149AEŠBG145*>A@;B c 1 eq>?B‡EG1T3 c I01TB…5X˜—1§FDE(I014B‡eq>?B:?FQBG1 c EG16:BG143":AFO9;¢>?B‡eq>?B}EG1TBGL±J0BG> {1 c E…5mFO93T9"<ˆ9?@;L8K01TB >Ae c >A@;B…571T5 `$Ì E~>A@0/2:€L3 “ 1P3[¢§BG1434E¢XFDeQEeq>?BR3T9"<;K0>0:"<ª~I0>ª/2>4ƒ?1T5L34E(I `Žš 9 E(I018>"E(I01TBCI?3T9;:t˜—E(I01 •Paac >A@0/O:[I?3§ƒ?1M3T9;:P5*I0>A@0/O:[I?3§ƒ?18:0>?9;1M38L@ c IPK;1XE…EH14B {>0K>Ae¹16:AFDEGFQ9;¢E(I01CLM3T9?@;5 c B…FOJAE `—a /DE(I0>A@n¢4I ’ 3T9;:?FDeq1TBE(I?369 “ 5uE(I01CL814L8K01TB…5‡>Ae;E(I01 16:AFDEG>?BGFQ3T/"K;>?3TBG:Ceq>?B¹E(I01§FOB  c >?905 c F2149AEGF2>A@;53T9;:BGFO¢4>?BG>A@;5Š1T:?F„EGFO9;¢"˜ FDE¹/2>0> “ 1T:gEG>CL81 AO BO CO AO BO CO · · = + + +2 OD OE OF OD OE OF

6™ 365mFDe-E(I01*<:AF2:8/QF„E…EG/21RLM>?BG1RE(I?3T9=:0>6~N9?/2>?36:ME(I01 c >?LJ0@?EG1TB ‹0/2145g3T9;:ˆ3":0:3RE…3"KA/21 >Ae c >?9AEG1T9AE…5g3T9;:8FO9;:0'1 3 `g‰ I01TBG13TBG 1 2@;5HE}EH>0>MLM3T9"<ME{<0J;>?5 Ç9;>"E(IAFQ9;¢RJ036BðEGF c @0/Q36B…/„< 5*14BGFO>A@;5§˜KA@?E 2@;5(Em~I?34EN>?9;1~>A@0/O:S'1 36Jn1 c Ee2BG>?L 3T9€>?9;‚H/QFQ9;1:0> c @;L8149AE*˜9;>"EN3 J0@nKA/QFQ5*I01T:EH41 3TE `u• >?BG1N571TB…F2>A@;5Š~N365E(I01 c /Q@;L5…< c BG>?5'5*‚(BG14eq14BG1T9 c FQ9;¢ `Š’ 369;:AFQeq14B}>Aeq‚ EH149[BG1§eq1TB…5gEG>P>"E(I01TB c >A/Q@;L905KT?9;1MI?365 I?369;:6< c >?LMJ?@?EH14B/QFQ9 “ 5X˜0KA@?EuFQ9 c >?9"ƒ?149?F2149AEuFQ93 66 ‚(J;36¢41K;>0> “n` .A3"¢419?@;L8K01TB…5X˜ >?B¹1*ƒ?1T9 c I?36JAEH14B¹9?@;LMK;14B…5§˜§57I0>A@0/2:gI?3Xƒ?1‡K016149gJ;BG>4ƒ?FO:;1T:meq>?B¹1T3 c I c BG>?5'5*‚(BG14eq14BG1T9 c 1 ` • >?BG16>4ƒ?1TB7˜nE(I01RFO9AEHBG>0:A@ c EH>?B{<ˆBG1TL36B “ 5me2BG>?L L369"<>Ae—E(I01 c >A/Q@;L905g9;161T:ˆBG1*ƒ?FQ5*‚ FO9;¢ `Ì 9=14575'3§< I01BG14Jn1434E…5 ~I?34EI015734F2:MFO9ME(I01J0BG1*ƒ?FO>A@;5gL8>?9AE(I - 5 c >A/D@;LM9 " FQ9 E(I01K0>0> “ E(I?34EL81436905uE(I?3§E c >?905*1 c @?EGF„ƒ?1J03"¢4145 c >?9AEG34FO9F2:01T9AEGF c 34/nJ;3TB…3"¢§B…3TJnI?5 `}Ì E ~>A@0/2:8I?3§ƒ?1RK;1T1T9ˆ1T3T5…<eq>?B‡3T9ˆ16:AFDEG>?BuEG>8:01T34/n~F„E(IE(I01T571LFQ9;>?B‡J0BG>0K?/O1TL5 ` Õ=1 5*I0>A@0/O:M41 3"J;1 c EuLM>?BG1eq>?Bu>A@;BuLM>?9;1*< `

¬ ?< !v(@0£ /DFO 3T 9ˆ "Ó
3XI¤ƒ?FD/†˜AJ0*@n KA /Q& FQ5* I01T :MžKT<=¡  .;BGFQ9« c 1X«EH>?&?9 £ ;  9?F ƒ? 1T  B…5( FDE{<= .; BG 14 575X˜ T"A‘ ̆’  Ž ‚ ™ c ‚ TA£T™ ‚  ˜;I?3TBG: c >Tƒ?14B*˜  ™ 3TFQFDFnJ;36¢41T5X˜ ;’" `  £ © 3 ª#     !       g¬    © ! 1ª"   *g¬  * (   





 



a L8>?9;¢}L Š14BHLM>?9AEIAFO¢4I5 c I0>0>A/O5—EG> ¢XF„ƒ?1}EG34/ “ 53"K0>A@?E-J;3TB…3":0>1361T5FQ9NJ;BG>0K?3"KAFQ/DFDE{< `—‰ I?3T9 “ 5-EH>(@0/DFO3T9ÓC3§ƒAFQ//- 5—~>?9;:01TB…eO@0/ K;>0> “ ˜ Ì - ƒ?1 2@;5HEeq>A@;9;:5*1*ƒ?14B…3T/n9;1X~Ž~N3X<05uEG>41 3"J0369;:LR</O1 c EG@;BG145 ` ‰ I01 c I?3TJ?EG1TB…5u>Ae ¬ !v£    "
:;143T/?J0BGFQLM3TBGFD/„ 361T5ŠFQ9J;BG>0K0‚ 3"KAFQ/DFDE{<ª369;: c >?L8KAFO903§EH>?B…F c 5§˜uKA@?E>"E(I01TBN'1 3T3TLMJ?/2145R36BG1ˆJ0@0/D/21T:ªFQ9ªe2BG>?L c 3T/ c @0/Q@;5X˜ L81 c I?369?F c 5X˜9?@;LMK;14BME(I016>?B{<¹˜N369;:Ž¢§36LM1€E(I01T>?B{< ` • 369" '1 36Jn14BGFO1T9 c 16:bLM3§E(I01TL34EGF c 5=BG143":01TB…5 `·‰ I01 ;FQB…E(I0:?3§<›.;BG>0KA/214L=˜N:;14B…369;¢414L8149AEG5 Ñ J;1TBGL@?E…34EGFO>?905bFO9 ~IAF c IùE(I014BG1œFQ5b9;>ù‹ 361T:žJ;>AFO9AEðÒH˜ ;@ ,—>?29 - 5b9;161T:?/O1?˜S3T9;: ‰ >?BGBGF c 1§/Q/D.F - 5 ‰ BG@;LJn1XE Ñ 34/O57> “ 9;>6~N9R3T5  36K0B…F21§./ - 5}I0>?BG9 €E(I01g5(@;BGe23 c 1g>Ae;BG1*ƒ?>A/Q@?EGFO>?9 eq>?B y = 1/x 3"K0>A@?EŠE(I01 x‚'3 36FQ5}eq>?B x ≥ 1Ò3T/D/nLM3 “ 1N3TJ;J;1T3TB…369 c 1T5 ` ‰ I01 c I?36JAEH14B…5>?9E(I01N5(/QFO¢4I"EG/„6~N9EG>?J0F c 5~14BG1CE(I0>?BG>A@n¢4IA/ <M1T9 {>4<;‚ 3"KA/21 `i‰ I01PL34E(I014LM3§EGF c 3T/Š~>?B “ FQ5J?FDE c I016:€34EE(I01/O1*ƒ?14/ >Ae 3T9€@;9;:;14BG¢§B…3":A@;34EG1 LM3§E(I01TL34EGF c 5 {>A@;BH9034/†˜—38:01 c FQ5'FO>?9ˆE(I?3§Em5*I0>6~N5~I6<[Ó3Xƒ?FD/FQ5C3ML365HEH14B EG1T3 c I01TB ` Ó3Xƒ?FD./ - 5 c >?9"ƒ?1TB…5'34EGFO>?903T/tEG>?9;1149AEGF c 1T5E(I01BG1T36:;14B FQ9AEH>ˆ3TBG1T3T5~IAF c IˆLM3X<=K011T9;‚ EH14B…E…3TFQ9?FO9;¢KA@?E*˜}3eq1¥~ c I?3TJ?EG1TB…5RFO9t˜}ÏA@0FDEG1 c >?9?eq>A@;9;:?FQ9;¢ `lj I01ˆ‹nB…5HE c I?3TJ?EG1TB>?9  ‰ I?BG1T1 ‰ 14909?FO5g.A36B…36:;1> 36145  FQ5E(IAF c'“ ~FDE(IR3T/O¢416K?B…3N3T9;:RE(I01C/O1*ƒ?14/n>AenLM3§E(I01TL34EGF c 5 EG3 “ 1T5m> ,=e2BG>?L E(I01TBG1 ` Ó3Xƒ?FD/t9;>"EH145 FO9ˆIAFQ5 FO9AEHBG>0:A@ c EGF2>?9E(I?3§EuE(I01:AF  c @0/DE{<M/O1*ƒ?14/ ¢41T9;14B…3T/D/„<=BGFQ5*145Ce2BG>?L c I?3TJ?EG1TB EH> c I?36JAEH14Bg3T9;:ˆE(I01MBG143":01TBmLM>Tƒ?145Ce2BG>?L 3T/O¢416K?B…3 369;:PK03T5'F c J;BG>0K?3"KAFQ/DFDE{<ˆE(I?BG>A@n¢4IEHB…F2¢4>?9;>?LM1§EHB{<¹˜:?F ,-14BG1T9AEGFQ3T/369;:FO9AEG16¢§B…34/ c 34/ c @n‚ /Q@;5X˜5716ÏA@n1T9 c 1T5R3T9;:ª5*14BGFO1T5X˜ c >?L8KAFO903§EH>?B…F c 5§˜¢4149;1TB…3§EGFO9;¢MeO@;9 c EGFO>?905§˜:?F ,-14BG1T9AEGFQ3T/ 16ÏA@;34EGFO>?905§˜-369;:PLM>0:?@0/Q36Bm36B…FDE(I?LM1§EGF c§` Ó3Xƒ?FD/Š>?J?EGFQLFQ5(EGF c 34/Q/ <8~NB…FDEG1T5gE(I?3§E *9;>?9;1 >Ae;FDEFO5}K01*?9;:3 c >?LMLFDE…EG16:R5*149?F2>?BŠIAFO¢4I5 c I0>0>A/05(EG@n:01T9AE*˜ K?@?E@;9?F„ƒ?14B…5'F„E†<0‚(/O1*ƒ?14/ 5(EG@n:01T9AE…5~F„E(Id57>?L81ªL34E(I014LM3§EGF c 3T/g41 3"J;1TB…F2149 c 1ªLFO¢4I"EMK;1XE…EH14B3TJ;J0BG1 c FQ34EG1[E(I01 36BðEGF c /O1T5 `gÑ†Ì ~FQ/D/-I?3Xƒ?1EG>~N3Xƒ?1LR<ˆI?369;:?5C>Tƒ?14B 57>?L8136BG14365eq>?B‡L<ˆIAF2¢4I85 c I0>0>A/ 3T@n:AF2149 c 145 ` Ò·Ó>6~1*ƒ?1TB7˜CÓ3Xƒ?FD/9;1*ƒ?14Beq>?BG¢41XEG5E(I?3§E143 c IÇJ0BG>0K?/O1TL 5HEH14LM5e2BG>?L 3=¢§BG1T3§EN5(EG>?B{<ª369;:S1*ƒ?1T9ª9;>TƒAF c 1T5~FD/Q/uK;1ˆ3§E…EHB…3 c EG16:SKT<[E(I01IAFQ5(EG>?BGFO1T5R>AeuE(I01 J;BG>0KA/214LM5X˜0>?9;1N3T5757> c FO3§EH1T:~FDE(I1T3 c I c I?36JAEH14B `

"‘ Ì 9CE(I01K;145(E c I?3TJ?EG1TB…5X˜AÓ3Xƒ?FD/TE…3 “ 1453}EH>?J?F c ~FDE(I~IAF c IL369"<BG143":01TB…5LF2¢4I"E K;1[e23TLFD/QFQ36B369;:d5'@;LLM3TBGF 0'1T58>?B:01T5 c BGFOK;145MF„Ee2BG>?L 3S9;1X~ 369;¢X/O1 `œ‰ I01SLM>?5(E 3 cHc 14575(F2KA/21L3§<K;1N–—I?3TJ?EG1TB >?9E(I0$1 ;FOBðE(I0:03X<=.;BG>0KA/214L ` • 369"<€>Ae}@;5R/O1T3TBH9;1T:S3"K0>A@?ECE(IAFO5RJ;BG>0KA/214Lê36514F„E(I01TBN3T9S36L30HFQ9;¢8e23 c E Ñ F„E EG3 “ 1T53¢§3§E(I01TB…FO9;¢‡>AeA>?9?/ < 23 J;16>?J?/21uEG>CI?3§ƒ?1‡3K01§E…EG1TBnE(I?369 50 c I?3T9 c 1 >AeAI?3§ƒAFO9;¢ Eð~>=J;16>?J?/21R~F„E(IˆE(I01573TL81K?FQB…E(I0:?3§<6Ò>?Bm3/2145757>?9FO9 c >?LJ0@?EGFQ9;¢J0BG>0K036K?FD/QF„EGF2145 ƒ?FQ3 c >?LJ0/O1TLM1T9AE…5 ` ÓC3§ƒAFQ/—'1 36J;3T9;:05E(I01RJ0BG>0K?/O1TL FO9MEð~>FO9AEG1TBG145(EGFQ9;¢R:?FQBG1 c EGFO>?905 ` FQB…5(E*˜tI01:?FQ57J;1T90571T5‡~FDE(I83LFQ575HEG3§EH14L8149AE>Ae—E(I01J0BG>0K?/O1TL dI0>6~L369"<ˆJ;16>?J?/21 36BG1M9;161T:;1T:=EH>PI?3§ƒ?1M3K01§E…EG1TBE(I?369 50 c I?3T9 c 18>Aeu57>?L81T>?9;18I?3§ƒAFO9;¢E(I01M573TL81 K?FQB…E(I0:?3§<83T5‡A@nÔ ÑQ‰ I01R3T905H~14B¥Š3/2>"E}L8>?BG1E(I?369 *;` Ò a eQEH14B‡J0BG>TƒAFO9;¢CE(I01@;5(@;3T/ BG1T5(@0/DE*˜uÓC3§ƒAFQ/¢41T9;14B…3T/DF 0'145E(I01 ;FQB…E(I0:?3§<Ç.;BG>0KA/214L·FQ9=ƒA3TBGFO>A@;5R:AFOBG1 c EGF2>?905 ` Ó>6~ LM3T9"?J?/21u36BG1}9;1T16:016:mFO9g3 ¢§BG>A@;JmEH>gI?3§ƒ?1E(I?BG1T1?˜Xeq>A@;B*˜4>?BnL8>?BG1Š~F„E(I E(I01}573TL81 K?FQB…E(I0:?3§<0Ô$ÓN>6~ LM3T9"?J0/O1M3TBG1M9;1T16:016:EG>PI?3Xƒ?1ML8>?BG1E(I?369Eð~>=K?FQB…E(I0:?3§Aen:03X<057Ô ‰ I01g3T905H~14B*˜§ƒ?FQ3R.A3T@0/nÓ34/OLM>?5§˜"K?BGFQ9;¢§5 FO9E(I01 a B…FDE(I?LM1§EGF cm• 14369 ¤  1T>?L81XEHBGF cm• 1T3T9 Ì 9;16ÏA@;3T/DFDE{< ` Ó3Xƒ?FD/ c >?9 c /D@n:;145 E(I01 c I?3TJ?EG1TB~F„E(IM3T9ˆ3TJ;J?/QF c 3§EGF2>?98>AetE(I01 ;FQB…E(I0:?3§0K;‚ /214L·EH>PFO:;149AEGFQeD6~N571TB…5369;: c >?LMJ?@?EH14B…5 `[‰ IAFQ5FQ5 2@;5HEN>?9;1'1 3T3TLMJ?/21 >Ae I0>6~ Ó3Xƒ?FD/ c >?909;1 c E…5IAFO5HEH>?B…F c 3T/J;BG>0KA/214LM5EH>[L8>0:01TBG9S36J0J0/DF c 34EGFO>?905 `©Ì 9[E(I01 c I?36JAEH14B>?9[.A36BGBG>?9;:;>  3TL8145 S~FQ909?FO9;¢¢§36LM1T5 E(I?3§E‡36BG1 c >?905(EHB…@ c EH1T:8e2BG>?LùEð~> /2>?5(FO9;¢¢§3TL8145 ”ÓC3§ƒAFQ/nL8149AEGF2>?905‡3R:01TLM>?905(EHB…3§EGF2>?9FO93  1X~
>?B “‰ FOLM1T5‡36BðEGF c /O1 e2BG>?L T"6 >AenI0>6~iEð~>N/O>?5'FQ9;¢5HEH> c'“ J;>?B…EGeq>A/DF2>?5 c 369K01 c >?L8KAFO9;1T:FQ9AEH>3 ~FQ909?FO9;¢ >?9;1 ` a 5i~F„E(IùE(I01K01T5HE c /O3T575'BG>0>?L EH143 c I014B…5§˜dÓC3§ƒAFQ// - 5Ž1T9AE(IA@;5(FO3T57Leq>?BÇE(I01 LM3§EH14BGFQ3T/ c 3TBHB…F2145NE(I01ˆBG143":01TBN1*ƒ?1T9PE(I?BG>A@n¢4IPE(I01MEH>A@n¢4Iª5'Jn>"E…5 `P‰ I01 c I?3TJ?EG1TBN>?9 I6<0J;1TBG:AFOLM1T905(F2>?905mFO5gÏ?@0F„EH15(EG@;909?FQ9;¢FO98FDE…5gBGFO¢4>A@;B*˜tK?@?ECÓC3§ƒAFQ/ c >?9AEGFQ9?@n1T5EG>ˆJ?@0/Q/ 1*ƒ?1TBLM>?BG136L3 0HFQ9;¢e23 c E…5C3T9;:ˆF2:01T9AEGF„EGF2145me2BG>?L IAFQ5K?3"¢>Ae-EHBGF c'“ 5 `Ì ÏA@n1T5HEGF2>?9;1T: 5*>?LM1>Ae"E(I01‡EG1 c I?9?FOÏ?@n145}ÓC3§ƒAFQ/6FQ9AEHBG>0:?@ c 16:I01TBG1‡eq>?BtE(I01§FOBt/O3 c'“ >Ae?J;I6<05(F c 3T/"L814369;‚ FO9;¢"˜?KA@?E Ì K01 c 36LM1C3NK014/DF21*ƒ?14B-~I01T9 Ì 573X~  1§/Qeq>?9;: - 5 c >?905(E…369AE Ñ e Ò(˜6E(I01  36LLM3 eO@;9 c EGF2>?9t˜?369;:E(I01N:;>A@nKA/21Ce23 c EG>?BGFQ3T/¹K014FQ9;¢CFQ9AEHBG>0:?@ c 16:EG>3N9;1X~Î34@n:?FO1T9 c 1 ` Ì - L 5(EGFD/Q/Š9;>"EC5'@;BG1 Ì @;9;:01TB…5HEG3T9;:PE(I01 c /2>?5(FO9;¢ c I?3TJ?EG1TB>?9 7>0I?9S–t>?9"~N3§< - 5 B…3 c EHB…369€J;BG>0¢§B…3TLMLFO9;¢8/O3T9;¢X@;3"¢41A˜}34eQEH14BN5*1*ƒ?14B…3T/}34E…EG1TLJ?E…5EH>S:;>ª5*>ª369;:€1*ƒ?1T9 3TeQEG1TBBG1T36:?FQ9;¢g–t>?9"~N3§< - 5u>6~N9R'1 36Jn>?5(FDEGFO>?9NFO9IAFQ5 $  «‹¬ "&   ˜§~NBGF„E…EH149~F„E(I tF c I?36BG:  @A< ` Ó3Xƒ?FD/n5*>>0KTƒ?FO>A@;5'/ <1T9 {>4<05uE(I01NL34EG1TB…FO34/†˜6E(I0>A@n¢4I¹˜AE(I?3§E Ì ~N369AEŠEG> “ 1614JEHB{?9–t>?9"~N3§< - 5 c I01 c'“ 1TBHK;>?3TBG:¢§3TL81gFQ5}L@ c I14365(F214BEG> :?FO¢41T5HE ` Ò Ó3Xƒ?FD./ - 5J0BG1T571T9AE…34EGFO>?9>?9 B…3 c EHB…369eq>A/D/2>6~N5  @A2< - 5   36BðEGF c /O1e2BG>?L ¤  *  &    %¤   ! A`=• >?5HEg>AeE(I01 c I?36JAEH14B…5'1 36J0/Q3TFQ9t˜—FO9AEG1TBGJ;BG1XE*˜3T9;:P5…A@;5(/„<8J0@nKA/QFQ5*I01T:ˆ3TB…EGF c /2145e2BG>?L >"E(I01TB‡34@?E(I0>?B…5 ` Ó3Xƒ?FD/ c F„EH145m3T/D/-IAFQ5m5*>A@;B c 1T5 E(I?BG>A@n¢4I0>A@?EE(I01CEG'1 34E*˜0K?@?EuL<>?9;1g~FQ5*IN~>A@0/2:I?3§ƒ?1NK;1T1T9eq>?Bu3@;9?F„EH1T:8KAF2KA/QFO>0¢4‚ B…36J;I6<M57>?L81¥~I01TBG1CFQ9E(I01K0>0> “¹` ¬ !v£    "
FQ5M3~>?9;:;14BGeO@0/ c >A/Q/O1 c EGFO>?9d>Aem365HEH>A@;9;:AFO9;¢=J;3TB…3":01> 361T5M3T9;: 5(EG@;909?FQ9;¢EG@;BH903TBG>A@;9;:05 `a /DE(I0>A@n¢4IˆE(I01ML34E(I014LM3§EGF c 5 c 369[¢41XEge234FOB…/„?BIAFQ5LM3§EH14BGFQ3T/BG14J;3X<05E(I01~>?B “ BG143":01TB…5L@;5HE FO9"ƒ?145(E ` 3T905>AeJ0@ 0 0H/O1?˜tJ;3TB…3":01> 3;˜—>?Bm¢§36LM1K;>0> “ 5C57I0>A@0/2:ˆ‹;9;:L@ c I8EG>1T9 {>4< ` Ì /O>0> “ eq>?Bð~N3TBG:EH>5*I?3TBGFQ9;¢3/2>"E‡>AetE(IAFO5}~FDE(I Š14BHLM>?9AE IAF2¢4IM5 c I0>0>A/t5HEG@n:;149AEG5FQ9 E(I01N9;'1 34Eeq1X~b







NxB CL KB =%L I>







 

ˆ # mn m 

Õ=1 J0BG1T571T9AE—369;:CJ;BG>4ƒ?1 39;1¥~dFO9;1TÏ?@;34/QF„E†A@?E¹EG>>,-14B—3T9 T3 /„EH14BH903§EGF„ƒ?1 3TJ;J0BG>?3 c ImEH>C57>A/„ƒAFO9;¢}3‡/O3TBG¢41 c /O3T575>Ae"FQ9;16ÏA@;3T/DFDEGFO1T5 ` Õ=I01T936J0J0/DF c 3"KA/21A˜ E(I01L81XE(I0>0:=3T/D/2>6~N5gBG16:A@ c FO9;¢3M5… 369FO9;1TÏ?@;34/QF„E†<FO9 2@;5HEux&  ƒA3TBGFQ3"KA/21 ` ‰ I01NL3TFQ9BG1T5(@0/DEŠFO5 j
 x %'"! |$¨T1XE ˜ ˜03T9;: K01N9;>?9;‚'9;1T¢§34EGF ƒ?1NBG1T34/¹9?@;L8K01TB…5 `Š‰ I01T9R~1I?3§ƒ?1CE(I01 eq>A/Q/O>6~FO9;¢mFO9;1TÏ?@;a34/QF„E†b< c ` Õ=1 57I?3T/D/6J0@?E¹E(IAFO55…<0LL81XEHBGF c FO9;1TÏ?@;34/QF„E†C369N1TÏ?@0F ƒA3T/O1T9AEteq>?BHL˜¥~IAF c I~1~FQ/D/ E(I01T9J;BG>4ƒ?1C3T9;:R@;571 `’ 1§E…EGFQ9;¢ S = a + b + c ˜ P = abc ˜A3T9;: Q = ab + bc + ca ˜ ~1N5HEG3TB…EuKTTƒ?1CFO9;1TÏ?@;34/QF„E†<365 ` (9SQ − 2S − 27P ) ≤ 4(S − 3Q) Ì eu~18E(I01T9ªEG3 “ 15*ÏA@;36BG1=BG>0>"E…5§˜9;>"EH18E(I?3§E ˜u3T9;:©14/DFOLFO903§EH18E(I01 3"K?5*>A/D@?EH1mƒA3T/D@n1 |9SQ − 2S − 27P | ˜6~1N>0SKAE…3TFQ−9E(3Q I011T≥Ï?@0F0ƒA3T/O1T9AEuBG1T5(@0/DE ` j
 x %'"! ( ¨T1XE a ˜ b ˜ c K;1M9;>?9;‚(9;16¢§3§EGF„ƒ?1MBG1T34/Š9?@;LMK;14B…5 `’ 1§E S = a + b + c ˜ ˜?369;: Q = ab + bc + ca `‰ I01T9R~1I?3Xƒ?1CE(I01N:;>A@nKA/21FQ9;16ÏA@;3T/DFDE{< P = abc ` 9SQ − 2S − 2(S − 3Q) ≤ 27P ≤ 9SQ − 2S + 2(S − 3Q) r "! ! ù¨T1§E 0 ≤ k < 1 `u’ 1XE x = 1 − 2√1 − k 3T9;: x = 1 + 2√1 − k ` ¨T1XE `‰ I0149 x ∈ [x , x ] ` 3k − 2 − 2(1 − k) ≤ x − 3x + 3kx ≤ 3k − 2 + 2(1 − k) ¡  « « * ?¦  & &  ’ 1§` E x‡1§‹n=9;1 1 − √1 − k 369;: x = 1 + √>?19 − k ` –n/214`Ž36B…/„Ì <¹EG5 ˜ x x < x f (x) :;14BGF <ƒA34EGxF ƒ?1 014−5m343xE x +3T9;3kx : x `‰ I0[x15(,F2x¢§9ˆ] >Ae f = 3(x − 2x + k) 5*I0>6~N5X˜365 ƒA36B…F2145Ne2BG>?L EH> ˜—E(I01T9Pe2BG>?L EG> ˜3T9;:P‹n9034/Q/ <=e2BG>?L EH> x ˜"E(I?34E xf (x) FQ9 c BG1T3T5*145§x˜6E(I01T9Mx:;1 c BG1T3T5*145§˜?3T9;:xE(I0149xFQ9 c BG1T3T5*145‡3"¢§34FO9 ` x ÓN149 c 1A˜?eq>?B3T9"< x ∈ [x , x ] ˜0>?9;1NI?365   ` min f (x ), f (x ) ≤ f (x) ≤ max f (x ), f (x ) (a + b + c)(2a2 + 2b2 + 2c2 − 5ab − 5bc − 5ca) + 27abc


2

≤ 4(a2 + b2 + c2 − ab − bc − ca)3

3

2

2

3

2

3

3

2

3/2

3

1

1

3/2

4

4

3/2

3

2

2

1

2

2

0

3

4

3/2

3 2

3

2

1

2

1

2

2

3

3

4

 ,

- ./! ""  #%$&"' ('

, -0$ 12  '3 !!""""  #%#%$&$&""'' ()()*4*!+% %("0 #',#% $3567

* -$8#&9%:05; <<'-=#2> 1

c

3

2

0

3

4

1

4

4



Õ=1 c /Q3TFQLŽE(I?34E f (x ) = f (x ) 369;: f (x ) = f (x ) `ŠÌ 9N>?BG:;14B¹EH>C1*ƒA3T/D@;34EG1 f (x ) 369;: ˜¹9;>"EH1E(I?3§E 369;: 3TBG1E(I01 0'14BG>01T5g>Ae x − 2x + (4k − 3) = 0 ` ‰ I01fu(xF ƒ?FQ5')FO>?9 a /2¢4>?B…FDE(I?L x J;BG>4ƒ?FO:;x145 2

4

4

f (x)

1

1

3

1

2

4

` `C’ FQLFD/O3TBG/ 0>"E…5

= x3 − 3x2 + 3kx
= x2 − 2x + (4k − 3) (x − 1) + (1 − k)x + (4k − 3)

ÓN149 c 1A˜ + (4k − 3) = 3k ` − 2 − 2(1 − k) ?> 9;1‹;9;:0f5u(xE(I?34)E =f (x(1)−=k)x 3k − 2 + 2(1 − k) ‰ >R1*ƒA34/Q@;3§EH1 369;: ˜§~1g9;>"EG1m‹;B…5(E—E(I?34E x 369;: x >Ae x − 2x + k =f0(x369;):RE(I?34E f (x ) 1

3/2

1

3/2

4

2

2

3

2

3

f (x) = x3 − 3x2 + 3kx = (x2 − 2x + k)(x − 1) − 2(1 − k)x + k

`

abc 3TBG14eO@0/ c I01 c'“ ƒ?1TB…FQ‹n1T5E(I01 c /O34FOL 369;:8E(I0149F„E eq>A/Q/O>6~N5E(I?3§E*˜teq>?B x ∈ [x , x ] ˜ >?9;1I?3T5 f (x ) ≤ f (x) ≤ f (x ) ˜?3T5 :;145'FQBG16: ` ¡ @; BðE(I0« 1TB7«˜—*~ 1"9; >" EH1E( I?& 3§E‰ E(I0I011 BGFO9;145'1T@0Ï?/„@;E-34K;/Q1§F„E†FO<=9;¢‡FQ5N>0KTI0ƒ?>?FOL8>A@;>05¢414eq>?9;B 16S>A@;=5 `M0‰ ˜XIA~@;1‡5X˜-3T57FDEg5(@;5(L8@  1 cS1T5 >Ñ KT0< ` BG1T5 c 34/QFQ9;¢*ÒmEH>ªJ;BG>4ƒ?1FDEg~I01T9 S = 3 ` Õ=13T/Q5*>ª571§E Q = 3k ˜ueq>?B57>?L819;>?9;‚ 9;16¢§3§EGF„ƒ?1BG1T34/—9?@;LMK;14B k `’ FQ9 c 1 0 ≤ Q ≤ S /3 ˜A~1R¢41§E 0 ≤ k ≤ 1 `‡‰ I01 c 3T5*1 Ñ ~I01TBG1 FQ5uE(I01NL13 3TFOL@;LÒ c >?BGBG1T5'Jn>?9;:?5}EG> 369;:E(I01 k BG1T=5(@0/D1EFQ5N>0KTƒAF2>AQ@;5 =` ÓNS149/3c 1A˜¹~1LM3X<=3T575(@;L81 0 ≤ k < 1 ` uaF ƒ?=FO:?FQb9;=¢KTc< 27 ˜¹~1 I?3§ƒ?1CEG>J0BG>Tƒ?1CE(I?3§E ` 3k − 2 − 2(1 − k) ≤ P ≤ 3k − 2 + 2(1 − k)  >6~Î571§E 369;: B = ab `‰ I01T9 A = 3 − c 369;: A=a+b ` B = Q − cA = 3k − c(3 − c) = c − 3c + 3k ’ @nK05HEGFDEG@?EGFQ9;¢ A = 3 − c 369;: B = c − 3c + 3k FO9AEG> A − 4B ≥ 0 :AFO5HEGFO9 c EBG>0>"E…5>AeAE(I01mÏ?@;36:0B…3§EGF c FO9 Ñ Ò E(I?3§E FO5X˜ 1 − 2√c 1 − k ≤ c ≤ 1 + 2√1 − k `’ FQ9 c 1 P = abc = Bc = c − 3c +" 3kc ˜ E(I01NBG1T5(@0/DEŠeq>A/Q/O>6~N5}e2BG>?L E(I01¨T14LML3 ` 

 E{Á @ :E  FG    À  È4À¹Å É ` –¹/O1T3TBG/ ?BHL34EGFO>?9 ˜a b ˜A369;: c 36BG1C3T/Q5*>NE(I01C5(F2:01T5‡>Aen3gEHBGFQ369;u¢X/O=1?˜TS~1g−/O1T3X3Qƒ?1gFDEŠ365}3T941 3601TB≤c FQ5*u1m≤EG> Sc I01 c'“ E(I?34E S /4 ≤ Q ≤ S /3 " E(IA@;5X˜"E(I01NB…3T9;¢41>Ae u BG1T:?@ c 1T5}EG> 0 ≤ u ≤ S /4 ` l  ! y  |; Ñ  Ò ’ @;J0Jn>?571E(I?3§E a ˜ b ˜n369;: c 3TBG1RJn>?5(FDEGF ƒ?1RBG143T/—9?@;L8K01TB…5 ` .;BG>Tƒ?1 E(I?34E 2(a + b + c ) 9(a + b + c) ` Ñ{ Ò + ≥ 33 1

1

4

4

2

2

3/2

3/2

2

2

2

2

3

2

2

2

2

2

3

2

3

abc

3

2

a2 + b 2 + c 2

 Ì 9;16ÏA@;3T/DFDE{< Ñ{ ÒFO5 I0>?LM>0¢41T9;1T>A@;5 " 57>~1L3§?9 Ñq Ò¹BG16:A@ c 145EH> 9P (1 − 3Q) ≤ (1 − 2Q)(1 − 3Q) `  >6~ ` Ì e 1 − 3Q = 0 ˜}E(I0149€~1P3TBG1ª:0>?9;1 `œÌ e 1 − 3Q˜;E(IA =FO5S − 3Q ≥ 0 ` ?< O / § 3 … E H E 4 1 B ' 5 Q F M L ? J Q / D F ¹ ‹ 4 1 5 EH> ˜t>?B 1 ‰ I0−1T>?3Q BG1TL > ˜T0FDE5'@  c 145EH>NJ;BG>4ƒ?1E(I?3§E 9Q9P− 2≤+12(1− −2Q3Q) 27P≤ 3≤−36Q−" 6Q E(I?34EFQ5§˜ `’ 1XE…EGFO9;¢ u = 1 − 3Q ˜A~157161E(I?3§EuE(IAFQ5mK;1 c >?LM1T5 2(1 − 3Q) ˜;>?≤ 5(1 − 3Q) √ B eq>?B 0 ≤ u ≤ 1 `‰ IAFQ5§˜?I0>6~1*ƒ?1TB7˜?FQ5uEHBGF ƒ?FQ3T/ ` 2u ≤ 5u u(5 − 2 u) ≥ 0 l  ! y  (  Ñ Ò‡¨T1XE x ˜ y ˜?369;: z K;19;>?9;‚'9;1T¢§34EGF ƒ?1NBG1T34/n9?@;L8K01TB…5 ` .;BG>Tƒ?1gE(I?34E
`Ñ Ò (x + y)(y + z)(z + x) ≥ xyz(2x + y + z)(2y + z + x)(2z + x + y) Ì 9;16ÏA@;3T/DFDE{< Ñ ÒFQ5 34/O57>I0>?LM>0¢41T9;1T>A@;5§˜?369;:R~1NLM3X<3"¢§34FO9EG3 “ 1 S = 1 `  >6~Ž~1 BG1X~NB…FDEG1 Ñ Ò5(@ c(c 1T5'5'F ƒ?14/ <365
˜ (1 − z)(1 − x)(1 − y) ≥ P (1 + x)(1 + y)(1 + z) ˜ (Q − P ) ≥ P (2 + Q + P ) ˜ Q ≥ P (2 + 3Q) ` 27Q ≥ 27P (2 + 3Q) ?< ‰ I01T>?BG1TL ˜6F„E5'@  c 145EH>NJ0BG>Tƒ?1 (2 + 3Q)(9Q − 2 + 2(1 ` ; 5(−FO9;3Q) ž E(I?34E-FQ5§˜ 2(3Q) + 2(1 − 3Q) + (3Q)(1 ¢ u = 1)−≤3Q27Q˜§~1 " − 3Q) ≤ 2 √ √ 5*1T1 E(I?3§E—E(IAFO5Š5(FOLJ0/DFQ‹n1T5-EH> ˜">?B eq>?B 0 ≤ u ≤ 1 ˜6~IAF c IFQ5uEHBG2u@n1 +` u ≥ 3u u(1 − u)(2 − u − u) ≥ 0 l  ! y  ª¨T1§E ˜ ˜ 3T9;: K01PJn>?5(FDEGF ƒ?1PBG143T/ 9?@;LMK;14B…5 `Ž‰ I01T9S~1[I?3Xƒ?1=E(I01 eq>A/Q/O>6~FO9;¢g9;>?9;‚7I0>?aLM>0b¢41T9;1T>A@;c5‡FQ9;16ÏA@;3T/DFDE{<— 2




3








2

3/2

3/2

3/2

2

2

2

2

2

3/2

3/2

2

3/2

5/2

3/2





1 1 1 + + a b c



(a + b + c)(2a + 2b + 2c + 3) + 1

≤ 9+

(a + b + c)2 (a + b + c + 1)2




`

3abc + (ab + bc + ca)



3 3 3 + + a b c



Ñ TÒ

Ì 9NEH14BHL5u>Ae P ˜ Q ˜6369;: S ˜4E(IAFO5FO5 Q S(2S + 3) + 1
≤ 9 + S (S + 1) + Q 3Q ˜ ~IAF c I c 369MK01NBG1436BGB…369;¢41T:M3T5 27PP≥ 9SQ − 2S − (S − 3Q)3P− (S − 3Q)P ` ?< ‰ I016>?BG14L ˜AF„Eu5(@  c 1T5uEH>J0BG>Tƒ?1gE(I?34E ` 9SQ − 2S − 2(S − 3Q) ≥ 9SQ − 2S − (S − 3Q) − (S − 3Q) ; 5'FQ9;¢ ˜;~1I?3Xƒ?1 ~F„E(I 0 ≤ u ≤ S ˜t3T9;:8E(IAFO5 }• Ì 9;16ÏA@;3T/DFDE{u< ` + u ≥ 2u I0>A/2:?5KTu
3

3

2

2

3/2

3

2

3/2

2

2

2

2

2

2

2

2

l  ! y 
¨T1§E a ˜ b ˜Š3T9;: c K018Jn>?5(FDEGF ƒ?18BG1T34/u9?@;LMK;14B…5 ` .;BG>4ƒ?1ME(I01Meq>A/Q/O>6~FO9;¢ FO9;1TÏ?@;34/QF„E†< (a + b) (b + c) (c + a) 8 ` Ñ{£ Ò + + ≥ (a + b + c) c a b 3 ’ FO9 c 1FO9;1TÏ?@;34/QF„E†< Ñ{£ ÒFO5 I0>?LM>0¢41T9;1T>A@;5§˜6~1gEG3 “ 1 S = 1 `‰ I01/21§eQEu5(F2:01N>Ae Ñ{£ ÒFO5 3

X (1 − a)3 a


3

3

2

 X 1 − 3 + 3a − a2 a


=



Q

=



− 9 + 3 − (1 − 2Q) =

Q

+ 2Q − 7

`

ÓN149 c 1A˜ˆFQ9;16ÏA@;3T/DFDE{< Ñ{£ ÒªK01 c >?L8145 (Q/P ) + 2Q ≥ 29/3 ˜=>?B€1TÏ?@0F ƒA3T/O1T9AEG/ < `  >"EG18E(I?34E − 6Q > 0 ˜u5(FO9 c 1 Q ≤ S = ` 27P ?< ‰ (29 I016>?BG−14L 6Q)˜AF„Eu≤5(@ 81Q  c 1T5uEH>J0BG>Tƒ?1gE(I?3429 E
` (29 − 6Q) 9Q − 2 + 2(1 − 3Q) ≤ 81Q ՈFDE(I u = 1 − 3Q ˜AE(IAFQ5 K;1 c >?LM1T5

˜ 29 − 2(1 − u) 3(1 − u) − 2 + 2u ≤ 27(1 − u) ~IAF c I5'FQLMJ?/QFD‹¹145uEG> 2u + 27u ≤ 3u + 26u ˜;>?B P

P

1 3

1 3

2

3/2

3/2

5/2

3/2

u 1−

2

√
√
u 26 + 2u − u ≥ 0

6˜ ~IAF c IFQ5uEHBG@n1 ` ÓN14BG1€FQ53€/DFO5HE8>AeFQ9;16ÏA@;3T/DFDEGFO1T58E(I?3§E c 369ŽK01SEHBG1434EG16:”5'FQLFD/O3TBG/ <”KT<”@;5(FO9;¢ ‰ I01T>?BG1TL 0` ¨T1§E a ˜ b ˜?3T9;: c K01N9;>?9;‚(9;16¢§3§EGF„ƒ?1NBG143T/n9?@;LMK;14B…5 ` .;BG>4ƒ?1CE(I?3§E • ` a (b + c) + b (c + a) + c (a + b) ≥ 2abc(a + b + c) ¨T1§E a ˜ b ˜?3T9;: c K01N9;>?9;‚(9;16¢§3§EGF„ƒ?1NBG143T/n9?@;LMK;14B…5 ` .;BG>4ƒ?1CE(I?3§E •

eq>?B

0≤u≤1

3

3

3

a2 (b + c)2 + b2 (c + a)2 + c2 (a + b)2 ≤ •

4 (a3 3

+ b3 + c3 )(a + b + c)

¨T1§E a ˜ b ˜?3T9;: c K01N9;>?9;‚(9;16¢§3§EGF„ƒ?1NBG143T/n9?@;LMK;14B…5 ` .;BG>4ƒ?1CE(I?3§E a4 + b4 + c4 + 3a2 b2 + 3b2 c2 + 3c2 a2 ≥

•

¨T1§E ˜ ˜"3T9;: K01g9;>?9;‚(9;16¢§3§EGF„ƒ?1mBG1T34/09?@;LMK;14B…5~FDE(I E(I?34E a b c

4 (a 27

+ b + c)4

a+b+c = 1

4 4 4 3 + + ≤ 1+a 1+b 1+c ab + bc + ca

`

`

` .;BG>4ƒ?1

`

•

¨T1§E a ˜ b ˜?3T9;: c K01NJ;>?5'F„EGF„ƒ?1BG143T/n9?@;L8K01TB…5 ` .;BG>4ƒ?1CE(I?3§E `

9 4(a + b + c)2 5 5 5 + ≥ + + a+b+c 3abc a b c

Ñ ‡Ò ¨T1XE a ˜ b ˜03T9;: c K;1NJ;>?5'F„EGF„ƒ?1CBG1T34/¹9?@;LMK;14B…5 ` .;BG>4ƒ?1CE(I?3§E 1 1 1 (a + b + c) ` + + ≤ a+b b+c c+a 6abc Ñ  Ò FO9;:RE(I01N5'LM34/Q/O1T5HEƒA34/Q@n1N>Ae α eq>?B~IAF c I • •

2

1 27

− xyz ≤ α

1

− (xy + yz + zx)



I0>A/2:?5}eq>?Bu34/Q/n9;>?9;‚(9;16¢§3§EGF„ƒ?1 x ˜ y ˜?3T9;: z ~IAF c I573§EGFO5(eD< x + y + z = 1 ` Ñ ™ Ò}¨T1XE ˜ ˜?3T9;: c K;1C9;>?9;‚'9;1T¢§34EGF ƒ?1BG1T34/n9?@;L8K01TB…5~F„E(I a + b + c = 1 ` • .;BG>Tƒ?1gE(I?3§aE b (1 − a)(1 − b)(1 − c) 1 ` ≤ 3

(1 − a2 )2 + (1 − b2 )2 + (1 − c2 )2

8

¨T1§E a ˜ b ˜-369;: c K01M9;>?9;‚(9;16¢§3§EGF„ƒ?1MBG143T/9?@;LMK;14B…5N573§EGFO5(eD?905'B…3TFQ9AE ` .;BG>Tƒ?1gE(I?34E a + b + c − 1 = ab + bc + ca   1 1 1 2(a + b + c − 1) ` (a + b + c) + + ≥ 9+ a b c 3abc  !  ÕB…FDEG1CE(I01 c >?905HEHB…3TFQ9AEu365 S − 3Q = 1 ` Ñ £ ҇¨T1XE a ˜ b ˜03T9;: c K;1CE(I015'FO:;145 >Aet3CEHBGFQ369;¢X/O1 `Š’ I0>6~E(I?3§E • Y abc(a + b + c) ` ≥ 2abc + (b + c − a) •

2

2

2

2

2

a2 + b 2 + c 2

•






¨T1§E ˜ ˜t369;: K;1J;>?5'F„EGF„ƒ?1BG143T/-9?@;L8K01TB…5~F„E(I .;BG>Tƒ?x1gE(yI?3§E z 15 + x + y + z xyz

+

1/x + 1/y + 1/z = 1

12(x2 + y 2 + z 2 ) x2 y 2 z 2

’ 1XE a = 1/x ˜ b = 1/y ˜ c = 1/z ` Ì 9369"<EHB…FO3T9;¢X/21m~F„E(I5'FO:;145 a ˜ b ˜?3T9;: c ˜6~1NI?3Xƒ?1 •  ! 

(a2 + b2 + c2 )(a + b + c) + 4abc (a + b + c)3

≥

≤

`

4 3

13 27

`

`

Ì 9369"<EHB…FO3T9;¢X/21m~F„E(I5'FO:;145 a ˜ b ˜?3T9;: c ˜?3T9;:FQ90B…3":AFQ@;5 r ˜6~1NI?3§ƒ?1 ` 12r (a + b + c) ≤ (a + b + c )  !  ‰ 3 “ 1 ˜0~IAF c I8L81436905 eq>?B}E(I01571TLF2‚(Jn14BGFQL81XEH14B s `°; 5*1 ÓN14BG>?29 - 5 >?BGSL@0=/Q32EG>¢41XE r = Q −sP=−11 ` Ñ ‘ Ò Ì e a ˜ b ˜?369;: c 36BG1Jn>?5(FDEGF ƒ?1BG1T34/¹9?@;LMK;14B…5§˜?J;BG>4ƒ?1CE(I?34E • a (b + c − a) b (c + a − b) c (a + b − c) ab + bc + ca ` + + ≤ b+c c+a a+b 2 ` ¨T1§E a ˜ b ˜?3T9;: c K01NJ;>?5'F„EGF„ƒ?1BG143T/n9?@;L8K01TB…5 5(@;LMLFO9;¢mEH> 3 .;BG>Tƒ?1gE(I?34E • •

2

2

2

2

2 2

2

2

2

2

a b c 135 9ab 9bc 9ac + + + + + + + 81 ≥ 82 bc ca ab abc c a b



1 1 1 + + a b c

`



h 
0xy  "!   `ùÌ 3TL ¢§B…3§EH1§eO@0/EG>€E(I01ªBG14eq14BG161[eq>?BIAFO5eO@0/Q/m3T575(FO5HEG3T9 c 1[3T9;: 5'@;J0Jn>?BðEŠ~IAF c ILM36:;1gE(I01N3TB…EGF c /21Jn>?5'5'FOK?/O1CFQ9FDE…5‡J;BG145*149AEueq>?BGL ` 



3

  %¤ *  &     & * ¤  *  &   ~¤ "  & ˜ &CÑ  ‘ ÒH˜?J `0£6 ˜ .;BG>0KA/214L 6  n`

  %¤ *  &     & * ¤  *  &   ~¤ "  & ˜ A‘gÑ{T" ÒH˜?J `06™  ˜ .;BG>0KA/214L T™ £;`  b¤ *  &         ˜ A‘ ˜ £T RÑq66 Ò(˜0J ` ‘ ˜t.;BG>0K?/O1TL ?‘?` – `

   ¤  *  &    "& * ¤ *  &   5¤ "  & ˜ " Ñ{T"6™ ÒH˜ J0J `  ˜ ˜t.;BG>0KA/214L •©*  `

   ¤  *  &    "& * ¤ *  &   5¤ "  & ˜ " Ñ{T"6™ ÒH˜ J0J `  ˜ c4 ˜t.;BG>0KA/214L *;`

   ¤  *  &    "& * ¤ *  &   5¤ "  & ˜ & Ñ{T"A‘ ÒH˜ J0J ` c4 ˜ c £ ˜t.;BG>0KA/214L 6"6;`

   ¤  *  &    "& * ¤ *  &   5¤ "  & ˜ & Ñ{T"A‘ ÒH˜ J0J `n& ˜  ˜t.;BG>0KA/214L 6 n`

 "%'   



 

£



™



‘



 



   

t>T< n3TB(K?36B…3 T¨ 16K?369;145*1 ; 9?F„ƒ?14B…5'F„E†<¹˜ 3 c @0/DE{<M>Ae ’6c F2149 c 1 36903TB c 36LJ0@;5X˜—‡14J;3TB…EHLM1T9AE}>Ae • 3§E(I01TL34EGF c 5 . `Dš` ¹1> 3  "™"£6™ 7:;1§F2:01§E Œ"/ • 1§EH9 ` ¨T16K?369;>?9

[email protected]





w Z^Q/[_a`7Z#W3TI_7SZI|7_/fR9\h\ ZSiQ7WR9jˆ[#jbT†i`aWxRU\I_ed3`a`W T†X#``7Ty7TUS[_/RR’TxTd_ed3ZS[U_‚QeXgSxS#f\8\/\h`7ZziR6Q7R9Wj­Z Q}†S_7SRUTk\ f_ed3\hS#ZUf,WˆZn
T9RXM}:`7 _e 6dr^Zs#[3s#_‚tvS u  WcSTU_7RU\/`T‡š TUZ^Q/[3_a`7Z#ƒmWSu y1dkf\hZiQ7RUj„`T†…^`7z^RUWg`aWbƒ3Wm…^Q `T‡dˆS#W Xb‰#\hRUWvy*dmŠ‹_edvR]ZŒxy/`S^Q QS#Wm…[S^…RTIZ|S#W3S3XmS u Že‰mW%\hR9`WTTUy1d?[R}:TV` Q Š’f‘U\hŠ‚R#y8“UR^Š X S#R]WvƒmXMW3…^”ŠIQ `TƒmdW3u’…^QŽe`WxTd{_edv}~R’` QTUZ^fQ/[3\h_aR`7y8Z#RWX

T Rb #T9‰mR#\hyhR9_aW`7Z#y1W3d3Š‹Š _eS#dW R X?f`a\hW%Zi`ThQ7T^R9[3j R9Tg}~•U` Q Š Q–iŠIR’—T9Š _aSS#_7W RXcXk`a˜9W Š _edRžQS#W3™…dv[^R6S^…RR]X#`7Z_7Z#|,\~_edv_edRžS#fW\/š#`ajˆT,›^S#R\1S#YIW|/œhRS_*S#[#\h\hyžR^X]™RUTU\*Z^\1Q/`7[3R9\~_a`7ZZ#|W_eu dRkŸW3`7z^RU\/T8`7_ YkZ|MZ#W_e\hRS^Q|/Z#\~_e\/S#WT8QSœ _a`7Z#WT’Z|,_edvRžf\hZiQ7R9jˆT^u 





(?)

Ígº Ë ` –—>?BGBG1 c EGFO>?9 ` ¡  3£   "  ‹„® ¦A¨g®  $ "    A©! *ª"    ¡$     
!    &  !; ` !  $    !"m£ !  !%¡  !    !& !    °¡ 3£    " ¨T1§E a ˜ a ˜ . . . ˜ a K01NJ;>?5'F„EGF„ƒ?1BG143T/n9?@;LMK;14B…5 ` .;BG>4ƒ?1CE(I?3§E X  Y X  ` a ≤ a 

1

2

n

n

n+1 2

n

n

n

k j

k

ÍgÍ» Ë ` ¡  3£    .;BG>Tƒ?1gE(I?34E

k=1

k=1

6ª

© 1ª

j=1

«

ž©

     " ! !  "   *               #

  1 1 1 1 1 ∞ ln 2 − ∞ + + ··· + 1 + + ··· + X X n+1 n+2 2n 2 n = n (2n + 1)(2n + 2) n=1 n=1

`

Õ=I?34EFQ5uE(IAFO5 c >?LL8>?9RƒA34/Q@n1TÔ ÍgÍ»º `¢¡  3£   ")%¤  ® * $ !   $    ª   &  ! 0` ¨T1§E ˜ ˜369;: :;149;>"EH1iE(I01”5*14LFQJn14BGFQL81XEH14B*˜E(I01iFO90B…36:?FD@;5§˜369;:$E(I01 c FOB c @;LB…3":AFQs@;5r>Aet3CEHBGFQ36R9;¢X/O1 ABC ˜?BG1T5'Jn1 c EGF„ƒ?1§/„< `Š’ I0>6~E(I?34E ˜ (s + r + 4Rr)(s + r + 2Rr) ≥ 4Rr(5s + r + 4Rr) 369;:M:01§EG1TBGLFQ9;1g~I01T9M1TÏ?@;34/QF„E†<I0>A/2:?5 ` ÍgÍ»Í ¢ ` ¡  3£   ")%¤  ® * $ !   $    ª   &  ! 0` ¨T1§E a ˜ b ˜?369;: c K;1Jn>?5(FDEGF ƒ?1BG1T34/n9?@;L8K01TB…5 `’ I0>6~E(I?34E Y Y

˜ 2(a + b) ≥ (a + s )(bc + s ) ~I01TBG1 s = a + b + c 3T9;: s = ab + bc + ca ` 2

2

2

2

2

3




1






2



1

2

2

`

£ gÍ Í» Ð `¢¡  3£   ")%¤  ® * $ !   $    ª   &  ! 0` ¨T1§E ˜ ˜ ˜ 0K 1mJn>?5(FDEGF ƒ?1mBG143T/?9?@;LMK;14B…5u369;:NF2:01T9AEGFDeD< a ~FDE(I a ` .;BG>Tƒ?1gE(I?34aE a . . . a X X ` a ≥ a a ÍgÍ»¼ ` ¡  3£   "   !       !     :¤  * :¡  ?¡     

 !& U«  ‹¤ #  © # ` .;BG>Tƒ?1gE(I?34E 1

2

n

n+1

n

n

3 k

k=1

tan

2π 6π + 4 sin 13 13

1

2 k k+1

k=1

4π + 4 sin 13 5π tan + 4 sin 13

= tan =

π 13 q √ 2π = 13 + 2 13 13

`

ÍgÍ»g¾ ` ¡  3£   "   !       !     :¤  * :¡  ?¡     

 !& U«  ‹¤ #  © # ` FQ9;:3 BG143T/69?@;LMK;14B 3T9;:J;>A/„?LFQ3T/Q5 f (x) ˜ g(x) ˜T369;: h(x) ~F„E(IgFQ9AEH1T¢41TB c >01  c FO1T9AE…5§˜?5(@ c IE(I?3§E t √ ˜ √ ˜ 369;: h(t) = √7 ` f (t) = 2 g(t) = 3 ÍgÍ»  ` ¡  3£   "¨ ! !†¡   *     ˆ© ! 1ª"   *         ^£     «    ! ¹` ŒA/DFOLFO903§EH1 θ e2BG>?LE(I015…<05HEH14L ˜ λ cos(2θ) = cos(θ + α) ` λ sin(2θ) = 2 sin(θ + α) ÍgÍ»  ` ¡  3£   "¨ ! !†¡   *     ˆ© ! 1ª"   *         ^£     «    ! ¹`  F ƒ?1T9 4ABC ˜?/21XE AD K01E(I01R34/DEGF„EG@n:;1EG> BC `Ì e AB : AC = 1 : √3 ˜ J;BG>4ƒ?1CE(I?34E AD ≤ BC ` Õ=I01T9M:;>0145 16ÏA@;3T/DFDE{<MI0>A/O:;Ô ÍgÍg» ² ¢ ` ¡  3£   ")!  ~¬   " $   "    &  ! ;` ¨T1§E α ˜ ⠘A369;: γ K01&‹ 3616:9;>?9;‚ 0'14BG>BG143T/n9?@;L8K01TB…5 `’ I0>6~E(I?34EŠE(I015…<05HEH14L ˜ αx + βy + γz = 1 ˜ xy + yz + zx = 1 I?365‡3N@;9?FOÏ?@n1N57>A/Q@?EGFO>?9eq>?B (x, y, z) FQet3T9;:M>?9?/„<FDe ˜ α + β + γ + 1 = 2(αβ + βγ + γα) 369;:¹˜?FQ9E(IAFO5 c 36571‹;9;:E(I?3§E@;9?F2ÏA@n157>A/Q@?EGFO>?9 ` 



√ 3 2

2

2

2

™ ÍgÍ Ë » ¢ ` ¡  3£   ")!  ~¬   " $   "    &  ! ;` ¨T1§E a ˜ b ˜?369;: c :;149;>"EH1A˜?365u@;5'@;34/†˜"E(I01C/O1T9;¢¥E(I?5 >Ae;E(I01N5(F2:01T5 BC ˜ CA ˜?3T9;: ` ¨T1XE K;1E(I01R571TLFOJ;1TB…FOLM1§EG1TB‡>Ae n ˜ G B 1T5'Jn1 c EGF„ƒ?1§/„<¹˜0FO9 ˜ E(I01 AB FO90B…36:?FD@;5§˜ h E(I01N34/DEGF„EG@n4ABC :;1gEH>5'FO:;1 BCs ˜?3T9;: r ˜ r ˜03T9;: r E(I01143"4ABC B…36:?FDF;EG> Ar ˜ B ˜ 369;: C ˜ABG1T5'Jn1 c EGF„ƒ?1§/„< ` Ñ 3§Ò ’ I0>6~E(I?34Eeq>?B x > 0 ˜6~1I?3Xƒ?1 h = 2s(s − a)x FDet369;:M>?9?/ <FQe x = r x + s(s − a) >?B x = r ` Ñ K6Ò ’ I0>6~ÎE(I?34E}eq>?B x > 0 ˜?~1RI?3§ƒ?1 h = 2(s − b)(s − c)x FDe-369;:ˆ>?9?/ <MFDe |x − (s − b)(s − c)| >?B x = r ` x=r ÍgÍ ËgË ¢ ` ¡  3£   ")%¤     $         !    !  A` ¨T1§E K;1N369FQ9AEH1T¢41TB~FDE(I n ≥ 2 `}’ @;J0Jn>?571E(I?3§Eueq>?B k = 0 ˜ 1 ˜ . . . ˜ n − 2 ~1I?3Xƒ?1 n   n−2 ` ≡ (−1) (k + 1) (mod n) k ’ I0>6~E(I?34E n FO5‡3NJ0BGFQL81 ` ÍgÍ Ë º ¢ ` ¡  3£   ")%¤     $         !    !  A` ¨T1§E K;138J;>?5'F„EGF„ƒ?1FO9AEG16¢414B c >?9;¢§BG@n149AEmEH> 1 LM>0:?@0/O> 6 `8’ I0>6~E(I?34E 3/n c 3T98K0141 3"J0nBG1T5'5*1T:3T5 a

a

a

b

c

b

2

c

a

2

a

k

1 1 1 + + ··· + a1 a2 ak a1 a2 . . . a k

eq>?B-5*>?LM1:?FQ5(EGFQ9 c E-Jn>?5(FDEGF ƒ?1‡FO9AEG16¢414B…5 ˜ ˜ ˜ ˜T369;:‹;9;:CE(I01LFO9?FQLM34/TƒA34/Q@n1 >Ae k ` ````````````````````````````````````````````````````````````````` Ígº  Ë ` –—>?BGBG1 c EGFO>?9 ` ¡  3£  ‹%® £   ‹l® ¦c¨g®  $ "    ?©! *ª"   3M® ¡  *    ! *±     !   $ "    !  ‡£ " ! †¡  !    !& !     V¡ 3£    " $      "&  !  "` ’ >AFDE a ˜ a ˜ . . . ˜ a :01T5‡9;>?LMK0BG145 B§1T¦ 14/Q5‡Jn>?5(FDEGFDe25 `¹• >?9AEHBG14B}Ï?@n1 X  Y X  ` a a ≤ 





1

2

n

n

ÍgÍ» Ë €` ¡  3£  ‹6® £ " • >?9AEHBG1TB}ÏA@n1

6ª

n+1 2

k

k=1

  

n

n

6© *ª 3® k=1

k j

j=1

 !  "                 

  1 1 1 1 1 ∞ ln 2 − ∞ + + ··· + 1 + + ··· + X X n+1 n+2 2n 2 n = n (2n + 1)(2n + 2) n=1 n=1

}@n14/D/21N1T5HE c 1§E…EG1gƒA34/21§@;B c >?LML@;9;1Ô 5

n

® 3©u`

 

`

‘ gÍ Í»º `{¡  3£  ‹]® £ " ¤ ! ‚®   $ !   $  vª   "&  !  "` ’ >AFDERBG1T5'Jn1 c EGF„ƒ?14L8149AE ˜ 1XE /O1[:;14LFO‚'J1T¦ B…FOLª1§¡ EHBG1A˜/O1PB…3§?9d:A@ c 14B c /21 FO905 c BGF„E}1§EŠ/21B…3§?9M:A@ c 1TB c /O1 sc FQBrc >?905 Rc BGF„E}:A@EHB…FO3T9;¢X/21 ABC `t• >?9AEHBG14B}Ï?@n1 ˜ (s + r + 4Rr)(s + r + 2Rr) ≥ 4Rr(5s + r + 4Rr) 1§E}:1§¦ EG1TBGLFQ9;1TB}ÏA@;369;:FQ/?<3ˆ1T¦ ¢§3T/DFDE§1 ¦ ` ÍgÍ»Í {` ¡  3£  ‹]® £ " ¤ ! ‚®   $ !   $  vª   "&  !  "` ’ >AFDE a ˜ b 1§E c EHBG>AFQ5‡9;>?L8K?BG1T5 B§16¦ 1§/O5‡J;>?5'F„EGFQe25 `¹• >?9AEHBG1TB}ÏA@n1 Y Y

˜ 2(a + b) ≥ (a + s )(bc + s ) >Š@ ¡ s = a + b + c 1§E s = ab + bc + ca ` ÍgÍ» Ð {` ¡  3£  ‹]® £ " ¤ ! ‚®   $ !   $  vª   "&  !  "` ’ >AFDE a ˜ a ˜ . . . ˜ a :01T59;>?LMK0BG145B§16¦ 1§/O5 Jn>?5(FDEGFDe25§˜01§EuFO:;149AEGFQ‹n1TB a 3§ƒ?1 c t ` • >?9AEHBG14B}ÏA@n1 a X X ` a ≥ a a ÍgÍ»¼ `l¡  3£  ‹]® £    !      !   ž¤ * ‹¡  c¡     

 !& U«  ž¤ #  ® ©` • >?9AEHBG1TB}ÏA@n1 2

2

2

2

2

3




1
















1



2

2

2

1

2

n

n+1

1

n

n

3 k

k=1

2 k k+1

k=1

 

tan

2π 6π + 4 sin 13 13

4π + 4 sin 13 5π tan + 4 sin 13

= tan =

π 13 q √ 2π 13 + 2 13 = 13

`

ÍgÍ»g¾ `l¡  3£  ‹]® £    !      !   ž¤ * ‹¡  c¡     

 !& U«  ž¤ #  ® ©` ‰ BG>A@Aƒ?1TB@;9M9;>?L8K?BG1B§16¦ 1§/ 1XE}:;145 Jn>A/ <09 >? LM1T5 f (x) ˜ g(x) 1XE h(x) 3 ¡ c >01§eq‚ ‹ c FO1T9AE…5149AEGF214B…5§˜0:01N57>?B…EG1NÏ?@n1 t √ ˜ √ ˜ 1§E h(t) = √7 ` f (t) = 2 g(t) = 3 ÍgÍ»  '` ¡  3£  ‹b® £ "x¨! !6¡   *     b© ! 1ª"   *         VU® £    *±      ! A` ŒA/DFOLFO9;14B θ :A@85…
 





ÍgÍ»  '` ¡  3£  ‹b® £ "x¨! !6¡   *     b© ! 1ª"   *         VU® £    *±      ! A` ˜65*>AF„E /Q3NI?34@?EH1§@;B36K034FO5'561T¦ 1g5'@;B/21 c>" E41 ¦ BC ` }36905@;9REHBGFQ369;¢X/O1 ’ F AB : AC = 1 : √3 ˜?4ABC ` 5}@;3T9;:R<3"‚GE(‚(FD/16¦ ¢§34/QF„E41Š¦ Ô L8>?9AEHBG14B}ÏA@n1 AD BC AD ≤ ÍgÍg» ² {` ¡  3£  ‹]® £ "    ~¬    $      "&  !  "` ’ >AFDE ˜ 1XE EHBG>AFQ5R9;>?L8K?BG1T5B§16¦ 1§/O5R9;>?9S9?@0/O5:;>?90914¦ 5 `‡• >?9AEHBG1TBÏA@n1ˆ/21 5…<05HE41T¡ LM1 α β 㠘 αx + βy + γz = 1 ˜ xy + yz + zx = 1 Jn>?5'561T¡ :;1C@;9;1N5714@0/O1N57>A/Q@?EGFO>?9 (x, y, z) 5(F¹1§Eu5714@0/O1TLM1T9AEu5(F ˜ α + β + γ + 1 = 2(αβ + βγ + γα) 1§E}:?36905 c 1 c 3T5§˜"EHBG>A@Aƒ?14B c 1§E…EG1N57>A/Q@?EGFO>?9 ` ÍgÍ Ë » {` ¡  3£  ‹]® £ "    ~¬    $      "&  !  "` ’ >AFDE-BG1457J;1 c EGF ƒ?1TLM1T9AE ˜ 1§E /2145/O>?9;¢X@n14@;B…5:01T5 c >" E41T¦ 5 ˜ CA 1§E AB :?@ EHBGFQ369;¢X/O1 ABC `?’ >AF„E s /O1:01TaLbF2‚(J14¦ cBGFQLª1X¡ EHBG1N:A@EHBGFQ369;¢X/O1 ABCBC ˜ /21CB…3X?9:;1C57>?9 c 14B c /21gFQ905 c B…FDE*˜ h /Q3NI?34@?EH1§@;B36K034FO5'561T¦ 1C:?@5*>?LL81XE A 5'@;B/21 c >" rE41 ¦ BC ˜"3TFQ905'F;ÏA@n1 ˜ 1§E /O1T5gB…3§?905:01T5 c 14B c /214541 36FQ905 c B…FDE…5 c >?BHBG1457J;>?9;:03T9AEG5CBG1457J;1 c EGF ƒ?1TLM1T9AE r3T& @ 3r5*>?LL8r1XEG5 A ˜ B 1XE C ` Ñ 3§Ò • >?9AEHBG1TBmÏA@n1J;>A@;B x > 0 ˜t>?9=3 h = 2s(s − a)x 5'F1§E5*1§@0/214L8149AEm5(F x + s(s − a) >A@ x = r ` x=r Ñ K6Ò • >?9AEHBG1TBmÏA@n1J;>A@;B x > 0 ˜—>?93 h = 2(s − b)(s − c)x 5'F1§E5*1§@0/21T‚ |x − (s − b)(s − c)| L8149AEu5'F x = r >A@ x = r ` ÍgÍ ËgË `{¡  3£  ‹]® £ " ¤     $         !
  !  A` ’ >AFDE @;91T9AEGFO1TBŠ3Xƒ?1 c `?š 95'@;J0Jn>?571Ï?@n1CJn>A@;B k = 0 ˜ 1 ˜ . . . ˜ n − 2 >?93 n  n ≥ 2 n−2 ` ≡ (−1) (k + 1) (mod n) • >?9AEHBG1TB}ÏA@n1 n 1T5HE@;9k9;>?L8K?BG1NJ;BG14LFO1TB ` ÍgÍ Ë º {` ¡  3£  ‹]® £ " ¤     $         !
  !  A` ’ >AFDE @;91T9AEGFO1TBJ;>?5'F„EGFQe c >?9;¢§BG@n149AEg3 ¡ 1 L8>0:A@0/2> 6 `;• >?9AEHBG1TBŠÏA@n1 3/n J;14@?E 1§ EHBG141 3"J0BGFQLªn1 ¦ c >?LL81 



√

3 2

2

2

2

a

a

b

c

a

b

2

c

a

2

a

k

1

+

1

+ ··· +

1

Jn>A@;B c 1TBðEG34FO905R149AEGF214B…5RJ;>?5'F„EGFQe2a5:?FQ5(EGaFQ9 c EG5 a ˜ a a˜ . . . ˜ a ˜1XEgEHBG>A@Aƒ?1TBg/O3ƒA34/21§@;B LFQ9?FOL3T/O1:01 k ` 1

2

1

2

k

k



¬ x£ ‚« ž£ 3ª ’£




b 
  




?£ ,£ £

      &  "    &  ! !        !   "                   !             ! !       !   ! ! !"   !       &  !

Ígº» Ë ` 66?‘  ˜  ¡  3£      !/¡ !+!  !  !    

&?£      «  6¬% ! &  &  «†¤ "    ! "& !—`  F ƒ?1T98Jn>?5(FDEGF ƒ?1FQ9AEH1T¢41TB…5 m 369;: n ˜ c >?905(F2:01TBuE(I01RBG143T/—LM>?9?F c J;>A/„?LFQ3T/Q5 P 369;: Q(x) = P b x ~FDE(I=9;>?9;‚(9;16¢§3§EGF„ƒ?1 c >01 c FO1T9AE…5 ` Õ=1 P (x) = ax 36BG1CFO9AEG1TBG145(EG16:FQ9R~I01§E(I014B P 369;: Q 5'34EGFQ5'eD<E(I01 c >?9;:?F„EGF2>?9 X ` P (x)Q(x) = x Ñ 3§Ò.;BG>Tƒ?1gE(I?3§EFQe m 369;: n 3TBG1K0>"E(I>0:;:¹˜AE(I014BG1N3TBG1N9;>5(@ c IJ;>A/„?LFQ3T/Q5 ` Ñ K6Ò=.;BG>Tƒ?1gE(I?3§EFQe m = n ˜"E(I014BG1N36BG19;>5'@ c IJ;>A/„?LFQ3T/Q5 ` Ñqc Ò ’ I0>6~E(I?3§Emeq>?BC143 c I m E(I01TBG1FO53T9=FO9?‹;9?FDEG1M571§Eg>AeƒA34/Q@n145>Ae n eq>?B ~IAF c I E(I01TBG1N:;>41 36FQ5(E5'@ c IJ;>A/„?LFQ3T/Q5 ` Ñ :6Ò=.;BG>Tƒ?1E(I?3§EuE(I01 c >01  c F2149AEG5 FQ9ˆ1*ƒ?14B{<85(@ c IMJ03TFQB‡>Ae-J;>A/„?LFQ3T/Q53TBG1R14F„E(I01TB >?B 1 ` 0 Ñ –t>?LMJ036BG1NJ;BG>0KA/214L 6™6™ FO9ˆŒ?:"~N3TBG:  ` n3TB(K01T34@—˜ • @;BGB…3§< ’;`  /Q36L “ FQ9t˜03T9;: ՈFQ/D/QFQ36L š`  `Ε >?571TB7˜ 1ª"   !   " ¤ *  &   

  !  ˜œ~I014BG1 `Ò m=n=5    !% ‹ ® !‹¬     © ! 1ª"      ,¯ "†¤       !  0` >?B s 3NJn>?5(FDEGF ƒ?1CFO9AEG16¢414B*˜A/21XE R (x) = P x `Š’ FQ9 c 1 ˜ (x − 1)R (x) = x −1 FDEeq>A/D/2>6~N5E(I?34E-E(I01gBG>0>"E…5}>Ae R 3TBG1gBG>0>"EG5u>Ae0@;9?F„E†< `uÌ e s FQ5u>0:;:¹˜TE(I01T9 R I?365 2@;5HE >?9;1BG1T34/tBG>0>"E*˜09036LM14/ < −1 " FQe s FO51*ƒ?1T9t˜?E(I0149834/Q/;E(I01BG>0>"E…5m>Ae R 36BG1 c >?LJ0/O'1 3 ` Ì e FQ53gBG143T/AL8>?9?F c J;>A/„?LFQ3T/6E(I?34E:?F ƒ?FO:;145 ˜4E(I01491T3 c IBG>0>"E>Ae FQ59;1 c 145*‚ 573TPBGFD/„0>"E>Ae0@;9?FDE{< `uÌ e e FO53 c >?LMJ?/241 3NBG>0>"
REŠ>Ae P ˜TE(I01T
9RFDE…5 c >?9 2@n¢§P3§EH1 e FQ5 3T/Q5*>g3BG>0>"E*˜43T9;: P (x) FO5:AF„ƒAFO5(F2KA/21 KT< x−e x−e = x −2(cos a)x+1 ` ÓN149 c 1A˜ e23 c EH>?B…5‡FO9AEG>M3RJ0BG>0:?@ c E >Ae¹E(I01T571RÏ?@;36:0B…3§EGF c Jn>A/ <09;>?LFO34/O5}EG>0¢41§E(I014B ~FDE(I¹˜?FDe¹PFDE}(x) I?365>0:0:8:016¢§BG1T1?˜AE(I01/QFQ9;1T3TBe23 c EH>?B x + 1 `u’ FO9 c 134/Q/0E(I01T571e23 c EG>?B…5 36BG1 5…<0LL81XEHBGF c ˜-57>ˆFQ5 P `MÌ 9P>"E(I01TB}~>?BG:05X˜tFQe P (x) = P a x ˜¹E(I01T9 a = a m

i

n

i

j

i=0

j

j=0

m+n

k

k=0

s

k

s

k=0

s+1

s

s

s

s

s

ai

ai

−ai

−ai

2

m

j

j=0

j

j

m−j

£6 eq>?B j = 0 ˜ . . . ˜ m `š e c >A@;B…5*1A˜AE(I01573TL81N:?FQ5 c @;5'5'FO>?93TJ;J?/QFO1T5uEH>RE(I01>"E(I014B}¢XF ƒ?1T9 Jn>A/ <09;>?LFO34/ Q(x) " FO9J036BðEGF c @0/Q36B7˜?5'FQ9 c 1 P 369;: Q 3TBG1NLM>?9?F c ˜ ` a = a = 1 = b = b Õ=1N3TBG1N9;>6~ÎBG143":6<eq>?BE(I01N3T905H~14B…5 ` Ñ 3XÒ Ì e 369;: 3TBG1K0>"E(Iˆ>0:;:¹˜;E(I01T9 369;: 57I0>A@0/2:ˆI?3§ƒ?1>?9;1RBG1T34/—BG>0>"E 1T3 c I " KA@?E mm+ n FQ5gn1*ƒ?1T9ˆ5*>E(I?3§E}E(I014FQB 5(@;PJ;J;>?5*1T:ˆQJ;BG>0:A@ c E R ~>A@0/O:I?3Xƒ?1 9;>BG1T34/¹BG>0>"E…5§˜A3 c >?9AEHB…3":AF c EGFO>?9 ` Ñ K6Ò Ì e m = n ˜nE(I01T98E(I01 c >01  c F2149AEm>Ae x FO9ME(I01J0BG>0:?@ c E P Q = R ~>A@0/2:K;1 X ˜ 1 = a b ≥ a b +a b = 2 3 c >?9AEHB…3":AF c EGFO>?9 ` Ñqc Ò ’ 1§E n = k(m + 1) ˜~I01TBG1 k FQ53T9Ç36BHK?F„EHB…36B{<©Jn>?5(FDEGF ƒ?1=FQ9AEH1T¢41TB `œÌ e P 369;: Q(x) = P x ˜"E(I0149 P (x) = x 0

m

n

0

m+n

m

m+n

m+n

j m−j

0 m

m 0

j=0

m

k

j

j=0

j(m+1)

j=0

k(m+1)+m

X

P (x)Q(x) =

xj = Rm+n (x)

˜

365 :;145'FQBG16: ` Ñ :6ÒՈF„E(I0>A@?ER/O>?575>Aem¢41T9;14B…3T/DFDE{01  c F2149AEt>Ae x FQ9 P QÒ `’ FQ9 c 1 a = b = 1 ˜ a b = a b =1 ~1M¢41XE P a b = 0 " L8>?BG1T>Tƒ?14B*˜—5'FQ9 c 1RE(I01 c >01  c FO1T9AE…5N>Ae P 3T9;: Q 36BG19;>?9;‚ 9;16¢§3§EGF„ƒ?1Neq>?B j = 1 ˜ . . . ˜ m ˜;34Eu/214365HE‡>?9;1L814L8K01TB‡>AetE(I01J;34FOB (a , b ) FO5 0 eq>?B 3T/D/ j ˜ 1 ≤ j ≤ m `  >6~$/21XE k K;1N3T9FO9AEG16¢414B*˜ 1 ≤ k ≤ m ˜03T9;:5(@;J;J;>?5*1gE(I?34E ÑÒ 1T3 c I a 3T9;:M1T3 c I b ~FDE(I 0 ≤ i ˜ j < k FO5 1§FDE(I014B 0 >?B 1 ` ‰ I0149E(I01 c >01  c F2149AE}>Ae x FO9 P Q FQ5 X X ` a b 1 = a b = a +b + ?

A@;B3T575(@;LMJAEGF2>?9 Ñ Ò(˜-143 c IJ0BG>0:?@ c E FQ9ˆE(I015(@;L FO51§FDE(I014B >?B eq>?B 1T3 c I j ˜ 1 ≤ j ≤ k − 1 `Ì e>?9;1>Ae-E(I014aL bFO5 1 ˜¹FDE eq>A/D/2>6~N5 E(I?34E3T/D/>"E(1I01TB‡EG01TBGLM5 FO9ME(I015(@;L 36BG1 0 " FO9ˆJ;3TB…EGF c @0/O3TB*˜0~1N~>A@0/O:I?3§ƒ?1 a = b = 0 `Nš E(I014Bð~FQ5*1 Ñ FDe 1T3 c I a b FQ5 0ÒH˜—~18¢41XE a + b = 1 369;:¹˜5'FQ9 c 1 a b = 0 ˜>?9;1ˆ>Ae a >?B FQ5 3T9;:E(I01>"E(I014BuFQ5 1 `‰ IA@;5§˜;KT<FQ9;:?@ c EGFO>?9~1N5*1T1CE(I?34E}1T3 c I c >01  c FO1T9AE}>Ae b FQ51§0FDE(I014B >?B ` uF„ƒAF2:AFO9;¢ KT< ˜;~15*1T1RE(I?3§E L@;5(EmI?3Xƒ?1FO9AEG16¢414B P c >01  c FO1T9AE…5 ` 1;@?Eu09;> b c 369M41 3 cR1T16: 1 Ñ FDEŠFOP5 K0>A@;9;:;1T:MKT<E(I0Q1 c >01  c FO1T9AE*˜?903TL81§/„< ˜0>Ae x FQ9 P QÒ " ~I0149 c 1A˜;143 c I b FQ5 14F„E(I01TB 1 >?B 0 ` 1 j=0

m

m

j j

m

m−j j

j=0

0

0

j=0

m

j j

j=1

j

j

j

j

k

k−1

k

j k−j

k

j k−j

k

j=1

j=0

j k−j

k

j k−j

k

k

k

m+n

j

j

j

k

k k

k

£ 
3
   HCT *1!H:*1H-:#];(  @)$  0 )1,  J8O, = <#S@) @J8# ;1  @)0)A 8$G ; 1* IN!8;(;1G#HR '1 )(#BSJ8@)$7O A MUGUIG-H TRL
DR#*-, 
@ U)(X,  J8O,X= #;@  , J8@( @#B#U5RA@)0 = .$

W$JN<W% JL

 

 "!#$&%')(*,+- ./-0/12%%&354687"9":<; (1=12%>:?6@;BAC% D %;BAC%E :F93C'$/ G/HEI HKLNM?OPLNQRHTSMUIWVXO KYM[Z5\  QFHEI HK]^X_`aM?]aIb`]  K/]5YXQFHEI HcIWVH Γ r A J J  IPK/]5d"H/]aI
Q?M?]5HeIP` KEI HT_`aM?]aIP\f`ag `]h`__`\WΓM[IPHT\WMRYH/\i`ag t J \ ZL V G/HEI K/]5Y Γ A B IWVK"I K/]5Y gB`ObC\j`J k=H_`\WM[IPM lHt OPH/K"QC]Z5k H"O \ K/]5Y A  G/HEI AB = mr AC = nr m n J f  p HK]^_5`aMF]aIm`ag YM[noH"OPH/]aIfgROP`k MF\ V`SqIWVK"I P Γ A cot ∠AP B + cot ∠AP J  KrLN`]\WI K]aImgB`OmK/Q[Qs\WZL Vt_`aM?]aIP\ K/]5YuY5H&INH"ONkMF]5HIWVaM?\LW`]\NIPK/]aIblaK"Q?ZHMF]IPH/COPk=\ P `ag K/]5Y  m n {% |r- D w~%;~6€o3aw</%& $Ww<'a9a'q'5%j:7x":[wR9"6=‚ƒ9 9„…9"w<(6 v 0EwR1w[:R9/t$&/: x;Fw<3$y(*z.E$s} %† E3C%j‡Exa%>:F9"ˆ
9/3s'u;BAC%!sW"!#$E%&> p HEI   K/]5Y p MF]5LNH8IWVH‰K]5d&QFH α = ∠CAP β = ∠AP B γ = ∠AP C H&I…ShHH"]2KtL V`OPYŠK]5Y=IPK/]5d"H/]aIcH/‹Z5K"QF\ VK"Q?g
IWVHrK]5d&QFHrK"IeIWVHLWH"]aINOPH\WZ IPH/]5YHY J ^tIWVHXL V`OPY  ShHh\ŒHH IWVK"I J mŽ VH ku`OPH/`/lH"O  Jp MF]5HtG"K&SK/_5_Q?MFHYXIN` AP = 2r sin α  ∠ABP = α − β d&M lH/\ 4AP B

ShVH/]5LNH 

sin(α − β) sin β sin α cos β − sin β cos α = = mr 2r sin α 2r sin α

p MFkM?Q?KO QU^  \ M?]5LWH ^MFH"QFY\



2 = cot β − cot α m  p

∠ACP = 180◦ − α − γ



IWVH

~>‘

MF]5H8G"KES’K__Q[MRH/Y2IP`

4AP C

sin γ sin(180◦ − α − γ) sin(α + γ) sin α cos γ + sin γ cos α = = = nr 2r sin α 2r sin α 2r sin α

ShVH/]5LNH 

2 = cot γ + cot α n



 ~ >‘

“ M?]K/Q[QU^  KY5Y ~>‘ K]5Y ~ >‘ IN`”` I K/M?] 2  ShVaMRL VM?\fKLW`]\NIPK/]aI  2 + cot β + cot γ = J m n K\TLNQFK"MFk=HY  v…v 0EwR1w[:R9/t$&/: x;Fw<3$€(* † 9a{ „…:R9a–•o3s%o„P— 3e/{* 6=˜5w?™›š
9&!w<'$ 6t‚œR62o0&žbˆ 93C' 'Ÿ a1zx/3s'0a *":F936fšsw?™9"6i¡/9;~&wF95  K\MF]Š\Œ`aQ?ZIPMF`]z£ ‘ K]5YuZ5\ŒHYaMFOPH/LPINH/Y2YM?\WI K]5LNH/\  ¢ Hr\ŒVK/Q[QsQRH&I α = ∠CAP  G/HEI 0 K]5Y 0 H€IWVHygBHH&IŠ`agIWVHƒ_5H/OP_H"]5YMFLNZQFK/O \ŠIN` gROP`k K/]5Y B CŽ J PA B C #  OPH/\ _H/LPIPMUlHEQU^ VH"] 

P A = 2r sin α  P B 0 = P A + AB 0 = 2r sin α + mr cos α P C 0 = P A − AC 0 = 2r sin α − nr cos α

BB 0 = mr sin α CC 0 = nr sin α



£" ‰ IA@;5X˜ cot ∠AP B + cot ∠AP C

= = =

P B0 P C0 + BB 0 CC 0 2r sin α + mr cos α 2r sin α − nr cos α + mr sin α nr sin α m+n 2 mn

`


3
   HCT *1!H:*1H-:#];(  @)$  0)1,  J8O, = <#S@) @J8# ;1  @)0)A 8$G ; 1* IN!8;(;1G#6HR '( )1# SJ8@)$7O A S1H-!$ 5 0 *%G-;(; !I H:5!EPR#OL *:LG(2 ,
,  S1 J?J8@J8, # 9 @<

@1 
,X # 5<W$@<, )A]$8M ::HDIX5<#[ , N 'J8, 51 '(= .$ J?)5(= @(= U)(,X J8O,X= # <W-
, )1#  R# U5RA!M%5-ID;(X5 /CLDE4G1X5<#P9 @JN@K Q,  , &.5(78.$1
#C= .0 )1O#"0JN  7? AS1H-!$ 5 0 9%G-H " !H  \ MUOI !\ # 8G(5 
\ @JN
0'( JN# M-J8,X Q&(F 4\ JN 1 @<#5W$@, )(A"RG-DS1S H:G-T PL+/!E%;(;# P@2+ )1#]0I # %5Q:A5!;1G1  ;(8/R (\ #=?'( )0= #5@J8@
O $

X Q&( Q#5@J8@
O #*%N)1, @ @)0  JQ Q&1(Q, )1@






 =

ua5"  "!#$&%'8(*Š+- ./-
0/12%%&3546o7"9":<; (1=12%>:?6e;BAs% D %;BA %E :F93C'$/ G/HEI HhIWVHrYM?Kk=HEIPH/OT`ag#Kr\jH"ktMFLNMFOPLNQFH  G/HEI HrK/]^Š_5`aMF]aI`]=IWVH AB J D J  K/]5YuQFHEI IPK/]5d"H/]aIeIP` KEI K/]5YuQ ^M?]5dX`]tIWVHX\ŒK/kuHh\ MFY5HXΓ`ag K\ HhIWVH ktMF Y W_`aM?]aIi`aΓg B
Ž VH\ŒHdEk=H/]aI \ K/]5Y M?]aINH"AB O \jH/LPI ΓgB`OsIWVH\jH/LWC `]5JY IPM?kuH K"ITIWVH_`aM?]aIP\ BDK]5Y  OPH/\ _H/LPIPMUlAC HEQU^  £Bg ADK]5Y K/OPHrΓIWVH_5OP` ~H/LPIPMR`]\`]aIP` K L M N   `ag K]5Y OPH"\Œ_5HLPIPM lH"Q ^ \jV`SqIWVK"I KL A B M L = LK = KN 0/:…x;Fw~38(*t%;<%&  o - r6i˜5w</:F9zo3aw</%& $Ww…;R*/6i¡/9‚€wR 9'a9" 6 žm6 #0>ž G/HEI H‰IWVH€\jH/LW`]5Y›_5`aMF]aI=`ag MF]aIPH/O \ŒHLPIPMF`P]z`aJ goIWVHrQ?M?]5H K]5YuIWVH D NB M LNMFOPLNQFH  Ž VH"] ∠BP L = ∠BAL K]5Y Γ \j`ŠIWVKEIcIWVH L ∠BP K = ∠BAK INOPM?K]5d&QFH/\ K/]5Y K/OPH‰\ M? k  LP N DAB    M?Q?KO H/]5LNH MF\2IWVH}kuH/YM?K] `ag K C  H/LNK"PZ5\jK X H MF\cIWVHtk=HYaMFK/] 4LP N J #Ž VaZ5\  AC  `ag 4DAB LK = KN N  G/HEI H IWVHktMF Y W_`aM?]aI#`ag XJ LK  p MF]5LNH  IWVH"] A B OL = OK OX LK O  ] IWVH–`IWVH"O2VK]5Y  M?\2IWV⊥ H}k MRY  O _`aM?]aIf`ag K]5Y M?\bKcINO K/_H  AB ` aMRYuSM[IWV8 O MRd"VIK]5ABN d&QFH/\K"M I K]5Y ShVH/]5LNH  MF\hK/Q?\j`2IWVH2kMRM Y _5`aMF]aI”N`ag  Γ  £RI XgB`aQ?QF`S”\IWVKEI  MVaN P Sh MRL VtLW`kt_QFHEIPH/\mIWVH_5OPM ``aLg  = KN ... ..............

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

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

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

 !"$#&%('*),+.-0/21435 ),1467-0/483:9<;2=>?@9BA A:C$D E4FG"H.A?@):A !$HJI!A?@>KLAD MGNP>BO 14C4QR1S 8T>G3 6 U-R+B9BV ?&/<MBDW$"H.XDWYW'Z[>$AH.A
\X"?5>A$H.AJ\X"$?^]$JMGD A_AM<#a`b"Hcd"dEBGdD MBAeN T;fg]B0T2]B8Th2? 3B"$%(AM<""i/MGD!"H.XD!Y '?PC$A$MBA$H.?j3B"$%(AM<$M(Ng),+.9L`7143k]B0SJ8+3B3G1(?:TR4FB"MG?jC(H.AM49<; ]1h3G34;0ST;2>0n;R?^+lo ]2op1(qD  D!rCB"HlH.A$H.D ?&=4Ae  AB#4 D!#$?*>eKLAeD M(NjT2+.9$8T2n;s]B8T2T;2>4;t9$ ),u;>e? /MGD!"H.XD!Y '[<>B"dD  !"d?L>G"D  W"d?<>KLAD MGNR9L`7+u92/2T S(+!>e?) D J4F$H.DL>G4FBY v<"HlMw>GY AGY"w/<MBDW$"H.XDWYW'e?GxeeKh D MB?

£*  R #U5RA-!M%5-ID;(X5 /CL0E4G1X5<#9 @JN@K Q, , &.[517.01
#$!= .$ ) O#"0JN  7? A:H:8$!!H:E ?LUG155# H0@)$78.$ M1@
 90 JN O#!#U5RA D "L:GR:9%G H0#"0J8@)$ 6M-J8@<, J8, Q#R!* A9 \ $;9 /RG
- T-\ # *-, & H0@W-,X O#  # %5Q:A 5!;1G1  ;(8/R (\ #=8'1 )$= #V5@J8@JO  $

X Q&( Q#V5@J8@
O # *%N)1, @ @)0  J Q Q&((, ) @A CG-; 5 $; :51*%G H"$# *$@O,X Q# T #U5RA G-E %E 5&TU;Q!# H0, &@#;@(=Q, @A ERD8G ; I5! #R=?'( )0= #4I@<, W% Q,-C2+ J8, 7@)V517.01
#PI@<, W% Q, #4IQ@, Y!@)AINI$ Z(9$CD!H0#D$2]\@ )$ ^ = , # HR2@)(, @AU@)0 = .$ !W$JN<W% JL







r/5 a E W"!#$E%/'u(*‚€wRAf9E{ :B*˜o%E3C„<‡>%&6f˜C 9/$&&6fš#1=93awR9  ShVH"OPH M?\fIWVHktK"INO M  K/Q[Q5`agaShV`\jHH/]aINO MRH"\mK/OPH  G/HEI  A J ∈ Mn×n ( )  z  p     Y5H"]5`INH ^ H&I IWV1H K]5YzQFHEI K/]5YzJgB`O b ∈  B = bJ k = 1 2 ... n Ak J k=KEINOPM ”` IPK"MF]5H/YhgROP`k ^XOPH"_Q?KLNM?]5dH"KL VXH"QFH/k=H/]aI#M?]XOP`S SM[IWVIWVH laK/Q[ZH  A k b J J 5OP`/lHIWVK"I 2

det(A + B) det(A − B) = (det A) −

n X

det Ak

k=1



!2

0"wF1w?:F9$E/:…x;Fw~3$(*8‚€w<„PAC%>: ˜C9;F9"w[:?:~%&6cš#x%&36P9/3s„N%>ˆ Aaw[!s
xj;Fw?$Œ6T‚€wF$ $Ex/ w 0x;BAs%&N3}0;R9;<%–
3w~/%E…$ w<;B*"6e.E"!:?wF36‚ i6o0&žbˆ 93C'}|h%E3N*yšsw<„…9W'56”‚–%'™/9 $ C/:[:<%>™% b D 6i˜CW4:B*/36 D 6To0&ž Ÿ /%& & ( )  QRH&I “`OrK]^ Y5H"]5`INHzIWVH OP`S `ag K]5Y@QRH&I K ∈ Mn×n i K  ( )  K (i) ¢ HhVK&lH (i)

U = (b b . . . b) = B

= det A(1) + B (1) , A(2) + B (2) , . . . , A(n) + B (n)
 = det A(1) + U, A(2) + U, . . . , A(n) + U

det(A + B)

p MF]5LNH”Kh\WM  H

n

i = 1 2 ... n

Y5H&INH"ONkMF]K/]aImMF\ K/]

n




WQ[MF]5H"KOfgFZ5]5LPIPMF`]u`agCMUIP\ OP`S”\  ShH”` PI K"MF] J 



det(A + B) = det A(1) , A(2) , . . . , A(n) + det U, A(2) , . . . , A(n)

 + det A(1) , U, . . . , A(n) + · · · + det A(1) , A(2) , . . . , U

ShVH/OPH2K8]Z5k "H OX`agmIPH/OPk=\  H"M?]5d2YHEIPH/OPktM?]K]aI \hSMUIWVzI…Sh`€OP`S”\rH‹aZ5K/QbIP` J VKElHlaK/]MF\ŒVHY J iŽ aV Z5\  det(A + B) = det(A) +

£~]IWVH\ŒK/kuHcS”KE^  SMUIWV

−U

MF]\NINH"KYt`ag

det Ak

n X

det Ak

 ShH”` k=1 I K/M?] J U

det(A − B) = det(A) −

Ž VH"OPH"gB`OPH 

n X







k=1

det(A + B) det(A − B) = (det A)2 −



U

n X

k=1

det Ak

!2



   ^W$"$#*%('x$;1 `; U0T2n?uL$H
Y Ae!"X?U6@) ?/>RNU`71B6bTf,T2+l9$0T2n;R?8),"$#$E$AH14$"H. 9L W"dEB" 92/R67f ?h]LHJB $ 'MB?467f4?/> "$Il$M<#0!FBY D!$M Nx; 143 >49L`b34;2>B]14T-L?4]LA$'$XDW#4"d?467f4?/>d2N ( ) A$ML#pY v<" (KLHJeK$")Ho



£ Ígº»¼ ` T"A‘  ˜ £ ¡  3£    G¤  ® * $ !  !  ¤ "  ! ¨ ! " $  vª  &  ! ;` ¨T1§E K01R3 c >?9"ƒ?1'3J;>A/„?9t˜n3T9;:M/21XE K01R369"<MFQ9AEH14BGFO>?B‡Jn>AFQ9AE >Ae—E(I01RJ;>A/„A?A9 ` · · >?· BAk = 1 ˜ 2 ˜ . . . ˜ n ˜;/O1§E G K;1NE(I01 Pc 149AEHBG>AF2:ˆ>AetE(I01RJn>A/ ?9 Ñ E(I01ªJn>A/ ?9”>0K"EG34FO9;1T:iKT4ƒ?FQ9;¢ƒ?1TBðEH143 A e2BG>?LA A· · A· A · · · AA Ò `uÌ ·e ·B· A FQ5uE(I01NBG1 ¹1 c EGF2>?9M>Ae G E(I?BG>A@n¢4IRE(I01NJn>AFQ9AE P ˜AJ;BG>4Aƒ?1 E(I?34EŠE(I01/DFO9;145 A B 3TBG1 c >?9 c @;BHBG149AEueq>?B i = 1 ˜ 2 ˜ . . . ˜ n ` o `      ! ?¤     $        !
  !   ! >?B k = 1 ˜ 2 ˜ . . . ˜ n ˜"~1NI?3Xƒ?1 X− 369;: 2→ − → → − − → − → (n − 1)G = A P = G +B " 1

2

n

k

1

2

k−1

1

k+1

2

n

n

i

k

k

k

i

k

I01T9 c 1?˜

j

k

k

j6=k

− → X− → → − (n − 1)Bk + Aj = 2(n − 1) P

‰ I014BG14eq>?BG1A˜

`

j6=k

n X − → − → → − − → (n − 1)Bk − Ak = 2(n − 1) P − Aj

3CƒA34/Q@n1CFO9;:01TJ;1T9;:01T9AE‡>Ae k `  6> ~›:014‹;9;1 U KT<

˜

j=1

−−→ n − 1 −−−→ A1 U = A1 B1 n−2

`‰ I01T9

→ − − → − → − → − → (n − 2) U = (n − 1)B1 − A1 = (n − 1)Bk − Ak

eq>?Bg3T/D/ k ˜ 1 ≤ k ≤ n `8‰ IA@;5§˜—E(I01MJn>AFQ9AE U /QFO1T5N>?9ª143 c I/QFQ9;1 A B ˜3T9;:=E(I01 BG1T5(@0/DEeq>A/D/2>6~N5 `   &  4` `[‰ I014BG18FO59;>[9;161T:ªeq>?B P EG>ªK;18FQ9AEH14BGFO>?BgEG>PE(I018Jn>A/ ?9t˜9;>?BCeq>?BgE(I01 Jn>A/ ?9EG>K01 c >?9"ƒ?'1 3 ` ;`uÌ e G :01T9;>"EG1T5uE(I01 c 1T9AEHBG>AFO:M>AenE(I01Jn>A/ ?9 A A · · · A ˜"E(I0149 X− → → − → − ` → − → − A = 2(n − 1) P − n G (n − 2) U = 2(n − 1) P − k

1

2

k

n

n

j

‰ IAFQ5 57I0>6~N5uE(I?34E U 34/O57>R/QFO1T5 >?9E(I01C/QFQ9;1 P G ` oðo `    !€¡      !   $   l© ! 1ª"   * ¦ %¤     #  ©  # ! Õ=1N~FD/Q/-¢41T9;14B…3T/DF 0'1E(I01BG145'@0/„E}EH> ˜t3 c /2>?5716:8Jn>A/ ?903T/ J;3§E(IFQ9 n‚7:AFOLM1T905(F2>?9034/¹57J03 c 1 ` ¨T1§E G K;Z1CE(=I01 Ac 1T9AAEHBG>A·FO·:8· >AAe—34/Q/ n Jn>AFQ9AEG5 `‰ I0149 G /QFO1T5 >?9E(I01C/QFQ9;1N5716¢§LM1T9AE A G 5'@ c IRE(I?3§E GG : GA = 1 : n − 1 ` ‰ I014BG14eq>?BG1A˜¹E(I01J;>A/„?9034/J034E(I FO5m5'FQLFD/O3TB}EH> ~F„E(I 1T3 c I5(F2:01 G G K014FQ9;¢J036B…34/Q/O14/0EG>RE(I0Z1N5(=F2:01 GAGA · · · Geq>?Bu34/Q/ i ˜ 1 ≤ i ≤Zn ` j=1

1

i

i

i

0

i

i+1

2

n

i

1

2

i

i+1

n

£6£ a eQEH14BmBG1 ¹1 c EGF2>?9ˆE(I?BG>A@n¢4IˆE(I01J;>AFO9AE ˜n~15'FQLFD/O3TBG/ <=I?3§ƒ?1RE(I01Jn>A/ ?903T/ J;3§E(I Z = B B · · · B c >?9;¢§B…@n1T9AE}EG> ZP~F„E(Iˆ143 c IM5(F2:01 G G K014FQ9;¢J;3TB(‚ 3T/D/21§/0EH>RE(I015'FO:;1 B B eq>?Bu34/Q/ i ˜ 1 ≤ i ≤ n ` ÓN149 c 1A˜ Z 3T9;: Z 3TBG1R5(FOLFQ/Q36BŠ~F„E(I81T3 c IM5'FO:;1 B B K014FQ9;¢J036B…34/Q/O14/nEG> ` 369;:MI?3Xƒ?FQ9;¢g/2149;¢¥E(I E(I?34E}>Ae;E(I01N5(F2:01 A A ‰ I014BG14eq>?BG1A˜TE(I014BG1mFQ53 c 1/(n 149AEHBG1C−>Ae01)5(FOLFQ/DFDEG@n:01gK01§Eð~1T1T9AE(I0A1m5'FQLFD/O3TB-Jn>A/ ?903T/ J;3§E(I?5 Z 3T9;: Z ` 00

1

2

i

0

i

0

n

i

i+1 00

i

i+1

00


 






i+1

i


 
X  M%:5$IC;1 5 /CL6E4G1X5<# 91@JNQ@K1 Q,X 
'(= , )  A 5 G\  L-GUI%#0)1,  J8O, @1 Q
Z('
, @<#% @J8@(7@<,X *$@O,X Q#U)T #!U5RA0@)0 = .$ !W$JN<W% J8L 1!W%, )$=   '1=U= . @(=U= .0 = J?2 7? )0= JN,X  @!W%
<&() 7O )$= JN,  %= .$  J = ,X7? !YD, = .6 -'@<
0YP Q, &.$= O# , , D= .0 7? )0= JN,X 7O )$= JN,  R= .$ 4 =R n O, O#-
, K 2+ = @<
0YD, JN L





i+1

i+1

4, &.F5(78.$1
#+= .0 )1O# "0JN  7O

#9$ )0
'( Q
@A DG ;51$;15(*%G-H("$#

R%, := .0@ D2+W% , &1
<'1&( ') ?@



”

r/5 a E W"!#$E%/'u(*‚€wRAf9E{ :B*˜o%E3C„<‡>%&6f˜C 9/$&&6fš#1=93awR9 G/HEI H”K”_`\WM[IPM lH MF]aIPHd"H"OmK/]5Y K”OPH/K"Q]Z5k H/O  5OP`/lHIWVK"I J nJ x 1 bxc + x + n 

2

2

n−1 + ··· + x + n 

2

= bnxc2

 ShVH"OPH M?gsK/]5Y=`]QU^rM[g YH/]5`IPH/\mIWVHhdEOPH"K"IPH/\NIbMF]aIPHd"H"ObgFZ5]5LPIPMR`]  bnxc = bxc b·c 0/:…x;Fw~38(* Ÿ 'a1zx3C'0 *E:R9/36išswF™/9/6i¡/9;<>wR9"6
12'ws%'u(*t;BAs%”%'w<;<> Ž VH _OP` QFH/k  K/\e\NIPKEINH/Y  MF\bM?]5LW`OPOPHLPI  ]H/K/\ ^LN`aZ5]aINH"OPH /K/k=_QRHMF\  x = 12  “MFO \NIiShJ H]5`INHIWVK"I `IWV J n=2     ~>‘ 1 2 n−1 2 2 2 bxc + x +

K]5Y

n

+ ··· + x +

= bnxc

n

~ >‘  ¢ H\j`aQ lH KOPHhH/‹Z5KEIPMR`]\eMF]rIWVHlaK/OPM?K QRH"\ K]5Y  ShVH/OPH K]5Y J n x n∈ +  x∈ IWVH/\ŒHXI…Sh`2H‹aZ5K"IPMF`]\ K]5Y8LN`k=_KOPHXIWVH\jH&IP\ `agi\j`aQ[ZIPMR`]\ £~]ŠIWVH_5OP`LNH/\ \  ShH H/\NIPK Q[MF\ŒVXIWVHhLN`]5YMUIPMR`]\eZ5]5YH/OiShVaMRL VXIWVHI…Sh`tH/‹Z5KEIPMR`]\ KOPH”H‹aZMUlaK"QRH"]aI  J G/HEI @Ž VH"]  \j`zIWVKEI  K/]5Y x0 = x − bxc 0 ≤ x0 < 1 0 ≤ nx0 < n  IWVH/OPHEgB`OPH     bnx0 c ∈ {0 1 . . . n − 1} £~]€_5K/O IPMFLNZQFK/O  M[gmK]5Y€`]Q ^zM[g M?gmK]5Y€`]Q ^zM[g bnx0 c = m m ≤ nx0 < m + 1  m/n ≤     IWVH"]  `xS 0  <M[g (m + 1)/n 0 bnxc = bxc

k ∈ {0 1 . . . n − bnx c − 1}

0 ≤ =

\j`hIWVK"I

j

k x0 + n

k

k

≤ x0 +

k

≤ x0 +

n n nx0 − bnx0 c + n − 1

=0



n

n − bnx0 c − 1

<

n 1+n−1 n

= 1



£6™ ’ FOLFQ/Q36B…/„<¹˜AFDe k ∈ {n − bnx c ˜ n − bnx c + 1 ˜ . . . ˜ n − 1} ˜AE(I0149 nx + n − bnx c k n−1 ˜ 1 ≤ ≤ x + ≤ x + < 2 n n n j k 5*>E(I?34E x + k = 1 ` ‰ IA@;5X˜ n 0

0

0

0

0

0

0



k x+ n



   k k 0 bxc + x + = bxc + x + n n ( bxc, k ∈ {0 1 . . . n − bnx0 c − 1} bxc + 1, k ∈ {n − bnx0 c n − bnx0 c + 1 . . . n − 1}



=

0

FDe ˜ ˜ ˜ FDe ˜ ÓN149 c 1A˜AE(I01C/21§eQEu5'FO:;1N>Ae—1TÏ?@;3§EGF2>?9 Ñ ÒFQ5 =

˜

˜ ˜



2 n − bnx0 c bxc2 + bnx0 c bxc + 1 = nbxc2 + 2bnx0 cbxc + bnx0 c 
 bnxc = n bxc + x0 = nbxc + bnx0 c

’ FO9 c 1

˜AF„EG5‡B…F2¢4I"Eu5'FO:;1CFQ5 ` bnxc = n bxc + 2nbxcbnx c + bnx c ‰ IA@;5X˜;1TÏ?@;3§EGF2>?9 Ñ Ò c 369MK;1m~NBGF„E…EH149365 2

2

2

0

`

`

0 2

` ‰ IAFQ5‡FQ5uEHBG@n1FDe n = 1 `Š’ @;J;J;>?5*1 n ≥ 2 `Š’ FQ9 c 1 bnx c − bnx c ≥ 0 ˜"E(I0149
` (n − 1)bxc nbxc + 2bnx c ≤ 0 Ì e bxc ≤ −2 ˜6E(I01T9 nbxc ≤ −2n 3T9;: 2bnx c ≤ 2n − 2 ˜?5*>E(I?34E
` (n − 1)bxc nbxc + 2bnx c > 0 ’ FOLFQ/Q36B…/„<¹˜AFDe bxc ≥ 1 ˜"E(I0149
` (n − 1)bxc nbxc + 2bnx c > 0 Ì e bxc = −1 ˜6E(I01T9M16ÏA@;34EGFO>?9 Ñ ÒŠK;1 c >?LM1T5 ˜ (n − 1)n − (2n − 1)bnx c + bnx c = 0 >?B

` (n − 1) − bnx c n − bnx c = 0 ’ FO9 c 1 n − bnx c > 0 ˜6E(I01T9 bnx c = n − 1 ˜T~IAF c I¢XF„ƒ?145 (n − 1)/n ≤ x < 1 ˜ >?B*˜  ˜
` x = bxc + x ∈ − 0 Ì e bxc = 0 ˜"E(I01T9M16ÏA@;34EGFO>?9 Ñ ÒŠK;1 c >?LM1T5 ˜ bnx c − bnx c = 0 (n − 1)nbxc2 + 2(n − 1)bnx0 cbxc + bnx0 c2 − bnx0 c = 0 0 2

0

0

0

0

0

0

0 2

0

0

0

0

0

1 n

0

0 2

0

£"‘ ~IAF c IªI?36J0Jn14905FQe

~ bnx I01T9 cc 1?=˜ 0 1/n ≤ x < 2/n " 0

>?B

bnx0 c = 1

0

`[‰ IAFO5R¢XF„ƒ?145

 x = bxc + x0 ∈ 0

0 ≤ x0 < 1/n

˜
`

>?B

2 n

Ì 9M5(@;LML36B{<¹˜;1TÏ?@;3§EGF2>?9 Ñ ŠÒ I?3T5 57>A/Q@?EGFO>?905 369;: (n, x) | n ∈ \ {1} ˜ x ∈ − ˜
` {(1, x) | x ∈ $ } 
 ’ FO9 c 1 ˜}16ÏA@;34EGFO>?9 Ñ{ Ò c 369©K;1 ~NBGF„E…EH149365 bnxc = n bxc + x = nbxc + bnx c 1 n

+

0

2 n

0

nbxc + bnx0 c = bxc

>?B

bnx0 c = (1 − n)bxc

`

‰ IAFQ5FO5 c /O1T3TBG/ <ªEHB…@n1eq>?B n = 1 `”’ @;J0Jn>?571 n ≥ 2 `”’ FO9 c 1 bnx c ≥ 0 ˜E(I01T9 ˜ ~IAFQ/O1 1 − n < 0 ˜g57>©E(I?34E bxc ≤ 0 ` Ì e bxc ≤ −2 ˜ (1 ≥ 0 E(I01T−9 (1n)bxc ˜‡~IAFD/21 bnx c ≤ (n − 1) `›‰ I01TBG1§eq>?BG1?˜ bxc c 3T9ª>?9?/„<[−K01 n)bxc  >6~−˜ 1) >0 ?B −1≥ ` 2(n FOLJ0/DF2145CE(I?3§E bnx c = 0 ˜t~IAF c I[¢XF ƒ?1T5 = 0
` ¨§F “ 1Xbxc ˜  ~ Q F * 5 A 1 ˜ FQLMJ?/QFO1T5E(I?3§E c = n − 1 ˜ `ŠÌ 95'=@;L−1LM3TB{?9 bnx Ñ{ җI?3T5Š57>A/Q@?EGFO>?905 ~x IA=F c INbxc¢XF„ƒ?14+5 xx =∈bxc0 + x ∈ − ˜ 0
bxc 3T9;: (n, x) | n ∈ \ {1} ˜ x ∈ − ˜
` {(1, x) | x ∈ R} –—>?905716ÏA@n1T9AEG/ ?905 Ñ Ò3T9;: Ñ{ Ò3TBG1€9;>"Eˆ1TÏ?@0F ƒA3T/O1T9AE*˜gFO9b¢4149;1TB…34/ ` ‰ I01*<3TBG11TÏ?@0F ƒA3T/O1T9AE@;9;:;14BŠE(I01N36:;:AFDEGFO>?903T/n365'5'@;LJ?EGFO>?9 x ∈/  ˜
eq>?B n ≥ 2 ` 0

0

0

1 n

0

0

1 n

0

1 n

+

1 n



1 n

2 n


 
X   5(G-S1/:G0I H%5(;DD4"R 1\ # 0)1,  J8O, =  B5@J8@JO #65@J8@JO #[*%N)1, @ @)0  JQ Q&((, ) @A 8$G ;]*1IN!8;(;1G#HR '( )1#+SJ8@)07? A $MF:RHDI( 5#  , N 'J8,51 '(= .$ J?) 5(= @(=

U)(,X J8O,X= # <W-
, )(#4 %#BU5RAC@)$ H:?$DH:EFOLCG155#H0@)07.0 M1@
690 JN O#P:#BU54LI<.0 JN

YP JN = . JN  !, )07?2BW-
X =X 4
X'1= ,X)1D<'1 2 , =8=X  $L


 


€s f" ‚ywFP9a'a9/6žm6T#0>ž

W"!#$E%/'}(*y%;~%E o- r6”˜5w~/:R9
3w~/%E…$ w<;B*"6”¡/9

  K]5Y /G HEIXK€LW`]lHƒ‹aZ5KYOPM[QFKEINH"O K/Q VKElH8MUIP\t\WMRYH/\ AP QC AP P Q    IPK/]5d"H/]aI
IN`KcktM?]5`OiLNM?OPLNZQFK/OoKOPL K"IsIWVHc_5`aMF]aI \ K]5Y OPH/\ _H/LPIPMUlQC HEQU^  ABC  A B C G/HEI HTIWVHc_OP` ~HLPIPMF`]r`ag `]aIP` G/HEI#Kc\jH"ktMFLNMFOPLNQFH SMUIWV K\bYM?Kk=HEIPH/O PQ  SM[IWV B H&I…ShAC  LNZI E J K"I K/]5Y HH"] K]5Y AC ¢ M[IWV`aHZIZ5\WMF]5KdtINO MRd"`]5H `k=J HEINO~^  _OP`/A lH=IWVK"I K MF\ŒHLPI \ K]5Y‰IWVKEI BE J ∠P EQ  M?\jH/LPIP\ PH J ∠AP B 0/:…x;Fw~38(* Ÿ 'a1zx3C'0 *E:R9/36išswF™/9/6i¡/9;<>wR9  G/H&I ¢ H8\ŒV`S IWVK"I HŠIWVH2gBHH&IX`ag IWVH 0 K/]5Y 0 ∠BEP = ∠BEQ #  p _H"ON_5H/]5YaMRLNZQ?KO \#gROP`k K/]5Y IP` M?]5LWHeIWVHTPL V`OPY Q k=J K H/\iH‹aZ5K/QK]5d&QFH/\ P

Q

AC

AC

£ ~FDE(IˆE(I01EG3T9;¢41T9AE…5 P A 369;: QC 34EFDE…5N1T9;:0‚'J;>AFO9AE…5§˜tE(I01MB…F2¢4I"EEHBGFQ369;¢X/O1T5 P AP 369;: QCQ 36BG15'FQLFD/O3TB*˜A57>RE(I?3§E PP PA ` = QQ QC ;@?EuEG3T9;¢41T9AE…5e2BG>?Lž3T9ˆ'1 34EH14BH9034/—J;>AFO9AE I?3Xƒ?116ÏA@;3T/¹/2149;¢¥E(I?5§˜;9036LM14/ < PA = PB 369;: QC = QB " c >?905716ÏA@n1T9AEG/ AC ˜E(I01*<©36BG1 J;3TB…3T/D/21§/?EH>>?9;13T9;>"PE(I0P1TBŠ369;BE : c @?EE(I01mEHB…36905…ƒ?14B…5734/O5 AC 3T9;: P Q J0BG>?Jn>?BðEGF2>?9034/Q/ < " 57J;1 c FD‹ c 3T/D/„< PB P E ` Ñ{ Ò = Œ0ÏA@;34EGFO>?905 Ñ Ò369;: Ñq ÒCFOLJ0/ <SE(I?QB 3§ENE(I01PBGFOQ¢4I"EEEHB…FO3T9;¢X/2145 P EP 3T9;: QEQ 36BG1 5'FQLFD/O3TB " ~I01T9 c 1?˜ ∠P EP = ∠QEQ ˜3T9;:[E(I014FQB c >?LJ0/O1TLM1T9AE…5§˜ ∠BEP 3T9;: 36BG1N16ÏA@;3T/{˜?365 :01T5(FOBG1T: ` ∠BEQ ‰ >g57I0>6~©E(I?3§E P H KAFO571 c E…5 ∠AP B ˜*~1u/O1§E M K01uE(I01}LFO:;‚(Jn>AFQ9AE—>Ae P Q 3T9;: K;1CE(I01Jn>AFQ9AEŠ~I014BG1E(I01N/DFO9;1CE(I?BG>A@n¢4I M J036B…34/Q/O14/;EG> P A LM161XEG5 AC `‰ I01T9 H `  ‰ IAFQ5 c /O34FOLœ~N3T5‡L3":01g~F„E(I0>A@?Eu3T9"< 2@;5(EGFD‹ c 34EGFO>?9t˜ M H = (P A + QC) K?@?ECE(IAFO51T:?F„EH>?BCe234FQ/Q5NEH>[5*1T1~I6<ªF„ENLF2¢4I"EK01=1§FDE(I014B>0KTƒAF2>A@;5>?Bm~1§/Q/ “ 9;>6~N9 " I0>6~1*ƒ?1TB7˜FDEFO51T3T5'FD/„<=ƒ?14BGFD‹¹1T:—¨T1§E C K018E(I01Jn>AFQ9AEC~I014BG18E(I01/DFO9;1ME(I?BG>A@n¢4I J036B…34/Q/O14/-EH> P A LM161XEG5 AC `8‰ I01T9 AP QC FQ53EHB…36J;"1 0'>AFO:ˆ~I0>?571MLF2:0‚(/DFO9;1 Q ` FO5-E(I01 3Xƒ?1TB…36¢41m>Ae6E(I01‡Eð~>CJ036B…34/Q/O14/"5(F2:01T5 E(I?3§E—FO5X˜ M • >?HBG16>4ƒ?1TB7˜ QC = QC K01 c 3T@;571iEHB…FO3T9;¢X/21 " QCC I?M3T5©H1TÏ?=@;34/(P3T9;A+QC ¢X/2145S34E C) 369;: C Ñ 5'FQ9 c 1 ∠QCC = ∠P AC 365CK;1§eq>?BG1?˜;~IAFQ/O1RE(I01/Q34E…EG1TB16ÏA@;3T/Q5 ∠CC Q K;1 c 34@;5*1ªE(I01*?BHBG1457J;>?9;:?FQ9;¢S369;¢X/O1T5ˆeq>?BGL81T:ŽKT?571T5HE A I ; @ § 5 ˜ ; K 1 4 3 ; @ * 5  1 „ F E Q F  5 ( E 0 I m 1 n J A > Q F A 9 E M c /O1N>?9=:?MFQ36LMP1§EG1TB P QHÑ ~FD=E(IHB…3":AFQ@;5 M P җFO9AEG1TB…571 c E…5 AC ` Õ=1 EH> HA ~=I014BG1g(PE(I0B1 +c FOB BQ) E(I01TBG1§eq>?BG1I?3§ƒ?1 0

0

0

0

0

0

0

0

0

0

0

0

0



1 2

0

0

0

0

0

0

0

0

0

0

1 2

0

0

0

0

0

1 2

0

∠HP B = ∠HP M

= = =

‰ IA@;5X˜ P H K?FQ5*1 c EG5

=

∠AP B

π − ∠P M H π − ∠P QC 0 = 2 2 π − (∠P QQ0 − ∠C 0 QQ0 ) 2 (π − ∠P QQ0 ) + ∠AP P 0 2 ∠P 0 P M + ∠AP P 0 ∠AP B = 2 2

˜?365 c /Q3TFQL81T: `

`


+
  V8$G ; *1!IN;1;(G#HR '( )1#PSJ8@)$7O ACM%:5$IC;1 5 /LCE4G1X5<#49 @JN@K Q,  4, &. 517.01
#D= .0 )1O#"0JN  7O AP5Q;(G   ;(8/R (\ #!=?'( )0= #45@J8@
O  $

X Q&( Q#45@J8@
O #*%N)1, @ @)$  J Q Q&1(Q, )1@
, )$7O2BW-
= 
'(= , )%L


£ Ígº»  ` 66?‘  ˜ £ ¡  3£   "     !       !   !  ! †¤       !        !  6¬R` FQ9;:ME(I01/Q36BG¢4145(E‡FQ9AEH1T¢41TB k 5'@ c IE(I?34E‡eq>?B 3T/D/-J;>?5'F„EGF„ƒ?1RBG1T34/-9?@;L8K01TB…5 a ˜ b ˜ ˜c 6~1NI?3§ƒ?1


` a + 3 b + 6 c + 12 > k(a + b + c) 3

3

3

©! ?*ª"£          «« ¡  ! * "  ž* ¡  *  !    '        !  ! * ]£  3# £ £         

&





!



3

        !  

a J;J?/„A# /O:;14B Ì 9;1TÏ?@;34/QF„E†<¹˜"~1NI?3Xƒ?1 (a3 + 3)1/3 (b3 + 6)1/3 (c3 + 12)1/3 = (a3 + 1 + 2)1/3 (2 + b3 + 4)1/3 (4 + 8 + c3 )1/3 √ √ √ √ √ 3 3 3 3 3 ≥ a 2 4 + b 8 + c 2 4 = 2(a + b + c)

` ÓN149 c 1A˜ (a + 3)(b + 6)(c + 12) ≥ 8(a + b + c) ` >?Bm16ÏA@;3T/DFDE{<ˆEG>=I0>A/2:¹˜ ~1L@;5(EI?3Xƒ?1 a = 2 = 4 369;: a = 2 = 4 `NÌ 9J036BðEGF c @0/Q36B7˜ a = 1 3T9;: ˜;~IAF c I81FO5mFQLMJ;b>?575(F2KA/281 `š 9ˆE(2I01>"E(I0414BmI?3Tc9;:t˜tFDe a = b = 1 3T9;: c2= 2 ˜ a = 1 E(I01T9 3

3

3

3

3

3

3

3

3

3

(a3 + 3)(b3 + 6)(c3 + 12)

‰ I014BG14eq>?BG1A˜ k = 8 `

`

= 4 × 7 × 20 = 560 < 576 = 9 × 43 = 9(a + b + c)3


V
  V?$G-; *1!IN;1;(G#4HR '( )1#SJ8@)$7O AH:8$!!H:E OLCG(55<#4H0@)07.0 M1@
 90 JN O#+:# %5Q:A /RG-G->…4!6;Q0# )(&]/R)(&#V1.1, ) @A 5Q;(G   !;1/: (\ #=8'1 )$= #5@J8@
O  $

Q&1 Q#5@J8@
O #!*%N)(, @ @)0 V JQ Q&1(Q, )1@
,



(a3 + 3)(b3 + 6)(c3 + 12) ≥ λ(a + b + c)3

Y!.0 JN

2





, C= .$ P')1, -'1 :W%O,X= ,X DJN11=0 0= .0 7 '( ,X7R -'@(= , )
@&()  &1  J8@ L

−1 89 ) − 7 λ=τ τ = 7 cos( 1 2 L 37?cos 2BWR'1=X 343 #R=X  7X, 2@<
1W-
@17? O#0Y!.1,
X P NC@)0 3 + 21x2 − 216 = 0 2x 7@<
X7N'
@1=X  λ =X @)0 W-
@(7O O#$JN ?W% 7N= ,Xλ Q=
 L 8.093 @&()  !
'(= , ) , 4 @  )+7O2BWR'(= J:@






 ”h

t5 a "’ "!#$&%'Š(* † 9"$ w[:<%P  j;<9 %>6c
3w~/%E…$ w<;B*tf:</w~%&$E;FwF6 ši1=9/3wF95     QRH&I G/HEI HK=LW`]lHtgFZ5]5LPIPMR`]z`]2K]ŠMF]aIPH/O~laK"Q  “`O f J I i = 1 2 ... n   H5]5H   5OP`/lHIWVK"I n 1 P

ai ∈ I

a=

n

ai

i=1

n  X X  n(n − 2) n a − aj f (a) + f (ai ) ≥ f a+ i 2 2(n − 1) n i=1

i6=j



™"

%¤

>?B

"` P I0149 c 1A˜;KT< 714905*1492- 5 Ì 9;16ÏA@;3T/DFDE{
    $          !    !

   !

i 6= j

/21XE

bj =

1 n−1

k

k6=j

`

    a − aj ai + (n − 1)bj 1 n−1 f a+ i = f ≤ f (ai ) + f (bj ) n

‰ IA@;5X˜

n

n

n

 X  n a − aj f a+ i 2(n − 1) n i6=j

≤ =

 X1 n n−1 f (ai ) + f (bj ) 2(n − 1) n n i6=j

n 2(n − 1)



n

n

i=1 n X

j=1

X X 1 n−1 (n − 1) f (ai ) + (n − 1) f (bj ) n n



` BG>?L 3=¢4149;1TB…34/QF 0(34EGFO>?9S>Aeg.?>?J;>TƒAF c FD@7- 5 Ì 9;1TÏ?@;34/QF„E†<ªJ0BG>Tƒ?1T:SKT?Jn>?571TB>Ae E(IAFO5‡J;BG>0KA/214Lù3T9;:J;BG145*149AEH1T:FO9E(IAFQ5 {>A@;BH9034/ Ñ 66A£  ¥¤  ÒH˜"~1NI?3§ƒ?1 X X ˜ (n − 1) f (b ) ≤ f (a ) + n(n − 2)f (a) 369;:E(I014BG14eq>?BG1 =

n 1X

2

n−1 2

f (ai ) +

i=1

n

f (bj )

j=1

n

j

j=1

i

i=1

 X  n a − aj f a+ i 2(n − 1) n i6=j

≤

n 1X 1 f (ai ) + 2 2 i=1

X n

i=1

f (ai ) + n(n − 2)f (a)





`


 
   ;(G
0!E4H:5 5 T$"RG ;!/: 5#=?'( )0= # 0)1,  J8O, = ] [1JN =X Q#  J8@K
, )1# "0JN  7O A$@)0 = .$ !W$JN<W% JL

”h

ta "’ "!#$&%'Š(* † 9"$ [w :<%P  j;<9 %>6c
3w~/%E…$ w<;B*tf:</w~%&$E;FwF6 ši1=9/3wF95 h  √ i  ShH”VKElH 5OP`/lHIWVK"I  gB`ObK/Q[QOPH/K"QC]Z5k H"O \   a1 a2 . . . an ∈ √12 , 2 J n n  X X 3 2 i=1

ai + 2ai+1

≥

ShVH/OPH IWVH\ Z j\ LWO MF_aIP\ KOPHIPKH/]tku`YaZQR` J

i=1

n



ai + ai+1

™

© *ª

«¤

   !  
  $   !   !   !  "        !"    !& !

t1X~NB…FDEG1CE(I01FQ9;16ÏA@;3T/DFDE{<3T5

%¡  &

"  !  ! 

 



n X

i=1

ai − ai+1 ≥ 0 (ai + 2ai+1 )(ai + ai+1 )

`

‡ 1T9;>"EG1E(I01/21§eQE 5(F2:01>Ae—E(IAFQ5FO9;1TÏ?@;34/QF„E†<ˆKT< L `‰ I0149t˜¹5'FQ9 c 1 ~1I?3Xƒ?1 6L

= =

n X

i=1 n X

n P ai − ai+1

i=1

ai ai+1

˜

=0

n X 6(ai − ai+1 ) ai − ai+1 − (ai + 2ai+1 )(ai + ai+1 ) ai ai+1 i=1

2

(ai − ai+1 ) (2ai+1 − ai ) ai ai+1 (ai + 2ai+1 )(ai + ai+1 )

`

X˜ ~1mI?3Xƒ?1 2a − a ≥ 0 eq>?B-34/Q/ i = 1 ˜ 2 ˜ . . . ˜ n ` Ó1T9 c 1?˜ ≤a ≤ 2 6 3 ; 9  : 6 1 A Ï ; @ T 3 D / D F { E M < 0 I >A/O:05}FQet3T9;:M>?9?/ <FDet34/Q/0E(I01 a 5 3TBG11TÏ?@;34/ ` L≥0 ’ FO9 c 1

i=1

1 √ 2

i

√

i+1

i

i



X   B= .$ !W$JN<W% JDYD, = .[ N )0= , @<

B= .0 4N@2+ 4
X'1= ,X)UL

 X  

r/a " W"!#$E%/'u(*h9/3=9/3s35*12x"$e!sW"!#$E%&>

/G HEI H”Kr‹aZ5KYOPM[QFKEINH"O K/Q5ShVaMFL VtM?\ MF]\ŒLWOPM H/YtMF]=KXLNMFOPLNQFH  “Z5O IWVH"O ABCD J Γ \ Z5__`\ŒHŠIWVKEI M[I \jHEQ?gTVK\rK/]ƒM?]5LNMFOPLNQFH  G/HEI J HŠIWVH8YM?Kk= HEIPH/Oh`ag ABCD EF J  ShVaMRL VuM?\_5H/OP_H"]5YMFLNZQFK/O IN` Q ^M?]5dr`]=IWVH\ Kk=Hr\WMRYH`ag K/\ Γ SM[IWV BD E  BD A MF]aIPH/O \ŒHLPI KEI K]5Y K"I G/HEI BD EF M AC S 5OP`/lHIWVK"I AS : SC = EM : M F

Ÿ ' eŽ VaM?\ _OP` QFH/k LNK/kuHXMF]aIP`=IWV H T]M lH/O \WM[I~^8`a g sHd&M?] K   \ yK"IW V sH/]aINO K"Q ShH \ MUINH  ZIfIWVH”]K/J kuH”`agIWVH”_OP`_`\ŒH/OmVK\ \WZ \ŒH‹aZH/]aIPQ ^ H/H/]QR`\NI   J J J J 0/:…x;Fw~38(*žf!#$E;</:[wF$ •b-
+%&1wF$ 6 † 9W&94%>w~u|cw?™A=0„PAC/:F6žc;BAC%E3$Œ 6 
W%%/„W% G/HEI HeIWVHLNH/]aINOPH`ag/IWVH MF]\ŒLWOPM H/Y”LNM?OPLNQRH K]5Y QFHEI    K]5Y HeIWVH IJ K  J   K/]5Y  OPH/L IPK/]5d"H/]5L<^2_`aM?]aIP\`ag
IWVHr\ MFY5H"\ \ _H/NLPIPMUlHEQU^  H“MFJ ]K"Q?Q ^  oŽ VHcMF]\ŒLWOPM H/YXK ]  Y5H ]5H IN` HIWVHcM?]aINH"O \jH/LPIPMR`AB ]X_5`aMF]aBC If`agQ?CD M?]5H/\ KDA ]5Y T J AD BC J d&QRH"\ K/]5Y KOPHuH/‹Z5K"Q H/LNK"Z5\jH 1 ∠BCD ∠BF D ∠ICN = ∠ICD = i  J ∠BCD  MUIrgB`aQ?QF`S”\rIWVK"I 2 K]5Y 1
`]\Œ∠BF H‹aZH/]aM IPQ ^  =IWV∠BF H›OPMFd"E VI2= INO MFK/2]5∠BF d&QRH"\ D K]5Y K/∠ICN OPH›\WMFkM?Q?K=O ∠BF ShVH/]5M LNH  ICN BF M  K/]5Y KOPHr\ M?ktM[QFK/O  K]5YtShHtd"H&I FM BM hp MFkM?Q?KO QU^  INO MFK/]5d&QRH"\ = IAH BEM CN IN  ShH”VKElH F M EM  `O EM BM bp M?]5LWHIWVHM?]O KYaM?M AH

=

IH

IN = IH

EM AH = FM CN



CN

=

AH

~>‘

™A E

A ........ . . . . ... ... ... ..... . . . . K.r... ... . . ... ... . . . ...H .. . r . . . ... .. . . . r . ... . O . . . . . I .. D ..... . r . B ....... Y H S .... M ......... . ......... .. ......... ...r. . . .......r.. .. I @ @ L .............. ..... ......... @

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

N

C

F

?> BgE(I01=>"E(I014BNB…3§EGF2>~1=>0K?5*14B{ƒ?18E(I?3§E nBGFQ369 c I0>?92- 5 ‰ TI016>?BG14Lê36J0J0/DF21T:[EG> FOLJ0/DF2145E(I?3§E FQ5C>?9 " E(I?34E‡FO5X˜nFQe3 c >?9?F cmÑ I014BG1RE(I01¢XF ƒ?1T9ˆFQ9 c FOB c /O1XÒ ABCD FO5FQ905 c B…F2K016:©FQ9Ç3ªÏ?@;36:0SB…3T9;¢X/21A˜‡HL E(I01 c I0>?BG:?5 {>AFQ9?FO9;¢ˆE(I01[J;>AFO9AE…5M>Ae EG3T9;¢41T9 c Ae >?J;J;>?5'F„EH1M5(F2:01T5 Ñ 903TL81§/„< HL 3T9;: KN Ò¢4>E(I?BG>A@n¢4IE(I01FO9AEG1TB…571 c EGFO>?9[J;>AFO9AEC>Ae E(I01gEð~>:?FQ3"¢4>?9034/O5 Ñ 903TL81§/„< S = AC ∩ BDÒ ` ?< • 1T9;1§/O34@;"5 - ‰ I016>?BG14Lù3TJ;J?/QFO16: EH>RE(I01gEHB…36905…ƒ?14B…5734/ HSL >Ae 4T AC ˜ TH

>?B*˜?5(FO9 c 1

TH = TL

369;:

HA

·

AS SC

CL = CN

˜

·

CL LT

= −1 "

AS AH = SC CN

`

Ñ{ Ò

‰ I01NBG145'@0/„Eeq>A/Q/O>6~N5ue2BG>?L 16ÏA@;34EGFO>?905 Ñ Ò369;: Ñq Ò `



  8$G ; *1!IN;1;(G#%HR '( )1#%SJ8@)$7O A05!;1G1  !;1/: (\ #$=?'( )0= #05@J8@
O  $

Q&1 Q#5@J8@JO #*%N)1, @ @)0  JQ Q&1(Q, )1@
 @1=8'JN  
'(= , ) JN2   1L

















h ƒ P /a85 
 /2W"!#$E%/'€(*X.E$sŠ {% ¡x"w?$+ { 9&‡hŒ˜C9/N %& Š93C'  93C„…wF$&„W 9":F9„…w~ $ TxEw 3s 3C%EW6b
3w~/%E…$ w<;R9;
/:[w…; %/ „P3w~„ 9'5% 9;R9E: x/3*"9"6
˜C9W„W%j:<39/60&!s9"wF3o   G/HEI   HXOPH/K"Q
]Z5k H/O \\ ZL VtIWVKEI gB`O a1 a2 . . . a n J J ak ≥ 1 1≤k ≤n 5OP`/lHIWVK"I v  u n n  Y X u ( ) 2 2k

ak n(n+1)

k=1

1/2

≤ exp t

k=1

ln ak 

™&

#£ ª

   ! 
    $    !"  ! !

’ 1§E…EGFQ9;¢



¢© *ª

xk = ln ak n Y

exp

k=1

>?B}1TÏ?@0F ƒA3T/O1T9AEG/
A¡    
 !   

     !   !  "   

    ! 

˜"E(I01CFO9;1TÏ?@;34/QF„E†<K;1 c >?LM1T5

s

2k n(n + 1)

xk

!

v  u n uX x2k  ≤ exp t

˜

k=1

v u n n X √ u n(n + 1) X kxk ≤ t x2k 2 k=1 k=1

˜

~IAF c Ieq>A/Q/O>6~N5ue2BG>?LE(I01–¹3T@ c I6<0‚ ’6c I6~N3TB 0 Ì 9;16ÏA@;3T/DFDE{<5'FQ9 c 1 n X √ kxk

k=1

!2

≤

Œ0ÏA@;3T/DFDE{<MI0>A/O:05}FQet3T9;:M>?9?/ <FDe

n X

k=1

k

!

n X

!

x2k

k=1

=

n n(n + 1) X

x1 x2 xn √ = √ = ··· = √ n 1 2

`

2

x2k

`

k=1


 
X  # 'O, )1& N )$= , @

 = .$ N@2+ 2+ = .01 #  E4?G *1;1G-T #FG ;5(8G  !MU*%G ;1;<#@)$ $DH:;(G15HCLE4XG#4)1&( Q
X 51= @1=X 0)1,  J8O, = <# 5@) )1&( Q
X# I C#+%5Q:A ?$G-;*1!IN;1;(G#UHR '( )1#USJ8@)$7O A08 !;\ T[*%G1(ZG#U*$J8@(#UHR2@)1, @
W$JN<W% JL











h 

r 5 E   "!#$&%'u(*‚€wRAf9E{ :B*t˜o%&3s„<‡>%>6f˜C 9/$&&6fš#1=93awR9  K ‘ /G HEI K/]5Y K]5YQRH&I H”_`\WM[IPM lH”OPH"K/Q]Z5k H/O \eSMUIWV
 ‘ G/HEI    H_`\WM[IPM lHcOPH/K"Q]Z5k H"O \iSM[IWV · · \W·IN< J ?K]5YŠQRaH&I 1 a2 . . . an J 2 < HtK=LW`]aIPM?]Z`aZ5J \ Q ^zYaM?n
H/OPaH"]a1IP< MFK aQFHt K/]5Y8 O MRLPaIPQ n^  J J f : [a1 , an ]f→ p ku`]5`IP`]5HgFZ5]5LPIPMF`] V`S›IWVK"I
IWVH"OPHcM?\bKcOPH/K"Q]Z5k H/O \ ZL V J c ∈ (a1 , an ) IWVK"I ! n n  X X ak f (c) =

k=1

ak f (ak )

k=1

™

%¤

"` >?B‡J;3TB…E Ñ 3§Ò~13T575(@;L81NE(I?3§E f FO5 c >?9AEGFO9?@n>A@;5m369;:ME(I?3§E f (a) 6= f (b) `Ì e af (a) + bf (b) ˜"E(I0149 m= a+b b(f (b) − f (a)) 369;: m − f (b) = a(f (a) − f (b)) ` m − f (a) = a+b ’ FO9 c 1 f (a) 6= f (b) ˜6~1C57161mE(I?34E m − f (a) 3T9;: m − f (b) I?3Xƒ?1N>?aJ0Jn+>?5(bFDEG1C5'FO¢§905 369;:CE(I?3§E m FQ5ŠK;1XEð~16149 f (a) 3T9;: f (b) `’ FO9 c 1 f FO5 c >?9AEGFO9?@n>A@;5X˜4FDEteq>A/Q/O>6~N5ŠKT?BG14L$E(I?34E m = f (c) eq>?B—5*>?LM1 c ∈ (a, b) ˜63T5BG16ÏA@0FOBG1T: ` >?BJ;3TB…E Ñ K6җFDEFQ5‡149;>A@n¢4IEG>R365'5'@;LM1mE(I?3§E f FO5 c >?9AEGFO9?@n>A@;5}3T9;:E(I?34E14F„E(I01TB >?B f (a ) > f (a ) > f (a ) eq>?B k = 2 ˜ . . . ˜ n − 1 ` f (a ) < f (a ) < f (a ) ¨T1§E…EGFQ9;¢ s = P a ˜ FDEReq>A/D/2>6~N5e2BG>?L 14F„E(I01TB>Ae E(I01[3"K0>Tƒ?1PFQ9;16ÏA@;3T/DFDEGFO1T5E(I?3§E FQ5ÇK;1XEð~16149 f (a ) 3T9;: f (a ) `  >"EG1bE(I?3§E a > 0 3T9;: 1 P a f (a ) s s P a P c ˆ ` ‰ O F N 5 3 ? > " 9 ? ƒ 41 3 c >?L8KAFO903§EGF2>?9[>Ae eq>?B 0 I 4 1 G B 4 1 q e ? > G B A 1 ˜ = 1 a f (a ) f (a ) s ˜ ˜ ˜ `”a ¢§3TFQ9t˜KT<€E(I01 Ì 9AEH14BHLM16:AFO3§EH1 34/Q@n1 ‰ I016>?BG14L=˜‡eq>?BR5*>?LM1 k = 1 2 ... n ˜6~1NI?3§ƒ?1 c ∈ (a , a ) 1 X ` a f (a ) = f (c)     $          !    !

   !

1

k

n

n

1

k

n

k

k=1

n

k

k

k=1 n

1

n

k

n

k

k

k=1

k

k

k=1

1

n

n

s

k

k

k=1




 
   HCT *1!H:*1H-:#+;1  @)0  0)1,  J8O, = <# S@) @J8# ;(  @)$)(A !M%5-ID;(X5 /CLE4G1X5<#[91@JNQ@K1 Q,X]4, &. 517.01
# != .$ ) O# "0JN  7? A MUGUIG-H TRLm !%#6*-,X
@U)(,X J8O,X= #+;@  , J8@( @#[:#+%5Q:AP@)$ ]= .0 [W$JN<W% JL :

4
X J86, )0 , 7@(=  = .1@1= = .0 , R JN )$= , @1 ,
, =  B= .$

8')$7N= ,X) , D)$1=U)0 7? NN@J? L




  

     "! #$%'&)(*# +-, #./! #0'&)( 13254 68759:13;=<9>4 6@?BA:CED:<F 23GIH>6J4 6@?BAONQPR9@?BHJAw254 <cB;=de<1R9>254 6@4 6J4x75CT9>A:DwCE<Xf 23D:GX<fH@23686J9@;yAgL84 ?xHMH8rt<EsuSRU UPjD:UBU3v%VXW3GGKAgY[rI7IAw‡?R<cBcR;=2523<498cB<‰9>4de6@ˆ-4x1RW5CT25GX4 cRDw6J75d5<Xf 9>}R23A:ŠuGXCEH@GI6873Dw75VXAOW3cRNjGKd5‚5Y‰}GO„M68c=