Solution 2. Since. 1. Sums and Products

Transcription

1 . Sus Prous I h, ver oe we hve soe eresg uers whh we woul le o her su or prou. elow we wll loo ew ehos or og hese operos. (Here we wll lso oser egrls whh we vew s sug uoul uers.) Prg Meho. Rell o S we s S 99 lso hee S 99 elg S r o pr he rs hope hs spl he prole. Eples. () ( APMO) F S Soluo. Noe S Se Ths sugges h hlg uers we Le we hve () ( HKMO Mh p) Epress os s or where re egers. Soluo. Rell os s os hve os s os s Tg os os s os s or S os os s os s os s os os or os os The os he s s os os we Soluo. Se os os s os Thereore, os s os os os os os s os () (98 Pu E) Evlue Soluo. Le r os el I I I s s s r s r os r os r os r os os s So s r s r s s s s os s os r r os r s r os he hge o vrle I Se wll Telesopg Meho. A prulrl sple pe o su evero o s o he or Ths pe o su s lle elesopg su. Slrl, here re elesopg prous, where he ors re o he or her prou s Soe suo or prou proles re o hese ors. So sug, we shoul r o see he ers e pu he or eples. Eples. () Spl s s s Here re soe

2 we Soluo. Rell s s s os s os os s s os os s os s os os So os s Soluo. (Use ople rhe) Le orul, os s os s I Now os os os Noe s Seg s s os we ge e Movre s s So or lrge Tg l s (6) Show h 7 7 we ge he swer s Soluo. (Noe e ul o he e su. We hve o ou he ers ro ove elow.) To ge elesopg ees, we use os os s s s s Sug ro o, he ulplg, we ge he equles. s s s os os So s os os s () (977 Pu E) Evlue Soluo. Noe h Also, os (Here s s l s ) ol Sus. For sus volvg ol oees, we wll rel o he ol heore soees o lulus o he swers. Trvll, ro ge seg seg he -h row o he Psl rgle s he lere su s. elow we wll loo ore eples. Eples. (7) Spl These equos epl wh he su o

3 Soluo. Se he su s he se s Noe hs su s he oee o Thereore, (Rer: loog he oees o o oh ses o he e we wll ge he ore geerl e (8) (96 Pu E) Evlue lose or Soluo. For!!! So The ses re esl hee o e he se. Soluo. (Use lulus) Dereg oh ses o we ge Mulplg oh ses he ereg oh ses g, we ge Seg we ge Fu s Prple. Whe we hve rows o uers, o he su o hese uers, we su eh row rs he up he row sus. Ths wll e he se s sug eh olu rs he up he olu sus. Ths sple s ow s Fu s Prple. There s slr see or he prou o he uers. I shor, we hve Hsorll he orgl Fu s prple ws ou erhgg he orer o oule egrls, el s egrle o he o, he Eples. (9) For hessor wh o, eh squre s wre or Le p e he prou o he uers he -h row q e he prou o he uers he -h olu. Show h p p p q q q Soluo. Fu s prple, p p p q q q Noe eh p or q s Suppose here re s s og p p p s og q q q The eher s re oh o or oh eve. Now p p p q q q s s s euse s s o. Noe: I wo eger vrles re eher oh o or oh eve, he we s he re o he se pr. 6

4 () (98 Pu E) Evlue Soluo. Ar Ar Ar Ar Ar u u u u u u u u u l The erhge s vl se he egr o he oule egrl s oegve ouous o he o he egrl s e. () Le e posve eger p e pre. F he hghes power o p vg!. Soluo. Wre he hghes power o p vg s he wre s olu elow he or (I p oes o ve elow s he hghes power o p vg! sus s ul, u sug he row sus s es. I he rs row, here s oe ever p oseuve egers, so he rs row su s [ p] I he seo row, here s oe ever p oseuve egers, so he seo row su s [ p ] Keep gog. The -h row su s [ p ] he [ p] For so o o wre.) The ol uer o s Sug he olu [ p ] Ths s he hghes power o p vg! () (987 IMO) Le P hve el e pos. Prove h h s o ove he peruo.) So he ol uer o s s [ p ] e he uer o peruos o 7 P! whh (A e po s uer Soluo. There re! peruos o ll he! Wre eh sepre row. For eh peruo, reple eh e po o reple ll oher uers he peruo. The he row su gves he uer o e pos o Now P s he su o he row sus, groupe org o he P rows h hve he se row su prple, hs s lso he su o he olu sus. For he -h olu, he uer o s s he uer o es s e po og he! peruos. I s e, he he uer o ws o perug he oher uers s P!! or eh o he olus. So he olu su s Thereore, he su o olu sus s. F o o. (988 Sgpore MO) opue. For posve eger Eerses! o 9. (99 Hugr MO) For posve eger s s. For posve eger show h! Ths s o 8 99 prove h o show h s [ ] *6. Prove h polol o egree hvg roos 8 99 s 89 Fu s P o o 89 s os (H: F 89 )

5 *7. (99 Ausr-Polsh Mh opeo) Le e eger le! e he! peruos o (Eh s eve uo ro o sel.) For eh peruo le us ee S F!! S. Iequles (Pr I) We oe opre uers or h epressos, suh s g or or pplg he swh heore. So we ee o ow soe useul equles. Here we wll loo soe o hese see how he e pple. *8. (99 MO) Le e e posve eger. F he su o ll posve egers wh he ollowg proper: I se, hs el gs ossg o s s. (The leos g o e.) AM-GM-HM Iequl. For AM G M H M Eher equl hols ol Eples. () he AM-GM equl, or equll ol wh () he AM-HM equl, he () I he u o Soluo. he AM-GM equl, So Thereore, he u s 9. () For posve eger Soluo. Le AM show h 9 wh equl ol G M he AM-GM equl, 9

6 Tg he () (96 IMO) Le Soluo. Le Ths s equvle o euse equl. -s power o oh ses, we ge e he ses o rgle. Prove h he The equl o e prove eoes 6 6 uh-shwr Iequl. For rel uers Equl hols ol Eples. or ll (6) F he u u o os Soluo. he uh-shwr equl, os s os So os s s os.e. o he le rgh equles. So he u s (7) (978 USAMO) For rel uers e 6 s There re wo suh s [ 6 6 he AM-GM whh s rue s where Equl hols ol e suh h orrespog he u s he u o e e 8 Soluo. he uh-shwr equl, So 8 e 6 Ths es e 6 equl, we see h whe vlue o 6 (8) (99 IMO) I e he equl e- Soluo. Susug oes Epg splg, we ge e e Eg he equl se o he uh-shwr e wll he u Now uh-shwr equl, we ge 6 he prove h Usg he ls equl he AM-GM equl, we ge 6 Rerrgee (or Peruo) Iequl. I he r r r he

7 r where r r s peruo o. hesev s Iequl. I he Eple. (9) (978 IMO) Le h e s posve egers. Prove Soluo. Le re s posve egers, we hve ge e he s rrge resg orer. Se he s Now, se he rerrgee equl, we () Reo eple (8) usg he rerrgee equl. Soluo. (Due o Ho Wg Yp) We ee s eple (). Whou loss o geerl, we ssue euse he equl s ser. The equl, Ag hese equles vg, we ge Applg he sple equl ses, he he AM-GM equl, we ge he rerrgee o he uerors o he rgh Eher equl hols ol Eples. () (97 USAMO) For Soluo. ser, we ssue hesev s equl, log log log log or prove h log log The esre equl ollows epoeo. () I or S S S log log Soluo. Whou loss o geerl, we ssue The hesev s equl, S The resul ollows. S he prove h The log log log

8 I h s well s sss, we oe ee o e verges (or es) o uers. Oher h AM, GM, HM, here re so-lle power es ser es, whh lue AM GM s spel ses. Power Me Iequl. For M s s s s Equl hols ol s s M Now M Rers. lerl, M AM M H M s lle he roo-e-squre (RMS) o he uers. I ppers sss phss. Also, g ls, e show h M s M AX s M I N hve M AX RM S AM G M H M M I N M s G M M So we Soluo. Le Now () I he he equl eoes M M M M M he prove h Soluo. The equl s equvle o he power e equl he ser e equl, M 8 8 M 8 S 8 S 6 S S 6 S 8 Mlur s Ser Me Iequl. For Mulplg o oh ses, we re oe. AM S S S where S s he verge o ll possle prous o A oe o he equles hols ol G M e e. () I Soluo. he ser e equl, he show h Rers. To e ler o he eg o S S S e 6 S I h se, we hve S S S S S S S S S S Eples. () Show h or pos- ve So S The S 6

9 p p Eerses. Nuer Theor. Reo eple ().. Reo eple ().. Reo eple ().. For. For 6. I 7. For show h show h 8 he show h he u o Dvsl. Deos. () I re egers suh h ves eoe hs () A posve eger p egers vg p uer. 6 (For eple, ves 6, so we wre he we s ) s pre uer p re he ol posve I posve eger s o pre, s opose There s ous proo o he h here re el pre uers. I goes s ollow. Suppose here re ol el pre uers, s he re p The he uer M p p p s greer h p p So M o e pre, hee here s pre uer p vg p M M oro. However, p lso ves M Hee p wll ve M 8. I 9. I. Le S he show h e he ses o rgle. Show h 7 he show h S Fuel Theore o Arhe (or Pre Foro Theore). Ever posve eger e wre s he prou o pre powers e e e 7 e p e w (eep or reorerg o he pres). Eples. 9 where he e s re oegve egers, oe ol oe 9 7 Quesos. Do posve rol uers hve pre oros? (Yes, epoes re llowe o e egers.) Do posve rel uers hve pre oros (llowg rol epoes)? (No, oes o.) orollres. () p p p ves p e pe p e ol e or () The uer e e p e hs el e e e posve vsors. () A posve eger s he -h power o posve eger (.e. ) ol he pre oro o e e e o Eples. () Se 9 vsors. The re hs 8 where pe ever e s ulple posve

10 [6 [6 [6 () Suppose s posve eger suh h hs 8 posve vsors hs posve vsors. How posve vsors oes 6 hve? Soluo. Wre e e p e e e e e 8, so us e or. I he e e 6 hs e e posve vsors. I e e The Now e e e he hve he uque soluo e e 8 e ves 8 whh e e e e I ollows whh hve o eger soluos sple heg. () (98 IMO) Gve se M o 98 s posve egers, oe o whh hs pre vsor greer h 6. Prove h M os les oe suse o our s elees whose prou s he ourh power o eger. Soluo. Le M e e e 98 e 9 possle pr (.e. o-eve) pers or he uers e og o he, here wll e wo (s Tg pre oros, suppose Se s he h pre uer, here re 9 e e Noe o hve pre vsor greer h 6. e9 Reove hese prs oe e. Se re les prs. oser he s or hese prs. There wll e wo (s l) suh h l l l The here Deos. Le () The grees oo vsor (or hghes oo or) o s he lrges posve eger vg ll o he. We eoe hs uer or g e egers, o ll eros. I he we s re opre or relvel pre. I prulr, wo opre egers hve o oo pre vsors! () The les oo ulple o whh s ulple o eh o he. We eoe hs uer [ l Eple. () s he les posve eger 6 Theore. I e 8 e 8] p e 6 he 9 8 e 9 e 8 9] p e 7 ] or So ) wh he se per. The For h uers.) Eple. () 6 [ [ ] ] 8 so e e p e (The ls equo ee o e rue or ore 6 8 8] Pre oro s ul or lrge uers. So o g s, we lso use he ollowg. Eule Algorh. I or posve egers r he r Eples. (6) re egers o oh eros, he 89 6 (7) (IMO 99) Prove h he ro Soluo I prulr, 6 6 s rreuleor ever url uer 7 The ollowg re soe useul s ou relvel pre egers. () For oegve egers o he or he here re egers Resos. lerl I he s rue or ll ses q r r egers q () I egers r p he suh h s he les posve eger where re egers. I prulr, o oh eros, suh h ser, we ssue ves posve uers o he or We wll u o or r r r q s rue. Se So he se r or p s so h he p he s (Ths s euse r hee Suppose hs where Eule lgor, he uve hpohess, here re s or soe ) I prulr, p s pre Fro hs, we ge h

11 r oulo r () I he eh o s he -h power o eger. (Ths ollows ro g pre oro o usg he seo pr o he ls.) Moulo Arhe. Dvso Algorh. Le e posve eger. For eger egers q q r r upo vso r suh h Reers re lws oegve.) here re (r s lle he reer o Noe h whe 9 s ve, he reer s, u whe 9 s ve, he reer s euse s ve, he respeve reers or he pero sequee 6 Whe he egers Deos. () We s s ogrue o oulo eoe hs o ol hve he se reer upo vso () For posve eger o egers r r (For eple, resues oulo or ever posve eger.) r oplee se o resues oulo s se r suh h ever eger s ogrue o el oe o s Properes. () o ol he eo o he ogrue relo.) or oplee se o (Oe hs s use s () I o o he o o o o P P or polol P wh eger oees. Eple. (8) F he reer o 978 upo vso Soluo o 8 6 o 6 Oher h g reers, oulo rhe s lso useul suos. For eple, (o ) s goo or pr he. The h uer s vsle ol he su o gs s vsle e esl eple o I worg wh squres, (o ) s useul og pr he se o o we hve o 9 org o o respevel. For ourh powers, or (o 6) org o s eve or o respevel. Also, s he resog ove or (o ), oe show h ever oegve eger se s ogrue o he su o s gs (o 9). To eere he us gs, we use (o ). or Slrl or ues, The ollowg s re ver useul elg wh soe proles. Furher Properes. () (ello Proper) I, he, so.e. o (Resos. he ello prop- (v) (Esee o Mulplve Iverse) I o suh h o Se here es egers o o er.) I he (v) For posve eger Se o we he h r s r s o r s (v) For oegve egers o s he here re egers o s We eoe hs so we lso hve suh h o he we ee r s o he here ess uque rs o oh eros, r suh h r Fer s Lle Theore. I p s pre (Ths es h p o p euse p p o o o or ll egers r o s he p o p Fro hs, r ) r r o p Euler s Theore. I o - uo s he uer o posve egers less h or equl o, whh re relvel pre o p p hee Euler s heore geerles Fer s lle heore.) he (For pre p where he Euler

12 r r r To uers he resos eh hese heores, we wll ee reue se o resues oulo o e se o egers r suh h ever eger relvel pre o s ogrue o el oe o r. To prove Euler s heore, we oe h r s reue se o resues oulo he r re relvel pre o r he lso or reue se o resues oulo. (Ths s euse r r ol r r (o ) he properes ove.) So eh r s ogrue o uque r oulo hee r r r r oulo o 7 7 Euler s heore. Now o 7 o so So he us g o 777 s. hese Reer Theore. Le or ever pr. The he equos o o o hve oo soluo. I, ever wo soluos re ogrue o we s he soluo s uque o e posve egers suh h r r r r r r r r r o Resos. Oe soluo e ou s ollow: le M he Se r r (o ). r pplg he ello proper, we ge Wlso s Theore. I p s pre uer, he p overse s lso rue, s opose, he!! (o p). (The (o ).) Resos. The se p s ler. Le p e pre p ples p or reue se o resues oulo p hee here s uque suh h p (o p). We hve ol p ves or p. Hee, or he p egers p prs p wh (o p). The p! p (o p). p p we or h s he For he overse o Wlso s heore, s opose, he le p e pre vg supoose p p so h p p p p ves! I p he p p p p so g ves p p p! The p whh ves Eples. (9) I p s pre, he re re;vel pre o p so p p (Ag, hs es Fer s lle heore s spel se o Euler s heore). For p p p p p p p p p pre, se he uers p re he ol uers less h p o relvel pre o p so p M s soluo se o or uque o or.e. or M so her l or M M M pl Euler s heore. Ne, o show he soluo s le o lso. The s oo ulple o he s. Se o opuo Foruls. I p e pe p e e or p The he, he p p M o ) Resos. For he rs see, oserve h he o upo vsos s re relvel pre o respevel he Eule lgorh. oversel, r s he r o s o hve uque soluo less h or equl o he hese reer heore. Thus, he prg r s oe-o-oe orrespoee. The seo see ollows ro he rs see he Eples. () r So he reers r s s p p p I () Le us he us g o 7 77 h or equl o relvel pre o. Se Noe 7 se ol,,7,9 re less so 7 () To solve he sse o 7 o he prgrph elow he hese reer heore o ge we use he orul 7 7

13 6 7 7 o (However, geerl he orul volve lrge uers.) Alervel, we solve s ollow: o 7 7 The o o o 7 (or 7 7 o ) o () (IMO 989) Prove h or eh posve eger here es oseuve posve egers, oe o whh s egrl power o pre uer. Soluo. Le p p hese reer heore, p p e s pre uers. Now he So, 978 (euse 978 s o) he les s. Le The prole ow s o loo or he les posve eger suh h 978 (.e. 978 o Se o Euler s heore. Thus he les suh s os. so 978 Suppose he les q r wh r 978 r 978 q 978 r 978 o epoe, r us e. The 978, so 978 o The ol suh s re ulples o. So or h 978 o o euse 978 o ples o ) So Thereore he les he les 6 whe Le Also, (Ths lso shows h 978 he However, eple (8) shows Se s o e he les suh o p p o p p o p p Eerses hve oo soluo. The eh o he uers s vsle wo ere pre uers. Hee eh o e pre power. () I q s pre or o q o he q q s soees useul, or eple eerses.) (Ths Soluo. Suppose q oes o ve q he q o q Fer s lle heore. Now q ples o q so o q The q oes o ve q q o q q slrl q s. The Le q org Fer s lle heore. So () (978 IMO) Le e url uers wh I her el represeos, he ls hree gs o 978 re equl, respevel, o he ls hree gs o 978 hs s les vlue. F suh h Soluo. Se he ls hree gs re equl, so o.e.. For eh o he ollowg sees, eere eh s rue or lse. I rue, gve eplo. I lse, prove ouereple. () I p s pre uer p () I () I he o he he p o. As he ls eerse, eere eh o he ollowg see s rue or lse. Prove reso or ouereple. () I p s o pre, he p p p () I p? he show h he sequee p us e ll s er he -h er. p How ou. I p s pre uer, show h p ves he ol oees p p!! p! or p. F ll ergers suh h o 6 o o

14 . opue he ls gs o 7 77 (H: oser (o ) (o ).) 6. (97 USAMO) Prove h or posve egers [ [ ] ][ ][ ] 7. Show h he grees power o pre uer p vg! s [ p ] [ p ] [ p ] [ p ] lso lle err s Posule euse ws eperell vere err or ro o eore hesev prove. Dvsl Proles Eples. (6) (998 IMO) Deere ll prs h 7 ves o posve egers suh Soluo. oserg he epressos s polols o reg s os, s url o eg s ollow. I 7 ves 7 ves he where [] s he grees eger less h or equl o (97 IMO 97) Le e rrr o-egve egers. Prove h!!!!! s eger. (H: Oe soluo uses he ls eerse. Aoher soluo s o ge reurree relo.) 9. Do here es oseuve posve egers eh o whh s vsle oe or ore pres p ro he ervl p?. Show h here re el pre uers o he or Mo he proo h here re el pre uers.). Show h here re el pre uers o he or Use eple.). (H: (H: Rers. There s ous heore lle Drhle s Theore o Pre Progresso, whh ses h or ever pr o relvel pre posve egers he rhe progresso o el pre uers. us Aoher ous heore ow s hesev s heore ssers h or ever here s lws pre uer p ewee Ths s 7 I 7 eger I 7 7 he 7 ves so I s es o he hese prs he or soe posve ss he oo. u hs ors I ves ves 7, whh ples or 9. I 9 ves ge re esl hee o e soluos. he he I he oro. So ol e or. I So Se 7 whh hs o eger soluo. Fll, (7) (988 IMO) Le e posve egers suh h Show h s he squre o eger. he ves we Soluo. Le u o pere squre. Aog ll suh ses, oser he se whe s les possle. Noe ples so h Hee, ser, we ssue hs s roo. The oher roo s he eger pl s posve eger. Assue here ess se s eger, 8 Now Now

15 Also, osqure eger or possle. Thereore, ll suh s re pere squres. Now s s les Rers. oserg o he roos o qur epresso s useul r soe uer heor proles! (8) ( IMO) Deere ll prs o posve egers s posve eger. suh h Soluo. Le e posve eger. The vro h s lso useul.) Mulplg opleg squres, we ge N he M N or M I I M I N (Noe s possle o oser roos. However, he ollowg s Le M he s eve M N I M he we ge N he wh eve eger. We esl he hese re soluos. he se N We hve M N N N Thus, we ge N whh ples hs ores e hee o e soluo or ever posve eger I M he M N whh ples N N 8 N So N N so Se he le se s lso oegve, The whh However, he rgh se equls oro. (9) (97 Pu E) Show h s eger greer h, he oes o ve 9 Soluo. Assue ves or soe eger so s o. Le p e he slles pre vsor o o p o p oo vsor o p us e euse ves p s he slles pre vsor o r s p oro. Fer s lle heore, p whh ples r p s Eerses. (99 IMO) F ll egers wh s vsor o Se The p ves The here re egers r s suh h o p s o, hee Now he grees suh h. (99 IMO) Deere ll orere prs o posve egers suh h s eger. (oes: There re 9 soluos.). Reo Eple 8 oserg roos o qur epresso s he soluo o Eple 7.. (999 IMO) Deere ll prs o posve egers suh h p s pre, p p s vsle p oser he slles pre vsor q o ) 6. ( HKMO) F ll pre uers p q suh h s eger. p 7. ( IMO) Le p e pre uer. Prove h here ess pre uer q suh h or ever eger p s o vsle q Noe o ll o he pre vsors o M p p p re ogrue o (o p ). Le q e suh pre vsor o M ) he uer p 7 p p pq 7 q p q (H: 8. (99 IMO) Deere ll egers suh h s eger. (H: Wre r wh or org r Show (H: For p

16 8 oserg (o 9). Show r vsor o r r ) oserg he slles pre org or So s he ol soluo. However, sple hegs show hese re o soluos. Dophe Equos Equos whh egrl soluos re sough. Eples. () (979 USAMO) Deere ll egrl soluos o 99 Soluo. We hve So, here e o soluo. 6 6 o 6 8 o 6, u 99 () (976 USAMO) Deere ll egrl soluos o Soluo. Suppose s soluo. I o, u o o, u or o o, u o o 6 re o, he o 6 I wo re o oe eve, he I oe s o wo eve, he So, Thereore, us ll e eve, s ge s o, he or o, u o us ll e eve g, s 6 us ll e eve. Fro hs we see uvel h e vsle power o. Thereore, I les oe o The So So () (99 APMO) Deere ll posve eger or whh he equo hs eger s soluo. Soluo. I s eve, he ers o he le se re oegve o ll e. So here wll o e eger soluo. I s he soluo. I s o les, he soluo us e eve, oherwse he le se s o. Suppose Ovousl o The he equo eoes We hve he So The we () Deere ll egrl soluos o Soluo. opleg squres, we hve or. I s eve, he org elg So I s o, he Susug o he equo splg we ge Thereore, he soluos re () Deere ll oero egrl soluos o Soluo. Epg, he splg he equo, we ge So 8 The equos Applg he qur orul, we ge us e pere squre, s 6 or So Fro hese, we ge he oero 8 7 () F ll posve egrl soluos o 8 so 8 or These le o he soluos (he: The soluos el he Soluo. We wll show here s eloe se o soluo, el. To spl he equo, we oser oulo. We hve o. I ollows h us e eve, s. The re o oh vsle, se her su s o vsle. So,. The, o o. Fro hese, we ge s o s eve. I, he or o 8, Now. 6

17 oro. So we ge.. The ples. Fll, Fll, we oe o he os ous Dophe equo. Le us ee Phgore rples o e rples o posve egers ssg. For eple, re Phgore rples. lerl,, he or posve eger. So, soluos o wh relvel pre (.e. hvg o oo pre vsors) re ore por. These re lle prve soluos. elow we wll eslsh ous heore gvg ll prve soluos. Theore. I u u he u u u gve prve soluo o. oversel, ever prve soluo s o hs or, wh possle peruo o. (For eple, u.) re relvel pre posve egers o oppose pr els Resos. For he rs see, u u. Suppose wo o hve oo pre vsor p, he he equo wll pl ll hree hve p s oo vsor. Noe p se u p relvel pre. So eg e relvel pre. Ths ors u re o. The p us For he seo see,, he or (o ). So, re lso relvel pre, he oe o or s o he oher s eve. Le us s s o s eve. The s o ollows re posve egers. Noe relvel pre ples p he p p ). Now. I ollows h oh re pere squres wh o oo pre vsors. Le us s u. The u u u. Se s o, u re o oppose pr. o hve oo pre vsor (or p Rer. The geerl soluos o re eher rvl wh or equls or orvl wh u u o he or u u where re s ove s posve. o he or Soluo. Le e soluo, he o o Se u u u ples u Se he wo ors o he rgh er us e powers o, we hve So re eve, s The re relvel pre, he heore o Phgore rples, 9. Show h whh gves 7 where u oe s o, he oher eve. Now The u. F ll egrl soluo(s) o rs.). F ll egrl soluo(s) o. Show h Eerses 9 hs o egrl soluos. 9 (H: oser 7 hs o egrl soluos. (H: Noe use eple.) o 9 *. Show h hs o posve egrl soluos. (H: Suppose s posve soluo wh les possle, he show here s oher soluo wh sller vlue usg he heore o Phgore rples.) The lso olue h hs o posve egrl soluo. Rers. The ous Fer s Ls Theore s he see h or ever eger he equo hs o posve egrl soluo. Fer le o hve proo ers go, u oo ou hs proo. Ol ew er go, Arew Wles ll prove. Hs proo ws pges log. Eple. (6) F ll posve egrl soluos o heore o Phgore rples. usg he

18 . oors oors s he su o oug oes. There re s, e useul prples h llow us o ou eel.. Ao Mulplo Prples. Ao Prple. Suppose eves A A respevel. I ll o hese ouoes re s, he he uer o ouoes ue o eve A or eve A or or eve A s A hve ouoes Rers. Whe ou ee o ou soehg, he ul s how o re up he hgs o groups h re es o ou. Eples. () Flppg o (eve A ) resuls wo ouoes: he or l. Tossg e (eve A ) resuls s ouoes:,,,,,6. So lppg o or ossg e resuls 6 8 ouoes. () F he uer o squres hvg ll her veres elogg o rr o equll spe os. Soluo. Eh suh squre hs uque rusre squre wh ses prllel o he ses o he rr! Use hese rusre squres s eves. I rusre squre s (h s, eh se hs os), he here re s squres (ouoes) sre hs rusre squre. For rusre squres o eso 9 usg he oruls So he swer s Mulplo Prple. Suppose eves A A ouoes respevel. The he uer o ouoes o eve A eve A..., ollowe eve A s 9 9 A hve here re 8 ollowe Eples. () Le e egers suh h () F he uer P (or P ) o peruos o s oes e e. Ths s he uer o ws o g o he oes oe er he oher whou replee so orerg s por. (I s o eoe, he he eul vlue s ) () F he uer (or or ) o oos o s oes e e. Ths s he uer o ws o g o he oes he se e so orerg s o por. Soluos () There re ouoes o g he rs oe, ollowe ouoes o g he seo oe,..., here re ouoes o g he -h oes. he ulplo prple, P!! () Dve ll peruos o oes e e o groups h hve he se oes. I eh group, he uer o ws he oes were e ws P! eh group s ol oue oe. Hee s he uer o groups, hee P!!!! Se orer s o por oos, he! ws () F he uer o posve vsors o posve eger N hvg pre oro e e p e o posve egers N F he uer o orere prs suh h l Soluo. Ever vsor o N s o he or A A p A A where or So he uer o ouoes he eve o llg A ollowe he eve o llg A... ollowe he eve o llg A s e e e Ths s he uer o posve vsors o N Le p p N ol e e... e oser he eve e here re e possle ouoes, el e e e e here re e ouoes or he eve e suh h l s e e e e e e 6 e The l e Slrl, So he uer o prs

19 A e suses o S. eo Prple. eo Prple. I here s oe-o-oe orrespoee ewee he ouoes o eve A ouoes o eve, he A hve he se uer o ouoes. Reer. Whe we re se o ou he uer o ouoes o eve, soees e possle o se up oe-o-oe orrespoee wh oher eve whose ouoes re eser o ou. Eples. () Le e posve eger. I how ws oe wre su o les wo posve egers h up o? oser he se se o egers wre ere orer s eg ere. (For eple, here re ws o epress s ) Soluo. (Frs he swer e sovere rg he ses oserve per!) We su he se wh or. Noe se eh su hs les wo ers, here s les o. We hve Eh o he su s oe-o-oe orrespoee where eh l squre s lle wh For geerl wh l squres. he ulplo prple, llg he l squres oe ollowe he oher, here re ws, u llg ll l squres wh s o llowe. So here re ws. we hve (6) How phs re here gog ro o o he oore ple suh h eher he or he oore o ever po o he ph s eger he oores o he ph re lws oeresg ever oe? Soluo. Noe suh ph s srse ro o wh orers le pos (.e. pos wh The eh pee s eher ovg le or up. proeg he ph o he -es, we see h he legh s us here re le up pees. There s oe-o-oe orrespoee ewee ph sequee o les ups soe orer. Hee, og he pees, epes where we e he les. Thereore, here re phs. egers). re he ph o u legh pees. 7 (7) Eh o he veres o regulr ogo (.e. 9-se polgo) hs ee olore eher re or lue. Prove h here es wo ogrue oohro (.e. veres hvg he se olor) rgles. Soluo. he pgeohole prple, here re les ve veres o he se olor, s re. So here re les rgles hvg re veres. For eh rgle A (veres lowse orer), e vere A, go rou he ogo he lowse reo ou he uer o ses o he ogo rvelle ro A o ro o ro o A There s oe-o- Arrge hese hree uers resg orer oe orrespoee ewee he ses o rgle A re posve egers, 9 here re 7 ouoes, el Se Se he pgeohole prple, here us e wo ogrue re rgles. (8) Le e egers greer h. Le S e se wh elees, le A A Assue h or wo elees S here s se A og eher or u o oh. Prove h Soluo. For eh elee S s he se A I he er soe oores. So he elees S orrespo o ere he ulplo prple, here re el possle s. Thereore, Pgeohole Prple. ee oherwse where Pgeohole Prple. I or ore oes re pu o oes, he les wo o he oes wll e he se o. More geerll, oes re pu o oes, he les oes wll e he se o. Eples. (9) (9 Pu E) Fve po re hose ro se o he u squre. Show h here re wo pos wh se os 8

20 So Soluo. Dve he se o he u squre o squres wh se pgeohole prple, here re wo pos he se squre. The her se s os he legh o he gol, whh s () Eleve uers re hose ro he s. he Show h he su o wo o Soluo. oser he ses prple, wo o he wll e ro he se se. Hee he su s. () Fro se o egers, where he su o whose elees s vsle Soluo. Le S 9 e he egers. oser he S S he pgeohole prple, here re S The S S o o whose elees s vsle () Suppose uers re hose ro o he suh h oe ves he oher. he pgeohole show h here us e suse S uers S wh suh h S S o s suse he su Show h here re wo Soluo. For eh o he uers o he or wh o. There re possles or wo o he hve he se or. The he oe wh he sller epoe ves he oe wh he lrger epoe el he pgeohole prple () Eh pr o 6 s pos re oe re or lue le sege. Show h here s re or lue rgle. (Noe 6 s reple, he prole wll o e rue s oe olor he eges o pego re he gols lue o ge ouereple.) Soluo. Te oe o he 6 pos, s A seges ro A here re o he se olor, s A A AD re re. Eher rgle D s lue or oe o he se, s s re, he rgle A s re. he pgeohole prple, og he Alerve Forulo. Aog 6 people, eher here re who ows eh oher or here re wh o pr ows eh oher. Soluo. Assoe eh perso po. Drw re sege og he pos he orrespog people ows eh oher, lue sege he o ow eh oher. The use he resul ove. Rers. There s ous heore ow s Rse s Theore, whh ssers h or posve egers p q R p suh h he ollowg see s rue: q here s slles posve eger I pos re gve wh o hree oller ll le seges oeg prs o he re olore re or lue, he eher here re p pos, ll seges oeg he re re or here re q pos, ll seges oeg he re lue. Eple () s he see R 6 p re ow. Also, Rse s heore e eee o ore h olors. rel, gve egers p here s les suh h ll seges oeg pos re olore olors he eher here re p pos wh seges ll olore or p Prple o Iluso Eluso. Ver ew Rse uers R or here re p pos wh seges ll olore. Prple o Iluso Eluso (PIE). Le S eoe he uer o elees se S A A A A A A q A A A A A A A A A A A A A A A 9

21 p I geerl, A A A A A A A A A Eples. () F he uer o posve egers os whh re vsle or or. Soluo. Le A e he se o posve egers os whh re vsle A A l l Se A A A A A 8 A A A A 7 6 l so PIE, A A A A 6 A A A l 7 8 l () (Dergee Prole) How ws leers e pu o evelopes so h o leer goes o he rgh evelope? Soluo. Toll here re! ws o pug leers o evelopes. We wll ou he oppose suo, where les oe leer goes o he rgh evelopes. Le A e ll possle ws o pug leers o evelopes suh h he -h leer goes o he rgh evelope. Se he oher leers go rol o he oher evelopes, A! A A! or s he oher leers e rol ple, e. PIE, A!! A!!! Slrl, A A!! A! A A So he uer o ws o leers go o he rgh evelopes s! A A A! whh s lose o e!!!! (Noe he prol h o leers wll go o he rgh evelopes s he ove epresso ve! whe!!! s lrge.) (6) (Euler p e pe -uo) For posve eger p e h re relvel pre o s s he pre oro o Soluo. Ise we ou he uer o egers relvel pre o p p Le A e he uer o egers p PIE, Hee A A p A A p p p p p p A p p A p p p A p show h he uer o egers p p p where h re o The hese egers re vsle les oe o he pres A p p p h re vsle A A p p

22 r r Reurree Relos Geerg Fuos. reurree relo s pll or- For eple, he ous F he Usg he reurree relo, Gve sequee ul or ers o Fo sequee s ee he l oos F reurree relo F F F or we see h he Fo sequee F s 89 For sequee he geerg uo o he sequee s (Noe hs seres o overge or ll rel ) We oe use reurree relos geerg uos o help oug hgs. Eples. (7) Le e he uer o regos ore o ple les, o wo o whh re prllel o hree ourre. F Soluo. lerl, 7 rwg pures. To solve he prole reurso, oserve h wo o he les us erse. Suppose les ore regos. The -h le wll erse he pos. These pos ve he -h le o prs eh pr us oe o he regos o wo. So Thereore, The 8 So where A So or r prl ro, we ge we hve r r Thereore, F orul ollows. r r r r r s s r s s Se r s s r r s s s A r Noe s s s r s e s Usg geerg uos, we oruls or ers o -h orer ler reurree relos (whh re o he or wh oss ) Theore. Le he sequee oss) or he hrers equo The here re oss suh h or ss hs roos r ( r (8) Show h he -h er o he Fo sequee s gve F Soluo. oser he geerg uo F F o he Fo sequee. Usg F F F F F F F F F F F F F F F (e s Forul) F F F we hve r r r r r r r r. Rers. Slr heore s rue or hgher orer ler reurree relo. I hs roos r r he here re oss suh h or sses r r r r r r r r r re s r r r r r r r r.

23 (9) (989 Pu E) Prove h here ess uque uo ee o suh h 6 Soluo. For ee or 6 equo 6 hs roos. So he whe s lrge, org 6 We ge 6 So Sple heg shows h Thereore, he uque uo s The The hrers I Se so sses () (99 hese Nol Seor Hgh Mh opeo) Le e he uer o posve egers hvg gs, ol su o gs equl (F reurree relo or ) show h s pere squre or Soluo. Le A e he se o ll suh egers. The A he uer ( g or) s A ol he uer A s A So or A. ge orul or usg he hrers equo eho, urs ou h orul wll o e o helpul o show s re pere squres. A So A Noe A Alhough we eore proeeg urher, we wll use he reurree relo o wre ou he ers o The sequee s Wh we oserve ro he sequee? Well, see o e rue. I rue, hese sh he prole or us euse wll pl ll s re pere squres uo. We ge 9 he sees o e ll pere squres. Now we he he relos hel uo. For he rs relo, he reurree relo, whh oplees he uo or he rs relo. For he seo relo, Suppose The Suppose The usg he reurree relo he rs relo, we ge whh oplees he uo or he seo relo. () Suppose egers suh h he uers Soluo. Le The So g g p The g Seg Show h s power o. so g g g g we ge re wo ere groups o posve g g g p re he se s he uers g g g Se or soe polol p p Thereore p p For he e ew eples, we wll ee o ulpl power seres. Oserve h 6

24 We For sequees o e he sequee where we ee her ovoluo s he oee sequee o he prou o he geerg uos or he wo sequees. We wll eoe he ovoluo O he ope ervl sel, we ge ereg he geoer seres.) I shor, Dereg oh ses, we ge Eples. () For we ulpl show h (Ths lso ollows ro 9 Soluo. Noe he le se s ovolue epresso. Now Ther prou s o s 9 6 whh s o egree 7 Now 6 So he oee where or ll rue or The ol heore ssers h or rel uer! I However, s o oegve eger (eg. oserg he ses!!! s oegve eger, hs s rue ), he s ol we hve! Eple. () (l Nuers) F he uer T o ws ove -se polgo e ve o rgles oerseg gols. (Here oerseg es o erseo se he polgo.) Soluo. The rs ew ses re T se ove polgo oe o he rgles wll e wh he polgo hs ses he polgo ses. These el T T rgulos o rgle or ro o T T T T T T ovolue epresso. (For he se we ge T oser he geerg uo hve T T T T T T T T T T oser F ege T 6 8 T T T - For o hese vsos, T T T hs hvg T shoul e se o.) Sug whh s T T T T For eh suh

25 9 So we ge Se or T T T T 6 Solvg or wh T he plus sg s reee. oprg oees, we ge T Eerses.!!!. (98 USSR Mh Olp) Le e o eger greer h. Show h oe o he uers s vsle (H: Noe o he s ogrue o o ). Show h here re egers h 6 suh h o ll ero, wh solue vlues less 6 (H: oser ll uers o he or r s see how he re srue.) 6. (976 USA Mh Olp) Eh squre o 7 or s olore whe or l. Prove h wh suh olorg, here s lws regle whose our orer squres re o he se olor. Is hs rue he or s 6?. For he gs o,, eh les oe? (H: Le A e he se o g uers, eh o s gs s or.) how g uers re here suh h eh g s, or. I group o people, suppose everoe ows les oher people he group. Show h here re hree people he group who ow eh oher. 9 where r s 6. (978 Ausr-Polsh Mh opeo) There re 978 lus. Eh lu hs eers. I s ow h ever pr o hese lus hs el oe oo eer. Show h here s oe eer who elogs o ever lu. 7. (96 IMO) Seveee people orrespo l wh eh oher. I her leers ol ere ops re susse. Eh leer els wh ol oe o hese ops. Show h here re les hree people who wre o eh oher o he se op. (H: Use 7 pos olors. Oe olor or eh op.) 8. Show h or egers p q he wh p q 6 or ever posve eger 9. Le e he uer o ws whh regle e ore ou o regles. F reurree relo o ers o he ers o. Le e he uer o srgs o sols eh o whh s eher, or suh h o wo oseuve s our. Show h ers o 8 (Noe. Show h here re ws o rrgg he egers row suh h eep or he leos, ever uer er ro soe uer o s le or (H: The rghos eger us e or ) F. I eple (), show h F F s he -h Fo uers. ) s eve s o, where F

26 . Fuol Equos A uol equo s equo whose vrles re rgg over uos. Hee, we re seeg ll possle uos ssg he equo. We wll le eoe he se o ll egers, or eoe he posve egers, eoe he oegve egers, eoes he rol uers, eoes he rel uers, eoe he posve rel uers eoe he ople uers. I sple ses, uol equo e solve roug soe susuos o el ore oros or ol equos. Eple. () F ll uos : or ll Soluo. Replg hve Se he: For, we hve suh h he gve equo, we hve, we Solvg or For er pe o uol equos, sr pproh o solvg he prole s o eere soe spel vlues (suh s or ), he uvel eere or use es o ll Eple. () F ll uos : or ll Soluo. Sep Tg ollow reprol vlues The ollowg re eples o suh pproh. we ge suh h (uh Equo) Sep We wll prove or Sep Tg or Sep Tg Sep For he: For So we ge we ge wh or hs s rue or Assue Tg I elg wh uos o sh he prole usg he ollowg. we ge uo. Ths s rue or Thereore, er g he uo o Thereore, wh we oe Des o Rol Nuers. For ever rel uer, here re rol uers p p rese o here re rol uers q q o q s We eoe hs p 9 erese I ollows ro el represeo o rel uers. For eple, s he ls o Eple. () F ll uos : or ll or Soluo. Sep eple, Sep I he So, s resg. Sep I suh h p p p So suh h or lso, he he es o rol uers, here re p q q s q p q As q So seps, p

27 q or ll So p he: For wh or q wll squeee wh Thereore, o. We ge Rers. () I eple, we reple he oo or s oooe, he he swer s essell he se, el wh or s reple s ouous, he seps eple re o eessr. We e rol p e l o p p p o ge se p Also, he oo () The uh equo or ll hs oouous soluos ( prulr, soluos o o he or ). Ths requres he oep o Hel ss o he veor spe over ro ler lger. The ollowg re soe useul s rele o he uh equo. F. Le A ll A or or or ll A [ I : A or ll sses he eher Proo. eple, we hve or ll or ll A se or ll we ge The I he es o rol uers, here re p p q s q q A he F. I uo : s oooe, he eher or ll Proo. For or. I A I Oherwse, we hve or he A whh ples s resg. Now or A q suh h p As s resg, we hve p Tg ls, he swh heore gves he p q or ll sses or ll or ll or here ess suh h Also, ples or ll I he or ll (se whh woul le o oro). g Dee g : l e g l e ples l e e The l e l Se s oooe, ollows h g s lso oooe. The g or ll Thereore, e g As pplo o hese s, we loo he ollowg eple. Eple. () ( IMO) F ll uos ro he se sel suh h or ll g or ll o rel uers o Soluo. (Due o Yu Ho Pu, Hog Kog IMO e eer, gol els) Suppose or ll or or ll or re soluos. or ll The he equo ples Reversg seps, we lso he Suppose he equo s sse oos uo or ll uo or ll Seg ero uo, or we ge I prulr, Seg ples I he ler se, seg Hee we ssue Seg or ll Dee he uo g : g we ge we hve So s eve. g whh ples we ge So ol e Seg or we ge he os Se s o he or ll g g whh The or ll

28 Ne s eve ples g equo, we ge or ll g g Tg squre roos leg g or ll g g or ll Seg, we hve g or ll ollows g or ll he: I he he equo reues o g Se we ge g he gve s eve, whh s well ow e esl e hee epso or see ro p q pq where p q The oep o e po s oher useul e g uol equos. Kowg ll he e pos re por er pes o uol equos. Deos. s e po o uo s he o o or he uo s lle he -h ere o Le () (98 IMO) F ll uos : ll s Soluo. Sep Tg se we ge Sep Tg o or ever we ge we ge The Sep Suppose hs e po po, s lso e po, So s e po o suh h So sep, Tg or s e po s lso e So s re e pos. Se s u Sep Suppose hs e po.e. hs e po po o Ths ors The So oes o hve e po org sep. So oes o hve e Sep Seps,, showe he ol e po o s or ll he: For (6) (996 IMO) F ll uos : or ll Soluo. Sep Tg or ever Tg we ge we ge Also he equo eoes Sep I s e po o ll lso e po. The ses As suh h.e. he we wll show Sep I s he ol e po o, he Ovousl, he ero uo s soluo. Oherwse, hs les e po re where r So r sep, we ge s e po o s e po o or re ow. Suppose s e po, he or ll so We wll show he ol e pos Suppose s e po. he vso lgorh, We hve r r r Se s he les posve e po, r 6 r r s sep. r

29 Se Sep For s e po or ll hve We he vso lgorh, r r he: For eh le uos r [ ] he ero uo) re ll he soluos. Wre r s The r s l r r r sep, r r r Le S e he se o e pos o he s lso e po o euse or S he Se g ples g o g re lso e pos o g we ge S S S S 8 s l r or soe We hve e rrr. The r s he reer o ve where ( r l s wh Oserve h s e po o So es S o sel. Also s eve o S euse g g Leg g Eple. (7) F ll uos : g suh h so he e pos 8 respevel, or ll Soluo. Assue suh ess. I urs ou S S re useul or hs prole. The e pos o re he roos o.e. S pos o re he roos o.e. S S S or ples s e po o I he Se perues S 7 The e Ths es h S s e se, he s peruo o S Le whh s o rue. So hee g oro. So o suh es. The The ove eples showe rol or ssel ws o solvg uol equos. The ollowg eples show soe oher pprohes o el wh hese equos. Eple. (8) F ll uos : or ll Soluo. lerl, he e uo s he ol soluo. To show h he s oro. Thereore, The s suppose Iuvel, suppose Thereore, or ll suh h s s soluo. We wll show h Le s hel uo. The (9) (987 IMO) Prove h here s o uo : 987 Soluo. Suppose here s suh uo 987 So suh h The s eve,.e. 987 Oserve Suppose sses el s vlues sses he s vlues s re s euse s eve.) Now suh h Se suh h sses ol he vlues o ohers. Se 987 sses he 987 vlues 987 hs s oro. or ll Ths shows The (The so here s he here s () (999 IMO) Deere ll uos : suh h he.e.

30 or ll Soluo. Le Seg Le A e he rge o he or euse Ne, we se Solvg or we ge Ths es A Now or rrr Se or he: For A :. F ll uos :. F ll uos :. F ll uos : A we ge hs gves : A : A le A e suh h so oh ses equl Eerses suh h suh h suh h 9 Hee, we hve 8 or ll or ll or ll So The or ll. F ll uos : () () l (H: For oser suh h or ll. (986 IMO) F ll uos : [ () () 6. Suppose : F or *7. (99 IMO) Le uo : or or s suh h ers o [ or ever suh h ) e he se o posve rol uers. osru suh h or ll *8. (99 IMO) Le S e he se o rel uers greer h : S S suh h () () s srl resg or (H: Show ol hve e po.) *9. (99 IMO) F ll uos : (H: Assue resg.) *. F ll uos : (H: Tr or ere or ll or suh h he show suh h 6 or ll or ) F ll uos S s

31 6. Iequles (Pr II) For equles volvg uos, he shpes o he grphs o he uos o ervls re ver por. Deos. A ouous uo s ove o ervl I or ever I Also, s srl ove o I s ove o I equl hols ol whe ove. A uo s ove o ervl I he uo (I h se, or ever srl ove o I s srl ove o I I s ove o I I Slrl, s Seo Dervve Tes. I o he ope ervl I he s ove o I o I he s srl ove o I or ove srl ove uos re slr reversg he equl sgs. The sees For uos ee o ervls og epos, ou he epos oegve seo ervves se he ervl re sue or he uos o e ove o he ervls. Slr sees or srl ove uos o suh ervls re rue. Usg he seo ervve es, we he h he ollowg re eples o srl ove uos o ervls: p o [ o or p p o or o [ or p The ollowg re eples o srl ove uos o ervls: p o [ or os o [ p log o ] 6 s o [ ] or The os por equl oerg hese uos s he ollowg. Jese s Iequl. I s ove o I I For srl ove uos, equl hols ol he Geerle Jese s Iequl. I s ove ouous o I I wh he (wh he se equl oo or srl ove uos). uos, ll equl sgs reverse. Eples. () For rgle A eere whe equl hols. Soluo. Se s A s s show h s A s s srl ove o [ Equl hols ol A () I Soluo. Noe srl ove o I euse 9 8 A A s.e. s A ] so he he u o Le 6 9 s For ove A s equlerl. o I or I he s

32 Jese s equl, 9 Thereore, he u s 9 () Prove he AM-GM equl. Soluo. I log Jese s equl, log e whe he se log log s srl ove o log Epoeg oh ses, we ge he AM-GM equl. log Rers. I we use he geerle Jese s equl se, we ge he weghe AM-GM equl, whh ses h ssg he wh equl ol () Prove Höler s equl, whh ses h p (Noe he se p Soluo. Le A q re rel uers, he q p p q q s he uh-shwr equl.) p q p q I A or s, he eher ll s or ll s re, whh wll e oh ses o he equl. So we ee ol 6 oser he se A Le A Se e s srl ove o equl, p q Ag hese or Thereore, l A p q p l we ge p q Le q p l p A p q he q he he geerle Jese s q l p p p Ne we wll roue geerlo o Jese s equl. Deo. I he we s ores ss he oos wre q q q Moro Iequl. I he uo s ove o he ervl I or I he 6 [ ]

33 For srl ove uos, equl hols ol or The sees or ove uos e oe reversg equl sgs. Eples. () For ue rgle A os A eere whe equl hols. os show h os Soluo. Whou loss o geerl, ssue A Se we hve srl ove o I A A ] A [ A The A Se he oro equl, os A os os os s For he rs equl, equl o hol (s wo o he gles o oh e rgh gles). For he seo equl, equl hols ol he rgle s equlerl. so we ge (7) F he u o Soluo. Noe he ouous uo he we l h o So he oro equl, The u vlue Ag I Thereore, s ove o [ Ths s euse s e whe o oh ses, ] euse I soe proles, he uos we ee o oser o e ove or ove o he ere ervls! Here s eple. (8) (999 IMO) Le e e eger, wh () Deere he les os suh h he equl (6) I rhe e o we ge Jese s equl.) Soluo. For he (Applg hs o he oro equl, we hve o show where s he Se hols or ll rel uers () For hs os Soluo. oser he se (.e. h eere whe equl hols. rs. Le ), he h 8 h h 6 66

34 wh equl ol h.e. Eerses For he se eoes le So [ ] or he I ers o s, he equl o e prove The le se e epe regroupe o gve Uoruel, 6 6 s srl ove ol o [ o Se he equl s ser he s, we ssue I equl, he se I he oro equl se he [ Se we hve ] s he oro 8. Use he eo o ove uo o gve proo o Jese s equl or he se o prove he se The use he se. Use Jese s equl o prove. Le Prove h [ ] eere whe equl hols.. Use Höler s equl o prove h he e suh h or ll *. Le P e po se rgle A suh h P A P P A (H: Appl se lw o hree rgles.) Show h 6. Suppose ue rgle A hs wo gles less h or equl o 6 h s A s s 7. Reo eple (8) usg AM-GM equl. 8. Le u o 9. Prove h 96 he s s 6 s Show F he u 8 Equl hols ol wo o he vrles re equl he oher vrles ll equl.. (998 l Mh Olp, se Prove h ) Le 67 68

35 7. Mhel Ges (Pr I) A vr s qu h oes o hge. A oovr s qu h eeps o resg or eeps o eresg. I soe hel ges, wg oe oes ro uersg he vrs or he oovrs h re orollg he ges. Eples. () (97 Kev Mh Olp) Nuers o or. You re llowe o reple wo o hese uers oe uer, whh s eher he su or he eree o hese uers. Show h er 97 es o perorg hs opero, he ol uer le o he or o e. 97 re wre Soluo. There re 987 o uers o he or he egg. Ever e he opero s perore, he uer o o uers le eher ss he se (whe he uers e ou re o oh o) or ereses wo (whe he uers e ou re oh o). So he uer o o uers le o he or er eh opero s lws o. So whe here s oe uer le, us e o, hee o e. () I 8 8 or, here re whe pees l pees, oe pee eh squre. I pler hge ll he whe pees o l ll he l pees o whe row or olu sgle ove, he s possle h er el oves, here wll e el oe l pee le o he or? Soluo. No. I here re el l pees row or olu eore ove s e o h row or olu, he er he ove, he uer o l pees he row or olu wll eoe l pees o he or. Se 8 s eve, he pr o he uer o l pees ss he se eore er he ove. Se he sr, here re l pees, here o e l pee le e. hge o () Four s ve o s re wre rou he rle rrr orer. I wo oseuve sols re he se, he ser ew ewee he, oherwse ser ew o ewee he. Reove he ol s o s. Keep o repeg hs opero. Is possle o ge e o s? 69 Soluo. I we le o he oe h oseuve sols re reple her prou. I we oser he prou P o ll e vlues eore er eh opero, we wll see h he ew P s he squre o he ol P Hee, P wll lws equl er opero. So e o s elg P ever hppe. () There re hree ples o soes uerg respevel. You re llowe o hoose wo ples rser oe soe ro eh o hese wo ples o he hr ples. Aer severl o hese operos, s possle h eh o he hree ples hs soes? Soluo. No. Le he uer o soes he hree ples e respevel. oser (o ) o hese uers. I he egg, he re opero, he eoe o er whh wo ples hve soes rser o he hr ple. So he reers re lws soe orer. Thereore, ll ples hvg soes re possle. () Two os pl he ollowg ge wh wo ples o es. I he rs ple, here re es he seo ple, here re es. Eh o es ur o e ove ossg o eg wo es ro oe o he ples or rserg ro he rs ple o he seo. The o who o e ove loses. Show h he o who ple seo o lose. he w? Soluo. oser S o e he uer o es he seo ple us he rs. Ill, S So S (o ) hs he per rs e ove, S (o ) woul lws e. Now o loses ol here re o es le he rs ple oe le he seo ple, he S o lose. Aer eh ove, S reses or ereses. Ever e er he o who ple So he o who ple seo lws e ove, hee Se eher he ol uer o es ereses or he uer o es he rs ple ereses, so eveull he ge us sop, so he o who ple seo us w. 7 Aer oe

36 (6) Eh eer o lu hs os hree eees he lu. (Here eees re uul.) Show h he eers e ve o wo groups so h eh eer eh group hs os oe ee he group. Soluo. I he egg, rol ve he eers o wo groups. oser he uer S o prs o eees he se group. I eer hs les wo eees he se group, he he eer hs os oe ee he oher group. Trserrg he eer o he oher group, we wll erese S les oe. Se S s oegve eger, o e erese orever. So er el rsers, eh eer hve os oe ee he se group. Rers. Ths eho o provg s ow s he eho o e ese. I showe h ou o lws erese qu whe ol hve el possle vlues. (7) (96 All-Russ Mh Olp) Rel uers re wre le. I s perssle o reverse he sgs o ll he uers row or olu. Prove h er uer o hese operos, we e he su o he uers log eh le (row or olu) oegve. Soluo. Le S e he su o ll he uers he le. Noe h er opero, eh uer ss he e or urs o s egve. Hee here re os les. So S ol hve el possle vlues. To e he su o he uers eh le oegve, us loo or le whose uers hve egve su. I o suh le ess, he we re oe. Oherwse, reverse he sg o ll he uers he le. The S reses. Se S hs el possle vlues, S rese el es. So eveull he su o he uers ever le us e oegve. (8) Gve pos he ple wh o hree o he oller. Show h he e ve o prs suh h he seges og eh pr o o erse. Soluo. I he egg rol pr he pos o he seges. Le S e he su o he leghs o he seges. (Noe h se here re el ws o oeg hese pos seges, here re el possle 7 vlues o S ) I wo seges A D erse O, he reple prs A D A D D AO O O O D A D he rgle equl, wheever here s erseo, og hs replee wll lws erese S so eveull here wll o e erseo. Se A Eerses. Ever uer ro o,, s reple he su o s gs. The resulg uers re repeel suee o he se opero ul ll he uers hve oe g. Wll he uer o oes he e e greer or less h he uer o wos? (H: Show h he su o gs o s ogrue o (o 9).). (989 Hugr Mh Olp) I he veres o squre, we ple soe hes. Ill here s oe h oe vere o h he oher hree veres. I oe ove, s llowe o reove uer o hes oe vere ple he wo e veres ol o we he uer o hes reove. Aer el oves, he uer o hes e oug lowse or ouerlowse he our veres? *. I eh squre o 8 8 le eger s wre. We hoose rrr or sule rese ll he uers oe. Is possle o o uers vsle hree ll squres o he 8 8 le er el suh operos? (H: Mr soe squres so h ever or sule wll over ulple o re squres.) *. The uers ehge wo e uers. Prove h o uer o suh ehges proues rrgee eessrl ere ro he l oe. re rrge soe orer o le. We. Severl uers re wre rou rle. I our oseuve uers peror hs opero ol el es. ss we ehge Prove h we 6. Eh e o ue hs uer wre o, o ll he uers re he se. Eh o he uers s reple he rhe e o he uers 7 Se here re ol el possle vlues or S

37 wre o he our e es. Is possle o o he l uers o he es g er les oe suh operos? 7. Fel squres o e squre gr rw o whe pper re pe l. A eh oe e olor o he or o he ollowg squres: he squre sel s op rgh-h eghors. Prove h soe e ler here wll e o l squre ll. eh squre es he 8. I he ple, here re pos, o hree oller, les, o wo re prllel. Prove h we rop perpeulr ro eh po o oe o he les, oe perpeulr per le, suh h o wo perpeulrs erse. 8. Mhel Ges (Pr II) There re hel ges volvg sreges o w or o ee. These ges volve uer heorel properes or geoerl eoposos or oorl resogs. Soe ges go o orever, whle soe ges oe o sop eveull. A wg sreg s shee h llows he pler o e oves o w he ge o er how he oppoe pls. A eesve sreg us o he oppoe s roues o wg. The ollowg eples llusre soe sr ehques. Eples. () There s le wh squre op. Two plers e ur pug ollr o o he le. The pler who o o so loses he ge. Show h he rs pler lws w. Soluo. The rs pler pus o he eer. I he seo pler e ove, he rs pler pu o he poso serll oppose he poso where he seo pler ple hs o wh respe o he eer o he le. Se he re o he vlle spe s eresg, he ge us e eveull. The rs pler wll w. () (he s Ge) Ill, here re heers o he le, where persos e ur o reove les os heers eh e ro he le. The ls perso who reove heer ws he ge. For wh vlues o wll he rs perso hve wg sreg? For wh vlues o wll he seo perso hve wg sreg? Two Soluo. esg sple ses o o he egg, he he rs perso hs wg sreg, oherwse he seo perso hs wg sreg. we esl see h s o ulple To prove hs, le e he uers o heers o he le. I q r wh r he he rs perso w reovg r heers eh e. (Noe r ever e he rs perso s ur se he egg s so he seo perso srs wh ulple o heers eh e ol reove o heers.) However, s ulple o he egg, he o er how heers he rs perso es, he seo perso ow w reovg r heers ever e. 7 7

38 () (Ge o N) There re ples o heers o he le. The rs, seo hr ples hve les oe o ll heers h ple eh e ro he le. The ls perso who reove heer ws he ge. Who hs wg sreg? heers respevel he egg, where Two persos e ur hoosg oe o he hree ples reovg Soluo. I se represeos, le N where (o ). (Here we hve o eros o he le o he se represeos o o esure he hve o he se uer o gs.) The rs perso hs wg sreg ol N s o,.e. o ll s re. To see hs, suppose N s o. The wg sreg s o reove heers so N eoes. Whe he s re o ll eros, loo he slles suh h oe o he equls s rs ple so h hs e e e heers le, where e The reove heers ro he (For eple, N he hge o ) Noe e er he ove so h he ew s less h he ol N eoes s ll he eore he ove wll eoe he eo o e Also, er he ove, So he rs perso lws e ove. The seo perso wll lws hve N hs ur g ove wll resul les oe o,.e. N o e ove wll lose he ge. As he uer o heers s eresg, eveull he seo perso () Twe grls re sg rou le re plg ge wh rs. Ill, oe grl hols ll he rs. I eh ur, les oe grl hols les wo rs, oe o hese grls us pss r o eh o her wo eghors. The ge es ol eh grl s holg os oe r. () Prove h he he ge o e. 7 () Prove h he he ge us e eveull. Soluo. () I holg les wo rs ever oe. So he ge o e. he pgeohole prple, here s les oe grl I he lel he grls G G le G e he grl holg ll he rs ll. Dee he urre vlue o r o e s eg hel G S G he lowse reo Le S e he ol vlue o he rs. Ill, oser eore er G psses r o eh o her eghors. I he S reses he S oes o hge euse I he S ereses euse 9 ulple o. Assue he ge wll e. The eh grl hols r S he ge o e. I So eore er oves, S s lws whh s o ulple o, oro. So () To see he ge us e le s hve he wo grls sg he r whe s he rs e oe o he psses r o he oher. Wheever oe grl psses r o her eghor, le s requre he grl o use he sge r ewee he pr vlle. So sge r wll e su ewee he pr who sge. I here wll e pr o eghors who ever sge r, hee ever ehge r. I he ge go o orever, reor he uer o es eh grl psse rs. Se he ge go o orever, o ever grl psse r el e. Also, se here re rs prs o grls sg e o eh oher, here s pr o grls who o o sg r, hee hve o ehge. Movg lowse oe grl e ro hs pr, eveull here s pr G G suh h G psse rs el es G psse rs el es. Ths s lerl possle se G wll eveull soppe pssg rs woul eep o reevg rs ro G () (996 Irsh Mh Olp) O 9 regulr hessor, he ollowg ge s ple. Ill, uer o ss re rol ple o soe o he squres, o squre og ore h oe s. A ur osss o ovg ll o he ss sue o he ollowg rules: 76