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1 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo Revst Eletronă MteInforo revstă de mtemtă lunră) MARTIE IN wwwmtenforo ARTICOLE : Other solutons for two prolems of REOIM nd L Get de l RME Asupr une proleme de mtemt dt l onursul nterjudeten Trdent- 5 Petru Morosn 9 Brl Formul lu Crdn pentru reolvre euţe de grdul 9 Reolvre CONJECTURII lu DHLEHMER refertore l ECUATIA k Aplţ le teoreme lu Lgrnge în demonstrre unor relţ trgonometre 4 Coordontor: Andre Otvn Dore E-ml pentru rtole: revsteletron@mtenforo dorendre@hooom
2 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo Other solutons for two prolems of REOIM nd L Get de l RME Ttu Zvonru Comăneşt nd Neul tnu Buău Prolem 9 from REOIM No 47 Novemre Ferero Our oluton: We denote: [ AOC ] [ COA ] [ BOA ] [ AOB ] [ COB ] [ BOC ] nd then: α β 4 γ 5 6 Also we denote: BA CB AC AC B A CB nd then the theorem of Cev we dedue tht: AC Beuse the trngles AOC nd BOC hs the sme hgh nd we otn tht CB 6 nd smlr 5 4 From the sme reson we hve [ ABA ] [ AAC ] whh s wrte: e 4 6 ) ) Anlogous we otn tht: [ BCB ] [ BAB ] e
3 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo 6 ) ) If γ β α then ) nd ) we otn suessvel tht: ) ) ) 4 6 ) ) ) ) ) ) ) ) ) ) ) ) )] ) )[ ) ) ) ) Therefore f γ β α then t lest one of numers re eul wth We ssume tht Then ) we otn tht: ) where we use the ft tht: Hene γ nd αβγ nd the proof s omplete Prolem 97 Vol 6 ) No pp 7-6 Our oluton: B Cuh-Bunkovsk-hwr s neult we hve:
4 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo 4 ) nd then s suffent to show tht: ) 6 ) 9 4 ne: t remns to show tht: ) ) ) whh s true We hve eult f nd onl f nd we re done
5 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo Asupr une proleme de mtemt dt l onursul nterjudeten Trdent-Petru Morosn 9 Brl Profesor ern George-Florn Leul Tehnolog Grgore Mosl Brl L ls 7- fost propus urmtore prolem de geometre: Fe trunghul ABC suttungh s D Є BC porul nltm dn A Bsetorele unghurlor ABC s ACB ntersete AD n E respetv F stfel nt BECF Artt trunghul ABC este sosel Vo preent n ontnure metode de reolvre este proleme Metod In Δ ABD s Δ ADC pl formul setore B AB* BD*os BE AB BD C AC * DC *os s CF AC DC dr BECF Δ ABD BDABosB Δ ADC DCACosC Dn T snusurlor vem ABRsnC ACRsnB Inloues BD DC AB s AC n formulele lu BE s CF B B *R *sn C * AB *os B *os 4RsnC os Bos BE RsnC os B AB os B) B B B os os os C *R *sn B *osc *os CF AC osc) C 4Rsn BosC os C C os os Rsn BosC C os Dr BECF 5
6 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo RsnC os B Rsn BosC B C os os de tgbos B tgcos C Fe funt F:Π/) R F) tgos Dervt lu F este os sn os sn F') os os > numtorul este m mre r os - snossn > mpl snostg < deoree ЄΠ/) / Є Π/4) < tg < de snostg << De funt F este strt restore pe Π/) de este njetv F B)F C) reult m B)m C) de ABAC Metod In reperul rten o Fe puntele A) D) B) s C ) In Δ ABD s Δ ADC pl teorem setore DE BD DF CD os B osc os B s osc E ) F ) AE AB AF AC os B osc BE * *os B *os B os B) os B) CF * *osc *osc osc) osc) dr BECF * *os B *os B * *osc *osc os B) os B) osc) osc) Imprt prn * os B *os B * osc *osc * os B) os B) * osc) osc) 6
7 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo tg os B *os B B tg osc *osc C os B) os B) osc) osc) os B ) os C ) sn B sn B os B) os B) sn C sn C osc) osc) os Bos B os B) sn Bsn B os B) os B) osc osc osc) snc sn C osc) osc) os Bos B Reult sn Bsn B os B) osc osc snc sn C osc) * Fe funt F:) R F) * ) ) Dervt funte F este ) * ) F') > pe )Reult F este strt restore pe * ) * ) ) ) ) de F este njetve FosB)FosC) B s C sunt unghur sutte reult m B)m C) de ABAC vem Δ ABC este sosel Metod In Δ ABD s Δ ADC pl teorem setore DE BD DF CD os B s osc AE AB AF AC In Δ ABD In Δ ACD os B) BD BA BE os B osc) CD CA CF osc Dr De BE CF BE CF Dr s BA BD DA CA CD DA os B) os B) De BD BD DA BE os B Anlog se otne osc) osc) CD CD DA CF osc 7
8 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo osc) DA CF CD osc BE BD Dr AD BC de * BD os B)os B) os B)) DA DA BD DA os B) os B) os B AD BE BD osb)osb) DA DA osb) osb) Anlog se otne CF CD BD osc)osc) DA DA osc) osc) osb)osb) DA DA osb) osb) CD osc)osc) DA DA osc) osc) mprt prn s otnem DA BD * BD os B *os B CD* CD osc *osc DA* DA os B) os B) DA* DA osc) osc) tg os B *os B B tg osc *osc C os B) os B) osc) osc) m otnut es formul s n metod ontnure este n metod ) Comentru: Δ ABC fnd sosel vem AD nltme reult AD este setore Folosm fptul ntr-un trungh setorele sun onurente Avem AD BE CF{I} I este entrul erulu nsrs puntul de ntersete l setorelor Δ ABC ) Reult IEF de puntele E s F ond Des dn enuntul proleme Bsetorele unghurlor ABC s ACB ntersete AD n E respetv F se ntelege puntele E s F sunt dstnte Blogrfe: uet dt l onursul nterjudeten Trdent-Petru Morosn 9 Brl 8
9 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo Formul lu Crdn pentru reolvre euţe de grdul Profesor Remve Dnel Florn grdul ddt Leul Tehnolog Aurel Vlu Lugoj jud Tmş Fe euţ p euţ de grdul u ) Notăm u v u v) u v uv u v) u v uv u v) Avem ă uv u v ) Pentru uv p ş u v uv p Pornnd de l sstemul u v oţnem ă p uv u v u v p 7 Notând u v ş folosnd relţle lu Vete oţnem ă sunt soluţle euţe p ) 7 Clulând Δ 4 p 7 euţ ) re soluţle 4 p 7 4 p 7 4 p 4 p 7 4 p 7 4 p Dn u reultă ă 9
10 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo ) ) 4 sn 4 os sn os sn os sn os u u u u u π π π π Anlog dn v reultă ă ) ) 4 sn 4 os sn os sn os sn os v v v v π π π π Tnând ont ă rădănle euţe ) sunt de form v u putem sre soluţle p p p p p p e întâlnes următorele stuţ: ) tun de ş ) Dă tun de r ) Dă tun ş Fe [ ) ) ) ) ϕ ϕ ϕ ϕ π ϕ ϕ ϕ sn os ) sn os ) sn os r r r sn os ϕ ϕ r sn os ϕ ϕ r
11 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo Cum Oţnem stfel p p r r ϕ ϕ r os sn ϕ ϕ r os sn ϕ ϕ r os rsn ϕ ϕ r os r sn ϕ ϕ ϕ ϕ r os rsn r os r sn ϕ r os p ϕ os π π os sn 4π 4π os sn Ş ele soluţ sunt rele 4π 4π p ϕ π os sn os π π p ϕ 4π os sn os Epresle rădănlor pot f eprmte prntr-o sngură formulă numtă formul lu Crdn p p Eemple: Eemplul ) 5 4 Avem ă p 5 4 folosnd notţle de m sus jungem l form euţe ) 4 7 Δ ± 47 oluţle euţe ) vor f 5 ± 47 4 ± ) Dn formul lu Crdn soluţle euţe nţle vor f
12 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo 4 ) 4 ) 8 π π 4π 4 ) os sn 4 ) 4π 4π 4π os sn 4 π π 4 ) os sn 4 ) os sn 4 Eemplul : Avem ă p folosnd notţle de m sus jungem l form euţe ) 4 5 Δ 4 ± 9 oluţle euţe ) vor f ± 9 Dn formul lu Crdn soluţle euţe nţle vor f 9 9 rem form trgonometră pentru osϕ snϕ) 9 r în re: 9 9 π r os ϕ snϕ ϕ ϕ kπ ϕ kπ ϕ kπ ϕ k Aşdr: u os sn 7os sn π ş 6 ϕ kπ ϕ kπ ϕ kπ ϕ kπ v os sn 7os sn k ϕ ϕ π ϕ 4π oluţle euţe dte vor f: 7 os 7 os 7 os Reduere une euţ de grdul l euţ de tpul ) Fe d
13 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo Reduem termenul de grd do prn shmre de vrlă form euţ duându-se l 9 Eemplul : 4 96 p ) Fem shmre de vrlă ş oţnem euţ 5 4 *) Avem p 5 4 folosnd notţle de m sus jungem l form euţe ) Δ 6 oluţle euţe ) vor f Y Y 5 Y 4Y 5 Dn formul lu Crdn soluţle euţe *) vor f Y De Y 5 Ţnând ont de susttuţ făută oţnem soluţle euţe nţle: 7 Eemplul 4: Fem shmre de vrlă Avem p 5 ş oţnem euţ 7 5 folosnd notţle de m sus jungem l form euţe ) 7 5 Y Y ± 69 Δ De Y ± 54 8 Dn formul lu Crdn soluţle euţe *) vor f Y Y
14 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo Reultă 5 69 u 5 69 v oluţle euţe în neunosut sunt Dn oţnem soluţle euţe nţle Blogrfe: ROŞCULEŢ M --Anlă mtemtă vol I Edtur Ddtă ş Pedgogă Buureşt 964 4
15 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo 4Reolvre CONJECTURII lu DHLEHMER refertore l ECUATIA k de NICODIM A NEGREA Leul Teoret Trn DEVA Mme mele Reproduem urmtorul rtol de pe INTERNET : Wolfrm Lrr Arhve MATHOURCE Ttle Computtonl Evdene for Lehmer s Totent Conjeture Author John Rene Orgnton : Wolfrm Reserh In Deprtment : Informton Reoures Revson dte Desrpton In 9 DHLehmer onsdered the euton for postve ntegers nd If s then the euton hs soluton wth He onjetured tht these re the onl solutons Conjeture : Let e postve nteger Then f nd onl f The onjeture remns open to ths d The urrent est result due to Cohen nd Hgs s tht must hve t lest 4 prme ftors nd e greter thn The ode n ths noteook rres out serh for ounter-emple to Lehmer s onjeture nd etends these lmts uet Mthemts > Numer Theor Downlods Lehmer Artle n kb ) Mthemt Noteook [ for Mthemt 5 ] [4] ) Au fost multe nerr de reolvre! de eemplu : n s ) 5
16 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo olutonre me pult n nu fost unnm eptt Inerm dn nou reolvre Conjetur nsprt de In prm prte flm multme solutlor une eut n numere ntreg re v f esentl n prte dou pentru reolvre Conjetur Notm pentru un numr ntreg f orere : { sunt numere ntreg } Pentru s otnem : unde unde s sunt tre numere ntreg de s de unde Dn otnem Reult po s sunt dou numere ntreg su su D tun : s { } Pentru otnem : unde unde sunt tre numere ntreg de 6
17 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo D tun : > D este un numr ntreg pentru re este devrt egltte : tun : Intr-devr unde 7
18 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo unde sunt tre numere ntreg > de s D s tun : Intr-devr dn s reult : numerele D n unde el putn unul dntre numerele r f pr tun : este pr este mpr este mpr sunt mpre sunt pre s este mpr In ontrr sunt mpre D unde tun tote u ees prtte unt num dou CAZURI : CAZUL : sunt mpre CAZUL : sunt pre s este mpr Nu m est lte ur n fr de s D unde tun numerele sunt n CAZUL or n CAZUL 8
19 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo D spunem : este pe pot este pe pot este pe pot este pe pot t este pe pot t Cele m m numere ntr-un CAZ ft dntre ele dou CAZURI posle orre r f est pe potle t t se note respetv u unde ) Otnem : n CAZUL : ; n CAZUL : In mele CAZURI : Intr-devr -) s -) ) Repro d tun : ) unde ; su CAZUL su CAZUL Fm un CAZ CAZUL or CAZUL ) po ontnum stfel : Pentru unde t dntr-o negltte nestrt lu otnem : Cel m m este d pentru re neglttle nestrte dn ultm prte sunt relte prn egltt de : Pentru unde t dntr-o negltte nestrt lu otnem : Cel m m este pentru re neglttle nestrte dn ultm prte sunt relte 9
20 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo prn egltt de : unde t s Proedm nlog ps u ps Otnem : Pentru unde este numr ntreg dntr-o negltte nestrt lu otnem Cel m m este unde pentru re neglttle nestrte dn ultm prte sunt relte prn egltt de : unde Pentru s otnem : s D tun : Numrul este determnt de unde Tote ondtonrle re nep u ) de pe prurs s tote ele sunt ndeplnte deoree prtenent este devrt Consttm : d ; d ;
21 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo Pentru un numr ntreg Otnem : d d CAZ In CAZUL : pentru In CAZUL : pentru Consrm nottle pentru este numere n ore CAZUL su CAZUL Nu m est lte CAZURI! ) Prurgem drumul n sens nvers n stut fr s shmm numerele u nottle onserte nteror ntr-unul dntre ele dou CAZURI orre r f CAZUL CAZUL su CAZUL ) Otnem : pentru s Dn este s reult : numrul este el m mre numr pe pot t re otnem : s onform relte Dn ultmele dou egltt s reult : Proedm nlog ps u ps Dn este el m mre numr pe pot t pentru
22 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo s otnem : Notm : este el m mre numr pe pot t s pentru numr s CAZUL orre r f m Reult : Dn s otnem : numrul este el m mre pe pot r unde pentru re otnem : Pentru su ) reult : Numrul este el m mre numr pe pot pentru re otnem : Dn ultm egltte otnem re este el m mre numr pe pot re este el m mre numr pe pot n CAZUL or n Numerele sunt tote smultn n els CAZ est CAZ CAZUL or CAZUL ) Aeste u fost lulte nteror Intr-un CAZ ft sunt s ele m mr s ele m de sunt pe potle respetv t t Reult : } { pentru Dn est s ele de pe tot prursul nlusv reult : TEOREMA D unde tun : nt : Presupunem est numere ntreg s stfel unde s sunt numere ntreg
23 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo Fm un s un stfel nt este devrt n ondtle mpuse pe re nu le shmm pn l sfrst orre r f este Pentru est est s sunt une urmtorele numere ntreg u proprettle lturte lor : numrul ntreg numerele d ) respetv numerele ntreg stfel nt Pentru otnem : este Presupunem r est el putn un numr ntreg stfel nt Pentru un stfel de dn s otnem : numrul r re ontre De presupunere fut fls In onsent s dn s otnem urmtorele tre egltt respetv : reult respetv : Presupunem Dn s tnnd sem de proprettle numerelor re sunt n otnem : Deoree dn s TEOREMA reult : D dn otnem re ontre De D dn otnem : unde re ontre este De Deoree este numr ntreg s stute n re dn otnem unde este unde
24 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo Dn otnem : unde Dn m otnut este s Aeste ondt sunt pentru Repro d este s tun : de k re este hr Am demonstrt ondtlor Otnem : TEOREMA Fe numerele ntreg s s d s num d este s d s num d este Am demonstrt Dn reult Prn TEOREMA m reolvt Eut DHLehmer refertore l est s Conjetur lu B I B L I O G R A F I E Vsle BOBANCU s olortor Dtonr de Mtemt Generle Edtur Enloped Romn 974 Mrn DEACONECU ON THE EQUATION m INTEGER : ELECTRONIC JURNAL OF COMBINATORIAL NUMBER THEORY 6 6) Nodm A NEGREA olutonre une onjetur dn 9 lu D H Lehmer n Revst Eletron MteInforo IN Noemre John RENZE Wolfrm Lrr Arhve A6 4
25 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo 5 Teorem lu Lgrnge ; Aplţ le teoreme lu Lgrnge în demonstrre unor relţ trgonometre Prof Vone-Ante Costă Leul Tehnolog Ele Rdu Botoşn Noţun teorete Teorem Teorem lu Lgrnge su teorem reşterlor fnte) Fe f o funţe Rolle pe un ntervl ompt [ ] tun estă ) f ) f ) f ) Demonstrţe: Consderăm funţ ulră onstntă relă pe re o determnăm stfel înât F) F) f ) f ) F ) F ) f ) k f ) k k stfel înât F :[ ] R F ) f ) k unde k este o f ) f ) Pentru k funţ F :[ ] R F ) f ) k verfă ondţle teoreme lu Rolle ) stfel înât F ) f ) f ) Dr F ) f ) k ) f ) k f ) k f ) Oservţe Interpretre geometră Teoreme lu Lgrnge) Fe f o funţe Rolle pe un ntervl ompt [ ] tun estă el puţn un punt ) stfel înât tngent l grful funţe f în f ) ) puntele f ) ) ş f ) ) este prlelă u ord determntă de O 5
26 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo Oservţ e pote pl teorem lu Lgrnge restrţe funţe f l ore suntervl [ ] [ ] unde < În est ) re depnde de stfel înât f ) f ) ) f ) Dă tun Osservţ Dă în formul reşterlor fnte notăm λ oţnem < λ < ş ) λ În est onlu teoreme lu Lgrnge se m sre: )) f ) f ) ) f λ u < λ < Consent funt u dervt nul D o funte re dervt nul pe un ntervl tun e este onstnt pe est ntervl Demostrte: Fe f : E R o funte dervl n puntele dn nterorul lu E s ontnu pe E E ntervl s E ft D E este rtrr tun onform teoreme lu Lgrnge plt funte f pe ntervlul [] su [] est un punt ) su ) stfel nt f ) f ) ) f ) Cum f ) vem f)f) orre r f dn E ee e rt f este onstnt pe E Consent funt u dervte egle D dou funt u dervtele egle pe un ntervl tun ele dfer prntr-o onstnt pe el ntervl Demonstrte: Fe f g : E R dervle pe nterorul lu E s ontnue pe EE fnd ntervl u f ) g ) E Aest ondte srs su form f g) ) E rt se pote pl onsent De est o onstnt k R stfel nt f ) g ) k E ltfel spus ele funt dfer prntr-o onstnt pe ntervlul E Consent monoton funtlor Fe f : E R E ntervl o funte dervl ) D f ) E tun f este restore pe E: ) D f ) E tun f este desrestore pe E; ) D f ) > E tun f este strt restore pe E; 4) D f ) < E tun f este strt desrestore pe E Demonstrte: Fe E; < s f ) o E Aplnd teorem lu Lgrnge pe ntervlul [ ] est ) stfel nt f ) ) ) ) f f ee e rt f ) f ) de funt este restore pe E Prm prte este teoreme pote f ntrt n sensul est funt stfel nt f ) E dr f s fe strt restore pe E eemplul uul fnd el l funte putere f : R R f ) f ) R dr funt f este o funte strt restore Asdr vem teorem: Funt f este strt restore d f ) E s multme puntelor n re dervt se nule nu nlude n un ntervl nedegenert Consent dervt une funt ntr-un punt Fe f : E R E ntervl s E D: ) f este ontnu n ; ) f este dervl pe E-{ }; 6
27 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo ) est lm f ) l R tun f re dervt n s f` )l Consenţ Fe f o funţe defntă pe o venătte V puntulu dervlă pe { } V \ ş ontnuă în Dă estă lmt λ lm f ) tun f ) estă ş f ) λ Dă λ este fntă tun f este dervlă în Demonstrţe Dn teorem lu Lgrnge pltă funţe f pe un ntervl [ ] V < ) stfel înât f ) f ) f ) f ) f ) Prn urmre: f ) f ) f lm s ) lm f ) λ deoree dă < < < ) Dn teorem lu Lgrnge pltă funţe f pe un ntervl [ ] V > ) stfel înât Prn urmre: f f ) f ) ) f ) f ) f lm d ) lm f ) λ deoree dă > > > ) Dn f s ) λ ş ) λ f d f ontnu f re dervtă în ş f ) λ Dă în plus λ este fnt defnte dervte f este dervlă în Aplț le teoreme lu Lgrnge în demonstrre unor relț trgonometre Aplt olute: Consderăm funţ o funţe dervlă u dervt de onform Consent funt u dervt nul f este onstntă Cum reultă ă dă 7
28 REVITA ELECTRONICĂ MATEINFORO IN MARTIE wwwmtenforo Anlog legem funțe dervlă u dervt de onform Consent funt u dervt nul f este onstntă Cum g)/ reultă ă g/ dă Aplt olute: Cum relţ dtă re sens Consderăm funţ de onform Consent funt u dervt nul f este onstntă Cum f) reultă ă f Aplt olute: Consderăm funţ Cum de onform Consent funt u dervt nul f este onstntă Cum reultă Blogrfe: Gh Guss O tăăşlă I to MATEMATICA - Elemente De Anl Mtemt - Cls I EDP 99 8
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