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CU TGITCU FQ TCEKQPCOGPVQ G UGO Q CWOGPVQ FG VCTKHC
GO CIQUVQ 2QFGUG GPVµQ EQPENWKT SWG Q EQPUWOQ FG
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Questões Matematica Fuvest 2002
- 1. 7 /#6'/Š6+%# Â 0WOC DCTTCEC FG HGKTC WOC RGUUQC EQORTQW OC¼µU DCPCPCU NCTCPLCU G RGTCU 2GNQ RTG¼Q PQTOCN FC DCTTCEC Q XCNQT RCIQ RGNCU OC¼µU DCPCPCU NCTCPLCU G RGTCU EQTTGURQPFGTKC C G FQ RTG¼Q VQVCN TGURGEVKXCOGPVG 'O XKTVWFG FG WOC RTQOQ¼µQ GUUC RGUUQC ICPJQW WO FGUEQPVQ FG PQ RTG¼Q FCU OC¼µU G FG PQ RTG¼Q FCU RGTCU 1 FGUEQPVQ CUUKO QDVKFQ PQ XCNQT VQVCN FG UWC EQORTC HQK FG C D E F G 1 NKOKVG FG EQPUWOQ OGPUCN FG GPGTIKC GN¾VTKEC FG WOC TGUKF¿PEKC UGO OWNVC HQK HKZCFQ GO M9J 2GNCU TGITCU FQ TCEKQPCOGPVQ UG GUVG NKOKVG HQT WNVTCRCUUCFQ Q EQPUWOKFQT FGXGT± RCICT C OCKU UQDTG Q GZEGUUQ #N¾O FKUUQ GO CIQUVQ C VCTKHC UQHTGW WO TGCLWUVG FG 5WRQPJC SWG Q XCNQT RCIQ RGNQ EQPUWOQ FG GPGTIKC GN¾VTKEC PQ O¿U FG QWVWDTQ VGPJC UKFQ OCKQT FQ SWG CSWGNG SWG VGTKC UKFQ RCIQ UGO CU TGITCU FQ TCEKQPCOGPVQ G UGO Q CWOGPVQ FG VCTKHC GO CIQUVQ 2QFGUG GPVµQ EQPENWKT SWG Q EQPUWOQ FG GPGTIKC GN¾VTKEC PQ O¿U FG QWVWDTQ HQK FG CRTQZKOCFCOGPVG C M9J D M9J E M9J F M9J G M9J 1U RQPVQU ( )# = G ( )$ = UµQ X¾TVKEGU EQPUGEWVKXQU FG WO RCTCNGNQITCOQ #$% UKVWCFQ PQ RTKOGKTQ SWCFTCPVG 1 NCFQ # ¾ RGTRGPFKEWNCT ¯ TGVC Z[ −= G Q RQPVQ RGTVGPEG ¯ EKTEWPHGT¿PEKC FG EGPVTQ PC QTKIGO G TCKQ 'PVµQ CU EQQTFGPCFCU FG % UµQ C ( ) D ( ) E ( ) F ( ) G ( ) 5GLC ( ) ZZH += 5G C G D UµQ VCKU SWG D HC H 4= RQFGUG CHKTOCT SWG C DC =+ D DC =+ E DC =− F DC =− G DC =− 1U RQPVQU ( ) G ( ) GUVµQ PQ IT±HKEQ FG WOC HWP¼µQ SWCFT±VKEC H 1 OÃPKOQ FG H ¾ CUUWOKFQ PQ RQPVQ FG CDUEKUUC 4 1 −=Z .QIQ Q XCNQT FG ( )H ¾ C D E F G # UQOC FCU TCÃGU FC GSWC¼µQ 0422 =− ZEQUZUGP SWG GUVµQ PQ KPVGTXCNQ [ ]π20 ¾ C π D π E π F π G π 7O UGPJQT HGWFCN EQPUVTWKW WO HQUUQ EKTEWPFCFQ RQT OWTQU GO XQNVC FG UGW ECUVGNQ EQPHQTOG C RNCPVC CDCKZQ EQO WOC RQPVG RCTC CVTCXGUU±NQ 'O WO EGTVQ FKC GNG FGW WOC XQNVC EQORNGVC PQ OWTQ GZVGTPQ CVTCXGUUQW C RQPVG G FGW WOC XQNVC EQORNGVC PQ OWTQ KPVGTPQ 'UUG VTCLGVQ HQK EQORNGVCFQ GO RCUUQU 0Q FKC UGIWKPVG GNG FGW FWCU XQNVCU EQORNGVCU PQ OWTQ GZVGTPQ CVTCXGUUQW C RQPVG G FGW WOC XQNVC EQORNGVC PQ OWTQ KPVGTPQ EQORNGVCPFQ GUUG PQXQ VTCLGVQ GO RCUUQU 2QFGUG EQPENWKT SWG C NCTIWTC . FQ HQUUQ GO RCUUQU ¾ C D E F G Â Â Â Â Â Â Â Â Â Â
- 2. 7 QKU VTK³PIWNQU EQPITWGPVGU EQO NCFQU EQNQTKFQU UµQ KPFKUVKPIWÃXGKU UG RQFGO UGT UQDTGRQUVQU FG VCN OQFQ SWG CU EQTGU FQU NCFQU EQKPEKFGPVGU UGLCO CU OGUOCU CFQU FQKU VTK³PIWNQU GSWKN±VGTQU EQPITWGPVGU ECFC WO FG UGWU NCFQU ¾ RKPVCFQ EQO WOC EQT GUEQNJKFC FGPVTG FWCU RQUUÃXGKU EQO KIWCN RTQDCDKNKFCFG # RTQDCDKNKFCFG FG SWG GUUGU VTK³PIWNQU UGLCO KPFKUVKPIWÃXGKU ¾ FG g  h  i  p  q   'O WO DNQEQ TGVCPIWNCT KUVQ ¾ RCTCNGNGRÃRGFQ TGVQ TGV³PIWNQ FG XQNWOG CU OGFKFCU FCU CTGUVCU EQPEQTTGPVGU GO WO OGUOQ X¾TVKEG GUVµQ GO RTQITGUUµQ IGQO¾VTKEC 5G C OGFKFC FC CTGUVC OCKQT ¾ C OGFKFC FC CTGUVC OGPQT ¾ C D E F G  7O DCPEQ FG CNVWTC TGIWN±XGN EWLQ CUUGPVQ VGO HQTOC TGVCPIWNCT FG EQORTKOGPVQ EO CRÉKCUG UQDTG FWCU DCTTCU KIWCKU FG EQORTKOGPVQ EO XGT HKIWTC %CFC DCTTC VGO VT¿U HWTQU G Q CLWUVG FC CNVWTC FQ DCPEQ ¾ HGKVQ EQNQECPFQUG Q RCTCHWUQ PQU RTKOGKTQU QW PQU UGIWPFQU QW PQU VGTEGKTQU HWTQU FCU DCTTCU XGT XKUµQ NCVGTCN FQ DCPEQ PC HKIWTC  # OGPQT CNVWTC SWG RQFG UGT QDVKFC ¾ C EO D EO E EO F EO G EO #U R±IKPCU FG WO NKXTQ OGFGO FO FG DCUG G FO+ FG CNVWTC 5G GUVG NKXTQ HQT RCTEKCNOGPVG CDGTVQ FG VCN HQTOC SWG Q ³PIWNQ GPVTG FWCU R±IKPCU UGLC C OGFKFC FQ ³PIWNQ α HQTOCFQ RGNCU FKCIQPCKU FCU R±IKPCU UGT± C D E F G 5G α GUV± PQ KPVGTXCNQ π G UCVKUHC EQUUGP =α−α GPVµQ Q XCNQT FC VCPIGPVG FG α ¾  C D E F G # HKIWTC CDCKZQ TGRTGUGPVC Q IT±HKEQ FG WOC HWP¼µQ FC HQTOC ( ) EDZ CZ ZH + + = RCTC Z ≤≤−   2QFGUG EQPENWKT SWG Q XCNQT FG D ¾  C D E F G
- 3. 7 CFQ Q RQNKPÊOKQ ( ) ( ) −−= ZZZZR Q IT±HKEQ FC HWP¼µQ ( )ZR[ −= ¾ OGNJQT TGRTGUGPVCFQ RQT 0C HKIWTC CQ NCFQ Q SWCFTKN±VGTQ #$% GUV± KPUETKVQ PWOC UGOKEKTEWPHGT¿PEKC FG EGPVTQ # G TCKQ 4##%#$ === # FKCIQPCN #% HQTOC EQO QU NCFQU $% G # ³PIWNQU α G β TGURGEVKXCOGPVG .QIQ C ±TGC FQ SWCFTKN±VGTQ #$% ¾ C ( )β+α UGPUGP 4 2 2 2 D ( )β+α UGPUGP 4 E ( )β+α 22 2 2 UGPEQU 4 F ( )β+α EQUUGP 4 2 2 G ( )β+α EQUUGP 4 2 2 2 5G Z [ ¾ UQNW¼µQ FQ UKUVGOC Â =+ =+ [ Z [ Z GPVµQ [ Z ¾ KIWCN C C Â D − Â E F − G − 0C HKIWTC CDCKZQ QU VTK³PIWNQU #$% G %' UµQ GSWKN±VGTQU FG NCFQ EQO $ % G ' EQNKPGCTGU 5GLC ( C KPVGTUGE¼µQ FG $ EQO #% 'PVµQ C ±TGC FQ VTK³PIWNQ $%( ¾ Â C D E F G 5G Z [ ¾ UQNW¼µQ FQ UKUVGOC =− = Z[ [ [Z Â RQFGUG CHKTOCT SWG C 0=Z QW NQIZ −−= D 1=Z QW 3 2 3 NQIZ += E 2=Z QW 3 2 3 NQIZ +−= F 2 32 INQ Z = QW 321 INQZ +−= G 3 2 2 NQIZ +−= QW 2 321 INQ Z +−=
- 4. 7 # HKIWTC CDCKZQ TGRTGUGPVC WOC RKT³OKFG FG DCUG VTKCPIWNCT #$% G X¾TVKEG 8 5CDGUG SWG #$% G #$8 UµQ VTK³PIWNQU GSWKN±VGTQU FG NCFQ G SWG / ¾ Q RQPVQ O¾FKQ FQ UGIOGPVQ #$ 5G C OGFKFC FQ ³PIWNQ %/8 ¾ GPVµQ Q XQNWOG FC RKT³OKFG ¾ C D E F G 1 OÉFWNQ Z FG WO PÐOGTQ TGCN Z ¾ FGHKPKFQ RQT ZZ = UG Z ≥ G ZZ −= UG Z CU CNVGTPCVKXCU CDCKZQ C SWG OGNJQT TGRTGUGPVC Q IT±HKEQ FC HWP¼µQ ( ) [[[I +−=Z ¾