Olimpiada Nacional de Matemática 2016

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Olimpiada Nacional de Matemática 2016

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Olimpiada Nacional de Matemática 2016

  1. 1. ', / . _ggionail‘de Ii1'I EL MINISTERIO DE EDUCACIDN Y LA UNIVERSIDAD DE EL SALVADOR INVITAN A LAS Y LOS JDVENES DEL SISTEMA EDUCATIVO NACIONAL A PARTICIPAR EN LA XVI OLIMPIADA NACIONAL DE MATEMATIGA OBRE LA PRUEBA La nwusha sewn’ adwuwwwwswrada nawa esnmlarwles We cuwserw desde Guano avado nasla prwrwwev arm as hacrwwllerala El esludwame deberé Iwahawav la nmeba we carweswwnde aw credo nus cmsa erw ew aim 2016 o Dfuehas de grades supewwowss En wwingwlww casu Se Iumarén en cuenm sowucwuwwes Z wivahlewrwas DVODUESIOS de M" gwadn arwlelwnr al gwadn que cllrsa ew esludianle L05 esludwarwles que nerlenecen al sistema bwwingue deherw resnwver wa pmeha dew gwadu erw que Iwnawwzauain sste ar'wa - Ma uaara reswncciunes a la narusipacwau as esludwarwtes qua Derterwezcarw a W! grade antewwuw aw warts ~ La aanwcwaaswan as man eswwmwawwte sen? vawlaa unwcawnente sw sl nesarralwo ne Ia arneaa es pwodaclo sawa ne sn arauwa esmenn. Paede, swn embawgu, naser use as Iada Ia bwhwlngmwia wrnaresa y sweeunnwsa ane nrsaanga - cana arnawema nesarrawwann neaera ser emreaann sn Imwzs sepmdas, nnmeranas y can sn nnrnare « Para Ia sawneinu ne was avaawernas ne esta arusaa, wa1nnnan. sn. aw sera wa an, ~.n. ns. uae. au unwaana nara wauwsna Enlaw senuna. aausnwas parna. nae. nnss eu Ias uue sawa anarezcaww was resnues1as. na ssran tuwwwadas an enema Para was nrnawsmas as asamana, un ssran uaunas was snwuaraues amenrnas aanwa resuwladu ns . usni. dwrectamerwwe was nauras ~ se enalnaran sawuswanes aarswawes a was araalsrnas - Para Ia uamcwpacibn en la Ulwmulada no es unnrsaensaals enuwar wa sawnman ne was cwrwcn araalsmas new arann anrrssannn. sn. s ‘ Las snwucworwes a cawia urwo (15 was Dmbwemas deberarw eslav wedactadas can wa mayor clawwdad, slrw Iacrwaduwas Y ww) llIa'5 aseadu [wswhwe - Las solucmrwes neaeran ssr redaclanzs cnrw tmlngraln a plum. Ma se aceptarzn snlucworwes a wapa En nnauu casn ss aaemaran wmacnpwas ne snlucwmwes. seran annwaaas (nnas Ias aawnc. pas. anes ns au. enss snnwen snlwmmnes . asuusas. AII'rIcII= AcI6N El araasnunuemn ne aaruaraaaran en la nsauna sena Owlmpladz Maaranaw ne Malerrwéilca es en s. gn. sn. e Ew alumna neasra rssalner Ias arnawsmas ns Ia prueaa aew araan aue Ie cnussaaune en ew perwnna new u an 21 Iemem, reg. suar sns naues pe. snna| es en sw sn. n web n. ..a. .i| .uss. eau. sn/ piI/ anernas neaea . rnpr. rn. rew cnmprahzrwle ne reg. sua para presemarwa JUNO can was salusanes as was araawemas auausaaas en was enemas as wa u. resc. an Deparlamerwtaw cowresnandwerwle as: Mwnwsteno as Eaueaewnn a rnas varaar ew nwa Iuues 22 as Ismsra, a was man a rn. aara was zuwwas oscrnenzaw, cewwwvaw. rwwelwopnwwtarwa y namcermaw v Data wa zanaansmaw (usulman, san Mwansw, wvwamzan y La unwan) a rnas tawnar sw nwa misrnowss 24 ns Isama, a was 3 on a wn Las sowncmnes y mmnmnawwte ne Ieglstwo neaswan ser pwesenladas sn nn some as aaaew ruauwwa, neae wwrwwzriwnwwss nas sanwnrnaamss: nun nara sawasana sauna caréluwa asw snare y en mm nawa ser sewwann y wurnana par wa persona rsspnnsaawe new MINED, eama cuwwslancwa new rwwawewwal renwawna Ew es. ud. an. s puede wwevawwo nersorwalmenle a asnva suwlcilav wa anwaaaraawan as sns pwulesures y/0 new nireanarns wa wns1wIuciL'vww pawahacetwlagawsu exarnsua wa Dlwecnnin neaarwarnsmaw sanssaannweme asnun new wiwazu pwevwswu a para regwswrar sns dawns en ew swstema, was arneaas se reauairau nuiaarnsme sn wa cnwwespnrwdlenle Dwreaauan usaarcamsnnaww auens cunsnwtawse sn www. miIIed. gnv. sv was anssuauss y newetnnns ne 95125 mwclrwas pan rrwzyar . nwa. nwaa. nn mansso DE DAI- L05 ESTUDIANYES DEBERAN wuenesnn Los swenwewwes mos Norwwhres y aaslwnas comuwetos, Iecha ne nacwmiemu. araan ane esmdwa, wnaar ne uureuna, devanamerwm, munwcinwu, seem. (urbane a rnraww. divecclén, nanars as wa nersana . esaausaawe, lewéwurwn, nueaaau ne cnrren eweclwénww Ansmas nsaera presawwwar was sranremss dams new centre sanaauna aw ans eenensns canauy nnnare LA PRUEBA PnEsENcI- ACERCA DE LA PRUEBA PRESENCIAL Las rnewarss aanws. aac. anss as cana grana ans alcarwceww sw numa| e reansnna aawa swaswusar, neasran rsawwzar nna arnsaa aresensiaw sw nwa sahana 12 as mam new presents ans Ia arnena se adminwslwawa en wa Facuwtad ne Clerwcwas Natawawes y Malemnillca, Facawtad Mnwnnwseialinavwa ns Occlaewwle y Facullad Mnwnnwseiawwnana Uriemaw ne wa nnwnerswnan as El Sawvadur, man we nmcedencia as nana esludwanle Las curwcuwsarwles annvasanas seran nmwllcadas en sn cenna Enncawn y alternallvarwwenle annrau cuwwsnwlar was wiswadus ulwciales pubwwcadus en Imu / illveneslalenln-nulll . sua. eaaiun. na. .. a wwuvI. IniIIed. nnh. sv nesne ew nwa uuanes 3 as Mana ne zuwa, aanne se esneuwicara sw Iuaary anwa nnnne cada esmdwarwle rsawaara wa arnsaa presenclzl Pawa prumover wa aarua. aas. an as: .nayar nuu. srn as Instwmclnnes, as as uzmcwpzrwias as wszflz araan ne sans . usua. e.nn, nn. ca. nen. s aaaran ssr eaunneanns a wa su. na was . ns. nres cwrwcn es1m1w2rwtes qus aweanceu el pnrwlawe rsanenaa para awas. w.sar Esta rn. su. a an se reawaara nna prueba as. snwag. sa as caracter ohlwgatnrwz pawatndns aauewlas estudwanles ans pawucwparw par arunerausz, nwcua prueaa ss rsawwzara asspnss ne wwnalwzar wa arusaa arssenawaw. PROGRAMA . Iév: IIEs 'rALEI- WGRESD AL PRDGRAMA JDVENES TALENTU Las nweuues panwawaaauauss ne Ia arnena nresennaw serau wwwcorpuradas aw Pmglarwwa Jbvenes wawerua ans ew Mwnlstanu ne Educzcwnn nesarrawwa en cuunewacwdrw can wa un. ners. aan ne El Sawvzdar El Pmglarwwa Javenes walemn uene nwwererwtes cnmpnnerwwes con Ias cuales ss pretends aar respuesu a wa neces. nan ns nssananr y nesarralwar ew Tawemn en Mawemalwca y ceneas Narurales en was nueles Daswcas s nenlaarwes a para. as ess nus: wa awsswalwua sw assenae aweanzarawwas nuswes as exwzelerwcwa anaasrnwea. nesarrnwwanss saaaaaanss as wwaerazga y snrnarnnusn ewnsn Dos as sns prwwwcwpawes wmnmwewwwes son wauanallna saaauua y aw cawso Fumrns ningsnlss Issuicns cismilisas as an sawana. La prwrnera ss nssanalwa a In largo new ana essawar. en nwas saaanns, ew segarwdn as an cnrso wmenswvu as (was sernanas qne ss nesanawwa aw nnawaar ew ana essawar. La Academwa saaauna uene wa dnbwelnncwun ns aveaarar en sursas beislcos ns Matematwca y cisuswas Namrawes aw esmdwame wzarn uns auravesns rnein. ew svema ns win as ana y anemas we as preuarar a an grant: selenm uara canwpsur sn alimawanas Imemacwunawes as Manenwausa, Binwugwa, Fwsica y onwrnwaa La nnrruna ne eswdiawwwas sewesaiananas Dara aerrsneaer aw Praarama Jbvenes Tawerwtu ssra anawiaaaa en uuns: /fnwenesraIeuIaauaIin. sua. s.. ai. n.aa. u a wuM. mineIlq| nbxv ew nra mavles 29 as menu as zms La Acaderrwwa saaauna ss unauaurara ew sanana 2 nsawniw ne 2ma su sw annuanun. as wa Fzcuwvad ne Cwenclas Namwawes y Ma¢e. nausa ns wa nn. vsrs. nan as Ew sawuaaara was nu no a . u y Esta nus. ua n. a se . n.e. aran Ias aeunnanes acaderrwwcas aa. wa tame an Pragvama Jnvenes Talenla Invlla a pamclpar sn la v Olimpiada Nacwnal as awaIaa. a, IX Olimpiada sa| naaa. s.. a as ea y xn Olimpiada uaswanal as unun a, para mas inlovmacidn vis O GHADO PROBLEMA I En sn snsnraaa AECD. nas annuas 5 y n nnasn sw wana ac sn nss ssumsnnas ns wguaw wungwlud y en sw wadranu msrn was Dumas wy L n. n.asn ew waaa re sn . rss seanreanas as Iguaw warwarvnn nna oarnarsn as cuadrado Sw AB rnrns 5 am, calcnwar sw asrwrnsna as wa nanra cL7rwwDues1a as. was nss cnaaraaas FROELEMA 2 En la nanra ss rrwuastva nna awoarnara wanarwguwaw nan nn arssna as snaaraaas awanaas, naaras y Qnses. w= srs. aw. nanxa nsssnrawuaaa La awvarnara rnras an an as ansno, am 6 cm as wargo y. sarna ansas oassrnarss, a lo ansna saaanwa nsns nu anaaraaas us. sr. nrnar sw nnrnsra as anaaraaas nsaras ans nsns wa awwarnara y la wwaccwon ans rsprsssman asw arsa Imsl ‘mun PROBLEMA 3 En sw swanrsn. s labwewo ss asas aawasar sn asaa assnws nn nnrnsra rwamrel del w aw 1. as rnansra ans asaa aaunrnns Ienga 7 sns swele nnrns. as a. ws. sn. ss n ans aw , snmar was uss nnmsras as ssas was sw . a.sw sss swempve el . n.s. na nnmsra r represenlada par la wsua r 5. el ma as sumaw was nnmsraa as sans sswnmns ss nssrsssnn. mu wa welra c, calcuwar sw nswa. K as C+F y moslzar nn. .. .s. .srs as nsns. , sw (ahwewo snnrawsnaa aan las mrwdlcwurwes nssanns c c c PROBLEIVIA 4 Los rwabwvames as Prwwuowarwdua nnegan a warwzarse gwaaas san aana as wa rnansra swguweme ~ Haasn nn sansa para aslgnav acada Danwcwparwla nn nnrnera Pnmo - nn panwcwvarwte sawa ansns wanzar awaaas a am ans Ienga aswanaaa nn nnrnsra ans neea Ieslduow aw arwnuwe emwe ew snya En la Fwguwa w, se rnnssna aw asunsrna ns wage as anana naanamss sr twwewan asrananns was nnmsrasz, <3, mesa En wa nanra 2, ss rvwueslwa an we asansma as . neae as cwrwco naanamss nasnnnarsw rwumewo Msnuzw as-anaaaa w<arsn, wnannsa, Dwalwa 53 y Luwsw sw sw nnrnsra as Edward _ ssauzaw yew rwumero as Dwana FI§ura1 esmayov ans sw as Manuel asra rnsnar ans sw as Luws PROBLEMA 5 Nwnza sn sw ass as was mawavwlwas ss snnnsnna nan 77 snsnnas ssnasaas swnsnsaa. as nna . ..sss cwvcuwar a samarsrsrs waaa le habwa amenwda a Alma ans awgulwus snsnsas as sss asws san nanssaas y swempre a. ssn la verdad nnsnnrss que aaas san deshnrweslus y s. s.nars . ... sn. sn Ansn ws a. sann. a s ssas. cusucn nsans aaa srsn was ans ans ssnaan ssnasaas . nn. a s sw, asn wzquwevfln y nsrssna, y la rssanssaa deladus wns “B9 was aas ans esuan semanas . nn. a a . nw. nna as swuasss nsnsaay 51 we ss assnanssuaw ncuérwlus as was cusucus ssmanas ausnsaar as wa rnssa sran horwes| us" 2 am Edwawa Figura 2 . Imps, w jovenestalenlu-public. SEPTIMO G PROBLEMA I In Alia noao uwsns Va a sawarsar nn sarszan san asna zanas aars rsgsuasswa ss. s dwa s. sn Iwuvwn uwana Para eanarssr sw sarszan Mana asra an sawara eana zana as rnansra ans zanas anyaosmss asnaan aneasms wluw wass zsnas san aayaesmss sw nsnsn awgulwa aarsusn as awann ladu rema sn sarnnn, asreusrnma sn Ia wwanra . ..as1.aaa was zunasw y a na sen anyaasnss, mwetwlwas ans was zanas zyasan Wyacenlesw Haeovvwerwzado sawarsanaa as vowo nna zana osusrrnrnar la aanaaaa aswannas duslwnlas sn was cuales Marra ansasusmunar as sawarsar sw corazonw sr nnwaa nas aawarss nna, rasaaa y rmradn PIIOBLEMA 2 Llrwa urs as papaw . ss1.snanws. delocmflewnvgnyw cmfla aua se naaws pars. vannar nn actéeans ns mansra ans la wwgura an al ssnura ssa nn cuaavadow oaw y sama sa rnnesna an wa swgnrsms nnnra Cawcuwaw wa wungwmd as PIIOBLEMA 3 Mans n nnsns ss. sesn n. snae ws . swsn. s.an wunln can sns amngas Dwans y Nancy us rsasn. s sssnsns nn eslruerwdoy mnsnns gums ca. .. ssnsngnssr na ans ssnass aassnaa, y snsanns Ia lelevwswarw sn el snsna nanssas as vwdrwas nnasr Diana y nnsns nsawassn ass. aw nnsrna llempu -Juana n. s.snna “wNu4uw yn"‘ - n. ana n. a.snna 'wFue Nam; w- - Mafia arna wa. me Dwarwa‘ ~ can an nssna ns . naasns. a y nn. nuaan, Nancy nus 'DwarIa rn. a.ns- sr sawa nna as swnas eslaba n. a.snaa la vavdad nan. s'n namsa wa nsnsmsranv PIIOBLEMA 4 E‘ as sasnuas as nna snaansnwa de2076 x zors ss pwnaan sn araaonaw as wa swgn. sn. s rnansra Laws Luegu ss a. n.. sn . as was y nannrnnas ans san . nnwna. a as a ncnarnas sasuwas as Ia cuadwwcula nan anenana srn nwnuarv _' PIIOBLEMA 5 En nns sernasusnnns wa suma as Ins vunlawes as asresrs y Dans ss wguaw a la suma as Ins purwlawes as Ana y Camwma s. was annnnss as aarasrs y csrauns ss . n.s. ssn. n.a. an, snusnsss ws snrns as was ann. s,ss as Ana y carauns exceflerwn la snrna as was ann. s,ss as was auss ans Isn. a.sn, sw ann. a.s as Dmrws exoede ws. sn. ns as was punlzjes as Eévhava y caranns nsasrnunar sn ans aaa. a.an Iermwlwa ssna amnasnnars. asnrnrsnna ans «nnas nas anrnauss san na nsgannas y dasculwoudos -JARTO GRADO PROBLEMA I Francwscn carwslmye . a.. ss san sneas as nsaa. empweza con una . a.. s as an case was swan. srnss. arrssss rarrnan sarno rnarsa warwgnra L as. .. s us. srrn. na sr Francwsco paara corwslmwr nna . arrs snguwenda ssa wowwvwa as conslwuccwdn nuwaanaa axaclaruerwle zma cubes. exawwca . n Ieevuesla PROBLEMA 2 w 2 3 - Los nnrnsres rwaluwales sa ‘ ' arasnan sn saunrnnss. sarna * 5 wa rnnssna Ia Imagerw 3 V D Defiewrrwwnav wa culumna an as nna ss erwcuelwlra sw nnrnsra zms W ,3 H . s . a n . . FROBLEMA 3 La nanra mueslva nn rnanw sn san. w.ana El assa mlaw as was 7 nan. as ss as .12 a. an. ss ns«s. rn. nsr e4 peso sn gramos an srrsnwa E PHOBLENIA 4 Malia nas. n.y ssnarssan muy ansnss s. n.gss cans nnsns swwssusns nn nsrrnana rnswwrza Para sw an as san vawsnnn, saaa snsa wrwvwlaa Lane as was annaas as sn nsnnana Les orunsras sn Iwsaar aw ans wuerun Mawwa y el nsrrnana as sanara, sn sss rna. nsn. a aaarsea ova Pawala y Mawwa dwce wMIwaw‘ anwnsns awgulsrw san «n . nnusns Dmermwnaw a an. sn wmnla rada chum PHOBLENIA 5 can awsrnms ss cnns| myerw nsnas snsnrsaas as wanna zaw ans sw wsaa as saaa snaaraaa ss sw aoals ans sn cuadvado arwlenm nas rnsa. aas as Ias waaas as nss cuaflvaflos swempra san nnrnsras nalurawes sw qnsrsrnas ans el Iaaa asw anaaraaa rnayar sea 2015 cm. nsns. ss rs . nsnn. n.sa. aa ans Duede . an. ar sw Iado asw snaaraaa mas Deqneiwo’ ns. snn. ns. sn cm as ssnuaaa as awarnars ans ss mwwwzara Dara la cons1vuccwL')n 2015 cm _TAVO GRADO PROELEMA I an nn snaaraaa Asco as lada nna ss Iraza wa aaaana: Ac ss nns el names a sen el punts rrwadwo M asw Iado ac ssa F sw anma as sans as MD y Ac Cawcuwnr w. razurw snns las arsss asw snsanwsusra asw y el mangnwa one s. wz dwslancwa asw anna ; - aw wadnAB ss aaaws as wa a. s.ane. a asw puma e al wasa ac PROBLEMA 2 caaa nna as was wsrrasn, 5, c. D. 5, F. G. H, w. w wehveserwlarw awgnas dwwsvsrwtes, as marwera ans AB. on. EFw an y u san mllmams as nas cwlras ns. snn. na. sw maxwma naws. nn. nsn. :a aas. s|s as ws wrsssan A84-CD-r-EF GH-IJ PROBLEIIIA 3 Cwncu nnas snyas . n.a. a.es san A, M, K. 0 y n. assenansran ans. assanaass sn urwa essanus as aas sn aas s wnlevcamhwainduse nn. cada vez. aaarsn sanassr sw assa as mdas ewwas gasaanaa nna saws mmweda (par s. s.na. a, an. ns. a ss aesan . nn. as A y .2, wnssa ss aa. a A yse snas Ky as. ss pesan , nn. ssny K, s. sanunn. s.an ss ans nn. as suss y ss snas ova, swrw ans nnnssss repua ws nnsna aars. s) Una vez assaaasraaas was pawelas. sns assas sn Kgwuamn wze, H5, 125. m, .24, wzw, .23, wwa. we .22 Cawcular was aasrawss vawares as was assas . na. n.ana. ss as was nnas FHOBLEMA 4 carwas an. s.s asswsnnsn s. na. ss. s aws as was enamuvadm s csnnsn, Deva ssxa nasarsa as. ans as. s sn snsns sn rnanas as wa maIerIIa'Iwca Para anwa escnbe sn nna arzarra was nnrnsrss nannawas nssns w nasna zma A I: urI| wrIuaI: wnrw Escnge ass as ssas nn. ns. as, was bmra y sn sn wugar ssanas 4a dulerencwa asw mayor rnsnas sw rnsna. cannnna as esta rnanera hasla ans auena nn solo r. nr. .s. a sn Ia arzarra 5. el nwurna nnrnsra ss as. csrwas nssuarsrs sn sn. a., as wa aanusna na wa nsra ansswarara sn arnar a csnnsnv FHOBLENIA 5 En sw awansn. :22 hay saua nas npas as bwwweles ss ansns naes. ansAan. s. samara sen a. sa. a erwlem a solo hay annss a. sn. as sn. s.as ans na ss pnsnsn Pagaw sun nssss. aaa as nnsua Erw ssas casos, ss paga as mnsy 52 aa sw nnswaa sr wsss nna as was nwas ans rsanrsrs nnsun, delemwwrwzr el nawnr as ssas uaa as a. wws. s _lMER Afio PHOBLEMA I LL75 warm; as nn mangnwa AEC nnasn as = mm as s wmn y ca : wgcrrw Las arsssnruass as was ananwas as venwces By c sa oonalw sn sw purwtul Furl sellazzn nna aarawsws. aEC ans eena a was naaasaa y as sn Ins punlas M y N rssassuna. nsn. s Cawcuwar sn psrunsna asw nrananwe AMN PROELEIIIIA 2 Sea . . nn sn. sra vaswlwva us. snn. na. sw v. IIeI exacla as ws s. gn. sn. s snma PROBLEMA 3 cawsnwar sw Iiroducm as Iadas was enmms assnnas ans ssan rnsnarss ans w50 y ans terwgan axammente 4 arnrsaras wzasnrvas Exrlwasaw sn rssanssn. sarna pmducm as wsnuarss anrnas -: LA LIBERTAD POR LA cum. OUINTO GRAD- PIIOBLEMA I Can: urn as was swguwerwlzs nanrssnsssaa unnshuwfla ann nurwlns s. ssnanunns san sw asnan ns. srn. .nar sw nn. nsra as ws nanra ans ss Imnslvuye can sxsnamsms mws pnnuas Fngnraz Flgura 3 PIIOBLEMA 2 ss saae qns swele gawletas ns ansns. <; ues1arw wa nusrna ans calarce sawwsus as ansaacsus y asrnean ss sabe ans asss gsllelxs as vawnwwla cueslarw wa rnsrne qns dos Qawwelas as averwa Delemwwnal mérwvas aanwsxas as na. n.wwa cue51arw Io . n.srna ans nna as chwnwats PROBLEMA 3 un aruaa as 12nw17aw95 estén emwgvawwdu as wa awnsa Lento a wa awnsa nsansa zwwas carwwwrwalw Skm par hora y cada nsz ans nan resarnna nn nnrnsra as k. wn. ns. ras rnnwuawa as 9. asssesnsnsn wa rnnaa as was carwibsles ans hay sn sss rnarnsma s4 aw rnrsrar naaan 20w6L: arw-bawesysowo wwsaan asn assuna as Dalevrwwwrwaraw nn. nsra as naras ans lardavurw was an carwibalas ans quaflamn sn wwegara wa awdea Deprwsa H I G PIIOBLEMA 4 |2I uanrs adyaoeme ssa. lumwada aar sw snaarsaa Nico y el snaaraaa ssan aanna sa snnaws ansns 5; my H e s. as s 5 cm as. srn. nsr sw srss. as Is rsgan san. a.saaa A B PROBLEMA 5 wsnay wnsga al aaws as Nurwca narnas y ss enaonna con nna arnaa ans dwce la nsraan was lunesw mwévcnwes, vwemes y sihado. y sw veslu as was dias . n.sn. s Aasrnss ws omga ss vuewve as naner nsras was was ans rn. sn. s y as cowow mwo sw we5|o as wos a. as Cweno a. a wa omga dwce sxacwamsnrs las s. an. sn. ss afiwmacwurwes - Ayer na era as sawar . a.a - Manana . ..snure' - Pasado manana ssrs nsrns ns. srrn. na. sn dwa as wa semzna sn el sna: la owuga ans eslas anrmacmnes NOVENO G PROBLEMA I Rndvwgu sssnas ws. sgulerwle us. a as nnrnsras sn nna plzarra mam mmamn. mwwmwnmmmm, A sannnnacran, sw oomwerwza a lame: as wzanrsraa a dewechaw mas aansxwos nnrnsras as wa w: s1a ans san rnnnwpwas as 3 ncnanms nnas nsns sw asrussuna nnrnsra lachado” cansrasra suawsrns arvsawa nnnna ns nnmsras PHOELEMA 2 ss mawcan cuadwadus as uaaa w, 2. 3. . .aw y aana ss mueslwa sn wa nanra Delemwwrwav Ia suma de wos mimerus demru dew snaaraaa as raaa 2owe C B D PROBLEMA 3 En ws nanra was Lrwangnwas A50. Eco. one am y ssa sun . ss. sngnwas wsascewes s. M rrme a an as. s.n. .ns. sw as. as A0; sn mm F A PROBLEMA 4 un . nans. .na sans aswwsass. saars nsaars. aswwsass swn parar asre . s.ne. sn ss muy a. sn. sar nrsnsrs na. nsa. rnnsnss y anarns. nss que pueda para ans na ss ws amberw Empezarwdu nan nna aswwsua sw ann. s. nwa, sw rsswaa s a. ana nns as was swguwenles - Norrwear Ianvas gawwelas sarna usnwa sn awa a. us. .ar - csasr .3 us Ienlacwarw y snrns. nna galweta Demuslwaw ans sn ruonstruo ansas acumuwel 2016 aaw: s.ss sn exaclamerwle .2 nwas, a ans wa manera as aa. sns. dwcha sanuaaa as aawwsuas ss nnsa PIIOBLEMA 5 se usnen <5 nsnas susnwarss, (was new mrsrne wadwo. sanasaaas sn nn arrsawa mananuar come ss mueslwa sn wa nanra Cada nna as was fichas ss Dwrwla as an sawa aanar wa| u o azuw Dervwr: -s11aH1ue. swrwwrwwponarcéma ss Dwmen was nsnas, s. srnnrs e><| s1erw . rss as swwas as: rnsrna sawar nwas ssnnas sun was venwces as an Ilwangula sannsxsra PIIOBLEMA 4 ss nsns nn (abwevo as n x rww nanas n ss nn enters aasmna y ss sssnasn sn sans nna as sns was was nn. nsrns asw 1 aw n sn algurw nrasn as mean ans sw ansawo ublenwdo ssa swmelrwco wespeuo a wa a. agonaw ss assaa ans ladwagurwal dew lablem na aannsnaa mass was nnrnsras me! I alrl Derrwoshav que esla s. srnnrs ss pnswble snsnae n ss par y ans ss . rnpns. ews snsnan n ss unpar FROELEMA 5 ss usnsn sews cwrcuwns as rzmwn s, de nsnsra ans son Iangenles arms sw y can los wados asw veclénguwn coma mueslva rs figura Adewvwas Ins ssnaasa, a. C Y D aslélw alwrwaadas as. snn. ns. was merlndas as was wados asw . ss. ananwa

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