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AMC 12 solutions 2013 PDF

12 Pages·2014·0.88 MB·English
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Solutions Pamphlet American Mathematics Competitions 64° Annual AMC 12 B “American Mathematics Contest 12.8 Wednesday, February 20, 2013 User etna het on sla cal ven ou hi yt gaa dows that llpchlenscat be sober wu senator ya stn ces ep odie stan reborn er goss ot spore cree come vanced ie slater "ac teony nes psi or ae he apache rane nay de, hapetl slufrr chs salen how slob Ladrations ofthe bin of nga acd in she os ine pobies nd ik eevee ge ‘nates cpocan loner dplsin preter eae tory inure the ines nls Slaeninaion sepia ep bees socal UC WEa on ‘or moa ary np din esprit sins open te Sew abat a tne Amercaa Mathenaricr auuzciiens Sng uf Neb HO. Box A160, lp, NE ESG-1 HG Fhe 02512 997 Ray AO 72S ma ase io dn Ie ee Prat Demande, Abr 2018 AMIEL Seivtions 2 1 Asorts (Gn Mice batt gh al Io enernes 1 ‘aggre, wo the sitar betwen cach of Ue ed he san eRe sine S deeeee ‘The ln tcaperatuse rar © Seneca 20 thar. “be aveage, 22% 2 user (A)¢ Ths sir is 2-18 — 20 four wide nd 2:20 — hot ts Haceo Me Grenu enyocle 30-40 = ON pounds of pevatos bs Ser te an et oe Pn there ely teciweafora 20) 198 are, 200s the 2 vases 108 the 9 maben, andi a (202—20 reavbes, Thtefre 3 i Lie (22 —84 neue, bic ithe 1 yam ‘Aneyer (B}+_ Let 2 oqul the ditoaee treed ty can ont. Then Rays ett oe 2 ile of gained Tons cae tea gales uf gsi, Theva Combine eri pr illo of gone ae Avwowre (Oe Wan Pathe ga 48-4 pe TA ‘sora fa the nae 6 Amewnr (18): soraplating the emaze te ecuston ca be eneniten a flows Anand bite Het fe SP iy Rar Thettore 2 = 9 and y= moat Ava (1): Rok to ture hy oping Tare a In Dior sovng 2 oumbere. tien Jo saying 3 mnabete, and so on. After sree ounpltn erten aes aging The aber Hie 12 rete TAMCUN Suton . sian he ea dn 9 fap. 28 mane ae bt aa. Thorfone ew ei A = ¥ Mowe mtacb3 nel So braid oe lean Anpwer (Dh: The sli se the pin oc rgcaios 41 ea B= feu) 2.1) The oie dav diane fens to Hoe A ae RAABO Ie SBC, 9. BO =2. mur i Cold ate tn the ne Bee ORGU idem eran petarepeay Aeeta eed 1 a Sine iin 1 Fe Min AC igs abaas LB ames (C0 rene 2 29.98 AN, a el pe List cides 1a ie 243°. 5° wiv kaw square sod ae Thee ee pnt S42 1 Arrer (Gh Aten Ales use rchains athe iubouth ain ehanges nt th etl oot, Alex bi: Ta (Ze) am lee aga gut mes seer tober. Yo ee exchanges ace pone shan herent 278 token afer tha bls taken. ‘Thies no seore nates a erie an uly if 2m > 74 aL fa oe a Balog eee be een Bet ines! —Bipds ae Ay wal age Herts cid eles eke ‘Note thal tb tome anges ped 2 ees tne ae a ed heeeee tenet a ee Acumr (A): Suppose thi the two 0 satt et the on snd that the baste drei afshe gad = uate anes orraqent the he {ost sort, abi up repel. Note iste bee ee Hang etter ton? hon ech ote. After bow A bs ted fe tik hen Yori 2 fet ona nal 2 fot up. is nition Weald Ite TE Oh a fare tine bee Bai Laws an 4 fot wn and et es mae tn Worle the point (3405 Thin pute las at ast vO= ORFF =n 7018 aNEIRB Seta ry Ate thin oeuert Ueo AE re eam Ho pit (9,24 sal ee Pel ese: wes he fie, ij Va ditaaae ee ama ne Rol VBA BST TE wads ene La A eling wost ode eeingv nthay anex Irfan re cech orb newer (D):_ Ce Canal Ye se rcade loss a uu vf Serve ho sepnoal casted 4D read and a astaad BAD ravl,opeoriey Esours hos eengsagad be Het ulin How vith verde, thea tage see 4 Ise ater ABD AUM, ALAND, ANNALS, 200 ADBDAB. bow st esl ronsonty seta route. sate Ihe Fy aera are 2 ey ede the esratr ebeingh alles Hr Pee ov We and tikes ax mine ere syplie Gs Tae is oer df gud DTaeadie Merwsraa Ip dire Amawer (DJs Tot the capris maseuzes laughs las seam fsb Fgste Ie oles of eangle na twice some end vr He recan ans kW", ae wen nt, oe TS) gk gil be ead ta BN By seasunay nh ole oF te aig ABD ad AC, win Pat 2 Bhasin 528 yO ad oe hy SAD ok eed se siclsue!s paves lest 20 Hoan sult bel tate felierg la 180 org S)—y9, inn Lameiiloes Bp iovich cage Sy = 5-418, or e-by=eTll pa rae 88 Neko Cane ie eat silky 40 he anes Taos = One tbe slant 18D), so chee 2 — Wea y= 1) Fro sop thos 60, Uy = 2-428, fm 80, end ths eaters imple Wo ABCD fag. 1A ~ 9.186) 1, 86100 12 Ty Uae y= iM el tho ssenee ie sigea a ABOD (eR? ua Te s0,1e. "25cm. suppune Pal ky TOL oy 138 Abang sretheceatenoe ufuighs in AWC amy IE L2Oee: = 5,78, 108, 5) Hayat Men e190 ond ei uch pee ‘Thorn, #6 i dati ie pet sie unehes f —28h eb Answer (C): Lee =30 ayo eoquetne et fag ae Py) ond acura wou doe sonatnity bet wy = by Ib entbitons cf a9 Seguros snl Hat 2018 ANCHE Snows 5 fe telatoy pt, 8 cee be. I ull bee BS bp—0% v.13. The winferunvalaco’ ¥ seals wt coang, by Bn ng 8 on ch case N= 1h oom 3 aa § Aun (1 The prvi bvteincn of 251 11-8, There ves 3 esr of GL tae emerson, xt >. Fore of wane a Clee of 50 aa 2013 cone vot, thve mae be co of 594 Lt Sewene wal fy nea, 6) 2126, su Fb sani walt ene he acioval cals fey 3. ed 9, Sith F nla, hc ius sal fatale hose seusuuusan 018 ~ Seana el Thue yy a3 Arma (A}t The sun ofthe iorn sso no pentegan AWEDY ie e180" = 5 vt he acum lela, eagles ae pines Pa ae al bps te EE" LO Mewione tho exppaneme cocles ay en The ferteor ufo a qual “RIE 08° = 72" lls that ae cage ‘aka up ths pointy of ke snr a igs “shmen we age amg BR)" and 30%, Label dhs a swe esta a hl Pre Ti ahowe armani ts 4 eae ssa that FO = aD CD aad tsar syressin or tae ee ewe pina of Ue sb Pees ee sei seviradey ccvala ACIS BOSBRESOA Ob4iatu'd BESE CS PACD 961 6 RAB = REY fuel ah the yasioun end naa vas met oma al ele ile Doves ‘Cho copaan: eotate Bet Anaver (1 25 AMIE eva the uatias, e4b a gard wi 12 ot Lee let atiaany Dod sonrbae aoe Les iy (aise > Gs ebut 20! Siete fas att oh ‘ise B92 cpu Ra for ail roa sles 2. Tus sens the dsc 403 —e}"—4. 2002-0) £0, ‘Suvi aud forcoving ies (2—10¢ 2) <0. So che mango valias oc 2 <c< 2. Both nuit wd niin ae altinatle by fein (x ho) (2.2-2) and (a,k6 = (23,8), Thecche Tae lifennee betwee Pe ‘maxinan ead mist possible as of is — (22) = Anawer (Be I the game starts wis 200 en sd Hema starts, dae he picks 3 el nl then no uatier age meiy Beriien choces, eam sooporvle ky keep Ue ane of resin cine east tal 5). het toa a Baviora pes 2 an she pro 1! Bachar picks. TWa mone tat on kar love so Toure wil kos eves sé thr sa ill win, Sided f Basen fav, en Jot ep tor hat Th nro: st ees ‘rset x28 fod}. Oh ae last Cara arbre 3 tits wa ae Feo Krab regs 3 ad tsa ll win ly ting be ete. tbe jase ety we 214 voiry ane Jenne ar thea se yieks 1 ofa sual wenees Si gave ln ty peevknie con 201d sa we ake we ‘er sata stat, seslets doi ‘hea nests wins Jen stacey Haan reepuu by heping tL tundra tera worneraet tot doh Te szourere Pint ox hat as: Ua Harhats nl haved coun se thy sor Ta sci go ist wll wa “be ge with 20-1 eal Anawer (B): he Pystugeraet Tsien appling 22 igh. Manges ABD pul ACH gies ABE UD" = AD? = ACE = CDN, thing BP BT = Le “1120S rom wake: “lbw tase BO HCD — 8 nel ADS 1D oes pes ABD ad oJ ate be ab ve wo nglringtioc LOA and AN Berane “AFH = /AD)P- 00% ilelngn we ADE inegel es Thea 2ABD~ 2 AUD ~ ASE bens wie ZA BT © ARE. There ah aieunle 08 td APE sors, Tense ews wich lee ral HS, Cousunally Ue BEF 7010 AMC22D Seasene 2 Ampwer (A}e Becnuas t 6r © 1 tse tl coe <1 in me Thread a tana, Notoner. cota = SES «sin Va Fi foun weress Ae th end 8 age loop n esr fone Pane me ears 6d RAB eral PB oc rine abt paca y — a eich that cap etl of and ate ea thay te snmtinore fC aad 2, Bebe yp st hes Ub Uae dpe of 299. lle thal Ue rs sale OF he ungeric vind he THe ond #8 has We she of QW te rmen tebe Foyer PS Toa, Movs by sk res 0 nat oun gens mir aon Pssarsenatr sing oy he mya ata tows tha ates = ns = sinseanes = Sung std iy tw fee that Ben seme — ha) ws ving “ie qlee din ewe ie 2y/1, Becoate “1 sin 2 < 1, te lyst iss) — Ts | cans? —2v2) = ere ic sel ep Anu ()e_Iftbe Saeco an palin ses hr po focu “lene, Uisu tu coreesoclang sorta merit Sa cate Tyo "lw ore anaes Tle seins ee ate ll fs Lotsa ae 9 Tnx Mircwen the cts are ocr set Zoe er lon to fines hon tLe mmenpor ng praelea 4o nl sec. Waly a pola eae bo The denser ote pata sae ose Fn) ocr ya fh tPod nny Cy, = OP © at fg) ne che ease wth enlln ‘std saste OF ac targel Ia both fans fe WP wal ee pale end © Tot Ei ere. thon olely each Siu mene) = aney ered sid 39 a 2 a Tint slr one resend Poe eects Cea! Ih 3 fd &, Meal gloves (0. Tp sure ia ave ef sol ga pull end Ob 2s ANCLR Socom 8 ‘Tis thee ne (2, pais of prshols wut che pairs thas dol ene ao ‘sowtiy she whim dcctaee ae theatrics ap alse ytencens ase hese san, Tuo a3 ice slopes nl =] ~ 6 ns fy arenas Use the some aga ahr tom {—8.—2, 11,23}, Bivase tie paso perso, las that inemcet do: at exactiy ti et a ne pS Ee al, Ff ther the lal wae ne ater on Lhe (Gye) Nite: It pict conse te Hw cise mg € wi tangent te Tsay fe lows: Lat Fh scr oft ale tere by he athe setsr spn ysl that ets O fr the atte ee i those acre parallel Os bean ters. Tet bet savant pee FO wit sespect ta und et othe Salento of fa te te Ot 0 = 14 then 1 be sum inaecetion of) ae a expense ine UO 1G © ove of he deine ies. UC pases tin © eal © we tae fay. fot Te ce pore a anagmey nf nl fy she Br 8 ole Bhoneny PE" = FOvFQ. The exe with easter Parl redis PO" PQ interests fv two pons Fal 1. Thy erect uf OT, wl OO we Me des 2 Auewer (A)s {ety camge of ase idety gives logy = hsm hag logy = byes Tho ea otto that — (a5 gag "gin ogn 22 MCU Sitios 8 ued cali lygsHa 9a Uhesarn ngry soon inepal ly dacerptonl 06 em ented boy tat Ingeatsl hie, Ulvens = Tages Hog’ — lg fo 9) Lee ne, hs pn af le asim. nz ya eee Sse dlfrot ements aera Then gay Sa lepit port pin De se = es TOE =E fav Sh entrocd nadwr brasmion ota Oh 9 faod teenie 74 ime te he gen =e 9 ed Siu bea epi idan tat Ahm niet DAP os ee AS Be I SoS and latte ou tncch) Rentaen 9a 13-3 thes 4 Ti 3 fred Ajanta A eeae SATE a Sev 22 — id web malls fives alle wa, Mia 92 L aed Gon AS ig a? 2 ge iphm = a 2 we EWN) 8, Tymeoce the wean fetal 8 =" ulin Uaqade wea = 2 and wae The ied: ane we ATS 1 Aumscen 88): Payal esl Fat ees ages inde teas Ds 2 69, we Jing sce Ayman, Besar eval ME) = a, 2438 i ae Baal a sated com Has nly 2 3D doee Therion ten he omen hm ahem tou. thee a ices Ie oo nve 80, tees Gages fae, Uy Lr vg ie Lisa 8 sepeteuistia uo AF sa oy ley Siete ss the at te dit fs heer A oqertcioy of 9 mney er ty UE asec, SS x, = (owed Ls but N=, (mo Yl a i de tN a the joe Fires 05 toe Ree SCAR N Ip aeee fet mfp Home £0 8 vod 3 lve a es esa uch chal 3° GB tag, Any = ay rl ion fans EM an = a8 Taal bh alney = Mignee 8 Ue tole that 8 ~ ON Le ate Inger Digwre sa UN, ey Sle, = Soy hay fond sap Pe Neha Font (od 39} aad ew By = Gy oases 8h Halle ta ieee eee een olsaer em rel eres °F bby Lay food 38 tapes bt yt ag bye dy =e imaod 947 a ea Bs Bn ih Time fe vb iat Ay Go 9, 8 Keon tucapr yan ot 8 ODD L3Oey SDR = 4s Peay 2 iggy 2 bd 130 a Lad ha, = 26 ey $y —2ey = 160, TO, Fag Gro 1 titan aIeeneh vison my 8) and Sta) — saad 2 FR" )Oy~ 1860 — 1608 a, ey = atl Sa OM ANCHE So ey es AN es Baa Oost = an by = ta 4) 1 fay oad LO). erase <0) € dane O< 4 < byl flume th ease 4 aera ate (4,8, smi (0,0), (2.0), 2h 8) ad rao eu ceased ith § cll valu of my W ene tee ‘soetly 25 Ginter bes ¥ vita tan wasn! ese Arawer (Ale Leta ZACH = ZNOW aul 2 = AN, Beenine AK W caisern i fll chit 2BKC = ONG ~ ID, ZCUR UI OF Sas ond <CIA = <BMC = HY a. Tow BABE © AUC od PANG ~/ADXC “Then we ne wo 2 ac Be = Be wo B= Vea ON ex _ox SC 7 BG = GF 0x2 VB te A {ot Pllotks s'dsohiscf X. Vocus AW in nquiated, us cae BPC ‘ernie wing wich 28/6 = Re Thea bythe Pete Fete 2 aR ort py These on betabina ti thee abun ine OX — (v4 he. then he Law ot aires app! ALOR yhoee 1-28-68 neat)

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American Mathematics Contest 12 B. Wednesday, February The low temperature was 8 degrees less than the average, so it that divides 12! is 210 - 34 - 52 which has square root 25 - 32 ~ 5. 2013 AMClZB Solutions 4. 12 If the game starts with 2014 coins and Jenna starts, then she picks 1 coin.
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