American Mathematics Competitions. 17 th Annual AMC 8. (American Mathematics Contest 8) Tuesday, NOVEMBER 13, 2001

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1 athmatical Aociation of Amrica Amrican athmatic Comtition Prnt by th Akamai Founation 17 th Annual AC 8 (Amrican athmatic Contt 8) Tuay, OVEBER 13, 2001 ISTRUCTIOS 1. DO OT OPE THIS BOOKLET UTIL YOUR PROCTOR TELLS YOU. 2. Thi i a twnty-fiv qution multil choic tt. Each qution i follow by anwr mark A, B, C, D an E. Only on of th i corrct. 3. Th anwr to th roblm ar to b mark on th AC 8 Anwr Form with a #2 ncil. Chck th blackn circl for accuracy an ra rror an tray mark comltly. Only anwr rorly mark on th anwr form will b gra. 4. Thr i no nalty for guing. Your cor on thi tt i th numbr of corrct anwr. 5. o ai ar rmitt othr than cratch ar, grah ar, rulr, rar, an calculator that ar acct for u on th SAT. o roblm on th tt will rquir th u of a calculator. 6. Figur ar not ncarily rawn to cal. 7. Bfor bginning th tt, your roctor will ak you to rcor crtain information on th anwr form. 8. Whn your roctor giv th ignal, bgin working on th roblm. You will hav 40 minut to comlt th tt. 9. Whn you finih th xam, ign your nam in th ac rovi on th Anwr Form. Th Committ on th Amrican athmatic Comtition rrv th right to r-xamin tunt bfor ciing whthr to grant official tatu to thir cor. Th Committ alo rrv th right to iqualify all cor from a chool if it i trmin that th rquir curity rocur wr not follow. Th ublication, rrouction, or communication of th roblm or olution of th AC 8 uring th rio whn tunt ar ligibl to articiat riouly joariz th intgrity of th rult. Dulication at any via coir, tlhon, -mail, Worl Wi Wb or mia of any ty i a violation of th coyright law. Coyright 2001, Committ on th Amrican athmatic Comtition, athmatical Aociation of Amrica

2 17th AC Cay ho cla i making a golf trohy. H ha to aint 300 iml on a golf ball. If it tak him 2 con to aint on iml, how many minut will h n to o hi job? (A) 4 (B) 6 (C) 8 (D) 10 (E) I m thinking of two whol numbr. Thir rouct i 24 an thir um i 11. What i th largr numbr? (A) 3 (B) 4 (C) 6 (D) 8 (E) Granny Smith ha $63. Elbrta ha $2 mor than Anjou an Anjou ha on-thir a much a Granny Smith. How many ollar o Elbrta hav? (A) 17 (B) 18 (C) 19 (D) 21 (E) Th igit 1, 2, 3, 4 an 9 ar ach u onc to form th mallt oibl vn fiv-igit numbr. Th igit in th tn lac i (A) 1 (B) 2 (C) 3 (D) 4 (E) 9 5. On a ark an tormy night Snooy unly aw a flah of lightning. Tn con latr h har th oun of thunr. Th of oun i 1088 ft r con an on mil i 5280 ft. Etimat, to th nart halfmil, how far Snooy wa from th flah of lightning. (A) 1 (B) (C) 2 (D) (E) 3 6. Six tr ar qually ac along on i of a traight roa. Th itanc from th firt tr to th fourth i 60 ft. What i th itanc in ft btwn th firt an lat tr? (A) 90 (B) 100 (C) 105 (D) 120 (E) 140

3 17th AC KITES O PARADE Problm 7, 8 an 9 ar about th kit. To romot hr chool annual Kit Olymic, Gnviv mak a mall kit an a larg kit for a bulltin boar ilay. Th kit look lik th on in th iagram. For hr mall kit Gnviv raw th kit on a on-inch gri. For th larg kit h tril both th hight an with of th ntir gri. 7. What i th numbr of quar inch in th ara of th mall kit? (A) 21 (B) 22 (C) 23 (D) 24 (E) Gnviv ut bracing on hr larg kit in th form of a cro conncting ooit cornr of th kit. How many inch of bracing matrial o h n? (A) 30 (B) 32 (C) 35 (D) 38 (E) Th larg kit i covr with gol foil. Th foil i cut from a rctangular ic that jut covr th ntir gri. How many quar inch of wat matrial ar cut off from th four cornr? (A) 63 (B) 72 (C) 180 (D) 189 (E) A collctor offr to buy tat quartr for 2000% of thir fac valu. At that rat how much will Bryn gt for hi four tat quartr? (A) $20 (B) $50 (C) $200 (D) $500 (E) $ Point A, B, C an D hav th coorinat: A(3, 2), B(3, 2), C( 3, 2) an D( 3, 0). Th ara of quarilatral ABCD i (A) 12 (B) 15 (C) 18 (D) 21 (E) 24

4 17th AC If a b = a + b, thn (6 4) 3 = a b (A) 4 (B) 13 (C) 15 (D) 30 (E) Of th 36 tunt in Richll cla, 12 rfr chocolat i, 8 rfr al, an 6 rfr blubrry. Half of th rmaining tunt rfr chrry i an half rfr lmon. For Richll i grah howing thi ata, how many gr houl h u for chrry i? (A) 10 (B) 20 (C) 30 (D) 50 (E) Tylr ha ntr a bufft lin in which h choo on kin of mat, two iffrnt vgtabl an on rt. If th orr of foo itm i not imortant, how many iffrnt mal might h choo? at: bf, chickn, ork Vgtabl: bak ban, corn, otato, tomato Drt: browni, chocolat cak, chocolat uing, ic cram (A) 4 (B) 24 (C) 72 (D) 80 (E) Homr bgan ling a il of 44 otato at th rat of 3 otato r minut. Four minut latr Chritn join him an l at th rat of 5 otato r minut. Whn thy finih, how many otato ha Chritn l? (A) 20 (B) 24 (C) 32 (D) 33 (E) A quar ic of ar, 4 inch on a i, i fol in half vrtically. Both layr ar thn cut in half aralll to th fol. Thr nw rctangl ar form, a larg on an two mall on. What i th ratio of th rimtr of on of th mall rctangl to th rimtr of th larg rctangl? (A) 1 3 (B) 1 2 (C) 3 4 (D) 4 5 (E) 5 6

5 17th AC For th gam how Who Want To B A illionair?, th ollar valu of ach qution ar hown in th following tabl (whr K = 1000). Qution Valu K 2K 4K 8K 16K 32K 64K 125K 250K 500K 1000K Btwn which two qution i th rcnt incra of th valu th mallt? (A) From 1 to 2 (B) From 2 to 3 (C) From 3 to 4 (D) From 11 to 12 (E) From 14 to Two ic ar thrown. What i th robability that th rouct of th two numbr i a multil of 5? (A) 1 36 (B) 1 18 (C) 1 6 (D) (E) Car travl at a contant for a givn. Thi i hown by th ah lin. Car travl at twic th for th am itanc. If Car an ar hown a oli lin, which grah illutrat thi? (A) (B) (C) (D) (E)

6 17th AC Kalana how hr tt cor to Quay, arty an Shana, but th othr k thir hin. Quay think, At lat two of u hav th am cor. arty think, I in t gt th lowt cor. Shana think, I in t gt th hight cor. Lit th cor from lowt to hight for arty (), Quay (Q) an Shana (S). (A) S,Q, (B) Q,,S (C) Q,S, (D),S,Q (E) S,,Q 21. Th man of a t of fiv iffrnt oitiv intgr i 15. Th mian i 18. Th maximum oibl valu of th largt of th fiv intgr i (A) 19 (B) 24 (C) 32 (D) 35 (E) On a twnty-qution tt, ach corrct anwr i worth 5 oint, ach unanwr qution i worth 1 oint an ach incorrct anwr i worth 0 oint. Which of th following cor i OT oibl? (A) 90 (B) 91 (C) 92 (D) 95 (E) Point R, S an T ar vrtic of an quilatral triangl, an oint X, Y an Z ar mioint of it i. How many noncongrunt triangl can b rawn uing any thr of th ix oint a vrtic? (A) 1 (B) 2 (C) 3 (D) 4 (E) 20 R Y S X Z T 24. Each half of thi figur i como of 3 r triangl, 5 blu triangl an 8 whit triangl. Whn th ur half i fol own ovr th cntrlin, 2 air of r triangl coinci, a o 3 air of blu triangl. Thr ar 2 r-whit air. How many whit air coinci? (A) 4 (B) 5 (C) 6 (D) 7 (E) Thr ar 24 four-igit whol numbr that u ach of th four igit 2, 4, 5 an 7 xactly onc. Only on of th four-igit numbr i a multil of anothr on. Which of th following i it? (A) 5724 (B) 7245 (C) 7254 (D) 7425 (E) 7542

7 SOLUTIOS Your School anagr will b nt at lat on coy of th 2001 AC 8 Solution Pamhlt. It i mant to b loan to tunt (but not ulicat). WRITE TO US Commnt about th roblm an olution for thi AC 8 houl b ar to: Prof. Joh W. Knny, AC 8 Chair / knnj@muohio.u Dartmnt of athmatic an Statitic, iami Univrity, Oxfor, OH Commnt about aminitrativ arrangmnt houl b ar to: Titu Anrcu, AA AC Dirctor / titu@amc.unl.u Amrican athmatic Comtition, Univrity of braka-lincoln P.O. Box 81606, Lincoln, E AC 10 & AC 12 Th AC 10 an AC 12 ar 25-qution, 75-minut contt with 5 choic of anwr for ach roblm (A through E). School with high coring tunt on th AC 8 will rciv an Invitation Brochur for th 2002 AC 10. Th bt way to rar for th ur lvl contt i to tuy xam from rviou yar. Orr for all ublication lit blow houl b ar to: Amrican athmatic Comtition ATT: Publication P.O. Box Lincoln, E PUBLICATIOS IIU ORDER: $10 (Bfor hanling f), PAYETS I US FUDS OLY. U.S.A. an Canaian orr mut b rai an will b hi UPS or AIRAIL. Shiing & Hanling charg for Publication Orr: Orr TOTAL Shiing Charg $ $30.00 $5.00 $ $40.00 $7.00 $ $50.00 $9.00 $ $75.00 $12.00 $ UP $15.00 ak chck ayabl to th Amrican athmatic Comtition; or giv your Via, atrcar or Amrican Exr numbr, xiration at an carholr hom ar. Intrnational Orr: Do OT ray; an invoic will b nt to you. Each ric i for an xam an it olution for on yar. Scify th yar you want an how many coi of ach xam. All ric ffctiv to Stmbr 1, AC 8 (Junior High/il School xam), , $1.00 r coy r yar. AC 10 & AC 12 (High School Exam), , $1.00 r coy r yar. Book (Exam an Solution) Problm Book I, AHSE ,...$10.00 Problm Book II, AHSE ,... $10.00 Problm Book III, AHSE ,...$13.00 Problm Book IV, AHSE ,...$13.00 Problm Book V, AHSE & AIE ,...$30.00 Problm Book VI, AHSE $24.00

8 2001 AC 8 DO OT OPE UTIL TUESDAY, OVEBER 13, 2001 **Aminitration On An Earlir Dat Will Diqualify Your School Rult** 1. All information (Rul an Intruction) n to aminitr thi xam i contain in th TEACHERS AUAL, which i outi of thi ackag. PLEASE READ THE AUAL BEFORE OVEBER 13, othing i n from ini thi ackag until ovmbr Your PRICIPAL or VICE-PRICIPAL mut vrify on th AC 8 CER- TIFICATIO Form that all rul aociat with th conuct of th xam wr follow. 3. Th Anwr Form mut b mail Firt Cla to th AC offic no latr than 24 hour following th xam. 4. THE AC 8 IS TO BE ADIISTERED DURIG A COVEIET 40 IUTE PERIOD. THE EXA AY BE GIVE DURIG A REGULAR ATH CLASS. 5. Th ublication, rrouction or communication of th roblm or olution of thi tt uring th rio whn tunt ar ligibl to articiat riouly joariz th intgrity of th rult. Dulication at any via coir, tlhon, -mail, Worl Wi Wb or mia of any ty i a violation of th coyright law. ajor Sonor athmatical Aociation of Amrica Th Akamai Founation Univrity of braka Contributor Amrican Statitical Aociation Caualty Actuarial Socity Socity of Actuari ational Council of Tachr of athmatic Amrican Socity of Pnion Actuari Amrican athmatical Socity Amrican athmatical Aociation of Two Yar Collg Pi u Eilon Conortium for athmatic an it Alication u Alha Thta ational Aociation of athmatician Kaa u Eilon School Scinc an athmatic Aociation Clay athmatic Intitut Intitut for Oration Rarch an th anagmnt Scinc