AoPS Community 2014 AMC 10 AMC102014 www.artofproblemsolving.com/community/c4812 by AlcumusGuy, El Ectric, djmathman, bestwillcui1, alex31415, niraekjs, brandbest1, Royalreter1, bobthes- martypants,sunny2000,flamefoxx99,professordad,Quadratic64,mathman523,rrusczyk – A – February4th 1 Whatis10·(cid:0)1 + 1 + 1 (cid:1)−1? 2 5 10 (A)3 (B)8 (C) 25 (D) 170 (E)170 2 3 2 Roy’s cat eats 1 of a can of cat food every morning and 1 of a can of cat food every evening. 3 4 BeforefeedinghiscatonMondaymorning,Royopenedaboxcontaining6cansofcatfood.On whatdayoftheweekdidthecatfinisheatingallthecatfoodinthebox? (A)Tuesday (B)Wednesday (C)Thursday (D)Friday (E)Saturday 3 Bridgetbakes48loavesofbreadforherbakery.Shesellshalfoftheminthemorningfor$2.50 each.Intheafternoonshesellstwothirdsofwhatshehasleft,andbecausetheyarenotfresh, shechargesonlyhalfprice.Inthelateafternoonshesellstheremainingloavesatadollareach. Eachloafcosts$0.75forhertomake.Indollars,whatisherprofitfortheday? (A)24 (B)36 (C)44 (D)48 (E)52 4 WalkingdownJaneStreet,Ralphpassedfourhousesinarow,eachpaintedadifferentcolor.He passedtheorangehousebeforetheredhouse,andhepassedthebluehousebeforetheyellow house. The blue house was not next to the yellow house. How many orderings of the colored housesarepossible? (A)2 (B)3 (C)4 (D)5 (E)6 5 Onanalgebraquiz,10%ofthestudentsscored70points,35%scored80points,30%scored90 points, and the rest scored 100 points. What is the difference between the mean and median scoreofthestudents’scoresonthisquiz? (A)1 (B)2 (C)3 (D)4 (E)5 6 Supposethatacowsgivebgallonsofmilkincdays.Atthisrate,howmanygallonsofmilkwill dcowsgiveinedays? (A) bde (B) ac (C) abde (D) bcde (E) abc ac bde c a de ©2022AoPSIncorporated 1 AoPS Community 2014 AMC 10 7 Nonzerorealnumbersx,y,a,andbsatisfyx < aandy < b.Howmanyofthefollowinginequal- itiesmustbetrue? (I)x+y < a+b (II)x−y < a−b (III)xy < ab (IV) x < a y b (A)0 (B)1 (C)2 (D)3 (E)4 8 Whichofthefollowingnumbersisaperfectsquare? 14!15! 15!16! 16!17! 17!18! 18!19! (A) (B) (C) (D) (E) 2 2 2 2 2 √ 9 The two legs of a right triangle, which are altitudes, have lengths 2 3 and 6. How long is the thirdaltitudeofthetriangle? (A)1 (B)2 (C)3 (D)4 (E)5 10 Fivepositiveconsecutiveintegersstartingwithahaveaverageb.Whatistheaverageof5con- secutiveintegersthatstartwithb? (A)a+3 (B)a+4 (C)a+5 (D)a+6 (E)a+7 11 A customer who intends to purchase an appliance has three coupons, only one of which may beused: Coupon1:10%offthelistedpriceifthelistedpriceisatleast$50 Coupon2:$20offthelistedpriceifthelistedpriceisatleast$100 Coupon3:18%offtheamountbywhichthelistedpriceexceeds$100 Forwhichofthefollowinglistedpriceswillcoupon1offeragreaterpricereductionthaneither coupon2orcoupon3? (A)$179.95 (B)$199.95 (C)$219.95 (D)$239.95 (E)$259.95 12 A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outsidethesectorsisshadedasshownWhatistheareaoftheshadedregion? ©2022AoPSIncorporated 2 AoPS Community 2014 AMC 10 √ √ √ √ √ (A)27 3−9π (B)27 3−6π (C)54 3−18π (D)54 3−12π (E)108 3−9π 13 Equilateral△ABChassidelength1,andsquaresABDE,BCHI,CAFGlieoutsidethetriangle. WhatistheareaofhexagonDEFGHI? F E A G D B C I H √ √ 12+3 3 9 √ 6+3 3 (A) (B) (C)3+ 3 (D) (E)6 4 2 2 14 The y-intercepts, P and Q, of two perpendicular lines intersecting at the point A(6,8) have a sumofzero.Whatistheareaof△APQ? (A)45 (B)48 (C)54 (D)60 (E)72 15 David drives from his home to the airport to catch a flight. He drives 35 miles in the first hour, butrealizesthathewillbe1hourlateifhecontinuesatthisspeed.Heincreaseshisspeedby 15milesperhourfortherestofthewaytotheairportandarrives30minutesearly.Howmany milesistheairportfromhishome? (A)140 (B)175 (C)210 (D)245 (E)280 ©2022AoPSIncorporated 3 AoPS Community 2014 AMC 10 16 In rectangle ABCD, AB = 1, BC = 2, and points E, F, and G are midpoints of BC, CD, and AD,respectively.PointH isthemidpointofGE.Whatistheareaoftheshadedregion? A B 1 H E G 1 D 1 F 1 C 2 2 √ √ √ 1 3 2 3 1 (A) (B) (C) (D) (E) 12 18 12 12 6 17 Threefairsix-sideddicearerolled.Whatistheprobabilitythatthevaluesshownontwoofthe dicesumtothevalueshownontheremainingdie? 1 13 7 5 2 (A) (B) (C) (D) (E) 6 72 36 24 9 18 Asquareinthecoordinateplanehasverticeswhosey-coordinatesare0,1,4,and5.Whatisthe areaofthesquare? (A)16 (B)17 (C)25 (D)26 (E)27 19 Four cubes with edge lengths 1, 2, 3, and 4 are stacked as shown. What is the length of the portionofXY containedinthecubewithedgelength3? √ √ 3 33 √ 2 33 √ (A) (B)2 3 (C) (D)4 (E)3 2 5 3 ©2022AoPSIncorporated 4 AoPS Community 2014 AMC 10 X 1 2 3 4 Y 20 Theproduct(8)(888...8),wherethesecondfactorhaskdigits,isanintegerwhosedigitshave asumof1000.Whatisk? (A)901 (B)911 (C)919 (D)991 (E)999 21 Positiveintegersaandbaresuchthatthegraphsofy = ax+5andy = 3x+bintersectthex-axis atthesamepoint.Whatisthesumofallpossiblex-coordinatesofthesepointsofintersection? (A)−20 (B)−18 (C)−15 (D)−12 (E)−8 22 In rectangle ABCD, AB = 20 and BC = 10. Let E be a point on CD such that ∠CBE = 15◦. WhatisAE? √ 20 3 √ √ (A) (B)10 3 (C)18 (D)11 3 (E)20 3 √ 23 Arectangularpieceofpaperwhoselengthis 3timesthewidthhasareaA.Thepaperisdivided into equal sections along the opposite lengths, and then a dotted line is drawn from the first dividertotheseconddividerontheoppositesideasshown.Thepaperisthenfoldedflatalong thisdottedlinetocreateanewshapewithareaB.WhatistheratioB : A? ©2022AoPSIncorporated 5 AoPS Community 2014 AMC 10 (A)1 : 2 (B)3 : 5 (C)2 : 3 (D)3 : 4 (E)4 : 5 24 Asequenceofnaturalnumbersisconstructedbylistingthefirst4,thenskippingone,listingthe next5,skipping2,listing6,skipping3,and,onthenthiteration,listingn+3andskippingn.The sequencebegins1,2,3,4,6,7,8,9,10,13.Whatisthe500,000thnumberinthesequence? (A)996,506 (B)996507 (C)996508 (D)996509 (E)996510 25 The number5867 is between22013 and22014. How manypairs of integers(m,n)are there such that1 ≤ m ≤ 2012and 5n < 2m < 2m+2 < 5n+1? (A)278 (B)279 (C)280 (D)281 (E)282 – B 1 1.Leahhas13coins,allofwhicharepenniesandnickels.Ifshehadonemorenickelthanshe hasnow,thenshewouldhavethesamenumberofpenniesandnickels.Incents,howmuchare Leah’scoinsworth? (A) 33 (B)35 (C)37 (D)39 (E)41 2 Whatis 23+23 ? 2−3+2−3 (A) 16 (B)24 (C)32 (D)48 (E)64 3 Randy drove the first third of his trip on a gravel road, the next 20 miles on pavement, and the remainingone-fifthonadirtroad.Inmiles,howlongwasRandy’strip? (A)30 (B) 400 (C) 75 (D)40 (E) 300 11 2 7 4 Susiepaysfor4muffinsand3bananas.Calvinspendstwiceasmuchpayingfor2muffinsand16 bananas.Amuffinishowmanytimesasexpensiveasabanana?(A) 3 (B) 5 (C) 7 (D)2 (E) 13 2 3 4 4 ©2022AoPSIncorporated 6 AoPS Community 2014 AMC 10 5 Doug constructs a square window using 8 equal-size panes of glass, as shown. The ratio of theheighttowidthforeachpaneis5 : 2,andthebordersaroundandbetweenthepanesare2 incheswide.Ininches,whatisthesidelengthothesquarewindow? (A)26 (B)28 (C)30 (D)32 (E)34 6 Orvinwenttothestorewithjustenoughmoneytobuy30balloons.Whenhearrived,hediscov- ered that the store had a special sale on balloons: buy 1 balloon at the regular price and get a secondat 1 offtheregularprice.WhatisthegreatestnumberofballoonsOrvincouldbuy? 3 (A)33 (B)34 (C)36 (D)38 (E)39 7 SupposeA > B > 0andAisx%greaterthanB.Whatisx? (A)100(cid:0)A−B(cid:1) (B)100(cid:0)A+B(cid:1) (C)100(cid:0)A+B(cid:1) (D)100(cid:0)A−B(cid:1) (E)100(cid:0)A(cid:1) B B A A B 8 Atrucktravels b feeteverytseconds.Thereare3feetinayard.Howmanyyardsdoesthetruck 6 travelin3minutes? (A) b (B) 30t (C) 30b (D) 10t (E) 10b 1080t b t b t 9 Forrealnumberswandz, 1 + 1 w z = 2014. 1 − 1 w z Whatis w+z ? w−z (A) −2014 (B) −1 (C) 1 (D)1 (E)2014 2014 2014 10 In the addition shown below A, B, C, and D are distinct digits. How many different values are possibleforD? ©2022AoPSIncorporated 7 AoPS Community 2014 AMC 10 ABBCB + BCADA DBDDD (A)2 (B)4 (C)7 (D)8 (E)9 11 For the consumer, a single discount of n% is more advantageous than any of the following discounts: (1) two successive 15% discounts (2) three successive 10% discounts (3) a 25% discount fol- lowedbya5%discount Whatisthesmallestpossiblepositiveintegervalueofn? (A) 27 (B)28 (C)29 (D)31 (E)33 12 Thelargestdivisorof2,014,000,000isitself.Whatisitsfifthlargestdivisor? (A)125,875,000 (B)201,400,000 (C)251,750,000 (D)402,800,000 (E)503,500,000 13 Sixregularhexagonssurroundaregularhexagonofsidelength1asshown.Whatistheareaof △ABC? B A C √ √ √ √ √ (A)2 3 (B)3 3 (C)1+3 2 (D)2+2 3 (E)3+2 3 14 Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, abc miles were displayed on the odometer, where abc is a 3-digit number with a ≥ 1 and a+b+c ≤ 7. At the end of the trip, where the odometer showed cba miles.Whatisa2+b2+c2? ©2022AoPSIncorporated 8 AoPS Community 2014 AMC 10 (A)26 (B)27 (C)36 (D)37 (E)41 15 In rectangle ABCD, DC = 2CB and points E and F lie on AB so that ED and FD trisect ∠ADC asshown.Whatistheratiooftheareaof△DEF totheareaofrectangleABCD? A E F B D C √ √ √ √ (A) 3 (B) 6 (C) 3 3 (D) 1 (E) 2 6 8 16 3 4 16 Fourfairsix-sideddicearerolled.Whatistheprobabilitythatatleastthreeofthefourdiceshow thesamevalue? (A) 1 (B) 7 (C) 1 (D) 5 (E) 1 36 72 9 36 6 17 Whatisthegreatestpowerof2thatisafactorof101002−4501? (A)21002 (B)21003 (C)21004 (D)21005 (E)21006 18 Alistof11positiveintegershasameanof10,amedianof9,andauniquemodeof8.Whatis thelargestpossiblevalueofanintegerinthelist? (A)24 (B)30 (C)31 (D)33 (E)35 19 Two concentric circles have radii 1 and 2. Two points on the outer circle are chosen indepen- dently and uniformly at random. What is the probability that the chord joining the two points intersectstheinnercircle? √ (A) 1 (B) 1 (C) 2− 2 (D) 1 (E) 1 6 4 2 3 2 20 Forhowmanyintegersisthenumberx4−51x2+50negative? (A)8 (B)10 (C)12 (D)14 (E)16 21 TrapezoidABCD hasparallelsidesAB orlength33andCD oflength21.Theothertwosides are of lengths 10 and 14. The angles at A and B are acute. What is the length of the shorter diagonalofABCD? ©2022AoPSIncorporated 9 AoPS Community 2014 AMC 10 √ √ √ (A)10 6 (B)25 (C)8 10 (D)18 2 (E)26 22 Eightsemicircleslinetheinsideofasquarewithsidelength2asshown.Whatistheradiusof thecircletangenttoallofthesesemicircles? √ √ √ √ √ 1+ 2 5−1 3+1 2 3 5 (A) (B) (C) (D) (E) 4 2 4 5 3 23 Asphereisinscribedinatruncatedrightcircularconeasshown.Thevolumeofthetruncated coneistwicethatofthesphere.Whatistheratiooftheradiusofthebottombaseofthetrun- catedconetotheradiusofthetopbaseofthetruncatedcone? √ √ 3 1+ 5 √ 3+ 5 (A) (B) (C) 3 (D)2 (E) 2 2 2 24 Thenumbers1,2,3,4,5aretobearrangedinacircle.Anarrangementisbadifitisnottruethat foreverynfrom1to15onecanfindasubsetofthenumbersthatappearconsecutivelyonthe circle that sum to n. Arrangements that differ only by a rotation or a reflection are considered thesame.Howmanydifferentbadarrangementsarethere? ©2022AoPSIncorporated 10