美国数学竞赛AMC题目及答案

1、2013 AMC 8 Problems/Problem 1ProblemD-ani匚刁 wants to arrange her me?d已I 匚ars in rows v-.:ith6 匚ars m 轻日gh ro She now has 23 mcdel carsis the smallest number of additional cars she must buy in order to be able to arrange all her cars th怜 way? 1(B) 2(C) 3(D) 4(E) 52.A sign at ttiE fish market says, &q

2、uot;5Oc/o off, today only: half-pound packages for just S3 per pa匚kage." What is th regular price for a full pound of fish, in dollars?(A) 6(B) 9(C) 10(D) 12(E) 153.What is the value ofI J : 1-1 -1 i1 H I?(A) -10(B) 0(C) 1(D) 500(E) 20004. Eight frie nds ate at a restaura nt and agreed to share

3、 the bill equally. Because Judi forgot her mon ey, each of her seven friends paid an extra $2.50 to cover her portion of the total bill. What was the total bill?(A) $120(B) 8128(C) S140(D) 8144(E) $lfi05.Hammie is in the 6'hgrade and weighs 106 pou nds. His quadruplet sisters are tiny babies and

4、 weigh 5,5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pou nds?(E)(A) median、by 60(B) median, by 20(C) average by 5(D) average, by 156. The n umber in each box below is the product of the n umbers in the two boxes that to

5、uch it in the row above. For example, 30 = 6 X 5. What is the missing number in the top row?21 / 17(A) 2(B) 3(C) 4(D) 5(E) &7. Trey and his mom stopped at a railroad cross ing to let a tra in pass. As the train bega n to pass, Trey cou nted 6 cars in the first 10 sec on ds. It took the train 2 m

6、in utes and 45 sec onds to clear the cross ing at a con sta nt speed. Which of the follow ing was the most likely n umber of cars in the train?(A) 60(B) SO (C) 100(D) 120(E) 1408. A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?(A) |(B) J (C)| (D)l (E)|9. The

7、 In credible Hulk can double the dista nee he jumps with each succeedi ng jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more tha n 1 kilometer?(A) 9th (B) 10th (C) 11th (D) 12th (E) 13th10. Wha

8、t is the ratio of the least com mon multiple of 180 and 594 to the greatest com mon factor of 180 and 594?(A) 110(B) 16$(C) 330(D) 625(E) 6&011. Ted's gran dfather used his treadmill on 3 days this week. He went 2 miles each day. On Mon day he jogged at a speed of 5 miles per hour. He walked

9、 at the rate of 3 miles per hour on Wedn esday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less?(A) 1(B) 2(C) 3(D) 4(E) 512. At the 2013 Winn ebago County Fair a ven dor is offeri ng a "

10、;fair special" on san dals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took adva ntage of the "fair special" to buy three pairs of san dals. What perce ntage of the $150 regula

11、r price did he save?(A) 25(B) 30(C) 33(D) 40(E) 4513. When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the follow ing might her in correct sum have differed from the correct one?(A) 45(B) 46(C) 47(D) 48(E) 4914. Let the two digits

12、 beGand &.The correct score was，卜:Iwhich factors into丄+ 込 Clara misinterpreted it asI k '':. The differenee between the two is.Therefore, since the differe nee is a multiple of 9, the only an swerchoice that is a multiple of 9 is(A) 4515. If 聲 +3" = 90, 2r -p 44 = 76, and + 6* = 142

13、1, what is the product of 戸，r , and 孚？(A) 27(B) 40(C) 50(D) 70(E) 9016. A n umber of stude nts from Fib on acci Middle School are tak ing part in a commu nity service project.The ratio of- -graders to-graders is> :工，and the the ratio of''-graders to7111 -graders isWhat is the smallest n u

14、mber of stude nts that could be participati ng in the project?(A) 16(B) 40(C) 55(D) 79(E) 8917. The sum of six consecutive positive integers is 2013. What is the largest of these six integers?(A) 335(C) 340(D) 345(E) 3S018. Isabella uses one-foot cubical blocks to build a rectangular fort that is 12

15、 feet Iong, 10 feet wide, and5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?-Arpanliku 16:22, 27 November 2013 (EST) Courtesy of Lord.of.AMC(A) 204(B) 280(C) 320(D) 340(E) 60019. Bridget, Cassie, and Hannah are discussing the results of their

16、last math test. Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show theirs to anyone. Cassie says, 'I did n't get the lowest score in our class,' and Bridget adds, 'I did n't get the highest score.' What is the ranking of the three girls from highe

17、st to lowest?(A) Hsnuiah. Cassie, Bridget (B) Hannah： Bridget, Cassie(C) CassieT Bridget, Hannah (D)Cassie, Hannah Bridget(E) Bridget. Cassie, Hannah20. A 1 X /recta ngle is in scribed in a semicircle with lon ger side on the diameter. What is the area of the semicircle?(A) 5 (B)ly (C)7T(D)丰 (E) y21

18、. Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the n ortheast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the no

19、rtheast corner, and the n bikes on streets to school. If her route is as short as possible, how many differe nt routes can she take?(A) 3(B) 6(C) 9(D) 12(E) 1822. Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?:一、are the

20、diameters of semicircles as(A) 1920(B) 1952(C) 1980(D) 2013(E) 393223. Angle 'of j 二 Jis a right an gle. The sides ofshow n. The area of the semicircle on.1 equalsWhat is the radius of the semicircle on'> 7, and the arc of the semicircle on(A) 7(B) 7.5<C) 8(D) 8.5(E> 924. Squares .

21、j1 f / , /.'' ;、, and f"=：are equal in area. Pointsand JJare the midpoints of sides/ J: and / / /:, respectively. What is the ratio of the area of the shaded pentagon./ ' ' / to the sum of theareas of the three squares?(A 片(B)l (C)l (D)| (E)AAB25. A ball with diameter 4 inches s

22、tarts at point A to roll along the track shown. The track is comprised of3 semicircular arcs whose radii are川 二 inches, -1 inches, and：'L 汽nches, respectively.B238肓240?r260tt280jt50(kThe ball always rema ins in con tact with the track and does not slip. What is the dista nee the cen ter of the b

23、all travels over the course from A to B?1.SolutionIn order to have her model 亡3的 in perfect, complete rows of 6, Danica must have白 number of cars that 誇 a muof 6, The snialles t multiple of 6 which is larger than 23 is 24so she'fl need tc buy(A) 1more model car.2.The 50% off price of half a poun

24、d of fish is $3, so the 100%, or the regular price, of a half pound of fish(D) 12is $6. Con seque ntly, if half a pou nd of fish costs $6, the n a whole pou nd of fish isdollars.3.Notice that we can pair up every two numbers to make a sum of 1:(-L + 2-3 + 4+ 1000)(-1 +2) + (-3 + 4) + + (-999 + 1000)

25、=500 5.The media n here is obviously less tha n the mean, so opti on (A) and (B) are out. Lining up the n umbers (5, 5, 6, 8, 106), we see that the media n weight is 6 pou nds.Therefore, the an swer is4 500= (E) 20004.Each of her seve n frie nds paid$2.50to cover Judi's portion. Therefore, Judi&

26、#39;s portion must be$2.50 7Since Judi was supposed to pay” of the total bill, the total bill must be8-72.50= (C) S1405+5 + 6 +呂+106130The average weight of the five kids is55Therefore, the average weight is bigger, by26 6 = 20pou nds, maki ng the an swer(E) average ly 206.Solution 1: Worki ng Backw

27、ardsLet the value in the empty box in the middle row be,r, and the value in the empty box in the top row be.A'is the answer we're looking for.We see that匚;1?丄;：maki ng -l|-'.Solution 2: Jumping Back to the StartAno ther way to do this problem is to realize what makes up the bottommost n

28、umber. This method does n't work quite as well for this problem, but in a larger tree, it might be faster. (In this case, Soluti on 1 would be faster since there's only two miss ing n umbers.)Agai n, let the value in the empty box in the middle row be , and the value in the empty box in the

29、toprow be . L is the answer we're looking for.We can write some equati ons:600 = 3Ot30 = 6-5t = 5yNow we can substitute into the first equation using the two others:600 = (6 5)(裁) =655 岁 =6- 25 *站600 = 150 |(C)4| = 6 cars3 £肛呂7.If Trey saw 10 SCOOIldK, then he saw 5 SCGOIlds .2 min utes and

30、 45 sec onds can also be expressed as； i .1 sec on ds.Trey's rate of see ing cars,i ears5 seconds ,can be multiplied byI ： f on the top and bottom (andpreserve the same rate):3 - 33 cars _99 <jars"'33 secoildis 16 J secuiKli . It follows that the most likely number of cars is(C) 100S

31、olution 22minutes andseconds is equal to 120+ 45 = 165 3<3OOlldsSince Trey probably cou nts arou ndFears every 1 卜 sec on ds, there are=16groups of cars thatTrey most likely cou nts. Since.|.、，the closest answer choice is(C) 1008.First, there are_!- ways to flip the coins, in order.The ways to ge

32、t two consecutive heads are HHT and THH.The way to get three con secutive heads is HHH.Therefore, the probability of flipp ing at least two con secutive heads is9. This is a geometric seque nce in which the com mon ratio is 2. To find the jump that would be over a 1000 meters, we note that0" =

33、1024.However, because the first term is10 - 0 + 1 = (C) 11th-.and notJ _ , the solution to the problem is10. To find either the LCM or the GCF of two numbers, always prime factorize first.The prime factorizatio n ofThe prime factorizatio n of180 = 3Z x 5 x 22Then, find the greatest power of all the

34、numbers there are; if one number is one but not the other, use it(this is.Multiply all of these to get 5940.For the GCF of 180 and 594, use the least power of all of the numbers that are in both factorizations and multiply. 3? X 2= 18.Thus the an swer =5940(c)aso18We start off with a similar approac

35、h as the original solution. From the prime factorizations, the GCF isIt is a well known fact thatgod(m,n) X lcm(m,Ti) = mn. s。we have,18 x lcmOSO, 594) = 594 x l»0Dividing byLSyields=同° X 10 = ®40.kin(180,594)5940讼Therefore,顽丽呵帀一11. We use that fact thatd = Tt. Let d= distance, r= rat

36、e or speed, and t=time. In this case, let鳥 represe nt the time.On Mon day, he was at a rate of2. soJ"For Wed nesday, he walked at a rate of2:. Therefore,一 -=hours4On Friday, he walked at a rate ofAdding up the hours yields22294-hows hours - hours - hours5+ 3+4=60We now find the amount of time G

37、randfather would have taken if he walked at- p per day. Set up34a： = 2 jnil-es x 3 days hoursthe equati on,_To find the amount of time saved, subtract the two amounts:94丽hours 辛 hours 盘 hours-2= wJ. To con vertthis to min utes, we multiply byThus, the soluti on to this problem is(D)412. First, find

38、the amount of money one will pay for three sandals without the discount. We have $50 x 3 sandals = 8150.50 + 0.6 x 50 + x 50 = S105The n, find the amou nt of money using the discou nt:1Finding the perce ntage yields105150100%-70%To find the perce nt saved, we have(B) 3013. Let the two digits bea a n

39、d &.The correct score was10ti + b. Clara misinterpreted it as106 + a . The differenee between the two isIwhich factors into丨9(“ 一 “|. Therefore, since the differenee is a multiple of 9, the only answerchoice that is a multiple of 9 is(A) 4514. The probability that both show a gree n bea n is1 11

40、 '-. The probability that both show a red beanis -1. Therefore the probability is1 -84+34 =90 +81 = W 15.浮一FTherefore, 戸=Zr + 44 = 762r = 32Therefore, r = &.53 + 6" = 1421125+fi = 12&6To most people, it would not be immediately evident thatJ ." , so we can multiply 6's unti

41、l we getthe desired n umber:6-6 = 366*36=2166 216 = 1296 =沪,so s = 4.2-54 = | (B) 40Therefore the an swer isI16. Solution 1: AlgebraWe multiply the first ratio by 8 on both sides, and the second ratio by 5 to get the same number for 8thgraders, in order that we can put the two ratios together:5:3 =

42、58) :3=40:24S: 5 = 8(5) : 5(5) =40:25Therefore, the ratio of 8th graders to 7th graders to 6th graders is40 ： 25 Z 24. Si nee the ratio is in lowest40+25 + 24= (E) 89terms, the smallest n umber of stude nts participati ng in the project is.Soluti on 2: Fakesolv ingThe number of 8th graders has to be

43、 a multiple of 8 and 5, so assume it is 40 (the smallest possibility).40* f = 2440* - = 25The n there are6th graders and&7 th graders. The n umbers of stude nts is40 + 24 + 25 = 1(E) 8917. Solution 1The mea n of these n umbers is67133 334t 335? 336? 337,338, so the answer is= 335.5.Therefore the

44、 n umbers are(B) 338Solution 2Let the丄叫umber be X". Then our desired number is- r ' 1 r 1 二 so we have that&r-3 = 2013 =k x= = 336 =+ 2 = | (B) 338Our in tegers areSolution 3Let the first term beE. Our integers are斗 忙 + 1, t + 2? i: + 3T j: + 4, t + 5.We have,&e + 15 = 2013 => 阪=

45、333 => x + 5 = I (B) S3&18. Solution 1There areU lcubeson the base of the box. Then, for each of the 4 layers above the bottom (assince each cube is 1 foot by 1 foot by 1 foot and the box is 5 feet tall, there are 4 feet left), there are L+ll+9+H = 40cubes. Hence, the answer is + 4 40 = |(B)2

46、80Solution 2We can just calculate the volume of the prism that was cut out of the original- Ibox. Eachin terior side of the fort will be-feet shorter tha n each side of the outside. Since the floor is-foot, theheight will be°feet. So the volume of the in terior box is10 X 8 X 4 = 320 ft3The vol

47、ume of the origi nal box is12 x 10 x 5 = 600ii ' Therefore, the number of blocks contained inthe fort is600 - 320 = (B) 28019. If Hannah did better tha n Cassie, there would be no way she could know for sure that she did n't getthe lowest score in the class. Therefore, Hannah did worse than

48、Cassie. Similarly, if Hannah did worse tha n Bridget, there is no way Bridget could have known that she did n't get the highest in the class.(D) Cassie, Hannah. BridgetTherefore, Hannah did better tha n Bridget, so our order is3,and the n umber of ways toget from City Park to school isThe n umbe

49、r of ways to get from Sama ntha's house to City Park is=6.Si nee there's one way to go through City Park (just walk ing straight through), the number of different ways to go from Samantha's house to City Park to school5-6=|(E)1822. There arevertical eolumns with a length of32 toothpicks,

50、 and there are3Shoriz on tal rows with alength ofvUtoothpicks. An effective way to verify this is to try a small case, i.e. a2 X 3 grid of toothpicks.61 >32 + 3360 = (E) 3932Thus, our an swer is23. Solution 1If the semicircle on AB were a full circle, the area would be 16pi. Therefore the diamete

51、r of the first circle is 8. The arc of the largest semicircle would no rmally have a complete diameter of 17. The Pythagorea n(B)theorem says that the other side has len gth 15, so the radius isSolution 2We go as in Soluti on 1, finding the diameter of the circle on AC and AB. Then, an exte nded ver

52、s ion of the theorem says that the sum of the semicircles on the left is equal to the biggest one, so the area of the289tt64 亓225 对largest is &, and the middle one is888, so the radius is24. FFirst let 号 三 (where商s the side length of the squares) for simplicity. We can extendun til it hits theex

53、te nsion ofAB .Call this poi nt. . The area of triangleAXJ then is 2 The area of recta ngleJ ' is - ' -. Thus, our desired area is-.Now, the ratio of the shaded area to thecomb ined area of the three squares isSolution 2BEDJ.Let the side len gth of each square beLet the in tersecti on ofa ndj be .Sincecon grue nt.