美国数学竞赛AMC题目及答案

1、2.A sign at the fish market says, "EO% oft today only: half-pound packages for just S3 per package/ What regular price for 3 full pound of fish, in dollars?(A) 6(B) 9(C) 10(D) 12(E) 153. What is the value of(_) - 3 - - - 1卜-"Jr - L.t!；4. Eight frie nds ate at a restaura nt and agreed to sh

2、are the bill equally. Because Judi forgot her money, each of her seven friends paid an extra $2.50 to cover her portion of the total bill. What was the total bill?5.Hammie is in the106 pou nds. His quadruplet sisters are tiny babies andweigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean

3、) weight of these five children or the media n weight, and by how many pou nds?6. The n umber in each box below is the product of the n umbers in the two boxes that touch it inthe row above. For example,： - C. What is the missing number in the top row?7. Trey and his mom stopped at a railroad cross

4、ing to let a train pass. As the tra in bega n to pass, Trey cou nted 6 cars in the first 10 sec on ds. It took the tra in 2 min 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 trai n?8. A fair coin is tossed 3 time

5、s. What is the probability of at least two consecutive heads?9. The In credible Hulk can double the dista nce he jumps with each succeed ing jump. If his first jump is 1 meter, the sec ond jump is 2 meters, the third jump is 4 meters, and so on, the n on which jump will he first be able to jump more

6、 tha n 1 kilometer?10. What is the ratio of the least com mon multiple of 180 and 594 to the greatest com mon factor of 180 and 594?11. Ted's grandfather 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 at the rate o

7、f 3 miles per hour on Wedn esday and at 4 miles per hour on Friday. If Gran dfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less?12. At the 2021 Winn ebago County Fair a ven dor is offer ing a "fair special" on san dals. If yo

8、u buy one pair of san dals at the regular price of $50, you get a sec ond pair at a 40% discou nt, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of san dals. What perce ntage of the $150 regular price did he save?13. When Clara t

9、otaled 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? and14. Let the two digits betwo is- '' dwhich factors into|9 b).Therefore, since the differe nee is a multipl

10、e of 9, theonly an swer choice that is a multiple of 9 is(A) 4515. If 阱 + 3° = 90,寥 + 44 = 76, and 5“ + 6° = 1421, what is the product of16. A n umber of stude nts from Fib on acci Middle School are tak ing part in a commu nity service b-graders to 林 -graders is-graders is ： ：. What is the

11、 smallest number of students that could be participating in theproject. The ratio of?, and the the ratio ofI.严 l-graders toIproject?17. The sum of six consecutive positive integers is 2021. What is the largest of these six integers?18. Isabella uses one-foot cubical blocks to build a rectangular for

12、t that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?-Arpanliku16:22, 27 November 2021 (EST) Courtesy of19. Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridge

13、t 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 highest to lowest?20. A . : c： recta ngl

14、e is in scribed in a semicircle with lon ger side on the diameter. What is the area of the semicircle?21. 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 no rth of the n ortheast corner of City Park. On school days she bik

15、es on streets to the southwest cor ner of City Park, the n takes a diag onal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many differe nt routes can she take?22. Toothpicks are used to make a grid that is 60 toothpicks

16、long and 32 toothpicks wide. How many toothpicks are used altogether?八.:-：i?'；7'are the diameters of semicircles二|and mare the midpointsAJICB局=100in ches,comprised of 3 semicircular arcs whose radii arein ches, respectively. The ball always rema ins in con tact with the track and does not sl

17、ip. What is the distance the center of the ball travels over the course from A to B?打)_ ：*打 nches, and人:，一-23. Angle：-；?of 八 s- ；?<?'is a right angle. The sides ofas show n. The area of the semicircle on'.-equalslen gth匸匸:.What is the radius of the semicircle on24. SquaresABCU, EFGH, and

18、SHIJare equal in area. Pointsof sides/ /and h k , respectively. What is the ratio of the area of the shaded pentagonto the sum of the areas of the three squares?25. A ball with diameter 4 inches starts at point A to roll along the track shown. The track isSolutionbuy (A) 1more model car.In order to

19、have her model cars in perfe匚E complete rows of 6, Danica must have a number of cars that is ; of 6. The smahest muttiple of 6 which is larger than 23 is 24, so she'll need to2. The 50% off price of half a pound of fish is $3, so the 100%, or the regular price, of a half pound(D) 12of fish is $6

20、. 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:(1 + 2 3 + 4 + 1000) = (1 4 2) + (3 + 4) + +(-999 + 1000)Therefore, the an swer is4*500 =(E)i 20004.Each of her seve n frie nds paid匸 ) . Sin

21、ce Judi was supposed to pay- Jto cover Judi's portion. Therefore, Judi's portion must beEl"of the total bill, the total bill must be8*7* $2 50= (C) $1405. The media n here is obviously less tha n the mea n, so opti on (A) and (B) are out.Lining up the n umbers (5, 5, 6, 8, 106), we see

22、that the media n weight is 6 pou nds.5 + 5 + 6+8 + 106130The average weight of the five kids isTherefore, the average weight is bigger, by_J pounds, making the answer(E) average, by 206.Solution 1: Worki ng BackwardsLet the value in the empty box in the middle row be row beE. Eis the answer we'r

23、e looking for.FT, and the value in the empty box in the topWe see that6IM=30rr,maki ng:3It follows that20 = 5ysoSolution 2: Jumping Back to the StartAno ther way to do this problem is to realize what makes up the bottommost n umber. This method does n't work quite as well for this problem, but i

24、n 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 be4, and the value in the empty box inthe top row be 巫. is the an swer we're look ing for.We can write some equat

25、i ons:Now we can substitute into the first equation using the two others:6 cars3 cars- r: 1' seco nds.7.If Trey sawimL , then he saw.：丄.2 min utes and 45 sec onds can also be expressed as(and preserve the same rate):5'33 secondsV9 cars165 seconds.It follows that the most likely n umber of ca

26、rs is(C) 100.Solution 2min utes andWSsec onds is equal to二I + '：1；，rm：".Since Trey probably cou nts arou nd - cars everyU；：sec on ds, there are165Io-=16groups of :cars that Trey most likely cou nts. Since工=沮：二m：, the closest answer choice is(C) 100.8.First, there are_、ways to flip the coins

27、, in order.The ways to get two con secutive 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 is21C = 10249.This is a geometric seque nce in which the com mon ratio is 2. To find the jump that would beover a 1

28、000 meters, we note thatHowever, because the first term is-.and not-,the solution to the problem is10 - 0 + 1 = (C) 11th180 = 32 x 5 x10. To find either the LCM or the GCF of two numbers, always prime factorize first.The prime factorizati on ofThe prime factorizati on of594 = 33 x 11 x 2).Multiply a

29、ll of these to get 5940.Then, find the greatest power of all the numbers there are; if one number is one but not the other, use it (this is、 < 1 _ < -factorizati ons and multiply.For the GCF of 180 and 594, use the least power of all of the n umbers that are in both 32 x 2= 18.Thus the an swer

30、 =5&40=We start off with a similar approach as the original solution. From the prime factorizations, the GCF is - 18 x lcm(180,594) = 594 x 180.Dividi ng byyields '门；宀一J X(C) 330lcm(180, 594) _ 594011. We use that fact thatTherefore, '- _ - - 'd 宀.Let d= dista nee, r= rate or speed,

31、and t=time. In this case, let眩represent the time.On Mon day, he was at a rate of' I：I J' . So,= 2 miles => a =叵oFor Wed nesday, he walked at a rate of' 1:1 1 . Therefore,. 23x = 2 miles =>= - hours3On Friday, he walked at a rate of4. = 2mil=>x=jhouraAddi ng up the hours yields22

32、294：-L+ - L+ I '=.1We now find the amount of time Grandfather would have taken if he walked at1:1 J1 per day.Set up the equati on,3= 2 miles x 3 days => x = - hoursTo find the amount of time saved, subtract the two amounts: IT.convert this to minutes, we multiply by3-一.ToThus, the soluti on t

33、o this problem is12. First, find the amount of money one will pay for three sandals without the discount. We have50 + 0.6x50 + x50 = $105$50 x 3 sandals = $150.Then, find the amou nt of money using the discou nt:Finding the perce ntage yieldsTo find the perce nt saved, we have100% - 70% = (B) 3013.

34、Let the two digits be and .two is|9(a 一叫.Therefore, since the differe nee is a multiple of 9, theinto3+34 =903P + 81 = 9015. ：'-Therefore,Therefore,To most people, it would not be immediately evide nt that64 = 1296,so we can multiply 6's un tilwe get the desired n umber:6216 =12fl6 = 64,so k

35、=4lTherefore the an swer is2-5 4= (B) 4016. Solution 1: AlgebraWe multiply the first ratio by 8 on both sides, and the second ratio by 5 to get the same number for 8th graders, in order that we can put the two ratios together:Therefore, the ratio of 8th graders to 7 th graders to 6th graders isI - -

36、. Si nee the ratio isin lowest terms, the smallest n umber of stude nts participat ing in the project is40 + 25 + 24 89Soluti on 2: Fakesolv ingThe number of 8th graders has to be a multiple of 8 and 5, so assume it is 40 (the smallestpossibility). The n there are6th graders and7 th graders. The n u

37、mbers40 + 24 + 25 =of stude nts is17. Solution 1The mea n of these n umbers is竽= 335.5.Therefore the n umbers are333,334. 335.33代 337.338so the an swer is(B) 338Solution 2Let then umber be:：；：.Then our desired number is3016(B) 338&r - 3 = 2021 => 2E- 丁=336 =骑 + 2 = oOur in tegers areso we hav

38、e thatSolution 3Let the first term beOur in tegers arexx 1,® + 2, h + 3北 l- 4,話 + 3.We have,61： + 15 = 2021 => x 333 jt + 5 = (13 j 33818. Solution 1There are.一 I - cubeson the base of the box. Then, for each of the 4 layers above thebottom (as since each cube is 1 foot by 1 foot by 1 foot a

39、nd the box is 5 feet tall, there are 4 feetleft), there are 9+11+9+11= 40cubes. Hence, the answer is120+4*®= I®)280Solution 2We can just calculate the volume of the prism that was cut out of the orig inal再 feet shorter tha n each side of the outside. Since the floor is12 x 10 x 5box.Each i

40、n terior side of the fort will be土 feet. So the volume of the in terior box isfoot, the height will be10x8x4 = 320 ft3:狂.Therefore, the number of blocksThe volume of the orig inal box iscontained in the fort is600 - 320 = (B) 28019. If Hannah did better than Cassie, there would be no way she could k

41、now for sure that she did n't get the lowest score in the class. Therefore, Hannah did worse tha n Cassie. Similarly, if Hannah did worse tha n Bridget, there is no way Bridget could have known that she did n't get thehighest in the class. Therefore, Hannah did better tha n Bridget, so our o

42、rder is(D) Cassie, Hamiah. BridgetA semicircle has symmetry, so the cen ter is exactly at the midpo int of the 2 side on the recta ngle.The area ismaking the radius, by the Pythagorean Theorem,The number of ways to get from Samantha's house to City Park isand the n umber ofways to get from City

43、Park to school is=6.Since there's one way to go through City Park(just walk ing straight through), the n umber of differe nt ways to go from Sama ntha's house tof 6 = |(E)18City Park to school.22. There are : Uvertical columns with a length of/'toothpicks, and there arewith a length of 用

44、'-toothpicks. An effective way to verify this is to try a small case, i.e. a61 32 + 33 -60= (E) 3932of toothpicks. Thus, our an swer is.33 horiz on tal rowsHagrid23. Solution 1(B) 7.5If the semicircle on AB were a full circle, the area would be 16pi. Therefore the diameter of the first circle is

45、 8. The arc of the largest semicircle would no rmally have a complete diameter of 17.The Pythagorea n theorem says that the other side has len gth 15, so the radius isSolution 2We go as in Solution 1, finding the diameter of the circle on AC and AB. Then, an extended version of the theorem says that

46、 the sum of the semicircles on the left is equal to the biggest one, so thearea of the largest is289tt8, and the middle one is289tt 64tt 225tt8" = T",so the radius is15224.(B)工 5A1：DFirst let '(where hits the exte nsion ofds the side len gth of the squares) for simplicity. We can exte nd-i '. Call this point亘.The area of tria ngleuntil it3 4- The area of1:-. Now, the ratio of the shadedrectangleQU is 二 Thus, our desired