AMC Senior 2008(澳大利亚数学竞赛AMC-E：11-12年级中英文历年真题)

THURSDAY 1 AUGUST 2013SENIOR DIVISIONAUSTRALIAN S CHOOL YEARS 11 and 12TIME ALLOWED: 75 MINUTES©AMT P ublishing 2013 AMTT liMiTed Acn 083 950 341A ustrAliAn M AtheMAtics c oMpetitionsponsored by the c oMMonweAlth b AnkAn AcTiviTy of The AusTrAliAn MATheMATics TrusTNAMEYEAR TEACHERA u s T r A l i A n M A T h e M A T i c s T r u s T姓 名： 年 级： 监考老师：意事项一般规定1.未获监考老师许可之前不可翻开此测验题本。

2.各种通讯器材一律不得携入考场，不准使用电子计算器、计算尺、对数表、数学公式等计算器具。

作答时可使用直尺与圆规，以及两面全空白的草稿纸。

3.题目所提供之图形只是示意图，不一定精准。

4.最前25题为选择题，每题有五个选项。

最后题要求填入的答案为000至999的正整数。

题目一般而言是依照越来越难的顺序安排，对于错误的答案不会倒扣分数。

5.本活动是数学竞赛而不同于学校测验，别期望每道题目都会作。

考生只与同地区同年级的其它考生评比，因此不同年级的考生作答相同的试卷将不作评比。

6.请依照监考老师指示，谨慎地在答案卡上填写您的基本数据。

若因填写错误或不详所造成之后果由学生自行负责。

7.进入试场后，须等待监考老师宣布开始作答后，才可以打开题本进行答题。

作答须知1.限用B 或2B 铅笔填写答案。

2.请用B 或2B 铅笔在答案卡上（不是在题本上）将您认为正确选项的圆圈涂满。

3.您的答案卡将由计算机阅卷，为避免计算机误判，请不要在答案卡上其它任何地方涂划任何记号。

填写答案卡时，若需要修改，可使用软性橡皮小心擦拭，并确定答案卡上无残留痕迹。

澳洲SACE（南澳大利亚州教育综合）是澳大利亚南澳大利亚州的高中证书体系。

在SACE的十二年级数学课程中，学生将学习高等数学的理论和实践，为日后进入大学或职业学校做准备。

以下是澳洲SACE 十二年级数学内容的详细介绍：一、数学方法1. 函数和图像在这一部分，学生将学习各种函数的定义和性质，包括线性函数、二次函数、指数函数、对数函数和三角函数等。

他们将学习如何绘制这些函数的图像，并理解函数的变化趋势。

2. 微积分的基本概念学生将开始接触微积分的基本概念，包括导数和积分。

他们将学习如何计算函数的导数和积分，以及如何应用这些概念来研究变化率和曲线下面积。

3. 统计学这一部分将介绍统计学的基本概念，包括数据收集、数据展示、概率和推断。

学生将学习如何分析和解释数据，并掌握统计学方法来做出推断和预测。

4. 离散数学学生将学习离散数学的基本概念，包括集合、排列组合、图论和逻辑推理。

他们将学习如何应用这些概念解决实际问题，并理解离散数学在计算机科学和信息技术中的应用。

二、专业数学1. 线性代数学生将学习线性代数的基本概念，包括向量、矩阵、线性方程组和向量空间等。

他们将学习如何解决线性代数相关的实际问题，并理解线性代数在工程、物理、经济等领域的应用。

2. 微积分的应用这一部分将深入探讨微积分的应用，包括曲线的长度、曲面的面积、体积和物理学中的应用等。

学生将学习如何应用微积分解决实际问题，并掌握微积分的工程和科学应用技巧。

3. 统计学进阶学生将深入研究统计学的高级方法，包括回归分析、方差分析、时间序列分析和统计推断的方法等。

他们将学习如何应用这些方法解决实际问题，并掌握统计学在商业、经济、金融等领域的应用技巧。

4. 计算数学这一部分将介绍计算数学的基本概念，包括差分方程、数值方法和优化算法等。

学生将学习如何应用计算数学方法解决实际问题，并掌握计算数学在科学工程和计算机科学领域的应用技巧。

三、高级数学1. 微积分的拓展学生将深入研究微积分的高级概念，包括多元函数、偏导数、重积分和曲线积分等。

This Solutions Pamphlet gives at least one solution for each problem on this year’s exam and shows that all the problems can be solved using material normally as-sociated with the mathematics curriculum for students in eighth grade or below. These solutions are by no means the only ones possible, nor are they necessarily superior to others the reader may devise.We hope that teachers will share these solutions with their students. However, the publication, reproduction, or communication of the problems or solutions of the AMC 8 during the period when students are eligible to participate seriously jeopardizes the integrity of the results. Dissemination at any time via copier, telephone, e-mail, World Wide Web or media of any type is a violation of the competition rules.Correspondence about the problems and solutions should be addressed to:Ms. Bonnie Leitch , AMC 8 Chair / bleitch@548 Hill Avenue, New Braunfels, TX 78130Orders for prior year Exam questions and solutions Pamphlets should be addressed to:Attn: Publications American Mathematics Competitions University of Nebraska-Lincoln P .O. Box 81606Lincoln, NE 68501-1606Copyright © 2008, The Mathematical Association of AmericaT he M aTheMaTical a ssociaTion of a MericaAmerican Mathematics Competitions 24th AnnualAMC 8(American Mathematics Contest 8)Solutions PamphletT uesday, NOvEMBER 18, 20081.Answer(B):Susan spent2×12=$24on rides,so she had50−12−24=$14to spend.2.Answer(A):Because the key to the code starts with zero,all the lettersrepresent numbers that are one less than their ing the key,C is 9−1=8,and similarly L is6,U is7,and E is1.BEST OF LUCK0123456789CLUE=86713.Answer(A):A week before the13th is the6th,which is thefirst Friday of themonth.Counting back from that,the5th is a Thursday,the4th is a Wednesday, the3rd is a Tuesday,the2nd is a Monday,and the1st is a Sunday.ORCounting forward by sevens,February1occurs on the same day of the week as February8and February15.Because February13is a Friday,February15is a Sunday,and so is February1.4.Answer(C):The area of the outer triangle with the inner triangle removedis16−1=15,the total area of the three congruent trapezoids.Each trapezoid =5.has area1535.Answer(E):Barney rides1661−1441=220miles in10hours,so his average=22miles per hour.speed is220106.Answer(D):After subdividing the central gray square as shown,6of the16congruent squares are gray and10are white.Therefore,the ratio of the area of the gray squares to the area of the white squares is6:10or3:5.7.Answer (E):Note that M45=35=3·95·9=2745,so M =27.Similarly,60N =35=3·205·20=60100,so N =100.The sum M +N =27+100=127.ORNote that M45=35,so M =35·45=27.Also 60N =35,so N60=53,andN =53·60=100.The sum M +N =27+100=127.8.Answer (D):The sales in the 4months were $100,$60,$40and $120.The average sales were 100+60+40+1204=3204=$80.ORIn terms of the $20intervals,the sales were 5,3,2and 6on the chart.Their sum is 5+3+2+6=16and the average is 164=4.The average sales were4·$20=$80.9.Answer (D):At the end of the first year,Tammy’s investment was 85%of the original amount,or $85.At the end of the second year,she had 120%of her first year’s final amount,or 120%of $85=1.2($85)=$102.Over the two-year period,Tammy’s investment changed from $100to $102,so she gained 2%.10.Answer (D):The sum of the ages of the 6people in Room A is 6×40=240.The sum of the ages of the 4people in Room B is 4×25=100.The sum of the ages of the 10people in the combined group is 100+240=340,so the average age of all the people is 34010=34.11.Answer (A):The number of cat owners plus the number of dog owners is20+26=46.Because there are only 39students in the class,there are 46−39=7students who have both.ORBecause each student has at least a cat or a dog,there are 39−20=19students with a cat but no dog,and 39−26=13students with a dog but no cat.So there are 39−13−19=7students with both a cat and a dog.13197DogCat12.Answer (C):The table gives the height of each bounce.Bounce 12345Height 23·2=23·43=23·89=23·1627=in Meters 2438916273281Because 1627>1632=12and 3281<3264=12,the ball first rises to less than 0.5meters on the fifth bounce.Note:Because all the fractions have odd denominators,it is easier to doublethe numerators than to halve the denominators.So compare 1627and 3281to their numerators’fractional equivalents of 12,1632and 3264.13.Answer (C):Because each box is weighed two times,once with each ofthe other two boxes,the total 122+125+127=374poundsis twice the combined weight of thethree boxes.The combined weight is 3742=187pounds.14.Answer (C):There are only two possible spaces for the B in row 1and onlytwo possible spaces for the A in row 2.Once these are placed,the entries in theremaining spaces are determined.The four arrangementsare:ORThe As can be placed eitheror In each case,the letter next to the top A can be B or C.At that point the rest of the grid is completely determined.So there are 2+2=4possible arrangements.15.Answer(B):The sum of the points Theresa scored in thefirst8games is37.After the ninth game,her point total must be a multiple of9between37and 37+9=46,inclusive.The only such point total is45=37+8,so in the ninth game she scored8points.Similarly,the next point total must be a multiple of 10between45and45+9=54.The only such point total is50=45+5,so in the tenth game she scored5points.The product of the number of points scored in Theresa’s ninth and tenth games is8·5=40.16.Answer(D):The volume is7×1=7cubic units.Six of the cubes have5square faces exposed.The middle cube has no face exposed.So the total surface area of thefigure is5×6=30square units.The ratio of the volume to the surface area is7:30.ORThe volume is7×1=7cubic units.There arefive unit squares facing each of six directions:front,back,top,bottom,left and right,for a total of30square units of surface area.The ratio of the volume to the surface area is7:30. 17.Answer(D):The formula for the perimeter of a rectangle is2l+2w,so2l+2w=50,and l+w=25.Make a chart of the possible widths,lengths,and areas,assuming all the widths are shorter than all the lengths.Width123456789101112Length242322212019181716151413Area24466684100114126136144150154156 The largest possible area is13×12=156and the smallest is1×24=24,for a difference of156−24=132square units.Note:The product of two numbers with afixed sum increases as the numbers get closer together.That means,given the same perimeter,the square has a larger area than any rectangle,and a rectangle with a shape closest to a square will have a larger area than other rectangles with equal perimeters.18.Answer(E):The length offirst leg of the aardvark’s trip is14(2π×20)=10πmeters.The third andfifth legs are each14(2π×10)=5πmeters long.Thesecond and sixth legs are each10meters long,and the length of the fourth leg is 20meters.The length of the total trip is10π+5π+5π+10+10+20=20π+40 meters.19.Answer(B):Choose two points.Any of the8points can be thefirst choice,and any of the7other points can be the second choice.So there are8×7=56ways of choosing the points in order.But each pair of points is counted twice,so there are 562=28possible pairs.A B CDEF G H Label the eight points as shown.Only segments AB ,BC ,CD ,DE ,EF ,F G ,GH and HA are 1unit long.So 8of the 28possible segments are 1unit long,and the probability that the points are one unit apart is 828=27.ORPick the two points,one at a time.No matter how the first point is chosen,exactly 2of the remaining 7points are 1unit from this point.So the probabilityof the second point being 1unit from the first is 27.20.Answer (B):Because 23of the boys passed,the number of boys in the class is a multiple of 3.Because 34of the girls passed,the number of girls in the classis a multiple of 4.Set up a chart and compare the number of boys who passed with the number of girls who passed to find when they are equal.Total boysBoys passed 326496Total girls Girls passed 4386The first time the number of boys who passed equals the number of girls who passed is when they are both 6.The minimum possible number of students is 9+8=17.ORBecause 23of the boys passed,the number of boys who passed must be a multiple of 2.Because 34of the girls passed,the number of girls who passed must be a multiple of 3.Because the same number of boys and girls passed,the smallestpossible number is 6,the least common multiple of 2and 3.If 6of 9boys and 6of 8girls passed,there are 17students in the class,and that is the minimum number possible.ORLet G =the number of girls and B =the number of boys.Then 23B =34G ,so 8B =9G .Because 8and 9are relatively prime,the minimum number of boysand girls is 9boys and 8girls,for a total of 9+8=17students.21.Answer (C):Using the formula for the volume of a cylinder,the bologna hasvolume πr 2h =π×42×6=96π.The cut divides the bologna in half.Thehalf-cylinder will have volume 96π2=48π≈151cm 3.Note:The value of πis slightly greater than 3,so to estimate the volume multiply 48(3)=144cm 3.The product is slightly less than and closer to answer C than any other answer.22.Answer (A):Because n3is at least 100and is an integer,n is at least 300andis a multiple of 3.Because 3n is at most 999,n is at most 333.The possible values of n are 300,303,306,...,333=3·100,3·101,3·102,...,3·111,so the number of possible values is 111−100+1=12.E F 323.Answer (C):Because the answer is a ratio,it doesnot depend on the side length of the square.Let AF =2and F E =1.That means square ABCE has side length 3and area 32=9square units.The area of BAF is equal to the area of BCD =12·3·2=3square units.Triangle DEF is an isosceles right triangle with leg lengths DE =F E =1.The area of DEF is 12·1·1=12square units.The area of BF D is equal to the area of the square minusthe areas of the three right triangles:9−(3+3+12)=52.So the ratio of the area of BF D to the area of square ABCE is 529=518.24.Answer (C):There are 10×6=60possible pairs.The squares less than60are 1,4,9,16,25,36and 49.The possible pairs with products equal to the given squares are (1,1),(2,2),(1,4),(4,1),(3,3),(9,1),(4,4),(8,2),(5,5),(6,6)and (9,4).So the probability is 1160.。

A u s t r A l i A n M At h e M At i c s c o M p e t i t i o na n a c t i v i t y o f t h e a u s t r a l i a n m a t h e m a t i c s t r u s tT H U R S D AY 31 J U LY 2008JUNIOR DIVISION COMPETITION PAPERA U S T R A L I A N S C H O O LY E A R S 7 A N D 8T I M E A L L O W E D : 75 M I N U T ESJunior DivisionQuestions 1to 10,3marks each1.The value of 2008+8002is (A)1010(B)4004(C)10008(D)8910(E)100102.Which of the following numbers has the largest value?(A)2.15(B)2.2(C)2.08(D)2.1(E)2.1853.The perimeter of the figure,in centimetres,is (A)8(B)10(C)12(D)16(E)20......................................................4cm2cm 4cm2cm4.One half of 19912is (A)9512(B)9534(C)9914(D)9912(E)99345.The value of x is(A)135(B)95(C)35(D)55(E)45...................................................................................................................135◦x ◦6.The value of 200×8200÷8is(A)1(B)8(C)16(D)64(E)2007.How many squares of any size are there in the diagram?(A)9(B)11(C)12(D)14(E)161111111122J28.A train left Fassifern at8:58am and arrived at Broadmeadow at9:34am on thesame day.The time taken,in minutes,was(A)82(B)22(C)36(D)38(E)789.The digits5,6,7,8and9can be arranged to form evenfive-digit numbers.Thetens digit in the largest of these numbers is(A)5(B)6(C)7(D)8(E)910.P QRS is a square and points E and F are outside the square so that P QE andQRF are equilateral triangles.The size of EQF,in degrees,is(A)60(B)90(C)120(D)150(E)180Questions11to20,4marks each11.A rectangle has an area of72square centimetres and the length is twice the width.The perimeter,in centimetres,of the rectangle is(A)34(B)36(C)42(D)48(E)5412.Marbles of three different colours are in a tin and 25of the marbles are red,13aregreen and the remaining12are yellow.The number of marbles in the tin is(A)30(B)45(C)54(D)60(E)9013.In the diagram,triangles P QR and LMN areboth equilateral and QSM=20◦.What is the value of x?(A)70(B)80(C)90(D)100(E)110....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................LNM PQRS20◦x◦J314.At half-time in a soccer match between Newcastle and Melbourne,the score wasNewcastle1,Melbourne0.Three goals were scored in the second half.Which of the following could not be the result of the match?(A)The match was drawn(B)Newcastle won by2goals(C)Melbourne won by2goals(D)Newcastle won by1goal(E)Newcastle won by4goals15.In how many ways can12be written as the sum of two or more different positivewhole numbers?(Changing the order of addition does not count as a different way.)(A)12(B)13(C)14(D)15(E)1616.How many different positive numbers are equal to the product of two odd one-digitnumbers?(A)25(B)15(C)14(D)13(E)1117.The perimeter of this rectangular paddock is700m.It is subdivided into sixidentical paddocks as shown.The perimeter,in metres,of each of the six smaller paddocks is(A)11613(B)300(C)200(D)150(E)60018.The student lockers at Euler High School are to be numbered consecutively from1to500using plastic digits which cost5cents each.The total cost of all the digits will be(A)$25(B)$63.65(C)$69.50(D)$69.60(E)$85J419.In the grid below,the squares are to befilled with the numbers1,2,3and4sothat they appear once only in each row,each column and each diagonal.123XYThe largest possible value of X+Y is(A)4(B)5(C)6(D)7(E)820.The average of one group of numbers is4.A second group contains twice asmany numbers and has an average of10.The average of both groups of numbers combined is(A)5(B)6(C)7(D)8(E)9Questions21to25,5marks each21.A cube with edge length2metres is cut up into cubes each with edge length5centimetres.If all these cubes were stacked one on top of the other to form a tower, the height of the tower would be(A)32km(B)160m(C)1600m(D)3.2km(E)320m22.A number is less than2008.It is odd,it leaves a remainder of2when divided by3and a remainder of4when divided by5.What is the sum of the digits of the largest such number?(A)26(B)25(C)24(D)23(E)2223.Farmer Taylor of Burra has two tanks.Water from the roof of his farmhouse iscollected in a100kL tank and water from the roof of his barn is collected in a 25kL tank.The collecting area of his farmhouse roof is200square metres while that of his barn is80square metres.Currently,there are35kL in the farmhouse tank and13kL in the barn tank.Rain is forecast and he wants to collect as much water as possible.He should:(A)empty the barn tank into the farmhouse tank(B)fill the barn tank from the farmhouse tank(C)pump10kL from the farmhouse tank into the barn tank(D)pump10kL from the barn tank into the farmhouse tank(E)do nothingJ 524.A fishtank with base 100cm by 200cm and depth 100cm contains water to a depthof 50cm.A solid metal rectangular prism with dimensions 80cm by 100cm by 60cm is then submerged in the tank with an 80cm by 100cm face on the bottom..................................................................................................................6.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................100100200501006080The depth of water,in centimetres,above the prism is then (A)12(B)14(C)16(D)18(E)2025.A strip of paper is folded in a line at an angle x ◦to the sides and then foldedunderneath forming an angle of 20◦as shown...............x ◦=⇒........................................................................................................................................................................................................................................................................................=⇒...............................................................................................................................................................................................................20◦The value of x is (A)60(B)65(C)70(D)75(E)80For questions 26to 30,shade the answer as an integer from 0to 999inthe space provided on the answer sheet.Question 26is 6marks,question 27is 7marks,question 28is 8marks,question 29is 9marks and question 30is 10marks.26.A two-digit number ab and its reversal ba are both prime.For example,13and 31are both prime.What is the largest possible sum of these two numbers ab and ba ?J627.Given a regular heptagon(7-sided polygon),how many obtuse-angled triangles arethere,where the vertices of each triangle are vertices of the heptagon?28.A rectangular prism6cm by3cm by3cm is made up by stacking1cm by1cmby1cm cubes.How many rectangular prisms,including cubes,are there whose vertices are vertices of the cubes,and whose edges are parallel to the edges of the original rectangular prism?(Rectangular prisms with the same dimensions but in different positions are different.)29.Let us call a sum of integers cool if thefirst and last terms are1and each termdiffers from its neighbours by at most1.For example,the sum1+2+3+4+3+ 2+3+3+3+2+3+3+2+1is cool.How many terms does it take to write2008as a cool sum if we use no more terms than necessary?30.A monument has been constructed from identical stone cubes.The views fromabove,the front f and the side s are shown.sfabove view front view side view What is the largest number of stones in the monument consistent with these views?***Junior 2008 Answers Question Answer 1E2B3D4E5E6D7D8C9A10D11B12B13B14D15C16C17B18D19C20D21D22A23D24B25E26176272128756298930106。

2008 AMC10美国数学竞赛A 卷1. A bakery owner turns on his doughnut machine at8:30 AM. At 11:10 AM the machine has completed one third of the day ’s job. At what time will the doughnut machine complete the job?(A) 1:50 PM (B) 3:00 PM (C) 3:30 PM (D) 4:30 PM (E) 5:50 PM2. A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is 2:1. The ratio of the rectangle ’s length to its width is 2:1. What percent of the rectangle ’s area is inside the square?(A) 12.5(B) 25 (C) 50 (D) 75 (E) 87.53. For the positive integer n, let <n> denote the sum of all the positive divisors of n with the exception of n itself. For example, <4>=1+2=3 and <12>=1+2+3+4+6=16. What is <<<6>>>?(A) 6(B) 12 (C) 24 (D) 32 (E) 364. Suppose that 2/3 of 10 bananas are worth as much as 8 oranges. How many oranges are worth as much as 1/2 of 5 bananas?(A) 2(B) 5/2 (C) 3 (D) 7/2 (E) 45. Which of the following is equal to the product 81216442008............481242004n n(A) 251(B) 502 (C) 1004 (D) 2008 (E) 40166. A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of 20 kilometers per hour, and runs at a rate of 10 kilometers per hour. Which of the following is closest to the triathlete ’s average speed, in kilometers per hour, for the entire race?(A) 3(B) 4 (C) 5 (D) 6 (E) 77. The fraction 20082200622007220052(3)(3)(3)(3)-- simplifies to which of the following? (A) 1(B) 9/4 (C) 3 (D) 9/2 (E) 98. Heather compares the price of a new computer at two different stores. Store A offers 15% off the sticker price followed by a $90 rebate, and store B offers 25% off the same sticker price with no rebate. Heather saves $15 by buying the computer at store A instead of store B. What is the sticker price of the computer, in dollars?(A) 750(B) 900 (C) 1000 (D) 1050 (E) 15009.Suppose that236x π- is an integer. Which of the following statements must be true about x ?(A) It is negative.(B) It is even, but not necessarily a multiple of 3.(C) It is multiple of 3, but not necessarily even.(D) It is a multiple of 6, but not necessarily a multiple of 12.(E) It is a multiple of 12.10. Each of the sides of a square s with area 16 is bisected, and a smaller square s is constructed using the bisection points as vertices. The same process is carried out on s to construct an even smaller square s. What is the area of s?(A) 1/2 (B) 1 (C) 2 (D) 3 (E) 411. While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing toward the shore at a constant rate of 4 miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking?(A) 2 (B) 4 (C) 6 (D) 8 (E) 1012. In a collection of red, blue, and green marbles, there are 25% more red marbles than blue marbles, and there are 60% more green marbles than red marbles. Suppose that there are r red marbles. What is the total number of marbles in the collection? (A) 2.85r(B) 3r(C) 3.4r(D) 3.85r(E) 4.25r13. Doug can paint a room in 5 hours. Dave can paint the same room in 7 hours. Dougand Dave paint the room together and take a one-hour break for lunch. Let t be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by t ? (A) 11()(1)157t ++= (B) 11()1157t ++= (C) 11()157t += (D) 11()(1)157t +-= (E) (57)1t +=14. Older television screens have an aspect ratio of 4:3. That is , the ratio of the width to the height is 4:3. The aspect ratio of many movies is not 4:3, so they are sometimes shown on a television screen by “letterboxing ”- darkening strips of equal height at the top and bottom of the screen, as shown. Suppose a movie has an aspect ratio of 2:1 and is shown on an older television screen with a 27-inch diagonal. What is the height, in inches, of each darkened strip?(A) 2(B) 2.25 (C) 2.5 (D) 2.7 (E) 315. Yesterday Han drove 1 hour longer than Ian at an average speed 5 miles per hour faster than Ian. Jan drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Han drove 70 miles more than Ian. How many more miles did Jan drive than Ian?(A) 120(B) 130 (C) 140 (D) 150 (E) 16016. Points A and B lie on a circle centered at O, and ∠AOB=60°. A second circle is internally tangent to the first and tangent to both OA and OB. What is the ratio of thearea of the smaller circle to that the larger circle?(A) 1/16 (B) 1/9 (C) 1/8 (D) 1/6 (E) 1/417. An equilateral triangle has side length 6. What is the area of the region containing all points that are outside the triangle but not more than 3 units from a point of the triangle?(A) 36+(B) 549π+(C) 546π+(D) 3)π(E) 1)π18. A right triangle has perimeter 32 and area 20. What is the length of its hypotenuse?(A) 57/4 (B) 59/4 (C) 61/4 (D) 63/4 (E) 65/419. Rectangle PQRS lies in a plane with PQ=RS=2 and QR=SP=6. The rectangle is rotated 90°clockwise about R, then rotated 90°clockwise about the point S moved to after the first rotation. What is the length of the path traveled by point P?(A)π(B) 6π(C) 3)π(D) π(E)20. Trapezoid ABCD has bases AB and CD and diagonals intersecting at K. Suppose that AB=9, DC=12, and the area of △ADK is 24. What is the area of trapezoid ABCD?(A) 924 (B) 94 (C)96 (D) 98 (E) 10021. A cube with side length 1 is sliced by a plane that passes through two diagonally opposite vertices A and C and the midpoints B and D of two opposite edges not containing A or C, as shown. What is the area of quadrilateral ABCD?(A) (B) 54(C) (D) 58 (E) 3422. Jacob uses the following procedure to write down a sequence of numbers. First he chooses the first term to be 6. To generate each succeeding term, he flips a fair coin. If it comes up heads, he doubles the previous term and subtracts 1. If it comes up tails, he takes half of the previous term and subtracts 1. What is the probability that the fourth term in Jacob ’s sequence is an integer? (A)16 (B) 13 (C) 12 (D) 58 (E) 3423. Two subsets of the set S=(a,b,c,d,e) are to be chosen so that their union is S and their intersection contains exactly two elements. In how many ways can this be done, assuming that the order in which the subsets are chosen does not matter?(A)20(B) 40 (C) 60 (D) 160 (E) 32024. Let 2200820082k =+. What is the units digit of 222k +?(A) 0 (B) 1 (C) 4 (D) 6 (E) 8C25. A round table has radius 4. Six rectangular place mats are placed on the table. Each place mat has width 1 and length x as shown. They are positioned so that each mat has two corners on the edge of the table, there two corners being end points of the same side of length x. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is x?(A) (B) 3(C)2(D)(E) 52+x 1。

美国数学竞赛AMC8 – 2008年真题解析(英文解析+中文解析)Problem 1Answer: BSolution:50-12-24=14中文解析：总共花的钱是：12+12*2=36元。

剩余50-36=14元。

答案是BProblem 2Answer: ASolution:We can derive that c=8,L=6, U=7,and E=1. Therefore, the answer is 8671.中文解析：这10个字母的对应关系是： B -0；E-1; S-2; ......K -9. 按照这个对应关系：C-8，L-6，U-7，E-1. 即8671. 答案是A。

Problem 3Answer: ASolution:We can go backwards by days, but we can also backwards by weeks. If we go backwards by weeks, we see that February 6 is a Friday. If we now go backwards by days, February 1 is a Sunday.中文解析：13日是周五，则13-7=6，即6日也是周五，则倒推2月1日是周日。

答案是A。

Problem 4Answer: CSolution:The area outside the small triangle but inside the large triangle is 16-1=15. This is equally distributed between the three trapezoids. Each trapezoid has an area of 15/3=5.中文解析：大三角形的面积等于小的等边三角形的面积加上3个梯形的面积。

据此，三个梯形的面积是16-1=15. 每个梯形的面积是15/3=5. 答案是C。

A u s t r A l i A n M At h e M At i c s c o M p e t i t i o na n a c t i v i t y o f t h e a u s t r a l i a n m a t h e m a t i c s t r u s tt h u r sd ay4a u g u s t2011senior Division Competition papera u st r a l i a n s c h o o l y e a r s11a n d12t i m e a l l o w e d:75m i n u t e sinstruCtions anD informationGeneraL1. Do not open the booklet until told to do so by your teacher.2. NO calculators, slide rules, log tables, maths stencils, mobile phones or other calculating aids arepermitted. Scribbling paper, graph paper, ruler and compasses are permitted, but are not essential.3. Diagrams are NOT drawn to scale. They are intended only as aids.4. There are 25 multiple-choice questions, each with 5 possible answers given and 5 questions thatrequire a whole number answer between 0 and 999. The questions generally get harder as you work through the paper. There is no penalty for an incorrect response.5. This is a competition not a test; do not expect to answer all questions. You are only competingagainst your own year in your own State or Region so different years doing the same paper are not compared.6. Read the instructions on the answer sheet carefully. Ensure your name, school name and schoolyear are entered. It is your responsibility to correctly code your answer sheet.7. When your teacher gives the signal, begin working on the problems.tHe ansWer sHeet1. Use only lead pencil.2. Record your answers on the reverse of the answer sheet (not on the question paper) by FULLYcolouring the circle matching your answer.3. Your answer sheet will be scanned. The optical scanner will attempt to read all markings evenif they are in the wrong places, so please be careful not to doodle or write anything extra on the answer sheet. If you want to change an answer or remove any marks, use a plastic eraser and be sure to remove all marks and smudges.inteGritY of tHe CompetitionThe AMT reserves the right to re-examine students before deciding whether to grant official status to their score.©amt P ublishing2011amtt limited acn083 950 341Senior DivisionQuestions 1to 10,3marks each1.The expression 3x (x −4)−2(5−3x )equals (A)3x 2−3x −14(B)3x 2−6x −10(C)3x 2−18x +10(D)3x 2−18x −10(E)9x 2−22x2.A coach notices that 2out of 5players in his club are studying at university.If there are 12university students in his club,how many players are there in total?(A)20(B)24(C)30(D)36(E)603.The value of 14÷0.4is (A)3.5(B)35(C)5.6(D)350(E)0.144.In the diagram,ABCD is a square.What is the value of x ?(A)142(B)128(C)48(D)104(E)52................................................................................................................ (52)◦x◦ABCD5.Which of the following is the largest?(A)210(B)210(C)102(D)201(E)2106.If m and n are positive whole numbers and mn =100,then m +n cannot be equalto (A)25(B)29(C)50(D)52(E)1017.P QRS is a square.T is a point on RS such that QT =2RT .The value of x is (A)100(B)110(C)120(D)150(E)160.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................P QRS T x ◦8.In my neighbourhood,90%of the properties are houses and 10%are shops.Today,10%of the houses are for sale and 30%of the shops are for sale.What percentage of the properties for sale are houses?(A)9%(B)80%(C)3313%(D)75%(E)25%9.The value of 12+14+182+4+8is(A)16(B)4(C)1(D)14(E)11610.Anne’s morning exercise consists of walking a distance of 1km at a rate of 5km/h,jogging a distance of 3km at 10km/h and fast walking for a distance of 2km at 6km/h.How long does it take her to complete her morning exercise?(A)30min(B)35min(C)40min(D)45min(E)50minQuestions 11to 20,4marks each11.The diagram shows a square of sidelength 12units divided into six triangles of equal area.What is the distance,in units,of T from the side P Q ?(A)4(B)3(C)2(D)1(E)√5......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................PS QRT12.Each of the first six prime numbers is written on a separate card.The cards areshu ffled and two cards are selected.The probability that the sum of the numbers selected is prime is(A)15(B)14(C)13(D)12(E)1613.Two tourists are walking 12km apart along a flat track at a constant speed of4km/h.When each tourist reaches the slope of a mountain,she begins to climb with a constant speed of 3km/h.✲✛12km✡✡✣✑✑✸✑✑✰k m What is the kilometres,the two tourists during the climb?(A)16(B)12(C)10(D)9(E)814.Lines parallel to the sides of a rectangle 56cm by 98cm and joining its oppositeedges are drawn so that they cut this rectangle into squares.The smallest number of such lines is(A)3(B)9(C)11(D)20(E)7515.What is the sum of the digits of the positive integer n for which n 2+2011is thesquare of an integer?(A)6(B)7(C)8(D)9(E)1016.Of the sta ffin an o ffice,15rode a pushbike to work on Monday,12rode on Tuesdayand 9rode on Wednesday.If 22sta ffrode a pushbike to work at least once during these three days,what is the maximum number of sta ffwho could have ridden a pushbike to work on all three days?(A)4(B)5(C)6(D)7(E)812 kmk m17.How many integer values of n make n 2−6n +8a positive prime number?(A)1(B)2(C)3(D)4(E)an infinite number18.If x 2−9x +5=0,then x 4−18x 3+81x 2+42equals(A)5(B)25(C)42(D)67(E)8119.The centre of a sphere of radius 1is one of the vertices of a cube of side 1.What is the volume of the combined solid?(A)7π6+1(B)7π6+56(C)7π6+43(D)7π8+1(E)π+120.In a best of five sets tennis match (where the first player to win three sets wins thematch),Chris has a probability of 23of winning each set.What is the probabilityof him winning this particular match?(A)23(B)190243(C)89(D)1927(E)6481Questions 21to 25,5marks each21.How many 3-digit numbers can be written as the sum of three (not necessarilydi fferent)2-digit numbers?(A)194(B)198(C)204(D)287(E)29622.A rectangular sheet of paper is folded along a single line so that one corner lieson top of another.In the resulting figure,60%of the area is two sheets thick and 40%is one sheet thick.What is the ratio of the length of the longer side of the rectangle to the length of the shorter side?(A)3:2(B)5:3(C)√2:1(D)2:1(E)√3:223.An irrational spider lives at one corner of a closed box which is a cube of edge 1metre.The spider is not prepared to travel more than √2metres from its home (measured by the shortest route across the surface of the box).Which of the following is closest to the proportion (measured as a percentage)of the surface of the box that the spider never visits?(A)20%(B)25%(C)30%(D)35%(E)50%24.Functions f ,g and h are defined byf (x )=x +2g (0)=f (1)g (x )=f (g (x −1))for x ≥1h (0)=g (1)h (x )=g (h (x −1))for x ≥1.Find h (4).(A)61(B)117(C)123(D)125(E)31325.A cone has base diameter 1unit and slant height 3units.From a point A halfwayup the side of the cone,a string is passed twice around it to come to a point B on the circumference of the base,directly below A .The string is then pulled until taut.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ABHow far is it from A to B along this taut string?(A)38(√29+√53)(B)3√72(C)3√32(D)94(E)3√1088For questions 26to 30,shade the answer as an integer from 0to 999inthe space provided on the answer sheet.Question 26is 6marks,question 27is 7marks,question 28is 8marks,question 29is 9marks and question 30is 10marks.26.Paul is one year older than his wife and they have two children whose ages are alsoone year apart.Paul notices that on his birthday in 2011,the product of his age and his wife’s age plus the sum of his children’s ages is 2011.What would have been the result if he had done this calculation thirteen years before?27.The diagram shows the net of a cube.On each face there is an integer:1,w ,2011,x ,y and z .................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... (x)yz2011w1If each of the numbers w ,x ,y and z equals the average of the numbers written on the four faces of the cube adjacent to it,find the value of x .28.Two beetles sit at the vertices A and H of a cube ABCDEF GH with edge length40√110units.The beetles start moving simultaneously along AC and HF with the speed of the first beetle twice that of the other one.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ADCBEF GH✈s✈s ..................................................................................................................................................................................................................................................................................................................................What will be the shortest distance between the beetles?29.A family of six has six Christmas crackers to pull.Each person will pull twocrackers,each with a di fferent person.In how many di fferent ways can this be done?30.A40×40white square is divided into1×1squares by lines parallel to its sides.Some of these1×1squares are coloured red so that each of the1×1squares, regardless of whether it is coloured red or not,shares a side with at most one red square(not counting itself).What is the largest possible number of red squares?Senior 2011 Answers Question Answer 1B2C3B4A5B6C7C8D9E10E11B12A13D14B15A16D17B18D19A20E21B22C23C24D25B269972780528440297030420。

AMC8（美国数学竞赛）历年真题、答案及中英文解析艾蕾特教育的AMC8 美国数学竞赛考试历年真题、答案及中英文解析：AMC8-2020年：真题 --- 答案---解析（英文解析+中文解析）AMC8 - 2019年：真题----答案----解析（英文解析+中文解析）AMC8 - 2018年：真题----答案----解析（英文解析+中文解析）AMC8 - 2017年：真题----答案----解析（英文解析+中文解析）AMC8 - 2016年：真题----答案----解析（英文解析+中文解析）AMC8 - 2015年：真题----答案----解析（英文解析+中文解析）AMC8 - 2014年：真题----答案----解析（英文解析+中文解析）AMC8 - 2013年：真题----答案----解析（英文解析+中文解析）AMC8 - 2012年：真题----答案----解析（英文解析+中文解析）析）AMC8 - 2010年：真题----答案----解析（英文解析+中文解析）AMC8 - 2009年：真题----答案----解析（英文解析+中文解析）AMC8 - 2008年：真题----答案----解析（英文解析+中文解析）AMC8 - 2007年：真题----答案----解析（英文解析+中文解析）AMC8 - 2006年：真题----答案----解析（英文解析+中文解析）AMC8 - 2005年：真题----答案----解析（英文解析+中文解析）AMC8 - 2004年：真题----答案----解析（英文解析+中文解析）AMC8 - 2003年：真题----答案----解析（英文解析+中文解析）AMC8 - 2002年：真题----答案----解析（英文解析+中文解析）AMC8 - 2001年：真题----答案----解析（英文解析+中文解析）AMC8 - 2000年：真题----答案----解析（英文解析+中文解析）析）AMC8 - 1998年：真题----答案----解析（英文解析+中文解析）AMC8 - 1997年：真题----答案----解析（英文解析+中文解析）AMC8 - 1996年：真题----答案----解析（英文解析+中文解析）AMC8 - 1995年：真题----答案----解析（英文解析+中文解析）AMC8 - 1994年：真题----答案----解析（英文解析+中文解析）AMC8 - 1993年：真题----答案----解析（英文解析+中文解析）AMC8 - 1992年：真题----答案----解析（英文解析+中文解析）AMC8 - 1991年：真题----答案----解析（英文解析+中文解析）AMC8 - 1990年：真题----答案----解析（英文解析+中文解析）AMC8 - 1989年：真题----答案----解析（英文解析+中文解析）AMC8 - 1988年：真题----答案----解析（英文解析+中文解析）析）AMC8 - 1986年：真题----答案----解析（英文解析+中文解析）AMC8 - 1985年：真题----答案----解析（英文解析+中文解析）◆AMC介绍◆AMC（American Mathematics Competitions) 由美国数学协会（MAA）组织的数学竞赛，分为 AMC8 、 AMC10、 AMC12 。

犀牛国际教育独家真题 澳大利亚AMC 真题D 级Intermediate DivisionQuestions 1 to 10 are worth 3 marks each.1-10题，每题3分1.A 40-minute lesson started at 10:50 am. Exactly half-way through the lesson the fire alarm went off.At what time did the fire alarm go off?一节时长为40分钟的课程从上午10:50开始上课。

当课程正好进行到一半时，火警报警器响了，那么火警报警器响起的时间是？（A ）10:30am （B ）11:00am （C ）11:10am （D ）11:20am （E ）11:30am2.Two rectangles have a vertex in common, as shown. What is the size of the angle marked x °between them?如下方图片，两个长方形共用一个顶点。

那么这两个长方形之间标记为x °角的度数是多少？（A ）10°（B ）20°（C ）30°（D ）40°（E ）50°3.What is the value of ++++789234? ++++789234的值是多少？ （A ）61（B ）72（C ）83（D ）94（E ）214.How many 25cm×25cm squares fit in a 50cm×1m rectangle?有多少个边长为25cm 的正方形可以完全放入宽50cm 长1m 的长方形？（A ）1（B ）2（C ）4（D ）6（E ）85.Which one of these is equal to 57×953?57×953的结果是？（A ）321（B ）4321（C ）54321（D ）654321（E ）76543216.A parallelogram PQRS has an area of 60 cm 2 and side PQ of length 10 cm.Which length is 6cm?如下图，平行四边形PQRS的面积为60cm2，边PQ的长度为10cm。