2013E试卷(澳大利亚数学竞赛AMC-E：11-12年级中英文历年真题)

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

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 it 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 2008SENIOR DIVISION COMPETITION PAPERINSTRUCTIO NS AND INFO RMATIO NGENERAL1. 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 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 filled in. It is your responsibility that the Answer Sheet is correctly coded. 7. When your teacher gives the signal, begin working on the problems.THE ANSWER SHEET 1. 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 read by a machine. The machine will see all markings even if they arein 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 AMC reserves the right to re-examine students before deciding whether to grant official status to their score.A U S T R A L I A N S C H O O L Y E A R S 11 A N D 12T I M E A L L O W E D : 75 M I N U T E SSenior DivisionQuestions 1to 10,3marks each1.The value of 8002−2008is (A)200(B)8(C)6006(D)1060(E)59942.The diﬀerence between 120and 210is (A)0(B)110(C)35(D)310(E)3203.In the diagram,x equals................................................................................................................................................................................ (x)◦100◦110◦80◦(A)100(B)110(C)120(D)130(E)1404.The value of 200×8200÷8is(A)1(B)8(C)16(D)64(E)2005.The smallest value that x 2−4x +3can have is (A)−1(B)−3(C)1(D)3(E)26.$3is shared between two people.One gets 50cents more than the other.The ratio of the larger share to the smaller share is (A)6:1(B)7:5(C)4:3(D)5:3(E)7:4S 27.When 10002008is written as a numeral,the number of digits written is (A)2009(B)6024(C)6025(D)8032(E)20128.A semicircle is drawn on one side of an equilateral triangle.The ratio of the area of the semicircle to the area of the triangle is(A)1:1(B)π:2√3(C)π:√3(D)√3:π(E)3:π................................................................................................................................................................................................................................................................................................................................................................................................................................................................9.Given that cos x =0.5and 0◦<x <90◦,which of the following has the greatestvalue?(A)cos 2x(B)cos x(C)0.75(D)sin x(E)tan x10.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)20Questions 11to 20,4marks each11.Which of the following numbers is the largest?(A)2500(B)3400(C)4300(D)5200(E)610012.A normal die is thrown 100times.The sum of the numbers obtained will mostlikely be(A)200(B)250(C)300(D)350(E)400S 313.What is the smallest whole number which gives a square number when multipliedby 2008?(A)2(B)4(C)251(D)502(E)200814.A cross is made up of ﬁve squares,each with side length 1unit.Two cuts aremade,the ﬁrst from X to Y and the second from Z to T ,so that ZT X is a right angle.The three pieces are then arranged to form a rectangle...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ZYX T IIIIII.................................................................... (II)IIIIWhat is the ratio of the length to the width of the rectangle?(A)3:1(B)√10:1(C)2:1(D)2√3:1(E)5:215.A function is said to be a toggle function on (p,q,r )if f (p )=q ,f (q )=r andf (r )=p .The function f (x )=ax 2+bx +c is a toggle function on (1,2,3).What is the value of c ?(A)−2(B)0(C)3(D)9(E)1416.Two conical rollers with perpendicu-lar axes touch on a line that is 30◦to the axis of the smaller roller and 60◦to the axis of the larger roller.If the larger roller makes 1revolution per sec-ond and there is no slipping,how many revolutions per second does the smaller roller make?(A)12(B)1(C)√2(D)√3(E)2...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................60◦30◦S 417.Consider the set X ={1,2,3,4,5,6}.How many subsets of X ,with at least one element,do not contain two consecutive integers?(A)16(B)18(C)20(D)21(E)2418.Farmer Taylor of Burra has two tanks.Water from the roof of his farmhouse iscollected in a 100kL tank and water from the roof of his barn is collected in a 25kL tank.The collecting area of his farmhouse roof is 200square metres while that of his barn is 80square metres.Currently,there are 35kL in the farmhouse tank and 13kL 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)pump 10kL from the farmhouse tank into the barn tank (D)pump 10kL from the barn tank into the farmhouse tank (E)do nothing19.A sequence {u 1,u 2,...,u n }of real numbers is deﬁned byu 1=√2,u 2=π,u n =u n −1−u n −2forn ≥3.What is u 2008?(A)−√2(B)2008(√2−2008π)(C)1003√2−1004π(D)π(E)√220.In the diagram,RU is equal in lengthto ST .What is the ratio of the area of QRU to the area of QST ?(A)√3:1(B)2:1(C)√6:1(D)√3:2(E)√6:2......................................................................................................................................................................................................................... (45)◦30◦UT Q RSQuestions 21to 25,5marks each21.P ,Q ,R ,S and T are consecutive vertices of a regular polygon.When extended,the lines P Q and T S meet at U with QUS =160◦.How many sides has the polygon?(A)36(B)42(C)48(D)52(E)54S522.How many numbers from1,2,3,4,...,2008have a cubic number other than1asa factor?(A)346(B)336(C)347(D)251(E)39323.The numbers828and313are3-digit palindromes where828−313=515,whichis also a palindrome.How many pairs(a,b)of3-digit palindromes are there witha>b and with a−b also a3-digit palindrome?(A)1972(B)1980(C)1988(D)1996(E)200824.The centres of all faces of a cube are joined to form an octahedron.The centresof all faces of this octahedron are now joined to form a smaller cube.What is the ratio of an edge of the smaller cube to an edge of the original cube?(A)1:√2(B)1:√3(C)1:2(D)1:3(E)1:425.In theﬁgure,all line segments are par-allel to one of the sides of the equi-lateral triangle P QR which has sidelength1unit.How long should P Xbe to maximise the smallest of the tenareas deﬁned?(A)13(B)4−√214(C)14(D)15(E)1√10.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................PQ RXFor questions26to30,shade the answer as an integer from0to999inthe space provided on the answer sheet.Question26is6marks,question27is7marks,question28is8marks, question29is9marks and question30is10marks.26.All possible straight lines joining the vertices of a cube with mid-points of its edgesare drawn.At how many points inside the cube do two or more of these lines meet?S 627.Let us call a sum of integers cool if the ﬁrst and last terms are 1and each termdiﬀers from its neighbours by at most 1.For example,the sum 1+2+3+4+3+2+3+3+3+2+3+3+2+1is cool.How many terms does it take to write 2008as a cool sum if we use no more terms than necessary?28.The positive integers x and y satisfy3x 2−8y 2+3x 2y 2=2008.What is the value of xy ?29.A point O is inside an equilateral triangleP QR and the perpendiculars OL ,OM and ON are drawn to the sides P Q ,QR and RP respectively.The ratios of lengths of the perpendiculars OL :OM :ON is 1:2:3.If area of LONP area of P QR =a b,where a and b are integers with no common factors,what is the value of a +b ?...............................................................................................................................................................................................................................................................................................................................................................................................................................................RPQLMNO.......................................................................................................................................................................................................................................................................................30.What is the smallest value that49+a 2−7√2a + a 2+b 2−√2ab +√50+b 2−10bcan have for positive real numbers a and b ?***Senior 2008 Answers Question Answer 1E2E3D4D5A6B7C8B9E10B11B12D13D14C15A16D17C18D19A20D21E22B23B24D25C26142789282829473013。

2013澳大利亚高考数学试题评分标准

3 ⌠ = 2π ⎡ 4e x ⎤ − 2π ⎮ xe x dx ⎣ ⎦1 ⌡ 1 3 3
3
Using integration by parts:
3 ⌠ 3 ⌠ x x xe dx = ⎡ xe ⎤ − ⎮ e x dx ⎮ ⎣ ⎦1 ⌡ ⎮ 1 ⌡1 3
(
)
= [ xe x − e x ]1
3
= ( 3e3 − e3 ) − ( e − e ) = 3e3 − e3 = 2e3 Hence V = 8π ( e3 − e ) − 2π 2e3 = 8π e3 − e − 4π e3 = 4π e3 − 8π e ∴ Volume is 4π e e 2 − 2 units3
z = x + iy ( x , y real ) z 2 + z 2 = ( x + iy ) + ( x − iy )
2 2
Marks 3 2 1
= x 2 + 2ixy − y 2 + x 2 − 2ixy − y 2 = 2 x 2 − 2 y2 = 2 x 2 − y2
(
)
x 2 y2 − ≤ 1. 4 4
= ⎮ cos2 θ (1 − cos2 θ ) sin θ dθ ⌡0 = ⎮ cos2 θ sin θ − cos4 θ sin θ dθ ⌡0 ⎡ cos3 θ = ⎢− + 3 ⎣
π 5 ⎤2 cos θ π ⌠2 π ⌠2 ππ Nhomakorabea5
⎥ ⎦0
⎡ 1 1⎤ = 0 − ⎢− + ⎥ ⎣ 3 5⎦ = 2 15
First use the t -substition, t = tan Then dx = 2

VCE考试数学2013试题1

Question 7 (6 marks) The probability distribution of a discrete random variable, X, is given by the table below. x Pr(X = x) a. 0 0.2 1 0.6p2 2 0.1 3 1-p 4 0.1 3 marks
b.
On the axes above, sketch the graph of g(x) = | f (x)| + 1, for x [-2, 2]. Label the end points with their coordinates.
2 marks
Question 9 – continued
6
Question 6 (3 marks) Let g: R → R, g(x) = (a – x)2, where a is a real constant. 31 The average value of g on the interval [–1, 1] is . 12 Find all possible values of a.
-x
y = f (x) P (x, f (x))
O
Q
x
a.
Find the area, A, of the triangle OQP in terms of x.
1 mark
Question 10 – continued
13
2013 MATHMETH (CAS) EXAM 1
b.
Find the maximum area of triangle OQP and the value of x for which the maximum occurs.

澳大利亚数学竞赛小学中年级(3--4年级)(2013年)

澳大利亚数学竞赛小学中年级（3—4）（2013年）1.请问公元 2013 年之后经过十年是公元多少年？（）（A）2003 （B）2013 （C）2014 （D）2023 （E）21132.请问一个正立方体共有多少条边？（）（A）4 （B）6 （C）8 （D）9 （E）123.小兰学校的操场跑道一圈的长度是 400 m，而小兰一共跑了三圈。

请问她一共跑了多长的距离？（）（A）300 m （B）600 m （C）800 m （D）1200 m （E）3000 m4.请问下图中矩形的几分之几被涂上阴影？（）（A）五分之一（B）五分之二（C）三分之二（D）三分之一（E）五分之三5.请问 9 和 3 之差的三倍等于多少？（）（A）6 （B）9 （C）18 （D）36 （E）816.小珍所戴的帽子上有「COTTON CLUB」的字样。

当小珍向镜子里看去，请问她所看到这顶帽子上的字样是什么？（）7.直线棋盘上的格子从左至右的编号为 1, 2, 3, …。

小莎的棋子向右移动 6 格，向左移动 4 格，然后向右移动 3 格。

假如棋子停留在编号为 7 的方格，请问开始时小莎的棋子所在格子上的编号是什么？（）（A）1 （B）2 （C）3 （D）4 （E）58.小伊位于一个边长为 10 m 正方形的迷宫之中心。

他知道只要沿着如下图所示螺旋状的路径便可走出这个迷宫。

已知这个迷宫共有A、B、C、D、E 五个出口，请问小伊将会从哪一个出口走出这个迷宫？（A）A （B）B （C）C （D）D （E）E9.用下面 3 张数码卡片拼成三位数，请问在所能拼出的数中，最大的数与最小的数相差多少？（）（A）198 （B）200 （C）202 （D）298 （E）30210.小柏在心中想着一个数，他将这个数乘以 2 之后再加 2，所得到的值是 14。

请问他心中原来想的这个数是什么？（）（A）6 （B）7 （C）8 （D）12 （E）3011.小艾有两枚 50 元硬币、三枚 20 元硬币与八枚 5 元硬币，小德有四枚 20 元硬币与六枚 10 元硬币。

2013年AMC8竞赛真题(英文版)

2013 AMC8 Problems1.Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the smallest number of additional cars she must buy in order to be able to arrange all her cars this way?2.A sign at the fish market says, "50% off, today only: half-pound packages for just $3 perpackage." What is the regular price for a full pound of fish, in dollars?What is the value of?3.4.Eight friends ate at a restaurant and agreed to share 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 thegrade and weighs 106 pounds. His quadruplet sisters are tiny babiesand 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 pounds?6.The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, . What is the missing number in the top row?7.Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass, Trey counted 6 cars in the first 10 seconds. It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed. Which of the following was the most likely number of cars in the train?8.A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?9.The Incredible Hulk can double the distance he jumps with each succeeding 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 than 1 kilometer?10.What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?11.Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday 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?12.At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. 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 advantage of the "fair special" to buy three pairs of sandals. What percentage of the $150 regular price did he save?13.When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one?14.Abe holds 1 green and 1 red jelly bean in his hand. Bea holds 1 green, 1 yellow, and 2 red jelly beans in her hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?15.If , , and , what is the product of , , and ?16.A number of students from Fibonacci Middle School are taking part in a community serviceproject. The ratio of -graders to -graders is , and the the ratio of -graders to-graders is . What is the smallest number of students that could be participating in the project?17.The sum of six consecutive positive integers is 2013. What is the largest of these six integers?18.Isabella uses one-foot cubical blocks to build a rectangular fort 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?19.Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show theirs to anyone. Cassie says, 'I didn't get the lowest score in our class,' and Bridget adds, 'I didn't get the highest score.' What is the ranking of the three girls from highest to lowest?20.A rectangle is inscribed in a semicircle with longer side on the diameter. What is thearea 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 north of the northeast 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 northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?22.Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?23.Angle of is a right angle. The sides of are the diameters of semicirclesas shown. The area of the semicircle on equals , and the arc of the semicircle onhas length . What is the radius of the semicircle on ?24.Squares , , and are equal in area. Points and are the midpointsof sides and , 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 iscomprised of 3 semicircular arcs whose radii are inches, inches, andinches, respectively. The ball always remains in contact with the track and does notslip. What is the distance the center of the ball travels over the course from A to B?。

澳大利亚数学竞赛小学高年级(5—6)(2013年)

澳大利亚数学竞赛小学高年级（5—6）（2013年）1.请问下图中矩形的几分之几被涂上阴影？（）（A）五分之一（B）五分之二（C）三分之二（D）三分之一（E）五分之三2. 请问下列哪一个数最接近 0？（）（A）0.03 （B）0.048 （C）0.009 （D）0.005 （E）0.023.一架波音 737 飞机的中间走道两侧每排各有 3 个座位。

若这架飞机150 位乘客，请问它的座位共有多少排？（）（A）50 （B）37 （C）33 （D）32 （E）254.小艾有两枚 50 元硬币、三枚 20 元硬币与八枚 5 元硬币，小德有四枚 2硬币与六枚 10 元硬币。

请问小艾的钱比小德的钱多了多少元？（）（A）40 元（B）60 元（C）80 元（D）140 元（E）2005.用下面 5 张数码卡片拼成五位数，请问在所能拼出的数中，最大的数与的数相差多少？（）（A）41967 （B）41976 （C）44444 （D）42024 （E）410766.在超市中，正常包装的薯条每包重量为 75 g。

有一种促销包，每包薯条量增加三分之一。

请问此促销包每包的重量是多少g？（）（A）50 （B）78 （C）100 （D）125 （E）1507. 请问下图中共有多少个不同位置的三角形？（）（A）9 （B）10 （C）13 （D）14 （E）178.小珍将一个数乘以2 后再加 2，接着再将所得的数除以 2 后再减 2，最所得到的结果是 6。

请问小珍原来的这个数是什么？（）（A ）1 （B ）6 （C ）7 （D ）14 （E ）169. 在下面的数线上，请问31应该位于哪里？（）（A ）在 0 与 0.3 之间；（B ）在 0.3 与 0.4 之间；（B ）在 0.4 与 0.7 之间（D ）在 0.7 与 0.8 之间；（D ）在 0.8 与 1 之间。

10. 下图中每个三角形都是正三角形。

请问阴影部分的面积占最大三角形面几分之几？（）（A ）41 （B ）6415 （C ）31 （D ）163 （E ） 327 11. 在下面的算式中，三个数码已经被□取代。

AMC Intermediate 2013AMC-D：9-10年级中文历年真题

2013INTERMEDIATE DIVISIONAUSTRALIAN S CHOOL YEARS 9 and 10TIME ALLOWED: 75 MINUTESINSTRUCTIONS 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 are permitted.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 that require awhole 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 competing against yourown year in your own country or Australian state 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 school year areentered. It is your responsibility to correctly code your answer sheet. 7. When your teacher gives the signal, begin working on the problems.THE ANSWER SHEET 1. Use only lead pencil.2. Record your answers on the reverse of the answer sheet (not on the question paper) by FULLY colouring thecircle matching your answer.3. Your answer sheet will be scanned. The optical scanner will attempt to read all markings even if they are inthe 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 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 TIntermediate DivisionQuestions1to10,3marks each1.2013+2014+2015equals(A)642(B)2016(C)6022(D)6032(E)60422.In the diagram below,what is the value,in degrees,of angle x?37◦85◦x◦(A)48(B)85(C)122(D)132(E)1433.If every digit of a whole number is either a3or a5,the number will always be(A)divisible by3(B)divisible by5(C)prime(D)even(E)odd4.The average of two numbers is twice the smaller number.The larger number is12.What is the smaller number?(A)2(B)3(C)4(D)6(E)85.The length of the base of a triangle is3times its perpendicular height and the areaof the triangle is24cm2.The sum of its base length and its perpendicular height, in centimetres,is(A)12(B)13(C)14(D)15(E)166.A regular icosahedron is a solid shape with twenty faces,where each face is directlyopposite another face.I label the faces from1to20so that,for all pairs of opposite faces,the two labels in any pair always add up to the same number.What number is on the face opposite the one labelled8?(A)11(B)12(C)13(D)14(E)157.If p=4b+26and b is a positive integer,then p could not be divisible by(A)2(B)4(C)5(D)6(E)78.My two dogs were running on the beach when I called them back.The faster dogwas100m away and the slower dog was70m away.The faster dog runs twice as fast as the slower dog.How far away was the second dog when theﬁrst dog reached me?(A)15m(B)20m(C)30m(D)40m(E)50m9.The value of x2+1x2when x=23is closest to(A)0(B)1(C)2(D)3(E)410.A piece of paper in the shape of an equilateral triangle has one corner folded over,as shown.What is the value of x?(A)60(B)70(C)80(D)90(E)100Questions11to20,4marks each11.Start with the number1and create the sequence1,2,4,8,16,22,24,28,...where each number is the sum of the previous number and itsﬁnal digit.How many numbers in the sequence are less than1000?(A)10(B)100(C)101(D)200(E)20112.A six-sided dice has the numbers1,2,2,3,3and3on its faces.Two such dice arerolled and a score is made by adding the numbers on the uppermost faces.The probability of rolling an odd score is(A)19(B)29(C)13(D)49(E)5913.If x2=x+3,then x3equals(A)x+6(B)2x+6(C)3x+9(D)4x+3(E)27x+914.The point T divides the side QR of the rectangle P QRS into two equal segments.The point U divides P Q such that P U:UQ=1:2.Point V divides SP such that SV:V P=1:3andﬁnally,point W divides RS such that RW:W S=1:4.Find the area of the quadrilateral T UV W if the area of P QRS equals120.V(A)67(B)70(C)72(D)75(E)7715.Three line segments of lengths1,a and2a are the sides of a triangle.Which ofthe following deﬁnes all possible values of a?(A)13<a<1(B)0<a<13(C)a<1(D)for all a>0(E)for no a16.The shaded segment in the circle below,centre O,has anarea of1cm2.The radiusof the circle,in centimetres,is(A) 4π(B)8π(C) 4π−2(D)4π(E)2√π17.Dan and Jane each have a measuring tape of length1m.Dan’s tape got stuck ina door and was extended by4cm.Jane left her tape in a pocket and it shrank by5cm after washing.However,the centimetre marks on both tapes remained evenly distributed.Measuring the schoolyard,Dan noted the length as23.75m.What length will Jane get measuring the same schoolyard with her tape?(A)23m(B)24m(C)25m(D)26m(E)27m18.In the regular hexagon pictured,the midpoints of the sides are joined to form theshaded regular hexagon.What fraction of the larger hexagon is shaded?(A)34(B)23(C)56(D)12(E)7819.A circular wheel of radius r rolls,without slipping,through half a revolution.Thepoint X is on the horizontal diameter at the start.XThe distance between the starting andﬁnishing position of the point X is(A)2πr(B)(π+2)r(C)(π−2)r(D)2(π+1)r(E)2(π−1)r20.The sport of bingbong involves two players.Each match consists of a number ofrounds and each round consists of a number of points.Theﬁrst player to win four points in a round wins the round.Theﬁrst player to win six rounds in a match wins the match.Suppose that after a match of bingbong,the winner has won W points while the loser has won L points.What is the largest possible value of L−W?(A)−6(B)−4(C)0(D)12(E)14Questions21to25,5marks each21.In how many ways can the numbers1,2,3,4,5,6be arranged in a row so thatthe product of any two adjacent numbers is even?(A)64(B)72(C)120(D)144(E)72022.Two circles,one of radius1and the other of radius2,touch externally at P.Astraight line through P cuts the area formed by these two circles in the ratio1:2.In what ratio does this line cut the area of the smaller circle?(A)1:2(B)2:5(C)1:3(D)2:7(E)1:423.How many positive integers n are there such that2n+1is a divisor of8n+46?(A)0(B)1(C)2(D)3(E)424.The rectangle P QRS shown has P Q=4,P S=12and centre C.The two shadedcircles have radius1and touch P S at U and V where P U=1and P V=4.The line CW divides the unshaded area in half.The length of P W isW(A)27(B)25(C)14(D)13(E)1225.In3013,King Warren of Australia isﬁnally deposed.Theﬁve remaining earls argueabout which one of them will be king,and which one of the others will be treasurer.Akaroa will be satisﬁed only if Darlinghurst or Erina is treasurer.Bairnsdale will be satisﬁed only if Claremont is treasurer.Claremont will be satisﬁed only if Darlinghurst is either king or treasurer.Darlinghurst will be satisﬁed only if Akaroa is either king or treasurer.Erina will be satisﬁed only if Akaroa is not king.It is not possible for allﬁve to be satisﬁed,so in the end they appoint king and treasurer so that the other three earls are satisﬁed.Who becomes king?(A)Akaroa(B)Bairnsdale(C)Claremont(D)Darlinghurst(E)ErinaFor questions26to30,shade the answer as an integer from0to999inthe space provided on the answer sheet.Question26is6marks,question27is7marks,question28is8marks, question29is9marks and question30is10marks.26.The4-digit number pqrs has the property that pqrs×4=srqp.If p=2,what isthe value of the3-digit number qrs?27.Three diﬀerent non-zero digits are used to form six diﬀerent3-digit numbers.Thesum ofﬁve of them is3231.What is the sixth number?28.A hockey game between two teams is‘relatively close’if the number of goals scoredby the two teams never diﬀer by more than two.In how many ways can theﬁrst 12goals of a game be scored if the game is‘relatively close’?29.How many pairs(a,b)of positive integers are there such that a and b are factorsof66and a is a factor of b?30.All the digits of the positive integer N are either0or1.The remainder afterdividing N by37is18.What is the smallest number of times that the digit1can appear in N?Intermediate 2013 Answers Question Answer 1E2C3E4C5E6C7B8B9D10C11E12D13D14A15A16C17D18A19B20E21D22D23D24A25B26178277652897229784305。

AMC Senior 2011(澳大利亚数学竞赛AMC-E：11-12年级中英文历年真题)

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。

2013A试卷AMC-A：澳大利亚数学3-4年级中文历年真题

THURSDAY 1 AUGUST 2013MIDDLE PRIMARY DIVISIONAUSTRALIAN S CHOOL YEARS 3 & 4TIME ALLOWED: 60 MINUTESINSTRUCTIONS AND INFORMATION GENERAL 1. Do not open the booklet until told to do so by your teacher. 2. You may use any teaching aids normally available in your classroom, such as MAB blocks, counters, currency, calculators, play money etc. You are allowed to work on scrap paper and teachers may explain the meaning of words in the paper. 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 that require 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 competing against 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 school year 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 SHEET 1. Use only lead pencil.2. Record your answers on the reverse of the answer sheet (not on the question paper) by FULLY colouring the circle matching your answer.3. Your answer sheet will be scanned. The optical scanner will attempt to read all markings even if 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 ublishing 2013 AMTT liMiTed Acn 083 950 341A ustrAliAn M AtheMAtics c oMpetitionsponsored by the c oMMonweAlth b AnkAn AcTiviTy of The AusTrAliAn MATheMATics TrusT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。

AMC8(美国数学竞赛)历年真题、答案及中英文解析

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 。

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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。

特别约定为确保竞赛之公平性及认证成绩优异学生，AMC 主办单位保留要求考生重测之权利。

注意事项一般规定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5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。

特别约定为确保竞赛之公平性及认证成绩优异学生，AMC 主办单位保留要求考生重测之权利。

高级卷1-10题，每题3分1.算式0.612等于（A ）5（B ）0.5（C ）0.05 （D ）0.005（E ）0.00052.请问以下哪一项是直角三角形？（A ）（B ）（C ）（D ）（E ）3.一块农地本季的产量增加20%而达到114公吨。

请问上一季的产量是多少公吨？（A ）90（B ）91 （C ）93 （D ）94（E ）954.请问201320132013的立方根接近于下列哪一个数？（A ）600（B ）5000（C ）6000（D ）50000（E ）600005.若p =4b +26且b 为正整数，则p 不可能被下列哪一个数整除？（A ）2 （B ）4 （C ）5 （D ）6 （E ）76.一个数列的前五项依序为：1、2、1、1−、2−、…已知从第二项之后，每一项都是由前面一项减去再前面一项而得到的。

请问这个数列的前42项之总和是多少？（A ）0 （B ）4 （C ）12 （D ）24（E ）30345.24681012.13372所示。

请问下列哪一项可能是这条直线的方程？（A ）512120x y +−=（B ）410x y ++=（C ）3770x y −+=（D ）3220x y −+=（E ）2330x y +−=8.小孔与小克进行100 m 的赛跑，小孔赢了5 m(即当小孔抵达终点时，小克距离终点还有5 m)；小克与小杰进行100 m 的赛跑，小克赢了10 m 。

请问若是小孔与小杰进行100 m 的赛跑，小孔应会赢多少m ？（A ）15.5（B ）15 （C ）14.5（D ）14（E ）13.59.在牧场中共有两种动物：骆驼与羊驼，而牧场也有许多位管理员。

已知骆驼与羊驼的数量比为2：3及羊驼与管理员的数量比为8：1，请问所有的动物与管理员的数量比是什么？（A ）16：3（B ）13：1（C ）12：1（D ）40：3（E ）20：310.从计算器上显示的数为0开始，我进行五个步骤的计算，每一个步骤都是加1或是乘以2。

请问不可能是计算结果之最小的数是什么？（A ）11 （B ）10 （C ）9 （D ）8 （E ）711-20题，每题4分11.在一个箱子中共有三个袋子。

一个袋子内有一颗白石子及三颗黑石子，另一个袋子内也是有一颗白石子及三颗黑石子，而第三个袋子内则有一颗白石子及四颗黑石子。

现从箱子中任选一个袋子后，再从袋子里任取一颗石子，请问取出白石子的机率是多少？（A ）730（B ）313（C ）19（D ）14（E ）113612.在坐标平面上，有一个由x 轴、y 轴、直线x =2与方程()y f x =所围成的区域，其中101()112x f x x ≤≤⎧⎪=⎨+<≤⎪⎩當當请问此区域的面积为多少平方单位？（A ）52（B ）14π+（C ）12π+（D ）2π+（E ）24π+13.将三个边长分别为3 cm 、5 cm 及8 cm 的正方形如下图所示的方式排列：请问阴影部分的梯形面积为多少cm 2？（A ）12（B ）736（C ）554（D ）14 （E ）25214.一个周长为20 cm 的正方形内接于一个周长为28 cm 的正方形。

请问一个在内部正方形的顶点与一个在外部正方形的顶点之最大距离是多少cm ？（A（B ）752（C ）8 （D（E）15.用三个非零且互不同的数码可以组成六个不同的二位数。

已知这六个数中的五个数之和为100，请问第六个数是什么？（A ）23（B ）32 （C ）45 （D ）54（E ）6716.掷三支飞镖进入一个3×3的方格靶内，每支飞镖都射入不同的小方格内。

经过每次投掷后，飞镖射入剩余的小方格之机率都相等。

请问最后三支飞镖所在的方格形成水平线、铅垂线或对角线的机率是什么？（A ）163（B ）221（C ）19（D ）142（E ）88117.一个箱子的每个面都是矩形且边长都是整数，若主对角线XY = 9，请问这个箱子的体积最大可能值是什么？（A ）32（B ）81 （C ）90 （D ）108（E ）11218.已知x 、y 都是整数且0x ≥，请问满足方程2222169x xy y −+=的不同数对(x ,y )共有多少组？（A ）2（B ）4 （C ）5 （D ）7 （E ）819.一个跑道所围成的区域是由一个矩形连接二个半圆所形成的，如下图所示。

若跑道全长必须恰好为400 m ，请问此矩形的最大面积为多少m 2？（A ）2160000(2)π+（B ）20000π（C ）30000π（D ）10000（E ）40000πXY20.请问将2013写成二个或二个以上的连续正整数之和的方式共有多少种？（A ）5 （B ）6 （C ）7 （D ）8 （E ）921-25题，每题5分21.若23x x =+，则5x 等于（A ）712x + （B ）127x + （C ）1717x +（D ）1921x + （E ）2119x +22.下图矩形PQRS 的中心为点C ，已知PQ = 4、PS = 12。

二个涂上阴影且半径都是1的圆分别与PS 相切于点U 与点V ，其中PU = 1、PV = 4。

若直线CW 将非阴影区域切为面积相等的二块区域，请问PW 的长度是什么？（A ）27（B ）25（C ）14（D ）13（E ）1223.若三个非零的实数x 、y+则111x y z ++之值一定等于（A ）0 （B ）1 （C ）1−（D ）xyz （E ）x +y +z24.黑板上写了一个二位数，对于这个数有五位学生分别做了以下叙述：A ：此数是质数。

B ：此数可以写成二个完全平方数之和。

C ：此数至少有一个数码是7。

D ：将此数的数码顺序颠倒后所得的数是奇数且是个合数。

E ：此数数与某个质数之差为2。

若恰有一个学生是错的，请问是哪一位？（A ）A（B ）B （C ）C （D ）D（E ）EPW25.半径分别为1与2的二个球互相外切于点P 。

一个经过点P 的平面将这二个球所围成的区域分割为体积比为1：2(图中无阴影部份：阴影部份)的二块。

请问此平面将小球分割成二块的体积比(图中阴影部份：无阴影部份)是什么？（A ）1：2（B ）4：9（C ）1：3（D ）4：11 （E ）2：5问题26～30的答案为000～999之间的整数，请将答案填在答案卡上对应的位置。

第26题占6分，第27题占7分，第28题占8分，第29题占9分，第30题占10分。

26.在一场曲棍球比赛中，如果在比赛中两支交手球队的进球数之差从未超过2时，则称这两支球队「实力相当」。

若两支球队共进12球且两支球队一直处于「实力相当」的情况，请问整个球赛共有多少种不同可能的赛况？27.已知点X 在正方形PQRS 内部使得点X 与点R 的距离为25 m 、与点S 的距离为51 m 、与点P 的距离为53 m 。

若点X 与各条边之距离都是整数m ，请问△PQX 的面积为多少m2？P255351X SR Q28.正整数N 的数码都是0或1，将N 除以37时所得的余数为18。

请问N 的所有数码中，最少有多少个1？29.如图所示，点X 、Y 及Z 位于△PQR 的边上使得QZ ：ZY ：YR = 1：2：3且PX ：XR = 4：5若QS = 11 cm ，请问ST 的长度是多少cm ？30.在坐标平面上，一个锐角三角形的三个顶点的坐标都是不同的整数且没有任何一条边与坐标轴平行。