AMC／AIME美国数学竞赛 试题真题

2000AMC12ProblemsProblem1In the year,the United States will host the International Mathematical Olympiad.Let and be distinct positive integers such that the product.What is the largest possible value of the sum?Problem2Problem3Each day,Jenny ate of the jellybeans that were in her jar at the beginning of that day. At the end of the second day,remained.How many jellybeans were in the jar originally?Problem4The Fibonacci sequence starts with two1s,and each term afterwards is the sum of its two predecessors.Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?Problem5If where thenProblem6Two different prime numbers between and are chosen.When their sum is subtracted from their product,which of the following numbers could be obtained?Problem7How many positive integers have the property that is a positive integer?Problem8Figures,,,and consist of,,,and non-overlapping squares.If the pattern continued,how many non-overlapping squares would there be in figure?Problem9Mrs.Walter gave an exam in a mathematics class of five students.She entered the scores in random order into a spreadsheet,which recalculated the class average after each score was entered.Mrs. Walter noticed that after each score was entered,the average was always an integer.The scores (listed in ascending order)were71,76,80,82,and91.What was the last score Mrs.Walters entered?Problem10The point is reflected in the-plane,then its image is rotatedby about the-axis to produce,and finally,is translated by5units in the positive-direction to produce.What are the coordinates of?Problem11Two non-zero real numbers,and satisfy.Which of the following is a possible value of?Problem12Let A,M,and C be nonnegative integers such that.What is the maximum value of+++?Problem13One morning each member of Angela’s family drank an8-ounce mixture of coffee with milk.The amounts of coffee and milk varied from cup to cup,but were never zero.Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee.How many people are in the family?Problem14When the mean,median,and modeof the listare arranged in increasing order,they form a non-constant arithmetic progression.What is the sum of all possible real values of?Problem15Let be a function for which.Find the sum of all values of for which.Problem16A checkerboard of rows and columns has a number written in each square,beginning in the upper left corner,so that the first row is numbered,the second row, and so on down the board.If the board is renumbered so that the left column,top to bottom, is,the second column and so on across the board,some squares have the same numbers in both numbering systems.Find the sum of the numbers in these squares (under either system).Problem17A centered at has radius and contains the point.The segment is tangent to the circle at and.If point lies on and bisects,thenProblem18In year,the day of the year is a Tuesday.In year,the day is also a Tuesday.On what day of the week did th day of year occur?Problem19triangle,,,.Let denote the midpointof and let denote the intersection of with the bisector of angle.Which of the following is closest to the area of the triangle?Problem20If and are positive numbers satisfyingThen what is the value of latex?Problem21Through a point on the hypotenuse of right triangle,lines are drawn parallel to the legs of the triangle so that the triangle is divided into asquare and two smaller right triangles.The area of one of the two small right triangles times the area of the square.The ratio of the area of the other small right triangle to the area of the square isProblem22The graph below shows a portion of the curve defined by the quarticpolynomial.Which of the following is the smallest?Problem23Professor Gamble buys a lottery ticket,which requires that he pick six different integers from through,inclusive.He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer.It so happens that the integers on the winning ticket have the same property—the sum of the base-ten logarithms is an integer.What is the probability that Professor Gamble holds the winning ticket?Problem24If circular arcs and centers at and,respectively,then there exists a circletangent to both and,and to.If the length of is,then the circumference of the circle isProblem25Eight congruent Equilateral triangle each of a different color,are used to construct a regular octahedron.How many distinguishable ways are there to construct the octahedron?(Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)Problem6For how many positive integers does there exist at least one positive integer n such that?infinitely manyProblem7A arc of circle A is equal in length to a arc of circle B.What is the ratio of circle A's area and circle B's area?Problem8Betsy designed a flag using blue triangles,small white squares,and a red center square,as shown. Let be the total area of the blue triangles,the total area of the white squares,and the area of the red square.Which of the following is correct?Problem9Jamal wants to save30files onto disks,each with1.44MB space.3of the files take up0.8MB, 12of the files take up0.7MB,and the rest take up0.4MB.It is not possible to split a file onto2different disks.What is the smallest number of disks needed to store all30files?Problem10Sarah places four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size.She then pours half the coffee from the first cup to the second and,after stirring thoroughly,pours half the liquid in the second cup back to the first.What fraction of the liquid in the first cup is now cream?Problem11Mr.Earl E.Bird gets up every day at8:00AM to go to work.If he drives at an average speed of40miles per hour,he will be late by3minutes.If he drives at an average speed of60milesper hour,he will be early by3minutes.How many miles per hour does Mr.Bird need to drive to get to work exactly on time?Problem12Both roots of the quadratic equation are prime numbers.The number of possible values of isProblem13Two different positive numbers and each differ from their reciprocals by.What is?Problem14For all positive integers,let.Let. Which of the following relations is true?Problem15The mean,median,unique mode,and range of a collection of eight integers are all equal to8. The largest integer that can be an element of this collection isProblem16Tina randomly selects two distinct numbers from the set{1,2,3,4,5},and Sergio randomly selects a number from the set{1,2,...,10}.What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?Problem17Several sets of prime numbers,such as use each of the nine nonzero digits exactly once.What is the smallest possible sum such a set of primes could have?Problem18Let and be circles definedby and respectively.What is the length of the shortest line segment that is tangent to at and to at?Problem19The graph of the function is shown below.How many solutions does theequation have?Problem20Suppose that and are digits,not both nine and not both zero,and the repeatingdecimal is expressed as a fraction in lowest terms.How many different denominators are possible?Problem21Consider the sequence of numbers:For,the-th term of the sequence is the units digit of the sum of the two previous terms.Let denote the sum of the first terms of this sequence.The smallest value of for which is:Problem22Triangle is a right triangle with as its right angle,, and.Let be randomly chosen inside,and extend to meet at. What is the probability that?Problem23In triangle,side and the perpendicular bisector of meet in point,and bisects.If and,what is the area of triangle?SAT II数学词汇表代数部分1.基础add，plus加subtract减difference差multiply times乘product积divide除divisible可被整除的divided evenly被整除dividend被除数divisor因子，除数quotient商remainder余数factorial阶乘power乘方radical sign,root sign根号round to四舍五入to the nearest四舍五入2.有关集合union并集proper subset真子集solution set解集3.有关代数式、方程和不等式algebraic term代数项like terms,similar terms同类项numerical coefficient数字系数literal coefficient字母系数inequality不等式triangle inequality三角不等式range值域original equation原方程equivalent equation同解方程等价方程linear equation线性方程(e.g.5x+6=22)4.有关分数和小数proper fraction真分数improper fraction假分数mixed number带分数vulgar fraction，common fraction普通分数simple fraction简分数complex fraction繁分数numerator分子denominator分母(least)common denominator(最小)公分母quarter四分之一decimal fraction纯小数infinite decimal无穷小数recurring decimal循环小数tenths unit十分位5.基本数学概念arithmetic mean算术平均值weighted average加权平均值geometric mean几何平均数exponent指数，幂base乘幂的底数,底边cube立方数，立方体square root平方根cube root立方根common logarithm常用对数digit数字constant常数variable变量inverse function反函数complementary function余函数linear一次的，线性的factorization因式分解absolute value绝对值，e.g.｜-32｜=32round off四舍五入6.有关数论natural number自然数positive number正数negative number负数odd integer奇整数,odd number奇数even integer,even number偶数integer,whole number整数positive whole number正整数negative whole number负整数consecutive number连续整数real number,rational number实数,有理数irrational(number)无理数inverse倒数composite number合数e.g.4,6,8,9,10,12,14,15……reciprocal 倒数common divisor公约数multiple倍数(least)common multiple(最小)公倍数(prime)factor(质)因子common factor公因子prime number质数e.g.2,3,5,7,11,13,15……ordinary scale,decimal scale十进制nonnegative非负的tens十位units个位mode众数median中数common ratio公比7.数列arithmetic progression(sequence)等差数列geometric progression(sequence)等比数列8.其它approximate近似(anti)clockwise(逆)顺时针方向cardinal基数ordinal序数direct proportion正比distinct不同的estimation估计，近似parentheses括号proportion比例permutation排列combination组合table表格trigonometric function三角函数unit单位,位几何部分1.所有的角alternate angle内错角corresponding angle同位角vertical angle对顶角central angle圆心角interior angle内角exterior angle外角supplementary angles补角complementary angle余角adjacent angle邻角acute angle锐角obtuse angle钝角right angle直角round angle周角straight angle平角included angle夹角2.所有的三角形equilateral triangle等边三角形scalene triangle不等边三角形isosceles triangle等腰三角形right triangle直角三角形oblique斜三角形inscribed triangle内接三角形3.有关收敛的平面图形，除三角形外semicircle半圆concentric circles同心圆quadrilateral四边形pentagon五边形hexagon六边形heptagon七边形octagon八边形nonagon九边形decagon十边形polygon多边形parallelogram平行四边形equilateral等边形plane平面square正方形，平方rectangle长方形regular polygon正多边形rhombus菱形trapezoid梯形4.其它平面图形arc弧line,straight line直线line segment线段parallel lines平行线segment of a circle弧形5.有关立体图形cube立方体，立方数rectangular solid长方体regular solid/regular polyhedron正多面体circular cylinder圆柱体cone圆锥sphere球体solid立体的6.有关图形上的附属物altitude高depth深度side边长circumference,perimeter周长radian弧度surface area表面积volume体积arm直角三角形的股cross section横截面center of acircle圆心chord弦radius半径angle bisector角平分线diagonal对角线diameter直径edge棱face of a solid立体的面hypotenuse斜边included side夹边leg三角形的直角边median of a triangle三角形的中线base底边，底数(e.g.2的5次方，2就是底数) opposite直角三角形中的对边midpoint中点endpoint端点vertex(复数形式vertices)顶点tangent切线的transversal截线intercept截距7.有关坐标coordinate system坐标系rectangular coordinate直角坐标系origin原点abscissa横坐标ordinate纵坐标Number line数轴quadrant象限slope斜率complex plane复平面8.其它plane geometry平面几何trigonometry三角学bisect平分circumscribe外切inscribe内切intersect相交perpendicular垂直Pythagorean theorem勾股定理congruent全等的multilateral多边的其它相关词汇cent美分penny一美分硬币nickel5美分硬币dime一角硬币dozen打(12个)score廿(20个)Centigrade摄氏Fahrenheit华氏quart夸脱gallon加仑(1gallon=4quart)yard码meter米micron微米inch英寸foot英尺minute分(角度的度量单位，60分=1度) square measure平方单位制cubic meter立方米pint品脱(干量或液量的单位)。

2018 AIME I ProblemsProblem 1Let be the number of ordered pairs ofintegers with and such that thepolynomial can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when is divided by .Problem 2The number can be written in base as , can be written inbase as , and can be written in base as , where . Find the base- representation of .Problem 3Kathy has red cards and green cards. She shuffles the cards and laysout of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy,but RRRGR will not. The probability that Kathy will be happy is ,where and are relatively prime positive integers. Find . Problem 4In and . Point lies strictlybetween and on and point lies strictly between and on sothat . Then can be expressed in the form ,where and are relatively prime positive integers. Find .Problem 5For each ordered pair of real numbers satisfyingthere is a real number such thatFind the product of all possible values of .Problem 6Let be the number of complex numbers with the propertiesthat and is a real number. Find the remainder when is divided by .Problem 7A right hexagonal prism has height . The bases are regular hexagons with side length . Any of the vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles).Problem 8Let be an equiangular hexagon suchthat , and . Denote the diameter of the largest circle that fits inside the hexagon. Find .Problem 9Find the number of four-element subsets of with the propertythat two distinct elements of a subset have a sum of , and two distinct elements of a subset have a sum of . Forexample, and are two such subsets.Problem 10The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point . At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along thepath , which has steps. Let be the number of paths with steps that begin and end at point . Find the remainder when is divided by .Problem 11Find the least positive integer such that when is written in base , its two right-most digits in base are .Problem 12For every subset of , let be the sum of the elements of , with defined to be . If is chosen at random among allsubsets of , the probability that is divisible by is , where and are relatively prime positive integers. Find .Problem 13Let have side lengths , , and .Point lies in the interior of , and points and are the incentersof and , respectively. Find the minimum possible areaof as varies along .Problem 14Let be a heptagon. A frog starts jumping at vertex . From any vertex of the heptagon except , the frog may jump to either of the two adjacentvertices. When it reaches vertex , the frog stops and stays there. Find the number of distinct sequences of jumps of no more than jumps that end at .Problem 15David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, , which can each be inscribed in a circle with radius . Let denote the measure of the acute angle made by the diagonals of quadrilateral , and define and similarly. Supposethat , , and . All three quadrilaterals have thesame area , which can be written in the form , where and are relatively prime positive integers. Find .2018 AMC 8 ProblemsProblem 1An amusement park has a collection of scale models, with ratio , of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its replica to the nearest whole number?Problem 2What is the value of the productProblem 3Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then Bob, and so forth. When the number contains a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and the counting continues. Who is the last one present in the circle?Problem 4The twelve-sided figure shown has been drawn on graph paper. What is the area of the figure in ?Problem 5What is the valueof ?Problem 6On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the highway as on the coastal road. If Anh spent 30 minutes driving on the coastal road, how many minutes did his entire trip take?Problem 7The -digit number is divisible by . What is the remainder when this number is divided by ?Problem 8Mr. Garcia asked the members of his health class how many days last week they exercised for at least 30 minutes. The results are summarized in the following bar graph, where the heights of the bars represent the number of students.What was the mean number of days of exercise last week, rounded to the nearest hundredth, reported by the students in Mr. Garcia's class?Problem 9Tyler is tiling the floor of his 12 foot by 16 foot living room. He plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will he use?Problem 10The of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?Problem 11Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture, as shown.If the seating positions are assigned randomly, what is the probability that Abby and Bridget are adjacent to each other in the same row or the same column?Problem 12The clock in Sri's car, which is not accurate, gains time at a constant rate. One day as he begins shopping he notes that his car clock and his watch (which is accurate) both say 12:00 noon. When he is done shopping, his watch says 12:30 and his car clock says 12:35. Later that day, Sri loses his watch. He looks at his car clock and it says 7:00. What is the actual time?Problem 13Laila took five math tests, each worth a maximum of 100 points. Laila's score on each test was an integer between 0 and 100, inclusive. Laila received the same score on the first four tests, and she received a higher score on the last test. Her average score on the five tests was 82. How many values are possible for Laila's score on the last test?Problem 14Let be the greatest five-digit number whose digits have a product of . What is the sum of the digits of ?Problem 15In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of square unit, then what is the area of the shaded region, in square units?Problem 16Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. How many ways are there to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together?Problem 17Bella begins to walk from her house toward her friend Ella's house. At the same time, Ella begins to ride her bicycle toward Bella's house. They each maintain a constant speed, and Ella rides 5 times as fast as Bella walks. The distancebetween their houses is miles, which is feet, and Bella covers feet with each step. How many steps will Bella take by the time she meets Ella?Problem 18How many positive factors does have?Problem 19In a sign pyramid a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid?Problem 20In a point is on with and Point ison so that and point is on so that What is the ratio of the area of to the area ofProblem 21How many positive three-digit integers have a remainder of 2 when divided by 6, a remainder of 5 when divided by 9, and a remainder of 7 when divided by 11?Problem 22Point is the midpoint of side in square and meets diagonal at The area of quadrilateral is What is the areaofProblem 23From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?Problem 24In the cube with opposite vertices and and are the midpoints of edges and respectively. Let be the ratio of the area of the cross-section to the area of one of the faces of the cube. What isProblem 25How many perfect cubes lie between and , inclusive?2018 AMC 10A ProblemsProblem 1What is the value ofProblem 2Liliane has more soda than Jacqueline, and Alice has more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have?Liliane has more soda than Alice.Liliane has more soda than Alice.Liliane has more soda than Alice.Liliane has more soda than Alice.Liliane has more soda than Alice.Problem 3A unit of blood expires after seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire?Problem 4How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)Problem 5Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of ?Problem 6Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of 0, and the score increases by 1 for each like vote and decreases by 1 for each dislike vote. At one point Sangho saw that his video had a score of 90, and that of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?Problem 7For how many (not necessarily positive) integer values of is the value of an integer?Problem 8Joe has a collection of 23 coins, consisting of 5-cent coins, 10-cent coins, and 25-cent coins. He has 3 more 10-cent coins than 5-cent coins, and the total value of his collection is 320 cents. How many more 25-cent coins does Joe have than 5-cent coins?Problem 9All of the triangles in the diagram below are similar to iscoceles triangle , inwhich . Each of the 7 smallest triangles has area 1, and has area 40. What is the area of trapezoid ?Problem 10Suppose that real number satisfies. What is the valueof ?Problem 11When fair standard -sided die are thrown, the probability that the sum of the numbers on the top faces is can be written as, where is a positive integer. What is ?Problem 12How many ordered pairs of real numbers satisfy the following system ofequations?Problem 13A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point falls on point . What is the length in inches of the crease?Problem 14What is the greatest integer less than or equal toProblem 15Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points and , as shown in the diagram. The distance can be written in the form , where and are relatively prime positive integers. What is ?Problem 16Right triangle has leg lengths and . Including and , how many line segments with integer length can be drawn from vertex to a point on hypotenuse ?Problem 17Let be a set of 6 integers taken from with the property that if and are elements of with , then is not a multiple of . What is the least possible values of an element inProblem 18How many nonnegative integers can be written in theformwhere for ?Problem 19A number is randomly selected from the set , and a number is randomly selected from . What is the probabilitythat has a units digit of ?Problem 20A scanning code consists of a grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of squares. A scanning code is called if its look does not change when the entire square is rotated by a multiple of counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?Problem 21Which of the following describes the set of values of for which thecurves and in the real -plane intersect at exactly points?Problem 22Let and be positive integers suchthat , , ,and . Which of the following must be a divisor of ?Problem 23Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplantedsquare so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from to the hypotenuse is 2 units. What fraction of the field is planted?Problem 24Triangle with and has area . Let be the midpointof , and let be the midpoint of . The angle bisectorof intersects and at and , respectively. What is the area of quadrilateral ?Problem 25For a positive integer and nonzero digits , , and , let be the -digit integer each of whose digits is equal to ; let be the -digit integer each of whose digits is equal to , and let be the -digit (not -digit) integer each of whose digits is equal to . What is the greatest possiblevalue of for which there are at least two values of such that ?2018 AMC 10B ProblemsProblem 1Kate bakes a 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain?Problem 2Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes?Problem 3In the expression each blank is to be filled in with one of the digits or with each digit being used once. How many different values can be obtained?Problem 4A three-dimensional rectangular box with dimensions , , and has faces whose surface areas are 24, 24, 48, 48, 72, and 72 square units. What is ?Problem 5How many subsets of contain at least one prime number?Problem 6A box contains 5 chips, numbered 1, 2, 3, 4, and 5. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds 4. What is the probability that 3 draws are required?Problem 7In the figure below, congruent semicircles are drawn along a diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let be the combined area of the small semicircles and be the area of the region inside the large semicircle but outside the small semicircles. The ratio is 1:18. What is ?Problem 8Sara makes a staircase out of toothpicks as shown:This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks?Problem 9The faces of each of 7 standard dice are labeled with the integers from 1 to 6. Let be the probability that when all 7 dice are rolled, the sum of the numbers on the top faces is 10. What other sum occurs with the same probability ?Problem 10In the rectangular parallelepiped shown, , , and . Point is the midpoint of . What is the volume of the rectangular pyramid with base and apex ?Problem 11Which of the following expressions is never a prime number when is a prime number?Problem 12Line segment is a diameter of a circle with . Point , not equal to or , lies on the circle. As point moves around the circle, the centroid (center of mass) of traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?Problem 13How many of the first numbers in the sequence are divisible by ?Problem 14A list of positive integers has a unique mode, which occurs exactly times. What is the least number of distinct values that can occur in the list?Problem 15A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point in the figure on the right. The box has base length and height . What is the area of the sheet of wrapping paper?Problem 16Let be a strictly increasing sequence of positive integers suchthat What is the remainderwhen is divided by ?Problem 17In rectangle , and . Points and lie on ,points and lie on , points and lie on , and points and lie on so that and the convex octagon is equilateral. The length of a side of this octagon can be expressed in the form , where , , and are integers and is not divisible by the square of any prime. What is ?Problem 18Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip?Problem 19Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?Problem 20A function is defined recursivelyby and for allintegers . What is ?Problem 21Mary chose an even -digit number . She wrote down all the divisors of in increasing order fromleft to right: . At some moment Mary wrote as a divisor of . What is the smallest possible value of the next divisor written to the right of ?Problem 22Real numbers and are chosen independently and uniformly at random from the interval . Which of the following numbers is closest to the probability that and are the side lengths of an obtuse triangle?Problem 23How many ordered pairs of positive integers satisfy theequation where denotes the greatest common divisor of and , and denotes their least common multiple?Problem 24Let be a regular hexagon with side length . Denote by , , and the midpoints of sides , , and , respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of and ?Problem 25Let denote the greatest integer less than or equal to . How many real numbers satisfy the equation ?2018 AMC 12A ProblemsProblem 1A large urn contains balls, of which are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be ? (No red balls are to be removed.)Problem 2While exploring a cave, Carl comes across a collection of -pound rocks worth each, -poundrocks worth each, and -pound rocks worth each. There are at least of each size. He can carry at most pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?Problem 3How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)Problem 4Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of ?Problem 5What is the sum of all possible values of for which the polynomials andhave a root in common?Problem 6For positive integers and such that , both the mean and the median ofthe set are equal to . What is ?Problem 7For how many (not necessarily positive) integer values of is the value of an integer?Problem 8All of the triangles in the diagram below are similar to iscoceles triangle , in which. Each of the 7 smallest triangles has area 1, and has area 40. What is the area of trapezoid ?Problem 9Which of the following describes the largest subset of values of within the closed interval forwhich for every between and , inclusive?How many ordered pairs of real numbers satisfy the following system of equations?Problem 11A paper triangle with sides of lengths 3,4, and 5 inches, as shown, is folded so that point falls on point . What is the length in inches of the crease?Problem 12Let be a set of 6 integers taken from with the property that if and are elements of with , then is not a multiple of . What is the least possible value of an element inProblem 13How many nonnegative integers can be written in the formwherefor ?Problem 14The solutions to the equation , where is a positive real number other thanor , can be written as where and are relatively prime positive integers. What is ?A scanning code consists of a grid of squares, with some of its squares colored black and therest colored white. There must be at least one square of each color in this grid of squares. A scanning code is called if its look does not change when the entire square is rotated by a multiple of counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?Problem 16Which of the following describes the set of values of for which the curves andin the real -plane intersect at exactly points?Problem 17Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted squareso that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from to the hypotenuse is 2 units. What fraction of the field is planted?Triangle with and has area . Let be the midpoint of, and let be the midpoint of . The angle bisector of intersects and atand , respectively. What is the area of quadrilateral ?Problem 19Let be the set of positive integers that have no prime factors other than , , or . The infinite sumof the reciprocals of the elements of can be expressed as , where and are relatively primepositive integers. What is ?Problem 20Triangle is an isosceles right triangle with . Let be the midpoint ofhypotenuse . Points and lie on sides and , respectively, so thatand is a cyclic quadrilateral. Given that triangle has area , the length canbe written as , where , , and are positive integers and is not divisible by the square of any prime. What is the value of ?Problem 21Which of the following polynomials has the greatest real root?Problem 22The solutions to the equations and whereform the vertices of a parallelogram in the complex plane. The area of thisparallelogram can be written in the form where and are positive integersand neither nor is divisible by the square of any prime number. What isProblem 23In and Points and lie on sidesand respectively, so that Let and be the midpoints of segmentsand respectively. What is the degree measure of the acute angle formed by linesandProblem 24Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between 0 and 1, and Bob announces that he will choose his number uniformly at random from all thenumbers between and Armed with this information, what number should Carol choose to maximize her chance of winning?Problem 25For a positive integer and nonzero digits , , and , let be the -digit integer each of whosedigits is equal to ; let be the -digit integer each of whose digits is equal to , and let bethe -digit (not -digit) integer each of whose digits is equal to . What is the greatest possible value of for which there are at least two values of such that ?2018 AMC 12B ProblemsProblem 1Kate bakes 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain?Problem 2Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes?Problem 3A line with slope 2 intersects a line with slope 6 at the point . What is the distance between the -intercepts of these two lines?Problem 4A circle has a chord of length , and the distance from the center of the circle to the chord is . What is the area of the circle?Problem 5How many subsets of contain at least one prime number?Suppose cans of soda can be purchased from a vending machine for quarters. Which of the following expressions describes the number of cans of soda that can be purchased for dollars, where 1 dollar is worth 4 quarters?Problem 7What is the value ofProblem 8Line segment is a diameter of a circle with . Point , not equal to or , lies on the circle. As point moves around the circle, the centroid (center of mass) of traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?Problem 9What isProblem 10A list of positive integers has a unique mode, which occurs exactly times. What is the least number of distinct values that can occur in the list?Problem 11。

2002 AMC 10A1、The ratio is closest to which of the following numbers?SolutionWe factor as . As , ouranswer is .2、For the nonzero numbers , , , define.Find .Solution. Ouranswer is then .Alternate solution for the lazy: Without computing the answer exactly,we see that , , and . The sumis , and as all the options are integers, the correct one is obviously .3、According to the standard convention for exponentiation,.If the order in which the exponentiations are performed is changed, how many other values are possible?SolutionThe best way to solve this problem is by simple brute force.It is convenient to drop the usual way how exponentiation is denoted,and to write the formula as , where denotes exponentiation. We are now examining all ways to add parentheses to this expression. There are 5 ways to do so:1.2.3.4.5.We can note that . Therefore options 1 and 2 are equal, and options 3 and 4 are equal. Option 1 is the one given in the problem statement. Thus we only need to evaluate options 3 and 5.Thus the only other result is , and our answer is .4、For how many positive integers does there exist at least one positive integer such that ?infinitely manySolutionSolution 1For any we can pick , we get , therefore theanswer is .Solution 2Another solution, slightly similar to this first one would be using Simon's Favorite Factoring Trick.Let , thenThis means that there are infinitely many numbers that can satisfythe inequality. So the answer is .5、Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region.SolutionThe outer circle has radius , and thus area . The littlecircles have area each; since there are 7, their total area is . Thus,our answer is .6、Cindy was asked by her teacher to subtract from a certain numberand then divide the result by . Instead, she subtracted and thendivided the result by , giving an answer of . What would heranswer have been had she worked the problem correctly?SolutionWe work backwards; the number that Cindy started with is. Now, the correct result is . Ouranswer is .7、If an arc of on circle has the same length as an arc of oncircle , then the ratio of the area of circle to the area of circle isSolutionLet and be the radii of circles A and B, respectively.It is well known that in a circle with radius r, a subtended arc oppositean angle of degrees has length .Using that here, the arc of circle A has length . The arcof circle B has length . We know that they are equal,so , so we multiply through and simplify to get . As all circles are similar to one another, the ratio of the areas is just thesquare of the ratios of the radii, so our answer is .8、Betsy designed a flag using blue triangles, small white squares, anda red center square, as shown. Let be the total area of the bluetriangles, the total area of the white squares, and the area of thered square. Which of the following is correct?SolutionThe blue that's touching the center red square makes up 8 triangles, or 4 squares. Each of the corners is 2 squares and each of the edges is 1, totaling 12 squares. There are 12 white squares, thus we have.9、There are 3 numbers A, B, and C, such that ,and . What is the average of A, B, and C?More than 1SolutionNotice that we don't need to find what A, B, and C actually are, just their average. In other words, if we can find A+B+C, we will be done.Adding up the equations gives soand the average is . Our answer is .10、Compute the sum of all the roots of.SolutionSolution 1We expand to get which isafter combining like terms. Using the quadratic partof Vieta's Formulas, we find the sum of the roots is . Solution 2Combine terms to get, hence the rootsare and , thus our answer is .11、Jamal wants to store computer files on floppy disks, each ofwhich has a capacity of megabytes (MB). Three of his files requireMB of memory each, more require MB each, and theremaining require MB each. No file can be split between floppydisks. What is the minimal number of floppy disks that will hold all the files?SolutionA 0.8 MB file can either be on its own disk, or share it with a 0.4 MB. Clearly it is not worse to pick the second possibility. Thus we will have 3 disks, each with one 0.8 MB file and one 0.4 MB file.We are left with 12 files of 0.7 MB each, and 12 files of 0.4 MB each.Their total size is MB. The total capacity of 9 disks is MB, hence we need at least 10 more disks. And wecan easily verify that 10 disks are indeed enough: six of them will carry two 0.7 MB files each, and four will carry three 0.4 MB files each.Thus our answer is .12、Mr. Earl E. Bird leaves his house for work at exactly 8:00 A.M. every morning. When he averages miles per hour, he arrives at hisworkplace three minutes late. When he averages miles per hour, hearrives three minutes early. At what average speed, in miles per hour, should Mr. Bird drive to arrive at his workplace precisely on time?SolutionSolution 1Let the time he needs to get there in be t and the distance he travelsbe d. From the given equations, we know that and. Setting the two equal, we have andwe find of an hour. Substituting t back in, we find . From, we find that r, and our answer, is .Solution 2Since either time he arrives at is 3 minutes from the desired time, the answer is merely the harmonic mean of 40 and 60. The harmonicmean of a and b is . In this case, a and b are 40 and 60,so our answer is , so .Solution 3A more general form of the argument in Solution 2, with proof:Let be the distance to work, and let be the correct average speed.Then the time needed to get to work is .We know that and . Summing these twoequations, we get: .Substituting and dividing both sides by , we get ,hence .(Note that this approach would work even if the time by which he is late was different from the time by which he is early in the other case - we would simply take a weighed sum in step two, and hence obtaina weighed harmonic mean in step three.)13、Give a triangle with side lengths 15, 20, and 25, find the triangle's smallest height.SolutionSolution 1This is a Pythagorean triple (a 3-4-5 actually) with legs 15 and 20. Thearea is then . Now, consider an altitude drawn to anyside. Since the area remains constant, the altitude and side to which it is drawn are inversely proportional. To get the smallest altitude, it must be drawn to the hypotenuse. Let the length be x; we have, so and x is 12. Our answer is then.Solution 2By Heron's formula, the area is , hence the shortest altitude'slength is .14、Both roots of the quadratic equation are prime numbers. The number of possible values of isSolutionConsider a general quadratic with the coefficient of being and theroots being and . It can be factored as which is just. Thus, the sum of the roots is the negative of the coefficient of and the product is the constant term. (In general, this leads to Vieta's Formulas).We now have that the sum of the two roots is while the product is. Since both roots are primes, one must be , otherwise the sumwould be even. That means the other root is and the product mustbe . Hence, our answer is .15、Using the digits 1, 2, 3, 4, 5, 6, 7, and 9, form 4 two-digit prime numbers, using each digit only once. What is the sum of the 4 prime numbers?SolutionOnly odd numbers can finish a two-digit prime number, and a two-digit number ending in 5 is divisible by 5 and thus composite,hence our answer is .(Note that we did not need to actually construct the primes. If we had to, one way to match the tens and ones digits to form four primes is , , , and .)16、Let . What is?SolutionLet . Since one ofthe sums involves a, b, c, and d, it makes sense to consider 4x. We have. Rearranging, we have , so .Thus, our answer is .17、Sarah pours four ounces of coffee into an eight-ounce cup and fourounces of cream into a second cup of the same size. She then transfers half the coffee from the first cup to the second and, after stirring thoroughly, transfers half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream?SolutionWe will simulate the process in steps.In the beginning, we have:▪ounces of coffee in cup▪ounces of cream in cupIn the first step we pour ounces of coffee from cup to cup ,getting:▪ounces of coffee in cup▪ounces of coffee and ounces of cream in cupIn the second step we pour ounce of coffee and ounces of cream from cup to cup , getting:▪ounces of coffee and ounces of cream in cup▪the rest in cupHence at the end we have ounces of liquid in cup , and outof these ounces is cream. Thus the answer is .18、A cube is formed by gluing together 27 standard cubicaldice. (On a standard die, the sum of the numbers on any pair of opposite faces is 7.) The smallest possible sum of all the numbers showing on the surface of the cube isSolutionIn a 3x3x3 cube, there are 8 cubes with three faces showing, 12 with two faces showing and 6 with one face showing. The smallest sum with three faces showing is 1+2+3=6, with two faces showing is 1+2=3, and with one face showing is 1. Hence, the smallest possiblesum is . Our answer is thus.19、Spot's doghouse has a regular hexagonal base that measures oneyard on each side. He is tethered to a vertex with a two-yard rope.What is the area, in square yards, of the region outside of the doghouse that Spot can reach?SolutionPart of what Spot can reach is of a circle with radius 2, whichgives him . He can also reach two parts of a unit circle, whichcombines to give . The total area is then , which gives .20、Points and lie, in that order, on , dividing it intofive segments, each of length 1. Point is not on line . Point lieson , and point lies on . The line segments andare parallel. Find .SolutionAs is parallel to , angles FHD and FGA are congruent. Also,angle F is clearly congruent to itself. From SSS similarity,; hence . Similarly, . Thus,.21、The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection isSolutionAs the unique mode is , there are at least two s.As the range is and one of the numbers is , the largest one can beat most .If the largest one is , then the smallest one is , and thus the meanis strictly larger than , which is a contradiction.If the largest one is , then the smallest one is . This means that wealready know four of the values: , , , . Since the mean of all thenumbers is , their sum must be . Thus the sum of the missing fournumbers is . But if is the smallest number,then the sum of the missing numbers must be at least ,which is again a contradiction.If the largest number is , we can easily find the solution. Hence, our answer is .NoteThe solution for is, in fact, unique. As the median must be , thismeans that both the and the number, when ordered by size,must be s. This gives the partial solution . For themean to be each missing variable must be replaced by the smallestallowed value.22、A sit of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?SolutionSolution 1The pattern is quite simple to see after listing a couple of terms.Solution 2Given tiles, a step removes tiles, leaving tiles behind. Now,, so in the next step tilesare removed. This gives , another perfect square.Thus each two steps we cycle down a perfect square, and insteps, we are left with tile, hence our answer is.23、Points and lie on a line, in that order, with and. Point is not on the line, and . The perimeterof is twice the perimeter of . Find .SolutionFirst, we draw an altitude to BC from E.Let it intersect at M. As triangle BEC is isosceles, we immediately get MB=MC=6, so the altitude is 8. Now, let . Using the Pythagorean Theorem on triangleEMA, we find . From symmetry,as well. Now, we use the fact that the perimeter of is twice the perimeter of .We have so. Squaring both sides, we havewhich nicely rearranges into. Hence, AB is 9 so our answer is .24、Tina randomly selects two distinct numbers from the setand Sergio randomly selects a number from the set. The probability that Sergio's number is larger than the sum of the two numbers chosen by Tina isSolutionThis is not too bad using casework.Tina gets a sum of 3: This happens in only one way (1,2) and Sergio can choose a number from 4 to 10, inclusive. There are 7 ways that Sergio gets a desirable number here.Tina gets a sum of 4: This once again happens in only one way (1,3). Sergio can choose a number from 5 to 10, so 6 ways here.Tina gets a sum of 5: This can happen in two ways (1,4) and (2,3). Sergio can choose a number from 6 to 10, so 2*5=10 ways here.Tina gets a sum of 6: Two ways here (1,5) and (2,4). Sergio can choose a number from 7 to 10, so 2*4=8 here.Tina gets a sum of 7: Two ways here (2,5) and (3,4). Sergio can choose from 8 to 10, so 2*3=6 ways here.Tina gets a sum of 8: Only one way possible (3,5). Sergio chooses 9 or 10, so 2 ways here.Tina gets a sum of 9: Only one way (4,5). Sergio must choose 10, so 1 way.In all, there are ways. Tina chooses twodistinct numbers in ways while Sergio chooses a number inways, so there are ways in all. Since , ouranswer is .25、In trapezoid with bases and , we have ,, , and . The area of isSolutionSolution 1It shouldn't be hard to use trigonometry to bash this and find the height, but there is a much easier way. Extend and to meet atpoint :Since we have , with the ratio ofproportionality being . Thus So the sides of are , which we recognize to be aright triangle. Therefore (we could simplify some of the calculation using that the ratio of areas is equal to the ratio of the sides squared),Solution 2Draw altitudes from points and :Translate the triangle so that coincides with . We getthe following triangle:The length of in this triangle is equal to the length of the original, minus the length of . Thus .Therefore is a well-known right triangle. Its area is, and therefore its altitude is.Now the area of the original trapezoid is.。

美国imo数学竞赛试题及答案问题1：代数问题设\( a, b, c \) 是正实数，满足 \( a + b + c = 1 \)。

证明：\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq 9 \]问题2：几何问题在三角形 \( ABC \) 中，点 \( D \) 和 \( E \) 分别是边 \( BC \) 和 \( AC \) 上的点，使得 \( AD \) 平行于 \( BE \)。

如果\( \angle A = 60^\circ \)，证明 \( \angle ADB = \angle BEC \)。

问题3：数论问题给定一个正整数 \( n \)，证明对于所有 \( n \) 的倍数 \( k \)，\( k \) 除以 \( n \) 的余数等于 \( k \) 除以 \( n+1 \) 的余数。

问题4：组合问题有 \( 2n \) 个不同的球和 \( n \) 个相同的盒子。

证明至少有一个盒子包含至少 \( 3 \) 个球。

问题5：不等式问题证明对于所有正实数 \( x \) 和 \( y \)，以下不等式成立：\[ \sqrt{x^2 + y^2} + \sqrt{2xy} \geq x + y \]答案问题1：代数问题由柯西不等式，我们知道：\[ (a + b + c)\left(\frac{1}{a} + \frac{1}{b} +\frac{1}{c}\right) \geq (1 + 1 + 1)^2 \]因为 \( a + b + c = 1 \)，所以：\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq 9 \]问题2：几何问题由于 \( AD \) 平行于 \( BE \)，根据相似三角形的性质，我们有\( \triangle ABD \sim \triangle CBE \)。

美国高中学生数学竞赛题1.(1995年文理)设(3x-1)6=a6x6+a5x5+a4x4+a3x3+a2x2+a1x+a0，求a6+a5+a4+a3+a2+a1+a0的值。

答案：64。

2.(1989年文)如果(1-2x)7=a0+a1x+a2x2+…+a7x7，那么a1+a2+…+a7的值等于()A.-2B.-1C.0D.2答案：(A)3.(1989年理)已知(1-2x)7=a0+a1x+a2x2+…+a7x7，那么a1+a2+…+a7=____。

答案：-2。

题源：(美28届10题)若(3x-1)7=a7x7+a6x6+…+a0，那么a7+a6+…+a0等于()A.0B.1C.64D.-64E.128答案：(E)改编点评：1题将指数7改为6，改为简答题；2题将底数(3x-1)改为(1-2x)，展开式改为x的升幂排列，所求结论中去掉了常数项a0，3题改编方法同2题，改为填空题。

4.(1990年文)已知f(x)=x5+ax3+bx-8，且f(-2)=10，那么f(2)等于()A.-26B.-18C.-10D.10答案：(A)题源：(美33届12题)设f(x)=ax7+bx3+cx-5，其中a.b和c是常数，如图f(-7)=7，那么f(7)等于()A.-17B.-7C.14D.21E.不能唯一确定答案：(A)改编点评：降低了次数，减少了一个字母系数，降低了难度。

5.(1990年文理)如果实数x、y满足等式(x-2)2+y2=3，那么的最大值是()A. B. C. D.答案：(D)题源：(美35届29题)在满足方程(x-3)2+(y-3)2=6的实数对(x，y)中，的最大值是()A.3+2B.2+C.3D.6E.6+2答案：(A)改编点评：圆方程中的圆心坐标、半径作了改变，题设的叙述方式也作了变化。

6.(1990年文理)函数y=+++的值域是()A.{-2，4}B.{-2，0，4}C.{-2，0，2，4}D.{-4，-2，0，4}答案：(B)题源：(美28届8题)非零实数的每一个三重组(a，b，c)构成一个数。

2000到2012年A M C10美国数学竞赛P 0 A 0B 0 C0 D 0 全美中学数学分级能力测验(AMC 10)2000年 第01届 美国AMC10 (2000年2月 日 时间75分钟)1. 国际数学奥林匹亚将于2001年在美国举办，假设I 、M 、O 分别表示不同的正整数，且满足I ?M ?O =2001，则试问I ?M ?O 之最大值为 。

(A) 23 (B) 55 (C) 99 (D) 111 (E) 6712. 2000(20002000)为 。

(A) 20002001 (B) 40002000 (C) 20004000 (D) 40000002000 (E) 200040000003. Jenny 每天早上都会吃掉她所剩下的聪明豆的20%，今知在第二天结束时，有32颗剩下，试问一开始聪明豆有 颗。

(A) 40 (B) 50 (C) 55 (D) 60 (E) 754. Candra 每月要付给网络公司固定的月租费及上网的拨接费，已知她12月的账单为12.48元，而她1月的账单为17.54元，若她1月的上网时间是12月的两倍，试问月租费是 元。

(A) 2.53 (B) 5.06 (C) 6.24 (D) 7.42 (E) 8.775. 如图M ，N 分别为PA 与PB 之中点，试问当P 在一条平行AB 的直在线移动时，下列各数值有 项会变动。

(a) MN 长 (b) △PAB 之周长 (c) △PAB 之面积 (d) ABNM 之面积 (A) 0项 (B) 1项 (C) 2项 (D) 3项 (E) 4项6. 费氏数列是以两个1开始，接下来各项均为前两项之和，试问在费氏数列各项的个位数字中， 最后出现的阿拉伯数字为 。

(A) 0 (B) 4 (C) 6 (D) 7 (E) 97. 如图，矩形ABCD 中，AD =1，P 在AB 上，且DP 与DB 三等分 ?ADC ，试问△BDP 之周长为 。

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?2013 AMC8 Problems/Solutions1. ProblemDanica 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?Solution:In order to have her model cars in perfect, complete rows of 6, Danica must have a number ofcars that is a multiple of 6. The smallest multiple of 6 which is larger than 23 is 24, so she'll need to buy more model car.2.A sign at the fish market says, "50% off, today only: half-pound packages for just $3 per package." What is the regular price for a full pound of fish, in dollars?ProblemSolution: The 50% off price of half a pound of fish is $3, so the 100%, or the regular price, of a half pound of fish is $6. Consequently, if half a pound of fish costs $6, then a whole pound of fish is dollars.What is the value of?3. ProblemNotice that we can pair up every two numbers to make a sum of 1:SolutionTherefore, the answer is .4. ProblemEight 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?Each of her seven friends paidto cover Judi's portion. Therefore, Judi's portion mustbe. Since Judi was supposed to payof the total bill, the total bill must be.Solution5.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 fivechildren or the median weight, and by how many pounds?ProblemLining up the numbers (5, 5, 6, 8, 106), we see that the median weight is 6 pounds. SolutionThe average weight of the five kids is .Therefore, the average weight is bigger, bypounds, making the answer.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?ProblemSolutionLet the value in the empty box in the middle row be , and the value in the empty box in the top row be . is the answer we're looking for.Solution 1: Working BackwardsWe see that, making.It follows that, so.Another way to do this problem is to realize what makes up the bottommost number. Thismethod doesn't work quite as well for this problem, but in a larger tree, it might be faster. (In this case, Solution 1 would be faster since there's only two missing numbers.)Solution 2: Jumping Back to the StartAgain, let the value in the empty box in the middle row be , and the value in the empty box in the top row be . is the answer we're looking for.We can write some equations:Now we can substitute into the first equation using the two others: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 clearthe crossing at a constant speed. Which of the following was the most likely number of cars inthe train?ProblemIf Trey saw, then he saw.Solution 12 minutes and 45 seconds can also be expressed asseconds.Trey's rate of seeing cars,, can be multiplied byon the top andbottom (and preserve the same rate):. It follows that the most likely number of cars is.2 minutes and 45 seconds is equal to.Solution 2Since Trey probably counts around 6 cars every 10 seconds, there are groups of 6cars that Trey most likely counts. Since, the closest answer choice is.8. A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?ProblemFirst, there areways to flip the coins, in order.Solution The ways to get two consecutive heads are HHT and THH. The way to get three consecutive heads is HHH.Therefore, the probability of flipping at least two consecutive heads is .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 onwhich jump will he first be able to jump more than 1 kilometer?ProblemThis is a geometric sequence in which the common ratio is 2. To find the jump that would be over a 1000 meters, we note that. SolutionHowever, because the first term isand not, the solution to the problem is10. What is the ratio of the least common multiple of 180 and 594 to the greatest common factorof 180 and 594?ProblemTo find either the LCM or the GCF of two numbers, always prime factorize first. Solution 1The prime factorization of . The prime factorization of .Then, find the greatest power of all the numbers there are; if one number is one but not the other, use it (this is ). Multiply all of these to get 5940.For the GCF of 180 and 594, use the least power of all of the numbers that are in bothfactorizations and multiply. = 18. Thus the answer = =.We start off with a similar approach as the original solution. From the prime factorizations, the GCF is 18.Similar SolutionIt is a well known fact that. So we have,.Dividing by 18 yields .Therefore, .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?ProblemWe use that fact that . Let d= distance, r= rate or speed, and t=time. In this case, letrepresent the time.SolutionOn Monday, he was at a rate of . So,.For Wednesday, he walked at a rate of . Therefore,.On Friday, he walked at a rate of. So,. Adding up the hours yields++=.We now find the amount of time Grandfather would have taken if he walked atperday. Set up the equation,.To find the amount of time saved, subtract the two amounts: -=.To convert this to minutes, we multiply by 60.Thus, the solution to this problem is12. 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?ProblemFirst, find the amount of money one will pay for three sandals without the discount. We have.SolutionThen, find the amount of money using the discount: .Finding the percentage yields .To find the percent saved, we have13. ProblemWhen 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?Let the two digits be and. SolutionThe correct score was . Clara misinterpreted it as. The difference between thetwo iswhich factors into. Therefore, since the difference is a multiple of 9,the only answer choice that is a multiple of 9 is.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?ProblemThe probability that both show a green bean is. The probability that both show ared bean is . Therefore the probability isSolution15. If ,, and , what is the product of, , and ?ProblemSolutionTherefore,.Therefore,.To most people, it would not be immediately evident that , so we can multiply 6'suntil we get the desired number:, so.Therefore the answer is16. 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 inthe project?ProblemSolutionWe multiply the first ratio by 8 on both sides, and the second ratio by 5 to get the same number for 8th graders, in order that we can put the two ratios together:Solution 1: AlgebraTherefore, the ratio of 8th graders to 7th graders to 6th graders is. Since the ratiois in lowest terms, the smallest number of students participating in the project is.The number of 8th graders has to be a multiple of 8 and 5, so assume it is 40 (the smallest possibility). Then there are 6th graders and7th graders. The numbers ofstudents isSolution 2: Fakesolving17. The sum of six consecutive positive integers is 2013. What is the largest of these six integers?ProblemThe mean of these numbers is. Therefore the numbers are, so the answer isSolution 1Let thenumber be . Then our desired number is.Solution 2Our integers are , so we have that.Let the first term be. Our integers are. We have,Solution 318.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?ProblemThere arecubes on the base of the box. Then, for each of the 4 layers abovethe bottom (as since each cube is 1 foot by 1 foot by 1 foot and the box is 5 feet tall, there are4 feet left), there arecubes. Hence, the answer is.Solution 1 We can just calculate the volume of the prism that was cut out of the originalbox. Each interior side of the fort will be 2 feet shorter than each side of the outside. Since thefloor is 1 foot, the height will be 4 feet. So the volume of the interior box is.Solution 2The volume of the original box is . Therefore, the number of blockscontained in the fort is19. 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?ProblemIf Hannah did better than Cassie, there would be no way she could know for sure that she didn't get the lowest score in the class. Therefore, Hannah did worse than Cassie. Similarly, ifHannah did worse than Bridget, there is no way Bridget could have known that she didn't getthe highest in the class. Therefore, Hannah did better than Bridget, so our order isSolution20. Arectangle is inscribed in a semicircle with longer side on the diameter. What is thearea of the semicircle?ProblemSolutionA semicircle has symmetry, so the center is exactly at the midpoint of the 2 side on the rectangle, making the radius, by the Pythagorean Theorem,. The area is21. ProblemSamantha 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?SolutionThe number of ways to get from Samantha's house to City Park is, and the number ofways to get from City Park to school is. Since there's one way to go through CityPark (just walking straight through), the number of different ways to go from Samantha's house to City Park to school22.Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?ProblemThere are 61 vertical columns with a length of 32 toothpicks, and there are 33 horizontal rowswith a length of 60 toothpicks. An effective way to verify this is to try a small case, i.e. a grid of toothpicks. Thus, our answer isSolution23.Angleof is a right angle. The sides ofare the diameters of semicircles as shown. The area of the semicircle on equals, and the arc of the semicircle onhas length . What is the radius of the semicircle on?ProblemIf the semicircle on AB were a full circle, the area would be 16pi. Therefore the diameter of the first circle is 8. The arc of the largest semicircle would normally have a complete diameter of 17. The Pythagorean theorem says that the other side has length 15, so the radius is.Solution 1We go as in Solution 1, finding the diameter of the circle on AC and AB. Then, an extended version of the theorem says that the sum of the semicircles on the left is equal to the biggest one, so the area of the largest is , and the middle one is , so the radius is .Solution 224. Squares, , andare equal in area. Pointsandare the midpointsof sidesand, respectively. What is the ratio of the area of the shaded pentagonto the sum of the areas of the three squares?ProblemSolution 1First let(whereis the side length of the squares) for simplicity. We can extenduntil it hits the extension of. Call this point. The area of trianglethen isThe area of rectangleis. Thus, our desired area is. Now, the ratio of the shaded area to the combined area of the three squares is.Solution 2Let the side length of each square be 1.Let the intersection ofandbe .Since, . Sinceand are vertical angles, theyare congruent. We also haveby definition.So we haveby congruence. Therefore,.Since andare midpoints of sides,. This combined withyields.The area of trapezoidis.The area of triangleis.So the area of the pentagon is .The area of the 3 squares is . Therefore, .Solution 3Let the intersection of andbe .Now we haveand .Because both triangles has a side on congruent squares therefore.Becauseand are vertical angles. Also bothand are right angles so .Therefore by AAS (Angle, Angle, Side) . Then translating/rotating the shadedinto the position ofSo the shaded area now completely covers the squareSet the area of a square asTherefore, .25.A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are inches, inches, andinches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from A to B?ProblemThe radius of the ball is 2 inches. If you think about the ball rolling or draw a path for the ball (see figure below), you see that in A and C it loses inches, and it gains inches on B.So, the departurefrom the length of the track means that the answer is .Solution 1The total length of all of the arcs is . Since we want the path fromthe center, the actual distance will be shorter. Therefore, the only answer choice less thanis . This solution may be invalid because the actual distance can be longer if the path the center travels is on the outside of the curve, as it is in the middle bump. Solution 2。

美国数学试卷九年级【含答案】专业课原理概述部分一、选择题（每题1分，共5分）1. 下列哪个数是素数？A. 21B. 23C. 27D. 302. 如果一个三角形的两边长分别是8cm和15cm，那么第三边的长度可能是多少？A. 3cmB. 10cmC. 23cmD. 17cm3. 下列哪个函数是增函数？A. y = -2x + 3B. y = x^2C. y = -x^2D. y = 1/x4. 一个圆的半径增加了50%，其面积增加了多少？A. 50%B. 100%C. 150%D. 200%5. 如果一个事件A的概率是0.2，那么事件A不发生的概率是多少？A. 0.2B. 0.8C. 1D. 0二、判断题（每题1分，共5分）1. 任何两个奇数之和都是偶数。

（）2. 平行四边形的对角线互相平分。

（）3. 两个负数相乘的结果是正数。

（）4. 任何一个大于2的偶数都可以表示为两个素数之和。

（）5. 对数函数是单调递增的。

（）三、填空题（每题1分，共5分）1. 如果一个等差数列的首项是3，公差是2，那么第10项是______。

2. 如果一个圆的直径是10cm，那么这个圆的面积是______cm²。

3. 如果一个事件A的概率是0.3，那么事件A发生3次的概率是______。

4. 如果一个函数的导数是2x + 3，那么这个函数是______。

5. 如果一个三角形的两个内角分别是30°和60°，那么第三个内角是______°。

四、简答题（每题2分，共10分）1. 解释什么是素数。

2. 什么是等差数列？给出一个等差数列的例子。

3. 解释什么是概率。

4. 什么是导数？给出一个函数的导数的例子。

5. 解释什么是相似三角形。

五、应用题（每题2分，共10分）1. 一个等差数列的前三项分别是2，5，8，求这个数列的第10项。

2. 一个圆的半径是7cm，求这个圆的面积。

3. 如果一个事件A的概率是0.5，那么事件A发生5次的概率是多少？4. 如果一个函数的导数是3x^2 2x + 1，那么这个函数是什么？5. 如果一个三角形的两个内角分别是45°和45°，那么这个三角形是什么类型的三角形？六、分析题（每题5分，共10分）1. 分析并解释为什么平行四边形的对角线互相平分。

2024年AIME2数学竞赛（英文）1、Among the 900 residents of Aimeville, there are 195 who own a diamond ring, 367 who own a set of golf clubs, and 562 who own a garden spade. In addition, each of the 900 residents owns a bag of candy hearts. There are 437 residents who own exactly two of these things, and 234 residents who own exactly three of these things. Find the number of residents of Aimeville who own all four of these things.2、A list of positive integers has the following properties:· The sum of the items in the list is 30.· The unique mode of the list is 9.· The median of the list is a positive integer that does not appear in the list itself.Find the sum of the squares of all the items in the list.3、Find the number of ways to place a digit in each cell of a 2×3 grid so that the sum of the two numbers formed by reading left to right is 999, and the sum of the three numbers formed by reading top to bottom is 99. The grid below is an example of such an arangement because 8+991=999 and 9+9+ 81=99.4、Let xx, yy, and zz be positive real numbers that satisfy the following system of equations:log2 (xx yyyy)=12log2 (yy xxyy)=13log2 (yy xxyy)=14Then the value of |log2 (xx4yy3zz2)| is mm nn, where mm and nn are relatively prime positive integers. Find mm+ nn.5、Let AAAAAAAAAAAA be a convex equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are extensions of segments AAAA, AAAA, and AAAA has side lengths 200, 240, and 300. Find the side length of the hexagon.6、Alice chooses a set AA of positive integers. Then Bob lists all finite nonempty sets AA of positive integers with the property that the maximum element of AA belongs to AA. Bob's list has 2024 sets. Find the sum of the elements of AA.7、Let NN be the greatest four-digit positive integer with the property that whenever one of its digits is changed to 1, the resulting number is divisible by 7. Let QQ and RR be the quotient and the remainder, respectively, when NN is divided by 1000. Find QQ+RR.8、Torus TT is the surface produced by revolving a circle with radius 3 around an axis in the plane of the circle that is a distance 6 from the center of the circle.Let SS be a sphere with radius 11. When TT rests on the inside of SS, it is internally tangent to SS along a circle with radius rr ii, and when TT rests on the outside of SS, it is externally tangent to SS along a circle with radius rr0. The difference rr1−rr0 can be written as mm nn, where mm and nn are relatively prime positive integers. Find mm+nn.9、There is a collection of 25 indistinguishable black chips and 25 indistinguishable white chips. Find the number of ways to place some of these chips in 25 unit cells of a 5×5 grid so that· each cell contains at most one chip,· all chips in the same row and all chips in the same column have the same color, and· any additional chip placed on the grid would violate one or more of the previous two conditions.10、Let △AAAAAA have incenter II, circumcenter OO, inradius 6, and circumradius 13. Suppose that II AA⊥OOII. Find AAAA⋅AAAA.11、Find the number of triples of nonnegative integers (aa,bb,cc) satisfying aa+bb+cc=300 and aa2bb+aa2cc+bb2aa+bb2cc+cc2aa+cc2bb=600000012、Let OO(0,0), AA(12,0), and AA(0,√32) be points in the coordinate plane. Let AA be the family of segments PPQQ of unit length lying in the first quadrant with PP on the x-axis and QQ on the y-axis. There is a unique point AA on AAAA, distinct from AA and AA, that does not belong to any segment from AA other than AAAA. Then OOAA2=pp qq, where pp and qq are relatively prime positive integers. Find pp+qq.13、Let ωω≠1 be a 13th root of unity. Find the remainder when∏12kk=0(2−2ωωkk+ωω2kk)is divided by 1000.14、Let bb⩾2 be an integer. Call a positive integer nn bb-eautiful if it has exactly two digits when expressed in base bb, and these two digits sum to √nn. For example, 81 is 13-eautiful because 6313 and 6+ 3=√81. Find the least integer bb⩾2 for which there are more than ten bb-eautiful integers.15、Find the number of rectangles inside a fixed regular dodecagon (12-gon) where each side of the rectangle lies on a side or on a diagonal of the dodecagon. The diagram below shows three of those rectangles.1 、【答案】073;【解析】记四种都有的人数为aa．由容斥原理有1×437+2×234+3×aa=195+367+562解得aa=073【标注】2 、【答案】236;【解析】由30<4×9知至多3个9，又由9为唯一众数知至少2个9.由中位数不在数列中，必为偶数个数．若恰有3个9，只能为3，9，9，9，中位数为9，不合，舍去．因此恰为2个9，其余数互不相同．若为6个数，1+2+3+4+9+9=28只能1，2，3，6，9，9或1，2，4，5，9，9中位数均不为整只能为4个数且9为较大两数，5，7，9，9合题52+72+92+92=236【标注】3 、【答案】045;【解析】由横向两数和为999知对应数位和为9则纵向形成的数均为9的倍数9aa+9bb+9cc=99aa+bb+cc=11正整数解有AA102=045【标注】4 、【答案】033;【解析】 设log 2 xx =aa ，log 2 yy =bb ， log 2 zz =cc 则有 ⎩⎪⎨⎪⎧aa −bb −cc =12−aa +bb −cc =13−aa −bb +cc =14 解得 ⎩⎪⎨⎪⎧aa =−724bb =−924cc =−524原式 =|4aa +3bb +2cc |=25825+8=033【标注】5 、【答案】 80; 【解析】 设六边形边长为 xx△PPAAAA ∽△PPRRQQPPPP PPBB =PPPP PPRR =23⇒PPAA =2xx 3 △RRAAAA ∽△RRPPQQPPRR RRAA =PPPP PPRR =56⇒RRAA =5xx 62xx 3+xx +5xx 6=200⇒xx =80【标注】 6 、【答案】 55;【解析】 AA 中以AA 中的元素kk 为最大元素的集合个数为 2kk−1只需将2024拆成互不相同的2的幂之和2024=2048−24=211−(24+23)=210+29+28+27+26+25+23则AA={11,10,9,8,7,6,4}，元素和为55【标注】7 、【答案】699;【解析】设题中的数为aabbccaa，则由题意有7|1bbccaa+aa1ccaa+aabb1aa+aabbcc1即7|3aabbccaa+1111，又1111≡5（mod 7），可得aabbccaa≡3（mod 7）则aabbccaa−aabbcc1≡3（mod 7）⇒aa−1≡3（mod 7）⇒aa=4则aabbccaa−aabb1aa≡3（mod 7）⇒10(cc−1)≡3（mod 7）⇒cc=2,9则aabbccaa−aa1ccaa≡3（mod 7）⇒100(bb−1)≡3（mod 7）⇒bb=6则aabbccaa−1bbccaa≡3（mod 7）⇒1000(aa−1)≡3（mod 7）⇒aa=5取最大，则为5694，694+5=699【标注】8 、【答案】127;【解析】TT内切于SS时，如图OOAA=11，OO1AA=3，OO1AA=6，OOOO1=11−3=8 OOOO1OORR=OO1PP RRBB⇒rr ii=AAAA=334TT外切于SS时，如图OOAA=11，OO2AA=3，OO2AA=6，OOOO2=11+3=14DDDD OO2AA=OODD OOOO2⇒rr0=AAAA=337rr ii−rr0=334−337=992899+28=127【标注】9 、【答案】902;【解析】（1）若棋子全黑或全白：2种（2）若棋子不同色原问题于确定5行5列分别为黑或白形成对应由题意，原问题的一种填法唯一确定行列的黑/白而当行列的黑白均确定，只能是：黑行黑列交叉点填黑、白行白列交叉点填白、其余位置不填同样唯一确定填法不为全黑/全白的行列选法有(25−2)2=900种综上，2+900=902【标注】10 、【答案】468;【解析】由垂径AAII=AAII=RRDD2AAAA=AAII=AAAA由托勒密AAAA⋅AAAA+AAAA⋅AAAA=AAAA⋅AAAA可得AAAA+AAAA=2AAAA，即c+b=2a再由三角形面积SS△RRPPBB=aaaaaa4PP=(aa+aa+aa)rr2代入RR=13，r=6aaaaaa52=3aa⋅62⇒bbcc=468【标注】11 、【答案】601;【解析】aa2bb+aa2cc+bb2aa+bb2cc+cc2aa+cc2bb=aabb(aa+bb)+bbcc(bb+cc)+ccaa(cc+aa)=aabb(300−cc)+bbcc(300−aa)+ccaa(300−bb)=300(aabb+bbcc+ccaa)−3aabbcc=6000000则100(aabb+bbcc+ccaa)−aabbcc=2000000=1002(aa+bb+cc)−1003 1003−1002⋅(aa+bb+cc)+100(aabb+bbcc+ccaa)−aabbcc=0(100−aa)(100−bb)(100−cc)=0aa，bb，cc均为100:1种aa，bb，cc恰有1个100：3×(201−1)=600【标注】12 、【答案】23;【解析】AAAA：yy=√32−√3xx可设AA 中直线为yy =sin θθ−tan θθ⋅xx （θθ 为直线与xx 轴所夹锐角）则AAAA 为θθ=ππ3 情况AAAA 与yy =sin θθ−tan θθ⋅xx 联立有√32−√3xx =sin θθ−tan θθ⋅xx ，得 xx =√3−2sin θθ2√3−2tan θθ 则当θθ→ππ3 时的极限即符合题意中不在AA 中另一直线上的要求洛必达则xx aa =18，得AA (18,3√38)，OOAA 2=716，7+16=23．【标注】 13 、【答案】 321;【解析】 ��2−2ωωkk +ωω2kk �12kk=0=���1−ωωkk �2+1�12kk=0=��1−ωωkk −ii�12kk=0�1−ωωkk +ii�=�[(1−ii )−ωωkk ]12kk=0[(1+ii )−ωωkk ] 由ωω 为13 次单位根，xx 13−1的分解为 ��xx −ωωkk �12kk=0 则原式 =[(1−ii )13−1][(1+ii )13−1]=[−64(1−ii )−1][−64(1+ii )−1]=(64ii −65)(−64ii −65)=652+642=83218321≡321（mod 1000） 【标注】 14 、【答案】 211;【解析】 考虑bb 进制两位数 xxyy aa则它需满足(xx +yy )2=bbxx +yy ，其中1⩽xx ⩽bb −1，0⩽yy ⩽bb −1 且 为整数．以上方程需至少有10组解． 首先， (xx +yy )2=bbxx +yy ⩽bb (bb −1)+(bb −1)=bb 2−1<bb 2∴ xx +yy ⩽bb −1 其次， (xx +yy )2=bbxx +yy =xx +yy +(bb −1)xx 则 (xx +yy )(xx +yy −1)=(bb −1)xx注意2⩽xx+yy⩽bb−1，1⩽xx⩽bb−1则转化为在bb−1以内可找到至少十组xx+yy⩾xx使得上式成立，以下分析bb−1数论性质记bb−1=bb′=Πnn ii=1pp ii kk ii（pp ii为质数）由bb′|(xx+yy)(xx+yy−1)且(xx+yy,xx+yy−1)=1则xx+yy与xx+yy−1无公共质因子但包含bb′全部质因子．考虑将bb′质因子分配给xx+yy与xx+yy−1方式需有十种则bb′至少4种质因子．bb′⩾2×3×5×7=210106×105=210×53，105×104=210×5270×69=210×23，141×140=210×9485×84=210×34，126×125=210×7591×90=210×39，120×119=210×6836×35=210×6，175×174=210×14521×20=210×2．故210+1=211【标注】15 、【答案】315;【解析】题中所述矩形可按边是否与正十二边形的边平行分为两类．（1）平行：如图所示（对称情况需×3）AA1AA5AA7AA11中有AA32AA52=30个矩形AA2AA4AA8AA10中也有AA32AA52=30个矩形重复了AA1AA2AA3AA4中的AA32AA32=9个此类共有(30+30−9)×3=153个（2）不平行：如图所示（对称情况需×3）ArrayAA1AA4AA7AA10中有AA42AA42=36个矩形AA5AA6AA11AA12中有AA62个矩形其中AA42个在AA1AA4AA7AA10中，还多AA62−AA42=9个此类共有(36+9+9)×3=162个总计153+162=315个【标注】第11页，共11页。

2.3.Whatisthevalueof4.5.Hammieisinthegradeandweighs106pounds.Hisquadrupletsistersaretinybabiesandweigh5,5,6,and8pounds.W hichisgreater,theaverage(mean)weightofthesefivechildrenorthemedianweight,andbyhowman ypounds6.Thenumberineachboxbelowistheproductofthenumbersinthetwoboxesthattouchitintherowab ove.Forexample,.Whatisthemissingnumberinthetoprow7.8.Afaircoinistossed3times.Whatistheprobabilityofatleasttwoconsecutiveheads9.TheIncredibleHulkcandoublethedistancehejumpswitheachsucceedingjump.Ifhisfirstjumpis1 meter,thesecondjumpis2meters,thethirdjumpis4meters,andsoon,thenonwhichjumpwillhefirst beabletojumpmorethan1kilometer10.Whatistheratiooftheleastcommonmultipleof180and594tothegreatestcommonfactorof180a nd59411.12.Atthe2013WinnebagoCountyFairavendorisofferinga"fairspecial"onsandals.Ifyoubuyonepai rofsandalsattheregularpriceof$50,yougetasecondpairata40%discount,andathirdpairathalfthe regularprice.Javiertookadvantageofthe"fairspecial"tobuythreepairsofsandals.Whatpercentag eofthe$150regularpricedidhesave13.WhenClaratotaledherscores,sheinadvertentlyreversedtheunitsdigitandthetensdigitofones core.Bywhichofthefollowingmightherincorrectsumhavedifferedfromthecorrectone14.Letthetwodigitsbe and.Thecorrectscorewas.Claramisinterpreteditas.Thedifferencebetweenthetwoiswhichfactorsinto.Therefore,sincethedifferenceisamultipleof9,theonlyanswerchoicethatisamultipleof9 is.15.If,,and,whatistheproductof,,and16.AnumberofstudentsfromFibonacciMiddleSchoolaretakingpartinacommunityserviceproject.Theratioof-gradersto-gradersis,andthetheratioof-gradersto-gradersis .Whatisthesmallestnumberofstudentsthatcouldbeparticipatingintheproject17.Thesumofsixconsecutivepositiveintegersis2013.Whatisthelargestofthesesixintegers18.--Arpanliku16:22,27November2013(EST)Courtesyof19.Bridget,Cassie,andHannaharediscussingtheresultsoftheirlastmathtest.HannahshowsBridg etandCassiehertest,butBridgetandCassiedon'tshowtheirstoanyone.Cassiesays,'Ididn'tgetthel owestscoreinourclass,'andBridgetadds,'Ididn'tgetthehighestscore.'Whatistherankingofthethr eegirlsfromhighesttolowest20.Arectangleisinscribedinasemicirclewithlongersideonthediameter.Whatistheareaofthesemicircle21.22.Toothpicksareusedtomakeagridthatis60toothpickslongand32toothpickswide.Howmanytoo thpicksareusedaltogether23.Angle of isarightangle.Thesidesofarethediametersofsemicirclesasshown.Theareaofthesemicircleon equals,andthearcofthesemicircleon haslength.Whatistheradiusofthesemicircleon24.Squares,,and areequalinarea.Points andarethemidpointsofsides and,respectively.Whatistheratiooftheareaoftheshadedpentagontothesumoftheareasofthethreesquares25.Aballwithdiameter4inchesstartsatpointAtorollalongthetrackshown.Thetrackiscomprisedof3semicirculararcswhoseradiiare inches,inches,and1.2.The50%offpriceofhalfapoundoffishis$3,sothe100%,ortheregularprice,ofahalfpoundoffishis$6.Consequently,ifhalfapoundoffishcosts$6,thenawholepoundoffishis dollars.3.Noticethatwecanpairupeverytwonumberstomakeasumof1:Therefore,theansweris.4.Eachofhersevenfriendspaid tocoverJudi'sportion.Therefore,Judi'sportionmustbe.SinceJudiwassupposedtopay ofthetotalbill,thetotalbillmustbe.5.Themedianhereisobviouslylessthanthemean,sooption(A)and(B)areout. Liningupthenumbers(5,5,6,8,106),weseethatthemedianweightis6pounds.Theaverageweightofthefivekidsis.Therefore,theaverageweightisbigger,by pounds,makingtheanswer.6.Solution1:WorkingBackwardsLetthevalueintheemptyboxinthemiddlerowbe,andthevalueintheemptyboxinthetoprowbe. istheanswerwe'relookingfor.Weseethat,making.Itfollowsthat,so.Solution2:JumpingBacktotheStartAnotherwaytodothisproblemistorealizewhatmakesupthebottommostnumber.Thismethoddoes n'tworkquiteaswellforthisproblem,butinalargertree,itmightbefaster.(Inthiscase,Solution1wou ldbefastersincethere'sonlytwomissingnumbers.)Again,letthevalueintheemptyboxinthemiddlerowbe,andthevalueintheemptyboxinthetoprowbe.istheanswerwe'relookingfor. Wecanwritesomeequations: Nowwecansubstituteintothefirstequationusingthetwoothers:7.IfTreysaw,thenhesaw.2minutesand45secondscanalsobeexpressedas seconds.Trey'srateofseeingcars,,canbemultipliedbyonthetopandbottom(andpreservethesamerate):.Itfollowsthatthemostlikelynumberofcarsis. Solution2minutesand secondsisequalto.SinceTreyprobablycountsaround carsevery seconds,thereare groupsofcarsthatTreymostlikelycounts.Since,theclosestanswerchoiceis.8.First,thereare waystoflipthecoins,inorder. ThewaystogettwoconsecutiveheadsareHHT andTHH. ThewaytogetthreeconsecutiveheadsisHHH.Therefore,theprobabilityofflippingatleasttwoconsecutiveheadsis.9.Thisisageometricsequenceinwhichthecommonratiois2.Tofindthejumpthatwouldbeovera1000meters,wenotethat.However,becausethefirsttermis andnot,thesolutiontotheproblemis10.TofindeithertheLCMortheGCFoftwonumbers,alwaysprimefactorizefirst. Theprimefactorizationof.Theprimefactorizationof.Then,findthegreatestpowerofallthenumbersthereare;ifonenumberisonebutnottheother,useit(t hisis).Multiplyallofthesetoget5940.FortheGCFof180and594,usetheleastpowerofallofthenumbersthatareinbothfactorizationsandmultiply.=18.Thustheanswer==. Westartoffwithasimilarapproachastheoriginalsolution.Fromtheprimefactorizations,theGCFis .Itisawellknownfactthat.Sowehave,.Dividingby yields.Therefore,.11.Weusethatfactthat.Letd=distance,r=rateorspeed,andt=time.Inthiscase,letrepresentthetime.OnMonday,hewasatarateof.So,.ForWednesday,hewalkedatarateof.Therefore,.OnFriday,hewalkedatarateof.So,.Addingupthehoursyields++=.WenowfindtheamountoftimeGrandfatherwouldhavetakenifhewalkedatperday.Setuptheequation,.Tofindtheamountoftimesaved,subtractthetwoamounts:-=.Toconvertthistominutes,wemultiplyby.Thus,thesolutiontothisproblemis12.First,findtheamountofmoneyonewillpayforthreesandalswithoutthediscount.Wehave.Then,findtheamountofmoneyusingthediscount:.Findingthepercentageyields.Tofindthepercentsaved,wehave13.Letthetwodigitsbe and.Thecorrectscorewas.Claramisinterpreteditas.Thedifferencebetweenthetwoiswhichfactorsinto.Therefore,sincethedifferenceisamultipleof9,theonlyanswerchoicethatisamultipleof9is.14.Theprobabilitythatbothshowagreenbeanis.Theprobabilitythatbothshowaredbeanis.Thereforetheprobabilityis15.Therefore,.Therefore,.Tomostpeople,itwouldnotbeimmediatelyevidentthat,sowecanmultiply6'suntilwegetthedesirednumber:,so.Thereforetheansweris.16.Solution1:AlgebraWemultiplythefirstratioby8onbothsides,andthesecondratioby5togetthesamenumberfor8thgra ders,inorderthatwecanputthetworatiostogether:Therefore,theratioof8thgradersto7thgradersto6thgradersis.Sincetheratioisinlowestterms,thesmallestnumberofstudentsparticipatingintheprojectis.Solution2:FakesolvingThenumberof8thgradershastobeamultipleof8and5,soassumeitis40(thesmallestpossibility).Thenthereare6thgradersand7thgraders.Thenumbersofstudentsis17.Solution1Themeanofthesenumbersis.Thereforethenumbersare,sotheanswerisSolution2Letthe numberbe.Thenourdesirednumberis.Ourintegersare,sowehavethat.Solution3Letthefirsttermbe.Ourintegersare.Wehave,18.Solution1Therearecubesonthebaseofthebox.Then,foreachofthe4layersabovethebottom(assinceeachcubeis1foot by1footby1footandtheboxis5feettall,thereare4feetleft),therearecubes.Hence,theansweris.Solution2Wecanjustcalculatethevolumeoftheprismthatwascutoutoftheoriginalbox.Eachinteriorsideofthefortwillbe feetshorterthaneachsideoftheoutside.Sincethefloorisfoot,theheightwillbe feet.Sothevolumeoftheinteriorboxis.Thevolumeoftheoriginalboxis.Therefore,thenumberofblockscontainedinthefortis.19.IfHannahdidbetterthanCassie,therewouldbenowayshecouldknowforsurethatshedidn'tgett helowestscoreintheclass.Therefore,HannahdidworsethanCassie.Similarly,ifHannahdidworseth anBridget,thereisnowayBridgetcouldhaveknownthatshedidn'tgetthehighestintheclass.Therefore,HannahdidbetterthanBridget,soourorderis.20.Asemicirclehassymmetry,sothecenterisexactlyatthemidpointofthe2sideontherectangle,makingtheradius,bythePythagoreanTheorem,.Theareais.21.ThenumberofwaystogetfromSamantha'shousetoCityParkis,andthenumberofwaystogetfromCityParktoschoolis.Sincethere'sonewaytogothroughCityPark(justwalkingstraightthrough),thenumberofdifferentwaystogofromSamantha'shousetoCityParktoschool.22.Thereare verticalcolumnswithalengthof toothpicks,andthereare horizontalrowswithalengthof toothpicks.Aneffectivewaytoverifythisistotryasmallcase,i.e.agridoftoothpicks.Thus,ouransweris.23.Solution1.Solution2WegoasinSolution1,findingthediameterofthecircleonACandAB.Then,anextendedversionofthet heoremsaysthatthesumofthesemicirclesontheleftisequaltothebiggestone,sotheareaofthelargestis,andthemiddleoneis,sotheradiusis.24.Firstlet(where isthesidelengthofthesquares)forsimplicity.Wecanextenduntilithitstheextensionof.Callthispoint.Theareaoftriangle thenis Theareaofrectangle is.Thus,ourdesiredareais.Now,theratiooftheshadedareatothecombinedareaofthethreesquaresis.Solution2Letthesidelengthofeachsquarebe.Lettheintersectionof and be.Since,.Since andareverticalangles,theyarecongruent.Wealsohave bydefinition.Sowehave by congruence.Therefore,.Since and aremidpointsofsides,.Thiscombinedwith yields.Theareaoftrapezoid is.Theareaoftriangle is.Sotheareaofthepentagon is.Theareaofthe squaresis.Therefore,.Solution3Lettheintersectionof and be.Nowwehave and.Becausebothtriangleshasasideoncongruentsquarestherefore.Because and areverticalangles.Alsoboth and arerightanglesso.ThereforebyAAS(Angle,Angle,Side).Thentranslating/rotatingtheshaded intothepositionofSotheshadedareanowcompletelycoversthesquareSettheareaofasquareasTherefore,.25.Solution1Theradiusoftheballis2inches.Ifyouthinkabouttheballrollingordrawapathfortheball(seefigurebelow),youseethatinAandCitloses inches,anditgains inchesonB.So,thedeparturefromthelengthofthetrackmeansthattheansweris.Solution2Thetotallengthofallofthearcsis.Sincewewantthepathfromthecenter,theactualdistancewillbeshorter.Therefore,theonlyanswerchoicelessthan is.Thissolutionmaybeinvalidbecausetheactualdistancecanbelongerifthepaththecent ertravelsisontheoutsideofthecurve,asitisinthemiddlebump.。