美国数学竞赛amc12

amc考试内容
AMC（American Mathematics Competitions）是由美国数学协
会（MAA）主办的一系列数学竞赛。

该系列竞赛主要分为以下几个阶段：
1. AMC 8：面向6-8年级的学生，考察基础数学知识和解决问
题的能力。

考试时间为40分钟，共25道选择题。

2. AMC 10：面向10年级以下的学生，考察基础数学知识和解决问题的能力。

考试时间为75分钟，共25道选择题。

3. AMC 12：面向12年级以下的学生，考察更高级的数学知识和解决问题的能力。

考试时间为75分钟，共25道选择题。

4. AIME（American Invitational Mathematics Examination）：AMC 10和AMC 12的前一部分考试，面向在AMC 10和
AMC 12中取得较高成绩的学生。

考试时间为3小时，共15
道填空题。

5. USAMO（United States of America Mathematics Olympiad）：AMC和AIME的后一部分考试，面向在AIME中取得较高成
绩的学生。

考试时间为9小时，共6道证明题。

AMC考试内容包括基本数学知识、解答问题的能力、证明题
以及一些创造性问题，旨在培养学生的数学思维、问题解决能力和创新能力。

19961The addition below is incorrect.What is the largest digit that can be changed to make the addition correct?641852+9732456(A)4(B)5(C)6(D)7(E)8Each day Walter gets $3for doing his chores or $5for doing them exceptionally well.After 10days of doing his chores daily,Walter has received a total of $36.On how many days did Walter do them exceptionally well?A.3B.4C.5D.6E.7(3!)!3!=(A)1(B)2(C)6(D)40(E)120Six numbers from a list of nine integers are 7,8,3,5,and 9.The largest possible value of the median of all nine numbers in this list is(A)5(B)(C)7(D)8(E)Given that 0<a <b <c <d ,which of the following is the largest?A.a +b c +d B.a +d b +c C.b +c a +d D.b +d a +c E.c +d a +bIf f (x )=x (x +1)(x +2)(x +3)then f (0)+f (−1)+f (−2)+f (−3)=A.−8/9B.0C.8/9D.1E.10/9A father takes his twins and a younger child out to dinner on the twins’birthday.The restaurant charges $4.95for the father and $0.45for each year of a child’s age,where age is defined as the age at the most recent birthday.If the bill is $9.45,which of the following could be the age of the youngest child?A.1B.2C.3D.4E.5If 3=k ·2r and 15=k ·4r ,then r =(A)−log 25(B)log 52(C)log 105(D)log 25(E)52Triangle P AB and square ABCD are in perpendicular planes.Given that P A =3,P B =4,and AB =5,what is P D ?A.5B.√34C.√41D.2√13E.8This file was downloaded from the AoPS −MathLinks Math Olympiad Resources Page Page 1http://www.mathlinks.ro/1996How many line segments have both their endpoints located at the vertices of a given cube?(A)12(B)15(C)24(D)28(E)56Given a circle of raidus 2,there are many line segments of length 2that are tangent to the circle at their midpoints.Find the area of the region consisting of all such line segments.(A)π4(B)4−π(C)π2(D)π(E)2πA function f from the integers to the integers is defined as follows:f (x )= n +3if n is odd n/2if n is evenSuppose k is odd and f (f (f (k )))=27.What is the sum of the digits of k ?(A)3(B)6(C)9(D)12(E)15Sunny runs at a steady rate,and Moonbeam runs m times as fast,where m is a number greater than 1.If Moonbeam gives Sunny a head start of h meters,how many meters must Moonbeam run to overtake Sunny?(A)hm (B)h h +m (C)h m −1(D)hm m −1(E)h +m m −1Let E (n )denote the sum of the even digits of n .For example,E (5681)=6+8=14.Find E (1)+E (2)+E (3)+···+E (100).(A)200(B)360(C)400(D)900(E)2250Two opposite sides of a rectangle are each divided into n congruent segments,and the endpoints of one segment are joined to the center to form triangle A .The other sides are each divided into m congruent segments,and the endpoints of one of these segments are joined to the center to form triangle B .[See figure for n =5,m =7.]What is the ratio of the area of triangle A to the area of triangle B ?(A)1(B)m/n (C)n/m (D)2m/n (E)2n/mA fair standard six-sided dice is tossed three times.Given that the sum of the first two tosses equal the third,what is the probability that at least one ”2”is tossed?(A)16(B)91216(C)12(D)815(E)712In rectangle ABCD ,angle C is trisected by CF and CE ,where E is on AB ,F is on AD ,BE =6,and AF =2.Which of the following is closest to the area of the rectangle ABCD ?(A)110(B)120(C)130(D)140(E)150A circle of radius 2has center at (2,0).A circle of radius 1has center at (5,0).A line is tangent to the two circles at points in the first quadrant.Which of the following is closest to the y -intercept of the line?(A)√4(B)8/3(C)1+√(D)2√(E)3The midpoints of the sides of a regular hexagon ABCDEF are joined to form a smaller hexagon.What fraction of the area of ABCDEF is enclosed by the smaller hexagon?1996(A)12(B)√33(C)23(D)34(E)√32In the xy-plane,what is the length of the shortest path from (0,0)to (12,16)that does not go inside the circle (x −6)2+(y −8)2=25?(A)10√3(B)10√5(C)10√3+5π3(D)40√33(E)10+5πTriangles ABC and ABD are isosceles with AB =AC =BD ,and BD intersects AC at E .If BD is perpendicular to AC ,then ∠C +∠D is[img]/Forum/albump ic.php ?pic i d =537[/img ](A)115◦(B)120◦(C)130◦(D)135◦(E)not uniquely determinedFour distinct points,A ,B ,C ,and D ,are to be selected from 1996points evenly spaced around a circle.All quadruples are equally likely to be chosen.What is the probability that the chord AB intersects the chord CD?(A)14(B)13(C)12(D)23(E)34The sum of the lengths of the twelve edges of a rectangular box is 140,and the distance from one corner of the box to the farthest corner is 21.The total surface area of the box is(A)776(B)784(C)798(D)800(E)812The sequence 1,2,1,2,2,1,2,2,2,1,2,2,2,2,1,2,2,2,2,2,1,2,...consists of 1s separated by blocks of 2s with n 2s in the nth block.The sum of the first 1234terms of this sequence is(A)1996(B)2419(C)2429(D)2439(E)2449Given that x 2+y 2=14x +6y +6,what is the largest possible value that 3x +4y can have?(A)72(B)73(C)74(D)75(E)76An urn contains marbles of four colors:red,white,blue,and green.When four marbles are drawn without replacement,the following events are equally likely:(a)the selection of four red marbles;(b)the selection of one white and three red marbles;(c)the selection of one white,one blue,and two red marbles;and (d)the selection of one marble of each color.What is the smallest number of marbles satisfying the given condition?(A)19(B)21(C)46(D)69Consider two solid spherical balls,one centered at (0,0,212)with radius 6,and the other centeredat (0,0,1)with radius 92.How many points (x,y,z )with only integer coordinates (lattice points)are there in the intersection of the balls?(A)7(B)9(C)11(D)13(E)15On a 4×4×3rectangular parallelepiped,vertices A ,B ,and C are adjacent to vertex D .The perpendicular distance from D to the plane containing A ,B ,and C is closest to(A)1.6(B)1.9(C)2.1(D)2.7(E)2.91996If n is a positive integer such that 2n has 28positive divisors and 3n has 30positive divisors,then how many positive divisors does 6n have?(A)32(B)34(C)35(D)36(E)38A hexagon inscribed in a circle has three consecutive sides each of length 3and three consecutive sides each of length 5.The chord of the circle that divides the hexagon into two trapezoids,one with three sides each of length 3and the other with three sides each of length 5,has length equal to m n ,where m and n are relatively prime positive integers.Find m +n .(A)309(B)349(C)369(D)389(E)409。

AMC2020 AProblem 1Carlos took of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left?Problem 2The acronym AMC is shown in the rectangular grid below with grid lines spaced unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMCProblem 3A driver travels for hours at miles per hour, during which her car gets miles per gallon of gasoline. She is paid per mile, and her only expense is gasoline at per gallon. What is her net rate of pay, in dollars per hour, after this expense?Problem 4How many -digit positive integers (that is, integersbetween and , inclusive) having only even digits are divisible byProblem 5The integers from to inclusive, can be arranged to form a -by- square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?Problem 6In the plane figure shown below, of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetryProblem 7Seven cubes, whose volumes are , , , , , ,and cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?Problem 8What is the median of the following list of numbersProblem 9How many solutions does the equation have on the intervalProblem 10There is a unique positive integer suchthat What is the sum of the digits ofProblem 11A frog sitting at the point begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length , and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square withvertices and . What is the probability that the sequence of jumps ends on a vertical side of the squareProblem 12Line in the coordinate plane has the equation . This line is rotated counterclockwise about the point to obtain line . What is the -coordinate of the -intercept of lineProblem 13There are integers , , and , each greater than 1, suchthat for all . What is ?Problem 14Regular octagon has area . Let be the area of quadrilateral . What isProblem 15In the complex plane, let be the set of solutions to and let be the set of solutions to . What is the greatest distance between a point of and a point ofA point is chosen at random within the square in the coordinate plane whose vertices are and . The probability that the point is within units of a lattice point is .(A point is a lattice point if and are both integers.) What is to the nearest tenthProblem 17The vertices of a quadrilateral lie on the graph of , and the -coordinates of these vertices are consecutive positive integers.The area of the quadrilateral is . What is the -coordinate of the leftmost vertex?Problem 18Quadrilateral satisfies , and . Diagonals and intersect at point ,and . What is the area of quadrilateral ?There exists a unique strictly increasing sequence of nonnegativeintegers such thatWhat isProblem 20Let be the triangle in the coordinate plane withvertices , , and . Consider the following five isometries (rigid transformations) of the plane: rotations of , , and counterclockwise around the origin, reflection across the -axis, and reflection across the -axis. How many ofthe sequences of three of these transformations (not necessarily distinct) will return to its original position? (For example, a rotation, followed by a reflection across the -axis, followed by a reflection across the -axis will return to its original position, but a rotation, followed by a reflection across the-axis, followed by another reflection across the -axis will not return to its original position.)How many positive integers are there such that is a multiple of , and the least common multiple of and equals times the greatest common divisor of andProblem 22Let and be the sequences of real numbers suchthat for all integers , where . WhatisProblem 23Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly . Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?Suppose that is an equilateral triangle of side length , with the property that there is a unique point inside the triangle such that , , and . What isProblem 25The number , where and are relatively prime positive integers, has the property that the sum of all realnumbers satisfying is , where denotes the greatest integer less than or equal to and denotes the fractional part of . What is2020 AMC 12A Answer Key 1. C2. C3. E4. B5. C6. D7. B8. C9. E10.E11.B12.B13.B14.B15.D16.B17.D18.D19.C20.A21.D22.B23.A24.B25.C。

全美数学竞赛流程全美数学竞赛（AMC）是美国数学协会（MAA）主办的一项数学竞赛，分为AMC 10和AMC 12两个级别。

以下是全美数学竞赛的流程：1. 注册报名：学生可以通过学校或个人报名参加AMC竞赛。

报名通常在每年10月开始，截止日期一般在11月初。

2. 竞赛日期：竞赛通常分为两个日期进行，AMC 10和AMC12的日期可能不同。

竞赛通常在每年2月进行，考试时间为75分钟。

3. 考试内容：AMC竞赛分为多个选择题，每个竞赛级别有25道题目。

AMC 10适用于初中和低年级高中学生，AMC 12适用于高年级高中学生。

题目类型包括代数、几何、概率、数论等数学知识点。

4. 答题方式：学生需要在答题卡上选择正确答案。

答题卡需要填写个人信息和参加的竞赛级别。

5. 判题与分数：竞赛结束后，答题卡会被寄回MAA进行判题。

每道题目的正确答案会公布在MAA网站上。

学生可以在个人平台上查询自己的分数和排名。

6. 参赛资格：根据竞赛成绩，学生有机会获得进入AMC竞赛的更高级别，如AIME（AMC的随机选取的约5%到接近2.5%的学生进入AIME竞赛）。

7. 成绩认可：AMC竞赛成绩被广泛用于许多数学竞赛和数学奖学金的选拔，包括全国数学奥林匹克、AMC奖学金、美国数学学会奖学金等。

8. 破解讲座：MAA常常会组织一些破解讲座，分享一些解题技巧和策略，帮助学生提高竞赛成绩和解题水平。

以上是全美数学竞赛的一般流程，具体流程可能会有一些差异，可以参考MAA的官方网站或者咨询相关责任人了解更多详细信息。

amc数学竞赛成绩划分摘要：一、前言二、AMC 数学竞赛简介1.竞赛背景2.竞赛级别与分类三、AMC 数学竞赛成绩划分1.等级划分2.评分标准四、成绩划分的影响1.对参赛者的影响2.对数学教育的影响五、总结正文：一、前言近年来，AMC 数学竞赛在我国逐渐受到关注，吸引了大量学生参与。

对于这样一项具有国际影响力的数学竞赛，了解其成绩划分规则具有重要意义。

本文将对AMC 数学竞赛成绩划分的相关内容进行详细介绍。

二、AMC 数学竞赛简介AMC，全称American Mathematics Competitions，是美国数学竞赛的简称。

该竞赛起源于1950 年，目前已经成为全球范围内最具影响力的数学竞赛之一。

AMC 竞赛分为多个级别，包括AMC8、AMC10、AMC12、AIME （美国数学邀请赛）等，分别针对不同年龄段的学生。

三、AMC 数学竞赛成绩划分1.等级划分AMC 数学竞赛的成绩主要分为以下几个等级：（1）杰出奖（Distinction）：成绩在前1% 的考生；（2）优秀奖（Merit）：成绩在前10% 的考生；（3）达标奖（Pass）：成绩在参赛者总数的50% 之前的考生；（4）未达标（Fail）：成绩在参赛者总数的50% 之后的考生。

2.评分标准AMC 数学竞赛的评分标准主要依据考生的答题正确率。

在竞赛中，每道题目分值不同，难度也有所区别。

考生在规定时间内完成答题后，根据答对的题目数量和题目难度计算得分。

最终，根据得分情况对考生进行等级划分。

四、成绩划分的影响1.对参赛者的影响AMC 数学竞赛成绩的划分对参赛者具有激励作用，能够激发学生的学习兴趣和竞争意识。

同时，成绩划分也为参赛者提供了自我评价和定位的依据，有助于他们明确自己的优势和不足，调整学习策略。

2.对数学教育的影响AMC 数学竞赛成绩的划分对我国数学教育产生了一定的影响。

一方面，AMC 竞赛的成绩可以作为选拔优秀学生的参考依据；另一方面，AMC 竞赛的题目和理念对课堂教学具有启示作用，有助于提高数学教育的质量。

AMC12 2014AProblem 1What isSolutionAt the theater children get in for half price. The price for adult tickets and child tickets is . How much would adult tickets and child tickets cost?SolutionWalking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?SolutionSuppose that cows give gallons of milk in days. At this rate, how many gallons of milk will cows give in days?SolutionOn an algebra quiz, of the studentsscored points, scored points, scored points, and the rest scored points. What is the difference between the mean and median score of the students' scores on this quiz?SolutionThe difference between a two-digit number and the number obtained by reversing its digits is times the sum of the digits of either number. What is the sum of the two digit number and its reverse?SolutionThe first three terms of a geometric progression are , , and . What is the fourth term?SolutionA customer who intends to purchase an appliance has three coupons, only one of which may be used:Coupon 1: off the listed price if the listed price is at leastCoupon 2: dollars off the listed price if the listed price is at leastCoupon 3: off the amount by which the listed price exceedsFor which of the following listed prices will coupon offer a greater price reduction than either coupon or coupon ?Five positive consecutive integers starting with have average . What is the average of consecutive integers that start with ?SolutionThree congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length . The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles?SolutionDavid drives from his home to the airport to catch a flight. He drives miles in the first hour, but realizes that he will be hour late if he continues at this speed. He increases his speed by miles per hour for the rest of the way to the airport and arrives minutes early. How many miles is the airport from his home?SolutionTwo circles intersect at points and . The minor arcs measure on one circle and on the other circle. What is the ratio of the area of the larger circle to the area of the smaller circle?A fancy bed and breakfast inn has rooms, each with a distinctive color-coded decor. One day friends arrive to spend the night. There are no other guests that night. The friends can room in any combination they wish, but with no morethan friends per room. In how many ways can the innkeeper assign the guests to the rooms?SolutionLet be three integers such that is an arithmetic progressionand is a geometric progression. What is the smallest possible value of ?SolutionA five-digit palindrome is a positive integer with respective digits , where is non-zero. Let be the sum of all five-digit palindromes. What is the sum of the digits of .SolutionThe product , where the second factor has digits, is an integer whose digits have a sum of . What is ?SolutionA rectangular box contains a sphere of radius and eight smaller spheres of radius . The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is ?SolutionThe domain of the function is an interval of length , where and are relatively prime positive integers. What is ?SolutionThere are exactly distinct rational numbers such that andhas at least one integer solution for . What is ?SolutionIn , , , and . Points and lieon and respectively. What is the minimum possible valueof ?SolutionFor every real number , let denote the greatest integer not exceeding , and let The set of all numbers suchthat and is a union of disjoint intervals. What is the sum of the lengths of those intervals?SolutionThe number is between and . How many pairs ofintegers are there such that andSolutionThe fraction where is the length of the period of the repeating decimal expansion. What is the sum ?SolutionLet , and for , let . For how many values of is ?The parabola has focus and goes through the points and . For how many points with integer coefficients is it truethat ?AMC 12 2013AProblem 1Square has side length . Point is on , and the areaof is . What is ?SolutionA softball team played ten games, scoring , and runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?SolutionA flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?SolutionWhat is the value ofSolutionTom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $, Dorothy paid $, and Sammy paid $. In order to share the costs equally, Tom gave Sammy dollars, and Dorothy gave Sammy dollars. What is ?SolutionIn a recent basketball game, Shenille attempted only three-point shots andtwo-point shots. She was successful on of her three-point shots and of her two-point shots. Shenille attempted shots. How many points did she score?SolutionThe sequence has the property that every term beginning with the third is the sum of the previous two. That is,Suppose that and . What is ?SolutionGiven that and are distinct nonzero real numbers such that , what is ?SolutionIn , and . Points and are onsides , , and , respectively, such that and are parallelto and , respectively. What is the perimeter of parallelogram ?SolutionLet be the set of positive integers for which has the repeating decimal representation with and different digits. What is the sum of the elements of ?SolutionTriangle is equilateral with . Points and are on and points and are on such that both and are parallel to . Furthermore, triangle and trapezoids and all have the same perimeter. What is ?SolutionThe angles in a particular triangle are in arithmetic progression, and the side lengths are . The sum of the possible values of x equals where , and are positive integers. What is ?SolutionLet points and .Quadrilateral is cut into equal area pieces by a line passing through . This line intersects at point , where these fractions are in lowest terms. What is ?SolutionThe sequence, , , ,is an arithmetic progression. What is ?SolutionRabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cotton-tail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done?Solution, , are three piles of rocks. The mean weight of the rocks in is pounds, the mean weight of the rocks in is pounds, the mean weight of the rocks in the combined piles and is pounds, and the mean weight of the rocks in the combined piles and is pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles and ?SolutionA group of pirates agree to divide a treasure chest of gold coins among themselves as follows. The pirate to take a share takes of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the pirate receive?SolutionSix spheres of radius are positioned so that their centers are at the vertices of a regular hexagon of side length . The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?SolutionIn , , and . A circle with center andradius intersects at points and . Moreover and have integer lengths. What is ?SolutionLet be the set . For , define to mean thateither or . How many ordered triples of elements of have the property that , , and ?SolutionConsider. Which of the following intervals contains ?SolutionA palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome is chosen uniformly at random. What is the probability that is also a palindrome?Solutionis a square of side length . Point is on such that . The square region bounded by is rotated counterclockwise with center , sweeping out a region whose area is , where , , and are positive integers and . What is ?SolutionThree distinct segments are chosen at random among the segments whoseend-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?SolutionLet be defined by . How many complexnumbers are there such that and both the real and the imaginary parts of are integers with absolute value at most ?AMC12 2012AProblem 1A bug crawls along a number line, starting at . It crawls to , then turns around and crawls to . How many units does the bug crawl altogether?SolutionCagney can frost a cupcake every seconds and Lacey can frost a cupcake every seconds. Working together, how many cupcakes can they frostin minutes?SolutionA box centimeters high, centimeters wide, and centimeters long canhold grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold grams of clay. What is ?SolutionIn a bag of marbles, of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red?SolutionA fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of pieces of fruit. There are twice as many raspberries as blueberries, three times as many grapes as cherries, and four times as many cherries as raspberries. How many cherries are there in the fruit salad?SolutionThe sums of three whole numbers taken in pairs are , , and . What is the middle number?SolutionMary divides a circle into sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?SolutionAn iterative average of the numbers , , , , and is computed in the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?SolutionA year is a leap year if and only if the year number is divisible by (such as ) or is divisible by but not by (such as ). The anniversary of the birth of novelist Charles Dickens was celebrated on February , , a Tuesday. On what day of the week was Dickens born?SolutionA triangle has area , one side of length , and the median to that side of length . Let be the acute angle formed by that side and the median. Whatis ?SolutionAlex, Mel, and Chelsea play a game that has rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For each round the probability that Alex wins is , and Mel is twice as likely to win as Chelsea. What is the probability that Alex wins three rounds, Mel wins two rounds, and Chelsea wins one round?SolutionA square region is externally tangent to the circle withequation at the point on the side . Vertices and are on the circle with equation . What is the side length of this square?SolutionPaula the painter and her two helpers each paint at constant, but different, rates. They always start at , and all three always take the same amount of time to eat lunch. On Monday the three of them painted of a house, quittingat . On Tuesday, when Paula wasn't there, the two helpers paintedonly of the house and quit at . On Wednesday Paula worked by herself and finished the house by working until . How long, in minutes, was each day's lunch break?SolutionThe closed curve in the figure is made up of congruent circular arcs each of length , where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side . What is the area enclosed by the curve?SolutionA square is partitioned into unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is the rotated clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black?SolutionCircle has its center lying on circle . The two circles meet at and . Point in the exterior of lies on circle and , ,and . What is the radius of circle ?SolutionLet be a subset of with the property that no pair of distinct elements in has a sum divisible by . What is the largest possible size of ?SolutionTriangle has , , and . Let denote the intersection of the internal angle bisectors of . What is ?SolutionAdam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?SolutionConsider the polynomialThe coefficient of is equal to . What is ?SolutionLet , , and be positive integers with such thatWhat is ?SolutionDistinct planes intersect the interior of a cube . Let be the unionof the faces of and let . The intersection of and consists of the union of all segments joining the midpoints of every pair of edges belonging to the same face of . What is the difference between the maximum and minimum possible values of ?SolutionLet be the square one of whose diagonals hasendpoints and . A point is chosen uniformly at random over all pairs of real numbers and suchthat and . Let be a translated copyof centered at . What is the probability that the square region determinedby contains exactly two points with integer coefficients in its interior?SolutionLet be the sequence of real numbers definedby , and in general,Rearranging the numbers in the sequence in decreasing order produces anew sequence . What is the sum of all integers , , such thatSolutionLet where denotes the fractional part of . Thenumber is the smallest positive integer such that the equationhas at least real solutions. What is ? Note: the fractional part of is a real number such that and is an integer.2014A1.C2.B3.B4.A5.C6.D7.A8.C9.B10.B11.C12.D13.B14.C15.B16.D17.A18.C19.E20.D21.A22.B23.B24.C25.B 2013A1. E2. C3. E4. C5. B6. B7. C8. D9. C10. D11. C12. A13. B14. B15. D16. E17. D18. B19. D20. B21. A22. E23. C24. E25. A2012A1. E2. D3. D4. C5. D6. D7. C8. C9. A10. D11. B12. D13. D14. E15. A16. E17. B18. A19. B20. B21. E22. C23. C24. C25. C。

amc数学竞赛成绩划分摘要：一、AMC数学竞赛简介二、AMC数学竞赛成绩划分及意义1.评分标准2.奖项设置3.全球排名三、如何提高AMC数学竞赛成绩1.学习策略2.解题技巧3.模拟练习四、总结正文：AMC数学竞赛是全球范围内最具影响力的数学竞赛之一，每年吸引着众多数学爱好者参加。

在我国，AMC竞赛同样具有很高的认可度和关注度。

本文将为大家介绍AMC数学竞赛的成绩划分及如何提高竞赛成绩。

一、AMC数学竞赛简介AMC数学竞赛分为两个阶段：AMC 8、AMC 10/12、AIME（美国数学邀请赛）。

参赛者需在规定时间内完成一定数量的数学题目。

题目难度逐渐递增，涵盖了从基础数学知识到高级数学技巧的各个方面。

二、AMC数学竞赛成绩划分及意义1.评分标准AMC数学竞赛采用积分制，正确答案得1分，部分正确或错误的答案不得分。

题目数量和分数决定了参赛者的总成绩。

2.奖项设置AMC竞赛设置有不同的奖项，如全球奖项、全国奖项、学校奖项等。

奖项的划分依据参赛者的总成绩在全球范围内的排名。

具体奖项设置如下：- 全球奖项：- 满分奖（Perfect Score）：全球排名前1%- 荣誉奖（Honor Roll）：全球排名前5%- 入围奖（Quick Start Award）：全球排名前10%- 全国奖项：- 一等奖（First Place）：全国排名前10名- 二等奖（Second Place）：全国排名前20名- 三等奖（Third Place）：全国排名前30名3.全球排名参赛者可以根据自己的成绩在全球排名中查找自己的位置。

全球排名有助于了解自己在国际范围内的数学水平，为今后的学术发展提供参考。

三、如何提高AMC数学竞赛成绩1.学习策略- 扎实掌握基础知识：AMC竞赛涉及的知识点较多，参赛者需要扎实掌握基础知识，才能在竞赛中游刃有余。

- 学习高级数学技巧：竞赛题目难度逐渐递增，掌握高级数学技巧对于解决难题至关重要。

- 总结经验：每次竞赛后，参赛者应总结自己的经验教训，找出不足之处，有针对性地进行提高。

2020年度美国数学竞赛AMC12A卷（带答案）AMC2020 AProblem 1Carlos took of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left?Problem 2The acronym AMC is shown in the rectangular grid below with grid lines spaced unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMCProblem 3A driver travels for hours at miles per hour, during which her car gets miles per gallon of gasoline. She is paid per mile, and her only expense is gasoline at per gallon. What is her net rate of pay, in dollars per hour, after this expense?Problem 4How many -digit positive integers (that is, integersbetween and , inclusive) having only even digits are divisible byProblem 5The integers from to inclusive, can be arranged to form a -by- square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?Problem 6In the plane figure shown below, of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetryProblem 7Seven cubes, whose volumes are , , , , , ,and cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?Problem 8What is the median of the following list of numbersProblem 9How many solutions does the equation have on the intervalProblem 10There is a unique positive integer suchthat What is the sum of the digits ofProblem 11A frog sitting at the point begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length , and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square withvertices and . What is the probability that the sequence of jumps ends on a vertical side of the squareProblem 12Line in the coordinate plane has the equation . This line is rotated counterclockwise about the point to obtain line . What is the -coordinate of the -intercept of lineProblem 13There are integers , , and , each greater than 1, suchthat for all . What is ?Problem 14Regular octagon has area . Let be the area of quadrilateral . What isProblem 15In the complex plane, let be the set of solutions to and let be the set of solutions to . What is the greatest distance between a point of and a point ofA point is chosen at random within the square in the coordinate plane whose vertices are and . The probability that the point is within units of a lattice point is .(A point is a lattice point if and are both integers.) What is to the nearest tenthProblem 17The vertices of a quadrilateral lie on the graph of , andthe -coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is . What isthe -coordinate of the leftmost vertex?Problem 18Quadrilateral satisfies , and . Diagonals and intersect at point ,and . What is the area of quadrilateral ?There exists a unique strictly increasing sequence of nonnegative integers suchthat What isProblem 20Let be the triangle in the coordinate plane withvertices , , and . Consider the following five isometries (rigid transformations) of the plane: rotations of , , and counterclockwise around the origin, reflection acrossthe -axis, and reflection across the -axis. How many ofthe sequences of three of these transformations (not necessarily distinct) will return to its original position? (For example, a rotation, followed by a reflection across the -axis, followed by a reflection across the -axis will return to its original position, but a rotation, followed by a reflection acrossthe -axis, followed by another reflection across the -axis will not return to its original position.)How many positive integers are there such that is a multiple of , and the least common multiple of and equals times the greatest common divisor of andProblem 22Let and be the sequences of real numbers suchthat for all integers , where . WhatisProblem 23Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly . Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?Suppose that is an equilateral triangle of side length , with the property that there is a unique point inside the triangle such that , , and . What isProblem 25The number , where and are relatively prime positive integers, has the property that the sum of all realnumbers satisfying is , where denotes the greatest integer less than or equal to and denotes the fractional part of . What is2020 AMC 12A Answer Key 1. C2. C3. E4. B5. C6. D7. B8. C9. E10.E11.B12.B13.B14.B15.D16.B17.D18.D19.C20.A21.D22.B23.A24.B25.C。

2020年美国数学竞赛(AMC12A)的试题与解答华南师范大学数学科学学院(510631)李湖南1.Carlos took 70%of a whole pie.Maria took one third of the remainder.What portion of the original pie was left?(A)10%(B)15%(C)20%(D)30%(E)35%译文卡洛斯取走了一整块派的70%,玛利亚取走了剩余的三分之一.问这块派还剩下多少?解(1−70%)×(1−13)=20%,故(C)正确.2.The acronym AMC is shown in the rectangular grid be-low with grid lines spaced 1unit apart.In units,what is the sum of the lengths of the line segments that form the acronym AMC?(A)17(B)15+2√2(C)13+4√2(D)11+6√2(E)21译文下图是矩形格子中的首字母缩写AMC ,其中小正方形格子的边长为1.则字母AMC 的线段长度之和是多少?解观察图形可知,直线段长度和为13,而小正方形对角线长度为√2,共4条,故总和为13+4√2,(C)正确.3.A driver travels for 2hours at 60miles per hour,during which her car gets 30miles per gallon of gasoline.She is paid $0.50per mile,and her only expense is gasoline at $2.00per gal-lon.What is her net rate of pay,in dollars per hour,after this expense?(A)20(B)22(C)24(D)25(E)26译文一位司机以60英里/小时的速度驾车2小时,她的车每跑30英里需要消耗1加仑汽油.她能获得0.50美元/英里的报酬,唯一的花费就是2美元/加仑的汽油.问她每小时除去消耗之后的净收益是多少美元?解1个小时她能跑60英里,获得60×0.50=30美元,汽油费为60÷30×2=4美元,故净收益为26美元,(E)正确.4.How many 4-digit positive integers (that is,integers be-tween 1000and 9999,inclusive)having only even digits are di-visible by 5?(A)80(B)100(C)125(D)200(E)500译文有多少个四位的正整数(也就是在1000和9999之间的整数)能被5整除且所有数字均为偶数?解依题意,符合条件的四位数的个位数只能是0,十位数和百位数可以是0,2,4,6,8,千位数只能是2,4,6,8,共有1×5×5×4=100种选择,故(B)正确.5.The 25integers from -10to 14,inclusive,can be arranged to form a 5-by-5square in which the sum of the numbers in each row,the sum of the numbers in each column,the sum of the num-bers along each of the main diagonals are all the same.What is the value of this common sum?(A)2(B)5(C)10(D)25(E)50译文将25个整数分别是从-10到14,放入5×5的格子中,使得格子里的每行、每列和两条对角线的数字和均相等.问这个数字和是多少?解这是一个5阶幻方问题,25个数字之和是(−10)+(−9)+···+13+14=50,分别放入5行,故每行的数字和是10,(C)正确.6.In the place figure shown below,3of the unit squares have been shaded.What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines ofsymmetry?(A)4(B)5(C)6(D)7(E)8译文如下图所示,3个单元格被涂成阴影部分.问至少还需要把多少个单元格涂成阴影才能使整个图有两条对称轴?解由于这是一个4×5的矩形,两条对称轴只可能是长和宽两条边的中垂线,从而至少有7个单元格需要填涂,如右图所示.故(D)正确.7.Seven cubes,whose volumes are 1,8,27,64,125,216,and 343cubic units,are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top.Ex-cept for the bottom cube,the bottom face of each cube lies com-pletely on top of the cube below it.What is the total surface area of the tower (including the bottom)in square units?(A)644(B)658(C)664(D)720(E)749译文七个立方体,体积分别是1,8,27,64,125,216,343个立方单位,依次按照体积大小由底到顶垂直地堆积成一座塔.除了最底部的立方体,每个立方体的底面都完全被下面的立方体的顶面覆盖.问这座塔的表面积(包括底面)是多少个平方单位?解这七个数都是立方数,则这七个立方体的棱长分别是1,2,3,4,5,6,7,从而塔的侧面积为4×(12+22+...+72)=560,而上、下底面积之和为2×72=98,共658,故(B)正确.8.What is the median of the following list of 4040numbers?1,2,3,...,2020,12,22,32,...,20202(A)1974.5(B)1975.5(C)1976.5(D)1977.5(E)1978.5译文下列4040个数:1,2,3,...,2020,12,22,32,...,20202的中位数是多少?解由于442=1936,452=2025,从而以上数列按递增排列的话,就成为:1,12,...,4,22,...,1936,442,...,1976,1977,...,2020,452,462,...,20202此时,1976成为第2020个数,所求中位数为1976+19772=1976.5,故(C)正确.9.How many solutions does the equation tan (2x )=cos (x 2)have on the interval [0,2π]?(A)1(B)2(C)3(D)4(E)5译文方程tan (2x )=cos (x 2)在区间[0,2π]上有多少个解?解y =tan 2x 是一个周期为π2、值域为R 的函数,在一个周期(−π4,π4)内严格单调递增,y =cos x2是一个周期为4π、值域为[−1,1]的函数,在区间[0,2π]上严格单调递减.如图示,在区间(π4,3π4),(3π4,5π4),(5π4,7π4)内,这是y =tan 2x 完整的周期,两条曲线均有一个交点;在区间[0,π4),(7π4,2π]上,这是y =tan 2x 的半周期,两条曲线刚好也有一个交点.故共有5个交点,(E)正确.10.There is a unique integer n such that log 2(log 16n )=log 4(log 4n ).What is the sum of the digits of n ?(A)4(B)7(C)8(D)11(E)13译文存在唯一的整数n 使得log 2(log 16n )=log 4(log 4n )成立,则n 的各个数位上的数字之和是多少?解log 4(log 4n )=log 2(log 16n )=log 22(log 16n )2,可得log 4n =(log 16n )2,使用换底公式有ln n ln 4=(ln n ln 16)2=ln 2n4ln 24,从而ln n 4ln 4=1⇒n =44=256.故(E)正确.11.A frog sitting at the point (1,2)begins a sequence of jumps,where each jump is parallel to one of the coordinate ax-es and has length 1,and the direction of each jump (up,down,left,right)is chosen independently at random.The sequence ends when the frog reaches a side of the square with vertices (0,0),(0,4),(4,0),and (4,4).What is the probability that the sequence ofjumps ends on a vertical side of the square?(A)12(B)58(C)23(D)34(E)78译文一只青蛙坐在点(1,2)上,开始一系列的跳跃,每次跳跃都平行于坐标轴且长度为1,方向(上、下、左、右)是随机的且独立,当青蛙到达由点(0,0),(0,4),(4,0),(4,4)构成的正方形的一条边的时候,跳跃终止.问跳跃终止于正方形竖直的两条边上的概率是多少?解如图示,青蛙在点F 1处,它可以向四个方向跳跃,概率均为14,向左跳跃,立刻达成目标;向上、向右、向下分别跳跃到点A 1,C,A 3处,再通过其它跳跃达成目标.根据对称性,青蛙由点A 1,A 2,A 3,A 4出发达成目标的概率是一样的,设为a ;青蛙由点B 1,B 2出发达成目标的概率是一样的,设为b ;青蛙由点F 1,F 2出发达成目标的概率是一样的,设为x ;青蛙由点C 出发达成目标的概率设为c .因此,P (青蛙由F 1出发达成目标)=P (青蛙向左)+P (青蛙向上)×P (青蛙由A 1出发达成目标)+P (青蛙向右)×P (青蛙由C 出发达成目标)+P (青蛙向下)×P (青蛙由A 3出发达成目标),即有x =14+a 2+c 4;同理,可得方程组x =14+a 2+c 4a =14+x 4+b 4b =a 2+c 4c =x 2+b 2成立,解得(x,a,b,c )=(58,12,38,12).故(B)正确.12.Line l in the coordinate plane has equation 3x −5y +40=0.This line is rotated 45◦counterclockwise about the point (20,20)to obtain line k .What is the x –coordinate of the x –intercept of line k ?(A)10(B)15(C)20(D)25(E)30译文坐标平面上的直线l 的方程为3x −5y +40=0,其绕点(20,20)作逆时针旋转45◦后得到直线k .则直线k 与x 轴交点的横坐标是多少?解如图示,直线k 与l 的斜率分别为tan ∠1和tan ∠2,依题意有∠1=∠2+45◦,tan ∠2=35,于是tan ∠1=tan (∠2+45◦)=tan ∠2+tan 45◦1−tan ∠2·tan 45◦=4,从而得到直线k 的方程为y −20=4(x −20),当y =0时,求得x =15.故(B)正确.13.There are integer a,b and c ,each greater than 1,such that a √N b √N c √N =36√N 25for all N >1.What is b ?(A)2(B)3(C)4(D)5(E)6译文设a,b,c 均是大于1的整数,且式子a √N b √N c √N =36√N 25对于N >1均成立.问b 是多少?解a √Nb √N c√N =N 1a N 1ab N 1abc =N bc +c +1abc 36√N 25=N 2536,依题意可得,bc +c +1abc =2536,解得a =2,b =3,c =6.故(B)正确.14.Regular octagon ABCDEF GH has area n .Let m bethe area of quadrilateral ACEG .What is mn?(A)√24(B)√22(C)34(D)3√25(E)2√23译文设正八边形ABCDEF GH 的面积为n ,四边形ACEG 的面积为m .则m n是多少?解如图,取正八边形的中心点O ,连OA ,OB .令OA =a ,则AC=√2a,∠AOB =45◦,于是m =S ACEG =(√2a )2=2a 2,n =S ABCDEF GH =8S ∆AOB =8×12a 2sin 45◦=2√2a 2,从而得m n =2a 22√2a 2=√22.故(B)正确.15.In the complex plane,let A be the set of solutions toz 3−8=0and let B be the set of solutions to z 3−8z 2−8z +64=0.What is the greatest distance between a point of A and a point of B ?(A)2√3(B)6(C)9(D)2√21(E)9+√3译文在复平面上,设A 是方程z 3−8=0的解集,B 是方程z 3−8z 2−8z +64=0的解集.问A 中一点到B 中一点的最远距离是多少?解解方程z 3−8=(z −2)(z 2+2z +4)=0,得A ={2,−1+√3i,−1−√3i };解方程z 3−8z 2−8z +64=(z −8)(z 2−8)=0,得B ={8,2√2,−2√2}.容易看出A 到B 的最远距离为d = (−1+√3i )−8 =√(−9)2+√32=2√21.故(D)正确.16.A point is chosen at random within the square in the co-ordinate plane whose vertices are (0,0),(2020,0),(2020,2020),and (0,2020).The probability that the point lies within d units ofa lattice point is 12.(A point (x,y )is a lattice point if x and y areboth integers.)What is d to the nearest tenth?(A)0.3(B)0.4(C)0.5(D)0.6(E)0.7译文坐标平面上有一个以(0,0),(2020,0),(2020,2020)和(0,2020)为顶点的正方形.在正方形内随机选择一个点,该点位于格点的d 个单位内的概率是12.(点(x,y )称为格点,若x 和y 均为整数.)则d 精确到十分位是多少?解如图示,以格点为圆心,d 为半径作一些圆,则正方形内的圆内部分就是符合条件的点集.因此,该点落在此区域的概率为P=该区域的面积正方形面积,即12=(20192+2019×2+1)πd220202=πd2,求得d=√12π≈0.4,故(B)正确.17.The vertices of a quadrilateral lie on the graph of y= ln x,and the x-coordinates of these vertices are consecutive pos-itive integers.The area of the quadrilateral is ln 9190.What is thex-coordinate of the leftmost vertex?(A)6(B)7(C)10(D)12(E)13译文一个四边形的顶点均在y=ln x的图像上,且它们的横坐标是连续正整数.该四边形的面积为ln 9190,则最左边顶点的横坐标是多少?解如图示,ABCD 是y=ln x上的四边形,过A作x轴的平行线,过C作x轴的垂线,交于点P,连结P B,P D,设点A坐标为(x,ln x),则有B(x+1,ln(x+1)),C(x+2,ln(x+2)),D(x+3,ln(x+3)),P(x+2,ln x),于是S ABCD=S∆P AB+S∆P BC+S∆P CD−S∆P AD=12×2×[ln(x+1)−ln x]+2×12×[ln(x+2)−ln x]×1−12×2×[ln(x+3)−ln x]=ln (x+1)(x+2)x(x+3)=ln9190.可得(x+1)(x+2)x(x+3)=9190,解得x=12或x=−15(舍去).故(D)正确.18.Quadrilateral ABCD satisfies∠ABC=∠ACD= 90◦,AC=20and CD=30.Diagonals AC and BD inter-sects at point E and AE=5.What is the area of Quadrilateral ABCD?(A)330(B)340(C)350(D)360(E)370译文四边形ABCD满足∠ABC=∠ACD=90◦, AC=20,CD=30.对角线AC和BD交于点E,且AE=5.求四边形ABCD 的面积是多少?解如图示,以AC为直径作一个圆,交BD与点F,依题意可得EC=15,ED=√EC2+CD2=15√5.设BE=x,依据相交弦定理AE·EC=BE·EF,则得EF=75x,DF=15√5−75x,DB=15√5+x;再由切割线定理DC2=DF·DB,得900=(15√5−75x)·(15√5+x),解得x=3√5或x=−5√5(舍去).而S∆ACD=12×20×30=300,S∆ABCS∆ACD=BEED=15,可得S∆ABC=60,故S ABCD=360,(D)正确.19.There exists a unique strictly increasing sequence of nonnegative integers a1<a2<···<a k such that2289+1217+1= 2a1+2a2+···+2a k.What is k?(A)117(B)136(C)137(D)273(E)306译文存在唯一严格递增的非负整数列a1<a2<···< a k使得2289+1217+1=2a1+2a2+···+2a k,则k是多少?解令217=x,则2289+1217+1=x17+1x+1=x16−x15+x14−x13+...+x2−x+1,而x16−x15=2272−2255=2271+2270+...+2255,同理x14−x13=2238−2221=2237+2236+ (2221)...,x2−x=234−217=233+232+ (217)从而2289+1217+1=20+(217+···+232+233)+···+(2255+···+2270+2271),共8×17+1=137项,故(C)正确.20.Let T be the triangle in the coordinate plane with vertices (0,0),(4,0),and(0,3).Consider the following five isometries (rigid transformations)of the plane:rotation of90◦,180◦,and 270◦counterclockwise around the origin,reflection across the x-axis,and reflection across the y-axis.How many of the125 sequences of three of these transformations(not necessarily dis-tinct)will return T to its original position?(For example,a180◦rotation,followed by a reflection across the x-axis,followed by a reflection across the y-axis will return T to its original position, but a90◦rotation,followed by a reflection across the x-axis,fol-lowed by another reflection across the x-axis will not return T to its original position.)(A)12(B)15(C)17(D)20(E)25译文设T是坐标平面上以(0,0),(4,0)和(0,3)为顶点的三角形.考虑以下五种平面上的等距变换(刚体变换):绕原点作90◦,180◦和270◦的逆时针旋转,关于x轴或y轴的反射.任选三种变换(不必不同)可以组成125种组合,有多少种组合将使得T 变回起始位置?(例如,一个关于y 轴的反射,接着一个关于x 轴的反射,再接着一个180◦的旋转,将会使得T 变回起始位置;但一个关于x 轴的反射,接着另一个关于x 轴的反射,再接着一个90◦的旋转,将不会使得T 变回起始位置.)解分两种情况:(1)全部由旋转组成:只要三次旋转的角度和为360◦或720◦即可满足要求,因此有90◦+90◦+180◦,90◦+180◦+90◦,180◦+90◦+90◦,270◦+270◦+180◦,270◦+180◦+270◦,180◦+270◦+270◦共6种组合;(2)由旋转和反射组合而成:有y 轴+x 轴+180◦,y 轴+180◦+x 轴,180◦+x 轴+y 轴,180◦+y 轴+x 轴,x 轴+180◦+y 轴,x 轴+y 轴+180◦,也是6种组合.故(A)正确.21.How many positive integers n are there such that n is a multiple of 5,and the least common multiple of 5!and n equals 5times the greatest common divisor of 10!And n ?(A)12(B)24(C)36(D)48(E)72译文有多少个正整数n ,使得n 是一个5的倍数,且n 与5!的最小公倍数是n 与10!的最大公因数的5倍?解由题意,[n,5!]=5×(n,10!),而5!=23×3×5,10!=28×34×52×7,可知n 不含除2,3,5,7以外的素因子,可设n =2a ×3b ×5c ×7d ,其中a,b,c,d ∈N ,且c 1.根据[2a ×3b ×5c ×7d ,23×3×5]=5×(2a ×3b ×5c×7d,28×34×52×7),以及最大公因数和最小公倍数的取法,可得3 a 8,1 b 4,c =3,0 d 1.故n 有6×4×1×2=48种取法,(D)正确.22.Let (a n )and (b n )be the sequence of real numbers suchthat (2+i )n =a n +b n i for all integers n 0,where i =√−1.What is ∞∑n =0a n b n7n?(A)38(B)716(C)12(D)916(E)47译文设(a n )和(b n )是使(2+i )n =a n +b n i 对所有的整数n 0均成立的实数列,其中i =√−1,则∞∑n =0a n b n7n是多少?解由(2+i )n =a n +b n i ,可得(2−i )n =a n −b n i,两式相加减得(2+i )n +(2−i )n =2a n ,(2+i )n −(2−i )n =2b n i从而a nb n =(2+i )n +(2−i )n 2·(2+i )n −(2−i )n2i =(3+4i )n −(3−4i )n4i 于是∞∑n =0a n b n 7n =∞∑n =017n ·(3+4i )n −(3−4i )n4i =14i ·∞∑n =0(3+4i )n −(3−4i )n 7n=14i·11−3+4i 7−11−3−4i 7=14i ·(74−4i −74+4i )=716故(B)正确.23.Jason rolls three fair standard six-sided dice.Then he looks at the rolls and chooses a subset of the dice (possibly emp-ty,possibly all three dice)to reroll.After rerolling,he wins if and only if the sum of the numbers faces up on the three dice is exactly 7.Jason always plays to optimize his chances of winning.What is the probability that he chooses to reroll exactly two of the dice?(A)736(B)524(C)29(D)1772(E)14译文詹森掷3颗标准、均匀的骰子,他看了结果之后会选择若干(可能是0,也可能是3)颗重掷.当3颗骰子正面朝上的数字和为7点的时候,他就赢了.詹森总是按照朝着他赢的最优策略去掷.问他刚好选择2颗骰子重掷的概率是多少?解掷1颗骰子得1,2,3,4,5,6点的概率均为16;掷2颗骰子得3点只有两种情况:12和21,概率为236,···;掷3颗骰子得7点有15种情况:115,151,511,124,142,214,241,412,421,133,313,331,223,232,322,概率为15216,···.经过计算,所有结果如下表所示:分类/概率/结果1234567掷1颗161616161616掷2颗136236336436536掷3颗1216321632161021615216因此,詹森要选择2颗骰子重掷,则上次掷的结果中,任意两颗骰子的数字和不能小于7点,否则他将选择重掷1颗骰子;且不能3颗骰子都是4点或者以上,要不然他将选择2019年全国高中数学联赛一试A 卷第10题的探究广东省佛山市乐从中学(528315)林国红一、题目呈现题目(2019年全国高中数学联赛一试(A 卷)第10题)在平面直角坐标系xoy 中,圆Ω与抛物线Γ:y 2=4x 恰有一个公共点,且圆Ω与x 轴相切于抛物线Γ的焦点F ,求圆Ω的半径.二、解法探究解法1(利用均值不等式)由题可知抛物线Γ的焦点为F (1,0),由对称性,不妨设圆Ω在x 轴上方与x 轴相切于F ,设圆Ω的半径为r .故圆Ω的方程为(x −1)2+(y −r )2=r 21⃝将x =14y 2代入1⃝并化简,得(y 24−1)2+y 2−2ry =0.显然y >0,故r =y 4+8y 2+1632y =(y 2+4)232y2⃝根据条件,2⃝恰有一个正数解y ,该y 值对应Ω与Γ的唯一公共点.考虑f (y )=(y 2+4)232y(y >0)的最小值.重掷3颗骰子.根据以上分析,满足条件的情况有:(1)掷出1点、6点、6点,3种情况;(2)掷出2点、5点、5点,3种情况;(3)掷出2点、5点、6点,6种情况;(4)掷出2点、6点、6点,3种情况;(5)掷出3点、4点、4点,3种情况;(6)掷出3点、4点、5点,6种情况;(7)掷出3点、4点、6点,6种情况;(8)掷出3点、5点、5点,3种情况;(9)掷出3点、5点、6点,6种情况;(10)掷出3点、6点、6点,3种情况.由加法原理,共42种情况,故所求概率为42216=736,(A)正确.24.Suppose that ∆ABC is an equilateral triangle of side length s ,with the property that there is a unique point P insidethe triangle such that AP =1,BP =√3,CP =2.What is s ?(A)1+√2(B)√7(C)83(D)√5+√5(E)2√2译文设∆ABC 是一个边长为s 的正三角形,内部有一点P ,使得AP =1,BP =√3,CP =2.问s 是多少?解如图,将∆AP C 绕点A 逆时针旋转60◦,得到∆ADB ,连结DP ,则AD =AP =1,DB =P C =2,∠DAP =60◦,因而∆ADP 是一个正三角形,可得DP =1,进而DP 2+BP 2=DB 2,所以∆DP B 是一个直角三角形,∠DP B =90◦,因此∠AP B =150◦.根据余弦定理,s 2=AB 2=AP 2+P B 2−2AP ·P B ·cos ∠AP B =1+3−2√3cos 150◦=7,即得s =√7.故(B)正确.25.The number a =pq,where p and q are relatively prime positive integers,has the property that the sum of all real num-bers x satisfying ⌊x ⌋·{x }=a ·x 2is 420,where ⌊x ⌋denotes the greatest integer less than or equal to x and {x }=x −⌊x ⌋denotes the dfractional part of x .What is p +q ?(A)245(B)593(C)929(D)1331(E)1332译文数a =pq满足性质:符合方程⌊x ⌋·{x }=a ·x 2的所有实数x 之和为420,其中p ,q 是互素的正整数,⌊x ⌋表示小于等于x 的最大整数,{x }=x −⌊x ⌋表示x 的小数部分.则p +q 是多少?解设⌊x ⌋=n ,{x }=r ,则x =n +r,0 r <1,代入方程⌊x ⌋·{x }=a ·x 2,整理得ar 2+(2a −1)nr +an 2=0,解得r =(1−2a )n ±√(1−4a )n 22a ,可知0<a 14.再由0 r =(1−2a )n −√(1−4a )n 22a<1,解得0 n <2a (1−2a )−√1−4ac .若c 是整数,则∑x =∑(n +r )=∑(n +nc)=c −1∑n =0(n +n c )=c 2−12=420,解得c =29,从而a =29900;若c 不是整数,则∑x =⌊c ⌋∑n =0(n +n c )=⌊c ⌋(⌊c ⌋+1)2·c +1c=420c 无解.故a =29900,p +q =929,(C)正确.。