2023年第25届AMC10A国际数学竞赛（高一及以下学生）试题

2023 MAA AMC 10A 1.Cities A and B are 45 miles apart.Alicia lives in A and Beth lives in B.Alicia bikes towards B at 18 miles per hour.Leaving at the same time,Beth bikes toward A at 12 miles per hour.How many miles from City A will they be when they meet? A城与B城相距45英里.Alicia住在A城，Beth住在B城.Alicia以每小 时18英里的速度骑自行车前往B城.Beth同时出发，以每小时12英里的速 度骑自行车前往A城，问当他们相遇时，距离A城有多少英里？ (A)20 (B)24 (C)25 (D)26 (E)27 .The weight otorperthf 3 the same as the weight ofof a large pizza together withcup oforange 1 slices.A cup of orange slices weighsof a pound.What is the weight,in pounds,of a large pizza? 一个大地萨饼的写与3写杯授片合在一起的重业与一个大比萨饼的量与司杯 3 1 橙片合在一起的重量相同.一杯橙片的重量是4磅。问一个大比萨饼的重量 是多少磅？ ω写 (B)2 o号 (D)3 阳号 1 2023 MAA AMC 10A 3.How many positive perfect squares less than 2023 are divisible by 5 在小于2023的正的完全平方数中，有多少个数可以被5整除？ (A)8 (B)9 (C)10 (D)11 (E)12 4.A quadrilateral has all integer side lengths,a perimeter of26,and one side of length 4.What is the greatest possible length of one side of this quadri- lateral? 一个四边形的各边长均为整数，周长为26，其中某一条边的长度为4.问这 个四边形的一条边最大可能的长度是多少？ (A)9 (B)10 (C)11 (D)12 (E)13 5.How many digits are in the base-ten representation of 85.510.155? 85.510.155的十进制表示中有多少个数字？ (A)14 (B)15 (C)16 (D)17 (E)18 2 2023 MAA AMC 10A 6.An integer is assigned to each vertex of a cube.The value of an edge is de fined to be the sum of the values of the two vertices it touches,and the value of a face is defined to be the sum of the values of the four edges surrounding it.The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is 21.What is the value of the cube? 立方体的每个顶点处指定了一个整数.每条边的值定义为它所接触的两个顶 点处的数之和，每个面的值定义为它的四条边的值之和.立方体的值定义为 它的六个面的值之和，假设各顶点处指定的整数之和为21.问立方体的值是 多少？ (A)42 (B)63 (C)84 (D)126 (E)252 7.Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls.What is the probability that at some point,her running total will equal 3 Janet先后抛掷了标准的6面骰子4次，并逐次记录她掷出数的总和，问在某 次抛掷后，她掷出数的总和等于3的概率是多少？ 49 (B) 216 (C (D) (E) 723码 3 2023 MAA AMC 10A 8.Barb the baker has developed a new temperature scale for her bakery called the Breadus scale,which is a linear function of the Fahrenheit scale.Bread rises at 110 degrees Fahrenheit,which is 0 degrees on the Breadus scale. Bread is baked at 350 degrees Fahrenheit,which is 100 degrees on the Breadus scale.Bread is done when its internal temperature is 200 degrees Fahrenheit. What is this in degrees on the Breadus scale? 面包师Barb为她的面包店开发了一种新的温标，称为Breadus温标，它是 华氏温标的线性函数.面包发酵的温度为1l0华氏度，在Breadus温标中是 0度.面包烘烤的温度为350华氏度，在Breadus温标中是100度.当面包 内部温度达到200华氏度时，面包就烤好了，问与此对应的Breadus温标是 多少度？ (A)33 (B)34.5 (C)36 (D)37.5 (E)39 9.A digital display shows the current date as an 8-digit integer consisting of a 4-digit year,followed by a 2-digit month,followed by a 2-digit date within the month.For example,Arbor Day this year is displayed as 20230428.For how many dates in 2023 does each digit appear an even number of times in the 8-digit display for that date? 电子显示器将当前日期显示为8位整数，其中包括表示年份的4位数字，后 面跟着表示月份的2位数字，接下来是表示日子的2位数字，例如，今年美 国的植树节显示为20230428.问2023年有多少个日期，使得每个数字在该 日期的8位显示中都出现偶数次？ (A)5 (B)6 (C)7 (D)8 (E)9 4 2023 MAA AMC 10A 10.Maureen is keeping track of the mean of her quiz scores this semester.If Maureen scores an 11 on the next quiz,her mean will increase by 1.If she scores an 11 on each of the next three quizzes,her mean will increase by 2. What is the mean of her quiz scores currently? Maureen一直关注着她本学期各次测验成绩的平均分，如果Maureen在下 一次测验中获得11分，那么她的平均分将增加1分，如果她在接下来的三次 测验中每次都获得11分，那么她的平均分将增加2分.问她目前的各次测验 成绩的平均分是多少分？ (A)4 (B)5 (C)6 (D)7 (E)8 11.A square of area 2 is inscribed in a square of area 3,creating four congruent triangles,as shown below.What is the ratio of the shorter leg to the longer leg in the shaded right triangle? 如下图所示，面积为2的正方形内接于面积为3的正方形，并由此形成了四 个全等的三角形.问阴影直角三角形中较短的直角边与较长的直角边之比是 多少？ (B) (C)2-V3 (D)3-V√2 (E)V2-1 2023 MAA AMC 10A 12.How many three-digit positive integers N satisfy the following properties? The number N is divisible by 7. The number formed by reversing the digits of Nis divisible by 5. 满足以下性质的三位正整数N有多少个？ ·数N可以被7整除 ·将组成N的数字顺序反转后所得的数能被5整除， (A)13 (B)14 (C)15 (D)16 (E)17 13.Abdul and Chiang are standing 48 feet apart in a field.Bharat is standing in the same field as far from Abdul as possible so that the angle formed by his lines of sight to Abdul and Chiang measures 60.What is the square of the distance(in feet)between Abdul and Bharat? Abdul和Chiang站在田地上，相距48英尺.Bharat站在同一块地上，他在 保持看Abdul与看Chiang的两条视线夹角为60°的情况下，离Abdul尽可 能远.问Abdul与Bharat之间距离英尺数的平方是多少？ (A)1728 (B)2601 (C)3072 (D)4608 (E)6912 2023 MAA AMC 10A 14.A number is chosen at random from among the first 100 positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by 11? 从前100个正整数中随机选取一个数，然后随机选取该数的一个正整数约数， 问所选的约数能被11整除的概率是多少？ 4 A)100 9 (B) 200 (C20 (D) 1 200 (E) 50 15.An even number of circles are nested,starting with a radius of 1 and increas ing by 1 each time,all sharing a common point.The region between every other circle is shaded,starting with the region inside the circle of radius 2 but outside the circle of radius 1.An example showing8circles is displayed below.What is the least number of circles needed to make the total shaded area at least2023π？ 偶数个圆嵌套在一起，他们的半径从1开始，逐一增加1，所有的圆有一个 公共，点.从半径2的圆中在半径1的圆外的区域开始，每隔一个区域，将相 邻两个圆之间的区域涂为阴影.下面显示了一个有8个圆嵌套的例子，为使 得阴影部分的总面积至少为2023π，最少需要多少个圆？ (A)46 (B)48 (C)56 (D)60 (E)64 2023 MAA AMC 10A 16.In a table tennis tournament every participant played every other partici- pant exactly once.Although there were twice as many right-handed play ers as left-handed players,the number of games won by left-handed play- ers was 40%more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.)What is the total number of games played? 在乒乓球锦标赛中，每位参赛者与其他参赛者恰好比赛一场.尽管右手执拍 的参赛者人数是左手执拍的参赛者人数的两倍，但左手执拍的参赛者获胜的 场次比右手执拍的参赛者获胜的场次多40%.（比赛没有平局，也没有参赛 者既是右手执拍又是左手执拍.)问一共进行了多少场比赛？ (A)15 (B)36 (C)45 (D)48 (E)66 17.Let ABCD be a rectangle with AB =30 and BC 28.Points P and Q lie on BC and CD,respectively,so that all sides of△ABP,△PCQ,and△QDA have integer lengths.What is the perimeter of AAPQ? 在矩形ABCD中，AB=30,BC=28.点P和Q分别位于边BC和CD上， 使得△ABP,△PCQ,△QDA的所有边的长度都是整数.问△APQ的周 长是多少？ (A)84 (B)86 (C)88 (D)90 (E)92 8 2023 MAA AMC 10A 18.A rhombic dodecahedron is a convex polyhedron where each of the 12 faces is a rhombus,and all of the faces are congruent to each other.The number of edges that meet at a vertex is either 3 or 4,depending on the vertex.What is the number of vertices at which exactly 3 edges meet? 如果一个凸多面体有12个面，并且所有面都是彼此全等的菱形，那么它称 为“菱形十二面体”.如果在菱形十二面体的每个顶点处交汇的边的数量均为 3或4.问恰好有3条边交汇的顶，点有多少个？ (A)5 (B)6 (C)7 (D)8 (E)9 19.The line segment from A(1,2)to B(3,3)can be transformed to the line seg- ment from A(3,1)to B(4,3),sending A to A'and B to B',by a rotation centered at the point P(s,t).What is s-t? 从A(1,2)到B(3,3)的线段可以通过以P(s,)为中心的旋转，变换为从 A'(3,1)到B(4,3)的线段，将A送到A',将B送到B.问s-t是多 少？ w好 B)2 D)4 (E)1 9 2023 MAA AMC 10A 20.Each square in a 3 x 3 grid of squares is colored red,white,blue,or green so that every 2 x 2 square contains one square of each color.One such coloring is shown on the right below.How many different colorings are possible? 33方格表的每个方格都涂为红色、白色、蓝色或绿色，使得每个2×2正 方形中包含每种颜色的方格各一个，右下图显示了一种这样的涂色方法，问 共有多少种不同的涂色方法？ B R B G G R R (A)24 (B)48 (C)60 (D)72 (E)96 10