2024年第26届AMC10A国际数学竞赛（高一及以下学生）试题

MAA AMERICAN MATHEMATICS ⑧ AMC COMPETITION MAA American Mathematics Competitions 26th Annual AMC 10 A Thursday,November 7,2024 INSTRUCTIONS 1.DO NOT TURN TO THE NEXT PAGE UNTIL YOUR COMPETITION MANAGER TELLS YOU TO BEGIN 未收到监考老师指示前，请不要翻开试卷。 2.This is a 25-question multiple-choice competition.For each question,only one answer choice among A,B,C,D and E is correct.If you don't answer the question,please select option "F". 试卷包含25道选择题，对于每道题，只有A、B、C、D和E中的一个选项是正确的。如果不 作答，请选择“F”选项。 3.Mark your answer to each problem on the answer sheet with a #2 pencil.Check blackened answers for accuracy and erase errors completely.Only answers that are properly marked on the answer sheet will be scored. 请将每道题目的答案用2B铅笔正确填涂在答题卡上，并用橡皮完全擦掉错误的答案和标记，只 有正确填涂在答题卡上的答案才会被评分。 4.SCORING:You will receive 6 points for each correct answer,1.5 points for each problem left unanswered,and 0 points for each incorrect answer. 计分规则：每题答对得6分，不答得1.5分，答错得0分。 5.Only blank scratch paper,rulers,compasses,and erasers are allowed as aids.No calculators, smartwatches.phones.or computing devices are allowed.No problems on the competition will require the use of a calculator. 考试只允许携带空白草稿纸、直尺、圆规和橡皮，不允许使用计算器、智能手表、手机或者其他 计算设备。没有任何题目必须使用计算器。 2024 Mathematical Association of America 6.Figures are not necessarily drawn to scale. 图形不一定按比例绘制。 7.You will have 75 minutes to complete the competition once your competition manager tells you to begin. 监考老师宣布考试开始后，你有75分钟的时间完成这套试卷。 8.When you finish with the competition,please follow the directions of your competition manager. 考试结束后，请听从监考老师指令。 The problems and solutions for this AMC 10 A were prepared by the MAA AMC 10/12 Editorial Board under the direction of Gary Gordon and Carl Yerger,co-Editors-in-Chief. The MAA AMC office reserves the right to disqualify scores from a school if it determines that the rules or the required security procedures were not followed. The publication,reproduction,or communication of the problems or solutions of this competition during the period when students are eligible to participate seriously jeopardizes the integrity of the results.Dissemination via phone,email,or digital media of any type during this period is a violation of the competition rules. Students who score well on this AMC 10 will be invited to take the 43rd annual American Invitational Mathematics Examination (AIME)on Wednesday,February 5,2025,or Wednesday,February 12,2025.More details about the AIME can be found at maa.org/AMC. 2024 Mathematical Association of America 2024 AMC 10 A Problems 3 1.What is the value of 101.9,901-99.10,101? 请问101·9,901-99·10,101的值是多少？ (A)2 (B)20 (C)21 (D)200 (E)2020 (F)Unanswered不作答 2.A model used to estimate the time it will take to hike to the top of a mountain on a trail is of the form T-aL +bG,where a and b are constants,T is the time in minutes,L is the length of the trail in miles,and G is the altitude gain in feet.The model estimates that it will take 69 minutes to hike to the top if a trail is 1.5 miles long and ascends 800 feet,as well as if a trail is 1.2 miles long and ascends 1100 feet.How many minutes does the model estimate it will take to hike to the top if the trail is 4.2 miles long and ascends 4000 feet? T=aL+bG是一个用于估算沿一条小径徒步至山顶所需时间的模型，其中a和b都是常数， T是用时（以分钟为单位），工是小径长度（以英里为单位），G是爬升高度（以英尺为单位）。 按此模型估算，在一座爬升高度为800英尺，小径长度为1.5英里的山中徒步，或者在一座爬升 高度为1100英尺，小径长度为1.2英里的山中徒步，徒步至山顶都需要69分钟。请通过此模 型估算在一座爬升高度为4000英尺，小径长度为4.2英里的山中徒步至山顶需要多少分钟？ (A)240 (B)246 (C)252 (D)258 (E)264 (F)Unanswered不作答 3.Let n be the least prime number that can be written as the sum of 5 distinct prime numbers. What is the sum of the digits of n? 设n是可以表示为5个不同质数之和的最小质数。请问n的各位数字之和是多少？ (A)5 (B)7 (C)8 (D)10 (E)11 (F)Unanswered不作答 4.The number 2024 is written as the sum of not necessarily distinct two-digit numbers.What is the least number of two-digit numbers needed to write this sum? 2024可以表示为一些两位数之和，这些两位数不一定不同。请问最少要写出多少个两位数才能 满足前述条件？ (A)20 (B)21 (C)22 (D)23 (E)24 (F)Unanswered不作答 2024 AMC 10 A Problems 4 5.The product of three integers is 60.What is the least possible positive sum of the three integers? 三个整数的乘积为60。请问这三个整数之和的最小可能正数值是多少？ (A)2 (B)3 (C)5 (D)6 (E)13 (F)Unanswered不作答 6.What is the least value of n such that n!is a multiple of 2024? 请问n的最小值是多少，才能使得n!是2024的倍数？ (A)11 (B)21 (C)22 (D)23 (E)253 (F)Unanswered不作答 7.What is the minimum number of successive swaps of adjacent letters in the string ABCDEF that are needed to change the string to FEDCBA?(For example,3 swaps are required to change ABC to CBA;one such sequence of swaps is ABC→BAC→BCA→CBA.) 请问字符串ABCDEF中相邻字母之间最少需要连续交换多少次才能得到字符串FEDCBA？(例 如，字符串ABC需要经历3次交换才能得到字符串CBA,其中一种交换方式是：ABC→BAC →BCA→CBA。) (A)6 (B)10 (C)12 (D)15 (E)24 (F)Unanswered不作答 8.Consider the following operation.Given a positive integer n,if n is a multiple of 3,then you replacenby Ifn is not a multiple of 3,then you replacenby+10.Then continue this process..For example,beginning with n=4,this procedure gives4→14→24→8→18→ 6→2→l2→·.Suppose you start with n=l00.What value results if you perform this operation exactly 100 times? 给定一个正整数m,如果n是3的倍数，则用号替代m:如果n不是3的倍数，则用n+10 替代n,以此类推。例如，初始设定n=4,则有4→14→24→8→18→6→2→12→ ·。假设初始设定n=100,那么按照上述规则恰好执行100次后最终结果是多少？ (A)10 (B)20 (C)30 (D)40 (E)50 (F)Unanswered不作答 2024 AMC 10 A Problems 5 9.Amy,Bomani,Charlie,and Daria work in a chocolate factory.On Monday Amy,Bomani,and Charlie started working at 1:00 PM and were able to pack 4,3,and 3 packages,respectively, every 3 minutes.At some later time,Daria joined the group,and Daria was able to pack 5 packages every 4 minutes.Together,they finished packing 450 packages at exactly 2:45 PM.At what time did Daria join the group? Amy、Bomani、Charlie、Daria在一家巧克力工厂工作。周一时，Amy、Bomani、Charlie从下午 l:00开始工作，每3分钟分别能打包4、3、3个包裹。晚些时候，Daria加入该工作小组。Daria 每4分钟能打包5个包裹。4个人通力合作刚好在下午2：45完成450个包裹。请问Daria是什 么时间加入工作小组的？ (A)1:25PM（下午) (B)1:35PM（下午) (C)1:45PM(下午) (D)1:55PM(下午) (E)2:05PM(下午) (F)Unanswered不作答 10.In how many ways can 6 juniors and 6 seniors form 3 disjoint teams of 4 people so that each team has 2 juniors and 2 seniors? 请问有多少种方式能够让6名初中生和6名高中生组成3个没有人员重叠的4人小组，且每个 小组中有2名初中生和2名高中生？ (A)720 (B)1350 (C)2700 (D)3280 (E)8100 (F)Unanswered不作答 11.Two transformations are said to commute if applying the first followed by the second gives the same result as applying the second followed by the first.Consider these four transformations of the coordinate plane: a translation 2 units to the right, a 90-rotation counterclockwise about the origin, a reflection across the z-axis,and a dilation centered at the origin with scale factor 2 Of the 6 pairs of distinct transformations from this list,how many commute? 2024 AMC 10 A Problems 6 如果两种变换的执行顺序不影响最终结果，那么称这两种变换为可交换的。给定坐标平面上的 4种变换如下： ·向右平移2个单位， ·绕原点逆时针旋转90°， ·关于X轴做对称变换， 。以原点为中心，按照比例放大2倍。 上述不同的变换两两组合成6对，请问其中有多少对是可交换的变换？ (A)1 (B)2 (C)3 (D)4 (E)5 (F)Unanswered不作答 12.How many ordered pairs of integers(m,n)satisfy vn2-49-m? 请问有多少个有序整数对（m,n)满足n2-49=m? (A)1 (B)2 (C)3 (D)4 (E)Infinitely many无限多个(F)Unanswered不作答 13.Zelda played the Adventures of Math game on August 1 and scored 1700 points.She continued to play daily over the next 5 days.The bar chart below shows the daily change in her score compared to the day before.(For example,Zelda's score on August 2 was 1700 +80 =1780 points.)What was Zelda's average score in points over the 6 days? 8月1日，Zelda在数学大冒险游戏中获得1700分。接下来5天中，她每天都玩这个游戏。下 方柱状图显示了她每天与前一天相比的得分变化。（例如，8月2日，Zelda的得分为1700+80 =1780。)请问Zelda这6天的平均得分是多少？ Daily Change in Score from August 2 to 6 8月2日-8月6日每日得分变化 +80 +60 -10 -40 -90 Aug 2 Aug 3 Aug 4 Aug 5 Aug 6 8月2日 8月3日 8月4日 8月5日 8月6日 (A)1700 (B)1702 (C)1703 (①D)1713 (E)1715 (F)Unanswered不作答 2024 AMC 10 A Problems 7 14.All of the rectangles in the figure below,which is drawn to scale,are similar to the enclosing rectangle.Each number represents the area of its rectangle.What is length AB 下图中所有矩形按比例绘制，且全部与最外围的矩形相似。每个数字表示对应矩形的面积。请问 AB的长度是多少？ 18 16 36 9 49 32 25 A B (A)4+4V5 (B)10v2 (C)5+5V5 (D)108 (E)20 (F)Unanswered不作答 15.One side of an equilateral triangle of height 24 lies on line e.A circle of radius 12 is tangent to e and is externally tangent to the triangle.The area of the region exterior to the triangle and the circle and bounded by the triangle,the circle,and line e can be written as avb-cm,where a,b,and c are positive integers and b is not divisible by the square of any prime.What is a+ b+c? 一个高为24的等边三角形中的一条边位于直线（上。一个半径为12的圆与直线相切且外切 于该等边三角形。在由三角形、圆、直线围成的区域中位于三角形和圆外的区域的面积可表示 为a√-cr,其中a、、c都是正整数且b不能被任何质数的平方整除。请问a+b+c的值 是多少？ (A)72 (B)73 (C)74 (D)75 (E)76 (F)Unanswered不作答 16.Let M be the greatest integer such that both M+1213 and M+3773 are perfect squares. What is the units digit of M? 设M是满足M+1213和M+3773都是完全平方数的最大整数。请问M的个位数是多少？ (A)1 (B)2 (C)3 (D)6 (E)8 (F)Unanswered不作答 2024 AMC 10 A Problems 8 17.The first three terms of a geometric sequence are the integers a,720,and b,where a<720<b. What is the sum of the digits of the least possible value of b? 一个等比数列的前三项都是整数，分别为a、720、b,其中a<720<b。请问b的最小可能值 中各位数字之和是多少？ (A)9 (B)12 (C)16 (D)18 (E)21 (F)Unanswered不作答 18.Two teams are in a best-two-out-of-three playoff:the teams will play at most 3 games,and the winner of the playoff is the first team to win 2 games.The first game is played on Team A's home field,and the remaining games are played on Team B's home field.Team A haschance of winning at home,and its probability of winning when playing away from home is p.Outcomes of the games are independent.The probability that Team A wins the playoff isThen pcan be written in the form(m-),where m and n are positive integers.What is m+n? 三局两胜制的季后赛中有两支队伍：这两支队伍至多进行三局比赛，最先赢得两局的队伍获胜。 第一局比赛在A队的主场展开，剩余两场比赛都在B队的主场展开。A队有三的概率主场获 胜，其客场作战获胜的概率为p。各局比赛的结果之间相互独立。A队在整个季后赛中获胜的概 率为。p可以表示为(m-√m),其中m和n都是正整数。请问m+n的值是多少？ (A)10 (B)11 (C)12 (D)13 (E)14 (F)Unanswered不作答 19.There are exactly K positive integers b with 5 <b<2024 such that the base-b integer 2024 is divisible by 16(where 16 is in base ten).What is the sum of the digits of K 当5≤b≤2024时，恰好有K个正整数b满足以b进制表示的整数2024畅可以被16整除（这 里的16是十进制数)。请问K中各位数字之和是多少？ (A)16 (B)17 (C)18 (D)20 (E)21 (F)Unanswered不作答 2024 AMC 10 A Problems 9 20.Let K be the kite formed by joining two right triangles with legs 1 and v3 along a common hypotenuse.Eight copies of K are used to form the polygon shown below.What is the area of triangle△ABC？ 设K是由两个以1和√3为直角边长的直角三角形沿一条公共斜边拼接而成的筝形。下方所示 多边形由8个相同的K组成。请问△ABC的面积是多少？ /3 K (A)2+3V3 (B)号V (C)10+8va 3 (D)8 (E)5v3 (F)Unanswered不作答 21.Let S be a subset of {1,2,3,...,2024}such that the following two conditions hold: If z and y are distinct elements of S,then x-y>2. If x and y are distinct odd elements of S,then z-y>6. What is the maximum possible number of elements in S? 设S为{1,2,3，，2024的一个子集，满足下列条件： ·如果x和y是S中的不同元素，那么x-刘>2。 ·如果x和y是S中的不同元素且都是奇数，那么x->6。 请问S中最多可能有多少个元素？ (A)436 (B)506 (C)608 (D)654 (E)675 (F)Unanswered不作答 2024 AMC 10 A Problems 10 22.The numbers,in order,of each row and the numbers,in order,of each column of a 5 x 5 array of integers form an arithmetic progression of length 5.The numbers in positions(5,5),(2,4), (4,3),and(3,1)are 0,48,16,and 12,respectively.What number is in position (1,2)? 一个5×5的整数数组中每一行和每一列中的数字按顺序排列都可以组成一个长度为5的等差 数列。数组中(5,5)、(2,4)、(4,3)、(3,1)位置上的数字分别是0、48、16、12，请问(1,2) 位置上的数字是多少？ 48 12 16 0 (A)19 (B)24 (C)29 (D)34 (E)39 (F)Unanswered不作答 23.Integers a,b,and c satisfy ab+c=100,bc+a =87,and ca+b=60.What is ab+bc ca? 整数ab、c满足ab+c=100,bc+a=87,ca+b=60。请问ab+bc+ca的值是多少？ (A)212 (B)247 (C)258 (D)276 (E)284 (F)Unanswered不作答 24.A bee is moving in three-dimensional space.A fair six-sided die with faces labeled A+,A,B, B,Ct,and C is rolled.Suppose the bee occupies the point (a,b,c).If the die shows A+,then the bee moves to the point (a+1,6,c),and if the die shows A-,then the bee moves to the point (a-1,6,c).Analogous moves are made with the other four outcomes.Suppose the bee starts at the point(0,0,0)and the die is rolled four times.What is the probability that the bee traverses four distinct edges of some unit cube? 一只蜜蜂在三维空间中飞行。投掷一枚质地均匀的六面骰子，骰子的各面用A+、A、B+、B、 C叶、C标记。假设蜜蜂落在点(a,b,c)处，如果投掷骰子后出现A+,那么蜜蜂飞向点(a+1,b,c: 如果投掷骰子后出现Aˉ，那么蜜蜂飞向点(a-1,b,c);投掷骰子出现其余4种结果的飞行方式以 此类推。假设蜜蜂从点(0,0,0)开始飞行，随后投掷4次骰子。请问蜜蜂遍历某个单位立方体 的4条不同边的概率是多少？ (A) B)ā (C)君 D)贵 (E)号 (F)Unanswered不作答