2024年第26届AMC10B国际数学竞赛（高一及以下学生）试题

MAA AMC AMERICAN MATHEMATICS COMPETITI©N MAA American Mathematics Competitions 26th Annual AMC 10 B Wednesday,November 13,2024 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 B 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 Thursday,February 6,2025,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 B Problems 3 1.What is10!-7l.6!? 请问10！-7！.6！是多少？ (A)-120 (B)0 (C)120 (D)600 (E)720 (F)Unanswered不作答 2.In a long line of people arranged left to right,the 1013th person from the left is also the 1010th person from the right.How many people are in the line? 一群人从左到右排成一个很长的队伍，从左边数起的第1013个人，也是从右边数起的第1010个 人。请问这个队伍中共有多少人？ (A)2021 (B)2022 (C)2023 (D)2024 (E)2025 (F)Unanswered不作答 3.For how many integer values of x is 2x<7 请问满足|2x≤7π的整数值x有多少个？ (A)16 (B)17 (C)19 (D)20 (E)21 (F)Unanswered不作答 4.Balls numbered 1,2,3,...are deposited in 5 bins,labeled A,B,C,D,and E,using the following procedure.Ball 1 is deposited in bin A,and balls 2 and 3 are deposited in bin B.The next 3 balls are deposited in bin C,the next 4 in bin D,and so on,cycling back to bin A after balls are deposited in bin E.(For example,balls numbered 22,23,...,28 are deposited in bin B at step 7 of this process.)In which bin is ball 2024 deposited? 编号为1,2,3，…的球按照以下操作步骤存放在5个分别标记为A,B,C,D,E的箱子中。 1号球存放在箱子A中，2号球和3号球存放在箱子B中。接下来的3个球存放在箱子C中， 再接下来的4个球存放在箱子D中，依此类推，球存放到箱子E中后，再循环回到箱子A。(例 如，编号为22,23，.，28的球在操作的第7步存放在箱子B中。)请问2024号球存放在哪个 箱子中？ (A)A (B)B (C)C (D)D (E)E (F)Unanswered不作答 2024 AMC 10 B Problems 4 5.A rectangle has integer length sides and an area of 2024.What is the least possible perimeter of the rectangle? 一个矩形的边长为整数且面积为2024。请问该矩形周长的最小可能值是多少？ (A)160 (B)180 (C)222 (D)228 (E)390 (F)Unanswered不作答 6.In the following expression,Melanie changed some of the plus signs to minus signs: 在下面的表达式中，Melanie将部分加号改为减号： 1+3+5+7+·+97+99 When the new expression was evaluated,it was negative.What is the least number of plus signs that Melanie could have changed to minus signs? 当对新的表达式求值时，得到的结果为负数。请问Melanie最少可能将多少个加号改为减号？ (A)14 (B)15 (C)16 (D)17 (E)18 (F)Unanswered不作答 7.What is the remainder when 72024+72025+72026 is divided by 19? 请问72024+72025+72026除以19的余数是多少？ (A)0 (B)1 (C)7 (D)11 (E)18 (F)Unanswered不作答 8.Let N be the product of all the positive integer divisors of 42.What is the units digit of N? 设N为42的所有正整数因数的乘积。请问N的个位数字是多少？ (A)0 (B)2 (C)4 (D)6 (E)8 (F)Unanswered不作答 2024 AMC 10 B Problems 9.Quadrilateral ABCD is a parallelogram,and E is the midpoint of the side AD.Let F be the intersection of lines EB and AC.What is the ratio of the area of quadrilateral CDEF to the area of△CFB? 四边形ABCD是平行四边形，E是边AD的中点。设F是直线EB和AC的交点。请问四边 形CDEF的面积与△CFB的面积之比是多少？ (A)5:4 (B)4:3 (C)3:2 (D)5:3 (E)2:1 (F)Unanswered不作答 10.Real numbers a,b,and c have arithmetic mean 0.The arithmetic mean of a2,b2,and c2 is 10. What is the arithmetic mean of ab,ac,and be 实数a,b,c的算术平均值为0。a2,b2,c2的算术平均值为10。请问ab,ac,bc的算术平均值是 多少？ (A)-5 (B)-9 (C)-9 (D)0 (®)9 (F)Unanswered不作答 11.In the figure below WXYZ is a rectangle with WX =4 and WZ=8.Point M lies on XY, point A lies on YZ,and LWMA is a right angle.The areas of triangles AWXM and AWAZ are equal.What is the area of AWMA 在矩形WXYZ中，WX=4且WZ=8。点M在XY上，点A在YZ上，且∠WMA为直 角。三角形△WXM与△WAZ的面积相等。请问△WMA的面积是多少？ W Z M (A)13 (B)14 (C)15 (D)16 (E)17 (F)Unanswered不作答 2024 AMC 10 B Problems 6 12.A group of 100 students from different countries meet at a mathematics competition.Each student speaks the same number of languages,and,for every pair of students A and B,student A speaks some language that student B does not speak,and student B speaks some language that student A does not speak.What is the least possible total number of languages spoken by all the students? 来自不同国家的100名学生共同参加一场数学竞赛。每名学生掌握的语种数量相同，且对于任意 两名学生A和B,A掌握某种B不会的语种，B也掌握某种A不会的语种。请问这些学生掌 握的语种总数最少可能是多少？ (A)9 (B)10 (C)12 (D)51 (E)100 (F)Unanswered不作答 13.Positive integers z and y satisfy the equation v+y=v1183.What is the minimum possible value of x+y 正整数x和y满足方程vE+V歹=V1183,请问x+y的最小可能值是多少？ (A)585 (B)595 (C)623 (D)700 (E)791 (F)Unanswered不作答 14.A dartboard is the region B in the coordinate plane consisting of points(,y)such that y< 8.A target T is the region where (22+32-25)2<49.A dart is thrown and lands at a random point in B.The probability that the dart lands in T can be expressed as m m,where m and n are relatively prime positive integers.What is m+n 在坐标平面上，由满足x+≤8的点(x,)组成靶盘区域B。由(x2+y2-25)2≤49界定 目标区域T。飞镖被投掷后随机落在B内的某个点上。飞镖落在T内的概率可以表示为·π， 其中m和n是互质的正整数。请问m+n是多少？ (A)39 (B)71 (C)73 (D)75 (E)135 (F)Unanswered不作答 2024 AMC 10 B Problems 7 15.A list of 9 real numbers consists of 1,2.2,3.2,5.2,6.2,and 7,as well as y,and z with <y<z. The range of the list is 7,and the mean and the median are both positive integers.How many ordered triples (y,2)are possible? 由9个实数组成的一列数字包含1、2.2、3.2、5.2、6.2、7，以及满足x≤y≤之的数字x、少、之。 此列数字的全距为7，且平均数和中位数都是正整数。请问有序三元组(x,y,之)有多少种可能？ (A)1 (B)2 (C)3 (D)4 (E)Infinitely many无穷多个(F)Unanswered不作答 16.In a race among 5 snails,there is at most one tie,but that tie can involve any number of snails For example,the result of the race might be that Dazzler is first;Abby,Cyrus,and Elroy are tied for second;and Bruna is fifth.How many different results of the race are possible? 在一场5只蜗牛参加的比赛中，最多只会出现一次平局，但这个平局可以涉及任意数量的蜗牛。 例如，比赛结果可能是Dazzler第一；Abby,Cyrus,Elroy并列第二；Bruna第五。请问可能 有多少种不同的比赛结果？ (A)180 (B)361 (C)420 (D)431 (E)720 (F)Unanswered不作答 17.Jerry likes to play with numbers.One day,he wrote all the integers from 1 to 2024 on the whiteboard.Then he repeatedly chose four numbers on the whiteboard,erased them,and replaced them by either their sum or their product.(For example,Jerry's first step might have been to erase 1,2,3,and 5,and then write either 11,their sum,or 30,their product,on the whiteboard.)After repeatedly performing this operation,Jerry noticed that all the remaining numbers on the whiteboard were odd.What is the maximum possible number of integers on the whiteboard at that time? Jery喜欢玩数字游戏。有一天，他在白板上写下从1到2024的所有整数。然后他反复执行以 下操作：从白板上选择四个数，擦去它们，然后写上它们的和或积（例如，Jery第一步可能擦 去1、2、3、5，然后写下它们的和11或积30)。在多次操作后，Jery发现白板上剩余的所有 数都是奇数。请问此时白板上最多可能有多少个整数？ (A)1010 (B)1011 (C)1012 (D)1013 (E)1014 (F)Unanswered不作答 2024 AMC 10 B Problems 8 18.How many different remainders can result when the 100th power of an integer is divided by 125? 当一个整数的100次方除以125时，可以得到多少种不同的余数？ (A)1 (B)2 (C)5 (D)25 (E)125 (F)Unanswered不作答 19.Three different pairs of shoes are placed in a row so that no left shoe is next to a right shoe from a different pair.In how many ways can these six shoes be lined up? 三双不同的鞋子排成一列，要求每只穿在左脚的鞋子不能和另一双鞋中穿在右脚的鞋子相邻。请 问这六只鞋共有多少种排列方式？ (A)60 (B)72 (C)90 (D)108 (E)120 (F)Unanswered不作答 20.In the following table,each question mark is to be replaced by "Possible"or "Not Possible" to indicate whether a nonvertical line with the given slope can contain the given number of lattice points(points both of whose coordinates are integers).How many of the 12 entries will be“Possible”? 在下表中，每个问号需要被替换为“可能”或“不可能”，以表示具有给定斜率的非竖直的直线 是否可能包含给定数量的格点（即横纵坐标均为整数的点）。请问这12个格中有多少个会替换 为“可能”？ exactly exactly zero more than one two two 零个 恰好一个 恰好两个 多于两个 zero slope ? 零斜率 nonzero rational 2 slope 非零有理数斜率 irrational slope ? ? ? 无理数斜率 (A)4 (B)5 (C)6 (D)7 (E)9 (F)Unanswered不作答 2024 AMC 10 B Problems 9 21.A group of 16 people will be partitioned into 4 indistinguishable 4-person committees.Each committee will have one chairperson and one secretary.The number of different ways to make these assignments can be written as 3"M,where r and M are positive integers and M is not divisible by 3.What is r? 将一组16个人分成4个不加区别的4人委员会。每个委员会将有一位主席和一位秘书。一共有 3rM种不同的分配方法，其中r和M是正整数，且M不能被3整除。请问r是多少？ (A)5 (B)6 (C)7 (D)8 (E)9 (F)Unanswered不作答 22.Two straight pipes (circular cylinders),with radii1 and lie parallel and in contact on a flat floor.The figure below shows a head-on view.What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both? 两根半径分别为1和号的圆柱形直管平行地放置在平坦的地面上并相互接触。下图显示的是正 视图。第三根直管平行放置在同一地面上，并且与这两根直管都接触。请问第三根直管可能的半 径之和是多少？ (A司 (B)1 (C9 D)号 (®)9 (F)Unanswered不作答 23.The Fibonacci numbers are defined by F1=1,F2-1,and Fn=Fn-1+Fn-2 for n >3.What is 斐波那契数列定义为乃=1，F2=1,且对于n≥3，Fm=Fnm-1+Fn-2。请问下方表达式的值 是多少？ +是++…+ F20 (A)318 (B)319 (C)320 (D)321 (E)322 (F)Unanswered不作答 2024 AMC 10 B Problems 10 24.Each of 27 bricks (right rectangular prisms)has dimensions a x b x c,where a,b,and c are pairwise relatively prime positive integers.These bricks are arranged to form a 3 x 3 x 3 block, as shown on the left below.A 28th brick with the same dimensions is introduced,and these bricks are reconfigured into a 2x 2 x 7 block,shown on the right.The new block is 1 unit taller, 1 unit wider,and 1 unit deeper than the old one.What is a+b+c? 有27块长方体形状的砖块，其尺寸均为a×b×c,其中a,b,c是两两互质的正整数。如左下 图所示，这些砖块按照3×3×3的样式堆砌成立体图形。现在添加一块相同尺寸的第28块砖， 如右下图所示，将所有砖块按照2×2×7的样式重新堆砌成新的立体图形。新的立体图形比原 来的高1个单位，宽1个单位，深1个单位。请问a+b+c是多少？ (A)88 (B)89 (C)90 (D)91 (E)92 (F)Unanswered不作答 25.Let 己知 州四-骨+g,g 8+ 8 How many of the values P(2022),P(2023),P(2024),and P(2025)are integers? 请问在P(2022),P(2023),P(2024),P(2025)这四个值中，有几个是整数？ (A)0 (B)1 (C)2 (D)3 (E)4 (F)Unanswered不作答