2024年AMC12A国际数学竞赛试卷(高三及以下学生)

2024 MAA AMC 12A Problem 1 What is the value of 9901.101-99.10101? 9901·101-99·10101的值是多少？ (A)2 (B)20 (C)200 (D)202 (E)2020 Problem 2 A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form T=aL+bG,where a and b are constants,Tis 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 estimates it will take to hike to the top if the trail is 4.2 miles long and ascends 4000 feet? 用于估算沿着山间小径登顶所儒时间的模型表示为T=aL十bG,其中a和b是 常数，T是以分钟为单位计的时间，L是以英里为单位计的小径长度，G是以 英尺为单位计的海拔增加量。该模型估算，沿着一条长1.5英里且海拔提升800 英尺的小径需要69分钟登顶，沿着一条长1.2英里且海拔提升1100英尺的小径 也同样需要69分钟登顶。根据该模型，沿着一条长4.2英里且海拔提升4000英 尺的小径需要多少分钟才能登顶？ (A)240 (B)246 (C)252 (D)258 (E)264 Problem 3 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? 将2024写成若千个两位数的和，这些两位数不一定互不相同。问写出这样的和 式最少需要多少个两位数？ (A)20 (B)21 (C)22 (D)23 (E)24 Problem 4 What is the least value of n such that n!is a multiple of 2024? 最小的n值是多少，使得n!是2024的倍数？ (A)11 (B)21 (C)22 (D)23 (E)253 Problem 5 A data set containing 20 numbers,some of which are 6,has mean 45.When all the 6s are removed,the data set has mean 66.How many 6s were in the original data set? 一个包含20个数的数据集中有一些数是6，均值是45.当移除所有的6后，数 据集的均值变为66.问数据集中原来有多少个6？ (A)4 (B)5 (C)6 D)7 (E8 Problem 6 The product of three integers is 60.What is the least possible positive sum of the three integers? 三个整数的乘积是60。如果它们的和是正数，那么这个正数最小可能是多少？ (A)2 (B)3 (C)5 (D)6 (©13 Problem 7 In△ABC,∠ABC=90°and BA=BC=V2.Points P1,P2,·,P2o24 lie on hypotenuse AC so that AP1=PP2=P2P3=.=P2023P2024=P2024C.What is the length of the vector sumBP1+BP2+BP3+.+BP2024? 在△ABC中，∠ABC=90°且BA=BC=V2.在斜边AC上有，点 P1,P2,…,P2024,使得A1=PP2=P2P3=…=P2023P2024=P2024C.问向量 和BP1+BP2十BP3十…+BP2024的长度是多少？ (A)1011 (B)1012 (C)2023 (D)2024 (E)2025 Problem 8 How many angles0with0≤θ≤2πsatisfy log(sin(30)+log(cos(2θ)= 0? 满足0≤0≤2m,以及log(sin(3θ)+log(cos(2θ)=0的角度θ有多少个？ (A)0 (B)1 (C)2 (D)3(E4 Problem 9 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是使得M+1213和M+3773都是完全平方数的最大整数。问M的个位数字 是几？ (A)1 (B)2 (C)3 (D)6 (E)8 Problem 10 Let a be the radian measure of the smallest angle in a 3-4-5 right triangle. Let B be the radian measure of the smallest angle in a 7-24-25right triangle.In terms of a,what is B? 设是三边长为3,4,5的直角三角形中最小角的孤度值，B是三边长为7,24， 25的直角三角形中最小角的孤度值.问B怎样用表示？ (A) ⑧a-日 (c)及-2a D (E)π-4a Problem 11 There are exactly K positive integers b with 5s b<2024 such that the base-b integer 2024p is divisible by 16(where 16 is in base ten).What is the sum of the digits of K? (A)16 (B)17(C)18 (D)20 (E)21 Problem 12 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是整数a,720,b,并且a<720<b。问b的最小可能值 的各位数字之和是多少？ (A)9 (B)12 (C)16 (D)18 (E)21 Problem 13 The graph of y=exti+e-x-2 has an axis of symmetry.What is the reflection of the point-1.over this axis? 函数y=e*+1+e-2的图像有一条对称轴，点(-1，)关于这条对称轴的反射 点是什么？ A(1,-引 (B)(-1,0) ©(-1，) D)(0,) @(3,) Problem 14 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)? 在5×5的整数数表中，每行的数按其现有顺序组成一个五项的等差数列，每列 的数也按其现有顺序组成一个五项的等差数列。已知在位置(5,5)，(2,4)，(4,3)， (3,1)处的数分别是0,48,16,12。问位置1,2)处的数是多少？ 2 12 16 (A)19 (B)24 (C)29 (D)34 (E)39 Problem 15 The roots of x3+2x2-x+3 are p,q,and r.What is the value of (p2+ 4)(q2+4)(r2+4)? 设x3+2x2-x+3的根为p,q,r.问(p2+4)(q2+4)(r2+4)的值是多少？ (A)64 (B)75 (C)100 (D)125 (E)144 Problem 16 A set of 12 tokens--3 red,2 white,1 blue,and 6 black-is to be distributed at random to 3 game players,4 tokens per player.The probability that some player gets all the red tokens,another gets all the white tokens,and the remaining player gets the blue token can be writen aswhere mandn are relatively prime positive integers.What is m+n? 现有3个红色筹码，2个白色筹码，1个蓝色筹码，6个黑色筹码共计12个筹码 要随机分给3名玩家，每人4个筹码.某个玩家获得全部红色筹码，另一个玩家 获得全部白色筹码，剩下的那个玩家获得蓝色筹码的概率可以表示为咒，其中m 和n是互质的正整数.问m十n是多少？ (A)387 (B)388 (C)389 (D)390 (E)391 Problem 17 Integers a,b,and c satisfy ab+c=100,bc+a=87,and ca+b=60 What is ab bc ca? 整数a,b,c满足ab+c=100,bc+a=87,ca+b=60.问ab+bc+ca是多 少？ (A)212 (B)247 (C)258 (D)276 (E)284 Problem 18 On top of a rectangular card with sides of length 1 and 2+v3,an identical card is placed so that two of their diagonals line up,as shown(AC,in this case).Continue the process,adding a third card to the second,and so on,lining up successive diagonals after rotating clockwise.In total,how many cards must be used until a vertex of a new card lands exactly on the vertex labeled B in the figure? 如图所示，在一个边长为1和2+V√3的矩形卡片上方，放置一个相同的卡片，使 它们的对角线重合(AC).继续这样的操作，在第二张卡片上放置第三张卡片， 顺时针旋转并使对角线重合，然后依此类推.问总共需要使用多少张卡片，才能 使新卡片的某个顶点恰好落在图中标记为B的顶，点上？ (A)6 B)8 (C)10 D)12 @)No new vertex will land on B|不会有新的顶，点落在B上 Problem 19 Cyclic quadrilateral ABCD has lengths BC=CD =3 and DA=5 with CDA =120.What is the length of the shorter diagonal of ABCD? 在圆内接四边形ABCD中，BC=CD=3,且DA=5,∠CDA=120。问ABCD的 较短对角线的长度是多少？ (A) (C)5 D)9 Problem 20 Points P and Q are chosen uniformly and independently at random on sides AB and AC respectively,of equilateral triangle A ABC Which of the following intervals contains the probability that the area of A APQ is less than half the area of△ABC? 在等边三角形△ABC的边AB和AC上分别均匀随机且独立地选取，点P和Q.问下列 哪个区间包含三角形△APQ的面积小于三角形△ABC的面积一半的概率？ ω层引 ®(G引©( D)(层引@(6,1 Problem 21 Suppose that a1=2 and the sequence (an)satisfies the recurrence relation for all n2.What is the greatest integer less than or equal to n-1(n-1)+1 100 ∑2 n= 假设a1=2,且对所有n≥2，序列(an)满足递推关系2=a-t出 问不超过 n-1 (n-1)+1 n=1 的最大整数是多少？ (A)338,550 (B)338,551 (C)338,552 (D)338,553 (E)338,554 Problem 22 The figure below shows a dotted grid 8 cells wide and 3 cells tall consisting of 1"x1"squares.Carl places 1-inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself.The numbers in the cells indicate the number of sides of that square that are to be covered by toothpicks,and any number of toothpicks are allowed if no number is written. In how many ways can Carl place the toothpicks? 下图显示了由1英寸×1英寸的方格组成的，宽是8个方格，高是3个方格的虚 线网格.Cal要沿着一些方格的边放置长为1英寸的牙签，以形成一个不自相交 的闭合折线.方格中的数表示该方格被牙签覆盖的边的个数；对于没有数的方格， 其被牙签覆盖的边的个数没有限制。问Cl有多少种不同的放置牙签的方法？ 11 (A)130 (B)144 (C)146 (D)162 (E)196 Problem 23 What is the value of tantanttntntn 16 16 16 tam2毫tam20 问tam2gtan2g+tan2是tan2g+tan2语tam2g+tam2gtam2g的债 16 16 是多少？ (A)28 (B)68 (C)70 (D)72 (E)84 Problem 24 A disphenoid is a tetrahedron whose triangular faces are congruent to one another.What is the least total surface area of a disphenoid whose faces are scalene triangles with integer side lengths? 如果一个四面体的所有三角形面都互相全等，那么称该四面体为“等面四面 体”·问每个面的三边长都互不相同，并且是整数的等面四面体的最小表面积是 多少？ (A)V3 (B)3V15 (C)15 (D)15v7 (E)24V6 Problem 25 A graph is symmetric about a line if the graph remains unchanged after reflection in that line.For how many quadruples of integers(a,b,c,d), where lal5 and cand dare not both 0,is the graph ofy cx+d symmetric about the line y=x? 如果一个图像经过一条直线反射后仍保持不变，那么称此图像关于该直线“对 称”。俊得画数y=招的图像关于直线y=x对称，并且满足al.clId]≤ 5,c和d不同时为0的四元整数组(a,b,c,d)有多少个？ (A)1282 (B)1292 (C)1310 (D)1320 (E)1330 2024 AMC 12A Answer Key 题目 2 3 4 5 6 7 8 10 11 12 13 答案 A B B D D B D A E C D E D 题目 14 15 16 17 18 19 20 21 22 23 24 25 答案 C D C D A D D B C B D B