2023年AMC12A国际数学竞赛试卷(高三及以下学生)

2023AMC12A 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Bth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)25D)26 E)27 2.The weight of 1 of a large pizza together with 3 cups of orange slices is the same as the 3 weight of 3 ofa large pizza together with 1 cup oforange slices.A cup of orange slices weighs 4 1 of apound.What is the weight,pounds,of a large pizza? 一个大比萨饼的}与3上杯橙片合在一起的重量与一个大比萨饼的？与上杯橙片合在一起的 3 42 重量相同一杯橙片的重量是1磅问一个大比萨饼的重量是多少磅？ (A)14 B)2 (©22 D)3 回)3 5 5 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)10D)11 E)12 4.How many digits are in the base-ten representation of 85.510.155? 8.5°.15的十进制表示中有多少个数字？ (A)14B)15(C)16D)17E)18 5.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，并持续记录她掷出数的总和.问在某个时刻，她掷出 数的总和等于3的概率是多少？ 号®) (C25 D)17 E)13 216 108 54 6.Points A and B lie on the graph of y=log,x.The midpoint of AB is(6,2).What is thepositive difference between the x-coordinates of A and B? 点A和B位于y=logx的图像上.AB的中点是(6,2).问A和B的x坐标之差（大减小）是多 少？ (A)2W11B)45(C)8D)45E)9 7.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)5B)6(C)7D)8E)9 8.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l1 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)4B)5 (C)6①)7E)8 9.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边与较长的边之比是多少？ W号四2-5o5-5四-1 4 10.Positive real numbers x and y satisfy y3=x2 and (y-x)2=4y2.What is x +y? 正实数x和y满足y=x2和(y-x)2=4y2.问x+y是多少？ A)12B)18(C)24D)36E)42 11.What is the degree measure of the acute angle formed by lines with slopes 2 andI? 3 斜率为2和1的两条直线形成的锐角是多少度？ 3 (A)30 B)37.5(C)45①)52.5E60 12.What is the value of the below expression? 下面表达式的值是多少？ 23-13+43-33+63-53+.+183-17 (A)2023B)2679(C)2941D)3159E)3235 13.In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players,the number of games won by left-handed players was 40%more than the number of games won by right-handed players.(There wereno 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)45D)48 E)66 14.How many complex numbers satisfy the equation 33=z,where is the conjugate of the complex number z? 令乙表示复数z的共轭，问满足方程z=乙的复数有多少个？ (A)2 B)3(C)5D)6E)7 15.Usain is walking for exercise by zigzagging across a 100-meter by 30-meter rectangular field, beginning at point A and ending on the segment BC.He wants to increase the distance walked by zigzagging as shown in the figure below(APQRS).What angle 0=/PAB=ZOPC=ZROB=... will produce a length that is 120 meters?(The figure is not drawn to scale.Do not assume that the zigzag path has exactly four segments as shown;it could be more or fewer.) Usain沿折线步行穿过长100米、宽30米的矩形场地，从A点开始，到BC边结束.下图中 的APQRS是折线路径的示意图，他想尽量增加步行的距离当整个路径的长度为120米时， 角度O=∠PAB=∠QPC=∠RQB=.是怎样的？（图形未按比例绘制.不要假设折线路径正好有 如图所示的四段，它的段数可能更多或更少) 5 (A)arccos 6 (B)arccos (C)arccos 4 4 ,(D)arcsin-月 (E)arcsin 5 10 5 6 16.Consider the set of complex numbers z satisfying+=4.The maximum value of the imaginary part of z can be written in the form mwheremandnare relatively prime positive n integers.What is m+n? 考虑满足+z+z=4的复数z组成的集合z的虚部的最大值可以写成m的形式，其中m n 和n是互质的正整数.问mtn是多少？ (A)20B)21(C)22 D)23E)24 17.Flora the frog starts at 0 on the number line and makes a sequence of jumps to the right.In any one jump,independent of previous jumps,Flora leaps a positive integer distance m with probability 1.What is the probability that Fora will eventually and at 10? 2 青蛙Floa从数轴上的0开始，向右进行一系列跳跃.对于任何一次跳跃，它均与之前的跳跃 相互独立，Flora跳跃正整数距离m的概率是1问Flora会落到10的概率是多少2 2m (A)5 B)45 (C) 127 12 D)511 1024 1024 1024 四 18.Cireles Cand Ceach have radius1,and the distanee between their centers is CireleCis 2 the largest circle internally tangent to both Ci and C2.Circle C4 is internally tangent to both Ci and C2 and externally tangent to C3.What is the radius of C4? 圆C1和C2的半径均为1，圆心之间的距离为.圆C3是与C1和C2内切的最大圆.圆C4与C 2 和C2内切，并且与C4外切问C4的半径是多少？ C (A) (C) 10 19.What is the product of all solutions to the equation 以下的方程所有解的乘积是多少？ 10g7:2023.l0g2s9x2023=1og23.2023? (A)(1og2237.log223289)2(B)log20237.log223289(C)1 (D)10g,2023.log2s92023(回(1og,2023.1og2sg2023)2 20.Rows 1,2,3,4,and 5 of a triangular array of integers are shown below.Each row after the first row is formed by placing a I at each end of the row,and each interior entry is 1 greater than the sum of the two numbers diagonally above it in the previous row.What is the units digit of the sum of the 2023 numbers in the 2023rd row? 整数三角形数阵的第1,2,3,4,5行如下图所示.第一行之后的每一行都是通过在把1放在 行的两端，并让中间的每个数等于前一行与它沿斜线相邻的两个数的和再加1而形成的问 第2023行的2023个数的总和的个位数字是多少？ 3 1 55 1 171171 (A)1 B)3(C)5D)7E)9 21.If A and B are vertices of a polyhedron,define the distance d(A,B)to be the minimum number of edges of the polyhedron one must traverse in order to connect A and B.For example,if AB is an edge of the polyhedron,then d(A,B)=1,but if AC and CB are edges and AB is not an edge,then d(A,B)=2.Let Q,R,and S be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles).What is theprobability that d(Q,R)>dR, S)2 设A和B是多面体的顶点，距离"”d(A,B)定义为由多面体的边来形成连接A与B的通路所需 的最少边数.例如，若AB是多面体的边，则d(A,B)=1,但如果AC和CB是多面体的边，并且AB 不是，那么d(A,B)=2.令Q,R,S为随机选择的正二十面体（由20个等边三角形组成的正多面 体)的不同顶点.问d(Q,R)>dR,S)的概率是多少？ 22.Let fbe the unique function defined on the positive integers such that f-1 for all positive integers n,where the sum is taken over all positive divisors of n.What is f(2023)? 令f为定义在正整数上的唯一函数，使得对于所有正整数n, ≥dr 其中求和取遍n的所有正约数，问f2023)是多少？ (A)-1536B)96(C)108D)116(E)144 23.How many ordered pairs of positive real numbers (a,b)satisfy the equation 满足下列方程的有序正实数对(a,b)有多少个？ (1+2a)(2+2b)(2a+b)=32ab? (A)0 B)1 (C)2D)3 E)an infinite number无穷多个 24.Let K be the number of sequences A1,A2,...,An such that n is a positive integer less than or equal to 10.each Ai is a subset of {1.2.3...10).and Ai-is a subset of Ai for each i between 2 and n.inclusive.For example,()(5,7),(2,5,7),(2,5,7),(2,5,6,7,9)is one such sequence,with n=5.What is the remainder when K is divided by 10? 考虑序列A1,A2,Am,其中n是小于或等于10的正整数，每个A:是{1,2,3,10}的子集 并且对于2与n之间（包括2和n)的每个i,A1是A的子集.例如，{，{5,7}，{2,5,7}.{2,5,7}. {2,5,6,7,9}就是n=5时一个这样的序列设K是所有这样的序列的数量，问K除以10的余数 是多少？ (A)1 B)3(C)5D)7(@)9 25.There is a unique sequence of integers a,az,a3,..,azo23 such that whenever tan 2023x is defined.What is azo23? 存在唯一的整数序列a1,a2,a3,,a2023,使得以下等式在tan2023x有定义时都成立.问a2o23 是多少？ tan 2023x=tan tand tando tan 1+d tanx+da tanx+doz tan2022x (A)-2023(B)-2022(C)-1 D)1E)2023 2023 AMC 12A Answer Key 1.E 2.A 3.A 4.E 5.B 6.D 7.E 8.D 9.C 10.D 11.c 12.D 13.B 14.E 15.A 16.B 17.E 18.D 19.C 20.C 21.A 2.B 23.B 24.C 25.C