2024年AMC12B国际数学竞赛试卷（高三及以下学生）

2024 MAA AMC 12B Problem 1 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.问这个队伍中共有多少人？ (A)2021 (B)2022 (C)2023 (D)2024 (E)2025 Problem 2 What is10！-7I·6!? 问10！-7·6！是多少？ (A)-120 (B)0 (C)120 (D)600 (E)720 Problem 3 For how many integer values of x is2xl≤7π？ 满足2x|≤7π的整数值x有多少个？ (A)16 (B)17 (C)19 (D)20 (E)21 Problem 4 Balls numbered 1,2,3,..are deposited in 5 bins,labeled A,B,C,D,and E using the following procedure.Ball l 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，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.)问2024号球存放在哪个箱子中？ (A)A (B)B (C)C (D)D (E)E Problem 5 In the following expression,Melanie changed some of the plus signs to minus signs: 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? 在下面的表达式中，Melanie将一些加号改为减号： 1+3+5+7+…+97+99 当对新的表达式求值时，结果为负数.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 Problem 6 The national debt of the United States is on track to reach 5.1013 dollars by 2033.How many digits does this number of dollars have when written as a numeral in base 5?(The approximation of log10 5 as 0.7 is sufficient for this problem.) 美国的国债预计到2033年将达到5×1013美元.当这个数用5进制表示时，有多 少位数？（在本题中，只需使用近似值10g105≈0.7.) (A)18 (B)20 (C)22 (D)24 (E)126 Problem 7 In the figure below WXYZ is a rectangle with WX =4 and WZ=8.Point M lies on XY ,point Alies on YZ,and WMA is a right angle.The areas of A WXM and△WAZ are equal.What is the area of△WMA? 在矩形WXYZ中，WX=4且WZ=8.,点M在X亚上，，点A在Yz上，且∠WMA为 直角.三角形△WXM与△WAZ的面积相等.问△WMA的面积是多少？ W (A)13 (B)14 (C)15 (D)16 (E)17 Problem 8 What value of x satisfies log2x-l0g3x -2? log2x+l0g3x 求满足方程132*03=2的x的取值？ log2x+l0g3x (A)25 (B)32 (C)36 (D)42 (E)48 Problem 9 A dartboard is the region B in the coordinate plane consisting of points (x,y)such that xl+lyl <8.A target T is the region where (x2+y2- 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.where m and n are relatively prime positive integers.What is m+n? 在坐标平面上，由满足x|+y川≤8的，点(x,y)组成靶盘区域B.由(x2+y2 25)2≤49界定目标区域T.飞镖被投掷后随机的落在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.问m+n是多少？ (A)39 (B)71 (C)73 (D)75 (E)135 Problem 10 A list of 9 real numbers consists of 1,2.2,3.2,5.2,6.2,and 7,as well as x,y, and z with x sy sz.The range of the list is 7,and the mean and the median are both positive integers.How many ordered triples (x,y,z)are possible? 由9个实数组成的数据列表包含1,2.2,3.2,5.2,6.2,7，以及满足x≤y≤z 的数x,y,z,此数据列表的全距为7，且平均数和中位数都是正整数.问有 序三元组(x,y,Z)有多少种可能？ (A)1 B)2(C)3 (D)4 ()infinitely many|无穷多个 Problem 11 Let xn sin2(n).What is the mean of x1,x2,x3,,x90? 设xn=sin2(n°)，问x1,x2,x3,…,x90的均值是多少？ 岩 (®)器 (C)180 89 D) (18 1 Problem 12 Suppose 3 is a complex number with positive imaginary part,with real part greater than 1,and with I=2.In the complex plane,the four points 0,3,32, and 33 are the vertices of a quadrilateral with area 15.What is the imaginary part of 3? 假设复数z的虚部为正，实部大于1，且|=2.在复平面上，以四个值0,3， z2,3为顶，点构成的四边形的面积为15.问3的虚部是多少？ (A子 (B)1 (C) D Problem 13 There are real numbers x,y,h and k that satisfy the system of equations x2+y2-6x-8y=h x2+y2-10x+4y=k What is the minimum possible value of h+k? 已知存在实数x,y,h,k满足下面的方程组.问h十k的最小可能值是多少？ x2+y2-6x-8y=h x2+y2-10x+4y=k (A)-54 (B)-46 (C)-34 (D)-16 (E)16 Problem 14 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 Problem 15 A triangle in the coordinate plane has vertices A(log2 1,log2 2), B(log2 3,log2 4),and C(log2 7,log2 8).What is the area of ABC? 在坐标平面上有一个三角形，其顶点为A(1og21,1og22),A(Iog21,log22), CIog27,log28).问△ABC的面积是多少？ A1ogz号 3 (B)1og2 (C)log2 7 D)l1o82方 国1g青 Problem 16 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? 现有一组共16人将被分成4个不加区别的委员会，每个委员会有4人，每个委 员会将有一位主席和一位秘书，所有不同的分配方法的总数可以写成3M,其中 r和M是正整数，且M不能被3整除.问r是多少？ (A)5 (B)6 (C)7 (D)8 (E)9 Problem 17 Integers a and b are randomly chosen without replacement from the set of integers with absolute value not exceeding 10.What is the probability that the polynomial x3+ax2 bx+6 has 3 distinct integer roots? 从绝对值不超过10的整数集合中随机选取两个不同的整数α和b,问多项式x3+ ax2+bx+6恰好有3个不同的整数根的概率是多少？ Problem 18 The Fibonacci numbers are defined by F1=1,F2 =1,and F=Fn-1+ B-2iorn≥3.What is景+管+号+-+e? F10 Fibonacci数列定义为R=1,B=1,且对于n≥3，R=Rn-1+R-2,问2+ +是十…十器是多少？ F2 F10 (A)318 (B)319 (C)320 (D)321 (E)322 Problem 19 Equilateral A ABC with side length 14 is rotated about its center by angle 0, where0<日<60°，to form△DEF.See the figure.The area of hexagon ADBECF is 91v3.What is tan0? 如图所示，边长为14的等边三角形△ABC绕中心旋转角度0形成△DEF,其中 0<0<60°.若六边形ADBECF的面积为91V3,问tan0的值是多少？ B (E) 73 13 Problem 20 Suppose A,B,and C are points in the plane with AB=40 and AC=42, and let x be the length of the line segment from A to the midpoint of BC. Define a function fby letting f(x)be the area of A ABC.Then the domain of f is an open interval (p,q),and the maximum value r of f(x)occurs at x=s.What is p+q+r+s? 假设平面上的点A,B,C满足AB=40且AC=42,令x为连接A和BC中，点的线 段的长度.定义函数f,使得f(x)表示△ABC的面积.若f的定义域是开区间(P,q), 且∫(x)的最大值r在x=s处取得.问p+q+r+s的值是多少？ (A)909 (B)910 (C)911 (D)912 (E)913 Problem 21 The measures of the smallest angles ofthree different right triangles sum to 90. All three triangles have side lengths that are primitive Pythagorean triples.Two of them are 3-4-5 and 5-12-13.What is the perimeter of the third triangle? 三个不同的直角三角形中最小角的角度之和为90°，这三个三角形的边长均为互 质的勾股三元数组.其中两个三角形的边长分别为3-4-5和5-12-13.问第 三个三角形的周长是多少？ (A)40 (B)126 (C)154 (D)176 (E)208 Problem 22 Let A ABC be a triangle with integer side lengths and the property that ZB= 22A.What is the least possible perimeter of such a triangle? 设△ABC是边长均为整数的三角形，且满足∠B=2LA.求满足条件的三角形的 最小可能周长？ (A)13 (B)14 (C)15 (D)16 (E17 Problem 23 A right pyramid has regular octagon ABCDEFGH with side length 1 as its base and apex V.Segments AV and DV are perpendicular.What is the square of the height of the pyramid? 正棱维的底面是边长为1的正八边形ABCDEFGH,项，点是V.线段AV和DV相互 垂直，问这个棱锥的高的平方是多少？ (A)1 (B) 1+v2 (C)V2 (D) (E) 2+V2 2 Problem 24 What is the number of ordered triples (a,b,c)of positive integers,with a s b≤c≤9，such that there exists a(non-degenerate)triangle△ABC with an integer inradius for which a,b,and c are the lengths of the altitudes from A to BC,B to AC,and C to AB,respectively?(Recall that the inradius of a triangle is the radius of the largest possible circle that can be inscribed in the triangle.) 考虑有序三元组(a,b,c):a,b,c均为正整数，且a≤b≤c≤9，存在一个（非 退化)的三角形△ABC,其内切圆半径为整数，且a,b,c分别为从A到BC,从 B到AC,从C到AB的高的长度.问具有上述性质的有序三元组（α，b,c)有多少个？ (注意：三角形的内切圆半径是指可以内接于该三角形的最大圆的半径.) (A)2 (B)3 (C)4 (D)5 E)6 Problem 25 Pablo will decorate each of 6 identical white balls with either a striped or a dotted pattern,using either red or blue paint.He will decide on the color and pattern for each ball by flipping a fair coin for each of the 12 decisions he must make.After the paint dries,he will place the 6 balls in an urn.Frida will randomly select one ball from the urn and note its color and pattern.The events "the ball Frida selects is red"and "the ball Frida selects is striped"may or may not be independent,depending on the outcome of Pablo's coin flips.The probability that these two events are independent can be written aswhere mand narerelatively prime positiveintegers.What is m(Recall that two events A and B are independent if P(A and B)=P(A).P(B). Pblo将装饰6个相同的白球，每个球都会用红色或蓝色油漆涂上条纹或圆，点图 案，他将通过抛掷均匀硬币的方式来决定每个球的颜色和图案，总共需要做12 次决定.油漆千燥后，他将这6个球放入罐中.Fida将从罐中随机选取一个球， 并记录其颜色和图案.事件「Fida选中的球是红色的」和「Frida选中的球是条 纹的」可能相互独立，也可能不独立，这取决于Pblo抛硬币的结果.这两个事 件相互独立的概率可以表示为只，其中m和肌互质。问m的值是多少？（注意： 两个事件A和B相互独立，当且仅当P(A和B同时发生)=P(A)·P(B).) (A)243 (B)245 (C)247 (D)249 (E)251 2024 AMC 12B Answer Key 题目 2 3 4 5 6 7 8 9 10 11 12 13 答案 B B E D B B C C B E D 题月 14 15 16 17 18 19 20 21 22 23 24 25 答案 B B C B B B B