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AMC10 2018 B

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AMC10 · 2018 (B)

Q1
Kate bakes a 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain?
Kate烤了一个20英寸×18英寸的玉米面包盘。玉米面包被切成2英寸×2英寸的块。这个盘子包含多少块玉米面包？
Q2
Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes?
Sam在90分钟内开车96英里。他前30分钟的平均速度是60 mph（英里每小时），第二30分钟的平均速度是65 mph。最后30分钟的平均速度是多少mph？
Q3
In the expression \(( \times ) + ( \times )\) each blank is to be filled in with one of the digits 1, 2, 3, or 4, with each digit being used once. How many different values can be obtained?
在表达式\(( \times ) + ( \times )\)中，每个空白处填入数字1、2、3或4各一次。能得到多少不同的值？
Q4
A three-dimensional rectangular box with dimensions \(X\), \(Y\), and \(Z\) has faces whose surface areas are 24, 24, 48, 48, 72, and 72 square units. What is \(X + Y + Z\)?
一个三维长方体盒子尺寸为\(X\)、\(Y\)和\(Z\)，各个面的表面积为24、24、48、48、72和72平方单位。求\(X + Y + Z\)？
Q5
How many subsets of \(\lbrace2, 3, 4, 5, 6, 7, 8, 9\rbrace\) contain at least one prime number?
集合\(\lbrace2, 3, 4, 5, 6, 7, 8, 9\rbrace\)有多少个子集至少包含一个质数？
Q6
A box contains 5 chips, numbered 1, 2, 3, 4, and 5. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds 4. What is the probability that 3 draws are required?
一个盒子里有5个标有数字1、2、3、4和5的筹码。随机依次不放回地抽取筹码，直到抽取的数字之和超过4。需要3次抽取的概率是多少？
Q7
In the figure below, \(N\) congruent semicircles are drawn along a diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let \(A\) be the combined area of the small semicircles and \(B\) be the area of the region inside the large semicircle but outside the small semicircles. The ratio \(A : B\) is \(1 : 18\). What is \(N\)?
如下图，沿着一个大半圆的直径画了$N$个全等的半圆，它们的直径覆盖了大半圆的直径，没有重叠。设$A$为小半圆的总面积，$B$为大半圆内部但小半圆外部的区域面积。比例$A:B=1:18$。$N$是多少？
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Q8
Sara makes a staircase out of toothpicks as shown: [This is a 3-step staircase and uses 18 toothpicks.] How many steps would be in a staircase that used 180 toothpicks?
Sara用牙签搭建了一个楼梯，如图所示：[这是一个3级楼梯，使用了18根牙签。] 使用180根牙签的楼梯有多少级？
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Q9
The faces of each of 7 standard dice are labeled with the integers from 1 to 6. Let \(p\) be the probability that when all 7 dice are rolled, the sum of the numbers on the top faces is 10. What other sum occurs with the same probability \(p\)?
有7个标准骰子，每个面标有1到6的整数。设$p$为当所有7个骰子掷出时，顶面数字之和为10的概率。哪一个其他和也以相同的概率$p$出现？
Q10
In the rectangular parallelepiped shown, \(AB = 3\), \(BC = 1\), and \(CG = 2\). Point \(M\) is the midpoint of \(\overline{FG}\). What is the volume of the rectangular pyramid with base \(BCHE\) and apex \(M\)?
在所示的直角平行六面体中，$AB=3$，$BC=1$，$CG=2$。点$M$是$\overline{FG}$的中点。底面$BCHE$、顶点$M$的直角锥体积是多少？
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Q11
Which of the following expressions is never a prime number when \(p\) is a prime number?
当 \(p\) 是质数时，下列哪个表达式永远不是质数？
Q12
Line segment \(\overline{AB}\) is a diameter of a circle with AB = 24. Point C, not equal to A or B, lies on the circle. As point C moves around the circle, the centroid (center of mass) of \(\triangle ABC\) traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?
线段 \(\overline{AB}\) 是圆的直径，AB = 24。点 C（不等于 A 或 B）位于圆上。当点 C 在圆周上移动时，\(\triangle ABC\) 的质心（形心）描出一条缺少两点的闭合曲线。该曲线围成的区域面积的四舍五入到最近正整数是多少？
Q13
How many of the first 2018 numbers in the sequence 101, 1001, 10001, ..., are divisible by 101?
数列 101、1001、10001、... 的前 2018 项中，有多少个数能被 101 整除？
Q14
A list of 2018 positive integers has a unique mode, which occurs exactly 10 times. What is the least number of distinct values that can occur in the list?
一个包含 2018 个正整数的列表有一个唯一众数，该众数恰好出现 10 次。该列表中可能出现的最少不同值的个数是多少？
Q15
A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point A in the figure on the right. The box has base length \(w\) and height \(h\). What is the area of the sheet of wrapping paper?
一个底部为正方形的闭合盒子要用一张正方形包装纸包裹。盒子置于包装纸中央，底面的顶点位于包装纸正方形边中点处（如左图所示）。包装纸的四个角要向上折叠覆盖盒子侧面，并在盒子顶面中心点 A（右图）相遇。盒子底边长 \(w\)，高 \(h\)。包装纸的面积是多少？
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Q16
Let \(a_1, a_2, \dots, a_{2018}\) be a strictly increasing sequence of positive integers such that \[a_1 + a_2 + \cdots + a_{2018} = 2018^{2018}.\] What is the remainder when \(a_1^3 + a_2^3 + \cdots + a_{2018}^3\) is divided by 6?
设 \(a_1, a_2, \dots, a_{2018}\) 是一个严格递增的正整数序列，使得 \[a_1 + a_2 + \cdots + a_{2018} = 2018^{2018}.\] 当 \(a_1^3 + a_2^3 + \cdots + a_{2018}^3\) 除以 6 的余数是多少？
Q17
In rectangle \(PQRS\), \(PQ = 8\) and \(QR = 6\). Points \(A\) and \(B\) lie on \(\overline{PQ}\), points \(C\) and \(D\) lie on \(\overline{QR}\), points \(E\) and \(F\) lie on \(\overline{RS}\), and points \(G\) and \(H\) lie on \(\overline{SP}\) so that \(AP = BQ < 4\) and the convex octagon \(ABCDEFEGH\) is equilateral. The length of a side of this octagon can be expressed in the form \(k + m\sqrt{n}\), where \(k, m,\) and \(n\) are integers and \(n\) is not divisible by the square of any prime. What is \(k+m+n\)?
在矩形 \(PQRS\) 中，\(PQ = 8\)，\(QR = 6\)。点 \(A\) 和 \(B\) 在 \(\overline{PQ}\) 上，点 \(C\) 和 \(D\) 在 \(\overline{QR}\) 上，点 \(E\) 和 \(F\) 在 \(\overline{RS}\) 上，点 \(G\) 和 \(H\) 在 \(\overline{SP}\) 上，使得 \(AP = BQ < 4\)，且凸八边形 \(ABCDEFEGH\) 是等边的。这个八边形的边长可以表示为 \(k + m\sqrt{n}\) 的形式，其中 \(k, m, n\) 是整数，且 \(n\) 不能被任何质数的平方整除。求 \(k+m+n\)？
Q18
Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip?
三个不同家庭的年轻兄弟姐妹对需要乘坐一辆面包车旅行。这六个孩子将占据面包车的第二排和第三排，每排有三个座位。为了避免干扰，同一排中兄弟姐妹不得紧挨着坐，且没有孩子可坐在其兄弟姐妹正前方。有多少种座位安排方式？
Q19
Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe’s age will be an integral multiple of Zoe’s age. What will be the sum of the two digits of Joey’s age the next time his age is a multiple of Zoe’s age?
乔伊、克洛伊和他们的女儿佐伊都有相同的生日。乔伊比克洛伊大 1 岁，佐伊今天正好 1 岁。今天是克洛伊年龄是佐伊年龄整数倍的 9 个生日中的第一个。下次乔伊年龄是佐伊年龄整数倍时，乔伊年龄的两数字之和是多少？
Q20
A function \(f\) is defined recursively by \(f(1) = f(2) = 1\) and \[f(n) = f(n-1) - f(n-2) + n\] for all integers \(n \geq 3\). What is \(f(2018)\)?
函数 \(f\) 通过如下递推定义：\(f(1) = f(2) = 1\)，且 \[f(n) = f(n-1) - f(n-2) + n\] 对所有整数 \(n \geq 3\)。求 \(f(2018)\)？
Q21
Mary chose an even 4-digit number \(n\). She wrote down all the divisors of \(n\) in increasing order from left to right: 1, 2, ..., \(n/2\), \(n\). At some moment Mary wrote 323 as a divisor of \(n\). What is the smallest possible value of the next divisor written to the right of 323?
Mary 选择了一个偶数四位数 \(n\)。她将 \(n\) 的所有除数按从小到大的顺序从左到右写下：1, 2, ..., \(n/2\), \(n\)。在某个时刻 Mary 写下了 323 作为 \(n\) 的一个除数。323 右侧下一个除数的最小可能值为多少？
Q22
Real numbers \(x\) and \(y\) are chosen independently and uniformly at random from the interval \([0, 1]\). Which of the following numbers is closest to the probability that \(x\), \(y\), and 1 are the side lengths of an obtuse triangle?
实数 \(x\) 和 \(y\) 从区间 \([0, 1]\) 中独立均匀随机选择。其中哪一个数最接近 \(x\)、\(y\) 和 1 作为钝三角形边长的概率？
Q23
How many ordered pairs \((a, b)\) of positive integers satisfy the equation \[a \cdot b + 63 = 20 \cdot \operatorname{lcm}(a, b) + 12 \cdot \operatorname{gcd}(a, b),\] where \(\operatorname{gcd}(a, b)\) denotes the greatest common divisor of \(a\) and \(b\), and \(\operatorname{lcm}(a, b)\) denotes their least common multiple?
有多少个正整数有序对 \((a, b)\) 满足方程 \[a \cdot b + 63 = 20 \cdot \operatorname{lcm}(a, b) + 12 \cdot \operatorname{gcd}(a, b),\] 其中 \(\operatorname{gcd}(a, b)\) 表示 \(a\) 和 \(b\) 的最大公约数，\(\operatorname{lcm}(a, b)\) 表示它们的最小公倍数？
Q24
Let \(ABCDEF\) be a regular hexagon with side length 1. Denote by \(X\), \(Y\), and \(Z\) the midpoints of sides \(AB\), \(CD\), and \(EF\), respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of \(\triangle ACE\) and \(\triangle XYZ\)?
设 \(ABCDEF\) 是边长为 1 的正六边形。设 \(X\)、\(Y\) 和 \(Z\) 分别是边 \(AB\)、\(CD\) 和 \(EF\) 的中点。凸六边形的面积是多少，其内部是 \(\triangle ACE\) 和 \(\triangle XYZ\) 内部的交集？
Q25
Let \(\lfloor x \rfloor\) denote the greatest integer less than or equal to \(x\). How many real numbers \(x\) satisfy the equation \[x^2 + 10{,}000 \lfloor x \rfloor = 10{,}000 x\]?
设 \(\lfloor x \rfloor\) 表示不超过 \(x\) 的最大整数。有多少个实数 \(x\) 满足方程 \[x^2 + 10{,}000 \lfloor x \rfloor = 10{,}000 x\]？
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