2018 AMC amc12b 真题 答案 详解

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中英双语真题

2018 AMC 12B

Problem 1

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 烤了一块 20 英寸乘以 18 英寸的玉米面包，这块玉米面包被切成 2 英寸乘以 2 英寸的多个小块面包。问这大块玉米面包包含多少块小面包？

(A) 90 (B) 100 (C) 180 (D) 200 (E) 360

Problem 2

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 分钟内开了 96 英里。他在前 30 分钟内的平均速度为 60 英里每小时，在第二个 30 分钟内的平均速度为 65 英里每小时。他在最后 30 分钟内的平均速度是多少英里每小时？

(A) 64 (B) 65 (C) 66 (D) 67 (E) 68

Problem 3

A line with slope 2 intersects a line with slope 6 at the point (40, 30). What is the distance between the x-intercepts of these two lines?

一条斜率为 2 的直线和一条斜率为 6 的直线交于点 (40, 30)。那么这两条直线的 x 截距的距离是多少？

(A) 5 (B) 10 (C) 20 (D) 25 (E) 50

Problem 4

A circle has a chord of length 10, and the distance from the center of the circle to the chord is 5. What is the area of the circle?

一个圆的一条弦长为 10，圆心到弦的距离是 5，那么这个圆的面积是多少？

(A) 25π (B) 50π (C) 75π (D) 100π (E) 125π

英文真题

2018 AMC 12B Problems

Problem 1

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?

(A) 90 (B) 100 (C) 180 (D) 200 (E) 360

Problem 2

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?

(A) 64 (B) 65 (C) 66 (D) 67 (E) 68

Problem 3

A line with slope 2 intersects a line with slope 6 at the point (40, 30). What is the distance between the x-intercepts of these two lines?

(A) 5 (B) 10 (C) 20 (D) 25 (E) 50

Problem 4

A circle has a chord of length 10, and the distance from the center of the circle to the chord is 5. What is the area of the circle?

(A) 25π (B) 50π (C) 75π (D) 100π (E) 125π

Problem 5

How many subsets of {2, 3, 4, 5, 6, 7, 8, 9} contain at least one prime number?

(A) 128 (B) 192 (C) 224 (D) 240 (E) 256

Problem 6

Suppose S cans of soda can be purchased from a vending machine for Q quarters. Which of the following expressions describes the number of cans of soda that can be purchased for D dollars, where 1 dollar is worth 4 quarters?

(A) (\frac{4DQ}{S}) (B) (\frac{4DS}{Q}) (C) (\frac{4Q}{DS}) (D) (\frac{DQ}{4S}) (E) (\frac{DS}{4Q})

Problem 7

What is the value of

[ \log_3 7 \cdot \log_5 9 \cdot \log_7 11 \cdot \log_9 13 \cdots \log_{21} 25 \cdot \log_{23} 27? ]

(A) 3 (B) (3 \log_7 23) (C) 6 (D) 9 (E) 10

Problem 8

Line segment 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 △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?

(A) 25 (B) 38 (C) 50 (D) 63 (E) 75

Problem 9

What is

[ \sum_{i=1}^{100} \sum_{j=1}^{100} (i+j)? ]

真题文字详解(英文)

Problem 1 The following problem is from both the 2018 AMC 12B #1 and 2018 AMC 10B #1, so both problems redirect to this page. 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? (A) 90 (B) 100 (C) 180 (D) 200 (E) 360 Solution 1 The area of the pan is $20\cdot18=360$. Since the area of each piece is $2\cdot2=4$, there are $\frac{360}{4}=\textbf{(A)}\ 90$ pieces. Solution 2 By dividing each of the dimensions by 2, we get a $10\times9$ grid that makes \textbf{(A)} 90 pieces. Problem 2 The following problem is from both the 2018 AMC 12B #2 and 2018 AMC 10B #2, so both problems redirect to this page. 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? (A) 64 (B) 65 (C) 66 (D) 67 (E) 68 Solution 1 Suppose that Sam's average speed during the last 30 minutes was $x$ mph. Recall that a half hour is equal to 30 minutes. Therefore, Sam drove $60\cdot0.5=30$ miles during the first half hour, $65\cdot0.5=32.5$ miles during the second half hour, and $x\cdot0.5$ miles during the last half hour. We have $$30+32.5+x\cdot0.5=96$$ $$x\cdot0.5=33.5$$ $$x=\textbf{(D)}\ 67.$$ ~Haha0201 ~MRENTHUSIASM Solution 2 Suppose that Sam's average speed during the last 30 minutes was $x$ mph. Note that Sam's average speed during the entire trip was $\frac{96}{3/2}=64$ mph. Since Sam drove at 60 mph, 65 mph, and $x$ mph for the same duration (30 minutes each), his average speed during the entire trip was the average of 60 mph, 65 mph, and $x$ mph. We have $$\frac{60+65+x}{3}=64$$ $$60+65+x=192$$ $$x=\textbf{(D)}\ 67.$$ ~coolmath_2018 ~MRENTHUSIASM Problem 3 A line with slope 2 intersects a line with slope 6 at the point $(40,30)$. What is the distance between the $x$-intercepts of these two lines? (A) 5 (B) 10 (C) 20 (D) 25 (E) 50 Solution 1 Using point-slope form, we get the equations $y-30=6(x-40)$ and $y-30=2(x-40)$. Simplifying, we get $6x-y=210$ and $2x-y=50$. Letting $y=0$ in both equations and solving for $x$ gives the $x$-intercepts: $x=35$ and $x=25$, respectively. Thus the distance between them is $35-25=\textbf{(B)}\ 10$. Solution 2 In order for the line with slope 2 to travel "up" 30 units (from $y=0$), it must have traveled $30/2=15$ units to the right. Thus, the $x$-intercept is at $x=40-15=25$. As for the line with slope 6, in order for it to travel "up" 30 units it must have traveled $30/6=5$ units to the right. Thus its $x$-intercept is at $x=40-5=35$. Then the distance between them is $35-25=\textbf{(B)}\ 10$. Problem 4 A circle has a chord of length 10, and the distance from the center of the circle to the chord is 5. What is the area of the circle? (A) $25\pi$ (B) $50\pi$ (C) $75\pi$ (D) $100\pi$ (E) $125\pi$ Solution

真题文字详解(中英双语)

Problem 1 The following problem is from both the 2018 AMC 12B #1 and 2018 AMC 10B #1, so both problems redirect to this page. 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? (A) 90 (B) 100 (C) 180 (D) 200 (E) 360 Solution 1 The area of the pan is $20\cdot18=360$. Since the area of each piece is $2\cdot2=4$, there are $\frac{360}{4}=\textbf{(A)}\ 90$ pieces. Solution 2 By dividing each of the dimensions by 2, we get a $10\times9$ grid that makes \textbf{(A)} 90 pieces. Problem 2 The following problem is from both the 2018 AMC 12B #2 and 2018 AMC 10B #2, so both problems redirect to this page. 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? (A) 64 (B) 65 (C) 66 (D) 67 (E) 68 Solution 1 Suppose that Sam's average speed during the last 30 minutes was $x$ mph. Recall that a half hour is equal to 30 minutes. Therefore, Sam drove $60\cdot0.5=30$ miles during the first half hour, $65\cdot0.5=32.5$ miles during the second half hour, and $x\cdot0.5$ miles during the last half hour. We have $$30+32.5+x\cdot0.5=96$$ $$x\cdot0.5=33.5$$ $$x=\textbf{(D)}\ 67.$$ ~Haha0201 ~MRENTHUSIASM Solution 2 Suppose that Sam's average speed during the last 30 minutes was $x$ mph. Note that Sam's average speed during the entire trip was $\frac{96}{3/2}=64$ mph. Since Sam drove at 60 mph, 65 mph, and $x$ mph for the same duration (30 minutes each), his average speed during the entire trip was the average of 60 mph, 65 mph, and $x$ mph. We have $$\frac{60+65+x}{3}=64$$ $$60+65+x=192$$ $$x=\textbf{(D)}\ 67.$$ ~coolmath_2018 ~MRENTHUSIASM Problem 3 A line with slope 2 intersects a line with slope 6 at the point $(40,30)$. What is the distance between the $x$-intercepts of these two lines? (A) 5 (B) 10 (C) 20 (D) 25 (E) 50 Solution 1 Using point-slope form, we get the equations $y-30=6(x-40)$ and $y-30=2(x-40)$. Simplifying, we get $6x-y=210$ and $2x-y=50$. Letting $y=0$ in both equations and solving for $x$ gives the $x$-intercepts: $x=35$ and $x=25$, respectively. Thus the distance between them is $35-25=\textbf{(B)}\ 10$. Solution 2 In order for the line with slope 2 to travel "up" 30 units (from $y=0$), it must have traveled $30/2=15$ units to the right. Thus, the $x$-intercept is at $x=40-15=25$. As for the line with slope 6, in order for it to travel "up" 30 units it must have traveled $30/6=5$ units to the right. Thus its $x$-intercept is at $x=40-5=35$. Then the distance between them is $35-25=\textbf{(B)}\ 10$. Problem 4 A circle has a chord of length 10, and the distance from the center of the circle to the chord is 5. What is the area of the circle? (A) $25\pi$ (B) $50\pi$ (C) $75\pi$ (D) $100\pi$ (E) $125\pi$