2018 AMC amc8 真题 答案 详解

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中英双语真题

2018 AMC8

Problem 1
An amusement park has a collection of scale models, with ratio (1:20), of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its replica to the nearest whole number?
一个游乐公园有一系列美国国内建筑和其他景观的模型，与实际建筑的比率为(1:20)。美国国会大厦的实际高为289英尺，那么它的模型复制品的高度是多少英尺？结果四舍五入到整数。

(A) 14 (B) 15 (C) 16 (D) 18 (E) 20

Problem 2
What is the value of the product
[ \left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right)? ]
下式的值为多少
[ \left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right)? ]

(A) (\frac{7}{6}) (B) (\frac{4}{3}) (C) (\frac{7}{2}) (D) 7 (E) 8

Problem 3
Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then Bob, and so forth. When the number contains a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and the counting continues. Who is the last one present in the circle?

Am, Bob, Cyd, Dan, Eve 和 Fon 以这样的顺序围成一圈。他们开始数数：Arn 先开始，然后 Bob，以此类推。当数字包含数字 7（例如 47）或者是 7 的倍数时，那个人就离开圆圈，数数继续进行。最后一个离开圆圈的是谁？

(A) Arn (B) Bob (C) Cyd (D) Dan (E) Eve

英文真题

Compiled on: January 1, 2021 AMC8 2018 Problems Page 14 of 109 Q1. An amusement park has a collection of scale models, with a ratio 1 : 20, of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its duplicate to the nearest whole number? A) 14 B) 15 C) 16 D) 18 E) 20 Q2. What is the value of the product (\left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right)?) A) (\frac{7}{6}) B) (\frac{4}{3}) C) (\frac{7}{2}) D) 7 E) 8 Q3. Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then Bob, and so forth. When the number contains a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and the counting continues. Who is the last one present in the circle? A) Arn B) Bob C) Cyd D) Dan E) Eve Q4. The twelve-sided figure shown has been drawn on 1 cm x 1 cm graph paper. What is the area of the figure in cm²? A) 12 B) 12.5 C) 13 D) 13.5 E) 14 Q5. What is the value of (1+3+5+\cdots +2017+2019-2-4-6-\cdots -2016-2018?) A) -1010 B) -1009 C) 1008 D) 1009 E) 1010 Q6. On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the highway as on the coastal road. If Anh spent 30 minutes driving on the coastal road, how many minutes did his entire trip take? A) 50 B) 70 C) 80 D) 90 E) 100 Q7. The 5-digit number 2018U is divisible by 9. What is the remainder when this number is divided by 8? A) 1 B) 3 C) 5 D) 6 E) 7 Q8. Mr. Garcia asked the members of his health class how many days last week they exercised for at least 30 minutes. The results are summarized in the following bar graph, where the heights of the bars represent the number of students.

真题文字详解(英文)

Problem 1 An amusement park has a collection of scale models, with a ratio of $1:20$ of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its duplicate to the nearest whole number?
(A) 14 (B) 15 (C) 16 (D) 18 (E) 20

Solution 1 (Good 'Ol' Ratios) You can see that since the ratio of real building's heights to the model building's height is $1:20$. We also know that the U.S. Capitol is 289 feet in real life, so to find the height of the model, we divide by 20. That gives us $14.45$ which rounds to 14. Therefore, to the nearest whole number, the duplicate is (A) 14.

Solution 2 We can compute $\frac{289}{20}$ and round our answer to get (A) 14. It is basically Solution 1 without the ratio calculation. However, Solution 1 is referring further to the problem.

Solution 3 We know that $20 \cdot 14 = 280$, and that $20 \cdot 15 = 300$. These are the multiples of 20 around 289, and the closest one of those is 280. Therefore, the answer is $\frac{280}{20} =$ (A) 14.

Problem 2 What is the value of the product
$\left(1+\frac11\right)\cdot\left(1+\frac12\right)\cdot\left(1+\frac13\right)\cdot\left(1+\frac14\right)\cdot\left(1+\frac15\right)\cdot\left(1+\frac16\right)$ ?
(A) $\frac76$ (B) $\frac43$ (C) $\frac72$ (D) 7 (E) 8

Solution By adding up the numbers in each of the 6 parentheses, we get:
$\begin{array}{ccccccc} 2 & 3 & 4 & 5 & 6 & 7 \ 1 & 2 & 3 & 4 & 5 & 6 \end{array}$
Using telescoping, most of the terms cancel out diagonally. We are left with $\frac71$ which is equivalent to 7. Thus, the answer would be (D) 7.

Also, you can make the fractions $\frac{7!}{6!}$, which also yields (D) 7.

Problem 3 Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then Bob, and so forth. When the number contains a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and the counting continues. Who is the last one present in the circle?
(A) Arn (B) Bob (C) Cyd (D) Dan (E) Eve

Solution 1 (Simulation) The five numbers which cause people to leave the circle are 7, 14, 17, 21, and 27. The most straightforward way to do this would be to draw out the circle with the people, and cross off people as you count. Assuming the six people start with 1, Arn counts 7 so he leaves first. Then Cyd counts 14 as there are 7 numbers to be counted from this point. Then, Fon, Bob, and Eve count 17, 21, and 27, respectively, so the last one standing is Dan. Hence, the answer would be (D) Dan.

Solution 2 Assign each person a position: Arn is in position 1, Bob in 2, etc. Note that if there are n people standing in a circle, a person will say a number that is congruent to their position modulo n. It follows that if there are k numbers to be said (with someone leaving on the kth), the person that leaves will stand in position $k \equiv 1 \pmod{n}$. The five numbers which cause people to leave the circle are 7, 14, 17, 21, and 27. Since there are initially 6 people in the circle, so the person who leaves stands in position $7 \equiv 1 \pmod{6}$ or position 1. Arn now leaves, and because Bob is the next person to say a number, we reassign the positions such that Bob is in position 1, Cyd in 2, and Fon in 5. There are 7 more numbers to be said, and someone leaves on the 7th. It follows that the person who leaves stands in position $7 \equiv 2 \pmod{5}$.

真题文字详解(中英双语)

Problem 1 An amusement park has a collection of scale models, with a ratio of (1:20) of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its duplicate to the nearest whole number?
(A) 14 (B) 15 (C) 16 (D) 18 (E) 20

Solution 1 (Good Ol' Ratios)
解法 1（老派的比率法） You can see that since the ratio of real building's heights to the model building's height is (1:20). We also know that the U.S Capitol is 289 feet in real life, so to find the height of the model, we divide by 20. That gives us 14.45 which rounds to 14. Therefore, to the nearest whole number, the duplicate is (\boxed{A}14).

可以看到，实际建筑的高度与模型建筑的高度之比为(1:20)。我们也知道美国国会大厦的实际高度为289英尺，因此要找出模型的高度，我们需要将其除以20。这样我们得到14.45，四舍五入后为14。因此，保留整数，复制品的高度为(\boxed{A}14)。

Solution 2
We can compute (\frac{289}{20}) and round our answer to get (\boxed{A}14) It is basically Solution 1 without the ratio calculation. However, Solution 1 is referring further to the problem.

我们可以计算出(\frac{289}{20})，并将答案四舍五入到(\boxed{A}14)。这基本上是解法1，但省略了比例计算。然而，解法1进一步指出了这个问题。

Solution 3
We know that (20 \cdot 14 = 280), and that (20 \cdot 15 = 300). These are the multiples of 20 around 289, and the closest one of those is 280. Therefore, the answer is (\frac{280}{20}=\boxed{A}14)

我们知道 (20 \cdot 14 = 280)，和 (20 \cdot 15 = 300)。它们是20在289附近的倍数，其中最接近的一个是280。因此，答案是(\frac{280}{20}=\boxed{A}14)。

Problem 2 What is the value of the product
这个乘积的值是多少
[ \left(1+\frac11\right)\cdot\left(1+\frac12\right)\cdot\left(1+\frac13\right)\cdot\left(1+\frac14\right)\cdot\left(1+\frac15\right)\cdot\left(1+\frac16\right)? ] (A) (\frac76) (B) (\frac43) (C) (\frac72) (D) 7 (E) 8

Solution
By adding up the numbers in each of the 6 parentheses, we get: 通过将每个括号内的数字加起来，我们得到： [ \begin{array}{ccccccc} 2 & 3 & 4 & 5 & 6 & 7 \ 1 & 2 & 3 & 4 & 5 & 6 \end{array} ]

Using telescoping, most of the terms cancel out diagonally. We are left with (\frac{7}{1}) which is equivalent to 7. Thus, the answer would be (\boxed{D}7)

使用望远镜法，大部分项沿对角线抵消。我们剩下的是(\frac{7}{1})，它相当于7。因此，答案是(\boxed{D}7)。

Also, you can make the fractions (\frac{7!}{6!}), which also yields (\boxed{D}7)

此外，你也可以将分数化为(\frac{7!}{6!})，这也得到(\boxed{D}7)。