2018 AMC amc10a 真题 答案 详解

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中英双语真题

2018 AMC10A Problem 1 What is the value of \left(\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1\right)? 下列算式的值是多少 \left(\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1\right)?
(A) \frac{5}{8} (B) \frac{11}{7} (C) \frac{8}{5} (D) \frac{18}{11} (E) \frac{15}{8}

Problem 2 Liliane has 50% more soda than Jacqueline, and Alice has 25% more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have? Liliane 有的汽水比 Jacqueline 多 50%，Alice 有的汽水比 Jacqueline 的多 25%。Liliane 和 Alice 拥有汽水量的关系是？
(A) Liliane has 20% more soda than Alice.|Liliane|有的汽水比 Alice 多 20%。
(B) Liliane has 25% more soda than Alice.|Liliane|有的汽水比 Alice 多 25%。
(C) Liliane has 45% more soda than Alice.|Liliane|有的汽水比 Alice 多 45%。
(D) Liliane has 75% more soda than Alice.|Liliane|有的汽水比 Alice 多 75%。
(E) Liliane has 100% more soda than Alice.|Liliane|有的汽水比 Alice 多 100%。

英文真题

2018 AMC 10A Problems

Problem 1 What is the value of $$\left(\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1\right)?$$
(A) \frac{5}{8} (B) \frac{11}{7} (C) \frac{8}{5} (D) \frac{18}{11} (E) \frac{15}{8}

Problem 2 Liliane has $50\%$ more soda than Jacqueline, and Alice has $25\%$ more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have?
(A) Liliane has $20\%$ more soda than Alice.
(B) Liliane has $25\%$ more soda than Alice.
(C) Liliane has $45\%$ more soda than Alice.
(D) Liliane has $75\%$ more soda than Alice.
(E) Liliane has $100\%$ more soda than Alice.

Problem 3 A unit of blood expires after $10! = 10 \cdot 9 \cdot 8 \cdots 1$ seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire?
(A) January 2 (B) January 12 (C) January 22 (D) February 11 (E) February 12

Problem 4 How many ways can a student schedule 3 mathematics courses - algebra, geometry, and number theory - in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)
(A) 3 (B) 6 (C) 12 (D) 18 (E) 24

Problem 5 Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let $d$ be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of $d$?
(A) $(0, 4)$ (B) $(4, 5)$ (C) $(4, 6)$ (D) $(5, 6)$ (E) $(5, \infty)$

Problem 6 Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of $0$, and the score increases by $1$ for each like vote and decreases by $1$ for each dislike vote. At one point Sangho saw that his video had a score of $90$, and that $65\%$ of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?
(A) 200 (B) 300 (C) 400 (D) 500 (E) 600

Problem 7 For how many (not necessarily positive) integer values of $n$ is the value of $4000 \cdot \left(\frac{2}{5}\right)^n$ an integer?
(A) 3 (B) 4 (C) 6 (D) 8 (E) 9

真题文字详解(英文)

Problem 1 What is the value of $$\left(\left(\left(2+1\right)^{-1}+1\right)^{-1}+1\right)^{-1}+1?$$
(A) \frac{5}{8} (B) \frac{11}{7} (C) \frac{8}{5} (D) \frac{18}{11} (E) \frac{15}{8}

Solution For all nonzero numbers (a), recall that (a^{-1}=\frac{1}{a}) is the reciprocal of (a). The original expression becomes
$$\begin{aligned} &\left(\left(\left(2+1\right)^{-1}+1\right)^{-1}+1\right)^{-1}+1 \ &= \left(\left(3^{-1}+1\right)^{-1}+1\right)^{-1}+1\ &= \left(\left(\frac{1}{3}+1\right)^{-1}+1\right)^{-1}\ &= \left(\left(\frac{4}{3}\right)^{-1}+1\right)^{-1}\ &= \left(\left(\frac{3}{4}+1\right)^{-1}+1\right)^{-1}\ &= \left(\left(\frac{7}{4}\right)^{-1}+1\right)^{-1}\ &= \frac{4}{7}+1\ &= \boxed{\text{(B)}\ \frac{11}{7}}. \end{aligned}$$

~MRENTHUSIASM

Problem 2 Liliane has $50\%$ more soda than Jacqueline, and Alice has $25\%$ more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have?
(A)Liliane has $20\%$ more soda than Alice.
(B)Liliane has $25\%$ more soda than Alice.
(C)Liliane has $45\%$ more soda than Alice.
(D)Liliane has $75\%$ more soda than Alice.
(E)Liliane has $100\%$ more soda than Alice.

Solution 1 Let's assume that Jacqueline has $1$ gallon(s) of soda. Then Alice has $1.25$ gallons and Liliane has $1.5$ gallons. Doing division, we find out that $\frac{1.5}{1.25}=1.2$, which means that Liliane has $20\%$ more soda. Therefore, the answer is $\boxed{\text{(A)}}$.

Solution 2 WLOG, lets use $4$ gallons instead of $1$. When you work it out, you get $6$ gallons and $5$ gallons. We have $6-5=1$ is $20\%$ of $5$. Thus, we reach $\boxed{\text{(A)}}$.

Solution 3 If Jacqueline has $x$ gallons of soda, Alice has $1.25x$ gallons, and Liliane has $1.5x$ gallons. Thus, the answer is $\frac{1.5}{1.25}=1.2\Rightarrow$ Liliane has $20\%$ more soda. Our answer is $\boxed{\text{(A)}}$.

Problem 3

真题文字详解(中英双语)

Problem 1 What is the value of 的值是多少 $$\left(\left(\left(2+1\right)^{-1}+1\right)^{-1}+1\right)^{-1}+1?$$ (A) \frac{5}{8} (B) \frac{11}{7} (C) \frac{8}{5} (D) \frac{18}{11} (E) \frac{15}{8}

Solution For all nonzero numbers a, recall that a^{-1}=\frac{1}{a} is the reciprocal of a. 对于所有非零数字，a是a^{-1}=\frac{1}{a}的倒数 The original expression becomes $$\left(\left(\left(2+1\right)^{-1}+1\right)^{-1}+1\right)^{-1}+1=\left(\left(3^{-1}+1\right)^{-1}+1\right)^{-1}+1$$ =\left(\left(\frac{1}{3}+1\right)^{-1}+1\right)^{-1}+1 =\left(\left(\frac{4}{3}\right)^{-1}+1\right)^{-1}+1 =\left(\left(\frac{3}{4}+1\right)^{-1}+1\right)^{-1}+1 =\left(\left(\frac{7}{4}\right)^{-1}+1\right)^{-1}+1 =\frac{4}{7}+1 =\boxed{\text{(B)}\frac{11}{7}}.$$

~MRENTHUSIASM 原始表达式变为 ~MRENTHUSIASM Problem 2 Liliane has 50% more soda than Jacqueline, and Alice has 25% more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have? Liliane 比 Jacqueline 多 50% 瓶汽水，Alice 比 Jacqueline 多 25% 瓶汽水。Liliane 和 Alice 的汽水数量之间有什么关系？ (A)Liliane has 20% more soda than Alice. (A)莉莉安有 20% 比爱丽丝更多的汽水。 (B)Liliane has 25% more soda than Alice. (B)莉莉安有 25% 比爱丽丝更多的汽水。 (C)Liliane has 45% more soda than Alice. (C)莉莉安有 45% 比爱丽丝更多的汽水。 (D)Liliane has 75% more soda than Alice. (D)莉莉安有 75% 比爱丽丝更多的汽水。 (E)Liliane has 100% more soda than Alice. (E)莉莉安有 100% 更多的汽水。

Solution 1