2024 AMC amc12b 真题 答案 详解

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中英双语真题

2024 AMC 12B Problems

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?

(A) 2021 (B) 2022 (C) 2023 (D) 2024 (E) 2025

Problem 2

What is $10! - 7! \cdot 6!$?

(A) -120 (B) 0 (C) 120 (D) 600 (E) 720

Problem 3

For how many integer values of x is $|2x| \leq 7\pi$?

(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 1 is deposited in bin A, and balls 2 and 3 are deposited in bin B. The next three 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, 22, 23, ..., 28 are deposited in bin B at step 7 of this process.) In which bin is ball 2024 deposited?

(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:

$$1 + 3 + 5 + 7 + \cdots + 97 + 99$$

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?

(A) 14 (B) 15 (C) 16 (D) 17 (E) 18

英文真题

2024 AMC 12B Problems 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? (A) 2021 (B) 2022 (C) 2023 (D) 2024 (E) 2025 Problem 2 What is $10! - 7! \cdot 6!?$ (A) -120 (B) 0 (C) 120 (D) 600 (E) 720 Problem 3 For how many integer values of $x$ is $\left|2x\right| \leq 7\pi ?$ (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 1 is deposited in bin A, and balls 2 and 3 are deposited in bin B. The next three 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, 22, 23, ..., 28 are deposited in bin B at step 7 of this process.) In which bin is ball 2024 deposited? (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: $$1 + 3 + 5 + 7 + \cdots + 97 + 99$$ 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? (A) 14 (B) 15 (C) 16 (D) 17 (E) 18 Problem 6 The national debt of the United States is on track to reach $5 \cdot 10^{13}$ dollars by 2033. How many digits does this number of dollars have when written as a numeral in base 5? (The approximation of $\log_{10} 5$ as 0.7 is sufficient for this problem.) (A) 18 (B) 20 (C) 22 (D) 24 (E) 26 Problem 7 In the figure below WXYZ is a rectangle with WX = 4 and WZ = 8. Point M lies XY, point A lies YZ, and ∠WMA is a right angle. The areas of △WX M and △WA Z are equal. What is the area of △WMA? (A) 13 (B) 14 (C) 15 (D) 16 (E) 17

真题文字详解(英文)

Problem 1 The following problem is from both the 2024 AMC 10B #1 and 2024 AMC 12B #1, so both problems redirect to this page. 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?
(A) 2021 (B) 2022 (C) 2023 (D) 2024 (E) 2025

Modified Problem in Certain China Testpapers
In a long line of people arranged left to right, the 1015th person from the left is also the 1010th person from the right. How many people are in line?
(A) 2021 (B) 2022 (C) 2023 (D) 2024 (E) 2025

Solution for Certain China Testpapers
If the person is the 1013th from the left, that means there is 1012 people to their left. If the person is the 1010th from the right, that means there is 1009 people to their right. Therefore, there are $1012 + 1 + 1009 = \boxed{\text{(B)}\ 2022}$ people in line.

Solution 1
If the person is the 1013th from the left, that means there is 1012 people to their left. If the person is the 1010th from the right, that means there is 1009 people to their right. Therefore, there are $1012 + 1 + 1009 = \boxed{\text{(B)}\ 2022}$ people in line.

Solution 2
The person is the 1013rd person from the left is also the 1010th person from the right, so the same person is counted twice.
Therefore, there are $1013 + 1010 + 1 = \boxed{\text{(B)}\ 2022}$ people in line.

Solution 3
We can look at a smaller case where it is the 4th person from the left and the 2nd person from the right. Listing out the people as numbers gives us a list of 1, 2, 3, 4, 5. We see that the total number of people is the position from the left + the position from the right - 1, which in the case above is $4 + 2 - 1 = 5$. Plugging in 1013 and 1010 gives us $1013 + 1010 - 1 = \boxed{\text{(B)}\ 2022}$.

Solution 4
Since the person is the 1013th person from the left and the 1010th person from the right, there are 1009 people after the 1013th person.
Therefore, there are $1013 + 1009 = \boxed{\text{(B)}\ 2022}$ people in line.
(Note: Similarly, you can perceive this information as 1012 people to the left of 1010 people and proceed.)

Solution 5
Think of a Pattern
If 4 people: p1, p2, p3, p4 and you are 2nd from right and third from left
So just add 1013+1010 and subtract 1 since you are overcounting yourself twice.
We then get, $1013 + 1009 = \boxed{\text{(B)}\ 2022}$.

Problem 2

真题文字详解(中英双语)

Problem 1 The following problem is from both the 2024 AMC 10B #1 and 2024 AMC 12B #1, so both problems redirect to this page. 以下问题来自2024年AMC 10B第1题和2024年AMC 12B第1题，因此这两个问题都指向这一页。

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? 在一排从左到右排列的人中，左边数第1013个人同时也是右边数第1010个人。这排有多少人？

(A) 2021 (B) 2022 (C) 2023 (D) 2024 (E) 2025

Modified Problem in Certain China Testpapers 某些中国考试中的修改问题 In a long line of people arranged left to right, the 1015th person from the left is also the 1010th person from the right. How many people are in line? 在一排从左到右排列的人中，左边数第1015个人同时也是右边数第1010个人。这排有多少人？

(A) 2021 (B) 2022 (C) 2023 (D) 2024 (E) 2025

Solution for Certain China Testpapers 某些中国考试题的解答 If the person is the 1013th from the left, that means there is 1012 people to their left. If the person is the 1010th from the right, that means there is 1009 people to their right. Therefore, there are 1012 + 1 + 1009 = (B) 2022 people in line.

如果这个人是从左边数的第1013位，这意味着有1012个人在他们左边。如果这个人是从右边数的第1010位，这意味着有1009个人在他们右边。因此，队伍中有1012 + 1 + 1009 = (B) 2022个人。

Solution 1 If the person is the 1013th from the left, that means there is 1012 people to their left. If the person is the 1010th from the right, that means there is 1009 people to their right. Therefore, there are 1012 + 1 + 1009 = (B) 2022 people in line.

如果这个人是从左边数的第1013位，这意味着有1012个人在他们左边。如果这个人是从右边数的第1010位，这意味着有1009个人在他们右边。因此，队伍中有1012 + 1 + 1009 = (B) 2022个人。

Solution 2 The person is 1013rd person from the left is also the 1010th person from the right, so the same person is counted twice. 从左边数的第1013位的人也是从右边数的第1010位的人，所以同一个人被计算了两次。 Therefore, there are 1013 + 1010 + 1 = (B) 2022 people in line.

因此，队伍中有1013 + 1010 + 1 = (B) 2022个人。

Solution 3