2024 AMC amc10b 真题 答案 详解

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中英双语真题

2024 AMC 10B 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? 在从左到右排列的长队中，从左边数第 1013 个人也是从右边数第 1010 个人。队伍中共有多少人？ (A) 2021 (B) 2022 (C) 2023 (D) 2024 (E) 2025 Problem 2 What is $10! - 7! \cdot 6!$ 的结果是？ (A) -120 (B) 0 (C) 120 (D) 600 (E) 720 Problem 3 For how many integer values of $x$ is $\left|2x\right|\leq7\pi$ 对于整数 $x$ 有多少个值使得 $\left|2x\right|\leq7\pi$ ? (A) 16 (B) 17 (C) 19 (D) 20 (E) 21 Problem 4 Balls numbered $1, 2, 3, \ldots$ 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 $3$ 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, balls numbered $22, 23, \ldots, 28$ are deposited in bin $B$ at step $7$ of this process.) In which bin is ball $2024$ deposited? 球编号为 $1, 2, 3, \ldots$ 的球存放在 $5$ 号箱子中，箱子分别标记为 $A、B、C、D$ 和 $E$ ，按照以下程序进行。球 $1$ 存放在 $A$ 号箱子里，球 $2$ 和 $3$ 存放在 $B$ 号箱子里。接下来的 $3$ 个球存放在 $C$ 号箱子里，接下来的 $4$ 个球存放在 $D$ 号箱子里，依此类推，在球存放在 $E$ 号箱子后，循环回到 $A$ 号箱子。（例如，球编号为 $22, 23, \ldots, 28$ 的球在第 $7$ 步的过程中存放在 $B$ 号箱子中。）球 $2024$ 存放在哪个箱子中？ (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? 在以下表达式中，Melanie 将一些加号改为了减号：当新的表达式被评估时，它是负数。Melanie 最少将多少个加号改为减号？ (A) 14 (B) 15 (C) 16 (D) 17 (E) 18 Problem 6 A rectangle has integer side lengths and an area of $2024$. What is the least possible perimeter of the rectangle? 一个矩形的边长为整数，面积为 $2024$ 。这个矩形的周长最小值是多少？ (A) 160 (B) 180 (C) 222 (D) 228 (E) 390

英文真题

2024 AMC 10B 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 $1, 2, 3, \ldots$ 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 $3$ 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, balls numbered $22, 23, \ldots, 28$ 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 A rectangle has integer side lengths and an area of $2024$. What is the least possible perimeter of the rectangle? (A) 160 (B) 180 (C) 222 (D) 228 (E) 390 Problem 7 What is the remainder when $7^{2024} + 7^{2025} + 7^{2026}$ is divided by $19$? (A) 0 (B) 1 (C) 7 (D) 11 (E) 18 Problem 8 Let $N$ be the product of all the positive integer divisors of $42$. What is the units digit of $N$? (A) 0 (B) 2 (C) 4 (D) 6 (E) 8 Problem 9 Real numbers $a, b,$ and $c$ have arithmetic mean $0$. The arithmetic mean of $a^2, b^2,$ and $c^2$ is $10$. What is the arithmetic mean of $ab, ac,$ and $bc$? (A) -5 (B) $-\frac{10}{3}$ (C) $-\frac{10}{9}$ (D) 0 (E) $\frac{10}{9}$

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 are 1012 people to their left. If the person is the 1010th from the right, that means there are 1009 people to their right. Therefore, there are 1012 + 1 + 1009 = (B) 2022 people in line. Solution 1 If the person is the 1013th from the left, that means there are 1012 people to their left. If the person is the 1010th from the right, that means there are 1009 people to their right. Therefore, there are 1012 + 1 + 1009 = (B) 2022 people in line. 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. Therefore, there are 1013 + 1010 + 1 = (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 = (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 = (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 = (B) 2022 Problem 2 The following problem is from both the 2024 AMC 10B #2 and 2024 AMC 12B #2, so both problems redirect to this page. What is 10! - 7! · 6!? (A) -120 (B) 0 (C) 120 (D) 600 (E) 720 [ONLY FOR CERTAIN CHINESE TESTPAPERS] What is 10! - 7! · 6! - 5! (A) -120 (B) 0 (C) 120 (D) 600 (E) 720 Solution 1 10! = 10 · 9 · 8 · 7! = 720 · 7! 6! · 7! = 720 · 7! Therefore, the equation is equal to 720 · 7! - 720 · 7! = (B) 0 [ONLY FOR CERTAIN CHINESE TESTPAPERS] 0 - 5! = (A) -120 Solution 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 某些中国试卷的解题方法 If the person is the 1013th from the left, that means there are 1012 people to their left. If the person is the 1010th from the right, that means there are 1009 people to their right. Therefore, there are 1012 + 1 + 1009 = (B) 2022 people in line.

如果这个人是左边数的第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 are 1012 people to their left. If the person is the 1010th from the right, that means there are 1009 people to their right. Therefore, there are 1012 + 1 + 1009 = (B) 2022 people in line.

如果这个人是左边数的第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 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 = (B) 2022 我们可以看一个较小的例子，它是从左边的第4个人，从右边的第2个人。将所有人编号列出，我们得到一个列表：1, 2, 3, 4, 5。我们看到总人数是左边位置+右边位置-1，在上面的例子中是4 + 2 - 1 = 5。将1013和1010代入得到1013 + 1010 - 1 = (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. 由于这个人是左边数的第1013个人，从右边数的第1010个人，所以他之后还有1009个人。 Therefore, there are 1013 + 1009 = (B) 2022 people in line. 因此，队伍中共有1013 + 1009 = (B) 2022个人。

(Note: Similarly, you can perceive this information