2023 AMC amc12a 真题 答案 详解

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中英双语真题

2023 AMC 12A Problems Problem 1 Cities A and B are 45 miles apart. Alicia lives in A and Beth lives in B. Alicia bikes towards B at 18 miles per hour. Leaving at the same time, Beth bikes toward A at 12 miles per hour. How many miles from City A will they be when they meet? (A) 20 (B) 24 (C) 25 (D) 26 (E) 27 Problem 2 The weight of (\frac{1}{3}) of a large pizza together with (3\frac{1}{2}) cups of orange slices is the same weight of (\frac{3}{4}) of a large pizza together with (\frac{1}{2}) cups of orange slices. A cup of orange slices weighs (\frac{1}{4}) of a pound. What is the weight, in pounds, of a large pizza? (A) (1\frac{4}{5}) (B) 2 (C) (2\frac{2}{5}) (D) 3 (E) (3\frac{3}{5}) Problem 3 How many positive perfect squares less than 2023 are divisible by 5? (A) 8 (B) 9 (C) 10 (D) 11 (E) 12 Problem 4 How many digits are in the base-ten representation of (8^5 \cdot 5^{10} \cdot 15^5) ? (A) 14 (B) 15 (C) 16 (D) 17 (E) 18 Problem 5 Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3? (A) (\frac{2}{9}) (B) (\frac{49}{216}) (C) (\frac{25}{108}) (D) (\frac{17}{72}) (E) (\frac{13}{54}) Problem 6 Points A and B lie on the graph of (y = \log_2 x). The midpoint of (\overline{AB}) is (6, 2). What is the positive difference between the x-coordinates of A and B? (A) (2\sqrt{11}) (B) (4\sqrt{3}) (C) 8 (D) (4\sqrt{5}) (E) 9 Problem 7 A digital display shows the current date as an 8-digit integer consisting of a 4-digit year, followed by a 2-digit month, followed by a 2-digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in 2023 will each digit appear an even number of times in the 8-digit display for that date? (A) 5 (B) 6 (C) 7 (D) 8 (E) 9 Problem 8 Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an 11 on the next quiz, her mean will increase by 1. If she scores an 11 on each of the next three quizzes, her mean will increase by 2. What is the mean of her quiz scores currently? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 Problem 9 A square of area 2 is inscribed in a square of area 3, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?

英文真题

2023 AMC 12A Problems Problem 1 Cities A and B are 45 miles apart. Alicia lives in A and Beth lives in B. Alicia bikes towards B at 18 miles per hour. Leaving at the same time, Beth bikes toward A at 12 miles per hour. How many miles from City A will they be when they meet? (A) 20 (B) 24 (C) 25 (D) 26 (E) 27 Problem 2 The weight of (\frac{1}{3}) of a large pizza together with (\frac{1}{2}) cups of orange slices is the same weight of (\frac{3}{4}) of a large pizza together with (\frac{1}{2}) cups of orange slices. A cup of orange slices weighs (\frac{1}{4}) of a pound. What is the weight, in pounds, of a large pizza? (A) (1\frac{4}{5}) (B) 2 (C) (2\frac{2}{5}) (D) 3 (E) (3\frac{3}{5}) Problem 3 How many positive perfect squares less than 2023 are divisible by 5? (A) 8 (B) 9 (C) 10 (D) 11 (E) 12 Problem 4 How many digits are in the base-ten representation of (8^5 \cdot 5^{10} \cdot 15^5) ? (A) 14 (B) 15 (C) 16 (D) 17 (E) 18 Problem 5 Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3? (A) (\frac{2}{9}) (B) (\frac{49}{216}) (C) (\frac{25}{108}) (D) (\frac{17}{72}) (E) (\frac{13}{54}) Problem 6 Points A and B lie on the graph of (y = \log_2 x). The midpoint of AB is (6, 2). What is the positive difference between the x-coordinates of A and B? (A) (2\sqrt{11}) (B) (4\sqrt{3}) (C) 8 (D) (4\sqrt{5}) (E) 9 Problem 7 A digital display shows the current date as an 8-digit integer consisting of a 4-digit year, followed by a 2-digit month, followed by a 2-digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in 2023 will each digit appear an even number of times in that date? (A) 5 (B) 6 (C) 7 (D) 8 (E) 9 Problem 8 Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an 11 on the next quiz, her mean will increase by 1. If she scores an 11 on each of the next three quizzes, her mean will increase by 2. What is the mean of her quiz scores currently? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 Problem 9 A square of area 2 is inscribed in a square of area 3, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle? (A) (\frac{1}{5}) (B) (\frac{1}{4}) (C) (2 - \sqrt{3}) (D) (\sqrt{3} - \sqrt{2}) (E) (\sqrt{2} - 1)

真题文字详解(英文)

Problem 1 The following problem is from both the 2023 AMC 10A #1 and 2023 AMC 12A #1, so both problems redirect to this page.

Problem 1 Cities A and B are 45 miles apart. Alicia lives in A and Beth lives in B. Alicia bikes towards B at 18 miles per hour. Leaving at the same time, Beth bikes toward A at 12 miles per hour. How many miles from City A will they be when they meet?
(A) 20 (B) 24 (C) 25 (D) 26 (E) 27

Solution 1 This is a d = st problem, so let x be the time it takes to meet. We can write the following equation:
$$12x + 18x = 45$$
Solving gives us x = 1.5. The 18x is Alicia so $18 \times 1.5 = \textbf{(E)} 27$

Solution 2 The relative speed of the two is 18 + 12 = 30, so $\frac{3}{2}$ hours would be required to travel 45 miles. d = st, so x = $18 \cdot \frac{3}{2} = \textbf{(E)} 27$

Solution 3 Since 18 mph is $\frac{3}{2}$ times 12 mph, Alicia will travel $\frac{3}{2}$ times as far as Beth. If x is the distance Beth travels,
$$\frac{3}{2}x + x = 45$$
$$\frac{5}{2}x = 45$$
$$x = 18$$
Since this is the amount Beth traveled, the amount that Alicia traveled was
$$45 - 18 = \textbf{(E)} 27$$

Solution 4 Alice and Barbara close in on each other at 30mph. Since they are 45 miles apart, they will meet in t = d/s = 45miles / 30mph = 3/2 hours. We can either calculate the distance Alice travels at 18mph or the distance Barbara travels at 12mph; since we want the distance from Alice, we go with the former. Alice (and Barbara) will meet in 1 1/2 hours at 18mph x 3/2 hours = 27 miles from A. (E) 27

Solution 5 (Under 20 seconds) We know that Alice approaches Beth at 18 mph and Beth approaches Alice at 12 mph. If we consider that if Alice moves 18 miles at the same time Beth moves 12 miles -> 15 miles left. Alice then moves 9 more miles at the same time that Beth moves 6 more miles. Alice has moved 27 miles from point A at the same time that Beth has moved 18 miles from point B, meaning that Alice and Beth meet 27 miles from point A.

Solution 6 (simple linear equations) We know that Beth starts 45 miles away from City A, let's create two equations: Alice-> 18t = d Beth-> -12t + 45 = d [-12 is the slope; 45 is the y-intercept] Solve the system:
$$18t = -12t + 45$$
$$30t = 45$$
$$t = 1.5$$
So, $18(1.5) = \textbf{(E)} 27$

Solution 7 Since Alicia and Beth's speeds are constant, they are directly proportional to their distances covered, so the ratio of their speeds is equal to the ratio of their covered distances. Since Alicia travels $\frac{18}{30} = \frac{3}{5}$ of their combined speed, she travels $\frac{3}{5} \cdot 45 = \textbf{(E)} 27$ of the total distance.
-Benedict T (countmath1)

Solution 8 Note that Alicia and Beth must have travelled 45 miles together in order for them to meet together. While traveling the 45 miles, Alicia came closer towards Beth and Beth came closer towards Alicia. Since Alicia is travelling faster than Beth, we know that they will meet slightly closer towards city B from the middle. Also, the distance remaining between Alicia and Beth as they bike toward each other is proportional to their combined velocity. The combined velocity is $18t + 12t = 30t$. The time it takes to travel 45 miles going at 30 miles per hour can be calculated using simple algebra. Let t be the time it takes for Alicia and Beth to meet together.

真题文字详解(中英双语)

Problem 1 The following problem is from both the 2023 AMC 10A #1 and 2023 AMC 12A #1, so both problems redirect to this page. 以下问题来自2023年AMC 10A #1和2023年AMC 12A #1，因此这两个问题都指向这一页。

Problem 1 Cities A and B are 45 miles apart. Alicia lives in A and Beth lives in B. Alicia bikes towards B at 18 miles per hour. Leaving at the same time, Beth bikes toward A at 12 miles per hour. How many miles from City A will they be when they meet? 城市 A 和 B 相距 45 英里。艾丽西亚住在 A，贝丝住在 B。艾丽西亚以每小时 18 英里的速度骑自行车前往 B。同时出发，贝丝以每小时 12 英里的速度骑自行车前往 A。当他们相遇时，距离城市 A 多远？ (A) 20 (B) 24 (C) 25 (D) 26 (E) 27 Solution 1 This is a d = st problem, so let x be the time it takes to meet. We can write the following equation: 这是一个 d=st 问题，所以设 x 为相遇所用的时间。我们可以写出以下方程： 12x + 18x = 45 Solving gives us x = 1.5. The 18x is Alicia so 18 × 1.5 = (E) 27 解得 x = 1.5。18x 是艾丽西娅，所以 18 × 1.5 = (E) 27 Solution 2 The relative speed of the two is 18 + 12 = 30, so 3/2 hours would be required to travel 45 miles. d = st, so x = 18 · 3/2 = (E) 27 两者的相对速度是 18 + 12 = 30，所以需要 3/2 小时才能行驶 45 英里。d = st，所以 x = 18 · 3/2 = (E) 27 Solution 3 Since 18 mph is 3/2 times 12 mph, Alicia will travel 3/2 times as far as Beth. If x is the distance Beth travels, 由于 18 英里每小时是 12 英里每小时的 3/2 倍，艾丽西亚将比贝丝多走 3/2 倍的路程。如果 x 是贝丝所走的距离， 3/2 x + x = 45 5/2 x = 45 x = 18 Since this is the amount Beth traveled, the amount that Alicia traveled was 由于这是贝丝所走的路程，艾丽西亚所走的路程是 45 - 18 = (E) 27 Solution 4 Alice and Barbara close in on each other at 30mph. Since they are 45 miles apart, they will meet in t = d/s = 45miles / 30mph = 3/2 hours. We can either calculate the distance Alice travels at 18mph or the distance Barbara travels at 12mph; since we want the distance from Alice, we go with the former. Alice (and Barbara) will meet in 1 1/2 hours at 18mph × 3/2 hours = 27 miles from A. 艾丽西娅和芭芭拉以每小时 30 英里的速度接近对方。由于他们相距 45 英里，他们将在 t = d/s = 45 英里 / 30 英里每小时 = 3/2 小时后相遇。我们可以计算艾丽西娅以 18 英里每小时的速率行驶的距离，或者芭芭拉以 12 英里每小时的速率行驶的距离；因为我们想要从艾丽西娅算起，我们选择前者。艾丽西娅（和芭芭拉）将在 1 1/2 小时后在距离 A 点 18 英里每小时 × 3/2 小时 = 27 英里的地方相遇。 (E) 27 Solution 5 (Under 20 seconds) 解答 5 （少于 20 秒） We know that Alice approaches Beth at 18 mph and Beth approaches Alice at 12 mph. If we consider that if Alice moves 18 miles at the same time Beth moves 12 miles -> 15 miles left. Alice then moves 9 more miles at the same time that Beth moves 6 miles. Alice has moved 27 miles from point A at the same time that Beth has moved 18 miles from point B, meaning that Alice and Beth meet 27 miles from point A. 我们知道艾丽西娅以每小时 18 英里的速度接近贝丝，贝丝以每小时 12 英里的速度接近艾丽西娅。如果我们考虑如果艾丽西娅移动 18