2023 AMC amc10a 真题 答案 详解

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中英双语真题

2023 AMC 10A 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?

城市 A 和 B 相距 45 英里。Alicia 居住在 A，Beth 居住在 B。Alicia 以每小时 18 英里的速度骑自行车前往 B。他们同时出发，Beth 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 有多远？

(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?

一个大型披萨的 (\frac{1}{3}) 加上 (3\frac{1}{2}) 杯橙子片与 (\frac{3}{4}) 个大型披萨加上 (\frac{1}{2}) 杯橙子片的重量相同。一杯橙子片的重量是 (\frac{1}{4}) 磅。一个大披萨的重量是多少磅？

(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?

有多少个小于 2023 的正完全平方数能被 5 整除？

(A) 8 (B) 9 (C) 10 (D) 11 (E) 12

Problem 4

A quadrilateral has all integer side lengths, a perimeter of 26, and one side of length 4. What is the greatest possible length of one side of this quadrilateral?

一个四边形的所有边长都是整数，周长为 26，其中一边的长为 4。这个四边形最长边的可能长度是多少？

(A) 9 (B) 10 (C) 11 (D) 12 (E) 13

Problem 5

How many digits are in the base-ten representation of (8^5 \cdot 5^{10} \cdot 15^5)?

在十进制中 (8^5 \cdot 5^{10} \cdot 15^5) 有多少位数字？

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

Problem 6

An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is 21. What is the value of the cube?

每个立方体的顶点都分配一个整数。边的值定义为它接触的两个顶点值的和，面的值定义为围绕它的四个边的值的和。立方体的值定义为它的六个面的值的和。假设分配给顶点的整数的和为 21。立方体的值是多少？

英文真题

2023 AMC 10A 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

A quadrilateral has all integer side lengths, a perimeter of 26, and one side of length 4. What is the greatest possible length of one side of this quadrilateral?

(A) 9 (B) 10 (C) 11 (D) 12 (E) 13

Problem 5

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 6

An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is 21. What is the value of the cube?

(A) 42 (B) 63 (C) 84 (D) 126 (E) 252

Problem 7

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 8

Barb the baker has developed a new temperature scale for her bakery called the Breadus scale, which is a linear function of the Fahrenheit scale. Bread rises at 110 degrees Fahrenheit, which is 0 degrees on the Breadus scale. Bread is baked at 350 degrees Fahrenheit, which is 100 degrees on the Breadus scale. Bread is done when its internal temperature is 200 degrees Fahrenheit. What is this, in degrees, on the Breadus scale?

(A) 33 (B) 34.5 (C) 36 (D) 37.5 (E) 39

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 $18 \times 1.5 = (E) 27$

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 = $18 \cdot \frac{3}{2} = (E) 27$

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
$$45 - 18 = (E) 27$$

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:
$$18t = -12t + 45$$
$$30t = 45$$
So, $18(1.5) = (E) 27$

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 $\frac{3}{5} \cdot 45 = (E) 27$ 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。

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 两人相对速度为 18+12=30，因此需要 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，因此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，
3/2x + x = 45
5/2x = 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 英里每小时的距离或芭芭拉以 12 英里每小时的距离；因为我们想要艾丽西娅的距离，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 英里