2023 AMC amc12b 真题 答案 详解

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中英双语真题

2023 AMC 12B Problems

Problem 1

Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only (\frac{1}{3}) full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice?

琼斯太太正在把橙汁倒入四个相同的杯子中，给她的四个儿子。她把前三个杯子倒满，但在第四个杯子只倒了(\frac{1}{3})时，橙汁就用完了。琼斯太太需要从前三个杯子里各倒出多少橙汁到第四个杯子里，才能使四个杯子里的橙汁量相同？

(A) (\frac{1}{12})

(B) (\frac{1}{4})

(C) (\frac{1}{6})

(D) (\frac{1}{8})

(E) (\frac{2}{9})

Problem 2

Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by 20% on every pair of shoes. Carlos also knew that he had to pay a 7.5% sales tax on the discounted price. He had $43 dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?

卡洛斯去一家体育用品店买跑步鞋。跑鞋正在打折，每双鞋的价格都降低了20%。卡洛斯还知道，他必须对折扣后的价格支付7.5%的销售税。他有$43美元。他能买得起的最贵鞋的原价（打折前）是多少？

(A) 46

(B) 50

(C) 48

(D) 47

(E) 49

Problem 3

A (3-4-5) right triangle is inscribed in circle A, and a (5-12-13) right triangle is inscribed in circle B. What is the ratio of the area of circle A to the area of circle B?

三边长为3,4,5的直角三角形内接于圆A，三边长为5,12,13的直角三角形内接于圆B.问圆A的面积与圆B的面积之比是多少？

(A) (\frac{9}{25})

(B) (\frac{1}{9})

(C) (\frac{1}{5})

(D) (\frac{25}{169})

(E) (\frac{4}{25})

Problem 4

Jackson's paintbrush makes a narrow strip with a width of 6.5 millimeters. Jackson has enough paint to make a strip 25 meters long. How many square centimeters of paper could Jackson cover with paint?

杰克逊的画笔画出一条宽度为6.5毫米的窄条带。杰克逊有足够的颜料画出一根25米长的条带。杰克逊能用颜料覆盖多少平方厘米的纸张？

(A) 162,500

(B) 162.5

(C) 1,625

(D) 1,625,000

(E) 16,250

Problem 5

You are playing a game. A (2\times 1) rectangle covers two adjacent squares (oriented either horizontally or vertically) of a (3\times 3) grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensure that at least one of your guessed squares is covered by the rectangle?

你在玩下面的游戏。一个(2\times 1)矩形覆盖了(3\times 3)方格表中的两个相邻方格(水平或垂直方向)，但你不知道哪两个方格被覆盖了。你的目标是找到至少一个被矩形覆盖的方格。一个“回合”是指你猜测某个方格，然后被告知这个方格是否被隐藏的矩形覆盖。为确保至少猜到一个被矩形覆盖的方格，你最少需要多少回合？

(A) 3

(B) 5

(C) 4

(D) 8

(E) 6

英文真题

2023 AMC 12B Problems

Problem 1

Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only (\frac{1}{3}) full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice?

[ \begin{align} (A)&\quad \frac{1}{12}\ (B)&\quad \frac{1}{4}\ (C)&\quad \frac{1}{6}\ (D)&\quad \frac{1}{8}\ (E)&\quad \frac{2}{9} \end{align} ]

Problem 2

Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by (20\%) on every pair of shoes. Carlos also knew that he had to pay a (7.5\%) sales tax on the discounted price. He had (43) dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?

[ \begin{align} (A)&\quad 46\ (B)&\quad 50\ (C)&\quad 48\ (D)&\quad 47\ (E)&\quad 49 \end{align} ]

Problem 3

A (3-4-5) right triangle is inscribed in circle A, and a (5-12-13) right triangle is inscribed in circle B. What is the ratio of the area of circle A to the area of circle B?

[ \begin{align} (A)&\quad \frac{9}{25}\ (B)&\quad \frac{1}{9}\ (C)&\quad \frac{1}{5}\ (D)&\quad \frac{25}{169}\ (E)&\quad \frac{4}{25} \end{align} ]

Problem 4

Jackson's paintbrush makes a narrow strip with a width of (6.5) millimeters. Jackson has enough paint to make a strip (25) meters long. How many square centimeters of paper could Jackson cover with paint?

[ \begin{align} (A)&\quad 162,500\ (B)&\quad 162.5\ (C)&\quad 1,625\ (D)&\quad 1,625,000\ (E)&\quad 16,250 \end{align} ]

Problem 5

You are playing a game. A (2\times 1) rectangle covers two adjacent squares (oriented either horizontally or vertically) of a (3\times 3) grid of squares, but you are not told which two squares are covered. Your goal is to find at least one square that is covered by the rectangle. A "turn" consists of you guessing a square, after which you are told whether that square is covered by the hidden rectangle. What is the minimum number of turns you need to ensure that at least one of your guessed squares is covered by the rectangle?

[ \begin{align} (A)&\quad 3\ (B)&\quad 5\ (C)&\quad 4\ (D)&\quad 8\ (E)&\quad 6 \end{align} ]

Problem 6

When the roots of the polynomial

[P(x)=(x-1)^{1}(x-2)^{2}(x-3)^{3}\cdots(x-10)^{10}]

are removed from the number line, what remains is the union of (11) disjoint open intervals. On how many of these intervals is (P(x)) positive?

[ \begin{align} (A)&\quad 3\ (B)&\quad 7\ (C)&\quad 6\ (D)&\quad 4\ (E)&\quad 5 \end{align} ]

Problem 7

For how many integers (n) does the expression

[\sqrt{\frac{\log(n^2)-( \log n )^{2}}{\log n -3}}]

represent a real number, where (\log) denotes the base (10) logarithm?

[ \begin{align} (A)&\quad 900\ (B)&\quad 3\ (C)&\quad 902\ (D)&\quad 2\ (E)&\quad 901 \end{align} ]

真题文字详解(英文)

Problem 1 The following problem is from both the 2023 AMC 10B #1 and 2023 AMC 12B #1, so both problems redirect to this page. Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only (\frac{1}{3}) full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice?
(A) (\frac{1}{12}) (B) (\frac{1}{4}) (C) (\frac{1}{6}) (D) (\frac{1}{8}) (E) (\frac{2}{9})

Solution 1 The first three glasses each have a full glass. Let's assume that each glass has "1 unit" of juice. It won't matter exactly how much juice everyone has because we're dealing with ratios, and that wouldn't affect our answer. The fourth glass has a glass that is one third. So the total amount of juice will be (1 + 1 + 1 + \frac{1}{3} = \frac{10}{3}). If we divide the total amount of juice by 4, we get (\frac{5}{6}), which should be the amount of juice in each glass. This means that each of the first three glasses will have to contribute (1 - \frac{5}{6} = \boxed{\text{(C)}\ \frac{1}{6}}) to the fourth glass.

Solution 2 (unnecessary numerical values) Given that the first three glasses are full and the fourth is only (\frac{1}{3}) full, let's represent their contents with a common denominator, which we'll set as 6. This makes the first three glasses (\frac{6}{6}) full, and the fourth glass (\frac{2}{6}) full. To equalize the amounts, Mrs. Jones needs to pour juice from the first three glasses into the fourth. Pouring (\frac{1}{6}) from each of the first three glasses will make them all (\frac{5}{6}) full. Thus, all four glasses will have the same amount of juice. Therefore, the answer is (\boxed{\text{(C)}\ \frac{1}{6}}).

Solution 3 (simple algebra) We let (x) denote how much juice we take from each of the first 3 children and give to the 4th child. We can write the following equation: (1 - x = \frac{1}{3} + 3x), since each value represents how much juice each child (equally) has in the end. (Each of the first three children now have (1 - x) juice, and the fourth child has (3x) more juice on top of their initial (\frac{1}{3})) Solving, we see that (x = \boxed{\text{(C)}\ \frac{1}{6}}).

Problem 2 The following problem is from both the 2023 AMC 10B #2 and 2023 AMC 12B #2, so both problems redirect to this page. Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by 20% on every pair of shoes. Carlos also knew that he had to pay a 7.5% sales tax on the discounted price. He had $43 dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?
(A) $46 (B) $50 (C) $48 (D) $47 (E) $49 Solution 1 (easy)

真题文字详解(中英双语)

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

Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses completely but runs out of juice when the fourth glass is only (\frac{1}{3}) full. What fraction of a glass must Mrs. Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice?

琼斯夫人正在为她的四个儿子倒橙汁到四个相同的杯子里。她倒满了前三个杯子，但在第四个杯子里只倒了一杯的零头。为了使四个杯子里的果汁量相同，琼斯夫人需要从前三个杯子中各倒入多少果汁到第四个杯子里？

(A) (\frac{1}{12}) (B) (\frac{1}{4}) (C) (\frac{1}{6}) (D) (\frac{1}{8}) (E) (\frac{2}{9})

Solution 1 The first three glasses each have a full glass. Let's assume that each glass has "1 unit" of juice. It won't matter exactly how much juice everyone has because we're dealing with ratios, and that wouldn't affect our answer. The fourth glass has a glass that is one third. So the total amount of juice will be (1 + 1 + 1 + \frac{1}{3} = \frac{10}{3}). If we divide the total amount of juice by 4, we get (\frac{5}{6}), which should be the amount of juice in each glass. This means that each of the first three glasses will have to contribute (1 - \frac{5}{6} = \left( C \right)\frac{1}{6}) to the fourth glass.

前三个杯子每个都是满的。假设每个杯子有“1单位”的果汁。因为我们处理的是比率，所以每个人有多少果汁并不重要，这不会影响我们的答案。第四个杯子有一杯是空的。所以果汁的总数量是(1+1+1+\frac{1}{3}=\frac{10}{3})。如果我们把果汁总量除以4，我们得到(\frac{5}{6})，这就是每个杯子应该有的果汁量。这意味着前三个杯子每个都要向第四个杯子贡献(1-\frac{5}{6}=\left(C\right)\frac{1}{6})。

Solution 2 (unnecessary numerical values) 解决方案2（不必要的数值）

Given that the first three glasses are full and the fourth is only (\frac{1}{3}) full, let's represent their contents with a common denominator, which we'll set as 6. This makes the first three glasses (\frac{6}{6}) full, and the fourth glass (\frac{2}{6}) full.

已知前三个杯子是满的，第四个杯子只有(\frac{1}{3})满，让我们用公倍数来表示它们的容量，我们将其设置为6。这样前三个杯子就(\frac{6}{6})满，第四个杯子(\frac{2}{6})满。

To equalize the amounts, Mrs. Jones needs to pour juice from the first three glasses into the fourth. Pouring (\frac{1}{6}) from each of the first three glasses will make them all (\frac{5}{6}) full. Thus, all four glasses will have the same amount of juice. Therefore, the answer is (\left(C\right)\frac{1}{6}).

为了使数量相等，琼斯夫人需要将前三个杯子里的果汁倒入第四个杯子。从每个前三个杯子中倒入(\frac{1}{6})果汁，会使它们都(\frac{5}{6})满。因此，四个杯子里的果汁量都相同。答案是(\left(C\right)\frac{1}{6})。