2023 AMC amc10b 真题 答案 详解

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中英双语真题

2023 AMC 10B 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? (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? (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? (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? (A) 162,500 (B) 162.5 (C) 1,625 (D) 1,625,000 (E) 16,250 Problem 5 Maddy and Lara see a list of numbers written on a blackboard. Maddy adds 3 to each number in the list and finds that the sum of her new numbers is 45. Lara multiplies each number in the list by 3 and finds that the sum of her new numbers is also 45. How many numbers are written on the blackboard? (A) 10 (B) 5 (C) 6 (D) 8 (E) 9

英文真题

2023 AMC 10B 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?
((A)\ \frac{1}{12} \quad (B)\ \frac{1}{4} \quad (C)\ \frac{1}{6} \quad (D)\ \frac{1}{8} \quad (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?
((A)\ $46 \quad (B)\ $50 \quad (C)\ $48 \quad (D)\ $47 \quad (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)? ((A)\ \frac{9}{25} \quad (B)\ \frac{1}{9} \quad (C)\ \frac{1}{5} \quad (D)\ \frac{25}{169} \quad (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?
((A)\ 162,500 \quad (B)\ 162.5 \quad (C)\ 1,625 \quad (D)\ 1,625,000 \quad (E)\ 16,250)

Problem 5 Maddy and Lara see a list of numbers written on a blackboard. Maddy adds (3) to each number in the list and finds that the sum of her new numbers is (45). Lara multiplies each number in the list by (3) and finds that the sum of her new numbers is also (45). How many numbers are written on the blackboard?
((A)\ 10 \quad (B)\ 5 \quad (C)\ 6 \quad (D)\ 8 \quad (E)\ 9)

Problem 6 Let (L_1 = 1), (L_2 = 3), and (L_{n+2} = L_{n+1} + L_n) for (n \geq 1). How many terms in the sequence (L_1, L_2, L_3, \cdots, L_{2023}) are even?
((A)\ 673 \quad (B)\ 1011 \quad (C)\ 675 \quad (D)\ 1010 \quad (E)\ 674)

Problem 7 Square (ABCD) is rotated (20^\circ) clockwise about its center to obtain square (EFGH), as shown below. What is the degree measure of (\angle EAB)?

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) We can create the equation: (0.8x \cdot 1.075 = 43) using the information given. This is because (x), the original price, got reduced by 20%, or multiplied by 0.8, and it also got multiplied by 1.075 on the discounted price. Solving that equation, we get (\frac{4}{5} \cdot x \cdot \frac{43}{40} = 43) (\frac{4}{5} \cdot x \cdot \frac{1}{40} = 1) (\frac{1}{5} \cdot x \cdot \frac{1}{10} = 1) (x = \boxed{50})

Solution 2 (One-Step Equation) The discounted shoe is 20% off the original price. So that means (1 - 0.2 = 0.8). There is also a 7.5% sales

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。她把前三个杯子装满了，但在第四个杯子里只倒了一杯子的零头。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( \text{(C)}\ \frac{1}{6} \right)) 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。因为我们在处理比率，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三分之一满的。如果果汁的总数量是(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中的每一个都要向第四个杯子贡献(1-\frac{5}{6}=\left( \text{(C)}\ \frac{1}{6} \right))。

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。这样前三个杯子就是(\frac{6}{6})满，第四个杯子是(\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 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出(\frac{1}{6})会使它们都变成(\frac{5}{6})满。因此，所有四个杯子里的果汁量都相同。答案是(\left( \text{(C)}\ \frac{1}{6} \right))。

Solution 3 (simple algebra) 解决方案3（简单的代数） We let (x) denote how much juice we take from each of the first 3 children and give to the 4th child. 我们用(x)表示我们从每个前3个孩子那里取多少果汁，并给第4个孩子。 We can write the following equation: (1-x=\frac{1}{3}+3x), since each value represents how much juice