2013 AMC amc12a 真题 答案 详解

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中英双语真题

2013 AMC12A Problem 1 Square ABCD has side length 10. Point E is on BC, and the area of △ABE is 40. What is BE? 正方形 ABCD 的边长为 10，点 E 在 BC 上，△ABE 的面积为 40，问 BE 的长是多少？ (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 Problem 2 A softball team played ten games, scoring 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score? 一支垒球队打了 10 场比赛，得分为 1, 2, 3, 4, 5, 6, 7, 8, 9 和 10 分，其中恰好 5 场比赛比对手少 1 分，在其他 5 场比赛中，得分是对方分数的 2 倍。那么他们的对手总共得分是多少？ (A) 35 (B) 40 (C) 45 (D) 50 (E) 55

英文真题

2013 AMC 12A Problems Problem 1 Square ABCD has side length 10. Point E is on BC, and the area of △ABE is 40. What is BE? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 Problem 2 A softball team played ten games, scoring 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score? (A) 35 (B) 40 (C) 45 (D) 50 (E) 55 Problem 3 A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations? (A) 15 (B) 30 (C) 40 (D) 60 (E) 70 Problem 4 What is the value of $$\frac{2^{2014} + 2^{2012}}{2^{2014} - 2^{2012}}$$ ? (A) -1 (B) 1 (C) \frac{5}{3} (D) 2013 (E) $2^{4024}$ Problem 5 Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $105, Dorothy paid $125, and Sammy paid $175. In order to share the costs equally, Tom gave Sammy t dollars, and Dorothy gave Sammy d dollars. What is t - d? (A) 15 (B) 20 (C) 25 (D) 30 (E) 35 Problem 6 In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on 20% of her three-point shots and 30% of her two-point shots. Shenille attempted 30 shots. How many points did she score? (A) 12 (B) 18 (C) 24 (D) 30 (E) 36 Problem 7 The sequence S_1, S_2, S_3, ..., S_{10} has the property that every term beginning with the third is the sum of the previous two. That is,

真题文字详解(英文)

Problem 1
Square ABCD has side length 10. Point E is on BC, and the area of △ABE is 40. What is BE?
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

Solution
We are given that the area of △ABE is 40, and that AB = 10.
The area of a triangle is:
A = (\frac{bh}{2})
Using AB as the height of △ABE,
40 = (\frac{10b}{2})
and solving for b,
b = 8, which is (E).

Problem 2
A softball team played ten games, scoring 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?
(A) 35 (B) 40 (C) 45 (D) 50 (E) 55

Solution
To score twice as many runs as their opponent, the softball team must have scored an even number.
Therefore we can deduce that when they scored an odd number of runs, they lost by one, and when they scored an even number of runs, they won by twice as much.
Therefore, the total runs by the opponent is (2 + 4 + 6 + 8 + 10) + (1 + 2 + 3 + 4 + 5) = 45, which is C.

Problem 3

真题文字详解(中英双语)

Problem 1
Square ABCD has side length 10. Point E is on BC, and the area of △ABE is 40. What is BE?
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

Solution
We are given that the area of △ABE is 40, and that AB = 10.
The area of a triangle is:
A = (\frac{bh}{2})
Using AB as the height of △ABE,
40 = (\frac{10b}{2})
and solving for b,
b = 8, which is (E).

Problem 2
A softball team played ten games, scoring 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?
(A) 35 (B) 40 (C) 45 (D) 50 (E) 55

Solution
To score twice as many runs as their opponent, the softball team must have scored an even number.
Therefore we can deduce that when they scored an odd number of runs, they lost by one, and when they scored an even number of runs, they won by twice as much.
Therefore, the total runs by the opponent is (2 + 4 + 6 + 8 + 10) + (1 + 2 + 3 + 4 + 5) = 45, which is C.

Problem 3