2017 AMC amc12a 真题 答案 详解

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中英双语真题

2017 AMC12A

Problem 1

Pablo buys popsicles for his friends. The store sells single popsicles for $1 each, 3-popsicle boxes for $2, and 5-popsicle boxes for $3. What is the greatest number of popsicles that Pablo can buy with $8?

Pablo 给他的朋友买棒冰，商店单根棒冰卖 1 美元一根，1 盒 3 根棒冰每盒卖 2 美元，1 盒 5 根棒冰每盒卖 3 美元，Pablo 有 8 美元，最多可以买多少根棒冰？

(A) 8 (B) 11 (C) 12 (D) 13 (E) 15

Problem 2

The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers?

两个非零实数的和是它们乘积的 4 倍，问这 2 个数的倒数之和是多少？

(A) 1 (B) 2 (C) 4 (D) 8 (E) 12

Problem 3

Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which one of these statements necessarily follows logically?

Carroll 女士承诺任何只要在即将到来的考试中做对所有的选择题，此次考试就能得到一个 A，由此，下面哪句话在逻辑上是对的？

(A) If Lewis did not receive an A, then he got all of the multiple choice questions wrong. | 如果 Lewis 没有得到 A，那么他的所有选择题做的都是错的。

(B) If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong. | 如果 Lewis 没有得到 A，那么他至少有一道选择题做的是错的。

(C) If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A. | 如果 Lewis 至少有一个选择题做错了，那么他就得不到 A.

(D) If Lewis received an A, then he got all of the multiple choice questions right. | 如果 Lewis 得到一个 A，那么他所有的选择题都做对了。

(E) If Lewis received an A, then he got at least one of the multiple choice questions right. | 如果 Lewis 得到一个 A，那么他至少做对了一道选择题。

英文真题

2017 AMC 12A Problems

Problem 1

Pablo buys popsicles for his friends. The store sells single popsicles for $1 each, 3-popsicle boxes for $2, and 5-popsicle boxes for $3. What is the greatest number of popsicles that Pablo can buy with $8?

(A) 8 (B) 11 (C) 12 (D) 13 (E) 15

Problem 2

The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers?

(A) 1 (B) 2 (C) 4 (D) 8 (E) 12

Problem 3

Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which one of these statements necessarily follows logically?

(A) If Lewis did not receive an A, then he got all of the multiple choice questions wrong.

(B) If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.

(C) If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A.

(D) If Lewis received an A, then he got all of the multiple choice questions right.

(E) If Lewis received an A, then he got at least one of the multiple choice questions right.

Problem 4

Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?

(A) 30% (B) 40% (C) 50% (D) 60% (E) 70%

Problem 5

At a gathering of 30 people, there are 20 people who all know each other and 10 people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?

(A) 240 (B) 245 (C) 290 (D) 480 (E) 490

Problem 6

Joy has 30 thin rods, one each of every integer length from 1 cm through 30 cm. She places the rods with lengths 3 cm, 7 cm, and 15cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?

(A) 16 (B) 17 (C) 18 (D) 19 (E) 20

Problem 7

Define a function on the positive integers recursively by ( f(1)=2 ), ( f(n)=f(n-1)+1 ) if n is even, and ( f(n)=f(n-2)+2 ) if n is odd and greater than 1. What is ( f(2017) )?

(A) 2017 (B) 2018 (C) 4034 (D) 4035 (E) 4036

Problem 8

The region consisting of all points in three-dimensional space within 3 units of line segment AB has volume ( 216\pi ). What is the length AB?

(A) 6 (B) 12 (C) 18 (D) 20 (E) 24

Problem 9

Let S be the set of points (x, y) in the coordinate plane such that two of the three quantities 3, x + 2, and y - 4 are equal and the third of the three quantities is no greater than the common value. Which of the following is a correct description of S?

真题文字详解(英文)

Problem 1 Pablo buys popsicles for his friends. The store sells single popsicles for $1 each, 3-popsicle boxes for $2, and 5-popsicle boxes for $3. What is the greatest number of popsicles that Pablo can buy with $8?
(A) 8 (B) 11 (C) 12 (D) 13 (E) 15

Solution We can take two 5-popsicle boxes and one 3-popsicle box with $8. Note that it is optimal since one popsicle is at the rate of $1 per popsicle, three popsicles at $2/3 per popsicle and finally five popsicles at $3/5 per popsicle, hence we want as many $3 sets as possible. It is clear that the above is the optimal method. D

Problem 2 The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers?
(A) 1 (B) 2 (C) 4 (D) 8 (E) 12

Solution Let ( x, y ) be our two numbers. Then ( x + y = 4xy ). Thus,
[ \frac{1}{x} + \frac{1}{y} = \frac{x+y}{xy} = 4 ]
C

Solution 2 We can let ( x = y ). Then, we have that ( 2x = 4x^2 ) making ( x = \frac{1}{2} ). The answer is ( \frac{2}{x} = 4 = C ). ~ Solasky (talk)

Problem 3 Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which one of these statements necessarily follows logically?
(A) If Lewis did not receive an A, then he got all of the multiple choice questions wrong.
(B) If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.
(C) If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A.
(D) If Lewis received an A, then he got all of the multiple choice questions right.
(E) If Lewis received an A, then he got at least one of the multiple choice questions right.

Solution Taking the contrapositive of the statement "if he got all of them right, he got an A" yields "if he didn't get an A, he didn't get all of them right", yielding the answer (B).

Problem 4 Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?
(A) 30% (B) 40% (C) 50% (D) 60% (E) 70%

Solution Let ( j ) represent how far Jerry walked, and ( s ) represent how far Sylvia walked. Since the field is a square, and Jerry walked two sides of it, while Sylvia walked the diagonal, we can simply define the side of the square field to be one, and find the distances they walked. Since Jerry walked two sides, ( j = 2 ). Since Sylvia walked the diagonal, she walked the hypotenuse of a 45, 45, 90 triangle with leg length 1. Thus, ( s = \sqrt{2} = 1.414... ). We can then take ( \frac{j - s}{j} \approx \frac{2 - 1.4}{2} = 0.3 = 30\%). A

Problem 5 At a gathering of 30 people, there are 20 people who all know each other and 10 people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?
(A) 240 (B) 245 (C) 290 (D) 480 (E) 490

Solution 1 (Basic) All of the handshakes will involve at least one person from the 10 who knows no one. Label these ten people ( A, B, C, D, E, F, G, H, I, J ). Person ( A ) from the group of 10 will initiate a handshake with everyone else (29 people). Person ( B ) initiates 28 handshakes plus the one already counted from person ( A ). Person ( C ) initiates 27 new hand

真题文字详解(中英双语)

Problem 1 Pablo buys popsicles for his friends. The store sells single popsicles for $1 each, 3-popsicle boxes for $2, and 5-popsicle boxes for $3. What is the greatest number of popsicles that Pablo can buy with $8?
(A) 8 (B) 11 (C) 12 (D) 13 (E) 15

Solution We can take two 5-popsicle boxes and one 3-popsicle box with $8. Note that it is optimal since one popsicle is at the rate of $1 per popsicle, three popsicles at $2/3 per popsicle and finally five popsicles at $3/5 per popsicle, hence we want as many $3 sets as possible. It is clear that the above is the optimal method. D

Problem 2 The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers?
(A) 1 (B) 2 (C) 4 (D) 8 (E) 12

Solution Let ( x, y ) be our two numbers. Then ( x + y = 4xy ). Thus,
[ \frac{1}{x} + \frac{1}{y} = \frac{x+y}{xy} = 4 ]
C

Solution 2 We can let ( x = y ). Then we have that ( 2x = 4x^2 ) making ( x = \frac{1}{2} ). The answer is ( \frac{2}{x} = 4 = C ). ~ Solanky (talk)

Problem 3 Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which one of these statements necessarily follows logically?
(A) If Lewis did not receive an A, then he got all of the multiple choice questions wrong.
(B) If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.
(C) If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A.
(D) If Lewis received an A, then he got all of the multiple choice questions right.
(E) If Lewis received an A, then he got at least one of the multiple choice questions right.

Solution Taking the contrapositive of the statement "if he got all of them right, he got an A" yields "if he didn't get an A, he didn't get all of them right", yielding the answer (B).

Problem 4 Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?
(A) 30% (B) 40% (C) 50% (D) 60% (E) 70%

Solution Let ( j ) represent how far Jerry walked, and ( s ) represent how far Sylvia walked. Since the field is a square, and Jerry walked two sides of it, while Sylvia walked the diagonal, we can simply define the side of the square field to be one, and find the distances they walked. Since Jerry walked two sides, ( j = 2 ). Since Sylvia walked the diagonal, she walked the hypotenuse of a 45-45-90 triangle with leg length 1. Thus, ( s = \sqrt{2} = 1.414... ). We can then take ( \frac{j-s}{j} \approx \frac{2 - 1.4}{2} = 0.3 = 30\%). A

Problem 5 At a gathering of 30 people, there are 20 people who all know each other and 10 people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?
(A) 240 (B) 245 (C) 290 (D) 480 (E) 490

Solution 1 (Basic) All of the handshakes will involve at least one person from the 10 who knows no one. Label these ten people ( A, B, C, D, E, F, G, H, I, J ). Person ( A ) from the group of 10 will initiate a handshake with everyone else (29 people). Person ( B ) initiates 28 handshakes plus the one already counted from person ( A ). Person ( C ) initiates 27 new handshakes plus the two