2015 AMC amc12b 真题 答案 详解

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中英双语真题

2015 AMC12B

Problem 1
What is the value of (2 - (-2)^{-2})?
(2 - (-2)^{-2}) 的值是多少?
(A) (-2) (B) (\frac{1}{16}) (C) (\frac{7}{4}) (D) (\frac{9}{4}) (E) 6

Problem 2
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?
Marie 连续做了3项耗时相同的任务，中间没有休息，她从下午1:00开始做第一个任务，在下午2:40完成第二个任务，她何时完成第三任务？
(A) 3:10 PM (B) 3:30 PM (C) 4:00 PM (D) 4:10 PM (E) 4:30 PM

Problem 3
Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, and one of the numbers is 28. What is the other number?
Isaac 把一个整数写了2遍，另一个整数写了3遍。这5个数之和为100，其中一个数是28，另一个数是多少?
(A) 8 (B) 11 (C) 14 (D) 15 (E) 18

英文真题

2015 AMC 12B Problems Problem 1 What is the value of $2-\left(-2\right)^{-2}$?
(A) -2 (B) $\frac{1}{16}$ (C) $\frac{7}{4}$ (D) $\frac{9}{4}$ (E) 6

Problem 2 Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?
(A) 3:10 PM (B) 3:30 PM (C) 4:00 PM (D) 4:10 PM (E) 4:30 PM

Problem 3 Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, and one of the numbers is 28. What is the other number?
(A) 8 (B) 11 (C) 14 (D) 15 (E) 18

Problem 4 David, Hikmet, Jack, Marta, Rand, and Todd were in a 12-person race with 6 other people. Rand finished 6 places ahead of Hikmet. Marta finished 1 place behind Jack. David finished 2 places behind Hikmet. Jack finished 2 places behind Todd. Todd finished 1 place behind Rand. Marta finished in 6th place. Who finished in 8th place?
(A) David (B) Hikmet (C) Jack (D) Rand (E) Todd

Problem 5 The Tigers beat the Sharks 2 out of the 3 times they played. They then played $N$ more times, and the Sharks ended up winning at least 95% of all the games played. What is the minimum possible value for $N$?
(A) 35 (B) 37 (C) 39 (D) 41 (E) 43

Problem 6 Back in 1930, Tillie had to memorize her multiplication facts from $0\times 0$ to $12\times 12$. The multiplication table she was given had rows and columns labeled with the factors, and the products formed the body of the table. To the nearest hundredth, what fraction of the numbers in the body of the table are odd?
(A) 0.21 (B) 0.25 (C) 0.46 (D) 0.50 (E) 0.75

Problem 7 A regular 15-gon has $L$ lines of symmetry, and the smallest positive angle for which it has rotational symmetry is $R$ degrees. What is $L+R$?
(A) 24 (B) 27 (C) 32 (D) 39 (E) 54

Problem 8 What is the value of $(625^{\log_562015})^\frac{1}{4}$?
(A) 5 (B) $\sqrt[4]{2015}$ (C) 625 (D) 2015 (E) $\sqrt[4]{5^{2015}}$

Problem 9 Larry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge. Larry throws first. The winner is the first person to knock the bottle off the ledge. At each turn the probability that a player knocks the bottle off the ledge is $\frac{1}{2}$, independently of what has happened before. What is the probability that Larry wins the game?
(A) $\frac{1}{2}$ (B) $\frac{3}{5}$ (C) $\frac{2}{3}$ (D) $\frac{3}{4}$ (E) $\frac{4}{5}$

真题文字详解(英文)

Problem 1 What is the value of $2-\left(-2\right)^{-2}$?
(A) -2 (B) $\frac{1}{16}$ (C) $\frac{7}{4}$ (D) $\frac{9}{4}$ (E) 6

Solution
$2-\left(-2\right)^{-2}=2-\frac{1}{\left(-2\right)^{2}}=2-\frac{1}{4}=\frac{8}{4}-\frac{1}{4}=\boxed{\text{(C)} \frac{7}{4}}$

Problem 2 Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?
(A) 3:10 PM (B) 3:30 PM (C) 4:00 PM (D) 4:10 PM (E) 4:30 PM

Solution
The first two tasks took $2:40 \, \text{PM} - 1:00 \, \text{PM} = 100$ minutes. Thus, each task takes $100 \div 2 = 50$ minutes. So the third task finishes at $2:40 \, \text{PM} + 50$ minutes $=\boxed{\text{(B)} 3:30 \, \text{PM}}$

Problem 3 Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, and one of the numbers is 28. What is the other number?
(A) 8 (B) 11 (C) 14 (D) 15 (E) 18

Solution
Let $a$ be the number written two times, and $b$ the number written three times. Then $2a+3b=100$. Plugging in $a=28$ doesn't yield an integer for $b$, so it must be that $b=28$, and we get $2a+84=100$. Solving for $a$, we obtain $a=\boxed{\text{(A)} 8}$.

Problem 4 Note This problem is also problem number 5 on the 2015 AMC 10B, just with different names. This is the link. Lian, Marzuq, Rafsan, Arabi, Nabeel, and Rahul were in a 12-person race with 6 other people. Nabeel finished 6 places ahead of Marzuq. Arabi finished 1 place behind Rafsan. Lian finished 2 places behind Marzuq. Rafsan finished 2 places behind Rahul. Rahul finished 1 place behind Nabeel. Arabi finished in 6th place. Who finished in 8th place?
(A) Lian (B) Marzuq (C) Rafsan (D) Nabeel (E) Rahul

Solution 1 Let — denote any of the 6 racers not named. Then the correct order looks like this:
—, Nabeel, Rahul, —, Rafsan, Arabi, —, Marzuq, —, Lian, —, —
Thus the 8th place runner is $\boxed{\text{(B) Marzuq}}$.

Solution 2 We list the people out vertically for clarity, starting with Marzuq and working from there.

真题文字详解(中英双语)

Problem 1 What is the value of $2-\left(-2\right)^{-2}$?
$2-\left(-2\right)^{-2}$ 的值是多少？
(A) -2 (B) $\frac{1}{16}$ (C) $\frac{7}{4}$ (D) $\frac{9}{4}$ (E) 6

Solution
$2-\left(-2\right)^{-2}=2-\frac{1}{\left(-2\right)^{2}}=2-\frac{1}{4}=\frac{8}{4}-\frac{1}{4}=\boxed{\text{(C)} \frac{7}{4}}$

Problem 2
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?
(A) 3:10 PM (B) 3:30 PM (C) 4:00 PM (D) 4:10 PM (E) 4:30 PM

Solution
The first two tasks took $2:40 \mathrm{PM}-1:00 \mathrm{PM}=100$ minutes. Thus, each task takes $100 \div 2=50$ minutes. So the third task finishes at $2:40 \mathrm{PM}+50$ minutes $=\boxed{\text{(B) } 3:30 \mathrm{PM}}$.

Problem 3
Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, and one of the numbers is 28. What is the other number?
(A) 8 (B) 11 (C) 14 (D) 15 (E) 18

Solution
Let $a$ be the number written two times, and $b$ the number written three times. Then $2a+3b=100$. Plugging in $a=28$ doesn't yield an integer for $b$, so it must be that $b=28$ and we get $2a+84=100$. Solving for $a$, we obtain $a=\boxed{\text{(A) } 8}$.

Problem 4
Note
This problem is also problem number 5 on the 2015 AMC 10B, just with different names. This is the link.
Lian, Marzuq, Rafsan, Arabi, Nabeel, and Rahul were in a 12-person race with 6 other people. Nabeel finished 6 places ahead of Marzuq. Arabi finished 1 place behind Rafsan. Lian finished 2 places behind Marzuq. Rafsan finished 2 places behind Rahul. Rahul finished 1 place behind Nabeel. Arabi finished in 6th place. Who finished in 8th place?
(A) Lian (B) Marzuq (C) Rafsan (D) Nabeel (E) Rahul

Solution 1
Let — denote any of the 6 racers not named. Then the correct order looks like this:
—, Nabeel, Rahul, —, Rafsan, Arabi, —, Marzuq, —, Lian, —, —
Thus the 8th place runner is $\boxed{\text{(B) } \text{Marzuq}}$.

Solution 2
We list the people out vertically for clarity, starting with Marzuq and working from there.