2016 AMC amc12b 真题 答案 详解

---

中英双语真题

2016 AMC12B

Problem 1

What is the value of (\frac{2a^{-1} + \frac{a^{-1}}{2}}{a}) when (a = \frac{1}{2})?

当(a=\frac{1}{2})时，(\frac{2a^{-1}+\frac{a^{-1}}{2}}{a})的值是多少？

(A) 1 (B) 2 (C) (\frac{5}{2}) (D) 10 (E) 20

Problem 2

The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of 1 and 2016 is closest to which integer?

两个数的调和平均值可以由它们乘积的2倍除以它们的和得到。那么1和2016的调和平均值最接近下面哪个整数？

(A) 2 (B) 45 (C) 504 (D) 1008 (E) 2015

Problem 3

Let (x = -2016). What is the value of (||x| - x| - |x| - x)?

令(x=-2016)，则(||x|-x|-|x|-x)的值是多少？

(A) -2016 (B) 0 (C) 2016 (D) 4032 (E) 6048

英文真题

2016 AMC 12B Problems
Problem 1
What is the value of (\frac{2a^{-1} + \frac{a^{-1}}{2}}{a}) when (a = \frac{1}{2})?
(A) 1 (B) 2 (C) (\frac{5}{2}) (D) 10 (E) 20

Problem 2
The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of 1 and 2016 is closest to which integer?
(A) 2 (B) 45 (C) 504 (D) 1008 (E) 2015

Problem 3
Let (x = -2016). What is the value of (\left|\left||x| - x\right| - |x|\right| - x) ?
(A) -2016 (B) 0 (C) 2016 (D) 4032 (E) 6048

Problem 4
The ratio of the measures of two acute angles is (5:4), and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles?
(A) 75 (B) 90 (C) 135 (D) 150 (E) 270

Problem 5
The War of 1812 started with a declaration of war on Thursday, June 18, 1812. The peace treaty to end the war was signed 919 days later, on December 24, 1814. On what day of the week was the treaty signed?
(A) Friday (B) Saturday (C) Sunday (D) Monday (E) Tuesday

Problem 6
All three vertices of (\triangle ABC) lie on the parabola defined by (y = x^2), with (A) at the origin and (\overline{BC}) parallel to the (x)-axis. The area of the triangle is 64. What is the length of (BC) ?
(A) 4 (B) 6 (C) 8 (D) 10 (E) 16

Problem 7
Josh writes the numbers 1, 2, 3, ..., 99, 100. He marks out 1, skips the next number (2), marks out 3, and continues skipping and marking out the next number to the end of the list. Then he goes back to the start of his list, marks out the first remaining number (2), skips the next number (4), marks out 6, skips 8, marks out 10, and so on to the end. Josh continues in this manner until only one number remains. What is that number?
(A) 13 (B) 32 (C) 56 (D) 64 (E) 96

Problem 8
A thin piece of wood of uniform density in the shape of an equilateral triangle with side length 3 inches weighs 12 ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of 5 inches. Which of the following is closest to the weight, in ounces, of the second piece?
(A) 14.0 (B) 16.0 (C) 20.0 (D) 33.3 (E) 55.6

真题文字详解(英文)

Problem 1 What is the value of (\frac{2a^{-1} + \frac{a^{-1}}{2}}{a}) when (a = \frac{1}{2})?
(A) 1 (B) 2 (C) (\frac{5}{2}) (D) 10 (E) 20

Solution By: Dragonfly We find that (a^{-1}) is the same as (2), since a number to the power of (-1) is just the reciprocal of that number. We then get the equation to be (\frac{2 \times 2 + \frac{2}{2}}{\frac{1}{2}}) We can then simplify the equation to get (D) 10

Problem 2 The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of (1) and (2016) is closest to which integer?
(A) 2 (B) 45 (C) 504 (D) 1008 (E) 2015

Solution Since the harmonic mean is (2) times their product divided by their sum, we get the equation (\frac{2 \times 1 \times 2016}{1+2016}) which is then (\frac{4032}{2017}) which is finally closest to (A) 2 dragonfly You can also think of (\frac{2 \times 1 \times 2016}{1+2016}) as (2 \times \frac{2016}{2017}) which it should be obvious that our number is most closest to 2 ((\left[\frac{2016}{2017}\right] = 1)), our answer here is (A)

Problem 3 Let (x = -2016). What is the value of (\left|\left||x| - x\right| - |x|\right| - x) ?
(A) −2016 (B) 0 (C) 2016 (D) 4032 (E) 6048

真题文字详解(中英双语)

Problem 1 What is the value of (\frac{2a^{-1} + \frac{a^{-1}}{2}}{a}) when (a = \frac{1}{2})? (\frac{2a^{-1} + \frac{a^{-1}}{2}}{a}) 的值是多少？当 (a = \frac{1}{2}) 时？ (A) 1 (B) 2 (C) (\frac{5}{2}) (D) 10 (E) 20 Solution By: Dragonfly We find that (a^{-1}) is the same as 2, since a number to the power of -1 is just the reciprocal of that number. We then get the equation to be (\frac{2 \times 2 + \frac{2}{2}}{\frac{1}{2}}) We can then simplify the equation to get (D) 10 Problem 2 The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of 1 and 2016 is closest to which integer? (A) 2 (B) 45 (C) 504 (D) 1008 (E) 2015 Solution Since the harmonic mean is 2 times their product divided by their sum, we get the equation (\frac{2 \times 1 \times 2016}{1 + 2016}) which is then (\frac{4032}{2017}) which is finally closest to (A) 2 -dragonfly You can also think of (\frac{2 \times 1 \times 2016}{1 + 2016}) as (2 \times \frac{2016}{2017}) which it should be obvious that our number is most closest to 2 ((\left\lceil \frac{2016}{2017} \right\rceil = 1)), our answer here is (A) Problem 3 Let (x = -2016). What is the value of (\left| |x| - x \right| - |x|)?