2016 AMC amc10b 真题 答案 详解

---

中英双语真题

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

Problem 2
If (n \heartsuit m = n^3m^2), what is (\frac{2 \heartsuit 4}{4 \heartsuit 2})?
如果(n \heartsuit m = n^3m^2)，(\frac{2 \heartsuit 4}{4 \heartsuit 2})是多少？
(A) (\frac{1}{4}) (B) (\frac{1}{2}) (C) 1 (D) 2 (E) 4

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

英文真题

2016 AMC 10B 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 If (n \heartsuit m = n^3m^2), what is (\frac{2 \heartsuit 4}{4 \heartsuit 2})? (A) (\frac{1}{4}) (B) (\frac{1}{2}) (C) 1 (D) 2 (E) 4 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 Zoey read 15 books, one at a time. The first book took her 1 day to read, the second book took her 2 days to read, the third book took her 3 days to read, and so on, with each book taking her 1 more day to read than the previous book. Zoey finished the first book on a Monday, and the second on a Wednesday. On what day of the week did she finish her 15th book? (A) Sunday (B) Monday (C) Wednesday (D) Friday (E) Saturday Problem 5 The mean age of Amanda's 4 cousins is 8, and their median age is 5. What is the sum of the ages of Amanda's youngest and oldest cousins? (A) 13 (B) 16 (C) 19 (D) 22 (E) 25 Problem 6 Isaac added two three-digit positive integers. All six digits in these numbers are different. Isaac's sum is a three-digit number (S). What is the smallest possible value for the sum of the digits of (S)? (A) 1 (B) 4 (C) 5 (D) 15 (E) 20 Problem 7 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 8 What is the tens digit of (2015^{2016} - 2017) ? (A) 0 (B) 1 (C) 3 (D) 5 (E) 8

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
Factorizing the numerator, (\frac{\frac{1}{a} \cdot (2 + \frac{1}{2})}{a}) then becomes (\frac{\frac{5}{2}}{a^2}) which is equal to (\frac{5}{2} \cdot 2^2) which is (D) 10.

Solution 2
Substituting (\frac{1}{2}) for (a) in (\frac{\frac{1}{a} \cdot (2 + \frac{1}{2})}{a}) gives us (D) 10.

Problem 2 If (n \heartsuit m = n^3m^2), what is (\frac{2 \heartsuit 4}{4 \heartsuit 2})?
(A) (\frac{1}{4}) (B) (\frac{1}{2}) (C) 1 (D) 2 (E) 4

Solution 1
(\frac{2^3(2^2)^2}{(2^2)^3 2^2} = \frac{2^7}{2^8} = \frac{1}{2}) which is (B).

Solution 2
We can replace 2 and 4 with (a) and (b) respectively. Then substituting with (n) and (m) we can get (\frac{a^3 b^2}{b^3 a^2} = \frac{a}{b}) and substitute to get (\frac{2}{4} = \left[\frac{1}{2}\right]) which is (B).

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

Solution 1
Substituting carefully, (\left|\left|2016 - (-2016)\right| - 2016\right| - (-2016)) becomes (|4032 - 2016| + 2016 = 2016 + 2016 = 4032) which is (D).

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 Factorizing the numerator, (\frac{\frac{1}{a} \cdot (2 + \frac{1}{2})}{a}) then becomes (\frac{5}{2} \cdot a^2) which is equal to (\frac{5}{2} \cdot 2^2) which is (D) 10. Solution 2 Substituting (\frac{1}{2}) for (a) in (\frac{\frac{1}{a} \cdot (2 + \frac{1}{2})}{a}) gives us (D) 10 Problem 2 If (n \heartsuit m = n^3m^2), what is (\frac{2 \heartsuit 4}{4 \heartsuit 2})? (A) (\frac{1}{4}) (B) (\frac{1}{2}) (C) 1 (D) 2 (E) 4 Solution 1 (\frac{2^3(2^2)^2}{(2^2)^3 2^2} = \frac{2^7}{2^8} = \frac{1}{2}) which is (B). Solution 2 We can replace 2 and 4 with (a) and (b) respectively. Then substituting with (n) and (m) we can get (\frac{a^3 b^2}{b^3 a^2} = \frac{a}{b}) and substitute to get (\frac{2}{4} = \left[\frac{1}{2}\right]) which is (B). 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 Solution 1