2000 AMC amc8 真题 答案 详解

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中英双语真题

2000 AMC8
Problem 1
Aunt Anna is 42 years old. Caitlin is 5 years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin?
Anna阿姨现在42岁，Caitlin比Brianna小5岁，Brianna的年龄是Anna阿姨的一半。问Caitlin多大？
(A) 15 (B) 16 (C) 17 (D) 21 (E) 37

Problem 2
Which of these numbers is less than its reciprocal?
下面哪个数比它的倒数小？
(A) -2 (B) -1 (C) 0 (D) 1 (E) 2

Problem 3
How many whole numbers lie in the interval between (\frac{5}{3}) and (2\pi) ?
在(\frac{5}{3})到(2\pi)之间有多少个整数？
(A) 2 (B) 3 (C) 4 (D) 5 (E) infinitely many

英文真题

Compiled on: January 1, 2021 AMC8 2000 Problems Page 96 of 109 Q1. Aunt Anna is 42 years old. Caitlin is 5 years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin? A) 15 B) 16 C) 17 D) 21 E) 37 Q2. Which of these numbers is less than its reciprocal? A) -2 B) -1 C) 0 D) 1 E) 2 Q3. How many whole numbers lie in the interval between (\frac{5}{3}) and (2\pi) ? A) 2 B) 3 C) 4 D) 5 E) infinitely many Q4. In 1960 only 5% of the working adults in Carlin City worked at home. By 1970 the 'at-home' work force had increased to 8%. In 1980 there were approximately 15% working at home, and in 1990 there were 30%. The graph that best illustrates this is: (A) % 30 + 20 + 10 + 1960 1970 1980 1990 (B) % 30 + 20 + 10 + 1960 1970 1980 1990 (C) % 30 + 20 + 10 + 1960 1970 1980 1990 (D) % 30 + 20 + 10 + 1960 1970 1980 1990 (E) % 30 + 20 + 10 + 1960 1970 1980 1990 Q5. Each principal of Lincoln High School serves exactly one 3-year term. What is the maximum number of principals this school could have during an 8-year period? A) 2 B) 3 C) 4 D) 5 E) 8 Q6. Figure ABCD is a square. Inside this square three smaller squares are drawn with the side lengths as labeled. The area of the shaded L-shaped region is

真题文字详解(英文)

Problem 1 Aunt Anna is 42 years old. Caitlin is 5 years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin?
(A) 15 (B) 16 (C) 17 (D) 21 (E) 37

Solution 1 If Brianna is half as old as Aunt Anna, then Brianna is ( \frac{42}{2} ) years old, or 21 years old.
If Caitlin is 5 years younger than Brianna, she is ( 21 - 5 ) years old, or 16.
So, the answer is (B).

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Problem 2 Which of these numbers is less than its reciprocal?
(A) -2 (B) -1 (C) 0 (D) 1 (E) 2

Solution 1 The number 0 has no reciprocal, and 1 and -1 are their own reciprocals. This leaves only 2 and -2. The reciprocal of 2 is ( \frac{1}{2} ), but 2 is not less than ( \frac{1}{2} ). The reciprocal of -2 is ( -\frac{1}{2} ), and -2 is less than ( -\frac{1}{2} ), so it is (A).

Solution 2 The statement "a number is less than its reciprocal" can be translated as ( x < \frac{1}{x} ).

Multiplication by ( x ) can be done if you do it in three parts: ( x > 0 ), ( x = 0 ), and ( x < 0 ). You have to be careful about the direction of the inequality, as you do not know the sign of ( x ).

If ( x > 0 ), the sign of the inequality remains the same. Thus, we have ( x^2 < 1 ) when ( x > 0 ). This leads to ( 0 < x < 1 ).

If ( x = 0 ), the inequality ( x < \frac{1}{x} ) is undefined.

If ( x < 0 ), the sign of the inequality must be switched. Thus, we have ( x^2 > 1 ) when ( x < 0 ). This leads to ( x < -1 ).

Putting the solutions together, we have ( x < -1 ) or ( 0 < x < 1 ), or in interval notation, ( (-\infty, -1) \cup (0, 1) ). The only answer in that range is (A) -2.

Solution 3 Starting again with ( x < \frac{1}{x} ), we avoid multiplication by ( x ). Instead, move everything to the left, and find a common denominator:

( x < \frac{1}{x} )

( x - \frac{1}{x} < 0 )

( \frac{x^2 - 1}{x} < 0 )

真题文字详解(中英双语)

Problem 1 Aunt Anna is 42 years old. Caitlin is 5 years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin? 安娜阿姨今年 42 岁。凯特琳比布莉安娜小 5 岁，而布莉安娜的年龄是安娜阿姨的一半。凯特琳多少岁？
(A) 15 (B) 16 (C) 17 (D) 21 (E) 37

Solution 1 If Brianna is half as old as Aunt Anna, then Brianna is (\frac{42}{2}) years old, or 21 years old. 如果布莉安娜的年龄是安娜阿姨的一半，那么布莉安娜是 (\frac{42}{2}) 岁，或者 21 岁。
If Caitlin is 5 years younger than Brianna, she is (21 - 5) years old, or 16. 如果凯特琳比布莉安娜小 5 岁，那么她 (21 - 5) 岁，或者 16 岁。
So, the answer is B. 所以，答案是 B.

Problem 2 Which of these numbers is less than its reciprocal? 这些数中哪个小于它的倒数？
(A) -2 (B) -1 (C) 0 (D) 1 (E) 2

Solution Solution 1 The number 0 has no reciprocal, and 1 and -1 are their own reciprocals. This leaves only 2 and -2. The reciprocal of 2 is 1/2, but 2 is not less than 1/2. The reciprocal of -2 is -1/2, and -2 is less than -1/2, so it is A. 数字 0 没有倒数，1 和 -1 是它们自己的倒数。这就只剩下 2 和 -2 了。2 的倒数是 1/2，但 2 不小于 1/2。-2 的倒数是 -1/2，而 -2 小于 -1/2，所以是 A。

Solution 2 The statement "a number is less than its reciprocal" can be translated as (x < \frac{1}{x}). “一个数小于它的倒数”这个陈述可以翻译为 (x < \frac{1}{x})。
Multiplication by (x) can be done if you do it in three parts: (x > 0), (x = 0), and (x < 0). You have to be careful about the direction of the inequality, as you do not know the sign of (x). 乘以 (x) 可以通过分成三部分：(x > 0)、(x = 0) 和 (x < 0) 来完成。你必须小心不等式的方向，因为你不知道 (x) 的符号。
If (x > 0), the sign of the inequality remains the same. Thus, we have (x^2 < 1) when (x > 0). 这导致 (0 < x < 1)。如果 (x > 0)，不等式的符号保持不变。因此，当 (x > 0) 时，我们有 (x^2 < 1)。