2005 AMC amc10a 真题 答案 详解

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英文真题

2005 AMC 10A Problems Problem 1 While eating out, Mike and Joe each tipped their server $2. Mike tipped 10% of his bill and Joe tipped 20% of his bill. What was the difference, in dollars between their bills? (A) 2 (B) 4 (C) 5 (D) 10 (E) 20 Problem 2 For each pair of real numbers a ≠ b, define the operation as (a * b) = \frac{a + b}{a - b} What is the value of ((1 * 2) * 3)? (A) -\frac{2}{3} (B) -\frac{1}{5} (C) 0 (D) \frac{1}{2} (E) This value is not defined. Problem 3 The equations 2x + 7 = 3 and bx - 10 = -2 have the same solution x. What is the value of b? (A) -8 (B) -4 (C) -2 (D) 4 (E) 8 Problem 4 A rectangle with a diagonal of length x is twice as long as it is wide. What is the area of the rectangle? (A) \frac{1}{4}x^2 (B) \frac{2}{5}x^2 (C) \frac{1}{2}x^2 (D) x^2 (E) \frac{3}{2}x^2 Problem 5 A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately? (A) 100 (B) 200 (C) 300 (D) 400 (E) 500 Problem 6 The average (mean) of 20 numbers is 30, and the average of 30 other numbers is 20. What is the average of all 50 numbers? (A) 23 (B) 24 (C) 25 (D) 26 (E) 27 Problem 7 Josh and Mike live 13 miles apart. Yesterday Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 Problem 8 In the figure, the length of side AB of square ABCD is \sqrt{50} and BE = 1. What is the area of the inner square EFGH?

真题文字详解(英文)

Problem 1

While eating out, Mike and Joe each tipped their server $2. Mike tipped $10\%$ of his bill and Joe tipped $20\%$ of his bill. What was the difference, in dollars between their bills?

(A) 2 (B) 4 (C) 5 (D) 10 (E) 20

Solution

Let ( m ) be Mike's bill and ( j ) be Joe's bill.

[ \frac{10}{100}m = 2 ]

[ m = 20 ]

[ \frac{20}{100}j = 2 ]

[ j = 10 ]

So the desired difference is ( m - j = 20 - 10 = 10 \Rightarrow \textbf{(D)} 10 )

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Problem 2

For each pair of real numbers ( a \neq b ), define the operation ( \star ) as:

[ (a \star b) = \frac{a + b}{a - b} ]

What is the value of ( ((1 \star 2) \star 3) )?

(A) ( -\frac{2}{3} ) (B) ( -\frac{1}{5} ) (C) 0 (D) ( \frac{1}{2} ) (E) This value is not defined.

Solution

[ ((1 \star 2) \star 3) = \left( \frac{1 + 2}{1 - 2} \right) \star 3 = (-3 \star 3) = \frac{-3 + 3}{-3 - 3} = 0 \Rightarrow \textbf{(C)} 0 )

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Problem 3

The following problem is from both the 2005 AMC 12A #2 and 2005 AMC 10A #3, so both problems redirect to this page.

The equations ( 2x + 7 = 3 ) and ( bx - 10 = -2 ) have the same solution. What is the value of ( b )?

(A) -8 (B) -4 (C) 2 (D) 4 (E) 8

Solution

[ 2x + 7 = 3 \Rightarrow x = -2 ]

[ -2b - 10 = -2 \Rightarrow -2b = 8 \Rightarrow b = \textbf{(B)} -4 ]

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Problem 4

A rectangle with a diagonal of length ( x ) is twice as long as it is wide. What is the area of the rectangle?

真题文字详解(中英双语)

Problem 1

While eating out, Mike and Joe each tipped their server $2. Mike tipped 10% of his bill and Joe tipped 20% of his bill. What was the difference, in dollars between their bills?

(A) 2 (B) 4 (C) 5 (D) 10 (E) 20

Solution

Let m be Mike's bill and j be Joe's bill.

设m为迈克的账单，j为乔的账单。

(\frac{10}{100}m = 2)

(m = 20)

(\frac{20}{100}j = 2)

(j = 10)

So the desired difference is (m - j = 20 - 10 = 10 \Rightarrow) (D) 10

所以所需的差异是 (m - j = 20 - 10 = 10 \Rightarrow) (D) 10

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Problem 2

For each pair of real numbers a ≠ b, define the operation ⭑ as

对于每一对实数a≠b，定义运算⭑为

((a \star b) = \frac{a + b}{a - b})

What is the value of ((1 ⭑ 2) ⭑ 3)?

((1 ⭑ 2) ⭑ 3) 的值是多少？

(A) (-\frac{2}{3}) (B) (-\frac{1}{5}) (C) 0 (D) (\frac{1}{2}) (E) This value is not defined.

Solution

(((1 \star 2) \star 3) = \left( \frac{1 + 2}{1 - 2} \right) \star 3 = (-3 \star 3) = \frac{-3 + 3}{-3 - 3} = 0 \Rightarrow) (C) 0

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Problem 3

The following problem is from both the 2005 AMC 12A #2 and 2005 AMC 10A #3, so both problems redirect to this page.

以下问题既来自2005年AMC 12A #2，也来自2005年AMC 10A #3，因此这两个问题都指向这一页。