2010 AMC amc10b 真题 答案 详解

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中英双语真题

2010 AMC 10B

Problem 1
What is (100(100-3)-(100\cdot100-3))?
(100(100-3)-(100\cdot100-3)) 的值是多少?
(A) -20,000 (B) -10,000 (C) -297 (D) -6 (E) 0

Problem 2
Markala attended two meetings during her 9-hour work day. The first meeting took 45 minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?
Markala 在她工作的 9 小时里参加了 2 次会议。第一场会议耗时 45 分钟，第二场会议时长是第一场的 2 倍。问她工作总时间的百分之多少是花在了会议上?
(A) 15 (B) 20 (C) 25 (D) 30 (E) 35

Problem 3
A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?
一张抽屉里有红色、绿色、蓝色和白色的袜子，每种颜色至少 2 只。问从抽屉里至少需要抽出多少只袜子才能保证存在一双颜色匹配的袜子?
(A) 3 (B) 4 (C) 5 (D) 8 (E) 9

Problem 4
For a real number (x), define (\heartsuit(x)) to be the average of (x) and (x^2). What is (\heartsuit(1)+\heartsuit(2)+\heartsuit(3))?
对于实数 (x)，定义 (\heartsuit(x)) 为 (x) 和 (x^2) 的平均值，(\heartsuit(1)+\heartsuit(2)+\heartsuit(3)) 的值是多少?
(A) 3 (B) 6 (C) 10 (D) 12 (E) 20

英文真题

2010 AMC 10B Problems Problem 1 What is $100(100-3)-(100\cdot100-3)$?
(A) -20,000 (B) -10,000 (C) -297 (D) -6 (E) 0

Problem 2 Makayla attended two meetings during her 9-hour work day. The first meeting took 45 minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?
(A) 15 (B) 20 (C) 25 (D) 30 (E) 35

Problem 3 A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?
(A) 3 (B) 4 (C) 5 (D) 8 (E) 9

Problem 4 For a real number $x$, define $\heartsuit(x)$ to be the average of $x$ and $x^2$. What is $\heartsuit(1)+\heartsuit(2)+\heartsuit(3)$?
(A) 3 (B) 6 (C) 10 (D) 12 (E) 20

Problem 5 A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

Problem 6 A circle is centered at $O$, $\overline{AB}$ is a diameter and $C$ is a point on the circle with $\angle COB=50^\circ$. What is the degree measure of $\angle CAB$?
(A) 20 (B) 25 (C) 45 (D) 50 (E) 65

Problem 7 A triangle has side lengths 10, 10, and 12. A rectangle has width 4 and area equal to the area of the triangle. What is the perimeter of this rectangle?
(A) 16 (B) 24 (C) 28 (D) 32 (E) 36

Problem 8 A ticket to a school play cost $x$ dollars, where $x$ is a whole number. A group of 9th graders buys tickets costing a total of $48, and a group of 10th graders buys tickets costing a total of $64. How many values for $x$ are possible?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

Problem 9 Lucky Larry's teacher asked him to substitute numbers for $a$, $b$, $c$, $d$, and $e$ in the expression $a-(b-(c-(d+e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry substituted for $a$, $b$, $c$, and $d$ were 1, 2, 3, and 4, respectively. What number did Larry substitute for $e$?
(A) -5 (B) -3 (C) 0 (D) 3 (E) 5

真题文字详解(英文)

Problem 1 What is $100(100-3)-(100\cdot100-3)?$
(A) -20,000 (B) -10,000 (C) -297 (D) -6 (E) 0

Solution
$100(100-3)-(100\cdot100-3)=10000-300-10000+3=-300+3=\boxed{(C)-297}.$

Problem 2 The following problem is from both the 2010 AMC 12B #1 and 2010 AMC 10B #2, so both problems redirect to this page.
Makarla attended two meetings during her 9-hour work day. The first meeting took 45 minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?
(A) 15 (B) 20 (C) 25 (D) 30 (E) 35

Solution
The total number of minutes in her 9-hour work day is $9\times60=540$. The total amount of time spend in meetings in minutes is $45+45\times2=135$. The answer is then $\frac{135}{540}=25\%$ or $\boxed{(C)}.$

Problem 3 A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?
(A) 3 (B) 4 (C) 5 (D) 8 (E) 9

Solution
After you draw 4 socks, you can have one of each color, so (according to the pigeonhole principle), if you pull $\boxed{(C)5}$ then you will be guaranteed a matching pair.

Problem 4 For a real number $x$, define $\heartsuit(x)$ to be the average of $x$ and $x^2$. What is $\heartsuit(1)+\heartsuit(2)+\heartsuit(3)?$
(A) 3 (B) 6 (C) 10 (D) 12 (E) 20

Solution
The average of two numbers, $a$ and $b$, is defined as $\frac{a+b}{2}$. Thus the average of $x$ and $x^2$ would be $\frac{x(x+1)}{2}$. With that said, we need to find the sum when we plug, 1, 2 and 3 into that equation. So:
$\frac{1(1+1)}{2}+\frac{2(2+1)}{2}+\frac{3(3+1)}{2}=\frac{2}{2}+\frac{6}{2}+\frac{12}{2}=1+3+6=\boxed{(C)10}.$

Problem 5 The following problem is from both the 2010 AMC 12B #4 and 2010 AMC 10B #5, so both problems redirect to this page.

真题文字详解(中英双语)

Problem 1 What is $100(100-3)-(100\cdot100-3)$?
(A) -20,000 (B) -10,000 (C) -297 (D) -6 (E) 0

Solution
$100(100-3)-(100\cdot100-3)=10000-300-10000+3=-300+3=\boxed{(C)-297}$.

Problem 2 The following problem is from both the 2010 AMC 12B #1 and 2010 AMC 10B #2, so both problems redirect to this page.
以下问题来自2010年AMC 12B #1和2010年AMC 10B #2，因此这两个问题都重定向到此页面。

Makarla attended two meetings during her 9-hour work day. The first meeting took 45 minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?
Makarla在她9小时的工作日里参加了两次会议。第一次会议持续了45分钟，第二次会议的时间是第一次的两倍。她参加会议的时间占她工作日的百分之多少？
(A) 15 (B) 20 (C) 25 (D) 30 (E) 35

Solution
The total number of minutes in her 9-hour work day is $9\times60=540$. The total amount of time spend in meetings in minutes is $45+45\times2=135$. The answer is then $\frac{135}{540}=\boxed{(C)25\%}$.

她9小时的工作日总共有$9\times60=540$分钟。她在会议中花费的总时间是$45+45\times2=135$分钟。因此答案是$\frac{135}{540}=\boxed{(C)25\%}$。

Problem 3 A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks that must be pulled from the drawer to guarantee a matching pair?
一个抽屉里有红、绿、蓝和白袜子，每种颜色至少有2双。要保证抽到一双相同的袜子，最少需要从抽屉里拿出多少只袜子？
(A) 3 (B) 4 (C) 5 (D) 8 (E) 9

Solution
After you draw 4 socks, you can have one of each color, so (according to the pigeonhole principle), if you pull $\boxed{(C)5}$ then you will be guaranteed a matching pair.
在你抽出4双袜子后，你可以拥有一双每种颜色的袜子，所以（根据鸽巢原理），如果你再抽出$\boxed{(C)5}$双，那么你将能保证配成一双。

Problem 4 For a real number $x$, define $\heartsuit(x)$ to be the average of $x$ and $x^2$. What is $\heartsuit(1)+\heartsuit(2)+\heartsuit(3)$?
对于一个实数$x$，定义$\heartsuit(x)$为$x$和$x^2$的平均值。$\heartsuit(1)+\heartsuit(2)+\heartsuit(3)$是多少？