amcdrill | AMC8 AMC10 AMC12 Practice

Q1

What is $(-1)^1 + (-1)^2 + \dots + (-1)^{2006}$?

什么是 $(-1)^1 + (-1)^2 + \dots + (-1)^{2006}$？

(A) -2006 -2006 (B) -1 -1 (C) 0 0 (D) 1 1 (E) 2006 2006

Q2

For real numbers $x$ and $y$, define $x \ast y = (x + y)(x - y)$. What is $3 \ast (4 \ast 5)$?

对于实数 $x$ 和 $y$，定义 $x \ast y = (x + y)(x - y)$。什么是 $3 \ast (4 \ast 5)$？

(A) -72 -72 (B) -27 -27 (C) -24 -24 (D) 24 24 (E) 72 72

Q3

A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score?

两队——美洲狮队和黑豹队进行了一场足球比赛。两队总共得了34分，美洲狮队赢了14分。黑豹队得了多少分？

(A) 10 10 (B) 14 14 (C) 17 17 (D) 20 20 (E) 24 24

Q4

Circles of diameter 1 inch and 3 inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area?

直径分别为1英寸和3英寸的两个圆心相同。小圆涂成红色，大圆内小圆外的部分涂成蓝色。蓝色的面积与红色的面积之比是多少？

(A) 2 2 (B) 3 3 (C) 6 6 (D) 8 8 (E) 9 9

Q5

A $2 \times 3$ rectangle and a $3 \times 4$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?

一个 $2 \times 3$ 矩形和一个 $3 \times 4$ 矩形不重叠内部点地包含在一个正方形内，且正方形的边与两个矩形的边平行。正方形的最小可能面积是多少？

(A) 16 16 (B) 25 25 (C) 36 36 (D) 49 49 (E) 64 64

Q6

A region is bounded by semicircular arcs constructed on the side of a square whose sides measure $2/\pi$, as shown. What is the perimeter of this region?

一个区域由在边长为$2/\pi$的正方形边上构造的半圆弧所包围，如图所示。该区域的周长是多少？

(A) $\frac{4}{\pi}$ $\frac{4}{\pi}$ (B) 2 2 (C) $\frac{8}{\pi}$ $\frac{8}{\pi}$ (D) 4 4 (E) $\frac{16}{\pi}$ $\frac{16}{\pi}$

Q7

Which of the following is equivalent to $\sqrt{\frac{x}{1-x^2}}$ when $x < 0$?

当$x < 0$时，下列哪个与$\sqrt{\frac{x}{1-x^2}}$等价？

(A) $-x$ $-x$ (B) $x$ $x$ (C) 1 1 (D) $\sqrt{\frac{x}{2}}$ $\sqrt{\frac{x}{2}}$ (E) $x\sqrt{-1}$ $x\sqrt{-1}$

Q8

A square of area 40 is inscribed in a semicircle as shown. What is the area of the semicircle?

一个面积为40的正方形内接于一个半圆中，如图所示。半圆的面积是多少？

(A) $20\pi$ $20\pi$ (B) $25\pi$ $25\pi$ (C) $30\pi$ $30\pi$ (D) $40\pi$ $40\pi$ (E) $50\pi$ $50\pi$

Q9

Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade?

Francesca使用100克柠檬汁、100克糖和400克水制作柠檬水。100克柠檬汁含25卡路里，100克糖含386卡路里。水不含卡路里。她的柠檬水中200克含有多少卡路里？

(A) 129 129 (B) 137 137 (C) 174 174 (D) 223 223 (E) 411 411

Q10

In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?

在一个具有整数边长的三角形中，一条边是第二条边的三倍，第三条边长为15。该三角形的最大可能周长是多少？

(A) 43 43 (B) 44 44 (C) 45 45 (D) 46 46 (E) 47 47

Q11

What is the tens digit in the sum $7! + 8! + 9! + \dots + 2006!$?

求和 $7! + 8! + 9! + \dots + 2006!$ 的十位数字是多少？

(A) 1 1 (B) 3 3 (C) 4 4 (D) 6 6 (E) 9 9

Q12

The lines $x = \frac{1}{4}y + a$ and $y = \frac{1}{4}x + b$ intersect at the point $(1,2)$. What is $a+b$?

直线 $x = \frac{1}{4}y + a$ 和 $y = \frac{1}{4}x + b$ 相交于点 $(1,2)$。求 $a+b$。

(A) 0 0 (B) $\frac{3}{4}$ $\frac{3}{4}$ (C) 1 1 (D) 2 2 (E) $\frac{9}{4}$ $\frac{9}{4}$

Q13

Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee?

Joe 和 JoAnn 各买了 12 盎司咖啡，装在 16 盎司的杯子里。Joe 喝了 2 盎司咖啡后加了 2 盎司奶油。JoAnn 先加了 2 盎司奶油，充分搅拌后喝了 2 盎司。Joe 的咖啡中奶油量与 JoAnn 的咖啡中奶油量的比值为多少？

(A) $\frac{6}{7}$ $\frac{6}{7}$ (B) $\frac{13}{14}$ $\frac{13}{14}$ (C) 1 1 (D) $\frac{14}{13}$ $\frac{14}{13}$ (E) $\frac{7}{6}$ $\frac{7}{6}$

Q14

Let $a$ and $b$ be the roots of the equation $x^2 - mx + 2 = 0$. Suppose that $a + (1/b)$ and $b + (1/a)$ are the roots of the equation $x^2 - px + q = 0$. What is $q$?

设方程 $x^2 - mx + 2 = 0$ 的根为 $a$ 和 $b$。假设 $a + (1/b)$ 和 $b + (1/a)$ 是方程 $x^2 - px + q = 0$ 的根。求 $q$。

(A) $\frac{5}{2}$ $\frac{5}{2}$ (B) $\frac{7}{2}$ $\frac{7}{2}$ (C) 4 4 (D) $\frac{9}{2}$ $\frac{9}{2}$ (E) 8 8

Q15

Rhombus ABCD is similar to rhombus BFDE. The area of rhombus ABCD is 24, and ∠BAD = 60°. What is the area of rhombus BFDE?

菱形 ABCD 与菱形 BFDE 相似。菱形 ABCD 的面积为 24，∠BAD = 60°。求菱形 BFDE 的面积。

(A) 6 6 (B) 4√3 4√3 (C) 8 8 (D) 9 9 (E) 6√3 6√3

Q16

Leap Day, February 29, 2004, occurred on a Sunday. On what day of the week will Leap Day, February 29, 2020, occur?

闰日，2004年2月29日，是星期日。2020年2月29日的闰日是星期几？

(A) Tuesday 星期二 (B) Wednesday 星期三 (C) Thursday 星期四 (D) Friday 星期五 (E) Saturday 星期六

Q17

Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process the contents of the two bags are the same?

Bob 和 Alice 各有一个袋子，里面各有一个蓝、绿、橙、红、紫色的球。Alice 随机从她的袋子里选一个球放入 Bob 的袋子。然后 Bob 随机从他的袋子里选一个球放入 Alice 的袋子。经过这个过程后，两个袋子的内容相同概率是多少？

(A) 1/10 1/10 (B) 1/6 1/6 (C) 1/5 1/5 (D) 1/3 1/3 (E) 1/2 1/2

Q18

Let $a_1, a_2, \dots$ be a sequence for which $a_1 = 2$, $a_2 = 3$, and $a_n = \frac{a_{n-1}}{a_{n-2}}$ for each positive integer $n \ge 3$. What is $a_{2006}$?

设序列 $a_1, a_2, \dots$ 满足 $a_1 = 2$，$a_2 = 3$，且对每个正整数 $n \ge 3$，$a_n = \frac{a_{n-1}}{a_{n-2}}$。 $a_{2006}$ 是多少？

(A) 1/2 1/2 (B) 2/3 2/3 (C) 3/2 3/2 (D) 2 2 (E) 3 3

Q19

A circle of radius 2 is centered at O. Square OABC has side length 1. Sides AB and CB are extended past B to meet the circle at D and E, respectively. What is the area of the shaded region in the figure, which is bounded by BD, BE, and the minor arc connecting D and E?

圆心为 O，半径为 2。正方形 OABC 边长为 1。边 AB 和 CB 分别向 B 外延长，与圆相交于 D 和 E。图中阴影区域由 BD、BE 和连接 D 和 E 的短弧所围成，其面积是多少？

(A) $\frac{\pi}{3} + 1 - \sqrt{3}$ $\frac{\pi}{3} + 1 - \sqrt{3}$ (B) $\frac{\pi}{2}(2 - \sqrt{3})$ $\frac{\pi}{2}(2 - \sqrt{3})$ (C) $\pi(2 - \sqrt{3})$ $\pi(2 - \sqrt{3})$ (D) $\frac{\pi}{6} + \frac{\sqrt{3} - 1}{2}$ $\frac{\pi}{6} + \frac{\sqrt{3} - 1}{2}$ (E) $\frac{\pi}{3} - 1 + \sqrt{3}$ $\frac{\pi}{3} - 1 + \sqrt{3}$

Q20

In rectangle ABCD, we have A = (6, -22), B = (2006, 178), and D = (8, y), for some integer y. What is the area of rectangle ABCD?

在矩形 ABCD 中，A = (6, -22)，B = (2006, 178)，D = (8, y)，y 为某整数。矩形 ABCD 的面积是多少？

(A) 4000 4000 (B) 4040 4040 (C) 4400 4400 (D) 40,000 40,000 (E) 40,400 40,400

Q21

For a particular peculiar pair of dice, the probabilities of rolling 1, 2, 3, 4, 5, and 6 on each die are in the ratio 1 : 2 : 3 : 4 : 5 : 6. What is the probability of rolling a total of 7 on the two dice?

对于一对特殊的骰子，每颗骰子上掷出1、2、3、4、5和6的概率之比为1 : 2 : 3 : 4 : 5 : 6。掷两个骰子总和为7的概率是多少？

(A) $\frac{4}{63}$ $\frac{4}{63}$ (B) $\frac{1}{8}$ $\frac{1}{8}$ (C) $\frac{8}{63}$ $\frac{8}{63}$ (D) $\frac{1}{6}$ $\frac{1}{6}$ (E) $\frac{2}{7}$ $\frac{2}{7}$

Q22

Elmo makes N sandwiches for a fundraiser. For each sandwich he uses B globs of peanut butter at 4¢ per glob and J blobs of jam at 5¢ per blob. The cost of the peanut butter and jam to make all the sandwiches is $2.53. Assume that B, J, and N are positive integers with N > 1. What is the cost of the jam Elmo uses to make the sandwiches?

Elmo为筹款活动做了N个三明治。对于每个三明治，他用了B团花生酱（每团4美分）和J团果酱（每团5美分）。制作所有三明治的花生酱和果酱总成本为2.53美元。假设B、J和N为正整数且N > 1。Elmo制作三明治所用果酱的成本是多少？

(A) $1.05 $1.05 (B) $1.25 $1.25 (C) $1.45 $1.45 (D) $1.65 $1.65 (E) $1.85 $1.85

Q23

A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7, as shown. What is the area of the shaded quadrilateral?

一个三角形通过从顶点向对边画两条线，被分割成三个三角形和一个四边形。三个三角形的面积分别为3、7和7，如图所示。阴影四边形的面积是多少？

(A) 15 15 (B) 17 17 (C) $\frac{35}{2}$ $\frac{35}{2}$ (D) 18 18 (E) $\frac{55}{3}$ $\frac{55}{3}$

Q24

Circles with centers at O and P have radii 2 and 4, respectively, and are externally tangent. Points A and B on the circle with center O and points C and D on the circle with center P are such that AD and BC are common external tangents to the circles. What is the area of the concave hexagon AOBCPD?

中心分别为O和P的圆，半径分别为2和4，且外部相切。在以O为中心的圆上的点A和B，以及以P为中心的圆上的点C和D，使得AD和BC是圆的公共外部切线。凹六边形AOBCPD的面积是多少？

(A) $18\sqrt{3}$ $18\sqrt{3}$ (B) $24\sqrt{2}$ $24\sqrt{2}$ (C) 36 36 (D) $24\sqrt{3}$ $24\sqrt{3}$ (E) $32\sqrt{2}$ $32\sqrt{2}$

Q25

Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. “Look, daddy!” she exclaims. “That number is evenly divisible by the age of each of us kids!” “That’s right,” replies Mr. Jones, “and the last two digits just happen to be my age.” Which of the following is not the age of one of Mr. Jones’s children?

Jones先生有八个不同年龄的孩子。在家庭旅行中，他最大的孩子9岁，看到一个四位数车牌号，其中两个数字各出现两次。“看，爸爸！”她叫道。“这个数字能被我们每个孩子的年龄整除！”“没错，”Jones先生回答，“而且最后两位数字正好是我的年龄。”以下哪项不是Jones先生孩子之一的年龄？

(A) 4 4 (B) 5 5 (C) 6 6 (D) 7 7 (E) 8 8

AMC10 2006 B