amcdrill | AMC8 AMC10 AMC12 Practice

Q1

Sandwiches at Joe's Fast Food cost $3 each and sodas cost $2 each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas?

Joe快餐店的三明治每个3美元，汽水每个2美元。购买5个三明治和8个汽水需要多少钱？

(A) 31 31 (B) 32 32 (C) 33 33 (D) 34 34 (E) 35 35

Q2

Define $x \otimes y = x^3 - y$. What is $h \otimes (h \otimes h)$?

定义$x \otimes y = x^3 - y$。求$h \otimes (h \otimes h)$的值？

(A) $-h$ $-h$ (B) 0 0 (C) $h$ $h$ (D) $2h$ $2h$ (E) $h^3$ $h^3$

Q3

The ratio of Mary's age to Alice's age is 3 : 5. Alice is 30 years old. How many years old is Mary?

Mary的年龄与Alice的年龄之比为3:5。Alice 30岁。Mary 多大岁数？

(A) 15 15 (B) 18 18 (C) 20 20 (D) 24 24 (E) 50 50

Q4

A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?

一个数字手表显示小时和分钟，并带有AM和PM标识。显示中各位数字之和的最大可能值是多少？

(A) 17 17 (B) 19 19 (C) 21 21 (D) 22 22 (E) 23 23

Q5

Doug and Dave shared a pizza with 8 equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half of the pizza. The cost of a plain pizza was \$8, and there was an additional cost of \$2 for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each then paid for what he had eaten. How many more dollars did Dave pay than Doug?

Doug和Dave分享一个分成8等份的披萨。Doug想要纯披萨，但Dave想要一半披萨加凤尾鱼。纯披萨的价格是8美元，在一半上加凤尾鱼额外收费2美元。Dave吃了所有凤尾鱼披萨片和一个纯片。Doug吃了剩下的。每人支付自己吃的部分的费用。Dave比Doug多付了多少钱？

(A) 1 1 (B) 2 2 (C) 3 3 (D) 4 4 (E) 5 5

Q6

What non-zero real value for $x$ satisfies $(7x)^{14} = (14x)^7$?

什么非零实数 $x$ 满足 $(7x)^{14} = (14x)^7$？

(A) $\dfrac{1}{7}$ $\dfrac{1}{7}$ (B) $\dfrac{2}{7}$ $\dfrac{2}{7}$ (C) 1 1 (D) 7 7 (E) 14 14

Q7

The 8 × 18 rectangle ABCD is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $y$?

8 × 18 的矩形 ABCD 被切成两个全等的六边形，如图所示，使得这两个六边形可以重新排列而不重叠形成一个正方形。$y$ 是多少？

(A) 6 6 (B) 7 7 (C) 8 8 (D) 9 9 (E) 10 10

Q8

A parabola with equation $y = x^2 + bx + c$ passes through the points (2,3) and (4,3). What is $c$?

抛物线方程 $y = x^2 + bx + c$ 经过点 (2,3) 和 (4,3)。$c$ 是多少？

(A) 2 2 (B) 5 5 (C) 7 7 (D) 10 10 (E) 11 11

Q9

How many sets of two or more consecutive positive integers have a sum of 15?

有多少组两个或更多连续正整数的和为 15？

(A) 1 1 (B) 2 2 (C) 3 3 (D) 4 4 (E) 5 5

Q10

For how many real values of $x$ is $\sqrt{120 - \sqrt{x}}$ an integer?

对于多少个实数 $x$，$\sqrt{120 - \sqrt{x}}$ 是整数？

(A) 3 3 (B) 6 6 (C) 9 9 (D) 10 10 (E) 11 11

Q11

Which of the following describes the graph of the equation $(x+y)^2 = x^2 + y^2$?

以下哪个描述了方程 $(x+y)^2 = x^2 + y^2$ 的图像？

(A) the empty set 空集 (B) one point 一点 (C) two lines 两条直线 (D) a circle 一个圆 (E) the entire plane 整个平面

Q12

Rolly wishes to secure his dog with an 8-foot rope to a square shed that is 16 feet on each side. His preliminary drawings are shown. Which of these arrangements gives the dog the greater area to roam, and by how many square feet?

Rolly 想用一根 8 英尺长的绳子将他的狗拴在每个边长 16 英尺的方形棚子上。他的初步图示如下。这些安排中哪一种给狗更大的活动面积，多出多少平方英尺？

(A) I, by $8\pi$ I，多 $8\pi$ (B) I, by $6\pi$ I，多 $6\pi$ (C) II, by $4\pi$ II，多 $4\pi$ (D) II, by $8\pi$ II，多 $8\pi$ (E) II, by $10\pi$ II，多 $10\pi$

Q13

A player pays $5 to play a game. A die is rolled. If the number on the die is odd, the game is lost. If the number on the die is even, the die is rolled again. In this case the player wins if the second number matches the first and loses otherwise. How much should the player win if the game is fair? (In a fair game the probability of winning times the amount won is what the player should pay.)

玩家支付 5 美元玩一个游戏。掷一个骰子。如果骰子上的数字是奇数，游戏输掉。如果是偶数，再掷一次骰子。在这种情况下，如果第二次数字与第一次相同则赢，否则输。如果游戏公平，玩家应该赢得多少？（在公平游戏中，赢的概率乘以赢得金额等于玩家支付的金额。）

(A) $12 $12 (B) $30 $30 (C) $50 $50 (D) $60 $60 (E) $100 $100

Q14

A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outside diameter of 20 cm. The outside diameter of each of the other rings is 1 cm less than that of the ring above it. The bottom ring has an outside diameter of 3 cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring?

许多相连的环，每个厚 1 厘米，挂在一个钉子上。最上面的环外径 20 厘米。其他每个环的外径比它上面的环少 1 厘米。最下面的环外径 3 厘米。从最上面环的顶部到底面最下面环的底部距离是多少厘米？

(A) 171 171 (B) 173 173 (C) 182 182 (D) 188 188 (E) 210 210

Q15

Odell and Kershaw run for 30 minutes on a circular track. Odell runs clockwise at 250 m/min and uses the inner lane with a radius of 50 meters. Kershaw runs counterclockwise at 300 m/min and uses the outer lane with a radius of 60 meters, starting on the same radial line as Odell. How many times after the start do they pass each other?

Odell 和 Kershaw 在一个圆形跑道上跑 30 分钟。Odell 顺时针以 250 米/分钟的速度跑，使用半径 50 米的内道。Kershaw 逆时针以 300 米/分钟的速度跑，使用半径 60 米的外道，从与 Odell 相同的径向线开始。起步后他们相遇多少次？

(A) 29 29 (B) 42 42 (C) 45 45 (D) 47 47 (E) 50 50

Q16

A circle of radius 1 is tangent to a circle of radius 2. The sides of $\triangle ABC$ are tangent to the circles as shown, and the sides $AB$ and $AC$ are congruent. What is the area of $\triangle ABC$?

一个半径为 1 的圆与一个半径为 2 的圆相切。$ riangle ABC$ 的三边如图所示与这些圆相切，且边 $AB$ 和 $AC$ 相等。求 $ riangle ABC$ 的面积。

(A) $\frac{35}{2}$ $\frac{35}{2}$ (B) $15\sqrt{2}$ $15\sqrt{2}$ (C) $\frac{64}{3}$ $\frac{64}{3}$ (D) $16\sqrt{2}$ $16\sqrt{2}$ (E) 24 24

Q17

In rectangle ADEH, points B and C trisect AD, and points G and F trisect HE. In addition, AH = AC = 2. What is the area of quadrilateral WXYZ shown in the figure?

在矩形 $ADEH$ 中，点 $B$ 和 $C$ 将 $AD$ 三等分，点 $G$ 和 $F$ 将 $HE$ 三等分。此外，$AH = AC = 2$。如图所示四边形 $WXYZ$ 的面积是多少？

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

Q18

A license plate in a certain state consists of 4 digits, not necessarily distinct, and 2 letters, also not necessarily distinct. These six characters may appear in any order, except that the two letters must appear next to each other. How many distinct license plates are possible?

某州的车牌由 4 个数字（不一定不同）和 2 个字母（也不一定不同）组成。这六个字符可以以任意顺序出现，但两个字母必须紧挨着。可能有多少种不同的车牌？

(A) $10^4 \cdot 26^2$ $10^4 \cdot 26^2$ (B) $10^3 \cdot 26^3$ $10^3 \cdot 26^3$ (C) $5 \cdot 10^4 \cdot 26^2$ $5 \cdot 10^4 \cdot 26^2$ (D) $10^2 \cdot 26^4$ $10^2 \cdot 26^4$ (E) $5 \cdot 10^3 \cdot 26^3$ $5 \cdot 10^3 \cdot 26^3$

Q19

How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression?

有多少个不成似的三角形，其角度度数是互不相同的正整数且形成等差数列？

(A) 0 0 (B) 1 1 (C) 59 59 (D) 89 89 (E) 178 178

Q20

Six distinct positive integers are randomly chosen between 1 and 2006, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of 5?

从 1 到 2006（包含）中随机选取 6 个不同的正整数。求其中某些一对整数的差是 5 的倍数的概率是多少？

(A) $\frac{1}{2}$ $\frac{1}{2}$ (B) $\frac{3}{5}$ $\frac{3}{5}$ (C) $\frac{2}{3}$ $\frac{2}{3}$ (D) $\frac{4}{5}$ $\frac{4}{5}$ (E) 1 1

Q21

How many four-digit positive integers have at least one digit that is a 2 or a 3?

有多少个四位正整数至少有一个数字是 2 或 3？

(A) 2439 2439 (B) 4096 4096 (C) 4903 4903 (D) 4904 4904 (E) 5416 5416

Q22

Two farmers agree that pigs are worth $300 and that goats are worth $210. When one farmer owes the other money, he pays the debt in pigs or goats, with "change" received in the form of goats or pigs as necessary. (For example, a $390 debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?

两位农民约定猪价值 $300$，羊价值 $210$。当一位农民欠另一位钱时，他用猪或羊支付债务，并必要时以羊或猪形式收到找零。（例如，$390$ 的债务可以用两头猪支付，并收到一头羊作找零。）可以用这种方式解决的最小正债务金额是多少？

(A) $5 $5 (B) $10 $10 (C) $30 $30 (D) $90 $90 (E) $210 $210

Q23

Circles with centers A and B have radii 3 and 8, respectively. A common internal tangent touches the circles at C and D, as shown. Lines AB and CD intersect at E, and AE = 5. What is CD?

圆心为 A 和 B 的圆分别半径 3 和 8。有公内切线触圆于 C 和 D，如图所示。直线 AB 和 CD 交于 E，且 $AE=5$。求 $CD$。

(A) 13 13 (B) $\dfrac{44}{3}$ $\dfrac{44}{3}$ (C) $\sqrt{221}$ $\sqrt{221}$ (D) $\sqrt{255}$ $\sqrt{255}$ (E) $\dfrac{55}{3}$ $\dfrac{55}{3}$

Q24

Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron?

单位正方体的相邻面的中心连线形成一个正八面体。这个八面体的体积是多少？

(A) $\frac{1}{8}$ $\frac{1}{8}$ (B) $\frac{1}{6}$ $\frac{1}{6}$ (C) $\frac{1}{4}$ $\frac{1}{4}$ (D) $\frac{1}{3}$ $\frac{1}{3}$ (E) $\frac{1}{2}$ $\frac{1}{2}$

Q25

A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?

一只虫子从正方体的一个顶点开始，按照以下规则沿正方体的边移动。在每个顶点，虫子会选择从该顶点发出的三条边中的一条。每条边被选择的概率相等，所有选择相互独立。七步后虫子恰好访问每个顶点一次的概率是多少？

(A) $\dfrac{1}{2187}$ $\dfrac{1}{2187}$ (B) $\dfrac{1}{729}$ $\dfrac{1}{729}$ (C) $\dfrac{2}{243}$ $\dfrac{2}{243}$ (D) $\dfrac{1}{81}$ $\dfrac{1}{81}$ (E) $\dfrac{5}{243}$ $\dfrac{5}{243}$

AMC10 2006 A