amcdrill | AMC8 AMC10 AMC12 Practice

Q1

Leah has 13 coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah’s coins worth?

Leah 有 13 枚硬币，全都是美分和五分镍币。如果她再多一枚五分镍币，那么她将有相同数量的美分和五分镍币。Leah 的硬币总价值多少美分？

(A) 33 33 (B) 35 35 (C) 37 37 (D) 39 39 (E) 41 41

Q2

What is $\frac{2^{3} + 2^{3}}{2^{-3} + 2^{-3}}$?

(A) 16 16 (B) 24 24 (C) 32 32 (D) 48 48 (E) 64 64

Q3

Randy drove the first third of his trip on a gravel road, the next 20 miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy’s trip?

Randy 的行程前三分之一在碎石路上行驶，接下来 20 英里在柏油路上行驶，剩余五分之一在土路上行驶。Randy 的行程总长多少英里？

(A) 30 30 (B) 40 40 (C) $\frac{75}{2}$ $\frac{75}{2}$ (D) 100 100 (E) $\frac{300}{7}$ $\frac{300}{7}$

Q4

Susie pays for 4 muffins and 3 bananas. Calvin spends twice as much paying for 2 muffins and 16 bananas. A muffin is how many times as expensive as a banana?

Susie 买了 4 个松饼和 3 个香蕉。Calvin 花费是 Susie 的两倍，买了 2 个松饼和 16 个香蕉。一个松饼比一个香蕉贵多少倍？

(A) $\frac{3}{2}$ $\frac{3}{2}$ (B) $\frac{5}{3}$ $\frac{5}{3}$ (C) $\frac{7}{4}$ $\frac{7}{4}$ (D) 2 2 (E) $\frac{13}{4}$ $\frac{13}{4}$

Q5

Doug constructs a square window using 8 equal-size panes of glass, as shown. The ratio of the height to width for each pane is 5 : 2, and the borders around and between the panes are 2 inches wide. In inches, what is the side length of the square window?

Doug 使用 8 块相同大小的玻璃板制作了一个方形窗户，如图所示。每块玻璃板的宽度与高度之比为 5 : 2，玻璃板之间和周围的边框宽 2 英寸。方形窗户的边长是多少英寸？

(A) 26 26 (B) 28 28 (C) 30 30 (D) 32 32 (E) 34 34

Q6

Orvin went to the store with just enough money to buy 30 balloons. When he arrived he discovered that the store had a special sale on balloons: buy 1 balloon at the regular price and get a second at $\frac{1}{3}$ off the regular price. What is the greatest number of balloons Orvin could buy?

Orvin 去商店时，正好有足够的钱买 30 个气球。他到达时发现商店正在搞气球特价活动：买一个气球按原价，第二个气球打 $\frac{1}{3}$ 折。Orvin 最多能买多少个气球？

(A) 33 33 (B) 34 34 (C) 36 36 (D) 38 38 (E) 39 39

Q7

Suppose $A > B > 0$ and $A$ is $x\%$ greater than $B$. What is $x$?

假设 $A > B > 0$，且 $A$ 比 $B$ 大 $x\%$。$x$ 是多少？

(A) $100\frac{A-B}{B}$ $100\frac{A-B}{B}$ (B) $100\frac{A+B}{B}$ $100\frac{A+B}{B}$ (C) $100\frac{A+B}{A}$ $100\frac{A+B}{A}$ (D) $100\frac{A-B}{A}$ $100\frac{A-B}{A}$ (E) $100\frac{A}{B}$ $100\frac{A}{B}$

Q8

A truck travels $\frac{b}{6}$ feet every $t$ seconds. There are 3 feet in a yard. How many yards does the truck travel in 3 minutes?

一辆卡车每 $t$ 秒行驶 $\frac{b}{6}$ 英尺。1 码 = 3 英尺。卡车在 3 分钟内行驶多少码？

(A) $\frac{b}{1080t}$ $\frac{b}{1080t}$ (B) $30\frac{t}{b}$ $30\frac{t}{b}$ (C) $30\frac{b}{t}$ $30\frac{b}{t}$ (D) $10\frac{t}{b}$ $10\frac{t}{b}$ (E) $10\frac{b}{t}$ $10\frac{b}{t}$

Q9

For real numbers $w$ and $z$, $\frac{\frac{1}{w} + \frac{1}{z}}{\frac{1}{w} - \frac{1}{z}} = 2014$. What is $\frac{w+z}{w-z}$?

对于实数 $w$ 和 $z$，$\frac{\frac{1}{w} + \frac{1}{z}}{\frac{1}{w} - \frac{1}{z}} = 2014$。求 $\frac{w+z}{w-z}$ 的值。

(A) -2014 -2014 (B) $\frac{-1}{2014}$ $\frac{-1}{2014}$ (C) $\frac{1}{2014}$ $\frac{1}{2014}$ (D) 1 1 (E) 2014 2014

Q10

In the addition shown below A, B, C, and D are distinct digits. How many different values are possible for D? $$ \begin{array}{r|r} ABBCB \\ + BCADA \\ \hline DBDDD \end{array} $$

如下加法所示，A、B、C、D 是不同的数字。D 有多少种不同的可能值？ $$ \begin{array}{r|r} ABBCB \\ + BCADA \\ \hline DBDDD \end{array} $$

(A) 2 2 (B) 4 4 (C) 7 7 (D) 8 8 (E) 9 9

Q11

For the consumer, a single discount of n% is more advantageous than any of the following discounts: (1) two successive 15% discounts (2) three successive 10% discounts (3) a 25% discount followed by a 5% discount. What is the smallest possible positive integer value of n?

对于消费者来说，单一的 n% 折扣比以下任何折扣更有利：(1) 连续两次 15% 折扣 (2) 连续三次 10% 折扣 (3) 先 25% 折扣后 5% 折扣。n 的最小正整数值是多少？

(A) 27 27 (B) 28 28 (C) 29 29 (D) 31 31 (E) 33 33

Q12

The largest divisor of 2,014,000,000 is itself. What is its fifth largest divisor?

2,014,000,000 的最大除数是它本身。它的第五大除数是多少？

(A) 125,875,000 125,875,000 (B) 201,400,000 201,400,000 (C) 251,750,000 251,750,000 (D) 402,800,000 402,800,000 (E) 503,500,000 503,500,000

Q13

Six regular hexagons surround a regular hexagon of side length 1 as shown. What is the area of $\triangle ABC$?

如图所示，六个正六边形围绕着一个边长为 1 的正六边形。\triangle ABC 的面积是多少？

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

Q14

Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, abc miles was displayed on the odometer, where abc is a 3-digit number with a ≥1 and a + b + c ≤7. At the end of the trip, the odometer showed cba miles. What is a² + b² + c²?

Danica 开着她的新车旅行了整数组小时，以平均 55 英里/小时的速度。在旅行开始时，里程表显示 abc 英里，其中 abc 是一个三位数，a ≥1 且 a + b + c ≤7。在旅行结束时，里程表显示 cba 英里。a² + b² + c² 是多少？

(A) 26 26 (B) 27 27 (C) 36 36 (D) 37 37 (E) 41 41

Q15

In rectangle $ABCD$, $DC=2CB$ and points $E$ and $F$ lie on $\overline{AB}$ so that $\overline{ED}$ and $\overline{FD}$ trisect $\angle ADC$ as shown. What is the ratio of the area of $\triangle DEF$ to the area of rectangle $ABCD$?

在矩形 $ABCD$ 中，$DC=2CB$，点 $E$ 和 $F$ 在 $\overline{AB}$ 上，使得 $\overline{ED}$ 和 $\overline{FD}$ 如图所示将 $\angle ADC$ 三等分。求 $\triangle DEF$ 的面积与矩形 $ABCD$ 的面积之比。

(A) $\frac{\sqrt{3}}{6}$ $\frac{\sqrt{3}}{6}$ (B) $\frac{\sqrt{6}}{8}$ $\frac{\sqrt{6}}{8}$ (C) $\frac{3\sqrt{3}}{16}$ $\frac{3\sqrt{3}}{16}$ (D) $\frac{1}{3}$ $\frac{1}{3}$ (E) $\frac{\sqrt{2}}{4}$ $\frac{\sqrt{2}}{4}$

Q16

Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value?

掷四个公平的六面骰子。四骰子中至少有三骰子显示相同数值的概率是多少？

(A) $\frac{1}{36}$ $\frac{1}{36}$ (B) $\frac{7}{72}$ $\frac{7}{72}$ (C) $\frac{1}{9}$ $\frac{1}{9}$ (D) $\frac{5}{36}$ $\frac{5}{36}$ (E) $\frac{1}{6}$ $\frac{1}{6}$

Q17

What is the greatest power of 2 that is a factor of $10^{1002} - 4^{501}$?

$10^{1002}-4^{501}$的最大2的幂因数是多少？

(A) $2^{1002}$ $2^{1002}$ (B) $2^{1003}$ $2^{1003}$ (C) $2^{1004}$ $2^{1004}$ (D) $2^{1005}$ $2^{1005}$ (E) $2^{1006}$ $2^{1006}$

Q18

A list of 11 positive integers has a mean of 10, a median of 9, and a unique mode of 8. What is the largest possible value of an integer in the list?

一个包含11个正整数的列表，平均数为10，中位数为9，唯一众数为8。列表中整数的最大可能值是多少？

(A) 24 24 (B) 30 30 (C) 31 31 (D) 33 33 (E) 35 35

Q19

Two concentric circles have radii 1 and 2. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?

两个同心圆，半径分别为1和2。在外圆上独立均匀随机选择两点。连接两点的弦与内圆相交的概率是多少？

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

Q20

For how many integers $x$ is the number $x^4 - 51x^2 + 50$ negative?

有整数$x$多少个，使得$x^4-51x^2+50$为负数？

(A) 8 8 (B) 10 10 (C) 12 12 (D) 14 14 (E) 16 16

Q21

Trapezoid ABCD has parallel sides $\overline{AB}$ of length 33 and $\overline{CD}$ of length 21. The other two sides are of lengths 10 and 14. The angles at A and B are acute. What is the length of the shorter diagonal of ABCD?

梯形ABCD有平行边 $\overline{AB}$ 长33和 $\overline{CD}$ 长21。另外两条边长分别为10和14。A和B处的角度是锐角。ABCD的较短对角线的长度是多少？

(A) $10\sqrt{6}$ $10\sqrt{6}$ (B) 25 25 (C) $8\sqrt{10}$ $8\sqrt{10}$ (D) $18\sqrt{2}$ $18\sqrt{2}$ (E) 26 26

Q22

Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?

如图，八个半圆沿边长为2的正方形内侧排列。求与所有这些半圆相切的圆的半径。

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

Q23

A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?

如图，一个球体内接于一个截锥（右圆锥的截断部分）。截锥的体积是球体体积的两倍。求截锥底面半径与顶面半径的比。

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

Q24

The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is bad if it is not true that for every n from 1 to 15 one can find a subset of the numbers that appear consecutively on the circle that sum to n. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?

数字1,2,3,4,5要排列成一个圆。对每一个从1到15的n，都能在圆上找到一段连续数字的子集其和为n。如果不能满足这个条件，则该排列为坏排列。只考虑旋转和平移不同的排列。有多少种不同的坏排列？

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

Q25

In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad N, 0 < N < 10, it will jump to pad N−1 with probability N/10 and to pad N + 1 with probability 1−N/10. Each jump is independent of the previous jumps. If the frog reaches pad 0 it will be eaten by a patiently waiting snake. If the frog reaches pad 10 it will exit the pond, never to return. What is the probability that the frog will escape being eaten by the snake?

一个小池塘中有11个依次排列的百合垫，标号0到10。青蛙初始在垫1上。当青蛙在垫N上时（0<N<10），它以概率N/10跳到N-1，以概率1-N/10跳到N+1。每跳独立。如果到达垫0会被蛇吃掉，到达垫10则逃出池塘。青蛙逃脱被吃掉的概率是多少？

(A) $\frac{32}{79}$ $\frac{32}{79}$ (B) $\frac{161}{384}$ $\frac{161}{384}$ (C) $\frac{63}{146}$ $\frac{63}{146}$ (D) $\frac{7}{16}$ $\frac{7}{16}$ (E) $\frac{1}{2}$ $\frac{1}{2}$

AMC10 2014 B