amcdrill | AMC8 AMC10 AMC12 Practice

Q1

What is the difference between the sum of the first 2003 even counting numbers and the sum of the first 2003 odd counting numbers?

前2003个偶数计数数的和与前2003个奇数计数数的和的差是多少？

(A) 0 0 (B) 1 1 (C) 2 2 (D) 2003 2003 (E) 4006 4006

Q2

Members of the Rockham Soccer League buy socks and T-shirts. Socks cost \$4 per pair and each T-shirt costs \$5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is \$2366, how many members are in the League?

Rockham足球联盟的成员购买袜子和T恤。一双袜子4美元，每件T恤比一双袜子贵5美元。每位成员需要一双袜子和一件用于主场比赛的球衣，以及另一双袜子和一件用于客场比赛的球衣。如果总花费为2366美元，联盟有多少成员？

(A) 77 77 (B) 91 91 (C) 143 143 (D) 182 182 (E) 286 286

Q3

A solid box is 15 cm by 10 cm by 8 cm. A new solid is formed by removing a cube 3 cm on a side from each corner of this box. What percent of the original volume is removed?

一个实心盒子尺寸为15 cm × 10 cm × 8 cm。从盒子每个角切除一个边长3 cm的立方体，形成一个新实心体。切除的体积占原体积的百分之多少？

(A) 4.5 4.5 (B) 9 9 (C) 12 12 (D) 18 18 (E) 24 24

Q4

It takes Mary 30 minutes to walk uphill 1 km from her home to school, but it takes her only 10 minutes to walk from school to home along the same route. What is her average speed, in km/hr, for the round trip?

Mary上坡从家走到学校1 km需要30分钟，但从学校沿同一路线走回家只需10分钟。她的往返平均速度是多少km/hr？

(A) 3 3 (B) 3.125 3.125 (C) 3.5 3.5 (D) 4 4 (E) 4.5 4.5

Q5

Let $d$ and $e$ denote the solutions of $2x^2 + 3x -5 = 0$. What is the value of $(d -1)(e -1)$?

设$d$和$e$是方程$2x^2 + 3x -5 = 0$的根。求$(d -1)(e -1)$的值。

(A) $-\frac{5}{2}$ $-\frac{5}{2}$ (B) 0 0 (C) 3 3 (D) 5 5 (E) 6 6

Q6

Define $x\heartsuit y$ to be $|x - y|$ for all real numbers $x$ and $y$. Which of the following statements is not true?

定义对于所有实数 $x$ 和 $y$，$x\heartsuit y$ 为 $|x - y|$。以下哪个陈述不正确？

(A) $x\heartsuit y = y\heartsuit x$ for all $x$ and $y$ 对所有 $x$ 和 $y$，$x\heartsuit y = y\heartsuit x$ (B) $2(x\heartsuit y) = (2x)\heartsuit(2y)$ for all $x$ and $y$ 对所有 $x$ 和 $y$，$2(x\heartsuit y) = (2x)\heartsuit(2y)$ (C) $x\heartsuit 0 = x$ for all $x$ 对所有 $x$，$x\heartsuit 0 = x$ (D) $x\heartsuit x = 0$ for all $x$ 对所有 $x$，$x\heartsuit x = 0$ (E) $x\heartsuit y > 0$ if $x \ne y$ 若 $x \ne y$，则 $x\heartsuit y > 0$

Q7

How many non-congruent triangles with perimeter 7 have integer side lengths?

周长为 7 的非全等整数边三角形有多少个？

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

Q8

What is the probability that a randomly drawn positive factor of 60 is less than 7?

随机抽取 60 的一个正因数小于 7 的概率是多少？

(A) $\frac{1}{10}$ $\frac{1}{10}$ (B) $\frac{1}{6}$ $\frac{1}{6}$ (C) $\frac{1}{4}$ $\frac{1}{4}$ (D) $\frac{1}{3}$ $\frac{1}{3}$ (E) $\frac{1}{2}$ $\frac{1}{2}$

Q9

Simplify $\sqrt[3]{\sqrt[3]{x^3\sqrt[3]{x\sqrt{x}}}}$ .

化简 $\sqrt[3]{\sqrt[3]{x^3\sqrt[3]{x\sqrt{x}}}}$ 。

(A) $\sqrt{x}$ $\sqrt{x}$ (B) $\sqrt[3]{x^2}$ $\sqrt[3]{x^2}$ (C) $\sqrt[3]{27x^2}$ $\sqrt[3]{27x^2}$ (D) $\sqrt[3]{54x}$ $\sqrt[3]{54x}$ (E) $\sqrt[3]{81x^{80}}$ $\sqrt[3]{81x^{80}}$

Q10

The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing?

图中实线围成的多边形由 4 个全等的正方形边对边连接而成。再在 9 个指示位置之一的边上附加一个全等的正方形。其中 9 个结果多边形有多少个可以折叠成缺一面立方体？

(A) 2 2 (B) 3 3 (C) 4 4 (D) 5 5 (E) 6 6

Q11

The sum of the two 5-digit numbers AMC10 and AMC12 is 123422. What is A + M + C?

两个5位数 AMC10 和 AMC12 的和是 123422。A + M + C 的值是多少？

(A) 10 10 (B) 11 11 (C) 12 12 (D) 13 13 (E) 14 14

Q12

A point $(x, y)$ is randomly picked from inside the rectangle with vertices $(0, 0)$, $(4, 0)$, $(4, 1)$, and $(0, 1)$. What is the probability that $x < y$?

从矩形内部随机选取一点 $(x, y)$，该矩形的顶点为 $(0, 0)$、$(4, 0)$、$(4, 1)$ 和 $(0, 1)$。$x < y$ 的概率是多少？

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

Q13

The sum of three numbers is 20. The first is 4 times the sum of the other two. The second is seven times the third. What is the product of all three?

三个数的和是 20。第一数是另外两个数之和的 4 倍。第二数是第三数的 7 倍。三者乘积是多少？

(A) 28 28 (B) 40 40 (C) 100 100 (D) 400 400 (E) 800 800

Q14

Let $n$ be the largest integer that is the product of exactly 3 distinct prime numbers, $d$, $e$ and $10d + e$, where $d$ and $e$ are single digits. What is the sum of the digits of $n$?

设 $n$ 是恰为 3 个不同素数 $d$、$e$ 和 $10d + e$ 的乘积的最大整数，其中 $d$ 和 $e$ 是单个数字。$n$ 的各位数字之和是多少？

(A) 12 12 (B) 15 15 (C) 18 18 (D) 21 21 (E) 24 24

Q15

What is the probability that an integer in the set $\lbrace 1, 2, 3, \dots , 100\rbrace$ is divisible by 2 and not divisible by 3?

集合 $\lbrace 1, 2, 3, \dots , 100\rbrace$ 中的整数能被 2 整除且不能被 3 整除的概率是多少？

(A) $\frac{1}{6}$ $\frac{1}{6}$ (B) $\frac{33}{100}$ $\frac{33}{100}$ (C) $\frac{17}{50}$ $\frac{17}{50}$ (D) $\frac{1}{2}$ $\frac{1}{2}$ (E) $\frac{18}{25}$ $\frac{18}{25}$

Q16

What is the units digit of $13^{2003}$?

$13^{2003}$ 的单位数字是多少？

(A) 1 1 (B) 3 3 (C) 7 7 (D) 8 8 (E) 9 9

Q17

The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle?

一个正三角形的周长（英寸数）等于其外接圆面积（平方英寸数）。求该圆的半径（英寸）。

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

Q18

What is the sum of the reciprocals of the roots of the equation $\frac{2003}{2004x + 1} + \frac{1}{x} = 0$?

方程 $\frac{2003}{2004x + 1} + \frac{1}{x} = 0$ 的根的倒数之和是多少？

(A) $-\frac{2004}{2003}$ $-\frac{2004}{2003}$ (B) $-1$ $-1$ (C) $\frac{2003}{2004}$ $\frac{2003}{2004}$ (D) 1 1 (E) $\frac{2004}{2003}$ $\frac{2004}{2003}$

Q19

A semicircle of diameter 1 sits at the top of a semicircle of diameter 2, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune.

直径为 1 的半圆位于直径为 2 的半圆顶部，如图所示。小半圆内、大半圆外的阴影区域称为月形区。求此月形区的面积。

(A) $\frac{1}{6}\pi - \frac{\sqrt{3}}{4}$ $\frac{1}{6}\pi - \frac{\sqrt{3}}{4}$ (B) $\frac{\sqrt{3}}{4} - \frac{1}{12}\pi$ $\frac{\sqrt{3}}{4} - \frac{1}{12}\pi$ (C) $\frac{\sqrt{3}}{4} - \frac{1}{24}\pi$ $\frac{\sqrt{3}}{4} - \frac{1}{24}\pi$ (D) $\frac{\sqrt{3}}{4} + \frac{1}{24}\pi$ $\frac{\sqrt{3}}{4} + \frac{1}{24}\pi$ (E) $\frac{\sqrt{3}}{4} + \frac{1}{12}\pi$ $\frac{\sqrt{3}}{4} + \frac{1}{12}\pi$

Q20

A base-10 three-digit number $n$ is selected at random. Which of the following is closest to the probability that the base-9 representation and the base-11 representation of $n$ are both three-digit numerals?

随机选取一个十进制的三位数 $n$。$n$ 的九进制表示和十一进制表示均为三位数表示法的概率最接近于以下哪一项？

(A) 0.3 0.3 (B) 0.4 0.4 (C) 0.5 0.5 (D) 0.6 0.6 (E) 0.7 0.7

Q21

Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How many different assortments of six cookies can be selected?

Pat 要从一个盘子里选择六块饼干，盘子里只有巧克力碎片、燕麦和花生酱饼干，每种至少有六块。有多少种不同的六块饼干组合可以选择？

(A) 22 22 (B) 25 25 (C) 27 27 (D) 28 28 (E) 729 729

Q22

In rectangle ABCD, we have AB = 8, BC = 9, H is on BC with BH = 6, E is on AD with DE = 4, line EC intersects line AH at G, and F is on line AD with GF $\perp$ AF. Find the length GF.

在矩形 ABCD 中，AB = 8，BC = 9，H 在 BC 上且 BH = 6，E 在 AD 上且 DE = 4，直线 EC 与直线 AH 相交于 G，F 在直线 AD 上且 GF $\perp$ AF。求 GF 的长度。

(A) 16 16 (B) 20 20 (C) 24 24 (D) 28 28 (E) 30 30

Q23

A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure we have 3 rows of small congruent equilateral triangles, with 5 small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of 2003 small equilateral triangles?

一个大正三角形由牙签构成小正三角形的行来构建。例如，图中 3 行小正三角形，底行有 5 个小三角形。如果底行由 2003 个小正三角形构成，需要多少牙签来构建这个大正三角形？

(A) 1,004,004 1,004,004 (B) 1,005,006 1,005,006 (C) 1,507,509 1,507,509 (D) 3,015,018 3,015,018 (E) 6,021,018 6,021,018

Q24

Sally has five red cards numbered 1 through 5 and four blue cards numbered 3 through 6. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?

Sally 有五张红牌，编号 1 到 5，和四张蓝牌，编号 3 到 6。她将牌堆叠，使得颜色交替，并且每张红牌的数字整除相邻蓝牌的数字。中间三张牌的数字之和是多少？

(A) 8 8 (B) 9 9 (C) 10 10 (D) 11 11 (E) 12 12

Q25

Let $n$ be a 5-digit number, and let $q$ and $r$ be the quotient and remainder, respectively, when $n$ is divided by 100. For how many values of $n$ is $q + r$ divisible by 11?

设 $n$ 是一个五位数，当 $n$ 除以 100 得到商 $q$ 和余数 $r$。有几个 $n$ 使得 $q + r$ 能被 11 整除？

(A) 8180 8180 (B) 8181 8181 (C) 8182 8182 (D) 9000 9000 (E) 9090 9090

AMC10 2003 A