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 $2003$ 个偶计数数之和与前 $2003$ 个奇计数数之和的差是多少？

(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。袜子每双 $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上衣用于主场比赛，另外还需要一双袜子和一件上衣用于客场比赛。若总费用为 $2366 美元，联盟共有多少名成员？

(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?

一个实心长方体盒子的尺寸为 $15$ cm $\times 10$ cm $\times 8$ cm。从这个盒子的每个角上都切去一个边长为 $3$ cm 的立方体，形成一个新的实心体。被切去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 Anna $30$ minutes to walk uphill $1$ km from her home to school, but it takes her only $10$ minutes to walk from school to her home along the same route. What is her average speed, in km/hr, for the round trip?

Anna 从家到学校沿同一路线上坡走 $1$ km 需要 $30$ 分钟，但从学校沿同一路线回家只需要 $10$ 分钟。她往返的平均速度是多少（单位：km/hr）？

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

Q5

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

两个五位数 $AMC10$ 和 $AMC12$ 的和为 $123422$。求 $A+M+C$。

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

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

(A) x♥y = y♥x for all x and y x♥y = y♥x for all x and y (B) 2(x♥y) = (2x)♥(2y) for all x and y 2(x♥y) = (2x)♥(2y) for all x and y (C) x♥0 = x for all x x♥0 = x for all x (D) x♥x = 0 for all x x♥x = 0 for all x (E) x♥y > 0 if x ≠ y x♥y > 0 if x ≠ y

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

A set $S$ of points in the $xy$-plane is symmetric about the origin, both coordinate axes, and the line $y=x$. If $(2,3)$ is in $S$, what is the smallest number of points in $S$?

在 $xy$ 平面上的点集 $S$ 关于原点、两条坐标轴以及直线 $y=x$ 都对称。若 $(2,3)$ 在 $S$ 中，则 $S$ 中点的最小个数是多少？

(A) 1 1 (B) 2 2 (C) 4 4 (D) 8 8 (E) 16 16

Q10

Al, Bert, and Carl are the winners of a school drawing for a pile of Halloween candy, which they are to divide in a ratio of $3:2:1$, respectively. Due to some confusion they come at different times to claim their prizes, and each assumes he is the first to arrive. If each takes what he believes to be the correct share of candy, what fraction of the candy goes unclaimed?

Al、Bert 和 Carl 是学校一次万圣节糖果堆抽奖的获胜者，他们将分别按 $3:2:1$ 的比例分糖果。由于一些混乱，他们在不同时间来领取奖品，并且每个人都以为自己是第一个到的。如果每个人都拿走他认为正确的那份糖果，那么有多少比例的糖果无人领取？

(A) \frac{1}{18} \frac{1}{18} (B) \frac{1}{6} \frac{1}{6} (C) \frac{2}{9} \frac{2}{9} (D) \frac{5}{18} \frac{5}{18} (E) \frac{5}{12} \frac{5}{12}

Q11

A square and an equilateral triangle have the same perimeter. Let $A$ be the area of the circle circumscribed about the square and $B$ the area of the circle circumscribed around the triangle. Find $A/B$.

一个正方形和一个正三角形具有相同的周长。设 $A$ 为外接该正方形的圆的面积，$B$ 为外接该三角形的圆的面积。求 $A/B$。

(A) \frac{9}{16} \frac{9}{16} (B) \frac{3}{4} \frac{3}{4} (C) \frac{27}{32} \frac{27}{32} (D) \frac{3\sqrt{6}}{8} \frac{3\sqrt{6}}{8} (E) 1 1

Q12

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 有五张红色卡片，编号 $1$ 到 $5$，以及四张蓝色卡片，编号 $3$ 到 $6$。她将卡片堆叠，使得颜色交替，并且每张红色卡片上的数字都能整除与其相邻的每张蓝色卡片上的数字。中间三张卡片上的数字之和是多少？

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

Q13

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?

图中由实线围成的多边形由 $4$ 个全等的正方形边对边连接而成。再将一个全等正方形附着在所示九个位置之一的边上。九种得到的多边形中，有多少种可以折叠成一个缺少一个面的立方体？

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

Q14

Points $K, L, M,$ and $N$ lie in the plane of the square $ABCD$ such that $AKB$, $BLC$, $CMD$, and $DNA$ are equilateral triangles. If $ABCD$ has an area of 16, find the area of $KLMN$.

点 $K, L, M,$ 和 $N$ 位于正方形 $ABCD$ 的平面内，使得 $AKB$、$BLC$、$CMD$ 和 $DNA$ 都是正三角形。若 $ABCD$ 的面积为 16，求 $KLMN$ 的面积。

(A) 32 32 (B) 16 + 16\sqrt{3} 16 + 16\sqrt{3} (C) 48 48 (D) 32 + 16\sqrt{3} 32 + 16\sqrt{3} (E) 64 64

Q15

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.

如图所示，一个直径为 $1$ 的半圆放在一个直径为 $2$ 的半圆的顶部。小半圆内部且大半圆外部的阴影区域称为月牙形（lune）。求该月牙形的面积。

(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

Q16

A point P is chosen at random in the interior of equilateral triangle $ABC$. What is the probability that $\triangle ABP$ has a greater area than each of $\triangle ACP$ and $\triangle BCP$?

在等边三角形 $ABC$ 的内部随机选择一点 $P$。$\triangle ABP$ 的面积大于 $\triangle ACP$ 和 $\triangle BCP$ 中每一个的概率是多少？

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

Q17

Square $ABCD$ has sides of length $4$, and $M$ is the midpoint of $\overline{CD}$. A circle with radius $2$ and center $M$ intersects a circle with radius $4$ and center $A$ at points $P$ and $D$. What is the distance from $P$ to $\overline{AD}$?

正方形 $ABCD$ 的边长为 $4$，$M$ 是 $\overline{CD}$ 的中点。以 $M$ 为圆心、半径为 $2$ 的圆与以 $A$ 为圆心、半径为 $4$ 的圆相交于点 $P$ 和 $D$。点 $P$ 到 $\overline{AD}$ 的距离是多少？

(A) 3 3 (B) \frac{16}{5} \frac{16}{5} (C) \frac{13}{4} \frac{13}{4} (D) 2\sqrt{3} 2\sqrt{3} (E) \frac{7}{2} \frac{7}{2}

Q18

Let $n$ be a $5$-digit number, and let $q$ and $r$ be the quotient and the 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 $5$ 位数，当 $n$ 除以 $100$ 时，商为 $q$，余数为 $r$。有多少个 $n$ 使得 $q+r$ 能被 $11$ 整除？

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

Q19

A parabola with equation $y=ax^2+bx+c$ is reflected about the $x$-axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of $y=f(x)$ and $y=g(x)$, respectively. Which of the following describes the graph of $y=(f+g)(x)?$

将方程为 $y=ax^2+bx+c$ 的抛物线关于 $x$ 轴反射。然后将该抛物线与其反射图像分别向相反方向水平平移 $5$ 个单位，得到 $y=f(x)$ 与 $y=g(x)$ 的图像。下列哪一项描述了 $y=(f+g)(x)$ 的图像？

(A) a parabola tangent to the x-axis a parabola tangent to the x-axis (B) a parabola not tangent to the x-axis a parabola not tangent to the x-axis (C) a horizontal line a horizontal line (D) a non-horizontal line a non-horizontal line (E) the graph of a cubic function the graph of a cubic function

Q20

How many $15$-letter arrangements of $5$ A's, $5$ B's, and $5$ C's have no A's in the first $5$ letters, no B's in the next $5$ letters, and no C's in the last $5$ letters?

由 $5$ 个 A、$5$ 个 B 和 $5$ 个 C 组成的 $15$ 字母排列中，有多少个满足：前 $5$ 个字母中没有 A，接下来的 $5$ 个字母中没有 B，最后 $5$ 个字母中没有 C？

(A) \sum_{k=0}^{5} \binom{5}{k} \sum_{k=0}^{5} \binom{5}{k} (B) 3^5 \cdot 2^5 3^5 \cdot 2^5 (C) 2^{15} 2^{15} (D) \frac{15!}{(5!)^3} \frac{15!}{(5!)^3} (E) 3^{15} 3^{15}

Q21

The graph of the polynomial \[P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e\] has five distinct $x$-intercepts, one of which is at $(0,0)$. Which of the following coefficients cannot be zero?

多项式 \[P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e\] 的图像有五个不同的 $x$ 截距，其中一个在 $(0,0)$。以下哪个系数不可能为零？

(A) a a (B) b b (C) c c (D) d d (E) e e

Q22

Objects $A$ and $B$ move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object $A$ starts at $(0,0)$ and each of its steps is either right or up, both equally likely. Object $B$ starts at $(5,7)$ and each of its steps is either to the left or down, both equally likely. Which of the following is closest to the probability that the objects meet?

物体 $A$ 和 $B$ 同时在坐标平面中通过一系列步长为 1 的步子移动。物体 $A$ 从 $(0,0)$ 出发，每一步要么向右要么向上，两种等可能。物体 $B$ 从 $(5,7)$ 出发，每一步要么向左要么向下，两种等可能。以下哪个最接近两物体相遇的概率？

(A) 0.10 0.10 (B) 0.15 0.15 (C) 0.20 0.20 (D) 0.25 0.25 (E) 0.30 0.30

Q23

How many perfect squares are divisors of the product $1!\cdot 2!\cdot 3!\cdots 9!$?

乘积 $1!\cdot 2!\cdot 3!\cdots 9!$ 的因数中，有多少个是完全平方数？

(A) 504 504 (B) 672 672 (C) 864 864 (D) 936 936 (E) 1008 1008

Q24

If $a\geq b > 1,$ what is the largest possible value of $\log_{a}(a/b) + \log_{b}(b/a)?$

若 $a\geq b > 1,$ 则 $\log_{a}(a/b) + \log_{b}(b/a)$ 的最大可能值是多少？

(A) -2 -2 (B) 0 0 (C) 2 2 (D) 3 3 (E) 4 4

Q25

Let $f(x)= \sqrt{ax^2+bx}$. For how many real values of $a$ is there at least one positive value of $b$ for which the domain of $f$ and the range of $f$ are the same set?

设 $f(x)= \sqrt{ax^2+bx}$。有多少个实数 $a$，使得存在至少一个正实数 $b$，使得 $f$ 的定义域与值域是同一个集合？

(A) 0 0 (B) 1 1 (C) 2 2 (D) 3 3 (E) infinitely many infinitely many

AMC12 2003 A