amcdrill | AMC8 AMC10 AMC12 Practice

Q1

In the year 2001, the United States will host the International Mathematical Olympiad. Let $I$, $M$, and $O$ be distinct positive integers such that the product $I \cdot M \cdot O = 2001$. What is the largest possible value of the sum $I + M + O$?

在2001年，美国将举办国际数学奥林匹克竞赛。设$I$、$M$和$O$是互不相同的正整数，使得乘积$I \cdot M \cdot O = 2001$。$I + M + O$的最大可能值为多少？

(A) 23 23 (B) 55 55 (C) 99 99 (D) 111 111 (E) 671 671

Q2

$2000(2000^{2000}) =$

$2000(2000^{2000}) = $

(A) $2000^{2001}$ $2000^{2001}$ (B) $4000^{2000}$ $4000^{2000}$ (C) $2000^{4000}$ $2000^{4000}$ (D) $4,000,000^{2000}$ $4,000,000^{2000}$ (E) $2000^{4,000,000}$ $2000^{4,000,000}$

Q3

Each day, Jenny ate 20% of the jellybeans that were in her jar at the beginning of that day. At the end of second day, 32 remained. How many jellybeans were in the jar originally?

每天，Jenny吃掉她罐子开始那天20%的果冻豆。到第二天结束时，还剩32颗。最初罐子里有多少果冻豆？

(A) 40 40 (B) 50 50 (C) 55 55 (D) 60 60 (E) 75 75

Q4

Chandra pays an on-line service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was $12.48, but in January her bill was $17.54 because she used twice as much connect time as in December. What is the fixed monthly fee?

Chandra向在线服务提供商支付固定的月费加上按小时计费的连接时间费。她的12月账单是$12.48，但1月账单是$17.54，因为她1月的连接时间是12月的两倍。固定的月费是多少？

(A) $2.53 $2.53 (B) $5.06 $5.06 (C) $6.24 $6.24 (D) $7.42 $7.42 (E) $8.77 $8.77

Q5

Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\triangle PAB$. As $P$ moves along a line that is parallel to side $AB$, how many of the four quantities listed below change? (a) the length of the segment $MN$ (b) the perimeter of $\triangle PAB$ (c) the area of $\triangle PAB$ (d) the area of trapezoid $ABNM$

点$M$和$N$分别是$ riangle PAB$的边$PA$和$PB$的中点。当$P$沿平行于边$AB$的直线移动时，下列四个量中有多少个会变化？(a) 线段$MN$的长度 (b) $\triangle PAB$的周长 (c) $\triangle PAB$的面积 (d) 梯形$ABNM$的面积

(A) 0 0 (B) 1 1 (C) 2 2 (D) 3 3 (E) 4 4

Q6

The Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, 21, \dots$ starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?

斐波那契数列 $1, 1, 2, 3, 5, 8, 13, 21, \dots$ 以两个 $1$ 开头，此后每一项是前两项之和。十个数字中，哪个数字是最晚出现在斐波那契数列中某个数的个位上的？

(A) 0 0 (B) 4 4 (C) 6 6 (D) 7 7 (E) 9 9

Q7

In rectangle ABCD, AD = 1, P is on AB, and DB and DP trisect $\angle ADC$. What is the perimeter of $\Delta BDP$?

在矩形 ABCD 中，AD = 1，P 在 AB 上，DB 和 DP 三等分 $\angle ADC$。求 $\Delta BDP$ 的周长？

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

Q8

At Olympic High School, 2/5 of the freshmen and 4/5 of the sophomores took the AMC 10. Given that the number of freshmen and sophomore contestants was the same, which of the following must be true?

在奥林匹克高中，$2/5$ 的新生和 $4/5$ 的二年级生参加了 AMC 10。已知新生和二年级参赛人数相同，下列哪项一定正确？

(A) There are five times as many sophomores as freshmen. 二年级生是新生数的五倍。 (B) There are twice as many sophomores as freshmen. 二年级生是新生数的两倍。 (C) There are as many freshmen as sophomores. 新生和二年级生人数相等。 (D) There are twice as many freshmen as sophomores. 新生是二年级生数的两倍。 (E) There are five times as many freshmen as sophomores. 新生是二年级生数的五倍。

Q9

If $|x - 2| = p$, where $x < 2$, then $x - p =$

若 $|x - 2| = p$，其中 $x < 2$，则 $x - p =$

(A) -2 -2 (B) 2 2 (C) 2 - 2p 2 - 2p (D) 2p - 2 2p - 2 (E) |2p - 2| |2p - 2|

Q10

The sides of a triangle with positive area have lengths 4, 6, and x. The sides of a second triangle with positive area have lengths 4, 6, and y. What is the smallest positive number that is not a possible value of |x - y|?

一个具有正面积的三角形的边长为 4、6 和 $x$。第二个具有正面积的三角形的边长为 4、6 和 $y$。最小的不可能是 $|x - y|$ 值的正数是多少？

(A) 2 2 (B) 4 4 (C) 6 6 (D) 8 8 (E) 10 10

Q11

Two different prime numbers between 4 and 18 are chosen. When their sum is subtracted from their product, which of the following number could be obtained?

从4和18之间选择两个不同的质数。当它们的和从它们的积中减去时，下列哪个数可能得到？

(A) 21 21 (B) 60 60 (C) 119 119 (D) 180 180 (E) 231 231

Q12

Figure 0, 1, 2, and 3 consist of 1, 5, 13, and 25 nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100?

图0、1、2和3分别由1、5、13和25个不重叠的单位正方形组成。如果图案继续，如何多不重叠的单位正方形在图100中？

Q13

There are 5 yellow pegs, 4 red pegs, 3 green pegs, 2 blue pegs, and 1 orange peg to be placed on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color?

有一个三角形钉板，要放置5个黄色钉、4个红色钉、3个绿色钉、2个蓝色钉和1个橙色钉。有多少种方法可以放置这些钉，使得没有（水平）行或（垂直）列包含相同颜色的两个钉？

(A) 0 0 (B) 1 1 (C) $5! \cdot 4! \cdot 3! \cdot 2! \cdot 1!$ $5! \cdot 4! \cdot 3! \cdot 2! \cdot 1!$ (D) $15!/(5! \cdot 4! \cdot 3! \cdot 2! \cdot 1!)$ $15!/(5! \cdot 4! \cdot 3! \cdot 2! \cdot 1!)$ (E) 15! 15!

Q14

Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were 71, 76, 80, 82, and 91. What was the last score Mrs. Walter entered?

沃尔特夫人给一个五名学生的数学班考试。她随机顺序将分数输入电子表格，每次输入一个分数后电子表格重新计算班级平均分。沃尔特夫人注意到，每次输入分数后，平均分总是整数。分数（按升序排列）是71、76、80、82和91。沃尔特夫人最后输入的分数是什么？

(A) 71 71 (B) 76 76 (C) 80 80 (D) 82 82 (E) 91 91

Q15

Two non-zero real numbers, a and b, satisfy $ab = a - b$. Find a possible value of $\frac{a}{b} + \frac{b}{a} - ab$.

两个非零实数$a$和$b$满足$ab=a-b$。求$\frac{a}{b}+\frac{b}{a}-ab$的一个可能值。

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

Q16

The diagram shows 28 lattice points, each one unit from its nearest neighbors. Segment AB meets segment CD at E. Find the length of segment AE.

图中显示了28个格点，每个点与其最近邻点相距一单位。线段AB与线段CD在点E相交。求线段AE的长度。

(A) $4\sqrt{5}/3$ $4\sqrt{5}/3$ (B) $5\sqrt{5}/3$ $5\sqrt{5}/3$ (C) $12\sqrt{5}/7$ $12\sqrt{5}/7$ (D) $2\sqrt{5}$ $2\sqrt{5}$ (E) $5\sqrt{65}/9$ $5\sqrt{65}/9$

Q17

Boris has an incredible coin changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly?

鲍里斯有一个神奇的找零机。当他投入一个25美分硬币时，机器返回五个5美分硬币；投入一个5美分硬币时，返回五个1美分硬币；投入一个1美分硬币时，返回五个25美分硬币。鲍里斯开始只有一个1美分硬币。经过反复使用机器后，他可能拥有下列哪种金额？

(A) $3.63 $3.63 (B) $5.13 $5.13 (C) $6.30 $6.30 (D) $7.45 $7.45 (E) $9.07 $9.07

Q18

Charlyn walks completely around the boundary of a square whose sides are each 5 km long. From any point on her path she can see exactly 1 km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number?

查琳完全绕着一个边长各为5千米的正方形的边界走一圈。从她路径上的任意一点，她都能在所有方向水平看清1千米。求查琳在行走过程中能看到的区域的面积（以平方千米为单位，四舍五入到最接近的整数）。

(A) 24 24 (B) 27 27 (C) 39 39 (D) 40 40 (E) 42 42

Q19

Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is $m$ times the area of the square. The ratio of the area of the other small right triangle to the area of the square is

在一个直角三角形的斜边上一点，作与两条直角边平行的直线，将三角形分割成一个正方形和两个更小的直角三角形。其中一个小的直角三角形的面积是正方形面积的$m$倍。另一个小直角三角形面积与正方形面积的比值为

(A) $\frac{1}{2m+1}$ $\frac{1}{2m+1}$ (B) $m$ $m$ (C) $1-m$ $1-m$ (D) $\frac{1}{4m}$ $\frac{1}{4m}$ (E) $\frac{1}{8m^2}$ $\frac{1}{8m^2}$

Q20

Let $A$, $M$, and $C$ be nonnegative integers such that $A + M + C = 10$. What is the maximum value of $A \cdot M \cdot C + A \cdot M + M \cdot C + C \cdot A$?

设$A$、$M$、$C$是非负整数且$A+M+C=10$。求$A\cdot M\cdot C+A\cdot M+M\cdot C+C\cdot A$的最大值。

(A) 49 49 (B) 59 59 (C) 69 69 (D) 79 79 (E) 89 89

Q21

If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true? I. All alligators are creepy crawlers. II. Some ferocious creatures are creepy crawlers. III. Some alligators are not creepy crawlers.

如果所有短吻鳄都是凶猛的生物，并且一些诡异的爬行动物是短吻鳄，那么以下哪项（些）声明必须为真？I. 所有短吻鳄都是诡异的爬行动物。II. 一些凶猛的生物是诡异的爬行动物。III. 一些短吻鳄不是诡异的爬行动物。

(A) I only 仅I (B) II only 仅II (C) III only 仅III (D) II and III only 仅II和III (E) None must be true 无须为真

Q22

One morning each member of Angela’s family drank an 8-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?

一个早晨，Angela 全家每个人都喝了一杯 8 盎司的咖啡与牛奶混合物。每个杯子中咖啡和牛奶的量各不相同，但都不为零。Angela 喝了总牛奶量的四分之一和总咖啡量的六分之一。全家有多少人？

(A) 3 3 (B) 4 4 (C) 5 5 (D) 6 6 (E) 7 7

Q23

When the mean, median, and mode of the list $10, 2, 5, 2, 4, 2, x$ are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real value of $x$?

当列表 $10, 2, 5, 2, 4, 2, x$ 的均值、中位数和众数按升序排列时，它们形成一个非恒等的等差数列。所有可能的实数 $x$ 的和是多少？

(A) 3 3 (B) 6 6 (C) 9 9 (D) 17 17 (E) 20 20

Q24

Let $f$ be a function for which $f(x/3) = x^2 + x + 1$. Find the sum of all values of $z$ for which $f(3z) = 7$.

设 $f$ 是一个函数，使得 $f(x/3) = x^2 + x + 1$。求所有满足 $f(3z) = 7$ 的 $z$ 的值之和。

(A) -$\frac{1}{3}$ -$\frac{1}{3}$ (B) -$\frac{1}{9}$ -$\frac{1}{9}$ (C) 0 0 (D) $\frac{5}{9}$ $\frac{5}{9}$ (E) $\frac{5}{3}$ $\frac{5}{3}$

Q25

In year $N$, the 300th day of the year is a Tuesday. In year $N + 1$, the 200th day is also a Tuesday. On what day of the week did the 100th day of year $N - 1$ occur?

在年份 $N$，当年的第 300 天是星期二。在年份 $N + 1$，第 200 天也是星期二。年份 $N - 1$ 的第 100 天是星期几？

(A) Thursday 星期四 (B) Friday 星期五 (C) Saturday 星期六 (D) Sunday 星期日 (E) Monday 星期一

AMC10 2000 A