amcdrill | AMC8 AMC10 AMC12 Practice

Q1

Ike and Mike go into a sandwich shop with a total of $\$30.00$ to spend. Sandwiches cost $\$4.50$ each and soft drinks cost $\$1.00$ each. Ike and Mike plan to buy as many sandwiches as they can, and use any remaining money to buy soft drinks. Counting both sandwiches and soft drinks, how many items will they buy?

Ike 和 Mike 进入一家三明治店，总共有 $\$$30.00$ 可供花费。三明治每个 $\$$4.50$，软饮料每个 $\$$1.00$。Ike 和 Mike 计划买尽可能多的三明治，用剩余的钱买软饮料。数三明治和软饮料，总共他们会买多少件物品？

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

Q2

Three identical rectangles are put together to form rectangle $ABCD$, as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is 5 feet, what is the area in square feet of rectangle $ABCD$?

三个相同的矩形组合成矩形 $ABCD$，如图所示。已知每个小矩形的短边长为 5 英尺，矩形 $ABCD$ 的面积（平方英尺）是多少？

(A) 45 45 (B) 75 75 (C) 100 100 (D) 125 125 (E) 150 150

Q3

Which of the following is the correct order of the fractions $\frac{15}{11},\frac{19}{15},$ and $\frac{17}{13},$ from least to greatest?

下列哪个是分数 $\frac{15}{11},\frac{19}{15},$ 和 $\frac{17}{13}$ 从小到大的正确顺序？

(A) \frac{15}{11}< \frac{17}{13}< \frac{19}{15} \frac{15}{11}< \frac{17}{13}< \frac{19}{15} (B) \frac{15}{11}< \frac{19}{15}<\frac{17}{13} \frac{15}{11}< \frac{19}{15}<\frac{17}{13} (C) \frac{17}{13}<\frac{19}{15}<\frac{15}{11} \frac{17}{13}<\frac{19}{15}<\frac{15}{11} (D) \frac{19}{15}<\frac{15}{11}<\frac{17}{13} \frac{19}{15}<\frac{15}{11}<\frac{17}{13} (E) \frac{19}{15}<\frac{17}{13}<\frac{15}{11} \frac{19}{15}<\frac{17}{13}<\frac{15}{11}

Q4

Quadrilateral $ABCD$ is a rhombus with perimeter $52$ meters. The length of diagonal $\overline{AC}$ is $24$ meters. What is the area in square meters of rhombus $ABCD$?

四边形 $ABCD$ 是边长相等的四边形，周长 52 米。对角线 $\overline{AC}$ 长 24 米。菱形 $ABCD$ 的面积（平方米）是多少？

(A) 60 60 (B) 90 90 (C) 105 105 (D) 120 120 (E) 144 144

Q5

A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance $d$ traveled by the two animals over time $t$ from start to finish?

一只乌龟向一只兔子发起赛跑挑战。兔子欣然同意并迅速领先，把行动迟缓的乌龟甩在后面。兔子自信会赢，便停下来小睡一会儿。与此同时，乌龟以缓慢稳定的步伐走完全程。兔子醒来冲向终点，却发现乌龟已先到。下列哪个图表匹配赛跑描述，显示两动物从起点到终点随时间 $t$ 行驶距离 $d$？

(A)

(B)

(C)

(D)

(E)

Q6

There are $81$ grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is in the center of the square. Given that point $Q$ is randomly chosen among the other $80$ points, what is the probability that the line $PQ$ is a line of symmetry for the square?

图中显示的正方形中有 $81$ 个均匀分布的网格点，包括边缘上的点。点 $P$ 位于正方形的中心。从其他 $80$ 个点中随机选择点 $Q$，$PQ$ 线是正方形的对称轴的概率是多少？

(A) \frac{1}{5} \frac{1}{5} (B) \frac{1}{4} \frac{1}{4} (C) \frac{2}{5} \frac{2}{5} (D) \frac{9}{20} \frac{9}{20} (E) \frac{1}{2} \frac{1}{2}

Q7

Shauna takes five tests, each worth a maximum of $100$ points. Her scores on the first three tests are $76$ , $94$ , and $87$ . In order to average $81$ for all five tests, what is the lowest score she could earn on one of the other two tests?

Shauna 参加了五次考试，每次满分 $100$ 分。前三次考试成绩分别是 $76$ 、 $94$ 和 $87$ 。为了五次考试平均分达到 $81$ 分，她在剩下两次考试中最低可能得分是多少？

(A) 48 48 (B) 52 52 (C) 66 66 (D) 70 70 (E) 74 74

Q8

Gilda has a bag of marbles. She gives $20\%$ of them to her friend Pedro. Then Gilda gives $10\%$ of what is left to another friend, Ebony. Finally, Gilda gives $25\%$ of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?

Gilda 有一个弹珠袋。她将其中 $20\%$ 给了朋友 Pedro。然后将剩下的 $10\%$ 给了另一个朋友 Ebony。最后将现在袋中剩下的 $25\%$ 给了弟弟 Jimmy。Gilda 自己剩下原来弹珠袋的百分之多少？

(A) 20 20 (B) 33\frac{1}{3} 33\frac{1}{3} (C) 38 38 (D) 45 45 (E) 54 54

Q9

Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are $6$ cm in diameter and $12$ cm high. Felicia buys cat food in cylindrical cans that are $12$ cm in diameter and $6$ cm high. What is the ratio of the volume of one of Alex's cans to the volume of one of Felicia's cans?

Alex 和 Felicia 都养猫作为宠物。Alex 买的猫粮罐是直径 $6$ cm、高 $12$ cm 的圆柱体。Felicia 买的猫粮罐是直径 $12$ cm、高 $6$ cm 的圆柱体。Alex 的一个罐的体积与 Felicia 的一个罐的体积的比是多少？

(A) 1:4 1:4 (B) 1:2 1:2 (C) 1:1 1:1 (D) 2:1 2:1 (E) 4:1 4:1

Q10

The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually $21$ participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?

图表显示了上周工作日足球练习的学生人数。计算均值和中位数后，教练发现周三实际有 $21$ 名参与者。修正后，均值和中位数的变化是哪种情况？

(A) The mean increases by $1$ and the median does not change. The mean increases by $1$ and the median does not change. (B) The mean increases by $1$ and the median increases by $1$. The mean increases by $1$ and the median increases by $1$. (C) The mean increases by $1$ and the median increases by $5$. The mean increases by $1$ and the median increases by $5$. (D) The mean increases by $5$ and the median increases by $1$. The mean increases by $5$ and the median increases by $1$. (E) The mean increases by $5$ and the median increases by $5$. The mean increases by $5$ and the median increases by $5$.

Q11

The eighth grade class at Lincoln Middle School has $93$ students. Each student takes a math class or a foreign language class or both. There are $70$ eighth graders taking a math class, and there are $54$ eighth graders taking a foreign language class. How many eighth graders take only a math class and not a foreign language class?

林肯中学八年级有$93$名学生。每名学生都上数学课或外语课或两者兼上。有$70$名八年级学生上数学课，有$54$名八年级学生上外语课。有多少名八年级学生只上数学课而不上外语课？

(A) 16 16 (B) 53 53 (C) 31 31 (D) 39 39 (E) 70 70

Q12

The faces of a cube are painted in six different colors: red $(R)$, white $(W)$, green $(G)$, brown $(B)$, aqua $(A)$, and purple $(P)$. Three views of the cube are shown below. What is the color of the face opposite the aqua face?

一个立方体的各个面涂有六种不同的颜色：红$(R)$、白$(W)$、绿$(G)$、棕$(B)$、水蓝$(A)$和紫$(P)$。下面展示了立方体的三个视图。水蓝面对面的颜色是什么？

(A) red red (B) white white (C) green green (D) brown brown (E) purple purple

Q13

A palindrome is a number that has the same value when read from left to right or from right to left. (For example, 12321 is a palindrome.) Let $N$ be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of $N$?

回文数是一个从左到右读或从右到左读值相同的数。（例如，12321是一个回文数。）设$N$为最小的一个不是回文数但等于三个不同两位回文数之和的三位整数。$N$的各位数字之和是多少？

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

Q14

Isabella has $6$ coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every $10$ days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the $6$ dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?

伊莎贝拉有$6$张可在Pete's Sweet Treats兑换免费冰淇淋甜筒的优惠券。为了让优惠券用得久一些，她决定每$10$天兑换一张，直到用完。她知道Pete's在周日休息，但当她在日历上圈出$6$个日期时，她意识到没有一个圈出的日期落在周日。那么伊莎贝拉第一张优惠券是在星期几兑换的？

(A) \text{Monday} \text{Monday} (B) \text{Tuesday} \text{Tuesday} (C) \text{Wednesday} \text{Wednesday} (D) \text{Thursday} \text{Thursday} (E) \text{Friday} \text{Friday}

Q15

On a beach $50$ people are wearing sunglasses and $35$ people are wearing caps. Some people are wearing both sunglasses and caps. If one of the people wearing a cap is selected at random, the probability that this person is also wearing sunglasses is $\frac{2}{5}$. If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?

海滩上有$50$人戴着太阳镜，$35$人戴着帽子。有些人同时戴着太阳镜和帽子。如果从戴帽子的人中随机选一人，该人同时戴太阳镜的概率是$\frac{2}{5}$。如果改从戴太阳镜的人中随机选一人，该人同时戴帽子的概率是多少？

(A) \frac{14}{85} \frac{14}{85} (B) \frac{7}{25} \frac{7}{25} (C) \frac{2}{5} \frac{2}{5} (D) \frac{4}{7} \frac{4}{7} (E) \frac{7}{10} \frac{7}{10}

Q16

Qiang drives $15$ miles at an average speed of $30$ miles per hour. How many additional miles will he have to drive at $55$ miles per hour to average $50$ miles per hour for the entire trip?

Qiang 以平均时速 30 英里/小时驾驶了 15 英里。要使整个行程的平均时速达到 50 英里/小时，他还需要以 55 英里/小时多驾驶多少英里？

(A) 45 45 (B) 62 62 (C) 90 90 (D) 110 110 (E) 135 135

Q17

What is the value of the product \[\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?\]

计算这个乘积的值 \[\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?\]

(A) \frac{1}{2} \frac{1}{2} (B) \frac{50}{99} \frac{50}{99} (C) \frac{9800}{9801} \frac{9800}{9801} (D) \frac{100}{99} \frac{100}{99} (E) 50 50

Q18

The faces of each of two fair dice are numbered $1$, $2$, $3$, $5$, $7$, and $8$. When the two dice are tossed, what is the probability that their sum will be an even number?

两个公平骰子的每个面都编号为 1、2、3、5、7 和 8。抛掷这两个骰子时，它们的和为偶数的概率是多少？

(A) \frac{4}{9} \frac{4}{9} (B) \frac{1}{2} \frac{1}{2} (C) \frac{5}{9} \frac{5}{9} (D) \frac{3}{5} \frac{3}{5} (E) \frac{2}{3} \frac{2}{3}

Q19

In a tournament there are six teams that play each other twice. A team earns $3$ points for a win, $1$ point for a draw, and $0$ points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for each of the top three teams?

锦标赛中有六支队伍，每两队交手两次。胜一场得 3 分，平局得 1 分，负一场得 0 分。所有比赛结束后，前三名队伍的总积分相同。前三名队伍每队可能的最大总积分是多少？

(A) 22 22 (B) 23 23 (C) 24 24 (D) 26 26 (E) 30 30

Q20

How many different real numbers $x$ satisfy the equation \[(x^{2}-5)^{2}=16?\]

方程 \[(x^{2}-5)^{2}=16?\] 有多少个不同的实数解 x？

(A) 0 0 (B) 1 1 (C) 2 2 (D) 4 4 (E) 8 8

Q21

What is the area of the triangle formed by the lines $y=5$, $y=1+x$, and $y=1-x$?

由直线 $y=5$、$y=1+x$ 和 $y=1-x$ 形成的三角形的面积是多少？

(A) 4 4 (B) 8 8 (C) 10 10 (D) 12 12 (E) 16 16

Q22

A store increased the original price of a shirt by a certain percent and then lowered the new price by the same amount. Given that the resulting price was $84\%$ of the original price, by what percent was the price increased and decreased$?$

一家商店将一件衬衫的原价提高了一定百分比，然后将新价格降低了相同的百分比。已知最终价格是原价的 $84\%$ ，价格提高了和降低了多少百分比？

(A) 16 16 (B) 20 20 (C) 28 28 (D) 36 36 (E) 40 40

Q23

After Euclid High School's last basketball game, it was determined that $\frac{1}{4}$ of the team's points were scored by Alexa and $\frac{2}{7}$ were scored by Brittany. Chelsea scored $15$ points. None of the other $7$ team members scored more than $2$ points. What was the total number of points scored by the other $7$ team members?

在 Euclid 高中最后一场篮球比赛后，确定球队得分中 Alexa 得了 $\frac{1}{4}$，Brittany 得了 $\frac{2}{7}$。Chelsea 得了 15 分。其他 7 名队员每人得分不超过 2 分。其他 7 名队员的总得分是多少？

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

Q24

In triangle $\triangle ABC$, point $D$ divides side $\overline{AC}$ so that $AD:DC=1:2$. Let $E$ be the midpoint of $\overline{BD}$ and let $F$ be the point of intersection of line $\overline{BC}$ and line $\overline{AE}$. Given that the area of $\triangle ABC$ is $360$, what is the area of $\triangle EBF$?

在三角形 $\triangle ABC$ 中，点 $D$ 将边 $\overline{AC}$ 分成 $AD:DC=1:2$。$E$ 是 $\overline{BD}$ 的中点，$F$ 是直线 $\overline{BC}$ 和直线 $\overline{AE}$ 的交点。已知 $\triangle ABC$ 的面积为 $360$，$\triangle EBF$ 的面积是多少？

(A) 24 24 (B) 30 30 (C) 32 32 (D) 36 36 (E) 40 40

Q25

Alice has $24$ apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?

Alice 有 24 个苹果。她有多少种方式与 Becky 和 Chris 分苹果，使得三人各至少有 2 个苹果？

(A) 105 105 (B) 114 114 (C) 190 190 (D) 210 210 (E) 380 380

AMC8 2019