Squares of side length 1, 1, 2, 3, and 5 are arranged to form the rectangle shown below. A curve is drawn by inscribing a quarter circle in each square and joining the quarter circles in order, from shortest to longest. What is the length of the curve?
In the figure below, each circle will be filled with a digit from 1 to 6. Each digit must appear exactly once. The sum of the digits in neighboring circles is shown in the box between them. What digit must be placed in the top circle?
The figure below shows a tiling of 1×1 unit squares. Each row of unit squares is shifted horizontally by half a unit relative to the row above it. A shaded square is drawn on top of the tiling. Each vertex of the shaded square is a vertex of one of the unit squares. In square units, what is the area of the shaded square?
Jami picked three equally spaced integer numbers on the number line. The sum of the first and the second numbers is 40, while the sum of the second and third numbers is 60. What is the sum of all three numbers?
Jami 在数轴上选取了三个等间距的整数。第一个数与第二个数之和为 40，第二个数与第三个数之和为 60。这三个数的总和是多少？
Elijah has a large collection of identical wooden cubes which are white on 4 faces and gray on 2 faces that share an edge. He glues some cubes together face-to-face. The figure below shows 2 cubes being glued together, leaving 3 gray faces visible. What is the fewest number of cubes that he could glue together to ensure that no gray faces are visible, no matter how he rotates the figure?
2026 AMC8 #17 · Counting, Probability & Statistics · ★★★
Four students are seated in a row. They chat with the people sitting next to them, then rearrange themselves so that they are no longer seated next to any of the same people. How many rearrangements are possible?
Miguel is walking with his dog, Luna. When they reach the entrance to a park, Miguel throws a ball straight ahead and continues to walk at a steady pace. Luna sprints toward the ball, which stops by a tree. As soon as the dog reaches the ball, she brings it back to Miguel. Luna runs 5 times faster than Miguel walks. What fraction of the distance from the entrance to the tree will Miguel have walked when Luna returns the ball to him?
Miguel 带着他的狗 Luna 散步。当他们走到公园入口时，Miguel 向前扔出一个球并继续匀速行走。Luna 冲向球，球停在一棵树旁。Luna 一拿到球就立刻跑回 Miguel 身边。Luna 的奔跑速度是 Miguel 步行速度的 5 倍。当 Luna 把球送回给 Miguel 时，Miguel 已经走了从入口到树这段距离的几分之几？
2026 AMC8 #20 · Counting, Probability & Statistics · ★★★
The land of Catania uses gold coins and silver coins. Gold coins are 1 mm thick and silver coins are 3 mm thick. In how many ways can Taylor make a stack of coins that is 8 mm tall using any arrangement of gold and silver coins, assuming order matters?
Catania 国使用金币和银币。金币厚 1 毫米，银币厚 3 毫米。Taylor 想用任意数量和顺序的金币与银币堆叠出一个总高为 8 毫米的硬币堆（顺序不同视为不同方式）。共有多少种堆法？
2026 AMC8 #21 · Counting, Probability & Statistics · ★★★
Charlotte the spider is walking along a web shaped like a 5-pointed star, shown in the figure below. The web has 5 outer points and 5 inner points. Each time Charlotte reaches a point, she randomly chooses a neighboring point and moves to that point. Charlotte starts at one of the outer points and makes 3 moves (re-visiting points is allowed). What is the probability she is now at one of the outer points of the star?
Charlotte 是一只蜘蛛，她在一张形如五角星的网上爬行（如下图所示）。这张网有 5 个外顶点和 5 个内顶点。每次 Charlotte 到达一个顶点时，会随机选择一个相邻顶点并移动过去（允许重复访问）。她从一个外顶点出发，共移动 3 次。此时她位于外顶点的概率是多少？
Atetromino consists of four squares connected along their edges. There are five possible tetromino shapes, I , O , L , T , and S , shown below, which can be rotated or flipped over. Three tetrominoes are used to completely cover a 3×4 rectangle. At least one of the tiles is an S tile. What are the other two tiles?
The region shown below consists of 24 squares, each with side length 1 centimeter. What is the area, in square centimeters, of the largest circle that can fit inside the region, possibly touching the boundaries?
2025 AMC8 #15 · Counting, Probability & Statistics · ★★★
Kei draws a 6 -by- 6 grid. He colors 13 of the unit squares silver and the remaining squares gold. Kei then folds the grid in half vertically, forming pairs of overlapping unit squares. Let m and M equal the least and greatest possible number of gold-on-gold pairs, respectively. What is the value of m+M ?
2025 AMC8 #16 · Counting, Probability & Statistics · ★★★
Five distinct integers from 1 to 10 are chosen, and five distinct integers from 11 to 20 are chosen. No two numbers differ by exactly 10 . What is the sum of the ten chosen numbers?
In the land of Markovia, there are three cities: A , B , and C . There are 100 people who live in A , 120 who live in B , and 160 who live in C . Everyone works in one of the three cities, and a person may work in the same city where they live. In the figure below, an arrow pointing from one city to another is labeled with the fraction of people living in the first city who work in the second city. (For example, 41 of the people who live in A work in B .) How many people work in A ?
The circle shown below on the left has a radius of 1 unit. The region between the circle and the inscribed square is shaded. In the circle shown on the right, one quarter of the region between the circle and the inscribed square is shaded. The shaded regions in the two circles have the same area. What is the radius R , in units, of the circle on the right?
Two towns, A and B , are connected by a straight road that is 15 miles long. Travelling from city A to town B , the speed limit changes every 5 miles: from 25 to 40 to 20 miles per hour (mph). Two cars, one at town A and one at town B , start moving toward each other at the same time. They drive at exactly the speed limit in each portion of the road. How far from town A , in miles, will the two cars meet?
Sarika, Dev, and Rajiv are sharing a large block of cheese. They take turns cutting off half of what remains and eating it: first Sarika eats half of the cheese, then Dev eats half of the remaining half, then Rajiv eats half of what remains, then back to Sarika, and so on. They stop when the cheese is too small to see. About what fraction of the original block of cheese does Sarika eat in total?
Rohan keeps a total of 90 guppies in 4 fish tanks.
• There is 1 more guppy in the 2nd tank than in the 1st tank.
• There are 2 more guppies in the 3rd tank than in the 2nd tank.
• There are 3 more guppies in the 4th tank than in the 3rd tank.
2024 AMC8 #16 · Counting, Probability & Statistics · ★★★
Minh enters the numbers 1 through 81 into the cells of a 9 × 9 grid in some order. She calculates the product of the numbers in each row and column. What is the least number of rows and columns that could have a product divisible by 3?
2024 AMC8 #17 · Counting, Probability & Statistics · ★★★
A chess king is said to attack all the squares one step away from it, horizontally, vertically, or diagonally. For instance, a king on the center square of a 3 × 3 grid attacks all 8 other squares, as shown below. Suppose a white king and a black king are placed on different squares of a 3 × 3 grid so that they do not attack each other. In how many ways can this be done?
Three concentric circles centered at O have radii of 1, 2, and 3. Points B and C lie on the largest circle. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angle ∠BOC, as shown in the figure below. Suppose the shaded and unshaded regions are equal in area. What is the measure of ∠BOC in degrees?
以点 O 为圆心的三个同心圆的半径分别为 1、2、3。点 B 和点 C 位于最大圆上。两个较小圆之间的区域被涂上阴影，两个较大圆之间且位于圆心角 ∠BOC 内的区域也被涂上阴影，如图所示。若阴影区域与非阴影区域面积相等，请问 ∠BOC 的度数是多少？
A group of frogs (called an army) is living in a tree. A frog turns green when in the shade and turns yellow when in the sun. Initially, the ratio of green to yellow frogs was 3:1. Then 3 green frogs moved to the sunny side and 5 yellow frogs moved to the shady side. Now the ratio is 4:1. What is the difference between the number of green frogs and the number of yellow frogs now?
NASA's Perseverance Rover was launched on July 30, 2020. After traveling 292,526,838 miles, it landed on Mars in Jezero Crater about 6.5 months later. Which of the following is closest to the Rover's average interplanetary speed in miles per hour?
The figure below shows a large white circle with a number of smaller white and shaded circles in its interior. What fraction of the interior of the large white circle is shaded?
Along the route of a bicycle race, 7 water stations are evenly spaced between the start and finish lines, as shown in the figure below. There are also 2 repair stations evenly spaced between the start and finish lines. The 3rd water station is located 2 miles after the 1st repair station. How long is the race in miles?
Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas would like to cover the package with a large number of stamps. Suppose he has a collection of 5-cent, 10-cent, and 25-cent stamps, with exactly 20 of each type. What is the greatest number of stamps Nicolas can use to make exactly $7.10 in postage?
(Note: The amount $7.10 corresponds to 7 dollars and 10 cents. One dollar is worth 100 cents.)
Viswam walks half a mile to get to school each day. His route consists of 10 city blocks of equal length and he takes 1 minute to walk each block. Today, after walking 5 blocks, Viswam discovers he has to make a detour, walking 3 blocks of equal length instead of 1 block to reach the next corner. From the time he starts his detour, at what speed, in mph, must he walk in order to get to school at his usual time?
The letters P, Q, and R are entered into a 20×20 table according to the pattern shown below. How many P's, Q's, and R's will appear in the completed table?
A regular octahedron has eight equilateral triangle faces with four faces meeting at each vertex. Jun will make the regular octahedron shown on the right by folding the piece of paper shown on the left. Which numbered face will end up to the right of Q?
Greta Grasshopper sits on a long line of lily pads in a pond. From any lily pad, Greta can jump 5 pads to the right or 3 pads to the left. What is the fewest number of jumps Greta must make to reach the lily pad located 2023 pads to the right of her starting position?
An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is 32 the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle?
2023 AMC8 #20 · Counting, Probability & Statistics · ★★★
Two integers are inserted into the list 3, 3, 8, 11, 28 to double its range. The mode and median remain unchanged. What is the maximum possible sum of the two additional numbers?
Henry the donkey has a very long piece of pasta. He takes a number of bites of pasta, each time eating 3 inches of pasta from the middle of one piece. In the end, he has $10$ pieces of pasta whose total length is 17 inches. How long, in inches, was the piece of pasta he started with?
驴 Henry 有一根很长的面条。他咬了几口面条，每次都是从一段面条的中间吃掉 3 英寸。最后，他还有 10 段面条，总长度为 17 英寸。问他开始吃的时候，那根面条的长度是多少英寸？
2022 AMC8 #12 · Counting, Probability & Statistics · ★★★
The arrows on the two spinners shown below are spun. Let the number N equal 10 times the number on Spinner A, added to the number on Spinner B. What is the probability that N is a perfect square number?
下图所示的两个转盘上的箭头可自由旋转。数 N 等于 10 乘以转盘 A 上箭头所指的数，再加上转盘 B 上箭头所指的数。问 N 是完全平方数的概率是多少？
How many positive integers can fill the blank in the sentence below? “One positive integer is _____ more than twice another, and the sum of the two numbers is 28 .”
2022 AMC8 #15 · Counting, Probability & Statistics · ★★★
Laszlo went online to shop for black pepper and found thirty different black pepper options varying in weight and price, shown in the scatter plot below. In ounces, what is the weight of the pepper that offers the lowest price per ounce?
Four numbers are written in a row. The average of the first two is 21, the average of the middle two is 26, and the average of the last two is 30. What is the average of the first and last of the numbers?
If n is an even positive integer, the double factorial notation n!! represents the product of all the even integers from 2 to n . For example, 8!!=2⋅4⋅6⋅8 . What is the units digit of the following sum? 2!!+4!!+6!!+⋯+2018!!+2020!!+2022!!
如果 n 是一个正整数，那么双阶乘记号 n！！代表从 2 到 n 的所有偶整数的乘积。例如， 8!!=2·4·6·8。问下面和式的个位数字是几？
2！！+4！！+6！！+ … +2018！！+2020！！+2022！！
2022 AMC8 #19 · Counting, Probability & Statistics · ★★★
Mr. Ramos gave a test to his class of 20 students. The dot plot below shows the distribution of test scores. Later Mr. Ramos discovered that there was a scoring error on one of the questions. He regraded the tests, awarding some of the students 5 extra points, which increased the median test score to 85 . What is the minimum number of students who received extra points? (Note that the median test score equals the average of the 2 scores in the middle if the 20 test scores are arranged in increasing order.)
The grid below is to be filled with integers in such a way that the sum of the numbers in each row and the sum of the numbers in each column are the same. Four numbers are missing. The number x in the lower left corner is larger than the other three missing numbers. What is the smallest possible value of x ?
After school, Maya and Naomi headed to the beach, 6 miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi's and Maya's average speeds?
For a positive integer n , the factorial notation n! represents the product of the integers from n to 1 . What value of N satisfies the following equation?
5!⋅9!=12⋅N!
对一个正整数 n，符号 n！表示从 n 到 1 的所有整数的乘积。例如， 6！=6·5·4·3·2·3.2·1 则 N 为何值时，满足下面的方程？
Jamal has a drawer containing 6 green socks, 18 purple socks, and 12 orange socks. After adding more purple socks, Jamal noticed that there is now a 60% chance that a sock randomly selected from the drawer is purple. How many purple socks did Jamal add?
2020 AMC8 #14 · Counting, Probability & Statistics · ★★★
There are 20 cities in the County of Newton. Their populations are shown in the bar chart below. The average population of all the cities is indicated by the horizontal dashed line. Which of the following is closest to the total population of all 20 cities?
2020 AMC8 #16 · Counting, Probability & Statistics · ★★★
Each of the points A,B,C,D,E, and F in the figure below represents a different digit from 1 to 6. Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is 47. What is the digit represented by B?
A number is called flippy if its digits alternate between two distinct digits. For example, 2020 and 37373 are flippy, but 3883 and 123123 are not. How many five-digit flippy numbers are divisible by $15?$
A scientist walking through a forest recorded as integers the heights of 5 trees standing in a row. She observed that each tree was either 2 times as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?
2020 AMC8 #21 · Counting, Probability & Statistics · ★★★
A game board consists of 64 squares that alternate in color between black and white. The figure below shows square P in the bottom row and square Q in the top row. A marker is placed at P. A step consists of moving the marker onto one of the adjoining white squares in the row above. How many 7 -step paths are there from P to Q? (The figure shows a sample path.)
一种游戏板由 64 个黑白相间的方块组成。下图显示了底行中的一个正方形 P 和顶行中的一个正方形 Q。在正方形 P 上放了一个标记，将标记移到上面一行中相邻的一个白色正方形上，这个操作称为一步。从 P 到 Q 有多少种 7 步路径？（图中显示了一种可能的示例路径）。
2019 AMC8 #11 · Counting, Probability & Statistics · ★★★
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?
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?
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 ?
回环数是指从左向右读和从右向左读，读数是一样的数（例如，12321 是个回环数）。N 是满足以下条件的最小三位整数：它不是个回环数，并且它是 3 个不同的两位回环数的和。问 N 的各个位上的数字之和是多少？
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?
Isabella 有 6 张优惠券，可用于在 Pete 甜品店里免费兑换冰激凌。为了最后使用优惠券，她决定每 10 天兑换一次直到所有的优惠券都使用完毕。已知 Pete 甜品店周日是不营业的，当她在日历上圈出兑换冰激凌的这 6 天时，发现没有哪天是周日。问 Isabella 兑换她的第一个优惠券是在周几？
2019 AMC8 #15 · Counting, Probability & Statistics · ★★★
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 52 . If instead, someone wearing sunglasses is selected at random, what is the probability that this person is also wearing a cap?
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?
2019 AMC8 #18 · Counting, Probability & Statistics · ★★★
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?
2019 AMC8 #19 · Counting, Probability & Statistics · ★★★
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?
2018 AMC8 #11 · Counting, Probability & Statistics · ★★★
Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture, as shown. If the seating positions are assigned randomly, what is the probability that Abby and Bridget are adjacent to each other in the same row or the same column?
The clock in Sri's car, which is not accurate, gains time at a constant rate. One day as he begins shopping, he notes that his car clock and his watch (which is accurate) both say 12:00 noon. When he is done shopping, his watch says 12:30 and his car clock says 12:35. Later that day, Sri loses his watch. He looks at his car clock and it says 7:00. What is the actual time?
Lalia took five math tests, each worth a maximum of 100 points. Laila's score on each test was an integer between 0 and 100, inclusive. Laila received the same score on the first four tests, and she received a higher score on the last test. Her average score on the five tests was 82. How many values are possible for Laila's score on the last test?
In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of 1 square unit, then what is the area of the shaded region, in square units?
2018 AMC8 #16 · Counting, Probability & Statistics · ★★★
Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. How many ways are there to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together?
Bella begins to walk from her house toward her friend Ella's house. At the same time, Ella begins to ride her bicycle toward Bella's house. They each maintain a constant speed, and Ella rides 5 times as fast as Bella walks. The distance between their houses is 2 miles, which is 10,560 feet, and Bella covers 221 feet with each step. How many steps will Bella take by the time she meets Ella?
Bella 开始从她家步行去她朋友 Ella 的家。同时，Ella 骑着自行车开始向 Bella 的家出发。她们均保持各自恒定的速度，并且 Ella 骑车的速度是 Bella 行走速度的 5 倍。她们家相距 2 英里，即 10,560 英尺，已知 Bella 每步长为 221 英尺，那么当 Bella 和 Ella 相遇时，Bella 走了多少步？
2018 AMC8 #19 · Counting, Probability & Statistics · ★★★
In a sign pyramid a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid?
In △ABC, a point E is on AB with AE=1 and EB=2. Point D is on AC so that DE∥BC and point F is on BC so that EF∥AC. What is the ratio of the area of CDEF to the area of △ABC?
在 △ABC 中，点 E 在边 AB 上，满足 AE=1，EB=2。点 D 在边 AC 上，满足 DE∥BC；点 F 在边 BC 上，满足 EF∥AC。则 CDEF 的面积与 △ABC 的面积之比是多少？
A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 37, how many tiles cover the floor?
The smallest positive integer greater than 1 that leaves a remainder of 1 when divided by 4, 5, and 6 lies between which of the following pairs of numbers?
2017 AMC8 #13 · Counting, Probability & Statistics · ★★★
Peter, Emma, and Kyler played chess with each other. Peter won 4 games and lost 2 games. Emma won 3 games and lost 3 games. If Kyler lost 3 games, how many games did he win?
Chloe and Zoe are both students in Ms. Demeanor's math class. Last night, they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only 80% of the problems she solved alone, but overall 88% of her answers were correct. Zoe had correct answers to 90% of the problems she solved alone. What was Zoe's overall percentage of correct answers?
2017 AMC8 #15 · Counting, Probability & Statistics · ★★★
In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the middle, a path allows only moves from one letter to an adjacent (above, below, left, or right, but not diagonal) letter. One example of such a path is traced in the picture.
在下图所示的字母和数字的排布中，拼写出 AMC8 有多少种方法？要求从中间的 A 开始，移动的路径只允许从一个字母移到和这个字母相邻的字母（上，下，左，右，但不允许是对角线）。一种可能的路径标示在图中。
Starting with some gold coins and some empty treasure chests, I tried to put 9 gold coins in each treasure chest, but that left 2 treasure chests empty. So instead I put 6 gold coins in each treasure chest, but then I had $3$ gold coins left over. How many gold coins did I have?
Determine how many two-digit numbers satisfy the following property: when the number is added to the number obtained by reversing its digits, the sum is 132.
Jefferson Middle School has the same number of boys and girls. 43 of the girls and 32 of the boys went on a field trip. What fraction of the students on the field trip were girls?
Jefferson 中学的男生和女生数目相同。 的女生和 的男生去郊游。那么这些去郊游的学生中，有多少比例是女生？
Karl's car uses a gallon of gas every 35 miles, and his gas tank holds 14 gallons when it is full. One day, Karl started with a full tank of gas, drove 350 miles, bought 8 gallons of gas, and continued driving to his destination. When he arrived, his gas tank was half full. How many miles did Karl drive that day?
Karl 的汽车每开 35 英里就需要 1 加仑汽油，且当他的油缸加满时，可以装 14 加仑的汽油。
一天，Karl 开始他的行程前，把油缸加满了汽油，接着开了 350 英里，之后买了 8 加仑汽油，然后继续开到目的地。当他到达目的地时，他的油缸只剩下一半汽油。问 Karl 那天总共开了多少英里？
Annie and Bonnie are running laps around a 400 -meter oval track. They started together, but Annie has pulled ahead, because she runs 25% faster than Bonnie. How many laps will Annie have run when she first passes Bonnie?
2016 AMC8 #17 · Counting, Probability & Statistics · ★★★
An ATM password at Fred's Bank is composed of four digits from 0 to 9 , with repeated digits allowable. If no password may begin with the sequence 9,1,1, then how many passwords are possible?
Fred 银行的取款机密码由 0 到 9 中的 4 个数字组成，且允许重复，如果密码不允许以序列 9,1,1 开头，那么一共有多少种可能的密码？
2016 AMC8 #18 · Counting, Probability & Statistics · ★★★
In an All-Area track meet, 216 sprinters enter a 100- meter dash competition. The track has $6$ lanes, so only 6 sprinters can compete at a time. At the end of each race, the five non-winners are eliminated, and the winner will compete again in a later race. How many races are needed to determine the champion sprinter?
The least common multiple of a and b is 12 , and the least common multiple of b and c is 15 . What is the least possible value of the least common multiple of a and c ?
a 和 b 的最小公倍数是 12，b 和 c 的最小公倍数是 15，问 a 和 c 的最小公倍数的最小可能值是多少？
2015 AMC8 #11 · Counting, Probability & Statistics · ★★★
In the small country of Mathland, all automobile license plates have four symbols. The first must be a vowel (A, E, I, O, or U), the second and third must be two different letters among the 21 non-vowels, and the fourth must be a digit (0 through 9). If the symbols are chosen at random subject to these conditions, what is the probability that the plate will read "AMC8"?
2015 AMC8 #15 · Counting, Probability & Statistics · ★★★
At Euler Middle School, 198 students voted on two issues in a school referendum with the following results: 149 voted in favor of the first issue and 119 voted in favor of the second issue. If there were exactly 29 students who voted against both issues, how many students voted in favor of both issues?
In a middle-school mentoring program, a number of the sixth graders are paired with a ninth-grade student as a buddy. No ninth grader is assigned more than one sixth-grade buddy. If 31 of all the ninth graders are paired with 52 of all the sixth graders, what fraction of the total number of sixth and ninth graders have a buddy?
Jeremy's father drives him to school in rush hour traffic in 20 minutes. One day there is no traffic, so his father can drive him 18 miles per hour faster and gets him to school in 12 minutes. How far in miles is it to school?
Jeremy 的爸爸在交通高峰期把他送到学校需要 20 分钟。这一天路上车不多，因此他爸爸开车的速度提高了 18 英里每小时，把他送到学校花了 12 分钟，那么从他家到学校的距离有多少英里？
An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, 2,5,8,11,14 is an arithmetic sequence with five terms, in which the first term is 2 and the constant added is 3 . Each row and each column in this 5×5 array is an arithmetic sequence with five terms. The square in the center is labelled X as shown. What is the value of X ?
Ralph went to the store and bought 12 pairs of socks for a total of $24 . Some of the socks he bought cost $1 a pair, some of the socks he bought cost $3 a pair, and some of the socks he bought cost $4 a pair. If he bought at least one pair of each type, how many pairs of $1 socks did Ralph buy?
2014 AMC8 #11 · Counting, Probability & Statistics · ★★★
Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks?
Jack 想从他家骑车到 Jill 家去，Jill 家位于 Jack 家向东 3 个街区，向北 2 个街区的位置。当 Jack 骑车经过每个街区后，他可以选择继续向东或者向北，但他必须绕开离他家向东 1 个街区，向北 1 个街区的一个危险的十字路口。那么他经过 5 个街区骑车到 Jill 的家一共有多少种路线？
2014 AMC8 #12 · Counting, Probability & Statistics · ★★★
A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What is the probability that a reader guessing at random will match all three correctly?
The circumference of the circle with center O is divided into 12 equal arcs, marked the letters A through L as seen below. What is the number of degrees in the sum of the angles x and y ?
如下图所示，圆心为 O 的圆被分成 12 个相等的圆弧，由字母 A 到 L 标识。那么角 x 和角 y 的度数之和是多少？
2014 AMC8 #16 · Counting, Probability & Statistics · ★★★
The "Middle School Eight" basketball conference has $8$ teams. Every season, each team plays every other conference team twice (home and away), and each team also plays 4 games against non-conference opponents. What is the total number of games in a season involving the "Middle School eight" teams?
George walks 1 mile to school. He leaves home at the same time each day, walks at a steady speed of 3 miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first 21 mile at a speed of only 2 miles per hour. At how many miles per hour must George run the last 21 mile in order to arrive just as school begins today?
George 每天同一时间以恒定的 3 英里每小时的速度从家步行 1 英里去学校，且恰好准时到校。今天他被好天气吸引，因此前 英里路程的速度仅为 2 英里每小时。那么后面 英里的路程他应该以多少英里每小时的速度奔跑才能准时到校？
2014 AMC8 #18 · Counting, Probability & Statistics · ★★★
Four children were born at City Hospital yesterday. Assume each child is equally likely to be a boy or a girl. Which of the following outcomes is most likely?
A cube with 3-inch edges is to be constructed from 27 smaller cubes with 1-inch edges. Twenty-one of the cubes are colored red and 6 are colored white. If the 3-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white?
Rectangle ABCD has sides CD=3 and DA=5 . A circle with a radius of 1 is centered at A , a circle with a radius of 2 is centered at B , and a circle with a radius of 3 is centered at C . Which of the following is closest to the area of the region inside the rectangle but outside all three circles?
长方形 ABCD 的边长 CD＝3，DA＝5．圆 A 的半径为 1，圆 B 的半径为 2，圆 C 的半径为
3．那么在长方形内部但在三个圆外部的区域的面积最接近下面哪个数？
Ted's grandfather used his treadmill on 3 days this week. He went 2 miles each day. On Monday he jogged at a speed of 5 miles per hour. He walked at the rate of 3 miles per hour on Wednesday and at 4 miles per hour on Friday. If Grandfather had always walked at 4 miles per hour, he would have spent less time on the treadmill. How many minutes less?
At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the $150 regular price did he save?
When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one?
当 Clara 把她的所有的考试分数相加时，她不小心把某个分数的十位数字和个位数字调换了。那么她不正确的总分和正确总分可能相差多少？
2013 AMC8 #14 · Counting, Probability & Statistics · ★★★
Abe holds 1 green and 1 red jelly bean in his hand. Bob holds 1 green, 1 yellow, and 2 red jelly beans in his hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match?
A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of 8th -graders to 6th -graders is 5:3 , and the the ratio of 8th -graders to 7th -graders is 8:5 . What is the smallest number of students that could be participating in the project?
Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?
2013 AMC8 #19 · Counting, Probability & Statistics · ★★★
Bridget, Cassie, and Hannah are discussing the results of their last math test. Hannah shows Bridget and Cassie her test, but Bridget and Cassie don't show theirs to anyone. Cassie says, 'I didn't get the lowest score in our class,' and Bridget adds, 'I didn't get the highest score.' What is the ranking of the three girls from highest to lowest?
2013 AMC8 #21 · Counting, Probability & Statistics · ★★★
Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?
Jamar bought some pencils costing more than a penny each at the school bookstore and paid $1.43 . Sharona bought some of the same pencils and paid $1.87 . How many more pencils did Sharona buy than Jamar?
2012 AMC8 #14 · Counting, Probability & Statistics · ★★★
In the BIG N, a middle school football conference, each team plays every other team exactly once. If a total of 21 conference games were played during the 2012 season, how many teams were members of the BIG N conference?
2012 AMC8 #16 · Counting, Probability & Statistics · ★★★
Each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 is used only once to make two five-digit numbers so that they have the largest possible sum. Which of the following could be one of the numbers?
A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square?
Marla has a large white cube that has an edge of 10 feet. She also has enough green paint to cover 300 square feet. Marla uses all the paint to create a white square centered on each face, surrounded by a green border. What is the area of one of the white squares, in square feet?
2011 AMC8 #11 · Counting, Probability & Statistics · ★★★
The graph shows the number of minutes studied by both Asha (black bar) and Sasha (grey bar) in one week. On the average, how many more minutes per day did Sasha study than Asha?
2011 AMC8 #12 · Counting, Probability & Statistics · ★★★
Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other?
Angie，Bridget，Carlos 和 Diego 随机围着一张方桌坐下，一人坐一边。那么 Angie 和 Carlos 面对面坐着的概率是多少？
Two congruent squares, ABCD and PQRS , have side length 15 . They overlap to form the 15 by 25 rectangle AQRD shown. What percent of the area of rectangle AQRD is shaded?
There are 270 students at Colfax Middle School, where the ratio of boys to girls is 5:4 . There are 180 students at Winthrop Middle School, where the ratio of boys to girls is 4:5 . The two schools hold a dance and all students from both schools attend. What fraction of the students at the dance are girls?
Let A be the area of the triangle with sides of length 25, 25 , and 30 . Let B be the area of the triangle with sides of length 25, 25, and 40 . What is the relationship between A and B ?
A 表示边长为 25，25，30 的三角形的面积，B 表示边长为 25，25，40 的三角形的面积，那么 A 和 B 的关系是什么？
2010 AMC8 #7 · Counting, Probability & Statistics · ★★★
Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than a dollar?
The top of one tree is 16 feet higher than the top of another tree. The heights of the two trees are in the ratio 3:4 . In feet, how tall is the taller tree?
Of the 500 balls in a large bag, 80% are red and the rest are blue. How many of the red balls must be removed so that 75% of the remaining balls are red?
The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shortest side is 30% of the perimeter. What is the length of the longest side?
A jar contains 5 different colors of gumdrops. 30% are blue, 20% are brown, 15% are red, 10% are yellow, and other 30 gumdrops are green. If half of the blue gumdrops are replaced with brown gumdrops, how many gumdrops will be brown?
The diagram shows an octagon consisting of 10 unit squares. The portion below PQ is a unit square and a triangle with base 5 . If PQ bisects the area of the octagon, what is the ratio QYXQ?
A decorative window is made up of a rectangle with semicircles on either end. The ratio of AD to AB is 3:2 , and AB is 30 inches. What is the ratio of the area of the rectangle to the combined areas of the semicircles?
The two circles pictured have the same center C . Chord AD is tangent to the inner circle at B , AC is 10 , and chord AD has length 16 . What is the area between the two circles?
下图中的两个圆是同心圆，圆心都是 C。弦 AD 与内圆切于 B，AC 长为 10，弦 AD 长为 16。那么这两个圆之间区域的面积是多少？
2010 AMC8 #20 · Counting, Probability & Statistics · ★★★
In a room, 2/5 of the people are wearing gloves, and 3/4 of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and gloves?
The hundreds digit of a three-digit number is 2 more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result?
The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $1.43. Some of the 30 sixth graders each bought a pencil, and they paid a total of $1.95. How many more sixth graders than seventh graders bought a pencil?
2009 AMC8 #12 · Counting, Probability & Statistics · ★★★
The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime?
Austin and Temple are 50 miles apart along Interstate 35. Bonnie drove from Austin to her daughter's house in Temple, averaging 60 miles per hour. Leaving the car with her daughter, Bonnie rode a bus back to Austin along the same route and averaged 40 miles per hour on the return trip. What was the average speed for the round trip, in miles per hour?
A recipe that makes 5 servings of hot chocolate requires 2 squares of chocolate, 41 cup sugar, 1 cup water and 4 cups milk. Jordan has $5$ squares of chocolate, 2 cups of sugar, lots of water, and 7 cups of milk. If he maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate he can make?
The positive integers x and y are the two smallest positive integers for which the product of 360 and x is a square and the product of 360 and y is a cube. What is the sum of x and y ?
x 和 y 是满足 360 和 x 之积为平方数，360 和 y 之积为立方数的最小正整数。那么 x 和 y 的和是多少？
The diagram represents a 7-foot-by-7-foot floor that is tiled with 1-square-foot black tiles and white tiles. Notice that the corners have white tiles. If a 15-foot-by-15-foot floor is to be tiled in the same manner, how many white tiles will be needed?
2008 AMC8 #11 · Counting, Probability & Statistics · ★★★
Each of the 39 students in the eighth grade at Lincoln Middle School has one dog or one cat or both a dog and a cat. Twenty students have a dog and 26 students have a cat. How many students have both a dog and a cat?
A ball is dropped from a height of 3 meters. On its first bounce it rises to a height of 2 meters. It keeps falling and bouncing to 32 of the height it reached in the previous bounce. On which bounce will it rise to a height less than 0.5 meters?
Mr. Harman needs to know the combined weight in pounds of three boxes he wants to mail. However, the only available scale is not accurate for weights less than 100 pounds or more than 150 pounds. So the boxes are weighed in pairs in every possible way. The results are 122 , 125 and 127 pounds. What is the combined weight in pounds of the three boxes?
2008 AMC8 #14 · Counting, Probability & Statistics · ★★★
Three A’s, three B’s, and three C’s are placed in the nine spaces so that each row and column contain one of each letter. If A is placed in the upper left corner, how many arrangements are possible?
把三个 A,三个 B 和三个 C 放置在九个空格中，使得每行和每列包含三个字母中的每一个字母。
如果 A 放置在左上角，那么一共有多少种安排方法？
In Theresa's first 8 basketball games, she scored 7, 4, 3, 6, 8, 3, 1 and 5 points. In her ninth game, she scored fewer than 10 points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than 10 points and her points-per-game average for the 10 games was also an integer. What is the product of the number of points she scored in the ninth and tenth games?
Ms.Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of 50 units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles?
Two circles that share the same center have radii 10 meters and 20 meters. An aardvark runs along the path shown, starting at A and ending at K . How many meters does the aardvark run?
两个同心圆的半径分别是 10 米和 20 米。一个土豚沿着图示路径运动，从 A 点开始到 K 点结束。则土豚总共运动了多少米？
2008 AMC8 #19 · Counting, Probability & Statistics · ★★★
Eight points are spaced around at intervals of one unit around a 2×2 square, as shown. Two of the 8 points are chosen at random. What is the probability that the two points are one unit apart?
The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and 43 of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class?
2007 AMC8 #11 · Counting, Probability & Statistics · ★★★
Tiles $I, II, III$ and IV are translated so one tile coincides with each of the rectangles A, B, C and D . In the final arrangement, the two numbers on any side common to two adjacent tiles must be the same. Which of the tiles is translated to Rectangle C ?
瓷砖 I，II，III 和 IV 被平移至与长方形 A，B，C，D 重合。排列完毕后，要求两块相邻的瓷砖的公共边上的数必须相等。那么哪块瓷砖平移到了长方形 C 上？
A unit hexagram is composed of a regular hexagon of side length 1 and its 6 equilateral triangular extensions, as shown in the diagram. What is the ratio of the area of the extensions to the area of the original hexagon?
2007 AMC8 #13 · Counting, Probability & Statistics · ★★★
Sets A and B, shown in the Venn diagram, have the same number of elements. Their union has 2007 elements and their intersection has 1001 elements. Find the number of elements in A.
如下图所示的文氏图中，集合 A 和集合 B 中的元素个数相等。它们的并集有 2007 个元素，且它们的交集有 1001 个元素。那么求集合 A 中的元素个数。
Amanda draws five circles with radii 1, 2, 3, 4 and 5. Then for each circle she plots the point (C,A), where C is its circumference and A is its area. Which of the following could be her graph?
A mixture of 30 liters of paint is 25% red tint, 30% yellow tint and 45% water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint in the new mixture?
The product of the two 99 -digit numbers 303,030,303,...,030,303 and 505,050,505,...,050,505 has thousands digit A and units digit B . What is the sum of A and B ?
两个 99 位数字 303,030,303,…,030,303 和 505,050,505,…,050,505 的乘积的千位数字为 A，个位数字是 B。那么 A 与 B 的和是多少？
Pick two consecutive positive integers whose sum is less than 100 . Square both of those integers and then find the difference of the squares. Which of the following could be the difference?
Before the district play, the Unicorns had won 45% of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all?
2007 AMC8 #21 · Counting, Probability & Statistics · ★★★
Two cards are dealt from a deck of four red cards labeled A , B , C , D and four green cards labeled A , B , C , D . A winning pair is two of the same color or two of the same letter. What is the probability of drawing a winning pair?
Antonette gets 70 % on a 10-problem test, 80 % on a 20-problem test and 90 % on a 30-problem test. If the three tests are combined into one 60-problem test, which percent is closest to her overall score?
Cassie leaves Escanaba at 8:30 AM heading for Marquette on her bike. She bikes at a uniform rate of 12 miles per hour. Brian leaves Marquette at 9:00 AM heading for Escanaba on his bike. He bikes at a uniform rate of 16 miles per hour. They both bike on the same 62-mile route between Escanaba and Marquette. At what time in the morning do they meet?
Problems 14, 15 and 16 involve Mrs. Reed's English assignment. A Novel Assignment The students in Mrs. Reed's English class are reading the same 760 -page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds. If Bob and Chandra both read the whole book, Bob will spend how many more seconds reading than Chandra?
Problems 14, 15 and 16 involve Mrs. Reed's English assignment. A Novel Assignment The students in Mrs. Reed's English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds. Chandra and Bob, who each have a copy of the book, decide that they can save time by "team reading" the novel. In this scheme, Chandra will read from page 1 to a certain page and Bob will read from the next page through page 760, finishing the book. When they are through they will tell each other about the part they read. What is the last page that Chandra should read so that she and Bob spend the same amount of time reading the novel?
Chandra 和 Bob 各自都有这本书，他们决定通过”团队阅读”这本小说来节省时间。在这种方案下，Chandra 会从第一页读到某一页，Bob 会从下一页一直读到第 760 页，把书读完。当他们都完成各自的任务，他们会告诉对方自己所读的内容。那么为了使得 Chandra 和 Bob 阅读的时间相同，Chandra 所读的最后一页是第几页？
Problems 14, 15 and 16 involve Mrs. Reed's English assignment. A Novel Assignment The students in Mrs. Reed's English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds. Before Chandra and Bob start reading, Alice says she would like to team read with them. If they divide the book into three sections so that each reads for the same length of time, how many seconds will each have to read?
在 Chandra 和 Bob 开始阅读前，Alice 说她想加入和他俩一起团队阅读。若他们把书分成 3 部分，使得每人阅读时间都相同，那么每人阅读时间是多少秒？
A cube with 3-inch edges is made using 27 cubes with 1-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white?
Triangle ABC is an isosceles triangle with AB=BC. Point D is the midpoint of both BC and AE, and CE is 11 units long. Triangle ABD is congruent to triangle ECD . What is the length of BD?
三角形 ABC 是个等腰三角形，。点 D 是线段的中点，同时也是线段的中点。
线段长度为 11 个单位。三角形 ABD 和三角形 ECD 全等的，则的长度是多少？
2006 AMC8 #20 · Counting, Probability & Statistics · ★★★
A singles tournament had six players. Each player played every other player only once, with no ties. If Helen won 4 games, Ines won 3 games, Janet won 2 games, Kendra won 2 games and Lara won 2 games, how many games did Monica win?
An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. The aquarium is filled with water to a depth of 37 cm. A rock with volume 1000cm3 is then placed in the aquarium and completely submerged. By how many centimeters does the water level rise?
The sales tax rate in Rubenenkoville is 6%. During a sale at the Bergville Coat Closet, the price of a coat is discounted 20% from its $90.00 price. Two clerks, Jack and Jill, calculate the bill independently. Jack rings up $90.00 and adds 6% sales tax, then subtracts 20% from this total. Jill rings up $90.00, subtracts 20% of the price, then adds 6% of the discounted price for sales tax. What is Jack's total minus Jill's total?
Big Al, the ape, ate 100 bananas from May 1 through May 5. Each day he ate six more bananas than on the previous day. How many bananas did Big Al eat on May 5?
2005 AMC8 #14 · Counting, Probability & Statistics · ★★★
The Little Twelve Basketball Conference has two divisions, with six teams in each division. Each team plays each of the other teams in its own division twice and every team in the other division once. How many conference games are scheduled?
2005 AMC8 #16 · Counting, Probability & Statistics · ★★★
A five-legged Martian has a drawer full of socks, each of which is red, white or blue, and there are at least five socks of each color. The Martian pulls out one sock at a time without looking. How many socks must the Martian remove from the drawer to be certain there will be 5 socks of the same color?
Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. The game ends when they stop on the same point. How many turns will this take?
Alice 和 Bob 玩了一个游戏，游戏中有一个圆的周长被 12 个等距点分成 12 等份。这些点按顺时针方向从 1 到 12 编号。两个人都从第 12 个点开始。Alice 顺时针移动，而 Bob 逆时针移动。
在游戏的每个回合，Alice 顺时针移动 5 个点，Bob 逆时针移动 9 个点。当他们停在同一点上时，比赛结束。则到游戏结束一共需要经过多少个回合？
2004 AMC8 #11 · Counting, Probability & Statistics · ★★★
The numbers -2, 4, 6, 9 and 12 are rearranged according to these rules:
1. The largest isn't first, but it is in one of the first three places.
2. The smallest isn't last, but it is in one of the last three places.
3. The median isn't first or last.
What is the average of the first and last numbers?
Niki usually leaves her cell phone on. If her cell phone is on but she is not actually using it, the battery will last for 24 hours. If she is using it constantly, the battery will last for only 3 hours. Since the last recharge, her phone has been on 9 hours, and during that time she has used it for 60 minutes. If she doesn’t use it any more but leaves the phone on, how many more hours will the battery last?
Thirteen black and six white hexagonal tiles were used to create the figure below. If a new figure is created by attaching a border of white tiles with the same size and shape as the others, what will be the difference between the total number of white tiles and the total number of black tiles in the new figure?
Two 600 mL pitchers contain orange juice. One pitcher is 1/3 full and the other pitcher is 2/5 full. Water is added to fill each pitcher completely, then both pitchers are poured into one large container. What fraction of the mixture in the large container is orange juice?
两个容量为 600 mL 的罐子里装有橙汁。一个罐子装了 31，另一个装了 52。向每个罐子加水使其装满，然后把两个罐子的液体都倒入一个大容器中。那么大容器里的混合液中橙汁占的比例是多少？
2004 AMC8 #18 · Counting, Probability & Statistics · ★★★
Five friends compete in a dart-throwing contest. Each one has two darts to throw at the same circular target, and each individual's score is the sum of the scores in the target regions that are hit. The scores for the target regions are the whole numbers 1 through 10 . Each throw hits the target in a region with a different value. The scores are: Alice 16 points, Ben 4 points, Cindy 7 points, Dave 11 points, and Ellen 17 points. Who hits the region worth 6 points?
A whole number larger than 2 leaves a remainder of 2 when divided by each of the numbers 3, 4, 5, and 6 . The smallest such number lies between which two numbers?
Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If there are 6 empty chairs, how many people are in the room?
2004 AMC8 #21 · Counting, Probability & Statistics · ★★★
Spinners A and B are spun. On each spinner, the arrow is equally likely to land on each number. What is the probability that the product of the two spinners' numbers is even?
将转盘 A 和 B 各旋转一次。已知在每个转盘上，指针落在每个数字上的可能性都相同，那么这两个转盘上（指针所指的）数字之积为偶数的概率是多少？
Business is a little slow at Lou's Fine Shoes, so Lou decides to have a sale. On Friday, Lou increases all of Thursday's prices by 10 percent. Over the weekend, Lou advertises the sale: "Ten percent off the listed price. Sale starts Monday." How much does a pair of shoes cost on Monday that cost 40 dollars on Thursday?
2003 AMC8 #12 · Counting, Probability & Statistics · ★★★
When a fair six-sided dice is tossed on a table top, the bottom face cannot be seen. What is the probability that the product of the faces that can be seen is divisible by 6 ?
Fourteen white cubes are put together to form the figure on the right. The complete surface of the figure, including the bottom, is painted red. The figure is then separated into individual cubes. How many of the individual cubes have exactly four red faces
A figure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a figure with the front and side views shown?
2003 AMC8 #16 · Counting, Probability & Statistics · ★★★
Ali, Bonnie, Carlo, and Dianna are going to drive together to a nearby theme park. The car they are using has $4$ seats: 1 Driver seat, 1 front passenger seat, and 2 back passenger seats. Bonnie and Carlo are the only ones who know how to drive the car. How many possible seating arrangements are there?
2003 AMC8 #17 · Counting, Probability & Statistics · ★★★
The six children listed in the table below are from two families of three siblings each. Each child has blue or brown eyes and black or blond hair. Children from the same family have at least one of these characteristics in common. Which two children are Jim's siblings?
下面列出的六名儿童来自两个家庭，每个家庭有三个兄弟姐妹。每个孩子都有蓝色或棕色的眼睛，和黑色或金色的头发。来自同一家庭的儿童至少有一个共同特征。哪两个孩子是 Jim 的兄弟姐妹？
2003 AMC8 #18 · Counting, Probability & Statistics · ★★★
Each of the twenty dots on the graph below represents one of Sarah's classmates. Classmates who are friends are connected with a line segment. For her birthday party, Sarah is inviting only the following: all of her friends and all of those classmates who are friends with at least one of her friends. How many classmates will not be invited to Sarah's party?
下图中的 20 个点中的每个点代表 Sarah 的一个同班同学。这些同班同学中，若某两人之间是朋友关系，则用线段将这两人相连。Sarah 只打算邀请符合下面条件的人参加她的生日聚会：
她的所有朋友，以及所有与她的至少一个朋友是朋友。有多少同学不会被邀请参加 Sarah 的聚会？
In the pattern below, the cat moves clockwise through the four squares, and the mouse moves counterclockwise through the eight exterior segments of the four squares.
If the pattern is continued, where would the cat and mouse be after the 247th move?
A ship travels from point A to point B along a semicircular path, centered at Island X . Then it travels along a straight path from B to C . Which of these graphs best shows the ship's distance from Island X as it moves along its course?
一艘船沿着以岛 X 为圆心的半圆形路径从点 A 航行到点 B。然后它沿着直线路径从 B 到 C。下面这些图表中哪一个最能显示当船沿航线行驶时，船与岛 X 的距离变化？
A sequence of squares is made of identical square tiles. The edge of each square is one tile length longer than the edge of the previous square. The first three squares are shown. How many more tiles does the seventh square require than the sixth?
For his birthday, Bert gets a box that holds 125 jellybeans when filled to capacity. A few weeks later, Carrie gets a larger box full of jellybeans. Her box is twice as high, twice as wide and twice as long as Bert's. Approximately, how many jellybeans did Carrie get?
Right isosceles triangles are constructed on the sides of a right triangle, as shown. A capital letter represents the area of each triangle. Which one of the following is true?
In a mathematics contest with ten problems, a student gains 5 points for a correct answer and loses 2 points for an incorrect answer. If Olivia answered every problem and her score was 29, how many correct answers did she have?
Gage skated 1 hr 15 min each day for 5 days and 1 hr 30 min each day for 3 days. How long would he have to skate the ninth day in order to average 85 minutes of skating each day for the entire time?
The area of triangle XYZ is 8 square inches. Points A and B are midpoints of congruent segments XY and XZ . Altitude XC bisects YZ . The area (in square inches) of the shaded region is
三角形 XYZ 的面积是 8 平方英寸。点 A 和点 B 分别是两条相等线段 XY 和 XZ 的中点，高 XC 平分 YZ。那么阴影部分的面积是多少平方英寸？
A corner of a tiled floor is shown. If the entire floor is tiled in this way and each of the four corners looks like this one, then what fraction of the tiled floor is made of darker tiles?
2001 AMC8 #13 · Counting, Probability & Statistics · ★★★
Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle's pie graph showing this data, how many degrees should she use for cherry pie?
2001 AMC8 #14 · Counting, Probability & Statistics · ★★★
Tyler has entered a buffet line in which he chooses one kind of meat, two different vegetables and one dessert. If the order of food items is not important, how many different meals might he choose?
Homer began peeling a pile of 44 potatoes at the rate of 3 potatoes per minute. Four minutes later Christen joined him and peeled at the rate of 5 potatoes per minute. When they finished, how many potatoes had Christen peeled?
A square piece of paper, 4 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle?
For the game show Who Wants To Be A Millionaire?, the dollar values of each question are shown in the table below (where K = 1000). Between which two questions is the percent increase of the value the smallest?
Car M traveled at a constant speed for a given time. This is shown by the dashed line. Car N traveled at twice the speed for the same distance. If Car M and Car N's speed and time are shown as solid line, which graph illustrates this?
汽车 M 在给定的时间内以恒定的速度行驶，其速度和时间曲线用虚线所示。汽车 N 以两倍的速度行驶了相同的距离。如果 N 车的速度和时间曲线用实线表示，则下面哪张图表说明了这一点？
2001 AMC8 #20 · Counting, Probability & Statistics · ★★★
Kaleana shows her test score to Quay, Marty and Shana, but the others keep theirs hidden. Quay thinks, "At least two of us have the same score." Marty thinks, "I didn't get the lowest score." Shana thinks, "I didn't get the highest score." List the scores from lowest to highest for Marty (M), Quay (Q) and Shana (S).
2000 AMC8 #5 · Counting, Probability & Statistics · ★★★
Each principal of Lincoln High School serves exactly one 3-year term. What is the maximum number of principals this school could have during an 8-year period?
2000 AMC8 #12 · Counting, Probability & Statistics · ★★★
A block wall 100 feet long and 7 feet high will be constructed using blocks that are 1 foot high and either 2 feet long or 1 foot long (no blocks may be cut). The vertical joins in the blocks must be staggered as shown, and the wall must be even on the ends. What is the smallest number of blocks needed to build this wall?
Triangles ABC , ADE , and EFG are all equilateral. Points D and G are midpoints of AC and AE , respectively. If AB = 4 , what is the perimeter of figure ABCDEFG ?
三角形 ABC，ADE 和 EFG 都是等边三角形。点 D 和 G 分别是形 ABCDEFG 的周长是多少？
In order for Mateen to walk a kilometer (1000m) in his rectangular backyard, he must walk the length 25 times or walk its perimeter 10 times. What is the area of Mateen's backyard in square meters?
Three circular arcs of radius 5 units bound the region shown. Arcs AB and AD are quarter-circles, and arc BCD is a semicircle. What is the area, in square units, of the region?
三段半径为 5 个单位长的圆弧围成图示区域。弧 AB 和 AD 均为四分之一圆，弧 BCD 是个半圆。
那么这个区域的面积是多少平方单位？
You have nine coins: a collection of pennies, nickels, dimes, and quarters having a total value of $1.02 , with at least one coin of each type. How many dimes must you have?
A cube has edge length 2 . Suppose that we glue a cube of edge length 1 on top of the big cube so that one of its faces rests entirely on the top face of the larger cube. The percent increase in the surface area (sides, top, and bottom) from the original cube to the new solid formed is closest to
1999 AMC8 #11 · Counting, Probability & Statistics · ★★★
Each of the five numbers 1, 4, 7, 10, and 13 is placed in one of the five squares so that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column. The largest possible value for the horizontal or vertical sum is
The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is 11/4 . To the nearest whole percent, what percent of its games did the team lose?
1999 AMC8 #13 · Counting, Probability & Statistics · ★★★
The average age of the 40 members of a computer science camp is 17 years. There are 20 girls, 15 boys, and 5 adults. If the average age of the girls is 15 and the average age of the boys is 16, what is the average age of the adults?
1999 AMC8 #15 · Counting, Probability & Statistics · ★★★
Bicycle license plates in Flatville each contain three letters. The first is chosen from the set {C,H,L,P,R}, the second from {A,I,O}, and the third from {D,M,N,T}. When Flatville needed more license plates, they added two new letters. The new letters may both be added to one set or one letter may be added to one set and one to another set. What is the largest possible number of ADDITIONAL license plates that can be made by adding two letters?
Tori's mathematics test had 75 problems: 10 arithmetic, 30 algebra, and 35 geometry problems. Although she answered 70% of the arithmetic, 40% of the algebra, and 60% of the geometry problems correctly, she did not pass the test because she got less than 60% of the problems right. How many more problems would she have needed to answer correctly to earn a 60% passing grade?
At Central Middle School the 108 students who take the AMC 8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: 121 cups of flour, 2 eggs, 3 tablespoons butter, 43 cups sugar, and 1 package of chocolate drops. They will make only full recipes, not partial recipes. Walter can buy eggs by the half-dozen. How many half-dozens should he buy to make enough cookies? (Some eggs and some cookies may be left over.)
中央中学 108 名参加 AMC 8 的学生晚上聚会讨论题目，平均每人吃两块曲奇。Walter 和 Gretel 今年正在烤 Bonnie's Best Bar Cookies。他们的配方可以做一盘 15 块曲奇，配料表如下：121 杯面粉、2 个鸡蛋、3 汤匙黄油、43 杯糖、1 包巧克力豆。他们只做完整的配方，不做部分配方。
Walter 可以按半打买鸡蛋。他应该买多少个半打鸡蛋才能做足够的曲奇？（可能会剩下一些鸡蛋和一些曲奇。）
At Central Middle School the 108 students who take the AMC8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: 121 cups flour, 2 eggs, 3 tablespoons butter, 43 cups sugar, and 1 package of chocolate drops. They will make only full recipes, not partial recipes. They learn that a big concert is scheduled for the same night and attendance will be down 25% . How many recipes of cookies should they make for their smaller party?
中央中学 108 名参加 AMC8 的学生晚上聚会讨论题目，平均每人吃两块曲奇。Walter 和 Gretel 今年正在烤 Bonnie's Best Bar Cookies。他们的配方可以做一盘 15 块曲奇，配料表如下：121 杯面粉、2 个鸡蛋、3 汤匙黄油、43 杯糖、1 包巧克力豆。他们只做完整的配方，不做部分配方。
At Central Middle School, the 108 students who take the AMC 8 meet in the evening to talk about problems and eat an average of two cookies apiece. Hansel and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists these items: 121 cups flour, 2 eggs, 3 tablespoons butter, 43 cups sugar, and 1 package of chocolate drops. They will make full recipes, not partial recipes. Hansel and Gretel must make enough pans of cookies to supply 216 cookies. There are 8 tablespoons in a stick of butter. How many sticks of butter will be needed? (Some butter may be leftover, of course.)
中央中学 108 名参加 AMC 8 的学生晚上聚会讨论题目，平均每人吃两块曲奇。Hansel 和 Gretel 今年正在烤 Bonnie's Best Bar Cookies。他们的配方可以做一盘 15 块曲奇，配料表如下：121 杯面粉、2 个鸡蛋、3 汤匙黄油、43 杯糖、1 包巧克力豆。他们只做完整的配方，不做部分配方。
Figure 1 is called a "stack map." The numbers tell how many cubes are stacked in each position. Fig. 2 shows these cubes, and Fig. 3 shows the view of the stacked cubes as seen from the front.
Which of the following is the front view for the stack map in Fig. 4?
In a far-off land three fish can be traded for two loaves of bread and a loaf of bread can be traded for four bags of rice. How many bags of rice is one fish worth?
An Annville Junior High School, 30% of the students in the Math Club are in the Science Club, and 80% of the students in the Science Club are in the Math Club. There are 15 students in the Science Club. How many students are in the Math Club?
As indicated by the diagram below, a rectangular piece of paper is folded bottom to top, then left to right, and finally, a hole is punched at X. What does the paper look like when unfolded?
如下图所示，将一张矩形纸从下往上折叠，再从左往右折叠，最后在 X 处打一个孔。展开后纸张是什么样子？
1998 AMC8 #19 · Counting, Probability & Statistics · ★★★
Tamika selects two different numbers at random from the set {8,9,10} and adds them. Carlos takes two different numbers at random from the set {3,5,6} and multiplies them. What is the probability that Tamika's result is greater than Carlos' result?
Tamika 从集合 {8,9,10} 中随机选两个不同的数并相加。Carlos 从集合 {3,5,6} 中随机选两个不同的数并相乘。Tamika 的结果大于 Carlos 的结果的概率是多少？
Let PQRS be a square piece of paper. P is folded onto R and then Q is folded onto S . The area of the resulting figure is 9 square inches. Find the perimeter of square PQRS .
设 PQRS 为一张正方形纸。将 P 折到 R 上，再将 Q 折到 S 上。所得图形的面积为 9 平方英寸。求正方形 PQRS 的周长。
Three bags of jelly beans contain 26, 28, and 30 beans. The ratios of yellow beans to all beans in each of these bags are 50% , 25% , and 20% , respectively. All three bags of candy are dumped into one bowl. Which of the following is closest to the ratio of yellow jelly beans to all beans in the bowl?
1997 AMC8 #14 · Counting, Probability & Statistics · ★★★
There is a set of five positive integers whose average (mean) is 5, whose median is 5, and whose only mode is 8. What is the difference between the largest and smallest integers in the set?
Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The ratio of the area of the inscribed square to the area of the large square is
Penni Precisely buys $100 worth of stock in each of three companies: Alabama Almonds, Boston Beans, and California Cauliflower. After one year, AA was up 20%, BB was down 25%, and CC was unchanged. For the second year, AA was down 20% from the previous year, BB was up 25% from the previous year, and CC was unchanged. If A, B, and C are the final values of the stock, then
A cube has eight vertices (corners) and twelve edges. A segment, such as x , which joins two vertices not joined by an edge is called a diagonal. Segment y is also a diagonal. How many diagonals does a cube have?
一个正方体有八个顶点和十二条棱。像 x 这样连接两个不被棱相连的顶点的线段称为对角线。线段 y 也是一条对角线。一个正方体有多少条对角线？
At the grocery store last week, small boxes of facial tissue were priced at 4 boxes for $5 . This week they are on sale at 5 boxes for $4 . The percent decrease in the price per box during the sale was closest to
1997 AMC8 #20 · Counting, Probability & Statistics · ★★★
A pair of 8-sided dice have sides numbered 1 through 8. Each side has the same probability (chance) of landing face up. The probability that the product of the two numbers that land face-up exceeds 36 is
In the fall of 1996, a total of 800 students participated in an annual school clean-up day. The organizers of the event expect that in each of the years 1997, 1998, and 1999, participation will increase by 50% over the previous year. The number of participants the organizers will expect in the fall of 1999 is
are placed in the squares in the figure shown so that the sum of the entries in the vertical column is 23 and the sum of the entries in the horizontal row is 12. The sum of the six digits used is
Figure OPQR is a square. Point O is the origin, and point Q has coordinates (2,2). What are the coordinates for T so that the area of triangle PQT equals the area of square OPQR ?
NOT TO SCALE
图形 OPQR 是一个正方形。点 O 是原点，点 Q 的坐标为 (2,2)。点 T 的坐标应为何值，才能使三角形 PQT 的面积等于正方形 OPQR 的面积？
Ana's monthly salary was $2000 in May. In June she received a 20% raise. In July she received a 20% pay cut. After the two changes in June and July, Ana's monthly salary was
Ana 五月份的月薪是 $2000。六月份她获得了 20% 的加薪。七月份她的薪水被削减了 20%。经过六月和七月的两次变动后，Ana 的月薪是
1996 AMC8 #19 · Counting, Probability & Statistics · ★★★
The pie charts below indicate the percent of students who prefer golf, bowling, or tennis at East Junior High School and West Middle School. The total number of students at East is 2000 and at West, 2500. In the two schools combined, the percent of students who prefer tennis is
以下饼图显示了 East 初中和 West 中学喜欢高尔夫、保龄球或网球的学生百分比。East 共有 2000 名学生，West 共有 2500 名。两所学校合计，喜欢网球的学生的百分比是
Suppose there is a special key on a calculator that replaces the number x currently displayed with the number given by the formula 1/(1-x) . For example, if the calculator is displaying 2 and the special key is pressed, then the calculator will display -1 since 1/(1-2)=-1 . Now suppose that the calculator is displaying 5. After the special key is pressed 100 times in a row, the calculator will display
Jane can walk any distance in half the time it takes Hector to walk the same distance. They set off in opposite directions around the outside of the 18-block area as shown. When they meet for the first time, they will be closest to
Jane 走任何距离所花的时间是 Hector 走同样距离所花时间的一半。他们从如图所示的 18 个街区区域的外围向相反方向出发。当他们第一次相遇时，他们最接近哪里？
A lucky year is one in which at least one date, when written in the form month/day/year, has the following property: The product of the month times the day equals the last two digits of the year. For example, 1956 is a lucky year because it has the date and 7×8=56 . Which of the following is NOT a lucky year?
A team won 40 of its first 50 games. How many of the remaining 40 games must this team win so it will have won exactly 70 % of its games for the season?
Students from three middle schools worked on a summer project. Seven students from Allen school worked for 3 days. Four students from Balboa school worked for 5 days. Five students from Carver school worked for 9 days. The total amount paid for the students' work was 774. Assuming each student received the same amount for a day's work, how much did the students from Balboa school earn altogether?
1995 AMC8 #17 · Counting, Probability & Statistics · ★★★
The table below gives the percent of students in each grade at Annville and Cleona elementary schools. Annville has 100 students and Cleona has 200 students. In the third grade, the number of students in Annville is what percent of the number of students in Cleona?
The area of each of the four congruent L-shaped regions of this 100-inch by 100-inch square is 3/16 of the total area. How many inches long is the side of the center square?
在这个 100 英寸 × 100 英寸的正方形中，四个全等 L 形区域各占总面积的 3/16。中心正方形的边长是多少英寸？
1995 AMC8 #19 · Counting, Probability & Statistics · ★★★
The graph shows the distribution of the number of children in the families of the students in Ms. Jordan's English class. The median number of children in the family for this distribution is
1995 AMC8 #20 · Counting, Probability & Statistics · ★★★
Diana and Apollo each roll a standard die obtaining a number at random from 1 to 6 . What is the probability that Diana's number is larger than Apollo's number?
Diana 和 Apollo 各自掷一个标准骰子，随机得到 1 到 6 中的一个数。Diana 的数大于 Apollo 的数的概率是多少？
1994 AMC8 #11 · Counting, Probability & Statistics · ★★★
Last summer 100 students attended basketball camp. Of those attending, 52 were boys and 48 were girls. Also, 40 students were from Jonas Middle School and 60 were from Clay Middle School. Twenty of the girls were from Jonas Middle School. How many of the boys were from Clay Middle School?
去年夏天 100 名学生参加了篮球夏令营。参加者中 52 名男生，48 名女生。另有 40 名来自 Jonas 中学，60 名来自 Clay 中学。女生中有 20 名来自 Jonas 中学。男生中有多少名来自 Clay 中学？
Each of the three large squares shown below is the same size. Segments that intersect the sides of the squares intersect at the midpoints of the sides. How do the shaded areas of these squares compare?
Two children at a time can play pairball. For 90 minutes, with only two children playing at time, five children take turns so that each one plays the same amount of time. The number of minutes each child plays is
The perimeter of one square is 3 times the perimeter of another square. The area of the larger square is how many times the area of the smaller square?
Pauline Bunyan can shovel snow at the rate of 20 cubic yards for the first hour, 19 cubic yards for the second, 18 for the third, etc., always shoveling one cubic yard less per hour than the previous hour. If her driveway is 4 yards wide, 10 yards long, and covered with snow 3 yards deep, then the number of hours it will take her to shovel it clean is closest to
Mike leaves home and drives slowly east through city traffic. When he reaches the highway he drives east more rapidly until he reaches the shopping mall where he stops. He shops at the mall for an hour. Mike returns home by the same route as he came, driving west rapidly along the highway and then slowly through city traffic. Each graph shows the distance from home on the vertical axis versus the time elapsed since leaving home on the horizontal axis. Which graph is the best representation of Mike's trip?
Mike 离开家，在城市交通中向东缓慢行驶。当他到达高速公路上时，他更快速地向东行驶，直到到达购物中心并停下来。他在购物中心购物一小时。Mike 按原路返回，沿高速公路快速向西行驶，然后缓慢通过城市交通到达家。每个图表的纵轴表示离家距离，横轴表示离家后经过的时间。哪张图能最好地表示 Mike 的行程？
Around the outside of a 4 by 4 square, construct four semicircles (as shown in the figure) with the four sides of the square as their diameters. Another square, ABCD , has its sides parallel to the corresponding sides of the original square, and each side of ABCD is tangent to one of the semicircles. The area of the square ABCD is
Let W,X,Y and Z be four different digits selected from the set {1,2,3,4,5,6,7,8,9}. If the sum XW+ZY is to be as small as possible, then XW+ZY must equal
1994 AMC8 #21 · Counting, Probability & Statistics · ★★★
A gumball machine contains 9 red, 7 white, and 8 blue gumballs. The least number of gumballs a person must buy to be sure of getting four gumballs of the same color is
1993 AMC8 #13 · Counting, Probability & Statistics · ★★★
The word "HELP" in block letters is painted in black with strokes 1 unit wide on a 5 by 15 rectangular white sign with dimensions as shown. The area of the white portion of the sign, in square units, is
Square corners, 5 units on a side, are removed from a 20 unit by 30 unit rectangular sheet of cardboard. The sides are then folded to form an open box. The surface area, in square units, of the interior of the box is
The five tires of a car (four road tires and a full-sized spare) were rotated so that each tire was used the same number of miles during the first 30,000 miles the car traveled. For how many miles was each tire used?
1992 AMC8 #13 · Counting, Probability & Statistics · ★★★
Five test scores have a mean (average score) of 90 , a median (middle score) of 91 and a mode (most frequent score) of 94 . The sum of the two lowest test scores is
On a trip, a car traveled 80 miles in an hour and a half, then was stopped in traffic for 30 minutes, then traveled 100 miles during the next 2 hours. What was the car's average speed in miles per hour for the 4 -hour trip?
1992 AMC8 #19 · Counting, Probability & Statistics · ★★★
The distance between the 5th and 26th exits on an interstate highway is 118 miles. If any two consecutive exits are at least 5 miles apart, then what is the largest number of miles there can be between two consecutive exits that are between the 5th and 26th exits?
1992 AMC8 #21 · Counting, Probability & Statistics · ★★★
Northside's Drum and Bugle Corps raised money for a trip. The drummers and bugle players kept separate sales records. According to the double bar graph, in what month did one group's sales exceed the other's by the greatest percent?
1991 AMC8 #14 · Counting, Probability & Statistics · ★★★
Several students are competing in a series of three races. A student earns 5 points for winning a race, 3 points for finishing second and 1 point for finishing third. There are no ties. What is the smallest number of points that a student must earn in the three races to be guaranteed of earning more points than any other student?
All six sides of a rectangular solid were rectangles. A one-foot cube was cut out of the rectangular solid as shown. The total number of square feet in the surface of the new solid is how many more or less than that of the original solid?
The 16 squares on a piece of paper are numbered as shown in the diagram. While lying on a table, the paper is folded in half four times in the following sequence:
1991 AMC8 #17 · Counting, Probability & Statistics · ★★★
An auditorium with 20 rows of seats has 10 seats in the first row. Each successive row has one more seat than the previous row. If students taking an exam are permitted to sit in any row, but not next to another student in that row, then the maximum number of students that can be seated for an exam is
1991 AMC8 #18 · Counting, Probability & Statistics · ★★★
The vertical axis indicates the number of employees, but the scale was accidentally omitted from this graph. What percent of the employees at the Gauss company have worked there for 5 years or more?
For every 3∘ rise in temperature, the volume of a certain gas expands by 4 cubic centimeters. If the volume of the gas is 24 cubic centimeters when the temperature is 32∘ , what was the volume of the gas in cubic centimeters when the temperature was 20∘ ?
1991 AMC8 #22 · Counting, Probability & Statistics · ★★★
Each spinner is divided into 3 equal parts. The results obtained from spinning the two spinners are multiplied. What is the probability that this product is an even number?
The numbers on the faces of this cube are consecutive whole numbers. The sum of the two numbers on each of the three pairs of opposite faces are equal. The sum of the six numbers on this cube is
1990 AMC8 #12 · Counting, Probability & Statistics · ★★★
There are twenty-four 4-digit numbers that use each of the four digits 2, 4, 5, and 7 exactly once. Listed in numerical order from smallest to largest, the number in the 17th position in the list is
One proposal for new postage rates for a letter was 30¢ for the first ounce and 22¢ for each additional ounce (or fraction of an ounce). The postage for a letter weighing 4.5 ounces was
1990 AMC8 #14 · Counting, Probability & Statistics · ★★★
A bag contains only blue balls and green balls. There are 6 blue balls. If the probability of drawing a blue ball at random from this bag is 41 , then the number of green balls in the bag is
A straight concrete sidewalk is to be 3 feet wide, 60 feet long, and 3 inches thick. How many cubic yards of concrete must a contractor order for the sidewalk if concrete must be ordered in a whole number of cubic yards?
1990 AMC8 #20 · Counting, Probability & Statistics · ★★★
The annual incomes of 1,000 families range from $8200 to $98,000 . In error, the largest income was entered on the computer as $980,000 . The difference between the mean of the incorrect data and the mean of the actual data is
The graph relates the distance traveled [in miles] to the time elapsed [in hours] on a trip taken by an experimental airplane. During which hour was the average speed of this airplane the largest?
Many calculators have a reciprocal key x1 that replaces the current number displayed with its reciprocal. For example, if the display is 00004 and the x1 key is depressed, then the display becomes 000.25 . If 00032 is currently displayed, what is the fewest number of times you must depress the x1 key so the display again reads 00032 ?
1989 AMC8 #19 · Counting, Probability & Statistics · ★★★
The graph below shows the total accumulated dollars (in millions) spent by the Surf City government during 1988 . For example, about 0.5 million had been spent by the beginning of February and approximately 2 million by the end of April. Approximately how many millions of dollars were spent during the summer months of June, July, and August?
下图显示了 1988 年 Surf City 政府累计支出总额（百万美元）。例如，到二月初已支出约 0.5 百万美元，到四月底约 2 百万美元。六月、七月和八月的夏季大约支出了多少百万美元？
The figure may be folded along the lines shown to form a number cube. Three number faces come together at each corner of the cube. What is the largest sum of three numbers whose faces come together at a corner?
Jack had a bag of 128 apples. He sold 25% of them to Jill. Next he sold 25% of those remaining to June. Of those apples still in his bag, he gave the shiniest one to his teacher. How many apples did Jack have then?
Jack 有一袋 128 个苹果。他将其中的 25% 卖给了 Jill。接着他将剩下苹果中的 25% 卖给了 June。然后从袋中剩下的苹果中，他把最亮的那一个送给了老师。那时 Jack 还有多少个苹果？
An artist has 14 cubes, each with an edge of 1 meter. She stands them on the ground to form a sculpture as shown. She then paints the exposed surface of the sculpture. How many square meters does she paint?
Suppose the estimated 20 billion dollar cost to send a person to the planet Mars is shared equally by the 250 million people in the U.S. Then each person's share is
1988 AMC8 #16 · Counting, Probability & Statistics · ★★★
Placing no more than one X in each small square, what is the greatest number of X 's that can be put on the grid shown without getting three X 's in a row vertically, horizontally, or diagonally?
Problem The glass gauge on a cylindrical coffee maker shows that there are 45 cups left when the coffee maker is 36% full. How many cups of coffee does it hold when it is full?
Tom's Hat Shoppe increased all original prices by 25% . Now the shoppe is having a sale where all prices are 20% off these increased prices. Which statement best describes the sale price of an item?
Tom's Hat Shoppe 将所有原价提高 25%。现在该店正在进行促销，所有价格在提高后的基础上打 20% 折扣。以下哪句话最能描述商品的促销价格？
Maria buys computer disks at a price of 4 for $5 and sells them at a price of 3 for $5 . How many computer disks must she sell in order to make a profit of $100 ?
The sale ad read: "Buy three tires at the regular price and get the fourth tire for $3. " Sam paid $240 for a set of four tires at the sale. What was the regular price of one tire?
促销广告写道："以原价购买三个轮胎，第四个轮胎仅需 $3。" Sam 在促销中以 $240 买了一套四个轮胎。一个轮胎的原价是多少？
Joyce made 12 of her first 30 shots in the first three games of this basketball season, so her seasonal shooting average was 40% . In her next game, she took 10 shots and raised her seasonal shooting average to 50% . How many of these 10 shots did she make?
1987 AMC8 #17 · Counting, Probability & Statistics · ★★★
Abby, Bret, Carl, and Dana are seated in a row of four seats numbered #1 to #4 . Joe looks at them and says: "Bret is next to Carl." "Abby is between Bret and Carl." However each one of Joe's statements is false. Bret is actually sitting in seat #3 . Who is sitting in seat #2 ?
Abby、Bret、Carl 和 Dana 坐在编号为 #1 到 #4 的四个座位上。Joe 看着他们说：
Half the people in a room left. One third of those remaining started to dance. There were then 12 people who were not dancing. The original number of people in the room was
A calculator has a squaring key x2 which replaces the current number displayed with its square. For example, if the display is 3 and the x2 key is depressed, then the display becomes 9 . If the display reads 2 , how many times must you depress the x2 key to produce a displayed number greater than 500 ?
Suppose n∗ means n1 , the reciprocal of n . For example, 5∗=51 . How many of the following statements are true? i) 3∗+6∗=9∗ ii) 6∗−4∗=2∗ iii) 2∗⋅6∗=12∗ iv) 10∗÷2∗=5∗
1987 AMC8 #23 · Counting, Probability & Statistics · ★★★
The table below gives the 1980 U.S. population for each of the four geographic regions and for each of two ethnic groups. To the nearest percent, what percent of the U.S. Black population lived in the South?
1986 AMC8 #12 · Counting, Probability & Statistics · ★★★
Problem The table below displays the grade distribution of the 30 students in a mathematics class on the last two tests. For example, exactly one student received a 'D' on Test 1 and a 'C' on Test 2 (see circled entry). What percent of the students received the same grade on both tests?
Sale prices at the Ajax Outlet Store are 50% below original prices. On Saturdays an additional discount of 20% off the sale price is given. What is the Saturday price of a coat whose original price is \textdollar180?
1986 AMC8 #16 · Counting, Probability & Statistics · ★★★
A bar graph shows the number of hamburgers sold by a fast food chain each season. However, the bar indicating the number sold during the winter is covered by a smudge. If exactly 25% of the chain's hamburgers are sold in the fall, how many million hamburgers are sold in the winter?
1986 AMC8 #18 · Counting, Probability & Statistics · ★★★
A rectangular grazing area is to be fenced off on three sides using part of a 100 meter rock wall as the fourth side. Fence posts are to be placed every 12 meters along the fence including the two posts where the fence meets the rock wall. What is the fewest number of posts required to fence an area 36 m by 60 m?
一块矩形放牧区需用围墙三面围起，以一段 100 米的石墙作为第四边。围墙柱每隔 12 米竖一根，包括围墙与石墙相接处的两根柱子。围起 36 m × 60 m 的区域最少需要多少根柱子？
At the beginning of a trip, the mileage odometer read 56,200 miles. The driver filled the gas tank with 6 gallons of gasoline. During the trip, the driver filled his tank again with 12 gallons of gasoline when the odometer read 56,560 . At the end of the trip, the driver filled his tank again with 20 gallons of gasoline. The odometer read 57,060 . To the nearest tenth, what was the car's average miles-per-gallon for the entire trip?
A piece of paper containing six joined squares labeled as shown in the diagram is folded along the edges of the squares to form a cube. The label of the face opposite the face labeled X is
一张纸上有六个相连的正方形，标记如图所示。将纸沿正方形的边折叠形成一个正方体。与面 X 相对的面上的标记是
If you walk for 45 minutes at a rate of 4 mph and then run for 30 minutes at a rate of 10 mph , how many miles will you have gone at the end of one hour and 15 minutes?
If your average score on your first six mathematics tests was 84 and your average score on your first seven mathematics tests was 85, then your score on the seventh test was
Nine copies of a certain pamphlet cost less than $10.00 while ten copies of the same pamphlet (at the same price) cost more than $11.00. How much does one copy of this pamphlet cost?