25. 9. 3. 오후 1:46 Art of Problem Solving Art of Problem Solving AoPS Online Beast Academy AoPS Academy 2025 AMC 8 Problems 2025 AMC 8 (Answer Key) Printable versions: • AoPS Resources (https://artofproblemsolvin g.com/community/contest/collection/c3413_amc_8/2025) • PDF (https://artofproblemsolving.com/community/contest/download/ c3413_amc_8/2025) Instructions 1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. 2. You will receive 1 point for each correct answer. There is no penalty for wrong answers. 3. No aids are permitted other than plain scratch paper, writing utensils, ruler, and erasers. In particular, graph paper, compass, protractor, calculators, computers, smartwatches, and smartphones are not permitted. Rules (https://amc-reg.maa.org/manual/AMC8.pdf) 4. Figures are not necessarily drawn to scale. 5. You will have 40 minutes working time to complete the test. 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 Contents 1 Problem 1 2 Problem 2 3 Problem 3 4 Problem 4 5 Problem 5 6 Problem 6 7 Problem 7 8 Problem 8 9 Problem 9 10 Problem 10 11 Problem 11 12 Problem 12 13 Problem 13 14 Problem 14 15 Problem 15 16 Problem 16 17 Problem 17 18 Problem 18 19 Problem 19 20 Problem 20 21 Problem 21 22 Problem 22 23 Problem 23 24 Problem 24 25 Problem 25 26 See Also Problem 1 The eight-pointed star, shown in the figure below, is a popular quilting pattern. What percent of the entire the star? https://artofproblemsolving.com/wiki/index.php/2025_AMC_8_Problems grid is covered by 1/10 25. 9. 3. 오후 1:46 Art of Problem Solving Solution Problem 2 The table below shows the Ancient Egyptian hieroglyphs that were used to represent different numbers. For example, the number was represented by the hieroglyphs combination of hieroglyphs? . What number is represented by the following Solution Problem 3 Buffalo Shuffle-o is a card game in which all the cards are distributed evenly among all players at the start of the game. When Annika and of her friends play Buffalo Shuffle-o, each player is dealt cards. Suppose more friends join the next game. How many cards will be dealt to each player? Solution Problem 4 Lucius is counting backward by s. His first three numbers are , , and . What is his th number? Solution Problem 5 Betty drives a truck to deliver packages in a neighborhood whose street map is shown below. Betty starts at the factory (labled and drives to location , then , then , before returning to . What is the shortest distance, in blocks, she can drive to complete the route? https://artofproblemsolving.com/wiki/index.php/2025_AMC_8_Problems ) 2/10 25. 9. 3. 오후 1:46 Art of Problem Solving Solution Problem 6 Sekou writes the numbers multiple of Which number did he erase? After he erases one of his numbers, the sum of the remaining four numbers is a Solution Problem 7 On the most recent exam on Prof. Xochi's class, students earned a score of at least students earned a score of at least students earned a score of at least students earned a score of at least How many students earned a score of at least and less than ? Solution Problem 8 Isaiah cuts open a cardboard cube along some of its edges to form the flat shape shown on the right, which has an area of square centimeters. What is the volume of the cube in cubic centimeters? https://artofproblemsolving.com/wiki/index.php/2025_AMC_8_Problems 3/10 25. 9. 3. 오후 1:46 Art of Problem Solving Solution Problem 9 Ningli looks at the pairs of numbers directly across from each other on a clock. She takes the average of each pair of numbers. What is the average of the resulting numbers? Solution Problem 10 In the figure below, is a rectangle with sides of length inches and = inches. Rectangle rotated clockwise around the midpoint of side to give a second rectangle. What is the total area, in square inches, covered by the two overlapping rectangles? is Solution Problem 11 A consists of four squares connected along their edges. There are five possible tetromino shapes, , , , , and , shown below, which can be rotated or flipped over. Three tetrominoes are used to completely cover a rectangle. At least one of the tiles is an tile. What are the other two tiles? https://artofproblemsolving.com/wiki/index.php/2025_AMC_8_Problems 4/10 25. 9. 3. 오후 1:46 and Art of Problem Solving and and and and Solution Problem 12 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? Solution Problem 13 Each of the even numbers of times each remainder occurs? is divided by . The remainders are recorded. Which histogram displays the number https://artofproblemsolving.com/wiki/index.php/2025_AMC_8_Problems 5/10 25. 9. 3. 오후 1:46 Art of Problem Solving Solution Problem 14 A number is inserted into the list . The mean is now twice as great as the median. What is ? Solution Problem 15 Kei draws a -by- grid. He colors 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 and equal the least and greatest possible number of gold-on-gold pairs, respectively. What is the value of ? Solution Problem 16 Five distinct integers from to are chosen, and five distinct integers from exactly . What is the sum of the ten chosen numbers? to are chosen. No two numbers differ by Solution Problem 17 In the land of Markovia, there are three cities: , , and . There are people who live in , who live in , and who live in . 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, of the people who live in work in .) How many people work in https://artofproblemsolving.com/wiki/index.php/2025_AMC_8_Problems ? 6/10 25. 9. 3. 오후 1:46 Art of Problem Solving Solution Problem 18 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 , in units, of the circle on the right? Solution Problem 19 Two towns, and , are connected by a straight road, miles long. Traveling from town to town , the speed limit changes every miles: from to to miles per hour (mph). Two cars, one at town and one at town , 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 , in miles, will the two cars meet? Solution Problem 20 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? Solution Problem 21 The Konigsberg School has assigned grades 1 through 7 to pods through , one grade per pod. Some of the pods are connected by walkways, as shown in the figure below. The school noticed that each pair of connected pods has been assigned grades differing by 2 or more grade levels. (For example, grades 1 and 2 will not be in pods directly connected by a walkway.) What https://artofproblemsolving.com/wiki/index.php/2025_AMC_8_Problems 7/10 25. 9. 3. 오후 1:46 Art of Problem Solving is the sum of the grade levels assigned to pods , and ? Solution Problem 22 A classroom has a row of 35 coat hooks. Paulina likes coats to be equally spaced, so that there is the same number of empty hooks before the first coat, after the last coat, and between every coat and the next one. Suppose there is at least 1 coat and at least 1 empty hook. How many different numbers of coats can satisfy Paulina's pattern? Solution Problem 23 How many four-digit numbers have all three of the following properties? (I) The tens and ones digit are both 9. (II) The number is 1 less than a perfect square. (III) The number is the product of exactly two prime numbers. Solution Problem 24 In trapezoid perimeter of , angles and measure and . The side lengths are all positive integers, and the is units. How many non-congruent trapezoids satisfy all of these conditions? https://artofproblemsolving.com/wiki/index.php/2025_AMC_8_Problems 8/10 25. 9. 3. 오후 1:46 Art of Problem Solving Solution Problem 25 Makayla finds all the possible ways to draw a path in a diamond-shaped grid. Each path starts at the bottom of the grid and ends at the top, always moving one unit northeast or northwest. She computes the area of the region between each path and the right side of the grid. Two examples are shown in the figures below. What is the sum of the areas determined by all possible paths? Solution See Also 2025 AMC 8 (Problems • Answer Key • Resources (http://www.artofproblemsolving.com/Forum/resources. php?c=182&cid=42&year=2025)) Preceded by 2024 AMC 8 Followed by 2026 AMC 8 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AJHSME/AMC 8 Problems and Solutions AMC 8 AMC 8 Problems and Solutions Resources for Mathematics Competitions These problems are copyrighted © by the Mathematical Association of America (https://www.maa.org), as part of the American Mathematics Competitions (https://maa.org/student-programs/amc/). Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2025_AMC_8_Problems&oldid=258632" https://artofproblemsolving.com/wiki/index.php/2025_AMC_8_Problems 9/10 25. 9. 3. 오후 1:46 Art of Problem Solving Copyright © 2025 Art of Problem Solving https://artofproblemsolving.com/wiki/index.php/2025_AMC_8_Problems 10/10
