Problems

Amy, Bill and Celine are friends with different ages. Exactly one of the following statements is true. I. Bill is the o…

The faces of a cube are painted in six different colors: red , white , green , brown , aqua , and purple . Three views o…

Bo, Coe, Flo, Jo, and Moe have different amounts of money. Neither Jo nor Bo has as much money as Flo. Both Bo and Coe h…

The six children listed below are from two families of three siblings each. Each child has blue or brown eyes and black…

Aaron, Darren, Karen, Maren, and Sharon rode on a small train that has five cars that seat one person each. Maren sat in…

Ravon, Oscar, Aditi, Tyrone, and Kim play a card game. Each person is given cards out of a set of cards numbered The…

Problems 8,9 and 10 use the data found in the accompanying paragraph and table: Juan organizes the stamps in his collec…

Jamie counted the number of edges of a cube, Jimmy counted the numbers of corners, and Judy counted the number of faces.…

At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are students in Mr…

Peter, Emma, and Kyler played chess with each other. Peter won 4 games and lost 2 games. Emma won 3 games and lost 3 gam…

The eighth grade class at Lincoln Middle School has students. Each student takes a math class or a foreign language cla…

Anna and Bella are celebrating their birthdays together. Five years ago, when Bella turned years old, she received a ne…

Rohan keeps 90 guppies in 4 fish tanks. There is 1 more guppy in the 2nd tank than in the 1st tank. There are 2 more gu…

Buzz Bunny is hopping up and down a set of stairs, one step at a time. In how many ways can Buzz Bunny start on the grou…

The one-way routes connecting towns and are shown in the figure below(not drawn to scale).The distances in kilometers…

Sergai skated around an ice rink, gliding along different paths. The gray lines in the figures below show four of the pa…

On the most recent exam on Prof. Xochi's class, students earned a score of at least , students earned a score of a…

A box contains red balls, green balls, yellow balls, blue balls, white balls, and black balls. What is the minimum…

Each square in a grid is either filled or empty, and has up to eight adjacent neighboring squares, where neighboring sq…

Aunt Anna is years old. Caitlin is years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Cai…

Each principal of Lincoln High School serves exactly one -year term. What is the maximum number of principals this schoo…

Figure is a square. Inside this square three smaller squares are drawn with the side lengths as labeled. The area of th…

Three dice with faces numbered through are stacked as shown. Seven of the eighteen faces are visible, leaving eleven f…

Five friends compete in a dart-throwing contest. Each one has two darts to throw at the same circular target, and each i…

How many different four-digit numbers can be formed by rearranging the four digits in ? ## Reference Solution _ _…

Ms. Hamilton’s eighth-grade class wants to participate in the annual three-person-team basketball tournament. Lance, Sa…

Ms. Hamilton's eighth-grade class wants to participate in the annual three-person-team basketball tournament. The losing…

Connie multiplies a number by and gets as her answer. However, she should have divided the number by to get the corre…

The area of polygon is 52 with , and . What is ? ```asy [asy] pair a=(0,9), b=(8,9), c=(8,0), d=(4,0), e=(4,4), f=(0,…

How many two-digit numbers have digits whose sum is a perfect square? ## Reference Solution There is integer whose…

On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5…

The letter T is formed by placing two inch rectangles next to each other, as shown. What is the perimeter of the T, in…

Susan had 50 dollars to spend at the carnival. She spent 12 dollars on food and twice as much on rides. How many dollars…

In the figure, the outer equilateral triangle has area , the inner equilateral triangle has area , and the three trapezo…

The five pieces shown below can be arranged to form four of the five figures shown in the choices. Which figure cannot b…

Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pa…

Bag A has three chips labeled 1, 3, and 5. Bag B has three chips labeled 2, 4, and 6. If one chip is drawn from each bag…

Paul owes Paula cents and has a pocket full of -cent coins, -cent coins, and -cent coins that he can use to pay her. Wh…

A -digit number is such that the product of the digits plus the sum of the digits is equal to the number. What is the un…

On her first day of work, Janabel sold one widget. On day two, she sold three widgets. On day three, she sold five widge…

Monica is tiling the floor of her 12-foot by 16-foot living room. She plans to place one-foot by one-foot square tiles t…

Four friends do yardwork for their neighbors over the weekend, earning 15, 25, respectively. They decide to split their…

Akash's birthday cake is in the form of a inch cube. The cake has icing on the top and the four side faces, and no icin…

In how many ways can the letters in be rearranged so that two or more s do not appear together? ## Reference Solutio…

Three positive integers are equally spaced on a number line. The middle number is and the largest number is times the…

Lola, Lolo, Tiya, and Tiyo participated in a ping pong tournament. Each player competed against each of the other three…

A chess king is said to attack all the squares one step away from it, horizontally, vertically, or diagonally. For insta…

Any three vertices of the cube , shown in the figure below, can be connected to form a triangle. (For example, vertices…

Four squares of side length and are arranged in increasing size order so that their left edges and bottom edges align.…

When Yunji added all the integers from to , she mistakenly left out a number. Her incorrect sum turned out to be a squa…

A consists of four squares connected along their edges. There are five possible tetromino shapes, , , , , and , shown b…

Buffalo Shuffle-o is a card game in which all the cards are distributed evenly among all players at the start of the gam…

Betty drives a truck to deliver packages in a neighborhood whose street map is shown below. Betty starts at the factory…

Ana and Bonita were born on the same date in different years, years apart. Last year Ana was times as old as Bonita. T…

For how many of the following types of quadrilaterals does there exist a point in the plane of the quadrilateral that is…

Consider the statement, "If is not prime, then is prime." Which of the following values of is a counterexample to thi…

How many -digit positive integers (that is, integers between and , inclusive) having only even digits are divisible by…

What is the value of ## Reference Solution An expert AMC tutor's solution to the problem. ### **English Version**…

There are people standing equally spaced around a circle. Each person knows exactly of the other people: the people…

Driving along a highway, Megan noticed that her odometer showed (miles). This number is a palindrome-it reads the same…

Portia's high school has times as many students as Lara's high school. The two high schools have a total of students.…

Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, and…

At a math contest, students are wearing blue shirts, and another students are wearing yellow shirts. The students are…

How many three-digit positive integers have an odd number of even digits? ## Reference Solution This problem asks fo…

A donkey suffers an attack of hiccups and the first hiccup happens at one afternoon. Suppose that the donkey hiccups re…

How many three-digit positive integers satisfy the following properties? The number is divisible by . The number form…

An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two…

A digital display shows the current date as an -digit integer consisting of a -digit year, followed by a -digit month, f…

You are playing a game. A rectangle covers two adjacent squares (oriented either horizontally or vertically) of a grid…

Let , and for . How many terms in the sequence are even? ## Reference Solution --- #### **Step 1: Write out the f…

What is the units digit of ? ## Reference Solution To find the units digit of , we need to find the units digit of e…

What is the sum of the digits of the smallest prime that can be written as a sum of distinct primes? ## Reference So…

What is the minimum number of successive swaps of adjacent letters in the string that are needed to change the string t…

In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the righ…

Jerry likes to play with numbers. One day, he wrote all the integers from to on the whiteboard. Then he repeatedly cho…

What is the degree measure of the smaller angle formed by the hands of a clock at 10 o'clock? ```asy [asy] draw(circle(…

A rectangular garden 60 feet long and 20 feet wide is enclosed by a fence. To make the garden larger, while using the sa…

The third exit on a highway is located at milepost 40 and the tenth exit is at milepost 160. There is a service center o…

Three flower beds overlap as shown. Bed A has 500 plants, bed B has 450 plants, and bed C has 350 plants. Beds A and B s…

The number has the property that it is divisible by its unit digit. How many whole numbers between 10 and 50 have this…

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…

In order for Mateen to walk a kilometer (1000m) in his rectangular backyard, he must walk the length 25 times or walk it…

You have nine coins: a collection of pennies, nickels, dimes, and quarters having a total value of 1.02\text{(A)}\ 1\qqu…

Casey's shop class is making a golf trophy. He has to paint 300 dimples on a golf ball. If it takes him 2 seconds to pai…

Tyler has entered a buffet line in which he chooses one kind of meat, two different vegetables and one dessert. If the o…

A square piece of paper, 4 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to…

I'm thinking of two whole numbers. Their product is 24 and their sum is 11. What is the larger number? ## Reference S…

The digits 1, 2, 3, 4 and 9 are each used once to form the smallest possible even five-digit number. The digit in the te…

Six trees are equally spaced along one side of a straight road. The distance from the first tree to the fourth is 60 fee…

A sequence of squares is made of identical square tiles. The edge of each square is one tile length longer than the edge…

In a mathematics contest with ten problems, a student gains 5 points for a correct answer and loses 2 points for an inco…

How many whole numbers between 99 and 999 contain exactly one 0? ## Reference Solution The problem asks for the numb…

How many different combinations of 2 bills can be used to make a total of \text {(A)}\ 2 \qquad \text {(B)}\ 3 \qquad \t…

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 thi…

The year 2002 is a palindrome (a number that reads the same from left to right as it does from right to left). What is t…

Carlos Montado was born on Saturday, November 9, 2002. On what day of the week will Carlos be 706 days old? ## Refer…

Problems 8,9 and 10 use the data found in the accompanying paragraph and table: Juan organizes the stamps in his collec…

Fourteen white cubes are put together to form the figure on the right. The complete surface of the figure, including the…

In this addition problem, each letter stands for a different digit. If T = 7 and the letter O represents an even numb…

Ali, Bonnie, Carlo, and Dianna are going to drive together to a nearby theme park. The car they are using has seats: D…

Each of the twenty dots on the graph below represents one of Sarah's classmates. Classmates who are friends are connect…

Which of the following numbers has the smallest prime factor? ## Reference Solution The smallest prime factor is , a…

In the pattern below, the cat moves clockwise through the four squares, and the mouse moves counterclockwise through th…

A group of children riding on bicycles and tricycles rode past Billy Bob's house. Billy Bob counted children and wheel…

Thirteen black and six white hexagonal tiles were used to create the figure below. If a new figure is created by attachi…

Three friends have a total of identical pencils, and each one has at least one pencil. In how many ways can this happen…

Joe had walked half way from home to school when he realized he was late. He ran the rest of the way to school. He ran 3…

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…

How many three-digit numbers are divisible by 13? ## Reference Solution --- #### **Step 1: Identify the range of th…

Soda is sold in packs of 6, 12 and 24 cans. What is the minimum number of packs needed to buy exactly 90 cans of soda?…

Initially, a spinner points west. Chenille moves it clockwise revolutions and then counterclockwise revolutions. In wh…

Tiles and are translated so one tile coincides with each of the rectangles and . In the final arrangement, the two nu…

Sets and , shown in the Venn diagram, have the same number of elements. Their union has elements and their intersectio…

What is the sum of the two smallest prime factors of ? ## Reference Solution The prime factorization of is . The sm…

A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a dif…

To complete the grid below, each of the digits 1 through 4 must occur once in each row and once in each column. What num…

Each of the students in the eighth grade at Lincoln Middle School has one dog or one cat or both a dog and a cat. Twent…

Ms.Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of units. All o…

The ten-letter code represents the ten digits , in order. What 4-digit number is represented by the code word ? ## R…

If February is a month that contains Friday the , what day of the week is February 1? ## Reference Solution We can g…

If , what is ? ## Reference Solution Separate into two equations and and solve for the unknowns. and , therefore…

Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie 3 apples,…

How many -digit positive integers have digits whose product equals ? ## Reference Solution Let the 3-digit positive…

The diagram represents a -foot-by--foot floor that is tiled with -square-foot black tiles and white tiles. Notice that t…

On the last day of school, Mrs. Wonderful gave jelly beans to her class. She gave each boy as many jelly beans as there…

The letters , , and represent digits. If and ,what digit does represent? ## Reference Solution Because , must b…

A sequence of numbers starts with , , and . The fourth number of the sequence is the sum of the previous three numbers i…

Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular p…

What is the sum of the prime factors of ? ## Reference Solution First, we must find the prime factorization of . . W…

Everyday at school, Jo climbs a flight of stairs. Jo can take the stairs , , or at a time. For example, Jo could climb…

Which of the following figures has the greatest number of lines of symmetry? ## Reference Solution An equila…

Margie bought apples at a cost of cents per apple. She paid with a 5-dollar bill. How much change did Margie receive?…

How many rectangles are in this figure? ```asy [asy] pair A,B,C,D,E,F,G,H,I,J,K,L; A=(0,0); B=(20,0); C=(20,20); D=(0,2…

Karl's rectangular vegetable garden is feet by feet, and Makenna's is feet by feet. Whose garden is larger in area?…

Students guess that Norb's age is , and . Norb says, "At least half of you guessed too low, two of you are off by one, a…

In how many ways can be written as the sum of two primes? ## Reference Solution For the sum of two numbers to be od…

What time was it minutes after midnight on January 1, 2011? ## Reference Solution The problem asks for the…

In a town of adults, every adult owns a car, motorcycle, or both. If adults own cars and adults own motorcycles, how…

What is the units digit of ? ## Reference Solution To find the units digit of , we only need to look at the units d…

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 t…

A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which…

In the country of East Westmore, statisticians estimate there is a baby born every hours and a death every day. To the…

On February 13 The Oshkosh Northwester listed the length of daylight as 10 hours and 24 minutes, the sunrise was , and t…

Peter's family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice equally with his brothe…

In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is n…

A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The ph…

The Fort Worth Zoo has a number of two-legged birds and a number of four-legged mammals. On one visit to the zoo, Margie…

Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the s…

Toothpicks are used to make a grid that is toothpicks long and toothpicks wide. How many toothpicks are used altogethe…

What is the value of ? ## Reference Solution --- #### **Step 1: Analyze the sum inside the parentheses by grouping…

The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example,…

The first AMC was given in and it has been given annually since that time. Samantha turned years old the year that sh…

If and are integers and is even, which of the following is impossible? and are even and are odd is even is…

The sum of two prime numbers is . What is the product of these two prime numbers? ## Reference Solution --- #### **…

Six rectangles each with a common base width of have lengths of , and . What is the sum of the areas of the six rectang…

How many pairs of parallel edges, such as and or and , does a cube have? ```asy [asy] import three; currentprojectio…

Which of the following integers cannot be written as the sum of four consecutive odd integers? ## Reference Solution…

An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous…

The Blue Bird High School chess team consists of two boys and three girls. A photographer wants to take a picture of the…

The longest professional tennis match ever played lasted a total of hours and minutes. How many minutes was this? #…

An ATM password at Fred's Bank is composed of four digits from to , with repeated digits allowable. If no password may…

In an All-Area track meet, sprinters enter a meter dash competition. The track has lanes, so only sprinters can compe…

The digits , , , , and are each used once to write a five-digit number . The three-digit number is divisible by , the…

The number is a two-digit number. • When is divided by , the remainder is . • When is divided by , the remainder is…

A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals…

The smallest positive integer greater than 1 that leaves a remainder of 1 when divided by 4, 5, and 6 lies between which…

In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in…

Starting with some gold coins and some empty treasure chests, I tried to put gold coins in each treasure chest, but tha…

Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer num…

What is the value of the expression ? ## Reference Solution This problem asks us to evaluate a fraction where the nu…

Let be a 6-digit positive integer, such as 247247, whose first three digits are the same as its last three digits taken…

Malcolm wants to visit Isabella after school today and knows the street where she lives but doesn't know her house numbe…

All of Marcy's marbles are blue, red, green, or yellow. One third of her marbles are blue, one fourth of them are red, a…

Let be the greatest five-digit number whose digits have a product of . What is the sum of the digits of ? ## Referen…

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 bel…

Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then B…

What is the value of ? ## Reference Solution The given expression is . --- #### **Step 1: Rearrange and group the…

On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times…

The -digit number is divisible by . What is the remainder when this number is divided by ? ## Reference Solution…

A palindrome is a number that has the same value when read from left to right or from right to left. (For example, 12321…

Isabella has coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons…

In a tournament there are six teams that play each other twice. A team earns points for a win, point for a draw, and…

Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of t…

Which of the following is the correct order of the fractions and from least to greatest? ## Reference Solution To…

Luka is making lemonade to sell at a school fundraiser. His recipe requires times as much water as sugar and twice as m…

For a positive integer , the factorial notation represents the product of the integers from to . What value of satisf…

A number is called flippy if its digits alternate between two distinct digits. For example, and are flippy, but and…

When a positive integer is fed into a machine, the output is a number calculated according to the rule shown below. ``…

Carrie has a rectangular garden that measures feet by feet. She plants the entire garden with strawberry plants. Carri…

Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon con…

Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice in…

How many integers between and have four distinct digits arranged in increasing order? (For example, is one integer.).…

Henry the donkey has a very long piece of pasta. He takes a number of bites of pasta, each time eating inches of pasta…

How many positive integers can fill the blank in the sentence below? “One positive integer is _____ more than twice ano…

Four numbers are written in a row. The average of the first two is the average of the middle two is and the average of…

If is an even positive integer, the notation represents the product of all the even integers from to . For example,…

When three positive integers , , and are multiplied together, their product is . Suppose . In how many ways can the num…

A cup of boiling water () is placed to cool in a room whose temperature remains constant at . Suppose the difference bet…

Along the route of a bicycle race, water stations are evenly spaced between the start and finish lines, as shown in the…

Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas wou…

The letters and are entered into a table according to the pattern shown below. How many s, s, and s will appear in th…

A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When u…

The numbers from to are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers…

A lake contains trout, along with a variety of other fish. When a marine biologist catches and releases a sample of fi…

What is the unit digit of: ## Reference Solution The problem asks for the units digit of the expression . --- ####…

Let the letters ,,,,, represent distinct digits. Suppose is the greatest number that satisfies the equation What is…

Minh enters the numbers through into the cells of a grid in some order. She calculates the product of the numbers in…

Jordan owns 15 pairs of sneakers. Three fifths of the pairs are red and the rest are white. Two thirds of the pairs are…

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…

Aaliyah rolls two standard 6-sided dice. She notices that the product of the two numbers rolled is a multiple of . Which…

A rectangle is covered without overlap by 3 shapes of tiles: , , and , shown below. What is the minimum possible number…

On Monday, Taye has 2 or doubles the amount of money he had on the previous day. How many different dollar amounts could…

All the marbles in Maria's collection are red, green, or blue. Maria has half as many red marbles as green marbles, and…

Each of the even numbers is divided by . The remainders are recorded. Which histogram displays the number of times each…

Five distinct integers from to are chosen, and five distinct integers from to are chosen. No two numbers differ by e…

The table below shows the ancient Egyptian hieroglyphs that were used to represent different numbers. For example, th…

A classroom has a row of 35 coat hooks. Paulina likes coats to be equally spaced, so that there is the same number of em…

Lucius is counting backward by s. His first three numbers are , , and . What is his th number? ## Reference Solution…

Sekou writes the numbers After he erases one of his numbers, the sum of the remaining four numbers is a multiple of Wh…

What is the hundreds digit of ## Reference Solution Because we know that is a factor of and , the last three digi…

Alicia had two containers. The first was full of water and the second was empty. She poured all the water from the firs…

What value of satisfies ## Reference Solution The given equation is . We need to find the value of . --- #### **S…

Bela and Jenn play the following game on the closed interval of the real number line, where is a fixed integer greater…

Carl has cubes each having side length , and Kate has cubes each having side length . What is the total volume of thes…

How many positive even multiples of less than are perfect squares? ## Reference Solution Any even multiple of is…

How many ways are there to place indistinguishable red chips, indistinguishable blue chips, and indistinguishable gre…

Tom has a collection of snakes, of which are purple and of which are happy. He observes that all of his happy snakes…

The ages of Jonie's four cousins are distinct single-digit positive integers. Two of the cousins' ages multiplied togeth…

Mr. Zhou places all the integers from to into a by grid. He places in the middle square (eighth row and eighth colu…

What is the value of ## Reference Solution To solve this problem, we will evaluate the expression by starting from t…

How many of the first ten numbers of the sequence are prime numbers? ## Reference Solution --- #### **Step 1: Anal…

Consider the following sets of elements each: \begin{align*} &\{1,2,3,\ldots,10\}, \\ &\{11,12,13,\ldots,20\},\\ &\{21…

Mrs. Jones is pouring orange juice into four identical glasses for her four sons. She fills the first three glasses comp…

Jackson's paintbrush makes a narrow strip with a width of millimeters. Jackson has enough paint to make a strip meters…

What is [ONLY FOR CERTAIN CHINESE TESTPAPERS] What is ## Reference Solution We are asked to compute the value…

The Fibonacci numbers are defined by and for What is ## Reference Solution The problem asks for the value of the…

Bicycle license plates in Flatville each contain three letters. The first is chosen from the set {C,H,L,P,R}, the second…

At Central Middle School the 108 students who take the AMC 8 meet in the evening to talk about problems and eat an avera…

When is divided by , the remainder is ## Reference Solution Note that the units digits of the powers of 9 have a p…

The diagram shows the miles traveled by bikers Alberto and Bjorn. After four hours, about how many more miles has Albert…

What is the units digit of ? ## Reference Solution --- #### **Step 1: Simplify the problem by focusing on the units…

Triangles , , and are all equilateral. Points and are midpoints of and , respectively. If , what is the perimeter of…

There are 24 four-digit whole numbers that use each of the four digits 2, 4, 5 and 7 exactly once. Only one of these fou…

To promote her school's annual Kite Olympics, Genevieve makes a small kite and a large kite for a bulletin board display…

A figure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum numbe…

What is the measure of the acute angle formed by the hands of the clock at 4:20 PM? ## Reference Solution --- ####…

Handy Aaron helped a neighbor hours on Monday, minutes on Tuesday, from 8:20 to 10:45 on Wednesday morning, and a half…

Two mL pitchers contain orange juice. One pitcher is full and the other pitcher is full. Water is added to fill each…

Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If th…

The Little Twelve Basketball Conference has two divisions, with six teams in each division. Each team plays each of the…

How many different isosceles triangles have integer side lengths and perimeter 23? ## Reference Solution --- #### *…

Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are…

What is the minimum number of small squares that must be colored black so that a line of symmetry lies on the diagonal…

A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are 6.1 cm, 8.2 cm and 9.7…

Suppose is a digit. For how many values of is ? ## Reference Solution thought I need to create a step-by-step solu…

Mindy made three purchases for dollars, dollars, and dollars. What was her total, to the nearest dollar? ## Refer…

Jorge's teacher asks him to plot all the ordered pairs of positive integers for which is the width and is the length…

An aquarium has a rectangular base that measures cm by cm and has a height of cm. The aquarium is filled with water t…

Points and are midpoints of the sides of the larger square. If the larger square has area 60, what is the area of the…

What is the product of  ? ## Reference Solution We are asked to find the product of the expression . --- #### **St…

For any positive integer , define to be the sum of the positive factors of . For example, . Find . ## Reference Sol…

The product of the two -digit numbers and has thousands digit and units digit . What is the sum of and ? ## Ref…

How many whole numbers between 1 and 1000 do not contain the digit 1? ## Reference Solution Note that this is the sa…

How many 4-digit positive integers have four different digits, where the leading digit is not zero, the integer is a mul…

In the BIG N, a middle school football conference, each team plays every other team exactly once. If a total of 21 confe…

When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of t…

Isabella uses one-foot cubical blocks to build a rectangular fort that is feet long, feet wide, and feet high. The fl…

Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven…

Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solvin…

Onkon wants to cover his room's floor with his favourite red carpet. How many square yards of red carpet are required to…

How many integers between 1000 and 9999 have four distinct digits? ## Reference Solution We need to count the number…

Karl's car uses a gallon of gas every miles, and his gas tank holds gallons when it is full. One day, Karl started wit…

Find the value of the expression ## Reference Solution The given expression is . --- #### **Step 1: Identify the pa…

What is the sum of the distinct prime integer divisors of ? ## Reference Solution **Problem Statement:** What is the…

Which of the following values is largest? ## Reference Solution --- #### **Step 1: Evaluate Expression (A)** We ar…

When is multiplied by the product is closest to which of the following? ## Reference Solution The problem asks to…

The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What…

Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. H…

Bella begins to walk from her house toward her friend Ella's house. At the same time, Ella begins to ride her bicycle to…

What is the value of the product ## Reference Solution --- #### **Step 1: Simplify each term in the parentheses**…

Ike and Mike go into a sandwich shop with a total of 30.00 each and soft drinks cost 1.00\textbf{(A) }6\qquad\textbf{(B)…

What is the value of the product ## Reference Solution The problem asks for the value of the product: P = \left(…

A game board consists of squares that alternate in color between black and white. The figure below shows square in the…

Ricardo has coins, some of which are pennies (-cent coins) and the rest of which are nickels (-cent coins). He has at l…

A bus takes minutes to drive from one stop to the next, and waits minute at each stop to let passengers board. Zia tak…

What is the value of ## Reference Solution The problem asks for the value of the product: \frac{1}{3}\cdot\frac{2}…

What is the value of ? ## Reference Solution The problem asks for the value of the expression . We will evaluate thi…

Harold made a plum pie to take on a picnic. He was able to eat only of the pie, and he left the rest for his friends. A…

Greta Grasshopper sits on a long line of lily pads in a pond. From any lily pad, Greta can jump pads to the right or p…

Alina writes the numbers on separate cards, one number per card. She wishes to divide the cards into groups of cards…

Wind chill is a measure of how cold people feel when exposed to wind outside. A good estimate for wind chill can be foun…

Malaika is skiing on a mountain. The graph below shows her elevation, in meters, above the base of the mountain as she s…

In January the Mauna Loa Observatory recorded carbon dioxide levels of ppm (parts per million). Over the years the av…

What is the value of this expression in decimal form? ## Reference Solution The problem asks for the decimal value o…

In the figure below, is a rectangle with sides of length inches and = 3 inches. Rectangle is rotated clockwise arou…

In the land of Markovia, there are three cities: , , and . There are 100 people who live in , 120 who live in , and 160…

How many four-digit numbers have all three of the following properties? (I) The tens and ones digit are both 9. (II) T…

What is the value of ## Reference Solution .\n ## Source 2019 AMC 10 Problem 1

A rectangular floor that is feet wide and feet long is tiled with one-foot square tiles. A bug walks from one corner…

How many positive integer divisors of are perfect squares or perfect cubes (or both)? ## Reference Solution Prime fa…

Melanie computes the mean , the median , and the modes of the values that are the dates in the months of . Thus her dat…

Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game…

What is the greatest number of consecutive integers whose sum is ## Reference Solution We might at first think that…

Two lines with slopes and intersect at . What is the area of the triangle enclosed by these two lines and the line…

What is the greatest three-digit positive integer for which the sum of the first positive integers is a divisor of th…

Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar the ratio of blue to…

What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than ? ##…

What is the sum of all real numbers for which the median of the numbers and is equal to the mean of those five number…

The base-ten representation for is , where , , and denote digits that are not given. What is ? ## Reference Solutio…

Let be the set of all positive integer divisors of How many numbers are the product of two distinct elements of ##…

In a high school with students, of the seniors play a musical instrument, while of the non-seniors do not play a musi…

All lines with equation such that form an arithmetic progression pass through a common point. What are the coordinates…

There is a positive integer such that . What is the sum of the digits of ? ## Reference Solution Solution 1\n \n…

Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either pieces of r…

What is the median of the following list of numbers ## Reference Solution --- #### **Step 1: Understand the defini…

The numbers and have an average (arithmetic mean) of . What is the average of and ? ## Reference Solution The ari…

For how many positive integers isnot divisible by ? (Recall that is the greatest integer less than or equal to .) #…

Let be the least positive integer greater than for which What is the sum of the digits of ? ## Reference Solutio…

A driver travels for hours at miles per hour, during which her car gets miles per gallon of gasoline. She is paid 0.5…

The integers from to inclusive, can be arranged to form a -by- square in which the sum of the numbers in each row, th…

A single bench section at a school event can hold either adults or children. When bench sections are connected end to…

Andy the Ant lives on a coordinate plane and is currently at facing east (that is, in the positive -direction). Andy mo…

Steve wrote the digits , , , , and in order repeatedly from left to right, forming a list of digits, beginning He the…

In a certain card game, a player is dealt a hand of cards from a deck of distinct cards. The number of distinct (unord…

The ratio of to is , the ratio of to is , and the ratio of to is . What is the ratio of to ## Reference Solut…

How many ordered pairs of integers satisfy the equation ## Reference Solution Solution 1\n Rearranging the terms…

What is the value of ## Reference Solution The problem asks for the value of the expression . --- #### **Step 1:…

For which of the following integers is the base- number not divisible by ? ## Reference Solution We have \n\nwhich…

In how many ways can the sequence be rearranged so that no three consecutive terms are increasing and no three consecut…

Hiram's algebra notes are pages long and are printed on sheets of paper; the first sheet contains pages and , the sec…

The sum of two natural numbers is . One of the two numbers is divisible by . If the units digit of that number is erased…

A cart rolls down a hill, travelling inches the first second and accelerating so that during each successive -second ti…

The quiz scores of a class with students have a mean of . The mean of a collection of of these quiz scores is . What i…

Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. C…

Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equa…

Let . What is the ratio of the sum of the odd divisors of to the sum of the even divisors of ? ## Reference Solution…

Let be a positive integer and be a digit such that the value of the numeral in base equals , and the value of the nu…

Suppose that is a finite set of positive integers. If the greatest integer in is removed from , then the average value…

In an after-school program for juniors and seniors, there is a debate team with an equal number of students from each cl…

Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is , and the aft…

On Halloween children walked into the principal's office asking for candy. They can be classified into three types: Som…

Mike cycled laps in minutes. Assume he cycled at a constant speed throughout. Approximately how many laps did he compl…

How many strings of length formed from the digits , , , , are there such that for each , at least of the digits are l…

The sum of three numbers is The first number is times the third number, and the third number is less than the second…

In some countries, automobile fuel efficiency is measured in liters per kilometers while other countries use miles per…

The least common multiple of a positive integer and is , and the greatest common divisor of and is . What is the sum…

A data set consists of (not distinct) positive integers: , , , , , and . The average (arithmetic mean) of the numbers…

Define to be for all real numbers and What is the value of ## Reference Solution The problem defines a new math…

Camila writes down five positive integers. The unique mode of these integers is greater than their median, and the medi…

All the high schools in a large school district are involved in a fundraiser selling T-shirts. Which of the choices belo…

Suppose that is a subset of such that the sum of any two (not necessarily distinct) elements of is never an element o…

Let be a sequence of numbers, where each is either or . For each positive integer , define Suppose for all . What…

Cities and are miles apart. Alicia lives in and Beth lives in . Alicia bikes towards at 18 miles per hour. Leaving…

Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an on the next quiz, her mean…

Let be the unique polynomial of minimal degree with the following properties: has a leading coefficient , is a root…

How many positive perfect squares less than are divisible by ? ## Reference Solution --- #### **Step 1: Understand…

A quadrilateral has all integer sides lengths, a perimeter of , and one side of length . What is the greatest possible l…

Barb the baker has developed a new temperature scale for her bakery called the Breadus scale, which is a linear function…

Suzanne went to the bank and withdrew 800 bills, 50 bills, with at least one of each denomination. How many different co…

Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by on every pair of…

How many distinct values of satisfy , where denotes the largest integer less than or equal to ? ## Reference Soluti…

Maddy and Lara see a list of numbers written on a blackboard. Maddy adds to each number in the list and finds that the…

The numbers and are a pair of consecutive positive squares whose difference is . How many pairs of consecutive positiv…

Consider the following operation. Given a positive integer , if is a multiple of , then you replace by . If is not a…

Zelda played the Adventures of Math game on August 1 and scored points. She continued to play daily over the next days…

The first three terms of a geometric sequence are the integers and where What is the sum of the digits of the least p…

Let be a subset of such that the following two conditions hold: If and are distinct elements of , then If and ar…

The number is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit n…

What is the least value of such that is a multiple of ? ## Reference Solution The problem asks for the least value…

Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at and…

A list of real numbers consists of , , , , , and , as well as , , and with . The range of the list is , and the m…

Let How many of the values , , , and are integers? ## Reference Solution The problem asks us to determine how many…

Balls numbered 1, 2, 3, ... are deposited in 5 bins, labeled A, B, C, D, and E, using the following procedure. Ball 1 is…

In the following expression, Melanie changed some of the plus signs to minus signs: When the new expression was evaluat…

What is the remainder when is divided by ? ## Reference Solution We are asked to find the remainder when is divide…

To make this statement true, the question mark between the 6 and the 3 should be replaced by ## Reference Solution…

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 i…

The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is . To the near…

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…

Tori's mathematics test had 75 problems: 10 arithmetic, 30 algebra, and 35 geometry problems. Although she answered 70%…

At Central Middle School the students who take the AMC8 meet in the evening to talk about problems and eat an average o…

At Central Middle School, the 108 students who take the AMC 8 meet in the evening to talk about food and eat an average…

Figure 1 is called a "stack map." The numbers tell how many cubes are stacked in each position. Fig. 2 shows these cubes…

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 ri…

Six squares are colored, front and back, (R = red, B = blue, O = orange, Y = yellow, G = green, and W = white). They are…

Ara and Shea were once the same height. Since then Shea has grown 20% while Ara has grown half as many inches as Shea. S…

The operation is defined for all nonzero numbers by . Determine . ## Reference Solution The problem defines a new b…

Which of these numbers is less than its reciprocal? ## Reference Solution The problem asks to find which of the give…

There is a list of seven numbers. The average of the first four numbers is , and the average of the last four numbers is…

The area of rectangle is units squared. If point and the midpoints of and are joined to form a triangle, the area o…

How many whole numbers lie in the interval between and ? ## Reference Solution --- #### **Step 1: Determine the l…

In 1960 only 5% of the working adults in Carlin City worked at home. By 1970 the "at-home" work force had increased to 8…

Three-digit powers of and are used in this "cross-number" puzzle. What is the only possible digit for the outlined squ…

A collector offers to buy state quarters for 2000% of their face value. At that rate how much will Bryden get for his fo…

Points , , and have these coordinates: , , and . The area of quadrilateral is ```asy [asy] for (int i = -4; i <= 4;…

If , then ## Reference Solution The problem defines a new binary operation, denoted by the symbol , as . We are ask…

Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the rem…

Homer began peeling a pile of 44 potatoes at the rate of 3 potatoes per minute. Four minutes later Christen joined him a…

For the game show Who Wants To Be A Millionaire?, the dollar values of each question are shown in the following table (w…

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…

Kaleana shows her test score to Quay, Marty and Shana, but the others keep theirs hidden. Quay thinks, "At least two of…

The mean of a set of five different positive integers is 15. The median is 18. The maximum possible value of the largest…

On a twenty-question test, each correct answer is worth 5 points, each unanswered question is worth 1 point and each inc…

Each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is…

Granny Smith has 2 more than Anjou and Anjou has one-third as much as Granny Smith. How many dollars does Elberta have?…

On a dark and stormy night Snoopy suddenly saw a flash of lightning. Ten seconds later he heard the sound of thunder. Th…