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…

A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks t…

Joey and his five brothers are ages , , , , , and . One afternoon two of his brothers whose ages sum to went to the mo…

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…

Let denote the sum of the digits of the positive integer . For example, and . For how many two-digit values of is ?…

A block of calendar dates is shown. First, the order of the numbers in the second and the fourth rows are reversed. The…

Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of…

In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whos…

Two integers have a sum of 26. When two more integers are added to the first two integers the sum is 41. Finally when tw…

Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house…

How many rearrangements of are there in which no two adjacent letters are also adjacent letters in the alphabet? For e…

David, Hikmet, Jack, Marta, Rand, and Todd were in a -person race with other people. Rand finished places ahead of Hik…

Mahdi practices exactly one sport each day of the week. He runs three days a week but never on two consecutive days. On…

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…

What is the difference between the sum of the first even counting numbers and the sum of the first odd counting number…

The grid shown contains a collection of squares with sizes from to . How many of these squares contain the black cente…

How many two-digit positive integers have at least one as a digit? ## Reference Solution Ten numbers have as the…

Sandwiches at Joe's Fast Food cost 3 each. How many dollars will it cost to purchase sandwiches and sodas? ## Refer…

A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display…

The numbers from to are placed at the vertices of a cube in such a manner that the sum of the four numbers on each fac…

How many even integers are there between and whose digits are all different and come from the set ? ## Reference So…

A square with side length 8 is cut in half, creating two congruent rectangles. What are the dimensions of one of these r…

Last year 100 adult cats, half of whom were female, were brought into the Smallville Animal Shelter. Half of the adult f…

A solid cube of side length is removed from each corner of a solid cube of side length . How many edges does the remai…

A softball team played ten games, scoring , , , , , , , , , and runs. They lost by one run in exactly five games. In…

Leah has coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have…

Mary thought of a positive two-digit number. She multiplied it by and added . Then she switched the digits of the resul…

Kate bakes a -inch by -inch pan of cornbread. The cornbread is cut into pieces that measure inches by inches. How many…

In the expression each blank is to be filled in with one of the digits or with each digit being used once. How many d…

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…

A set of numbers is called sum-free if whenever and are (not necessarily distinct) elements of the set, is not an ele…

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…

Using the digits 1, 2, 3, 4, 5, 6, 7, and 9, form 4 two-digit prime numbers, using each digit only once. What is the sum…

The positive integers and are all prime numbers. The sum of these four primes is ## Reference Solution Since and…

The arithmetic mean of the nine numbers in the set is a -digit number , all of whose digits are distinct. The number d…

Suppose July of year has five Mondays. Which of the following must occur five times in the August of year ? (Note: Both…

Let be the largest integer that is the product of exactly 3 distinct prime numbers , , and , where and are single dig…

What is the units digit of ? ## Reference Solution \n Since : \n \n Therefore, the units digit is \n "],[" Sinc…

Sally has five red cards numbered through and four blue cards numbered through . She stacks the cards so that the col…

There are players in a single tennis tournament. The tournament is single elimination, meaning that a player who loses…

A clock chimes once at minutes past each hour and chimes on the hour according to the hour. For example, at there is o…

How many distinct four-digit numbers are divisible by and have as their last two digits? ## Reference Solution Sol…

You and five friends need to raise dollars in donations for a charity, dividing the fundraising equally. How many doll…

At a party, each man danced with exactly three women and each woman danced with exactly two men. Twelve men attended th…

Bertha has 6 daughters and no sons. Some of her daughters have 6 daughters, and the rest have none. Bertha has a total…

A grocer stacks oranges in a pyramid-like stack whose rectangular base is oranges by oranges. Each orange above the f…

A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one…

Each row of the Misty Moon Amphitheater has seats. Rows through are reserved for a youth club. How many seats are res…

A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row abov…

Patty has coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have…

At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her…

A store normally sells windows at 100\textbf{(A) } 100\qquad \textbf{(B) } 200\qquad \textbf{(C) } 300\qquad \textbf{(D)…

A scout troop buys candy bars at a price of five for dollars. They sell all the candy bars at the price of two for do…

Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth o…

A number of linked rings, each cm thick, are hanging on a peg. The top ring has an outside diameter of cm. The outside…

How many four-digit positive integers have at least one digit that is a or a ? ## Reference Solution Since we are a…

Leap Day, February , , occurred on a Sunday. On what day of the week will Leap Day, February , , occur? ## Reference…

A rectangle and a rectangle are contained within a square without overlapping at any point, and the sides of the squar…

Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but…

For each positive integer , let denote the sum of the digits of For how many values of is ## Reference Solution…

Isabella's house has bedrooms. Each bedroom is feet long, feet wide, and feet high. Isabella must paint the walls of…

A cryptographic code is designed as follows. The first time a letter appears in a given message it is replaced by the le…

A bakery owner turns on his doughnut machine at . At the machine has completed one third of the day's job. At what time…

For the positive integer , let denote the sum of all the positive divisors of with the exception of itself. For examp…

A basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers…

One can holds ounces of soda, what is the minimum number of cans needed to provide a gallon ( ounces) of soda? ## Re…

Each morning of her five-day workweek, Jane bought either a -cent muffin or a -cent bagel. Her total cost for the week w…

What is the remainder when is divided by 8? ## Reference Solution Solution 1\n The sum of any four consecutive pow…

Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages? ##…

Marvin had a birthday on Tuesday, May 27 in the leap year . In what year will his birthday next fall on a Saturday? #…

A , such as , is a number that remains the same when its digits are reversed. The numbers and are three-digit and four…

A month with days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the fi…

A ticket to a school play cost dollars, where is a whole number. A group of 9th graders buys tickets costing a total o…

Let X and Y be the following sums of arithmetic sequences: \begin{eqnarray*}X &=& 10+12+14+\cdots+100,\\ Y &=& 12+14+1…

Set has elements, and set has elements. What is the smallest possible number of elements in ? ## Reference Solut…

In multiplying two positive integers and , Ron reversed the digits of the two-digit number . His erroneous product was…

A year is a leap year if and only if the year number is divisible by 400 (such as 2000) or is divisible by 4 but not 100…

Chubby makes nonstandard checkerboards that have squares on each side. The checkerboards have a black square in every…

Let and . What is the smallest possible degree measure for ? ## Reference Solution and share ray . In order to min…

In a bag of marbles, of the marbles are blue and the rest are red. If the number of red marbles is doubled and the numb…

Each third-grade classroom at Pearl Creek Elementary has students and pet rabbits. How many more students than rabbits…

When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same wit…

Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid 125, and Sa…

Jo and Blair take turns counting from to one more than the last number said by the other person. Jo starts by saying ,…

Mr. Green measures his rectangular garden by walking two of the sides and finds that it is steps by steps. Each of Mr.…

When counting from to , is the number counted. When counting backwards from to , is the number counted. What is ?…

Ann made a -step staircase using toothpicks as shown in the figure. How many toothpicks does she need to add to complet…

How many terms are in the arithmetic sequence , , , , , ? ## Reference Solution , so the amount of terms in the sequ…

Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at PM and finis…

Isaac has written down one integer two times and another integer three times. The sum of the five numbers is , and one o…

Four siblings ordered an extra large pizza. Alex ate , Beth , and Cyril of the pizza. Dan got the leftovers. What is th…

Ximena lists the whole numbers through once. Emilio copies Ximena's numbers, replacing each occurrence of the digit b…

All the numbers are written in a array of squares, one number in each square, in such a way that if two numbers are co…

Zoey read books, one at a time. The first book took her day to read, the second book took her days to read, the third…

Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-dig…

Define a sequence recursively by and the remainder when is divided by for all Thus the sequence starts What is…

There are horses, named Horse , Horse , . . . , Horse . They get their names from how many minutes it takes them to run…

Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the fiv…

Pablo buys popsicles for his friends. The store sells single popsicles for 13 each, and -popsicle boxes for 3? ## Ref…

How many integers between 100 and 999, inclusive, have the property that some permutation of its digits is a multiple of…

Tamara has three rows of two -feet by -feet flower beds in her garden. The beds are separated and also surrounded by -fo…

At a gathering of people, there are people who all know each other and people who know no one. People who know each o…

How many of the base-ten numerals for the positive integers less than or equal to contain the digit ? ## Reference S…

Let be a set of integers taken from with the property that if and are elements of with , then is not a multiple o…

A function is defined recursively by and for all integers . What is ? ## Reference Solution For all integers note…

Sara makes a staircase out of toothpicks as shown: ```asy [asy] size(150); defaultpen(linewidth(0.8)); path h = ellipse…

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…

A Pascal-like triangle has as the top row and followed by as the second row. In each subsequent row the first number…

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…

Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up…

Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtrac…

Let be a positive integer such that is an integer. Which of the following statements is not true: ## Reference So…

Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 mo…

Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs…

Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, i…

A subset of the set of integers from to , inclusive, has the property that no two elements of sum to . What is the ma…

A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third…

Doug and Dave shared a pizza with equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half th…

What is the tens digit in the sum ## Reference Solution Since is divisible by , any factorial greater than is als…

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

In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of th…

Which of the following is equal to the product ## Reference Solution . \n "],[" Notice that everything cancels ou…

A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least 700,000…

What is the sum of the digits of the square of ? ## Reference Solution Using the standard multiplication algorithm,…

As shown below, convex pentagon has sides , , , , and . The pentagon is originally positioned in the plane with vertex…

A particular -hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to displa…

By inserting parentheses, it is possible to give the expression several values. How many different values can be obtai…

A book that is to be recorded onto compact discs takes minutes to read aloud. Each disc can hold up to minutes of read…

A cell phone plan costs dollars each month, plus cents per text message sent, plus cents for each minute used over h…

A small bottle of shampoo can hold milliliters of shampoo, whereas a large bottle can hold milliliters of shampoo. Jas…

What is ## Reference Solution \n \nNote: This exact problem was reused in 2013 AMC 10B: \n https://artofproblemso…

At a store, when a length is reported as inches that means the length is at least inches and at most inches. Suppose…

On Halloween Casper ate of his candies and then gave candies to his brother. The next day he ate of his remaining can…

Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many c…

In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one…

The point in the -plane with coordinates is reflected across the line . What are the coordinates of the reflected poin…

A taxi ride costs 1.50 per mile traveled. How much does a -mile taxi ride cost? ## Reference Solution There are fiv…

Alice is making a batch of cookies and needs cups of sugar. Unfortunately, her measuring cup holds only cup of sugar.…

What is ? ## Reference Solution This expression is equivalent to \n For the exact same problem but with answer choi…

Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for each. In the afternoon she s…

What is the tens digit of ## Reference Solution Notice that, for , is congruent to when is even and when is od…

What is the value of ? ## Reference Solution Notice this is the term in a recursive sequence, defined recursively a…

Mia is "helping" her mom pick up toys that are strewn on the floor. Mia’s mom manages to put toys into the toy box eve…

What is the largest number of solid blocks that can fit in a box? ## Reference Solution We find that the volume of…

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…

Consider the sequence of positive integers What is the th term in the sequence? ## Reference Solution **What we kn…

The instructions on a -gram bag of coffee beans say that proper brewing of a large mug of pour-over coffee requires gra…

Jerry wrote down the ones digit of each of the first positive squares: . What is the sum of all the numbers Jerry wrote…

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…