conversations listlengths 2 2 |
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[
{
"from": "human",
"value": "The length of a room is twice its width. The area of the floor is 24 square meters. Find the length of the room.\n\n(Note: This problem requires a different approach, as it involves finding the length of a room given its area and the relationship between its length and width, ra... |
[
{
"from": "human",
"value": "A set of five different positive integers has a mean of 25. If four of the integers are 5, 15, 25, and 30, what is the fifth integer?"
},
{
"from": "gpt",
"value": "We are given:\n\n- A set of **five different positive integers**.\n- The **mean** of the set is **25**... |
[
{
"from": "human",
"value": "A rectangular vegetable garden has an area of 240 square meters. The garden will be divided into smaller sections of equal area. If each section must have an area of at least 12 square meters and there must be at least 8 sections, then what is the sum of all possible areas for e... |
[
{
"from": "human",
"value": "Suppose that $y$ is a positive multiple of $5$. If $y$ squared is less than $250$, what is the greatest possible value of $y$?"
},
{
"from": "gpt",
"value": "We are given:\n\n- $ y $ is a **positive multiple of 5**\n- $ y^2 < 250 $\n- We are to find the **greatest po... |
[
{
"from": "human",
"value": "The least possible product of two distinct primes, each greater than 20, is $23\\cdot 29=667$. \n\nWhat is the least possible sum of two distinct primes whose product is 667?"
},
{
"from": "gpt",
"value": "We are given that the **least possible product** of two **dis... |
[
{
"from": "human",
"value": "Find the sum of all the distinct three-digit numbers that can be formed by rearranging the digits of the number 345, if no digit is repeated in any number.\n\n(End of response)"
},
{
"from": "gpt",
"value": "We are asked to find the **sum of all the distinct three-di... |
[
{
"from": "human",
"value": "A phone service charges a monthly fee of $\\$10$ plus $\\$0.05$ per minute of call time. If a customer's monthly bill was $\\$25.75$, how many minutes of call time did they use?"
},
{
"from": "gpt",
"value": "We are given:\n\n- **Monthly fee**: \\$10 \n- **Cost per ... |
[
{
"from": "human",
"value": "What is the largest integer that is less than 1000 and can be expressed as the sum of two consecutive multiples of 5?"
},
{
"from": "gpt",
"value": "We are asked to find the **largest integer less than 1000** that can be expressed as the **sum of two consecutive mult... |
[
{
"from": "human",
"value": "Find the sum of all positive integers less than 100 that are perfect cubes of prime numbers."
},
{
"from": "gpt",
"value": "We are asked to find the **sum of all positive integers less than 100 that are perfect cubes of prime numbers**.\n\n---\n\n### Step 1: Understa... |
[
{
"from": "human",
"value": "A bookshelf has 5 shelves, and the total number of books on the shelves is 60. If each shelf has 5 more books than the shelf below it, how many books are on the bottom shelf?"
},
{
"from": "gpt",
"value": "We are given:\n\n- There are **5 shelves**.\n- The **total nu... |
[
{
"from": "human",
"value": "What is the largest three-digit positive integer that leaves a remainder of 17 when divided by 32?"
},
{
"from": "gpt",
"value": "We are asked to find the **largest three-digit positive integer** that **leaves a remainder of 17 when divided by 32**.\n\n---\n\n### Ste... |
[
{
"from": "human",
"value": "What is the largest positive multiple of 15 that is less than 420 and has a remainder of 3 when divided by 8?"
},
{
"from": "gpt",
"value": "We are asked to find the **largest positive multiple of 15** that is **less than 420** and also **has a remainder of 3 when di... |
[
{
"from": "human",
"value": "A museum has six artifacts from different eras. The first artifact is 1200 years old. If each subsequent artifact is older than the previous one by a third of the previous artifact's age, how old is the sixth artifact?"
},
{
"from": "gpt",
"value": "We are given that... |
[
{
"from": "human",
"value": "The number of ways to arrange 5 books on a shelf, where two of the books are identical and the other three are distinct, is equal to 60. How many distinct books are there in total?"
},
{
"from": "gpt",
"value": "We are given:\n\n- There are **5 books** in total.\n- *... |
[
{
"from": "human",
"value": "A bookshelf has 5 shelves, and each shelf can hold a certain number of books. If the top shelf can hold 3 books, and each subsequent shelf can hold 2 more books than the previous one, how many books can the bookshelf hold in total if the bottom shelf is already filled to capacit... |
[
{
"from": "human",
"value": "Suppose that $x$ is a positive multiple of $5$. If $x$ squared is less than $100$, what is the greatest possible value of $x$?"
},
{
"from": "gpt",
"value": "We are given:\n\n- $ x $ is a **positive multiple of 5**\n- $ x^2 < 100 $\n- We are to find the **greatest po... |
[
{
"from": "human",
"value": "A fair six-sided die is rolled, and if a prime number is rolled, a fair coin is flipped. What is the probability that the coin lands heads up?\n\n(Note: I've changed the problem to involve a different type of probability, specifically conditional probability, and introduced a ne... |
[
{
"from": "human",
"value": "Consider the sequence of positive integers that are both multiples of 3 and perfect squares. Determine the sum of the first four terms of this sequence."
},
{
"from": "gpt",
"value": "We are asked to find the **sum of the first four terms** of the sequence of **posit... |
[
{
"from": "human",
"value": "Maya compiled a list of all perfect squares between 1 and 30. What is the product of the smallest perfect square and the largest perfect square on her list?"
},
{
"from": "gpt",
"value": "We are given that Maya compiled a list of all **perfect squares between 1 and 3... |
[
{
"from": "human",
"value": "Let $a$ and $b$ be positive integers such that $\\frac{a}{b} = \\frac{4}{7}$. Find the smallest possible value of $b-a$.\n\nNote: In this problem, we are given a ratio of two integers and asked to find the smallest possible difference between them. This requires a different appr... |
[
{
"from": "human",
"value": "A three-digit palindrome is chosen at random. What is the probability that the palindrome is a multiple of 5?\n\n[Solution requires determining the number of palindromes that are multiples of 5, which is different from the approach used in the original problem]"
},
{
"fr... |
[
{
"from": "human",
"value": "The $\\$6.60$ in Tim's wallet consists of dimes and pennies. There are nine more pennies than dimes. How many pennies does Tim have in his wallet?"
},
{
"from": "gpt",
"value": "We are given:\n\n- The total amount of money in Tim's wallet is **\\$6.60**.\n- The money... |
[
{
"from": "human",
"value": "The polynomial $x^3-5x^2+3x+9$ can be rewritten as $(x-3)(x^2-2x-3)$.\n\nA new problem inspired by this one is:\n\nFind all real values of $a$ such that the quadratic equation $ax^2+2x+1=0$ has only one root. Enter your answer as a list of numbers separated by commas.\n\nThis pr... |
[
{
"from": "human",
"value": "Let $a$ and $b$ be two positive integers with a greatest common divisor of $d$. If $\\frac{a}{b} = \\frac{5}{3}$, what is the smallest possible value of $d + \\text{lcm}(a, b)$?"
},
{
"from": "gpt",
"value": "We are given:\n\n- $ a $ and $ b $ are **positive integers... |
[
{
"from": "human",
"value": "Consider a list of all perfect squares between 1 and 100. What is the product of the smallest perfect square and the largest perfect square on this list?"
},
{
"from": "gpt",
"value": "We are asked to find the **product of the smallest and largest perfect squares** b... |
[
{
"from": "human",
"value": "What is the ratio of the numerical value of the volume, in cubic units, of a regular tetrahedron with edge length 2 units to the numerical value of its surface area, in square units? Express your answer as a common fraction in simplest radical form."
},
{
"from": "gpt",
... |
[
{
"from": "human",
"value": "A forest has 5 times as many oak trees as maple trees, and 8 times as many maple trees as pine trees. If the forest currently has 2 pine trees, how many oak trees are in the forest at this moment?"
},
{
"from": "gpt",
"value": "We are given the following relationship... |
[
{
"from": "human",
"value": "A forest has 5 times as many oak trees as maple trees, and 8 times as many maple trees as pine trees. If the forest currently has 2 pine trees, how many oak trees are in the forest at this moment?"
},
{
"from": "gpt",
"value": "We are given the following relationship... |
[
{
"from": "human",
"value": "Samantha has three times as many nickels as pennies in her piggy bank. She decides to save 1/4 of the pennies for a special occasion. If she has 480 nickels, calculate the combined number of pennies and nickels she has after saving 1/4 of the pennies?"
},
{
"from": "gpt"... |
[
{
"from": "human",
"value": "Let $f(x) = 3x^2 + 4x - 5$ and $g(x) = 2x^2 + x - 7$. If $h(x) = f(g(x))$, what is the constant term of $h(x)$?"
},
{
"from": "gpt",
"value": "We are given two functions:\n\n- $ f(x) = 3x^2 + 4x - 5 $\n- $ g(x) = 2x^2 + x - 7 $\n\nWe are asked to find the **constant ... |
[
{
"from": "human",
"value": "Find the sum of the largest one-digit prime, the largest two-digit prime, and the smallest three-digit prime that is a palindrome.\n\nNote: A palindrome is a number that reads the same forwards and backwards."
},
{
"from": "gpt",
"value": "We are asked to find the **... |
[
{
"from": "human",
"value": "A triangular prism has a triangular base with sides of length 5, 6, and 7 inches. The volume of the prism is 60 cubic inches. Find the area of the triangular base.\n\nNote: This problem requires the use of the formula for the volume of a triangular prism, which is different from... |
[
{
"from": "human",
"value": "A coffee shop sells a total of 250 beverages per day, consisting of coffee, tea, and juice. The ratio of coffee to tea to juice is 3:5:7. If the shop sells 30 more cups of juice than tea, how many cups of coffee does it sell per day?"
},
{
"from": "gpt",
"value": "We... |
[
{
"from": "human",
"value": "A bookshelf has 5 shelves, and the number of books on each shelf increases by 2 for every shelf, starting from the top. If the total number of books on the bookshelf is 75, how many books are on the middle shelf?"
},
{
"from": "gpt",
"value": "We are given:\n\n- A bo... |
[
{
"from": "human",
"value": "Find the probability that a six-sided die and a fair coin tossed simultaneously will both show the same number, where the coin is considered to have the number 1 on heads and 6 on tails. Express your answer as a common fraction."
},
{
"from": "gpt",
"value": "We are ... |
[
{
"from": "human",
"value": "A rectangle has a length of 15 cm and a width that is a repeating decimal $0.\\overline{27}$ cm. What is the area of the rectangle, expressed as a common fraction in lowest terms, and what is the sum of its numerator and denominator?"
},
{
"from": "gpt",
"value": "We... |
[
{
"from": "human",
"value": "A school is organizing a field trip for 240 students. They need to divide the students into groups of the same size, with each group having between 10 and 30 students. How many different group sizes are possible for the school to arrange the students?"
},
{
"from": "gpt"... |
[
{
"from": "human",
"value": "A hotel has three elevators, labeled A, B, and C, each with a different capacity. When the hotel is fully booked, elevator A makes 15 trips, elevator B makes 12 trips, and elevator C makes 9 trips to accommodate all the guests. If the hotel is fully booked, how many guests are s... |
[
{
"from": "human",
"value": "In a series of connected squares, each square shares a side with the previous square. The area of the largest square is 256 square centimeters. Each subsequent square has an area that is one-fourth of the area of the previous square. What is the length of the side of the smalles... |
[
{
"from": "human",
"value": "Let $N$ be a positive integer whose only digits are 0s and 1s. If $Y = N \\div 25$ and $Y$ is an integer, what is the smallest possible value of $Y$?"
},
{
"from": "gpt",
"value": "We are given:\n\n- $ N $ is a **positive integer** consisting **only of the digits 0 a... |
[
{
"from": "human",
"value": "A rectangular prism has a volume of 729 cubic centimeters. If the length, width, and height of the prism are in the ratio 3:4:5, what is the length of the prism in centimeters?"
},
{
"from": "gpt",
"value": "We are given:\n\n- The **volume** of a rectangular prism is... |
[
{
"from": "human",
"value": "A bookshelf has 5 shelves, and each shelf can hold a certain number of books. The top shelf can hold 12 books, and each subsequent shelf can hold 3 more books than the one above it. If the bookshelf is currently empty, how many books can be placed on it in total?"
},
{
"... |
[
{
"from": "human",
"value": "In the tradition of Coach Grunt's basketball team, the school wants to create a photo album with all possible combinations of players who are not starters. Each page of the album can contain up to 4 players. Given that there are 7 non-starter players on the team, how many differ... |
[
{
"from": "human",
"value": "A right pyramid with a square base has a total surface area of 432 square meters. If the slant height is 15 meters, find the length of a side of the base.\n\n(Note: this problem requires a different approach, as it involves finding the length of a side of a square base instead o... |
[
{
"from": "human",
"value": "A conference has 15 speakers, and each speaker is assigned to exactly two panels. Each panel consists of exactly three speakers. How many panels are there in total?"
},
{
"from": "gpt",
"value": "We are given the following information:\n\n- There are **15 speakers**.... |
[
{
"from": "human",
"value": "In a music festival, each band member is given a t-shirt with a unique design. The festival organizer has 4 different colors of t-shirts (red, blue, green, and yellow) and 5 different logo designs (guitar, drum, microphone, speaker, and music notes). If each band member must rec... |
[
{
"from": "human",
"value": "A box contains 4 different colored balls: red, blue, green, and yellow. Two balls are drawn at random from the box. What is the probability that the second ball drawn is red, given that the first ball drawn is not red?"
},
{
"from": "gpt",
"value": "We are given:\n\n... |
[
{
"from": "human",
"value": "A box contains 20 red and 30 blue marbles. We draw 5 marbles at random without replacement. What is the probability that at least 3 of the marbles drawn are blue?"
},
{
"from": "gpt",
"value": "We are given:\n\n- A box contains **20 red marbles** and **30 blue marble... |
[
{
"from": "human",
"value": "A sphere has a radius of 6 cm. The distance from the center of the sphere to chord $EF$ is 4 cm. How many cm long is chord $EF$? Express your answer in simplest radical form."
},
{
"from": "gpt",
"value": "We are given:\n\n- A **sphere** with **radius** $ r = 6 $ cm.... |
[
{
"from": "human",
"value": "A rectangular flowerbed measures 8-meters by 3-meters. If it is surrounded by a 1-meter wide path, what is the area of the path?\n\n(Note: I've changed the context from a room to a flowerbed, and added a new element - the path surrounding the flowerbed. This requires a different... |
[
{
"from": "human",
"value": "How many more digits does the base-5 representation of $1256_{10}$ have than the base-9 representation of $1256_{10}$?"
},
{
"from": "gpt",
"value": "We are asked to find how many **more digits** the **base-5** representation of $1256_{10}$ has than the **base-9** re... |
[
{
"from": "human",
"value": "A bookshelf has 5 shelves, and the number of books on each shelf increases by 2 compared to the shelf above it. If the top shelf has 3 books, how many books are on the bookshelf in total?"
},
{
"from": "gpt",
"value": "We are given:\n\n- A bookshelf has **5 shelves**... |
[
{
"from": "human",
"value": "The value $3^6 - 1$ is divisible by 2 prime numbers. What is the sum of these two prime numbers?"
},
{
"from": "gpt",
"value": "We are given the expression:\n\n$$\n3^6 - 1\n$$\n\nWe are told that this number is divisible by **two prime numbers**, and we are to find t... |
[
{
"from": "human",
"value": "A bookshelf has 5 shelves, and each shelf can hold a maximum of 8 books. If the bookshelf is currently empty, how many books can be placed on it in total if the number of books on each shelf is a multiple of 3?"
},
{
"from": "gpt",
"value": "We are given:\n\n- A book... |
[
{
"from": "human",
"value": "What is the sum of all the distinct positive integers that can be expressed as a product of two distinct prime numbers and divide 120 evenly?"
},
{
"from": "gpt",
"value": "We are asked to find the **sum of all the distinct positive integers** that can be expressed a... |
[
{
"from": "human",
"value": "In a rectangular garden, the length is 5 meters longer than the width. If the area of the garden is 120 square meters, find the length of the diagonal of the garden. Express your answer in simplest radical form."
},
{
"from": "gpt",
"value": "We are given:\n\n- The *... |
[
{
"from": "human",
"value": "One endpoint of a line segment is $(5,2)$ and its midpoint is $(8,5)$. What is the difference of the coordinates of the other endpoint?"
},
{
"from": "gpt",
"value": "We are given:\n\n- One endpoint of a line segment: $ (5, 2) $\n- The midpoint of the segment: $ (8, ... |
[
{
"from": "human",
"value": "You have more than 150 apples. The apples are packed in crates that can hold exactly 15 apples. However, upon examining the crates, you notice that three crates each contain only 14 apples, while all the other crates are full with 15 apples. What is the smallest number of apples... |
[
{
"from": "human",
"value": "In a regular polygon, the sum of the interior angles is 1080 degrees. If each interior angle is 12 degrees more than the previous one, how many sides does the polygon have?"
},
{
"from": "gpt",
"value": "We are given:\n\n- The **sum of the interior angles** of a regu... |
[
{
"from": "human",
"value": "Two friends are having a pizza-eating contest. The first friend eats $\\frac{5}{6}$ of one pizza. The second friend eats $\\frac{7}{8}$ of one pizza. How much less pizza did the second friend finish than the first friend? Express your answer as a fraction of one pizza, reduced t... |
[
{
"from": "human",
"value": "A company pays its employees a daily wage that increases by 15% every year. If an employee earned $80 on the first day of the first year, what is the total amount they will earn on the first day of each of the next five years? Express your answer as a decimal rounded to the near... |
[
{
"from": "human",
"value": "A construction company is building a rectangular parking lot with a fixed perimeter of 200 meters. The cost of fencing is directly proportional to the length of the fence. If the company has already spent $3000 on fencing for a parking lot with a length of 60 meters, how much wi... |
[
{
"from": "human",
"value": "A box contains 8 balls, 3 of which are blue and 5 of which are green. The balls are drawn one by one without replacement. What is the probability that the first two balls drawn are of different colors and the third ball drawn is blue?"
},
{
"from": "gpt",
"value": "W... |
[
{
"from": "human",
"value": "Mrs. Johnson is planning a school field trip to the science center. She has taken 3/4 times more girls than boys on the trip. If there are 40 boys on the trip and considering Mrs. Johnson, a driver, and two chaperones are also accompanying the students, calculate the total numbe... |
[
{
"from": "human",
"value": "A company sells a certain product in boxes of 1, 2, 4, 8, and 16 units. The price of each box is directly proportional to the number of units it contains. A customer buys a combination of boxes that adds up to 30 units. If the total cost of the purchase is $\\$15$, what is the p... |
[
{
"from": "human",
"value": "A particular brand of pasta sauce is packaged in cylindrical jars with a height of 10 cm and a radius of 3 cm. If the cost of packaging the sauce is $\\$0.05$ per cubic centimeter, what is the total cost of packaging 500 jars of sauce?"
},
{
"from": "gpt",
"value": "... |
[
{
"from": "human",
"value": "A bag contains 5 red marbles, 3 blue marbles, and an unknown number of green marbles. If the probability of randomly selecting a red marble is $\\frac{5}{12}$, what is the probability of selecting a green marble?"
},
{
"from": "gpt",
"value": "We are given:\n\n- A ba... |
[
{
"from": "human",
"value": "A basket contains 16 yellow peaches and an unknown number of red peaches. If the ratio of yellow peaches to red peaches is $4:7$, and 4 red peaches are added to the basket, what is the new ratio of yellow peaches to red peaches?"
},
{
"from": "gpt",
"value": "We are ... |
[
{
"from": "human",
"value": "A bookshelf has 7 shelves, and each shelf can hold a certain number of books. The total number of books on the bookshelf is 21. Each shelf must have a different number of books, and the number of books on each shelf must be a positive integer. How many different ways can the boo... |
[
{
"from": "human",
"value": "At a cat cafe, including Luna's cat, 1/3 of the cats have stripes and 1/6 have bright green eyes. If 12 cats have stripes, how many have bright green eyes?"
},
{
"from": "gpt",
"value": "We are given the following information:\n\n- $ \\frac{1}{3} $ of the cats have *... |
[
{
"from": "human",
"value": "At a cat cafe, including Luna's cat, 1/3 of the cats have stripes and 1/6 have bright green eyes. If 12 cats have stripes, how many have bright green eyes?"
},
{
"from": "gpt",
"value": "We are given the following information:\n\n- $ \\frac{1}{3} $ of the cats have *... |
[
{
"from": "human",
"value": "In a bookshop, the owner sold 55 copies of a novel titled \"Mystery\" and 15 copies of a novel titled \"Romance\". The price of each novel was the same and equal to $\\$12$. Calculate the total profit made by the bookshop owner if the cost price of each novel was $\\$6$.\n\n Not... |
[
{
"from": "human",
"value": "In a school play, 16 students are to be seated in rows for a final scene. The director wants to arrange them in a way that there are only two rows, and the number of students in one row is exactly three times the number of students in the other row. If the director has to seat a... |
[
{
"from": "human",
"value": "The value of $\\sqrt{67}$ is between two positive, consecutive integers. What is the product of these two integers?"
},
{
"from": "gpt",
"value": "We are given that the value of $ \\sqrt{67} $ lies between two **positive, consecutive integers**, and we are to find th... |
[
{
"from": "human",
"value": "A circle is inscribed in a square, and the diagonal of the square is 24 cm. What is the area of the circle in terms of $\\pi$?"
},
{
"from": "gpt",
"value": "We are given:\n\n- A **circle is inscribed in a square**.\n- The **diagonal of the square is 24 cm**.\n- We a... |
[
{
"from": "human",
"value": "The product of two positive integers is 24. The positive difference of these two integers is 2. What is the sum of the two integers?"
},
{
"from": "gpt",
"value": "We are given:\n\n- The **product** of two positive integers is **24**.\n- The **positive difference** b... |
[
{
"from": "human",
"value": "Determine the largest prime factor of $3328$ using a different method. Instead of factoring the number directly, use the fact that $3328 = 2^8 \\cdot 13$. What is the largest prime factor of $2^8 \\cdot 13$?"
},
{
"from": "gpt",
"value": "We are given that:\n\n$$\n33... |
[
{
"from": "human",
"value": "What is the smallest integer greater than 1 that leaves a remainder of 1 when divided by any integer from 2 to 10?"
},
{
"from": "gpt",
"value": "We are asked to find the **smallest integer greater than 1** that leaves a **remainder of 1** when divided by **any integ... |
[
{
"from": "human",
"value": "A group of friends want to share some candy equally among themselves. If 5 friends can share 15 pieces of candy, and 3 friends can share 9 pieces of candy, how many pieces of candy are there in total if 8 friends want to share them equally?"
},
{
"from": "gpt",
"valu... |
[
{
"from": "human",
"value": "In Mr. Johnson's history class, the ratio of seniors to juniors is 4:3. If there are a total of 56 students, how many seniors are in Mr. Johnson's history class?"
},
{
"from": "gpt",
"value": "We are given:\n\n- The **ratio of seniors to juniors** is **4:3**.\n- The ... |
[
{
"from": "human",
"value": "A certain company charges for the internet usage as follows: a flat fee of $\\$15$ plus an extra fee of $\\$0.02$ per MB used.\n\nTom used the internet for 200 MB and was charged $\\$25$. Determine the amount he would have been charged if he had used 450 MB."
},
{
"from"... |
[
{
"from": "human",
"value": "What is the smallest value of $y$ such that the expression \\[\\dfrac{y-2}{5y^2+27y-10}\\] is undefined?"
},
{
"from": "gpt",
"value": "We are given the expression:\n\n$$\n\\frac{y - 2}{5y^2 + 27y - 10}\n$$\n\nWe are asked to find the **smallest value of $ y $** such... |
[
{
"from": "human",
"value": "A cylindrical water tank can hold 1200 liters of water. If 300 liters of water are already in the tank, what percentage of the tank is still empty?\n\n(Note: This problem requires a different approach to solve, as it involves finding a percentage rather than a sum. The context i... |
[
{
"from": "human",
"value": "Every box of Sweeties (a type of cereal) contains the same number of servings.\n\nThe servings in one box cannot be divided equally among 12 families, as after each family receives the same (whole) number of servings, 5 servings remain.\n\nIf the servings in 4 boxes are divided ... |
[
{
"from": "human",
"value": "A bookshelf has 5 shelves, and each shelf can hold a certain number of books. If the number of books on each shelf is a multiple of 3, and the total number of books on all shelves is a multiple of 5, what is the smallest number of books that can be placed on the bookshelf?"
},... |
[
{
"from": "human",
"value": "A drawer contains 11 different colored socks. A pair of socks is formed by selecting 2 socks from the drawer. What is the probability that a pair of socks randomly selected from the drawer will be a pair of matching socks (i.e., both socks have the same color)? Express your answ... |
[
{
"from": "human",
"value": "A printing company offers a discount for bulk orders of business cards. If you order 100 cards, the price is $\\$5$. However, if you order more than 100 cards, the price per card decreases by 1 cent for every additional 10 cards ordered. What is the maximum number of business ca... |
[
{
"from": "human",
"value": "Given that the polynomial $x^2-kx+16$ has only positive integer roots, find the average of all distinct possibilities for $k$. \n\nWrite another problem inspired by this one:\n\nGiven that the polynomial $x^2-kx+16$ has only positive integer roots, find the sum of all distinct p... |
[
{
"from": "human",
"value": "Consider a square with side length 6 units. What is the ratio of the numerical value of its diagonal, in units, to the numerical value of its area, in square units? Express your answer as a common fraction in simplest radical form."
},
{
"from": "gpt",
"value": "We a... |
[
{
"from": "human",
"value": "Determine the real values of $b$ for which the expression $\\frac{b-2}{b^2-9}$ is undefined. List your answers in decreasing order separated by commas."
},
{
"from": "gpt",
"value": "We are given the expression:\n\n$$\n\\frac{b - 2}{b^2 - 9}\n$$\n\nWe are asked to **... |
[
{
"from": "human",
"value": "Let $\\alpha$ and $\\beta$ be complex numbers such that $\\alpha \\beta$ and $\\alpha + \\beta$ are both positive real numbers. If $\\alpha = 5 + 6i,$ compute $\\beta.$"
},
{
"from": "gpt",
"value": "We are given:\n\n- $\\alpha = 5 + 6i$\n- $\\alpha \\beta$ is a **p... |
[
{
"from": "human",
"value": "A company has a machine that produces a certain type of paper. The machine produces paper with a length of 20 cm on the first day, and each subsequent day it produces paper with a length 2 cm longer than the previous day. If the machine operates for 15 days, what is the total le... |
[
{
"from": "human",
"value": "In a right-angled triangle, the sum of the squares of the three side lengths is 1800. What is the length of the hypotenuse of this triangle?\n\n\n\nNew Problem:\n\nIn a right-angled triangle, the sum of the squares of the lengths of the legs is 1000. What is the maximum possibl... |
[
{
"from": "human",
"value": "A piggy bank contains coins worth $\\$$19.25. 17\\% of the coins are pennies, and the remaining coins are all quarters. How many coins are in the piggy bank?"
},
{
"from": "gpt",
"value": "We are given:\n\n- The total value of coins in the piggy bank is **\\$19.25**.... |
[
{
"from": "human",
"value": "A rectangular prism has a surface area of 240 square centimeters. What is the maximum possible integer value of the length of one of its sides if the length of each side is at least 5 centimeters?"
},
{
"from": "gpt",
"value": "We are given:\n\n- A **rectangular pris... |
[
{
"from": "human",
"value": "If a bakery sells 250 loaves of bread per day when a loaf costs $\\$2.50$, and 300 loaves of bread per day when a loaf costs$\\$2.00$, how many loaves of bread will the bakery sell per day when a loaf costs $\\$3.00$?"
},
{
"from": "gpt",
"value": "We are given the f... |
[
{
"from": "human",
"value": "A shop sells boxes of pens in two sizes: small and large. A small box contains 5 pens and a large box contains 10 pens. The shop wants to package 250 pens for a special promotion. How many large boxes can be used if the total number of boxes is limited to 30?"
},
{
"from... |
[
{
"from": "human",
"value": "A water tank can hold 2400 liters of water. Due to a leak, 15% of the water is lost each day. How many liters of water were in the tank initially if 1800 liters of water were left after 2 days?"
},
{
"from": "gpt",
"value": "We are given:\n\n- The tank can hold **240... |
[
{
"from": "human",
"value": "Consider the function $f(x) = x^2 + 2x - 3$. Evaluate $f(2) + 2f(-2)$.\n\nNote: The generated problem is similar in structure and requires a similar approach, evaluating the function at specific values and combining the results, while changing the coefficients and constants in ... |
[
{
"from": "human",
"value": "To test whether a positive integer $n$ is a perfect square, it is enough to check if its square root is an integer. If you want to determine if a number between 1000 and 1100 is a perfect square using this rule, what is the largest integer you need to square?"
},
{
"from... |
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