simonycl/amc_aime_training_positive_sequence_cot_gpt-4.1-prompt

messages listlengths 1 1
[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 2 nor 5?", "role": "user" } ]
[ { "content": "Question:\nLet a and b be real numbers such that a + b = 5 and ab = 6. Find the value of a^3 + b^3.", "role": "user" } ]
[ { "content": "Question:\nFind the number of ordered pairs of integers (x, y) such that 1 ≤ x ≤ 10, 1 ≤ y ≤ 10, and x + y is a multiple of 3.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all positive integers n < 100 such that the remainder when n^2 is divided by 8 is 1. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, angle C is a right angle, and AC = 8, BC = 15. Find the length of AB.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be positive integers such that x^2 + y^2 = 2024. What is the sum of all possible values of x?", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has a side length of 6. What is the area of the triangle formed by connecting every other vertex of the hexagon?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of integers between 1 and 1000 inclusive that are divisible by neither 2 nor 5. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nHow many ordered triples (a, b, c) of positive integers satisfy a + b + c = 15 and a ≥ 2, b ≥ 3, c ≥ 4?", "role": "user" } ]
[ { "content": "Question:\nLet f(x) = x^2 - 4x + 7. Let S be the set of all real numbers x such that f(f(x)) = 15. What is the sum of all elements in S?", "role": "user" } ]
[ { "content": "Question:\nLet x be a real number such that x^2 - 5x + 6 = 0. What is the value of x^4 - 10x^2 + 24?", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has a side length of 4. What is the area of the triangle formed by joining every other vertex of the hexagon?", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 2 nor 5?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all positive integers less than or equal to 100 such that the sum of the digits of n is divisible by 7. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nLet a and b be positive integers such that a^2 + b^2 = 2024. What is the sum of all possible values of a + b?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that the sum 1 + 2 + 3 + ... + n is greater than 1000. Find the value of n.", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit numbers are there such that the sum of their digits is 6?", "role": "user" } ]
[ { "content": "Question:\nA triangle has side lengths 7, 24, and 25. What is the area of the triangle?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of integers between 1 and 1000 inclusive that are divisible by neither 2 nor 5. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nIf x and y are real numbers such that x^2 + y^2 = 10 and x + y = 4, find the value of xy.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 12 and x^2 + y^2 = 80. What is the value of xy?", "role": "user" } ]
[ { "content": "Question:\nIn how many ways can the letters of the word \"COMMITTEE\" be arranged so that no two M's are adjacent?", "role": "user" } ]
[ { "content": "Question:\nA right triangle has integer side lengths and its area is 210. What is the smallest possible value of its hypotenuse?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of positive integers less than 1000 that are divisible by neither 2 nor 5. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nThe arithmetic mean of three positive numbers is 10, and their geometric mean is 8. If two of the numbers are equal, what is the third number?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that when n is divided by 6, the remainder is 4, and when n is divided by 8, the remainder is 5. What is the value of n?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 8, AC = 6, and the area of triangle ABC is 24. Find the length of BC.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all three-digit numbers whose digits add up to 15. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nSuppose x and y are real numbers satisfying x^2 + y^2 = 10 and x + y = 4. Find the value of x^3 + y^3.", "role": "user" } ]
[ { "content": "Question:\nLet n be the number of integers between 1 and 1000 (inclusive) that are divisible by neither 2 nor 5. Find n.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x^2 + y^2 = 25 and x + y = 7. What is the value of xy?", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit positive integers have digits that sum to 10?", "role": "user" } ]
[ { "content": "Question:\nA triangle has side lengths 7, 24, and 25. What is the area of the triangle?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all positive integers less than 1000 that are divisible by neither 2 nor 5. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nSuppose a fair six-sided die is rolled three times. What is the probability that the sum of the three numbers rolled is exactly 10?", "role": "user" } ]
[ { "content": "Question:\nLet the sum of three positive integers be 30. What is the maximum possible value of their product?", "role": "user" } ]
[ { "content": "Question:\nLet \\( ABCD \\) be a square with side length 10. Points \\( E \\) and \\( F \\) are on sides \\( AB \\) and \\( BC \\), respectively, such that \\( AE = BF = 3 \\). Find the area of triangle \\( AEF \\).", "role": "user" } ]
[ { "content": "Question:\nFind the number of ordered pairs of positive integers \\((x, y)\\) such that \\( x^2 + y^2 = 2024 \\).", "role": "user" } ]
[ { "content": "Question:\nA sequence \\( a_1, a_2, a_3, \\ldots \\) is defined recursively by \\( a_1 = 2 \\), \\( a_2 = 5 \\), and \\( a_{n+2} = a_{n+1} + 2a_n \\) for \\( n \\geq 1 \\). Find \\( a_6 \\).", "role": "user" } ]
[ { "content": "Question:\nSuppose \\( x \\) and \\( y \\) are real numbers such that \\( x^2 + y^2 = 25 \\) and \\( x + y = 7 \\). Compute the value of \\( xy \\).", "role": "user" } ]
[ { "content": "Question:\nLet x and y be positive integers such that x^2 + y^2 = 2024. What is the value of x + y?", "role": "user" } ]
[ { "content": "Question:\nA triangle has side lengths 13, 14, and 15. What is the area of the triangle?", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit numbers have digits that are strictly increasing from left to right?", "role": "user" } ]
[ { "content": "Question:\nLet S be the sum of all positive integers less than 100 that are divisible by either 3 or 7. What is S?", "role": "user" } ]
[ { "content": "Question:\nIf a and b are real numbers such that a + b = 6 and ab = 5, what is the value of a^3 + b^3?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be positive integers such that x^2 + y^2 = 100. Find the number of ordered pairs (x, y) that satisfy this equation.", "role": "user" } ]
[ { "content": "Question:\nA rectangle has a perimeter of 40 units. If the length and width are both integers and the area is maximized, what is the area of the rectangle?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of integers between 1 and 1000, inclusive, that are divisible by neither 2 nor 5. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 7, BC = 8, and AC = 9. Find the length of the altitude from B to AC.", "role": "user" } ]
[ { "content": "Question:\nThe sum of all positive integers n less than 100 for which n^2 - 1 is divisible by 24 is S. Find S.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of positive integers less than 100 that are divisible by 3 or 5 but not both. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nA triangle has sides of lengths 13, 14, and 15. What is the area of the triangle?", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit numbers have the property that the sum of their digits is 7?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 6 and x^2 + y^2 = 20. Find the value of x^3 + y^3.", "role": "user" } ]
[ { "content": "Question:\nIn a regular hexagon with side length 1, what is the length of a diagonal that is not a side?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be positive integers such that 3x + 7y = 100. How many ordered pairs (x, y) satisfy this equation?", "role": "user" } ]
[ { "content": "Question:\nA triangle has side lengths 7, 24, and 25. What is the area of the triangle?", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit numbers are there such that the product of their digits is 36?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of integers between 1 and 1000, inclusive, that are divisible by neither 2 nor 5. How many elements does S have?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that the sum 1 + 2 + ... + n is greater than 1000. Find the value of n.", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that the sum 1 + 2 + 3 + ... + n is a perfect square. Find the value of n.", "role": "user" } ]
[ { "content": "Question:\nHow many integer values of x satisfy the equation x^2 - 7x + 12 ≤ 0?", "role": "user" } ]
[ { "content": "Question:\nA triangle has side lengths 5, 12, and 13. What is the area of the circle inscribed in this triangle?", "role": "user" } ]
[ { "content": "Question:\nFind the number of positive integers less than 1000 that are divisible by 7 but not by 5.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all integers between 1 and 1000 (inclusive) that are not divisible by 2 or 5. Find the sum of the elements of S.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 8 and x^2 + y^2 = 34. What is the value of xy?", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 3 nor 5?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 7, AC = 9, and BC = 10. Let D be the point on BC such that the length of AD is minimized. What is the length of AD?", "role": "user" } ]
[ { "content": "Question:\nLet N be the smallest positive integer such that when N is divided by 5, the remainder is 2, and when N is divided by 7, the remainder is 3. What is N?", "role": "user" } ]
[ { "content": "Question:\nHow many ordered triples of positive integers (a, b, c) satisfy a + b + c = 12?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all three-digit numbers whose digits sum to 18. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 7, AC = 8, and BC = 9. Let G be the centroid of triangle ABC. What is the length of AG?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be positive real numbers such that x + y = 10 and x^2 + y^2 = 58. Find the value of xy.", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 2 nor 5?", "role": "user" } ]
[ { "content": "Question:\nA fair six-sided die is rolled three times. What is the probability that the product of the three numbers rolled is a perfect square? Express your answer as a common fraction.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of positive integers less than 1000 that are divisible by 3 or 5 but not both. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 6 and x^2 + y^2 = 20. Find the value of x*y.", "role": "user" } ]
[ { "content": "Question:\nThe sum of the interior angles of a regular polygon is 1980 degrees. How many sides does the polygon have?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that the decimal representation of 7n ends with three consecutive zeros. Find n.", "role": "user" } ]
[ { "content": "Question:\nHow many ordered pairs of integers (a, b) satisfy the equation a^2 + b^2 = 2024?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of integers between 1 and 1000 inclusive that are divisible by 3 or 5 but not by both. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nA circle with center O passes through points A(2,4) and B(8,10). If the center O lies on the line y = x, find the radius of the circle.", "role": "user" } ]
[ { "content": "Question:\nThe sum of three consecutive positive integers is equal to the product of the smallest and the largest. What is the sum of these three integers?", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit positive integers have digits that are strictly increasing from left to right?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 7 and x^2 + y^2 = 37. Compute the value of x^3 + y^3.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be positive integers such that x^2 + y^2 = 2024. What is the sum of all possible values of x?", "role": "user" } ]
[ { "content": "Question:\nA triangle has side lengths 8, 15, and 17. What is the area of the triangle?", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit integers are there such that the sum of their digits is 10?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all integers n between 1 and 1000 (inclusive) for which n^3 + 2n is divisible by 3. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nEvaluate the sum \\( \\sum_{k=1}^{10} \\frac{1}{k(k+1)} \\).", "role": "user" } ]
[ { "content": "Question:\nLet the sum of the digits of a positive integer n be 2024. What is the smallest possible value of n?", "role": "user" } ]
[ { "content": "Question:\nA triangle has side lengths 7, 15, and x, where x is an integer. If the area of the triangle is also an integer, what is the greatest possible value of x?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all positive integers less than 1000 that are divisible by 3 or 5 but not by 15. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nSuppose a and b are real numbers such that a^2 + b^2 = 10 and ab = 3. Find the value of (a + b)^4 - (a - b)^4.", "role": "user" } ]
[ { "content": "Question:\nHow many ways are there to arrange the letters of the word 'STATISTICS' such that no two identical letters are adjacent?", "role": "user" } ]
[ { "content": "Question:\nLet f(x) = x^2 + bx + c be a quadratic polynomial where b and c are integers. If f(1) = 7 and f(3) = 19, what is f(5)?", "role": "user" } ]
[ { "content": "Question:\nA rectangle has side lengths in the ratio 3:4. If its area is 192 square units, what is its perimeter?", "role": "user" } ]
[ { "content": "Question:\nHow many integers between 100 and 999 inclusive have the property that the sum of their digits is 15?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, the side AB = 7, AC = 8, and angle BAC = 60°. What is the length of side BC?", "role": "user" } ]
[ { "content": "Question:\nA bag contains red, blue, and green marbles. There are 5 red, 4 blue, and 3 green marbles. If three marbles are drawn at random without replacement, what is the probability that all three are of different colors? Express your answer as a reduced fraction.", "role": "user" } ]

[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 2 nor 5?", "role": "user" } ]

[ { "content": "Question:\nLet a and b be real numbers such that a + b = 5 and ab = 6. Find the value of a^3 + b^3.", "role": "user" } ]

[ { "content": "Question:\nFind the number of ordered pairs of integers (x, y) such that 1 ≤ x ≤ 10, 1 ≤ y ≤ 10, and x + y is a multiple of 3.", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all positive integers n < 100 such that the remainder when n^2 is divided by 8 is 1. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, angle C is a right angle, and AC = 8, BC = 15. Find the length of AB.", "role": "user" } ]

[ { "content": "Question:\nLet x and y be positive integers such that x^2 + y^2 = 2024. What is the sum of all possible values of x?", "role": "user" } ]

[ { "content": "Question:\nA regular hexagon has a side length of 6. What is the area of the triangle formed by connecting every other vertex of the hexagon?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of integers between 1 and 1000 inclusive that are divisible by neither 2 nor 5. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nHow many ordered triples (a, b, c) of positive integers satisfy a + b + c = 15 and a ≥ 2, b ≥ 3, c ≥ 4?", "role": "user" } ]

[ { "content": "Question:\nLet f(x) = x^2 - 4x + 7. Let S be the set of all real numbers x such that f(f(x)) = 15. What is the sum of all elements in S?", "role": "user" } ]

[ { "content": "Question:\nLet x be a real number such that x^2 - 5x + 6 = 0. What is the value of x^4 - 10x^2 + 24?", "role": "user" } ]

[ { "content": "Question:\nA regular hexagon has a side length of 4. What is the area of the triangle formed by joining every other vertex of the hexagon?", "role": "user" } ]

[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 2 nor 5?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all positive integers less than or equal to 100 such that the sum of the digits of n is divisible by 7. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nLet a and b be positive integers such that a^2 + b^2 = 2024. What is the sum of all possible values of a + b?", "role": "user" } ]

[ { "content": "Question:\nLet n be the smallest positive integer such that the sum 1 + 2 + 3 + ... + n is greater than 1000. Find the value of n.", "role": "user" } ]

[ { "content": "Question:\nHow many four-digit numbers are there such that the sum of their digits is 6?", "role": "user" } ]

[ { "content": "Question:\nA triangle has side lengths 7, 24, and 25. What is the area of the triangle?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of integers between 1 and 1000 inclusive that are divisible by neither 2 nor 5. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nIf x and y are real numbers such that x^2 + y^2 = 10 and x + y = 4, find the value of xy.", "role": "user" } ]

[ { "content": "Question:\nLet x and y be real numbers such that x + y = 12 and x^2 + y^2 = 80. What is the value of xy?", "role": "user" } ]

[ { "content": "Question:\nIn how many ways can the letters of the word \"COMMITTEE\" be arranged so that no two M's are adjacent?", "role": "user" } ]

[ { "content": "Question:\nA right triangle has integer side lengths and its area is 210. What is the smallest possible value of its hypotenuse?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of positive integers less than 1000 that are divisible by neither 2 nor 5. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nThe arithmetic mean of three positive numbers is 10, and their geometric mean is 8. If two of the numbers are equal, what is the third number?", "role": "user" } ]

[ { "content": "Question:\nLet n be the smallest positive integer such that when n is divided by 6, the remainder is 4, and when n is divided by 8, the remainder is 5. What is the value of n?", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, AB = 8, AC = 6, and the area of triangle ABC is 24. Find the length of BC.", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all three-digit numbers whose digits add up to 15. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nSuppose x and y are real numbers satisfying x^2 + y^2 = 10 and x + y = 4. Find the value of x^3 + y^3.", "role": "user" } ]

[ { "content": "Question:\nLet n be the number of integers between 1 and 1000 (inclusive) that are divisible by neither 2 nor 5. Find n.", "role": "user" } ]

[ { "content": "Question:\nLet x and y be real numbers such that x^2 + y^2 = 25 and x + y = 7. What is the value of xy?", "role": "user" } ]

[ { "content": "Question:\nHow many four-digit positive integers have digits that sum to 10?", "role": "user" } ]

[ { "content": "Question:\nA triangle has side lengths 7, 24, and 25. What is the area of the triangle?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all positive integers less than 1000 that are divisible by neither 2 nor 5. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nSuppose a fair six-sided die is rolled three times. What is the probability that the sum of the three numbers rolled is exactly 10?", "role": "user" } ]

[ { "content": "Question:\nLet the sum of three positive integers be 30. What is the maximum possible value of their product?", "role": "user" } ]

[ { "content": "Question:\nLet \\( ABCD \\) be a square with side length 10. Points \\( E \\) and \\( F \\) are on sides \\( AB \\) and \\( BC \\), respectively, such that \\( AE = BF = 3 \\). Find the area of triangle \\( AEF \\).", "role": "user" } ]

[ { "content": "Question:\nFind the number of ordered pairs of positive integers \\((x, y)\\) such that \\( x^2 + y^2 = 2024 \\).", "role": "user" } ]

[ { "content": "Question:\nA sequence \\( a_1, a_2, a_3, \\ldots \\) is defined recursively by \\( a_1 = 2 \\), \\( a_2 = 5 \\), and \\( a_{n+2} = a_{n+1} + 2a_n \\) for \\( n \\geq 1 \\). Find \\( a_6 \\).", "role": "user" } ]

[ { "content": "Question:\nSuppose \\( x \\) and \\( y \\) are real numbers such that \\( x^2 + y^2 = 25 \\) and \\( x + y = 7 \\). Compute the value of \\( xy \\).", "role": "user" } ]

[ { "content": "Question:\nLet x and y be positive integers such that x^2 + y^2 = 2024. What is the value of x + y?", "role": "user" } ]

[ { "content": "Question:\nA triangle has side lengths 13, 14, and 15. What is the area of the triangle?", "role": "user" } ]

[ { "content": "Question:\nHow many four-digit numbers have digits that are strictly increasing from left to right?", "role": "user" } ]

[ { "content": "Question:\nLet S be the sum of all positive integers less than 100 that are divisible by either 3 or 7. What is S?", "role": "user" } ]

[ { "content": "Question:\nIf a and b are real numbers such that a + b = 6 and ab = 5, what is the value of a^3 + b^3?", "role": "user" } ]

[ { "content": "Question:\nLet x and y be positive integers such that x^2 + y^2 = 100. Find the number of ordered pairs (x, y) that satisfy this equation.", "role": "user" } ]

[ { "content": "Question:\nA rectangle has a perimeter of 40 units. If the length and width are both integers and the area is maximized, what is the area of the rectangle?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of integers between 1 and 1000, inclusive, that are divisible by neither 2 nor 5. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, AB = 7, BC = 8, and AC = 9. Find the length of the altitude from B to AC.", "role": "user" } ]

[ { "content": "Question:\nThe sum of all positive integers n less than 100 for which n^2 - 1 is divisible by 24 is S. Find S.", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of positive integers less than 100 that are divisible by 3 or 5 but not both. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nA triangle has sides of lengths 13, 14, and 15. What is the area of the triangle?", "role": "user" } ]

[ { "content": "Question:\nHow many four-digit numbers have the property that the sum of their digits is 7?", "role": "user" } ]

[ { "content": "Question:\nLet x and y be real numbers such that x + y = 6 and x^2 + y^2 = 20. Find the value of x^3 + y^3.", "role": "user" } ]

[ { "content": "Question:\nIn a regular hexagon with side length 1, what is the length of a diagonal that is not a side?", "role": "user" } ]

[ { "content": "Question:\nLet x and y be positive integers such that 3x + 7y = 100. How many ordered pairs (x, y) satisfy this equation?", "role": "user" } ]

[ { "content": "Question:\nA triangle has side lengths 7, 24, and 25. What is the area of the triangle?", "role": "user" } ]

[ { "content": "Question:\nHow many four-digit numbers are there such that the product of their digits is 36?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of integers between 1 and 1000, inclusive, that are divisible by neither 2 nor 5. How many elements does S have?", "role": "user" } ]

[ { "content": "Question:\nLet n be the smallest positive integer such that the sum 1 + 2 + ... + n is greater than 1000. Find the value of n.", "role": "user" } ]

[ { "content": "Question:\nLet n be the smallest positive integer such that the sum 1 + 2 + 3 + ... + n is a perfect square. Find the value of n.", "role": "user" } ]

[ { "content": "Question:\nHow many integer values of x satisfy the equation x^2 - 7x + 12 ≤ 0?", "role": "user" } ]

[ { "content": "Question:\nA triangle has side lengths 5, 12, and 13. What is the area of the circle inscribed in this triangle?", "role": "user" } ]

[ { "content": "Question:\nFind the number of positive integers less than 1000 that are divisible by 7 but not by 5.", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all integers between 1 and 1000 (inclusive) that are not divisible by 2 or 5. Find the sum of the elements of S.", "role": "user" } ]

[ { "content": "Question:\nLet x and y be real numbers such that x + y = 8 and x^2 + y^2 = 34. What is the value of xy?", "role": "user" } ]

[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 3 nor 5?", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, AB = 7, AC = 9, and BC = 10. Let D be the point on BC such that the length of AD is minimized. What is the length of AD?", "role": "user" } ]

[ { "content": "Question:\nLet N be the smallest positive integer such that when N is divided by 5, the remainder is 2, and when N is divided by 7, the remainder is 3. What is N?", "role": "user" } ]

[ { "content": "Question:\nHow many ordered triples of positive integers (a, b, c) satisfy a + b + c = 12?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all three-digit numbers whose digits sum to 18. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, AB = 7, AC = 8, and BC = 9. Let G be the centroid of triangle ABC. What is the length of AG?", "role": "user" } ]

[ { "content": "Question:\nLet x and y be positive real numbers such that x + y = 10 and x^2 + y^2 = 58. Find the value of xy.", "role": "user" } ]

[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 2 nor 5?", "role": "user" } ]

[ { "content": "Question:\nA fair six-sided die is rolled three times. What is the probability that the product of the three numbers rolled is a perfect square? Express your answer as a common fraction.", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of positive integers less than 1000 that are divisible by 3 or 5 but not both. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nLet x and y be real numbers such that x + y = 6 and x^2 + y^2 = 20. Find the value of x*y.", "role": "user" } ]

[ { "content": "Question:\nThe sum of the interior angles of a regular polygon is 1980 degrees. How many sides does the polygon have?", "role": "user" } ]

[ { "content": "Question:\nLet n be the smallest positive integer such that the decimal representation of 7n ends with three consecutive zeros. Find n.", "role": "user" } ]

[ { "content": "Question:\nHow many ordered pairs of integers (a, b) satisfy the equation a^2 + b^2 = 2024?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of integers between 1 and 1000 inclusive that are divisible by 3 or 5 but not by both. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nA circle with center O passes through points A(2,4) and B(8,10). If the center O lies on the line y = x, find the radius of the circle.", "role": "user" } ]

[ { "content": "Question:\nThe sum of three consecutive positive integers is equal to the product of the smallest and the largest. What is the sum of these three integers?", "role": "user" } ]

[ { "content": "Question:\nHow many four-digit positive integers have digits that are strictly increasing from left to right?", "role": "user" } ]

[ { "content": "Question:\nLet x and y be real numbers such that x + y = 7 and x^2 + y^2 = 37. Compute the value of x^3 + y^3.", "role": "user" } ]

[ { "content": "Question:\nLet x and y be positive integers such that x^2 + y^2 = 2024. What is the sum of all possible values of x?", "role": "user" } ]

[ { "content": "Question:\nA triangle has side lengths 8, 15, and 17. What is the area of the triangle?", "role": "user" } ]

[ { "content": "Question:\nHow many four-digit integers are there such that the sum of their digits is 10?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all integers n between 1 and 1000 (inclusive) for which n^3 + 2n is divisible by 3. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nEvaluate the sum \\( \\sum_{k=1}^{10} \\frac{1}{k(k+1)} \\).", "role": "user" } ]

[ { "content": "Question:\nLet the sum of the digits of a positive integer n be 2024. What is the smallest possible value of n?", "role": "user" } ]

[ { "content": "Question:\nA triangle has side lengths 7, 15, and x, where x is an integer. If the area of the triangle is also an integer, what is the greatest possible value of x?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all positive integers less than 1000 that are divisible by 3 or 5 but not by 15. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nSuppose a and b are real numbers such that a^2 + b^2 = 10 and ab = 3. Find the value of (a + b)^4 - (a - b)^4.", "role": "user" } ]

[ { "content": "Question:\nHow many ways are there to arrange the letters of the word 'STATISTICS' such that no two identical letters are adjacent?", "role": "user" } ]

[ { "content": "Question:\nLet f(x) = x^2 + bx + c be a quadratic polynomial where b and c are integers. If f(1) = 7 and f(3) = 19, what is f(5)?", "role": "user" } ]

[ { "content": "Question:\nA rectangle has side lengths in the ratio 3:4. If its area is 192 square units, what is its perimeter?", "role": "user" } ]

[ { "content": "Question:\nHow many integers between 100 and 999 inclusive have the property that the sum of their digits is 15?", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, the side AB = 7, AC = 8, and angle BAC = 60°. What is the length of side BC?", "role": "user" } ]

[ { "content": "Question:\nA bag contains red, blue, and green marbles. There are 5 red, 4 blue, and 3 green marbles. If three marbles are drawn at random without replacement, what is the probability that all three are of different colors? Express your answer as a reduced fraction.", "role": "user" } ]