simonycl/amc_aime_training_positive_sequence_cot_gpt-4.1-prompt

messages listlengths 1 1
[ { "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:\nA regular hexagon has vertices A, B, C, D, E, and F in order. If points P and Q are chosen randomly and independently from the set of vertices, what is the probability that segment PQ is a diagonal of the hexagon?", "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:\nLet a, b, and c be positive real numbers such that abc = 8. Find the minimum value of a + 2b + 4c.", "role": "user" } ]
[ { "content": "Question:\nThe sum of the first n positive integers is equal to 2016. What is the value of n?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all two-digit positive integers whose digits sum to 10. Find the sum of all elements in S.", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 5, AC = 7, and the area of triangle ABC is 14. Find the length of BC.", "role": "user" } ]
[ { "content": "Question:\nHow many distinct 4-digit numbers are there such that the sum of their digits is 9?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that both n and n+1 are divisible by either 2, 3, or 5. Find n.", "role": "user" } ]
[ { "content": "Question:\nLet x be a real number such that x^2 + 4x + 7 = 0. Find the value of x^4 + 8x^3 + 30x^2 + 56x + 49.", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that the sum of the digits of n is 2024. What is the sum of the digits of n + 1?", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has side length 2. A point is chosen at random inside the hexagon. What is the probability that its distance to the center is less than 1?", "role": "user" } ]
[ { "content": "Question:\nFind the number of positive integers less than 1000 that are divisible by neither 2 nor 5.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 6 and x^2 + y^2 = 20. Compute all possible values of xy.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all four-digit numbers that use each of the digits 1, 2, 3, and 4 exactly once. Compute the sum of all numbers in S.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x^2 + y^2 = 25 and x + y = 7. Find the value of xy.", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers n less than 100 are there such that n^2 + 2n + 1 is a perfect square?", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has vertices labeled A, B, C, D, E, F in order. Let P be the intersection point of diagonals AC and BD. If the side length of the hexagon is 6, what is the distance from P to vertex E?", "role": "user" } ]
[ { "content": "Question:\nHow many distinct four-digit numbers can be formed using the digits 1, 2, 3, 4, and 5, each at most once, such that the number is even?", "role": "user" } ]
[ { "content": "Question:\nA fair six-sided die is rolled three times. What is the probability that the sum of the outcomes is 10?", "role": "user" } ]
[ { "content": "Question:\nLet a positive integer n have exactly 6 positive divisors, including 1 and n. How many such integers n less than 100 are there?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, angle BAC = 60°, AB = 8, and AC = 6. Find the length of BC.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all positive integers less than 1000 that are divisible by 6 but not by 9. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nLet a and b be real numbers such that a + b = 5 and a^2 + b^2 = 13. Find the value of ab.", "role": "user" } ]
[ { "content": "Question:\nFind the smallest positive integer n such that when n is divided by 7, the remainder is 4, and when n is divided by 9, the remainder is 5.", "role": "user" } ]
[ { "content": "Question:\nLet a and b be positive integers such that a^2 + b^2 = 2024. What is the largest possible value of ab?", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has side length 6. A point P is chosen uniformly at random inside the hexagon. What is the probability that the distance from P to the center of the hexagon is less than 3?", "role": "user" } ]
[ { "content": "Question:\nLet S be the sum of all positive integers less than 100 that leave a remainder of 2 when divided by 7. Find the value of S.", "role": "user" } ]
[ { "content": "Question:\nHow many ways are there to arrange the letters in the word 'PROBLEM' such that no two vowels are adjacent?", "role": "user" } ]
[ { "content": "Question:\nIf x and y are real numbers satisfying x + y = 7 and x^2 + y^2 = 37, compute the value of x^3 + y^3.", "role": "user" } ]
[ { "content": "Question:\nLet a and b be positive integers such that a^2 + b^2 = 2024. What is the greatest possible value of ab?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, angle C is a right angle and AB = 13, AC = 5. Find the length of BC.", "role": "user" } ]
[ { "content": "Question:\nA fair six-sided die is rolled three times. What is the probability that the numbers rolled are in strictly increasing order?", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit integers have the property that the sum of their digits is 10?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that the sum 1 + 2 + ... + n is greater than 1000. Find n.", "role": "user" } ]
[ { "content": "Question:\nLet x be a positive real number such that x + 1/x = 5. Find the value of x^3 + 1/x^3.", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 13, AC = 15, and the altitude from A to BC has length 12. Find the area of triangle ABC.", "role": "user" } ]
[ { "content": "Question:\nHow many integers n between 1 and 1000 (inclusive) are divisible by neither 2 nor 5?", "role": "user" } ]
[ { "content": "Question:\nLet S be the sum of all integers k such that 1 ≤ k ≤ 100 and the remainder when k is divided by 7 is 3. Find S.", "role": "user" } ]
[ { "content": "Question:\nLet a and b be positive integers such that a^2 + b^2 = 2024. Find the number of ordered pairs (a, b) that satisfy this equation.", "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 + y?", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has a side length of 6. What is the area of the triangle formed by joining every other vertex of the hexagon?", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit numbers have digits that strictly increase from left to right?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all positive integers less than 100 that are relatively prime to 100. What is the sum of the elements of S?", "role": "user" } ]
[ { "content": "Question:\nEvaluate the sum \\( \\sum_{n=1}^{10} (2n-1)^2 \\).", "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:\nIn triangle ABC, AB = AC and the altitude from A to BC is 6. If BC = 10, find the area of triangle ABC.", "role": "user" } ]
[ { "content": "Question:\nFind the number of ordered triples (x, y, z) of positive integers such that x + y + z = 12.", "role": "user" } ]
[ { "content": "Question:\nLet a and b be positive integers such that a^2 + b^2 = 2024. Find the number of such pairs (a, b).", "role": "user" } ]
[ { "content": "Question:\nA fair six-sided die is rolled four times. What is the probability that the sum of the results is 14?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x^2 + y^2 = 10 and x + y = 4. Find the value of x^3 + y^3.", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon and a regular pentagon have equal perimeters. If the length of a side of the hexagon is 6, what is the length of a side of the pentagon?", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit positive integers are there in which the digits are strictly increasing from left to right?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of integers from 1 to 50 inclusive. How many subsets of S contain exactly 3 elements whose sum is divisible by 3?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 13, AC = 15, and BC = 14. Find the length of the altitude from A to BC.", "role": "user" } ]
[ { "content": "Question:\nLet n be a positive integer such that the sum of the digits of n is 2019. What is the smallest possible value of n?", "role": "user" } ]
[ { "content": "Question:\nA circle passes through the points (0,0), (6,0), and (2,4). What is the radius of the circle?", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 2 nor 5?", "role": "user" } ]
[ { "content": "Question:\nLet f(x) = x^2 + bx + c, where b and c are integers. If f(1) = 7 and f(3) = 15, what is f(5)?", "role": "user" } ]
[ { "content": "Question:\nIn how many ways can the letters in the word 'PROBLEM' be arranged so that no two vowels are adjacent?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of integers from 1 to 1000, inclusive. How many numbers in S are divisible by neither 2 nor 5?", "role": "user" } ]
[ { "content": "Question:\nTriangle ABC has side lengths AB = 13, BC = 14, and AC = 15. Let D be the foot of the altitude from B to AC. What is the length of BD?", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers n less than 100 satisfy that n^2 + 4n is divisible by 5?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 6 and x^2 + y^2 = 20. What is the value of x^3 + y^3?", "role": "user" } ]
[ { "content": "Question:\nIn how many ways can the letters of the word MATHCOMP be arranged so that no two vowels are adjacent?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of positive integers less than 1000 that are divisible by 3 or 7 but not by 21. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nA triangle has side lengths 7, 8, and 9. What is the area of the triangle?", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 can be written as the sum of two distinct perfect squares?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x^2 + y^2 = 25 and x + y = 7. Find the value of x*y.", "role": "user" } ]
[ { "content": "Question:\nA fair six-sided die is rolled three times. What is the probability that the sum of the numbers rolled is exactly 10?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 7 and x^2 + y^2 = 29. What is the value of xy?", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has side length 2. A point is chosen at random inside the hexagon. What is the probability that the point is closer to the center than to any of the hexagon's sides? Express your answer as a common fraction.", "role": "user" } ]
[ { "content": "Question:\nLet S be the sum of all positive integers n less than 100 such that n^2 + 1 is divisible by 5. Find S.", "role": "user" } ]
[ { "content": "Question:\nIf a and b are integers such that 3a + 7b = 100, how many ordered pairs (a, b) satisfy this equation?", "role": "user" } ]
[ { "content": "Question:\nLet f(x) = x^4 - 4x^2 + 3. Find the sum of all real solutions to the equation f(x) = 0.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all positive integers less than 100 that are divisible by either 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 = 7 and x^2 + y^2 = 37. Compute the value of xy.", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, angle C is a right angle, and AC = 6, BC = 8. Find the radius of the circle inscribed in triangle ABC.", "role": "user" } ]
[ { "content": "Question:\nFind the number of positive integers less than 1000 that leave a remainder of 2 when divided by 5 and a remainder of 3 when divided by 7.", "role": "user" } ]
[ { "content": "Question:\nLet S be the sum of all integers n between 1 and 100 inclusive such that n^2 + n is divisible by 6. Compute S.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 8 and x^2 + y^2 = 34. Find 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 right triangle has integer side lengths and a hypotenuse of length 65. How many such triangles exist?", "role": "user" } ]
[ { "content": "Question:\nThe sum of the first n positive integers is equal to the sum of the next k consecutive positive integers. If n > 1 and k > 1, find the smallest possible value of n.", "role": "user" } ]
[ { "content": "Question:\nLet a and b be positive integers such that the greatest common divisor of a^2 + b^2 and a + b is 13. Find the smallest possible value of a + b.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of positive integers less than 100 that are divisible by either 3 or 5 but not both. How many elements are in S?", "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 integers are there such that the sum of their digits is 23?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x^2 + y^2 = 10 and x + y = 4. What is the value of x^3 + y^3?", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has side length 6. A point is chosen at random inside the hexagon. What is the probability that its distance from the center is less than 3?", "role": "user" } ]
[ { "content": "Question:\nLet a, b, and c be positive integers such that a + b + c = 15 and a < b < c. How many ordered triples (a, b, c) satisfy these conditions?", "role": "user" } ]
[ { "content": "Question:\nIf x and y are real numbers such that x + y = 7 and x^2 + y^2 = 37, find the value of xy.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all positive integers n less than 100 such that n^2 + 2n + 2 is a perfect square. Find the sum of all elements of S.", "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:\nHow many integers between 1 and 1000 (inclusive) are divisible by neither 2 nor 5?", "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:\nA regular hexagon has side length 4. What is the area of the region inside the hexagon that is closer to the center than to any of the sides?", "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 integers between 1 and 1000 inclusive that are not perfect squares. What is the sum of the elements of S?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, angle C is a right angle, and the lengths of the sides are integers with AB = 17. What is the number of possible distinct values for the length of BC?", "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:\nA regular hexagon has vertices A, B, C, D, E, and F in order. If points P and Q are chosen randomly and independently from the set of vertices, what is the probability that segment PQ is a diagonal of the hexagon?", "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:\nLet a, b, and c be positive real numbers such that abc = 8. Find the minimum value of a + 2b + 4c.", "role": "user" } ]

[ { "content": "Question:\nThe sum of the first n positive integers is equal to 2016. What is the value of n?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all two-digit positive integers whose digits sum to 10. Find the sum of all elements in S.", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, AB = 5, AC = 7, and the area of triangle ABC is 14. Find the length of BC.", "role": "user" } ]

[ { "content": "Question:\nHow many distinct 4-digit numbers are there such that the sum of their digits is 9?", "role": "user" } ]

[ { "content": "Question:\nLet n be the smallest positive integer such that both n and n+1 are divisible by either 2, 3, or 5. Find n.", "role": "user" } ]

[ { "content": "Question:\nLet x be a real number such that x^2 + 4x + 7 = 0. Find the value of x^4 + 8x^3 + 30x^2 + 56x + 49.", "role": "user" } ]

[ { "content": "Question:\nLet n be the smallest positive integer such that the sum of the digits of n is 2024. What is the sum of the digits of n + 1?", "role": "user" } ]

[ { "content": "Question:\nA regular hexagon has side length 2. A point is chosen at random inside the hexagon. What is the probability that its distance to the center is less than 1?", "role": "user" } ]

[ { "content": "Question:\nFind the number of positive integers less than 1000 that are divisible by neither 2 nor 5.", "role": "user" } ]

[ { "content": "Question:\nLet x and y be real numbers such that x + y = 6 and x^2 + y^2 = 20. Compute all possible values of xy.", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all four-digit numbers that use each of the digits 1, 2, 3, and 4 exactly once. Compute the sum of all numbers in S.", "role": "user" } ]

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

[ { "content": "Question:\nHow many positive integers n less than 100 are there such that n^2 + 2n + 1 is a perfect square?", "role": "user" } ]

[ { "content": "Question:\nA regular hexagon has vertices labeled A, B, C, D, E, F in order. Let P be the intersection point of diagonals AC and BD. If the side length of the hexagon is 6, what is the distance from P to vertex E?", "role": "user" } ]

[ { "content": "Question:\nHow many distinct four-digit numbers can be formed using the digits 1, 2, 3, 4, and 5, each at most once, such that the number is even?", "role": "user" } ]

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

[ { "content": "Question:\nLet a positive integer n have exactly 6 positive divisors, including 1 and n. How many such integers n less than 100 are there?", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, angle BAC = 60°, AB = 8, and AC = 6. Find the length of BC.", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all positive integers less than 1000 that are divisible by 6 but not by 9. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nLet a and b be real numbers such that a + b = 5 and a^2 + b^2 = 13. Find the value of ab.", "role": "user" } ]

[ { "content": "Question:\nFind the smallest positive integer n such that when n is divided by 7, the remainder is 4, and when n is divided by 9, the remainder is 5.", "role": "user" } ]

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

[ { "content": "Question:\nA regular hexagon has side length 6. A point P is chosen uniformly at random inside the hexagon. What is the probability that the distance from P to the center of the hexagon is less than 3?", "role": "user" } ]

[ { "content": "Question:\nLet S be the sum of all positive integers less than 100 that leave a remainder of 2 when divided by 7. Find the value of S.", "role": "user" } ]

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

[ { "content": "Question:\nIf x and y are real numbers satisfying x + y = 7 and x^2 + y^2 = 37, compute the value of x^3 + y^3.", "role": "user" } ]

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

[ { "content": "Question:\nIn triangle ABC, angle C is a right angle and AB = 13, AC = 5. Find the length of BC.", "role": "user" } ]

[ { "content": "Question:\nA fair six-sided die is rolled three times. What is the probability that the numbers rolled are in strictly increasing order?", "role": "user" } ]

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

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

[ { "content": "Question:\nLet x be a positive real number such that x + 1/x = 5. Find the value of x^3 + 1/x^3.", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, AB = 13, AC = 15, and the altitude from A to BC has length 12. Find the area of triangle ABC.", "role": "user" } ]

[ { "content": "Question:\nHow many integers n between 1 and 1000 (inclusive) are divisible by neither 2 nor 5?", "role": "user" } ]

[ { "content": "Question:\nLet S be the sum of all integers k such that 1 ≤ k ≤ 100 and the remainder when k is divided by 7 is 3. Find S.", "role": "user" } ]

[ { "content": "Question:\nLet a and b be positive integers such that a^2 + b^2 = 2024. Find the number of ordered pairs (a, b) that satisfy this equation.", "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 + y?", "role": "user" } ]

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

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

[ { "content": "Question:\nLet S be the set of all positive integers less than 100 that are relatively prime to 100. What is the sum of the elements of S?", "role": "user" } ]

[ { "content": "Question:\nEvaluate the sum \\( \\sum_{n=1}^{10} (2n-1)^2 \\).", "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:\nIn triangle ABC, AB = AC and the altitude from A to BC is 6. If BC = 10, find the area of triangle ABC.", "role": "user" } ]

[ { "content": "Question:\nFind the number of ordered triples (x, y, z) of positive integers such that x + y + z = 12.", "role": "user" } ]

[ { "content": "Question:\nLet a and b be positive integers such that a^2 + b^2 = 2024. Find the number of such pairs (a, b).", "role": "user" } ]

[ { "content": "Question:\nA fair six-sided die is rolled four times. What is the probability that the sum of the results is 14?", "role": "user" } ]

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

[ { "content": "Question:\nA regular hexagon and a regular pentagon have equal perimeters. If the length of a side of the hexagon is 6, what is the length of a side of the pentagon?", "role": "user" } ]

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

[ { "content": "Question:\nLet S be the set of integers from 1 to 50 inclusive. How many subsets of S contain exactly 3 elements whose sum is divisible by 3?", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, AB = 13, AC = 15, and BC = 14. Find the length of the altitude from A to BC.", "role": "user" } ]

[ { "content": "Question:\nLet n be a positive integer such that the sum of the digits of n is 2019. What is the smallest possible value of n?", "role": "user" } ]

[ { "content": "Question:\nA circle passes through the points (0,0), (6,0), and (2,4). What is the radius of the circle?", "role": "user" } ]

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

[ { "content": "Question:\nLet f(x) = x^2 + bx + c, where b and c are integers. If f(1) = 7 and f(3) = 15, what is f(5)?", "role": "user" } ]

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

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

[ { "content": "Question:\nTriangle ABC has side lengths AB = 13, BC = 14, and AC = 15. Let D be the foot of the altitude from B to AC. What is the length of BD?", "role": "user" } ]

[ { "content": "Question:\nHow many positive integers n less than 100 satisfy that n^2 + 4n is divisible by 5?", "role": "user" } ]

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

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

[ { "content": "Question:\nLet S be the set of positive integers less than 1000 that are divisible by 3 or 7 but not by 21. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nA triangle has side lengths 7, 8, and 9. What is the area of the triangle?", "role": "user" } ]

[ { "content": "Question:\nHow many positive integers less than 1000 can be written as the sum of two distinct perfect squares?", "role": "user" } ]

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

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

[ { "content": "Question:\nLet x and y be real numbers such that x + y = 7 and x^2 + y^2 = 29. What is the value of xy?", "role": "user" } ]

[ { "content": "Question:\nA regular hexagon has side length 2. A point is chosen at random inside the hexagon. What is the probability that the point is closer to the center than to any of the hexagon's sides? Express your answer as a common fraction.", "role": "user" } ]

[ { "content": "Question:\nLet S be the sum of all positive integers n less than 100 such that n^2 + 1 is divisible by 5. Find S.", "role": "user" } ]

[ { "content": "Question:\nIf a and b are integers such that 3a + 7b = 100, how many ordered pairs (a, b) satisfy this equation?", "role": "user" } ]

[ { "content": "Question:\nLet f(x) = x^4 - 4x^2 + 3. Find the sum of all real solutions to the equation f(x) = 0.", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all positive integers less than 100 that are divisible by either 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 = 7 and x^2 + y^2 = 37. Compute the value of xy.", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, angle C is a right angle, and AC = 6, BC = 8. Find the radius of the circle inscribed in triangle ABC.", "role": "user" } ]

[ { "content": "Question:\nFind the number of positive integers less than 1000 that leave a remainder of 2 when divided by 5 and a remainder of 3 when divided by 7.", "role": "user" } ]

[ { "content": "Question:\nLet S be the sum of all integers n between 1 and 100 inclusive such that n^2 + n is divisible by 6. Compute S.", "role": "user" } ]

[ { "content": "Question:\nLet x and y be real numbers such that x + y = 8 and x^2 + y^2 = 34. Find 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 right triangle has integer side lengths and a hypotenuse of length 65. How many such triangles exist?", "role": "user" } ]

[ { "content": "Question:\nThe sum of the first n positive integers is equal to the sum of the next k consecutive positive integers. If n > 1 and k > 1, find the smallest possible value of n.", "role": "user" } ]

[ { "content": "Question:\nLet a and b be positive integers such that the greatest common divisor of a^2 + b^2 and a + b is 13. Find the smallest possible value of a + b.", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of positive integers less than 100 that are divisible by either 3 or 5 but not both. How many elements are in S?", "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 integers are there such that the sum of their digits is 23?", "role": "user" } ]

[ { "content": "Question:\nLet x and y be real numbers such that x^2 + y^2 = 10 and x + y = 4. What is the value of x^3 + y^3?", "role": "user" } ]

[ { "content": "Question:\nA regular hexagon has side length 6. A point is chosen at random inside the hexagon. What is the probability that its distance from the center is less than 3?", "role": "user" } ]

[ { "content": "Question:\nLet a, b, and c be positive integers such that a + b + c = 15 and a < b < c. How many ordered triples (a, b, c) satisfy these conditions?", "role": "user" } ]

[ { "content": "Question:\nIf x and y are real numbers such that x + y = 7 and x^2 + y^2 = 37, find the value of xy.", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all positive integers n less than 100 such that n^2 + 2n + 2 is a perfect square. Find the sum of all elements of S.", "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:\nHow many integers between 1 and 1000 (inclusive) are divisible by neither 2 nor 5?", "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:\nA regular hexagon has side length 4. What is the area of the region inside the hexagon that is closer to the center than to any of the sides?", "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 integers between 1 and 1000 inclusive that are not perfect squares. What is the sum of the elements of S?", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, angle C is a right angle, and the lengths of the sides are integers with AB = 17. What is the number of possible distinct values for the length of BC?", "role": "user" } ]