simonycl/amc_aime_training_positive_sequence_cot_gpt-4.1-prompt

messages listlengths 1 1
[ { "content": "Question:\nLet n be the smallest positive integer such that the decimal representation of 7n ends with the digits 2024. What is the value of n?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 7, AC = 9, and BC = 10. The angle bisector of angle BAC meets BC at D. What is the length of BD?", "role": "user" } ]
[ { "content": "Question:\nHow many ordered pairs of integers (x, y) satisfy x^2 + y^2 = 100?", "role": "user" } ]
[ { "content": "Question:\nLet S be the sum of all positive integers less than 1000 that are divisible by neither 2 nor 5. Find S.", "role": "user" } ]
[ { "content": "Question:\nIf x, y, and z are real numbers such that x + y + z = 6 and x^2 + y^2 + z^2 = 20, what is the maximum value of x?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that 2n + 1 is divisible by 13. Find the value of n.", "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 all positive integers less than 1000 that are divisible by either 3 or 5 but not both. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nGiven that x and y are real numbers such that x^2 + y^2 = 10 and x + y = 4, compute the value of x^3 + y^3.", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that the sum 1^2 + 2^2 + ... + n^2 is divisible by 2016. 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 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 n such that the sum of the digits of n is 10 and n < 100. Find the sum of all elements in S.", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, angle A is 60°, AB = 10, and AC = 6. If BC = x, what is the value of x^2?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that n! is divisible by 10000. Find n.", "role": "user" } ]
[ { "content": "Question:\nLet x be a real number such that x^2 - 6x + 10 = 0. What is the value of x^4 + x^2?", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit integers have the property that the sum of their digits is 10?", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has sides of length 2. What is the area of the region inside the hexagon but outside the circle of radius 1 centered at the hexagon's center?", "role": "user" } ]
[ { "content": "Question:\nFind the smallest positive integer n such that 2^n ends with the digits 888.", "role": "user" } ]
[ { "content": "Question:\nIf a and b are real numbers such that a^2 + b^2 = 13 and a + b = 3, find the value of ab.", "role": "user" } ]
[ { "content": "Question:\nIf x and y are real numbers such that x + y = 7 and x^2 + y^2 = 37, what is the value of xy?", "role": "user" } ]
[ { "content": "Question:\nA triangle has side lengths 7, 8, and x, where x is a positive integer. For how many values of x is the area of the triangle also an integer?", "role": "user" } ]
[ { "content": "Question:\nWhat is the sum of all positive integers n less than 100 for which n^2 + 3n + 2 is divisible by 5?", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit numbers are there such that the sum of their digits is 21?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all integers between 1 and 1000 inclusive that are divisible by 3 or 5 but not both. How many elements are in S?", "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 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 xy.", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has side length 2. What is the area of the region inside the hexagon but outside the circle of radius 1 centered at the hexagon's center?", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit positive integers have digits that sum to 10?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that the decimal representation of 3^n ends with the digits 001. Find n.", "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:\nLet ABC be a triangle with AB = 13, BC = 14, and AC = 15. The angle bisector of angle BAC meets BC at D. What is the length of BD?", "role": "user" } ]
[ { "content": "Question:\nA fair coin is tossed 10 times. What is the probability that exactly 4 of the tosses result in heads and no two heads are consecutive? Express your answer as a common fraction.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 10 and x^2 + y^2 = 58. What is the value of \|x - y\|?", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 3 nor 5?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers satisfying x^2 + y^2 = 10 and x + y = 2. What is the value of x^3 + y^3?", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers n less than 1000 have the property that the sum of the digits of n is 10?", "role": "user" } ]
[ { "content": "Question:\nLet triangle ABC have AB = 13, AC = 15, and the area of triangle ABC is 84. Find the length of BC.", "role": "user" } ]
[ { "content": "Question:\nA fair coin is flipped 8 times. What is the probability that no two consecutive flips are both heads?", "role": "user" } ]
[ { "content": "Question:\nWhat is the sum of all positive integers k less than 100 such that k^2 + 6k + 5 is a perfect square?", "role": "user" } ]
[ { "content": "Question:\nWhat is the sum of all positive integers less than 1000 that are divisible by either 3 or 5?", "role": "user" } ]
[ { "content": "Question:\nLet f(x) = x^2 - 6x + 10. Find the minimum value of f(x) for real numbers x.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all four-digit numbers whose digits are strictly increasing from left to right. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nA triangle has sides of lengths 5, 12, and 13. Find the radius of the circle inscribed in the triangle.", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers n less than 100 satisfy that both n and n+1 are divisible by either 2 or 3?", "role": "user" } ]
[ { "content": "Question:\nLet the sum of all positive integers less than 100 that are divisible by neither 2 nor 5 be S. Find the value of S.", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has side length 6. Find the area of the region inside the hexagon but outside the circle of radius 3 centered at the hexagon's center.", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 have digits that sum to 9?", "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:\nLet a, b, and c be the sides of a triangle such that a + b + c = 24 and ab + bc + ca = 128. If a, b, and c are positive integers, find the value of c.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 7 and x^2 + y^2 = 37. What is the value of xy?", "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 four-digit positive 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 + 3 + ... + n is greater than 1000. Find n.", "role": "user" } ]
[ { "content": "Question:\nThe sum of the squares of three consecutive positive integers is 365. What is the largest of these integers?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all positive integers less than 1000 that are divisible by 7 but not by 5. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nLet x be a real number such that x^2 - 6x + 8 = 0. What is the value of (x-2)^3 + (x-4)^3?", "role": "user" } ]
[ { "content": "Question:\nLet a and b be integers such that 2a + 3b = 100 and 0 ≤ a ≤ 50. How many ordered pairs (a, b) satisfy this equation?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, the angle at C is a right angle. If AC = 5 and BC = 12, what is the length of the altitude from C to hypotenuse AB?", "role": "user" } ]
[ { "content": "Question:\nA bag contains 6 red, 5 blue, and 4 green balls. If three balls are drawn at random without replacement, what is the probability that all three are of different colors? Express your answer as a simplified fraction.", "role": "user" } ]
[ { "content": "Question:\nLet x be a real number such that x^2 + 4x = 21. What is the value of (x + 2)^2?", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has a side length of 2. What is the area of the triangle formed by connecting every other vertex of the hexagon?", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit integers are divisible by 9 and have all digits distinct?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of positive integers less than 1000 that are perfect squares and also perfect cubes. 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 maximum possible value of ab?", "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 xy?", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has a side length of 4. What is the area of the region inside the hexagon but outside the circle inscribed within it?", "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 sum of all three-digit numbers whose digits are all odd. What is the value of S?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, angle BAC = 60°, side AB = 5, and side AC = 7. What is 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 bag contains 6 red balls and 4 blue balls. Two balls are drawn at random without replacement. What is the probability that both balls are of the same color? Express your answer as a reduced fraction.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of integers n such that 1 ≤ n ≤ 100 and n is divisible by 3 or 5 but not both. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 13, AC = 15, and BC = 14. What is the area of triangle ABC?", "role": "user" } ]
[ { "content": "Question:\nLet N be the number of positive integers less than 1000 that are divisible by neither 2 nor 5. Find N.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be positive integers such that x + y = 20 and the greatest common divisor of x and y is 4. How many ordered pairs (x, y) satisfy these conditions?", "role": "user" } ]
[ { "content": "Question:\nLet ABC be a triangle with AB = 8, AC = 6, and the area of triangle ABC is 24. Find the length of BC.", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit numbers are divisible by 9 and have all digits distinct?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all positive integers less than 1000 that are divisible by 7 but not by 5. Find the sum of the elements of S.", "role": "user" } ]
[ { "content": "Question:\nSuppose a sequence a_1, a_2, a_3, ..., a_n is defined by a_1 = 2, and a_{k+1} = 3a_k + 2 for all k ≥ 1. Find the smallest n such that a_n > 1000.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 8 and x^2 + y^2 = 34. Compute 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 triangle has side lengths 7, 24, and x, where x is a positive integer. For how many values of x does the triangle have an integer area?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that the decimal representation of 2024^n ends with the digits 024. Find n.", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, angle BAC = 60°, AB = 5, and AC = 7. Find the length of BC in simplest radical form.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all integers between 1 and 1000 inclusive that are divisible by neither 2 nor 5. How many elements does S have?", "role": "user" } ]
[ { "content": "Question:\nA triangle has side lengths 13, 14, and 15. Compute the area of the triangle.", "role": "user" } ]
[ { "content": "Question:\nFind the number of positive integers less than 1000 that can be expressed as the sum of exactly three consecutive positive integers.", "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:\nThe sum of all positive integers n such that 1 ≤ n ≤ 100 and n^2 + 4 is divisible by n + 2 can be written as N. Find the value of N.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 10 and x^2 + y^2 = 58. What is the value of xy?", "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 all positive integers less than 1000 that are multiples of both 4 and 6 but not of 9. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nThe sum of the squares of three consecutive positive integers is 365. What is the smallest of these integers?", "role": "user" } ]
[ { "content": "Question:\nFind the number of ordered quadruples (a, b, c, d) of positive integers such that a + b + c + d = 12 and a ≥ 2, b ≥ 3, c ≥ 1, d ≥ 1.", "role": "user" } ]
[ { "content": "Question:\nA positive integer n leaves a remainder of 3 when divided by 8 and a remainder of 4 when divided by 11. What is the smallest possible value of n?", "role": "user" } ]
[ { "content": "Question:\nLet triangle ABC have AB = 13, BC = 14, and AC = 15. The altitude from B to AC meets AC at point D. What is the length of BD?", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit positive integers are there whose digits sum to 10?", "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:\nFind the number of ordered triples of positive integers (a, b, c) such that a < b < c and a + b + c = 15.", "role": "user" } ]

[ { "content": "Question:\nLet n be the smallest positive integer such that the decimal representation of 7n ends with the digits 2024. What is the value of n?", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, AB = 7, AC = 9, and BC = 10. The angle bisector of angle BAC meets BC at D. What is the length of BD?", "role": "user" } ]

[ { "content": "Question:\nHow many ordered pairs of integers (x, y) satisfy x^2 + y^2 = 100?", "role": "user" } ]

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

[ { "content": "Question:\nIf x, y, and z are real numbers such that x + y + z = 6 and x^2 + y^2 + z^2 = 20, what is the maximum value of x?", "role": "user" } ]

[ { "content": "Question:\nLet n be the smallest positive integer such that 2n + 1 is divisible by 13. Find the value of n.", "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 all positive integers less than 1000 that are divisible by either 3 or 5 but not both. How many elements are in S?", "role": "user" } ]

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

[ { "content": "Question:\nLet n be the smallest positive integer such that the sum 1^2 + 2^2 + ... + n^2 is divisible by 2016. 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 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 n such that the sum of the digits of n is 10 and n < 100. Find the sum of all elements in S.", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, angle A is 60°, AB = 10, and AC = 6. If BC = x, what is the value of x^2?", "role": "user" } ]

[ { "content": "Question:\nLet n be the smallest positive integer such that n! is divisible by 10000. Find n.", "role": "user" } ]

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

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

[ { "content": "Question:\nA regular hexagon has sides of length 2. What is the area of the region inside the hexagon but outside the circle of radius 1 centered at the hexagon's center?", "role": "user" } ]

[ { "content": "Question:\nFind the smallest positive integer n such that 2^n ends with the digits 888.", "role": "user" } ]

[ { "content": "Question:\nIf a and b are real numbers such that a^2 + b^2 = 13 and a + b = 3, find the value of ab.", "role": "user" } ]

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

[ { "content": "Question:\nA triangle has side lengths 7, 8, and x, where x is a positive integer. For how many values of x is the area of the triangle also an integer?", "role": "user" } ]

[ { "content": "Question:\nWhat is the sum of all positive integers n less than 100 for which n^2 + 3n + 2 is divisible by 5?", "role": "user" } ]

[ { "content": "Question:\nHow many four-digit numbers are there such that the sum of their digits is 21?", "role": "user" } ]

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

[ { "content": "Question:\nA regular hexagon has side length 2. What is the area of the region inside the hexagon but outside the circle of radius 1 centered at the hexagon's center?", "role": "user" } ]

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

[ { "content": "Question:\nLet n be the smallest positive integer such that the decimal representation of 3^n ends with the digits 001. Find n.", "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:\nLet ABC be a triangle with AB = 13, BC = 14, and AC = 15. The angle bisector of angle BAC meets BC at D. What is the length of BD?", "role": "user" } ]

[ { "content": "Question:\nA fair coin is tossed 10 times. What is the probability that exactly 4 of the tosses result in heads and no two heads are consecutive? Express your answer as a common fraction.", "role": "user" } ]

[ { "content": "Question:\nLet x and y be real numbers such that x + y = 10 and x^2 + y^2 = 58. What is the value of |x - y|?", "role": "user" } ]

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

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

[ { "content": "Question:\nHow many positive integers n less than 1000 have the property that the sum of the digits of n is 10?", "role": "user" } ]

[ { "content": "Question:\nLet triangle ABC have AB = 13, AC = 15, and the area of triangle ABC is 84. Find the length of BC.", "role": "user" } ]

[ { "content": "Question:\nA fair coin is flipped 8 times. What is the probability that no two consecutive flips are both heads?", "role": "user" } ]

[ { "content": "Question:\nWhat is the sum of all positive integers k less than 100 such that k^2 + 6k + 5 is a perfect square?", "role": "user" } ]

[ { "content": "Question:\nWhat is the sum of all positive integers less than 1000 that are divisible by either 3 or 5?", "role": "user" } ]

[ { "content": "Question:\nLet f(x) = x^2 - 6x + 10. Find the minimum value of f(x) for real numbers x.", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all four-digit numbers whose digits are strictly increasing from left to right. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nA triangle has sides of lengths 5, 12, and 13. Find the radius of the circle inscribed in the triangle.", "role": "user" } ]

[ { "content": "Question:\nHow many positive integers n less than 100 satisfy that both n and n+1 are divisible by either 2 or 3?", "role": "user" } ]

[ { "content": "Question:\nLet the sum of all positive integers less than 100 that are divisible by neither 2 nor 5 be S. Find the value of S.", "role": "user" } ]

[ { "content": "Question:\nA regular hexagon has side length 6. Find the area of the region inside the hexagon but outside the circle of radius 3 centered at the hexagon's center.", "role": "user" } ]

[ { "content": "Question:\nHow many positive integers less than 1000 have digits that sum to 9?", "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:\nLet a, b, and c be the sides of a triangle such that a + b + c = 24 and ab + bc + ca = 128. If a, b, and c are positive integers, find the value of c.", "role": "user" } ]

[ { "content": "Question:\nLet x and y be real numbers such that x + y = 7 and x^2 + y^2 = 37. What is the value of xy?", "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 four-digit positive 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 + 3 + ... + n is greater than 1000. Find n.", "role": "user" } ]

[ { "content": "Question:\nThe sum of the squares of three consecutive positive integers is 365. What is the largest of these integers?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all positive integers less than 1000 that are divisible by 7 but not by 5. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nLet x be a real number such that x^2 - 6x + 8 = 0. What is the value of (x-2)^3 + (x-4)^3?", "role": "user" } ]

[ { "content": "Question:\nLet a and b be integers such that 2a + 3b = 100 and 0 ≤ a ≤ 50. How many ordered pairs (a, b) satisfy this equation?", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, the angle at C is a right angle. If AC = 5 and BC = 12, what is the length of the altitude from C to hypotenuse AB?", "role": "user" } ]

[ { "content": "Question:\nA bag contains 6 red, 5 blue, and 4 green balls. If three balls are drawn at random without replacement, what is the probability that all three are of different colors? Express your answer as a simplified fraction.", "role": "user" } ]

[ { "content": "Question:\nLet x be a real number such that x^2 + 4x = 21. What is the value of (x + 2)^2?", "role": "user" } ]

[ { "content": "Question:\nA regular hexagon has a side length of 2. What is the area of the triangle formed by connecting every other vertex of the hexagon?", "role": "user" } ]

[ { "content": "Question:\nHow many four-digit integers are divisible by 9 and have all digits distinct?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of positive integers less than 1000 that are perfect squares and also perfect cubes. 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 maximum possible value of ab?", "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 xy?", "role": "user" } ]

[ { "content": "Question:\nA regular hexagon has a side length of 4. What is the area of the region inside the hexagon but outside the circle inscribed within it?", "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 sum of all three-digit numbers whose digits are all odd. What is the value of S?", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, angle BAC = 60°, side AB = 5, and side AC = 7. What is 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 bag contains 6 red balls and 4 blue balls. Two balls are drawn at random without replacement. What is the probability that both balls are of the same color? Express your answer as a reduced fraction.", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of integers n such that 1 ≤ n ≤ 100 and n is divisible by 3 or 5 but not both. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, AB = 13, AC = 15, and BC = 14. What is the area of triangle ABC?", "role": "user" } ]

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

[ { "content": "Question:\nLet x and y be positive integers such that x + y = 20 and the greatest common divisor of x and y is 4. How many ordered pairs (x, y) satisfy these conditions?", "role": "user" } ]

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

[ { "content": "Question:\nHow many four-digit numbers are divisible by 9 and have all digits distinct?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all positive integers less than 1000 that are divisible by 7 but not by 5. Find the sum of the elements of S.", "role": "user" } ]

[ { "content": "Question:\nSuppose a sequence a_1, a_2, a_3, ..., a_n is defined by a_1 = 2, and a_{k+1} = 3a_k + 2 for all k ≥ 1. Find the smallest n such that a_n > 1000.", "role": "user" } ]

[ { "content": "Question:\nLet x and y be real numbers such that x + y = 8 and x^2 + y^2 = 34. Compute 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 triangle has side lengths 7, 24, and x, where x is a positive integer. For how many values of x does the triangle have an integer area?", "role": "user" } ]

[ { "content": "Question:\nLet n be the smallest positive integer such that the decimal representation of 2024^n ends with the digits 024. Find n.", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, angle BAC = 60°, AB = 5, and AC = 7. Find the length of BC in simplest radical form.", "role": "user" } ]

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

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

[ { "content": "Question:\nFind the number of positive integers less than 1000 that can be expressed as the sum of exactly three consecutive positive integers.", "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:\nThe sum of all positive integers n such that 1 ≤ n ≤ 100 and n^2 + 4 is divisible by n + 2 can be written as N. Find the value of N.", "role": "user" } ]

[ { "content": "Question:\nLet x and y be real numbers such that x + y = 10 and x^2 + y^2 = 58. What is the value of xy?", "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 all positive integers less than 1000 that are multiples of both 4 and 6 but not of 9. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nThe sum of the squares of three consecutive positive integers is 365. What is the smallest of these integers?", "role": "user" } ]

[ { "content": "Question:\nFind the number of ordered quadruples (a, b, c, d) of positive integers such that a + b + c + d = 12 and a ≥ 2, b ≥ 3, c ≥ 1, d ≥ 1.", "role": "user" } ]

[ { "content": "Question:\nA positive integer n leaves a remainder of 3 when divided by 8 and a remainder of 4 when divided by 11. What is the smallest possible value of n?", "role": "user" } ]

[ { "content": "Question:\nLet triangle ABC have AB = 13, BC = 14, and AC = 15. The altitude from B to AC meets AC at point D. What is the length of BD?", "role": "user" } ]

[ { "content": "Question:\nHow many four-digit positive integers are there whose digits sum to 10?", "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:\nFind the number of ordered triples of positive integers (a, b, c) such that a < b < c and a + b + c = 15.", "role": "user" } ]