simonycl/amc_aime_training_positive_sequence_cot_gpt-4.1-prompt

messages listlengths 1 1
[ { "content": "Question:\nLet S be the set of 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:\nIn triangle ABC, AB = 7, AC = 8, and BC = 9. What is the area of triangle ABC?", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers n less than or equal to 1000 satisfy the condition that both n and n+1 are perfect squares?", "role": "user" } ]
[ { "content": "Question:\nLet x be a real number such that x + 1/x = 4. Find the value of x^5 + 1/x^5.", "role": "user" } ]
[ { "content": "Question:\nSuppose a fair coin is flipped 8 times. What is the probability that no two consecutive flips are both heads? Express your answer as a common fraction.", "role": "user" } ]
[ { "content": "Question:\nLet f(x) = x^2 - 10x + 29. Find all real values x such that f(f(x)) = 0.", "role": "user" } ]
[ { "content": "Question:\nA circle is inscribed in a square. If the area of the circle is 25π, what is the perimeter of the square?", "role": "user" } ]
[ { "content": "Question:\nHow many integers between 1 and 1000 (inclusive) are divisible by neither 2 nor 5?", "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:\nIf the sum of the first n positive integers is 378, find the value of n.", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that the sum 1 + 2 + 3 + ... + n exceeds 1000. Find n.", "role": "user" } ]
[ { "content": "Question:\nLet a, b, and c be positive real numbers such that a + b + c = 6 and ab + bc + ca = 9. Find the maximum possible value of abc.", "role": "user" } ]
[ { "content": "Question:\nA fair coin is flipped 7 times. What is the probability that no two consecutive flips result in heads? Express your answer as a common fraction.", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, angle ABC = 90°, AB = 6, and BC = 8. A circle is inscribed in triangle ABC. Find the radius of the inscribed circle.", "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 100 that are divisible by either 2 or 5 but not both. How many elements does S contain?", "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 triangle has side lengths 5, 12, and 13. What is the area of the triangle?", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit positive integers have digits that sum to 6?", "role": "user" } ]
[ { "content": "Question:\nEvaluate the sum \\( \\sum_{k=1}^{5} \\frac{1}{k(k+1)} \\).", "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 x*y.", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 2 nor 5?", "role": "user" } ]
[ { "content": "Question:\nLet triangle ABC have AB = 13, BC = 14, and AC = 15. Let P be the point inside the triangle such that the distances from P to AB, BC, and AC are all equal. Find the distance from P to each side.", "role": "user" } ]
[ { "content": "Question:\nHow many ordered triples (x, y, z) of positive integers satisfy x + y + z = 12 and x ≥ 2, y ≥ 3, and z ≥ 4?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of integers from 1 to 20 inclusive. If two distinct integers are chosen at random from S, what is the probability that their sum is divisible by 5? Express your answer as a reduced fraction.", "role": "user" } ]
[ { "content": "Question:\nLet the sum of the squares of three consecutive integers be 365. What is the middle integer?", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has side length 6. What is the area of the triangle formed by connecting every other vertex of the hexagon?", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by exactly two distinct prime numbers?", "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 ordered pairs of integers (a, b) satisfy 1 ≤ a < b ≤ 20 such that a + b is divisible by 5?", "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:\nTriangle ABC has AB = 13, BC = 14, and CA = 15. Find the length of the altitude from B to AC.", "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:\nA fair 6-sided die is rolled four times. What is the probability that at least two of the rolls show the same number? Express your answer as a common fraction.", "role": "user" } ]
[ { "content": "Question:\nLet x be a real number such that x^3 - 3x + 1 = 0. Find the value of x^4 - 4x^2 + 6.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be positive integers such that x^2 + y^2 = 2024. Find the number of ordered pairs (x, y) that satisfy this equation.", "role": "user" } ]
[ { "content": "Question:\nThe sum of three consecutive positive integers is equal to the sum of the squares of the next two consecutive integers. What is the smallest such integer?", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has side length 6. Find the area of the region inside the hexagon but outside the circle inscribed in 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 integers from 1 to 100 inclusive. Find the number of subsets of S whose elements sum to a multiple of 5.", "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 x^3 + y^3?", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers n less than 1000 are divisible by neither 3 nor 5?", "role": "user" } ]
[ { "content": "Question:\nA circle with center O has diameter AB = 10. Point C lies on the circle such that triangle ABC has area 20. What is the length of AC?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of integers between 1 and 1000 (inclusive) that are not divisible by 2, 3, or 5. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nFind the smallest positive integer n such that the sum 1 + 2 + ... + n is divisible by 100.", "role": "user" } ]
[ { "content": "Question:\nLet $x$ be a real number such that $x^2 + 4x = 21$. What is the value of $x^3 + 8x$?", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has side length $6$. What is the area of the triangle formed by connecting 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 $a, b, c$ be positive real numbers such that $abc = 8$ and $a + b + c = 6$. Find the minimum value of $a^2 + b^2 + c^2$.", "role": "user" } ]
[ { "content": "Question:\nSuppose the sum of the digits of a positive integer $N$ is $2024$. What is the remainder when $N$ is divided by $9$?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that 3n is a perfect square and 2n is a perfect cube. Find the value of n.", "role": "user" } ]
[ { "content": "Question:\nA triangle has side lengths 6, 8, and x, where x is an integer. How many possible values of x are there?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all 3-digit positive integers whose digits are all odd. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nEvaluate the sum \\( \\sum_{k=1}^{4} (k^3 + 3k^2 + 3k + 1) \\).", "role": "user" } ]
[ { "content": "Question:\nLet a and b be positive integers such that a^2 + b^2 = 2025. What is the value of a + b?", "role": "user" } ]
[ { "content": "Question:\nLet the sum of the squares of three consecutive positive integers be 365. What is the smallest of these integers?", "role": "user" } ]
[ { "content": "Question:\nA circle has center O and a chord AB with length 8. If the distance from O to AB is 3, what is the radius of the circle?", "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 = 7 and x^2 + y^2 = 37. Find the value of xy.", "role": "user" } ]
[ { "content": "Question:\nHow many ways are there to arrange the letters of the word 'PROBLEM' so that no two vowels are adjacent?", "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 circle passes through the points (0,0) and (6,0) and is tangent to the line y = 4. What is the radius of the circle?", "role": "user" } ]
[ { "content": "Question:\nIf the sum of the positive divisors of 60 is S, what is S?", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit numbers have digits that sum to 10?", "role": "user" } ]
[ { "content": "Question:\nA fair coin is flipped 8 times. What is the probability that heads never occurs on two consecutive flips? Express your answer as a common fraction.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 7 and x^2 + y^2 = 37. Find the value of x·y.", "role": "user" } ]
[ { "content": "Question:\nA positive integer n leaves a remainder of 2 when divided by 5 and a remainder of 3 when divided by 7. What is the smallest such n?", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit positive integers have digits that sum to 10?", "role": "user" } ]
[ { "content": "Question:\nLet triangle ABC have AB = 10, AC = 8, and angle BAC = 60°. Find the area of triangle ABC.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all permutations of the numbers 1, 2, 3, 4, 5. How many of these permutations have exactly two fixed points?", "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:\nIn a group of 10 students, how many ways can a committee of 4 be chosen such that no two committee members have consecutive student numbers (labeled 1 through 10)?", "role": "user" } ]
[ { "content": "Question:\nA circle passes through the vertices A and B of a square ABCD and is tangent to side CD at its midpoint. If the side length of the square is 8, what is the radius of the circle?", "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 remainder when n is divided by 9?", "role": "user" } ]
[ { "content": "Question:\nIf sin θ + cos θ = \\frac{5}{4}, where 0 < θ < \\frac{π}{2}, what is the value of sin 2θ?", "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 integers n between 1 and 1000 (inclusive) are divisible by neither 2 nor 5?", "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:\nA fair six-sided die is rolled three times. What is the probability that the sum of the outcomes is exactly 10?", "role": "user" } ]
[ { "content": "Question:\nFind the smallest positive integer n such that the sum 1 + 2 + ... + n is divisible by 100.", "role": "user" } ]
[ { "content": "Question:\nLet n be a positive integer such that the sum of the digits of n is 23 and n is divisible by 9. What is the smallest possible value of n?", "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 are divisible by neither 2 nor 5?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all integers between 1 and 1000, inclusive. For how many values of n in S is the remainder when n^2 is divided by 7 equal to 1?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 4 and x^2 + y^2 = 20. Find the value of x^3 + y^3.", "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:\nA regular hexagon has a side length of 2. 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 sum of all positive integers n less than 100 for which n^2 + n + 1 is divisible by 7. Find S.", "role": "user" } ]
[ { "content": "Question:\nLet a and b be positive real numbers such that a + b = 6 and ab = 5. Compute the value of a^3 + b^3.", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that the decimal representation of 7n ends with three consecutive digits 777. Find n.", "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 positive integers less than 1000 are divisible by neither 2 nor 5?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of integers from 1 to 100, inclusive. How many subsets of S have a sum that is divisible by 5?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x^2 + y^2 = 20 and x + y = 6. Find the value of x^3 + y^3.", "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 x*y.", "role": "user" } ]
[ { "content": "Question:\nA circle passes through the vertices A(0,0) and B(6,0) of a triangle ABC and is tangent to the line x = 2. Find 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:\nThe sum of the first n positive integers is 2016. Find the value of n.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all ordered triples (a, b, c) of positive integers such that a + b + c = 10. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of 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:\nIn triangle ABC, AB = 7, AC = 8, and BC = 9. What is the area of triangle ABC?", "role": "user" } ]

[ { "content": "Question:\nHow many positive integers n less than or equal to 1000 satisfy the condition that both n and n+1 are perfect squares?", "role": "user" } ]

[ { "content": "Question:\nLet x be a real number such that x + 1/x = 4. Find the value of x^5 + 1/x^5.", "role": "user" } ]

[ { "content": "Question:\nSuppose a fair coin is flipped 8 times. What is the probability that no two consecutive flips are both heads? Express your answer as a common fraction.", "role": "user" } ]

[ { "content": "Question:\nLet f(x) = x^2 - 10x + 29. Find all real values x such that f(f(x)) = 0.", "role": "user" } ]

[ { "content": "Question:\nA circle is inscribed in a square. If the area of the circle is 25π, what is the perimeter of the square?", "role": "user" } ]

[ { "content": "Question:\nHow many integers between 1 and 1000 (inclusive) are divisible by neither 2 nor 5?", "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:\nIf the sum of the first n positive integers is 378, find the value of n.", "role": "user" } ]

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

[ { "content": "Question:\nLet a, b, and c be positive real numbers such that a + b + c = 6 and ab + bc + ca = 9. Find the maximum possible value of abc.", "role": "user" } ]

[ { "content": "Question:\nA fair coin is flipped 7 times. What is the probability that no two consecutive flips result in heads? Express your answer as a common fraction.", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, angle ABC = 90°, AB = 6, and BC = 8. A circle is inscribed in triangle ABC. Find the radius of the inscribed circle.", "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 100 that are divisible by either 2 or 5 but not both. How many elements does S contain?", "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 triangle has side lengths 5, 12, and 13. What is the area of the triangle?", "role": "user" } ]

[ { "content": "Question:\nHow many four-digit positive integers have digits that sum to 6?", "role": "user" } ]

[ { "content": "Question:\nEvaluate the sum \$ \\sum_{k=1}^{5} \\frac{1}{k(k+1)} \$.", "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 x*y.", "role": "user" } ]

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

[ { "content": "Question:\nLet triangle ABC have AB = 13, BC = 14, and AC = 15. Let P be the point inside the triangle such that the distances from P to AB, BC, and AC are all equal. Find the distance from P to each side.", "role": "user" } ]

[ { "content": "Question:\nHow many ordered triples (x, y, z) of positive integers satisfy x + y + z = 12 and x ≥ 2, y ≥ 3, and z ≥ 4?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of integers from 1 to 20 inclusive. If two distinct integers are chosen at random from S, what is the probability that their sum is divisible by 5? Express your answer as a reduced fraction.", "role": "user" } ]

[ { "content": "Question:\nLet the sum of the squares of three consecutive integers be 365. What is the middle integer?", "role": "user" } ]

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

[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by exactly two distinct prime numbers?", "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 ordered pairs of integers (a, b) satisfy 1 ≤ a < b ≤ 20 such that a + b is divisible by 5?", "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:\nTriangle ABC has AB = 13, BC = 14, and CA = 15. Find the length of the altitude from B to AC.", "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:\nA fair 6-sided die is rolled four times. What is the probability that at least two of the rolls show the same number? Express your answer as a common fraction.", "role": "user" } ]

[ { "content": "Question:\nLet x be a real number such that x^3 - 3x + 1 = 0. Find the value of x^4 - 4x^2 + 6.", "role": "user" } ]

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

[ { "content": "Question:\nThe sum of three consecutive positive integers is equal to the sum of the squares of the next two consecutive integers. What is the smallest such integer?", "role": "user" } ]

[ { "content": "Question:\nA regular hexagon has side length 6. Find the area of the region inside the hexagon but outside the circle inscribed in 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 integers from 1 to 100 inclusive. Find the number of subsets of S whose elements sum to a multiple of 5.", "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 x^3 + y^3?", "role": "user" } ]

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

[ { "content": "Question:\nA circle with center O has diameter AB = 10. Point C lies on the circle such that triangle ABC has area 20. What is the length of AC?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of integers between 1 and 1000 (inclusive) that are not divisible by 2, 3, or 5. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nFind the smallest positive integer n such that the sum 1 + 2 + ... + n is divisible by 100.", "role": "user" } ]

[ { "content": "Question:\nLet $x$ be a real number such that $x^2 + 4x = 21$. What is the value of $x^3 + 8x$?", "role": "user" } ]

[ { "content": "Question:\nA regular hexagon has side length $6$. What is the area of the triangle formed by connecting 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 $a, b, c$ be positive real numbers such that $abc = 8$ and $a + b + c = 6$. Find the minimum value of $a^2 + b^2 + c^2$.", "role": "user" } ]

[ { "content": "Question:\nSuppose the sum of the digits of a positive integer $N$ is $2024$. What is the remainder when $N$ is divided by $9$?", "role": "user" } ]

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

[ { "content": "Question:\nA triangle has side lengths 6, 8, and x, where x is an integer. How many possible values of x are there?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all 3-digit positive integers whose digits are all odd. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nEvaluate the sum \$ \\sum_{k=1}^{4} (k^3 + 3k^2 + 3k + 1) \$.", "role": "user" } ]

[ { "content": "Question:\nLet a and b be positive integers such that a^2 + b^2 = 2025. What is the value of a + b?", "role": "user" } ]

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

[ { "content": "Question:\nA circle has center O and a chord AB with length 8. If the distance from O to AB is 3, what is the radius of the circle?", "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 = 7 and x^2 + y^2 = 37. Find the value of xy.", "role": "user" } ]

[ { "content": "Question:\nHow many ways are there to arrange the letters of the word 'PROBLEM' so that no two vowels are adjacent?", "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 circle passes through the points (0,0) and (6,0) and is tangent to the line y = 4. What is the radius of the circle?", "role": "user" } ]

[ { "content": "Question:\nIf the sum of the positive divisors of 60 is S, what is S?", "role": "user" } ]

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

[ { "content": "Question:\nA fair coin is flipped 8 times. What is the probability that heads never occurs on two consecutive flips? Express your answer as a common fraction.", "role": "user" } ]

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

[ { "content": "Question:\nA positive integer n leaves a remainder of 2 when divided by 5 and a remainder of 3 when divided by 7. What is the smallest such n?", "role": "user" } ]

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

[ { "content": "Question:\nLet triangle ABC have AB = 10, AC = 8, and angle BAC = 60°. Find the area of triangle ABC.", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all permutations of the numbers 1, 2, 3, 4, 5. How many of these permutations have exactly two fixed points?", "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:\nIn a group of 10 students, how many ways can a committee of 4 be chosen such that no two committee members have consecutive student numbers (labeled 1 through 10)?", "role": "user" } ]

[ { "content": "Question:\nA circle passes through the vertices A and B of a square ABCD and is tangent to side CD at its midpoint. If the side length of the square is 8, what is the radius of the circle?", "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 remainder when n is divided by 9?", "role": "user" } ]

[ { "content": "Question:\nIf sin θ + cos θ = \\frac{5}{4}, where 0 < θ < \\frac{π}{2}, what is the value of sin 2θ?", "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 integers n between 1 and 1000 (inclusive) are divisible by neither 2 nor 5?", "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:\nA fair six-sided die is rolled three times. What is the probability that the sum of the outcomes is exactly 10?", "role": "user" } ]

[ { "content": "Question:\nFind the smallest positive integer n such that the sum 1 + 2 + ... + n is divisible by 100.", "role": "user" } ]

[ { "content": "Question:\nLet n be a positive integer such that the sum of the digits of n is 23 and n is divisible by 9. What is the smallest possible value of n?", "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 are divisible by neither 2 nor 5?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all integers between 1 and 1000, inclusive. For how many values of n in S is the remainder when n^2 is divided by 7 equal to 1?", "role": "user" } ]

[ { "content": "Question:\nLet x and y be real numbers such that x + y = 4 and x^2 + y^2 = 20. Find the value of x^3 + y^3.", "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:\nA regular hexagon has a side length of 2. 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 sum of all positive integers n less than 100 for which n^2 + n + 1 is divisible by 7. Find S.", "role": "user" } ]

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

[ { "content": "Question:\nLet n be the smallest positive integer such that the decimal representation of 7n ends with three consecutive digits 777. Find n.", "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 positive integers less than 1000 are divisible by neither 2 nor 5?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of integers from 1 to 100, inclusive. How many subsets of S have a sum that is divisible by 5?", "role": "user" } ]

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

[ { "content": "Question:\nA circle passes through the vertices A(0,0) and B(6,0) of a triangle ABC and is tangent to the line x = 2. Find 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:\nThe sum of the first n positive integers is 2016. Find the value of n.", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all ordered triples (a, b, c) of positive integers such that a + b + c = 10. How many elements are in S?", "role": "user" } ]