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 = 6 and x^2 + y^2 = 20. Find the value of x^3 + y^3.", "role": "user" } ]
[ { "content": "Question:\nA triangle has sides of lengths 5, 12, and 13. What is the area of the circle inscribed in this triangle?", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 3 nor 5?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all integers between 1 and 1000 inclusive that leave a remainder of 2 when divided by 5. Find the sum of all elements in S.", "role": "user" } ]
[ { "content": "Question:\nLet a, b, and 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:\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:\nLet triangle ABC have AB = 7, AC = 8, and BC = 9. Find the radius of the circle inscribed in triangle ABC.", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit positive integers have digits that add up to 10?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 6 and x^2 + y^2 = 20. Find the value of x^3 + y^3.", "role": "user" } ]
[ { "content": "Question:\nIn how many distinct ways can the letters in the word 'PROBLEM' be arranged so that no two vowels are adjacent?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that when n is divided by 7, 9, and 11, the remainder is 5 each time. Find n.", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, angle C is a right angle. If AC = 8 and BC = 15, find the length of the altitude from C to AB.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all positive integers less than 1000 that are divisible by exactly two distinct prime numbers. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nSuppose x and y are real numbers such that x^2 + y^2 = 25 and x + y = 7. Find the greatest possible value of xy.", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit positive integers have digits that sum to 10 and are strictly increasing from left to right?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 7 and x^2 + y^2 = 37. What is the value of (x - y)^2?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 10, AC = 8, and the altitude from A to BC has length 6. Find the length of BC.", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit positive integers are there whose digits are strictly increasing from left to right?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that the sum 1 + 2 + ... + n is divisible by 1000. Find n.", "role": "user" } ]
[ { "content": "Question:\nThe numbers 1 through 8 are arranged in a row. How many such arrangements are there in which no two consecutive numbers are consecutive integers?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of integers between 1 and 1000, inclusive, that are divisible by neither 2 nor 5. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 7, AC = 9, and the area of triangle ABC is 21. Find the length of BC.", "role": "user" } ]
[ { "content": "Question:\nLet x be a real number such that x^2 + 4x + 7 = 0. Find the value of (x + 2)^4.", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 have exactly three distinct prime divisors?", "role": "user" } ]
[ { "content": "Question:\nA fair six-sided die is rolled three times. What is the probability that the product of the three numbers rolled is divisible by 5?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that 2^n ends with the digits 888. What is the value of n?", "role": "user" } ]
[ { "content": "Question:\nA triangle has sides of lengths 5, 7, and x, where x is an integer. How many possible values are there for x?", "role": "user" } ]
[ { "content": "Question:\nA fair six-sided die is rolled three times. What is the probability that the sum of the three numbers is exactly 10?", "role": "user" } ]
[ { "content": "Question:\nLet S be the sum of all positive integers less than 1000 that are divisible by 3 or 5. Find the value of S.", "role": "user" } ]
[ { "content": "Question:\nFind the number of positive integers less than or equal to 1000 that are not divisible by 2, 3, or 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 $x^3 + y^3$?", "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:\nHow many four-digit positive integers have digits that strictly increase from left to right?", "role": "user" } ]
[ { "content": "Question:\nLet $n$ be the smallest positive integer such that $n!$ is divisible by $2016$. Find $n$.", "role": "user" } ]
[ { "content": "Question:\nThe sum of five consecutive odd integers is $275$. What is the smallest of these integers?", "role": "user" } ]
[ { "content": "Question:\nLet a triangle have 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 sum of all three-digit numbers whose digits add up to 10. Find the value of S.", "role": "user" } ]
[ { "content": "Question:\nIn how many ways can the letters of the word 'STATISTICS' be arranged such that no two identical letters are adjacent?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be positive real numbers such that x + y = 6 and x^2 + y^2 = 20. Find the value of xy.", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that the sum of the digits of 2024n is 2024. What is the remainder when n is divided by 9?", "role": "user" } ]
[ { "content": "Question:\nA right triangle has integer sides and its area is numerically equal to its perimeter. Find the smallest possible perimeter of such a triangle.", "role": "user" } ]
[ { "content": "Question:\nLet a and b be real numbers such that a^2 + b^2 = 10 and ab = 3. Compute the value of (a + b)^4.", "role": "user" } ]
[ { "content": "Question:\nA fair six-sided die is rolled three times. What is the probability that the sum of the three rolls is exactly 10?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, angle ABC is a right angle. If AB = 6 and BC = 8, compute the length of the altitude from B to AC.", "role": "user" } ]
[ { "content": "Question:\nLet x be a real number such that x^2 + 4x + 7 = 0. What is the value of x^4 + 8x^2 + 49?", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 3 nor 5?", "role": "user" } ]
[ { "content": "Question:\nLet triangle ABC have AB = 13, BC = 14, and AC = 15. Find the length of the altitude from B to AC.", "role": "user" } ]
[ { "content": "Question:\nHow many 5-digit numbers contain exactly two distinct digits, both of which are odd?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of integers n such that 1 ≤ n ≤ 100 and the greatest common divisor of n and 84 is 1. Compute the number of elements in 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:\nA regular hexagon has side length 4. 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 3 or 5 but not both?", "role": "user" } ]
[ { "content": "Question:\nSuppose a, b, and c are positive integers such that a + b + c = 20 and a < b < c. How many ordered triples (a, b, c) satisfy these conditions?", "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 a triangle have side lengths 7, 8, and 9. What is the area of the triangle?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of integers between 1 and 100 inclusive 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 = 6 and x^2 + y^2 = 20. Find the value of x*y.", "role": "user" } ]
[ { "content": "Question:\nLet N be the number of four-digit positive integers whose digits sum to 10. Find N.", "role": "user" } ]
[ { "content": "Question:\nThe numbers 1 to 10 are arranged in a row. How many ways are there to arrange them so that no two consecutive numbers are consecutive integers?", "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 x*y?", "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, 8, and 9. What is the area of the triangle?", "role": "user" } ]
[ { "content": "Question:\nLet S be the sum of all positive integers n such that 1 ≤ n ≤ 100 and the greatest common divisor of n and 100 is 1. Find S.", "role": "user" } ]
[ { "content": "Question:\nA fair six-sided die is rolled three times. What is the probability that the sum of the three rolls is exactly 10?", "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 triangle has side lengths 13, 14, and 15. What is the area of the triangle?", "role": "user" } ]
[ { "content": "Question:\nHow many integers n between 1 and 1000 inclusive satisfy that the sum of the digits of n is divisible by 5?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all positive integers less than 100 that are divisible by 3 or 7. What is the sum of all elements in S?", "role": "user" } ]
[ { "content": "Question:\nLet a, b, and c be the roots of the equation x^3 - 6x^2 + 11x - 6 = 0. Compute the value of ab + bc + ca.", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that the decimal representation of 7n ends with the digits 2024. Find n.", "role": "user" } ]
[ { "content": "Question:\nThe sum of the interior angles of a convex polygon is 2880 degrees. How many sides does the polygon have?", "role": "user" } ]
[ { "content": "Question:\nLet a and b be positive integers such that a + b = 20 and the least common multiple of a and b is 96. Find the value of ab.", "role": "user" } ]
[ { "content": "Question:\nSuppose f(x) = x^2 - 6x + 11. The sequence a_1, a_2, a_3, ... is defined by a_1 = 2 and a_{n+1} = f(a_n) for all n ≥ 1. Find a_4.", "role": "user" } ]
[ { "content": "Question:\nHow many ordered triples of positive integers (x, y, z) satisfy x + y + z = 15 and x ≤ y ≤ z?", "role": "user" } ]
[ { "content": "Question:\nLet x be a real number such that x^2 + 6x + 5 = 0. What is the value of x^4 + 12x^2 + 28?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 13, BC = 14, and AC = 15. Find the area of triangle ABC.", "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 such that n < 100 and both n and n+1 are perfect squares. Find S.", "role": "user" } ]
[ { "content": "Question:\nSuppose a fair six-sided die is rolled four times. What is the probability that exactly two of the rolls show a 3? Express your answer as a simplified fraction.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be positive integers such that x^2 + y^2 = 2024. How many ordered pairs (x, y) satisfy this equation?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, angle C is a right angle. If AB = 13, BC = 5, and AC = 12, find the area of the circle inscribed in triangle ABC.", "role": "user" } ]
[ { "content": "Question:\nA fair six-sided die is rolled three times. What is the probability that the sequence of numbers obtained is strictly increasing?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all positive integers less than 1000 that are divisible by 3 or 7 but not by 21. Find the number of elements in S.", "role": "user" } ]
[ { "content": "Question:\nSuppose a and b are real numbers such that a + b = 4 and a^2 + b^2 = 20. Find the value of ab.", "role": "user" } ]
[ { "content": "Question:\nLet x be a real number such that x^2 - 6x + 10 = 0. What is the value of x^4 + 1?", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has a perimeter of 24 units. What is the area of the hexagon?", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit integers have the property that the sum of their digits is 8?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all 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:\nLet a, b, and c be positive real numbers such that abc = 1. Find the minimum value of a + b + c given that ab + bc + ca = 4.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be positive integers such that x^2 - y^2 = 45. Find the number of ordered pairs (x, y) that satisfy this equation.", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit numbers are there such that the sum of their digits is 9?", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has side length 6. Find the area of the region inside the hexagon that is closer to the center than to any of its sides.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all integers n such that 1 ≤ n ≤ 100 and n is divisible by 3 or 5 but not by 7. Find the sum of all elements in S.", "role": "user" } ]
[ { "content": "Question:\nLet a, b, and c be positive real numbers such that a + b + c = 6 and ab + bc + ca = 9. Compute the maximum possible value of abc.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all three-digit positive integers whose digits sum to 9. 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:\nAn equilateral triangle has side length 6. A point P is chosen uniformly at random inside the triangle. What is the probability that the distance from P to each side is less than 2?", "role": "user" } ]
[ { "content": "Question:\nFind the number of integers n between 1 and 1000 (inclusive) such that n and n+1 are both divisible by either 2 or 3.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set {1,2,3,...,10}. How many subsets of S contain no two consecutive integers?", "role": "user" } ]

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

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

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

[ { "content": "Question:\nLet S be the set of all integers between 1 and 1000 inclusive that leave a remainder of 2 when divided by 5. Find the sum of all elements in S.", "role": "user" } ]

[ { "content": "Question:\nLet a, b, and 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:\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:\nLet triangle ABC have AB = 7, AC = 8, and BC = 9. Find the radius of the circle inscribed in triangle ABC.", "role": "user" } ]

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

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

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

[ { "content": "Question:\nLet n be the smallest positive integer such that when n is divided by 7, 9, and 11, the remainder is 5 each time. Find n.", "role": "user" } ]

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

[ { "content": "Question:\nLet S be the set of all positive integers less than 1000 that are divisible by exactly two distinct prime numbers. How many elements are in S?", "role": "user" } ]

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

[ { "content": "Question:\nHow many four-digit positive integers have digits that sum to 10 and are strictly increasing from left to right?", "role": "user" } ]

[ { "content": "Question:\nLet x and y be real numbers such that x + y = 7 and x^2 + y^2 = 37. What is the value of (x - y)^2?", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, AB = 10, AC = 8, and the altitude from A to BC has length 6. Find the length of BC.", "role": "user" } ]

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

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

[ { "content": "Question:\nThe numbers 1 through 8 are arranged in a row. How many such arrangements are there in which no two consecutive numbers are consecutive integers?", "role": "user" } ]

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

[ { "content": "Question:\nIn triangle ABC, AB = 7, AC = 9, and the area of triangle ABC is 21. Find the length of BC.", "role": "user" } ]

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

[ { "content": "Question:\nHow many positive integers less than 1000 have exactly three distinct prime divisors?", "role": "user" } ]

[ { "content": "Question:\nA fair six-sided die is rolled three times. What is the probability that the product of the three numbers rolled is divisible by 5?", "role": "user" } ]

[ { "content": "Question:\nLet n be the smallest positive integer such that 2^n ends with the digits 888. What is the value of n?", "role": "user" } ]

[ { "content": "Question:\nA triangle has sides of lengths 5, 7, and x, where x is an integer. How many possible values are there for x?", "role": "user" } ]

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

[ { "content": "Question:\nLet S be the sum of all positive integers less than 1000 that are divisible by 3 or 5. Find the value of S.", "role": "user" } ]

[ { "content": "Question:\nFind the number of positive integers less than or equal to 1000 that are not divisible by 2, 3, or 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 $x^3 + y^3$?", "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:\nHow many four-digit positive integers have digits that strictly increase from left to right?", "role": "user" } ]

[ { "content": "Question:\nLet $n$ be the smallest positive integer such that $n!$ is divisible by $2016$. Find $n$.", "role": "user" } ]

[ { "content": "Question:\nThe sum of five consecutive odd integers is $275$. What is the smallest of these integers?", "role": "user" } ]

[ { "content": "Question:\nLet a triangle have 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 sum of all three-digit numbers whose digits add up to 10. Find the value of S.", "role": "user" } ]

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

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

[ { "content": "Question:\nLet n be the smallest positive integer such that the sum of the digits of 2024n is 2024. What is the remainder when n is divided by 9?", "role": "user" } ]

[ { "content": "Question:\nA right triangle has integer sides and its area is numerically equal to its perimeter. Find the smallest possible perimeter of such a triangle.", "role": "user" } ]

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

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

[ { "content": "Question:\nIn triangle ABC, angle ABC is a right angle. If AB = 6 and BC = 8, compute the length of the altitude from B to AC.", "role": "user" } ]

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

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

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

[ { "content": "Question:\nHow many 5-digit numbers contain exactly two distinct digits, both of which are odd?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of integers n such that 1 ≤ n ≤ 100 and the greatest common divisor of n and 84 is 1. Compute the number of elements in 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:\nA regular hexagon has side length 4. 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 3 or 5 but not both?", "role": "user" } ]

[ { "content": "Question:\nSuppose a, b, and c are positive integers such that a + b + c = 20 and a < b < c. How many ordered triples (a, b, c) satisfy these conditions?", "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 a triangle have side lengths 7, 8, and 9. What is the area of the triangle?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of integers between 1 and 100 inclusive 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 = 6 and x^2 + y^2 = 20. Find the value of x*y.", "role": "user" } ]

[ { "content": "Question:\nLet N be the number of four-digit positive integers whose digits sum to 10. Find N.", "role": "user" } ]

[ { "content": "Question:\nThe numbers 1 to 10 are arranged in a row. How many ways are there to arrange them so that no two consecutive numbers are consecutive integers?", "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 x*y?", "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, 8, and 9. What is the area of the triangle?", "role": "user" } ]

[ { "content": "Question:\nLet S be the sum of all positive integers n such that 1 ≤ n ≤ 100 and the greatest common divisor of n and 100 is 1. Find S.", "role": "user" } ]

[ { "content": "Question:\nA fair six-sided die is rolled three times. What is the probability that the sum of the three rolls is exactly 10?", "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 triangle has side lengths 13, 14, and 15. What is the area of the triangle?", "role": "user" } ]

[ { "content": "Question:\nHow many integers n between 1 and 1000 inclusive satisfy that the sum of the digits of n is divisible by 5?", "role": "user" } ]

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

[ { "content": "Question:\nLet a, b, and c be the roots of the equation x^3 - 6x^2 + 11x - 6 = 0. Compute the value of ab + bc + ca.", "role": "user" } ]

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

[ { "content": "Question:\nThe sum of the interior angles of a convex polygon is 2880 degrees. How many sides does the polygon have?", "role": "user" } ]

[ { "content": "Question:\nLet a and b be positive integers such that a + b = 20 and the least common multiple of a and b is 96. Find the value of ab.", "role": "user" } ]

[ { "content": "Question:\nSuppose f(x) = x^2 - 6x + 11. The sequence a_1, a_2, a_3, ... is defined by a_1 = 2 and a_{n+1} = f(a_n) for all n ≥ 1. Find a_4.", "role": "user" } ]

[ { "content": "Question:\nHow many ordered triples of positive integers (x, y, z) satisfy x + y + z = 15 and x ≤ y ≤ z?", "role": "user" } ]

[ { "content": "Question:\nLet x be a real number such that x^2 + 6x + 5 = 0. What is the value of x^4 + 12x^2 + 28?", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, AB = 13, BC = 14, and AC = 15. Find the area of triangle ABC.", "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 such that n < 100 and both n and n+1 are perfect squares. Find S.", "role": "user" } ]

[ { "content": "Question:\nSuppose a fair six-sided die is rolled four times. What is the probability that exactly two of the rolls show a 3? Express your answer as a simplified fraction.", "role": "user" } ]

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

[ { "content": "Question:\nIn triangle ABC, angle C is a right angle. If AB = 13, BC = 5, and AC = 12, find the area of the circle inscribed in triangle ABC.", "role": "user" } ]

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

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

[ { "content": "Question:\nSuppose a and b are real numbers such that a + b = 4 and a^2 + b^2 = 20. Find the value of ab.", "role": "user" } ]

[ { "content": "Question:\nLet x be a real number such that x^2 - 6x + 10 = 0. What is the value of x^4 + 1?", "role": "user" } ]

[ { "content": "Question:\nA regular hexagon has a perimeter of 24 units. What is the area of the hexagon?", "role": "user" } ]

[ { "content": "Question:\nHow many four-digit integers have the property that the sum of their digits is 8?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all 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:\nLet a, b, and c be positive real numbers such that abc = 1. Find the minimum value of a + b + c given that ab + bc + ca = 4.", "role": "user" } ]

[ { "content": "Question:\nLet x and y be positive integers such that x^2 - y^2 = 45. Find the number of ordered pairs (x, y) that satisfy this equation.", "role": "user" } ]

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

[ { "content": "Question:\nA regular hexagon has side length 6. Find the area of the region inside the hexagon that is closer to the center than to any of its sides.", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all integers n such that 1 ≤ n ≤ 100 and n is divisible by 3 or 5 but not by 7. Find the sum of all elements in S.", "role": "user" } ]

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

[ { "content": "Question:\nLet S be the set of all three-digit positive integers whose digits sum to 9. 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:\nAn equilateral triangle has side length 6. A point P is chosen uniformly at random inside the triangle. What is the probability that the distance from P to each side is less than 2?", "role": "user" } ]

[ { "content": "Question:\nFind the number of integers n between 1 and 1000 (inclusive) such that n and n+1 are both divisible by either 2 or 3.", "role": "user" } ]

[ { "content": "Question:\nLet S be the set {1,2,3,...,10}. How many subsets of S contain no two consecutive integers?", "role": "user" } ]