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 3 or 5 but not both. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, angle C is a right angle. If AC = 8 and BC = 15, what is the length of AB?", "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 four-digit positive integers have the property that their digits are strictly increasing from left to right?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that 2^n ends with the digits 888. Find n.", "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 does S contain?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 7, AC = 8, and angle BAC = 60°. Find the area of triangle ABC.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 4 and x^2 + y^2 = 10. Find the value of x^3 + y^3.", "role": "user" } ]
[ { "content": "Question:\nFind the number of 5-digit positive integers that have digits in strictly increasing order.", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that 2^n ends with the digits 888. Find n.", "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:\nA square and an equilateral triangle have equal perimeters. If the area of the square is 36, what is the area of the triangle?", "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 circle passes through the vertices A(0,0) and B(6,0) of a rectangle ABCD and is tangent to the side CD. If the rectangle has width 4, what is the radius of the circle?", "role": "user" } ]
[ { "content": "Question:\nHow many 4-digit positive integers have the property that the sum of their digits is 10?", "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 right triangle has integer side lengths and the length of the hypotenuse is 17. How many such triangles exist?", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 2 nor 5?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, the angle bisector of angle A meets side BC at D. If AB = 6, AC = 8, and BC = 10, find the length of segment BD.", "role": "user" } ]
[ { "content": "Question:\nThe sum of three consecutive positive integers is equal to the sum of the next two consecutive positive integers. What is the smallest possible value of the largest of these five integers?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that 7n leaves a remainder of 3 when divided by 19. Find n.", "role": "user" } ]
[ { "content": "Question:\nA triangle has side lengths 7, 8, and x, where x is an integer. How many possible values are there for x?", "role": "user" } ]
[ { "content": "Question:\nLet S be the sum of all positive integers less than 1000 that are divisible by 7 or 11 but not both. Find S.", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit positive integers have the property that their digits are strictly increasing from left to right?", "role": "user" } ]
[ { "content": "Question:\nLet a and b be real numbers such that a + b = 4 and a^2 + b^2 = 10. Compute the value of a^3 + b^3.", "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 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 positive integers less than 1000 are divisible by neither 2 nor 5?", "role": "user" } ]
[ { "content": "Question:\nIf a, b, and c are positive real numbers such that abc = 8 and a + b + c = 6, find the minimum possible value of ab + bc + ca.", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, angle C is a right angle. If AC = 8 and BC = 15, what is the length of AB?", "role": "user" } ]
[ { "content": "Question:\nLet n be a positive integer such that n^2 + 2n + 19 is a perfect square. Find the value of n.", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by exactly two distinct primes?", "role": "user" } ]
[ { "content": "Question:\nA triangle has side lengths 5, 12, and 13. Find the sum of the radii of its incircle and circumcircle.", "role": "user" } ]
[ { "content": "Question:\nA fair die is rolled three times. What is the probability that the sum of the numbers rolled is exactly 10?", "role": "user" } ]
[ { "content": "Question:\nLet a and b be real numbers such that a^2 + b^2 = 10 and a + b = 4. Find the value of ab.", "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 is inscribed in a circle of radius 10. 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, and c be positive real numbers such that abc = 1. Find the minimum value of (a + 1)(b + 1)(c + 1).", "role": "user" } ]
[ { "content": "Question:\nA fair six-sided die is rolled six times. What is the probability that each face appears exactly once?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 8 and x^2 + y^2 = 34. What is the value of xy?", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon is inscribed in a circle of radius 6. What is the area of the triangle formed by joining every other vertex of the hexagon?", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 2 nor 5?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all three-digit numbers whose digits are in strictly increasing order. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nIf a and b are positive integers such that a^2 - b^2 = 105, how many ordered pairs (a, b) are there?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be positive integers such that x + y = 20 and x > y. What is the value of x^2 - y^2?", "role": "user" } ]
[ { "content": "Question:\nA circle passes through the points (0,0) and (4,0) and is tangent to the line y = 3. What is the radius of the circle?", "role": "user" } ]
[ { "content": "Question:\nHow many ordered pairs of integers (m, n) satisfy the equation m^2 + n^2 = 65?", "role": "user" } ]
[ { "content": "Question:\nLet a and b be real numbers such that a + b = 7 and ab = 10. What is the value of a^3 + b^3?", "role": "user" } ]
[ { "content": "Question:\nHow many distinct 4-digit numbers have digits that sum to 10 and contain no zeros?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that the sum 1 + 2 + 3 + ... + n is greater than 2024. Find the value of n.", "role": "user" } ]
[ { "content": "Question:\nLet x be a real number such that x^2 + 4x + 7 = 0. Compute the value of x^4 + 8x^3 + 30x^2 + 56x + 49.", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 2 nor 5?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all positive integers less than or equal to 1000 that are multiples of 3 or 7. Find the number of elements in S.", "role": "user" } ]
[ { "content": "Question:\nLet ABC be a triangle with AB = 13, BC = 14, and AC = 15. Find the length of the altitude from B to AC.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of positive integers less than 100 that are divisible by either 2 or 5 but not both. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nLet a and b be real numbers such that a + b = 7 and ab = 12. Find the value of a^2 + b^2.", "role": "user" } ]
[ { "content": "Question:\nLet triangle ABC have AB = 9, AC = 11, and BC = 6. The angle bisector of angle BAC meets BC at D. Find the length of BD.", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit positive integers are there such that their digits are strictly increasing from left to right?", "role": "user" } ]
[ { "content": "Question:\nFind the number of ordered pairs of positive integers (x, y) such that 2x + 3y = 100.", "role": "user" } ]
[ { "content": "Question:\nLet f(x) = x^2 + 6x + 13. Find the sum of all integer values of x for which f(x) is a perfect square.", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has side length 2. What is the area of the region inside the hexagon that is closer to the center than to any of its sides?", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 3 nor 5?", "role": "user" } ]
[ { "content": "Question:\nLet a, b, and c be real numbers such that a + b + c = 6 and ab + bc + ca = 9. Find the maximum possible value of a.", "role": "user" } ]
[ { "content": "Question:\nA fair six-sided die is rolled three times. What is the probability that the sum of the three numbers rolled is 15?", "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:\nIn triangle ABC, angle ABC is 90 degrees, and the lengths of sides AB and BC are 8 and 15, respectively. Find the length of the altitude from B to AC.", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 2 nor 5?", "role": "user" } ]
[ { "content": "Question:\nLet x be a real number such that (x^2 - 4x + 7)^2 = 25. Find all possible values of x.", "role": "user" } ]
[ { "content": "Question:\nA bag contains 5 red balls and 7 blue balls. How many ways are there to choose 4 balls so that at least two of them are red?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that the sum of the digits of n is 2024. What is the sum of the digits of n + 1?", "role": "user" } ]
[ { "content": "Question:\nA circle has center O and a chord AB such that OA = 8 and OB = 6. If the length of AB is k, find the value of k.", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit integers are there such that the sum of their digits is 9?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 5 and x^2 + y^2 = 13. Find x^3 + y^3.", "role": "user" } ]
[ { "content": "Question:\nA fair six-sided die is rolled three times. What is the probability that the sum of the three results is exactly 10?", "role": "user" } ]
[ { "content": "Question:\nLet n be the sum of all positive integers less than 100 that leave a remainder of 1 when divided by 5. What is n?", "role": "user" } ]
[ { "content": "Question:\nHow many ways are there to arrange the letters in the word 'PROBLEM' so that no two vowels are adjacent?", "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 triangle has side lengths 7, 8, and x, where x is an integer. For how many values of x is the area of the triangle also an integer?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all integers between 1 and 1000 (inclusive) that are not divisible by 2 or 5. Find the sum of the elements of S.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be positive integers such that x^2 + y^2 = 2024. What is the largest possible value of x + y?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = AC = 13 and BC = 10. Find the area of triangle ABC.", "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:\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 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 $x$ and $y$ be real numbers such that $x + y = 6$ and $x^2 + y^2 = 20$. Compute the value of $xy$.", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit positive integers have digits in strictly increasing order from left to right?", "role": "user" } ]
[ { "content": "Question:\nLet $S$ be the set of integers from $1$ to $1000$ inclusive. How many elements of $S$ are divisible by neither $2$ nor $5$?", "role": "user" } ]
[ { "content": "Question:\nIn triangle $ABC$, $AB = 8$, $AC = 15$, and $BC = 17$. Let $h$ be the length of the altitude from $A$ to $BC$. Compute $h$.", "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 n be the smallest positive integer such that the sum of the digits of 9n is equal to 27. What is n?", "role": "user" } ]
[ { "content": "Question:\nA cube has a side length of 6. A sphere is inscribed inside the cube. What is the volume of the portion of the sphere that lies above the plane passing through the center of the cube and parallel to one of its faces?", "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 positive real numbers such that x + y = 10 and x^2 + y^2 = 58. What is the value of xy?", "role": "user" } ]
[ { "content": "Question:\nFind the number of ordered triples (a, b, c) of positive integers such that a + b + c = 12 and a \\geq 2, b \\geq 3, c \\geq 4.", "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:\nIn an arithmetic sequence, the fifth term is 15 and the twelfth term is 43. What is the sum of the first 20 terms?", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit positive integers are there such that all digits are odd and no digit repeats?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 13, BC = 14, and AC = 15. Let h be the length of the altitude from B to AC. What is h?", "role": "user" } ]
[ { "content": "Question:\nA box contains 8 red balls and 6 blue balls. How many ways are there to select 4 balls so that at least 2 are red?", "role": "user" } ]

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

[ { "content": "Question:\nIn triangle ABC, angle C is a right angle. If AC = 8 and BC = 15, what is the length of AB?", "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 four-digit positive integers have the property that their digits are strictly increasing from left to right?", "role": "user" } ]

[ { "content": "Question:\nLet n be the smallest positive integer such that 2^n ends with the digits 888. Find n.", "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 does S contain?", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, AB = 7, AC = 8, and angle BAC = 60°. Find the area of triangle ABC.", "role": "user" } ]

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

[ { "content": "Question:\nFind the number of 5-digit positive integers that have digits in strictly increasing order.", "role": "user" } ]

[ { "content": "Question:\nLet n be the smallest positive integer such that 2^n ends with the digits 888. Find n.", "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:\nA square and an equilateral triangle have equal perimeters. If the area of the square is 36, what is the area of the triangle?", "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 circle passes through the vertices A(0,0) and B(6,0) of a rectangle ABCD and is tangent to the side CD. If the rectangle has width 4, what is the radius of the circle?", "role": "user" } ]

[ { "content": "Question:\nHow many 4-digit positive integers have the property that the sum of their digits is 10?", "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 right triangle has integer side lengths and the length of the hypotenuse is 17. How many such triangles exist?", "role": "user" } ]

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

[ { "content": "Question:\nIn triangle ABC, the angle bisector of angle A meets side BC at D. If AB = 6, AC = 8, and BC = 10, find the length of segment BD.", "role": "user" } ]

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

[ { "content": "Question:\nLet n be the smallest positive integer such that 7n leaves a remainder of 3 when divided by 19. Find n.", "role": "user" } ]

[ { "content": "Question:\nA triangle has side lengths 7, 8, and x, where x is an integer. How many possible values are there for x?", "role": "user" } ]

[ { "content": "Question:\nLet S be the sum of all positive integers less than 1000 that are divisible by 7 or 11 but not both. Find S.", "role": "user" } ]

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

[ { "content": "Question:\nLet a and b be real numbers such that a + b = 4 and a^2 + b^2 = 10. Compute the value of a^3 + b^3.", "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 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 positive integers less than 1000 are divisible by neither 2 nor 5?", "role": "user" } ]

[ { "content": "Question:\nIf a, b, and c are positive real numbers such that abc = 8 and a + b + c = 6, find the minimum possible value of ab + bc + ca.", "role": "user" } ]

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

[ { "content": "Question:\nLet n be a positive integer such that n^2 + 2n + 19 is a perfect square. Find the value of n.", "role": "user" } ]

[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by exactly two distinct primes?", "role": "user" } ]

[ { "content": "Question:\nA triangle has side lengths 5, 12, and 13. Find the sum of the radii of its incircle and circumcircle.", "role": "user" } ]

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

[ { "content": "Question:\nLet a and b be real numbers such that a^2 + b^2 = 10 and a + b = 4. Find the value of ab.", "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 is inscribed in a circle of radius 10. 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, and c be positive real numbers such that abc = 1. Find the minimum value of (a + 1)(b + 1)(c + 1).", "role": "user" } ]

[ { "content": "Question:\nA fair six-sided die is rolled six times. What is the probability that each face appears exactly once?", "role": "user" } ]

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

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

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

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

[ { "content": "Question:\nIf a and b are positive integers such that a^2 - b^2 = 105, how many ordered pairs (a, b) are there?", "role": "user" } ]

[ { "content": "Question:\nLet x and y be positive integers such that x + y = 20 and x > y. What is the value of x^2 - y^2?", "role": "user" } ]

[ { "content": "Question:\nA circle passes through the points (0,0) and (4,0) and is tangent to the line y = 3. What is the radius of the circle?", "role": "user" } ]

[ { "content": "Question:\nHow many ordered pairs of integers (m, n) satisfy the equation m^2 + n^2 = 65?", "role": "user" } ]

[ { "content": "Question:\nLet a and b be real numbers such that a + b = 7 and ab = 10. What is the value of a^3 + b^3?", "role": "user" } ]

[ { "content": "Question:\nHow many distinct 4-digit numbers have digits that sum to 10 and contain no zeros?", "role": "user" } ]

[ { "content": "Question:\nLet n be the smallest positive integer such that the sum 1 + 2 + 3 + ... + n is greater than 2024. Find the value of n.", "role": "user" } ]

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

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

[ { "content": "Question:\nLet S be the set of all positive integers less than or equal to 1000 that are multiples of 3 or 7. Find the number of elements in S.", "role": "user" } ]

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

[ { "content": "Question:\nLet S be the set of positive integers less than 100 that are divisible by either 2 or 5 but not both. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nLet a and b be real numbers such that a + b = 7 and ab = 12. Find the value of a^2 + b^2.", "role": "user" } ]

[ { "content": "Question:\nLet triangle ABC have AB = 9, AC = 11, and BC = 6. The angle bisector of angle BAC meets BC at D. Find the length of BD.", "role": "user" } ]

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

[ { "content": "Question:\nFind the number of ordered pairs of positive integers (x, y) such that 2x + 3y = 100.", "role": "user" } ]

[ { "content": "Question:\nLet f(x) = x^2 + 6x + 13. Find the sum of all integer values of x for which f(x) is a perfect square.", "role": "user" } ]

[ { "content": "Question:\nA regular hexagon has side length 2. What is the area of the region inside the hexagon that is closer to the center than to any of its sides?", "role": "user" } ]

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

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

[ { "content": "Question:\nA fair six-sided die is rolled three times. What is the probability that the sum of the three numbers rolled is 15?", "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:\nIn triangle ABC, angle ABC is 90 degrees, and the lengths of sides AB and BC are 8 and 15, respectively. Find the length of the altitude from B to AC.", "role": "user" } ]

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

[ { "content": "Question:\nLet x be a real number such that (x^2 - 4x + 7)^2 = 25. Find all possible values of x.", "role": "user" } ]

[ { "content": "Question:\nA bag contains 5 red balls and 7 blue balls. How many ways are there to choose 4 balls so that at least two of them are red?", "role": "user" } ]

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

[ { "content": "Question:\nA circle has center O and a chord AB such that OA = 8 and OB = 6. If the length of AB is k, find the value of k.", "role": "user" } ]

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

[ { "content": "Question:\nLet x and y be real numbers such that x + y = 5 and x^2 + y^2 = 13. Find x^3 + y^3.", "role": "user" } ]

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

[ { "content": "Question:\nLet n be the sum of all positive integers less than 100 that leave a remainder of 1 when divided by 5. What is n?", "role": "user" } ]

[ { "content": "Question:\nHow many ways are there to arrange the letters in the word 'PROBLEM' so that no two vowels are adjacent?", "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 triangle has side lengths 7, 8, and x, where x is an integer. For how many values of x is the area of the triangle also an integer?", "role": "user" } ]

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

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

[ { "content": "Question:\nIn triangle ABC, AB = AC = 13 and BC = 10. Find the area of triangle ABC.", "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:\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 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 $x$ and $y$ be real numbers such that $x + y = 6$ and $x^2 + y^2 = 20$. Compute the value of $xy$.", "role": "user" } ]

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

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

[ { "content": "Question:\nIn triangle $ABC$, $AB = 8$, $AC = 15$, and $BC = 17$. Let $h$ be the length of the altitude from $A$ to $BC$. Compute $h$.", "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 n be the smallest positive integer such that the sum of the digits of 9n is equal to 27. What is n?", "role": "user" } ]

[ { "content": "Question:\nA cube has a side length of 6. A sphere is inscribed inside the cube. What is the volume of the portion of the sphere that lies above the plane passing through the center of the cube and parallel to one of its faces?", "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 positive real numbers such that x + y = 10 and x^2 + y^2 = 58. What is the value of xy?", "role": "user" } ]

[ { "content": "Question:\nFind the number of ordered triples (a, b, c) of positive integers such that a + b + c = 12 and a \\geq 2, b \\geq 3, c \\geq 4.", "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:\nIn an arithmetic sequence, the fifth term is 15 and the twelfth term is 43. What is the sum of the first 20 terms?", "role": "user" } ]

[ { "content": "Question:\nHow many four-digit positive integers are there such that all digits are odd and no digit repeats?", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, AB = 13, BC = 14, and AC = 15. Let h be the length of the altitude from B to AC. What is h?", "role": "user" } ]

[ { "content": "Question:\nA box contains 8 red balls and 6 blue balls. How many ways are there to select 4 balls so that at least 2 are red?", "role": "user" } ]