simonycl/amc_aime_training_positive_sequence_cot_gpt-4.1-prompt

messages listlengths 1 1
[ { "content": "Question:\nLet n be the smallest positive integer such that n^2 + 7n + 12 is divisible by 24. What is the value of n?", "role": "user" } ]
[ { "content": "Question:\nA circle of radius 5 is tangent to the sides AB and AC of triangle ABC at points D and E, respectively. If AB = 13, AC = 14, and the distance from A to the center of the circle is 8, find the length of BC.", "role": "user" } ]
[ { "content": "Question:\nIf x and y are positive integers such that x^2 - y^2 = 45 and x + y < 20, how many ordered pairs (x, y) satisfy these conditions?", "role": "user" } ]
[ { "content": "Question:\nHow many permutations of the digits 1 through 6 are there such that no two consecutive digits in the permutation are both odd?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of integers between 1 and 1000, inclusive, that are divisible by either 4 or 6 but not both. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all integers between 10 and 99, inclusive. How many numbers in S have the property that the sum of their digits is 10?", "role": "user" } ]
[ { "content": "Question:\nA right triangle has integer side lengths and a hypotenuse of length 13. Find the area of this triangle.", "role": "user" } ]
[ { "content": "Question:\nLet x be a real number such that x + 1/x = 3. Compute the value of x^5 + 1/x^5.", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit numbers are there such that all four digits are distinct and the number is divisible by 5?", "role": "user" } ]
[ { "content": "Question:\nTwo fair six-sided dice are rolled. What is the probability that the product of the numbers rolled is a perfect square?", "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 xy?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, angle C is a right angle, and the lengths of the legs are AC = 8 and BC = 15. A circle is inscribed in triangle ABC. What is the radius of the inscribed circle?", "role": "user" } ]
[ { "content": "Question:\nA five-digit number is formed using the digits 1, 2, 3, 4, and 5 exactly once. How many such numbers are divisible by 6?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all positive integers n such that 1 ≤ n ≤ 100 and the remainder when n is divided by 7 is equal to the remainder when n is divided by 11. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nLet a and b be positive integers such that a^2 + b^2 = 2020. What is the number of ordered pairs (a, b) that satisfy this equation?", "role": "user" } ]
[ { "content": "Question:\nLet $x$ be a real number such that $x^2 - 6x + 10 = 0$. What is the value of $x^4 + x^2$?", "role": "user" } ]
[ { "content": "Question:\nLet $ABC$ be a triangle with $AB = 7$, $BC = 8$, and $CA = 9$. Find the length of the altitude from $A$ to side $BC$.", "role": "user" } ]
[ { "content": "Question:\nFor positive integers $n$, define $S(n)$ as the sum of the digits of $n$. Find the number of integers $n$ between $100$ and $999$ (inclusive) such that $S(n) = 12$.", "role": "user" } ]
[ { "content": "Question:\nLet $a$ and $b$ be real numbers such that $a + b = 4$ and $a^2 + b^2 = 10$. Find the value of $a^3 + b^3$.", "role": "user" } ]
[ { "content": "Question:\nSuppose five fair coins are flipped. What is the probability that no two heads appear on consecutive flips? Express your answer as a common fraction.", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that 3^n ends with the digits 001. 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 different four-digit positive integers have digits that sum to 10?", "role": "user" } ]
[ { "content": "Question:\nLet S be the sum of all positive integers less than 1000 that are divisible by neither 2 nor 5. Find S.", "role": "user" } ]
[ { "content": "Question:\nLet x be a real number such that x^2 - 6x + 13 = 0. What is the value of x^4 + x^2 + 1?", "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^3 + y^3.", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit positive integers have digits that sum to 10?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, the sides have lengths AB = 5, BC = 7, and CA = 8. What is the area of triangle ABC?", "role": "user" } ]
[ { "content": "Question:\nA bag contains 7 red balls and 5 blue balls. Two balls are drawn at random without replacement. What is the probability that both balls are red?", "role": "user" } ]
[ { "content": "Question:\nFind the smallest positive integer n such that the sum of the digits of n and the sum of the digits of 2n are both equal to 10.", "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 a positive integer. For how many values of x is the area of the triangle also an integer?", "role": "user" } ]
[ { "content": "Question:\nThe sum of three consecutive positive integers is equal to one-fourth the square of the smallest of the three. Find the value of the smallest integer.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all three-digit integers whose digits sum to 9. Find the number of elements in S that are divisible by 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 x and y be real numbers such that x^2 + y^2 = 25 and x + y = 7. Find the value of xy.", "role": "user" } ]
[ { "content": "Question:\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:\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 positive integers n such that the sum of the digits of n is 12 and n < 100. Find the sum of all elements in S.", "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 a multiple of 5?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of positive integers less than 100 that are multiples of 6 or 8 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^3 + y^3.", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has area 54. What is the length of one 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 positive real numbers such that abc = 8 and a + b + c = 6. What is the minimum possible 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 multiples of 3 or 5. What is the sum of all the elements of S?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 7, AC = 8, and angle BAC = 60°. Find the area of triangle ABC in simplest radical form.", "role": "user" } ]
[ { "content": "Question:\nA function f satisfies f(x) + 2f(1 - x) = x^2 for all real x. Find f(2).", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit positive integers are there such that the sum of the digits is 9?", "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) with a > b.", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that 5n is a perfect square and 8n is a perfect cube. What is the value of n?", "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:\nA positive integer is selected at random from the integers 1 to 1000, inclusive. What is the probability that its base-10 representation contains exactly two digits that are the digit 7?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all ordered triples (x, y, z) of positive integers such that x + y + z = 10. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nLet f(x) = x^4 - 8x^2 + 16. Find the sum of all real values of x for which f(x) = 0.", "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^3 + y^3.", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit positive integers have digits that sum to 10?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, angle BAC = 60°, AB = 8, and AC = 6. Find the length of BC.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of 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:\nIf a, b, and c are integers such that 2^a + 2^b + 2^c = 56, where a > b > c > 0, find the value of a + b + c.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be positive integers such that x + y = 20 and the greatest common divisor (gcd) of x and y is 4. How many ordered pairs (x, y) satisfy these conditions?", "role": "user" } ]
[ { "content": "Question:\nA right triangle has legs of integer length, and its perimeter is 30. What is the length of the hypotenuse?", "role": "user" } ]
[ { "content": "Question:\nIn how many ways can 5 identical balls be placed into 3 distinct boxes so that no box is empty?", "role": "user" } ]
[ { "content": "Question:\nThe sum of the squares of three consecutive positive integers is 365. What is the largest of the three integers?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that 2^n ≡ 1 (mod 65). Find the value of n.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 5 and x^2 + y^2 = 13. Find the value of x^3 + y^3.", "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 possible value of n?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 7, AC = 9, and angle BAC = 60°. Find the area of triangle ABC in simplest form.", "role": "user" } ]
[ { "content": "Question:\nLet S be the sum of all three-digit numbers that leave a remainder of 1 when divided by 5. Find the value of S.", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit positive integers are there such that the sum of their digits is 4?", "role": "user" } ]
[ { "content": "Question:\nLet a and b be positive integers such that a^2 - b^2 = 77. How many ordered pairs (a, b) satisfy this equation?", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has side length 6. What is the area of the region inside the hexagon but outside the circle centered at the hexagon's center with radius 3?", "role": "user" } ]
[ { "content": "Question:\nHow many integers between 100 and 999 (inclusive) have digits that sum to exactly 19?", "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:\nEvaluate the sum \\( \\sum_{k=1}^{5} \\frac{1}{k(k+1)} \\) and 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. What is the sum of all possible values of x + y?", "role": "user" } ]
[ { "content": "Question:\nA triangle has side lengths 7, 24, and x, where x is an integer. What is the number of possible values of x that result in a triangle with integer area?", "role": "user" } ]
[ { "content": "Question:\nHow many integers n between 1 and 1000 inclusive are such that the sum of the digits of n is divisible by 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 = 20. How many elements of S are there?", "role": "user" } ]
[ { "content": "Question:\nLet f(x) = x^4 - 8x^2 + 16. What is the sum of all real values of x for which f(x) = 0?", "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:\nA rectangle has a perimeter of 60 units. If the length is twice the width, what is the area of the rectangle?", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit positive integers have their digits in strictly increasing order?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that 7n is a perfect square and n is divisible by 10. Find n.", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 13, BC = 14, and AC = 15. Find the length of the altitude from B to AC.", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 13, AC = 15, and the altitude from A to BC has length 12. Find the length of BC.", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit positive integers are there such that the sum of their digits is 10?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be positive integers satisfying 3x + 4y = 100. Find the number of ordered pairs (x, y) that satisfy this equation.", "role": "user" } ]
[ { "content": "Question:\nEvaluate the sum of the squares of all positive integers less than 100 that are divisible by 7.", "role": "user" } ]
[ { "content": "Question:\nLet a, b, and c be real numbers such that a + b + c = 9 and ab + bc + ca = 26. Find the maximum possible value of c.", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that 3n is a perfect square and 4n is a perfect cube. Find the value of n.", "role": "user" } ]
[ { "content": "Question:\nA circle passes through the vertices A and B of a square ABCD and is tangent to the side CD at its midpoint. If the side length of the square is 6, 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 by 7. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nIf x and y are real numbers such that x^2 + y^2 = 10 and x + y = 2, find the value of x^3 + y^3.", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 have the property that the sum of their digits is 10?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of integers from 1 to 1000, inclusive. How many numbers in S are divisible by neither 4 nor 6?", "role": "user" } ]
[ { "content": "Question:\nThe sum of three consecutive positive integers is 99. What is the product of the smallest and largest of these integers?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 13, AC = 15, and BC = 14. Let D be the point on BC such that AD bisects angle BAC. Find the length of BD.", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 have digits that sum to 10?", "role": "user" } ]
[ { "content": "Question:\nLet x be a real number such that x + 1/x = 4. Compute x^3 + 1/x^3.", "role": "user" } ]

[ { "content": "Question:\nLet n be the smallest positive integer such that n^2 + 7n + 12 is divisible by 24. What is the value of n?", "role": "user" } ]

[ { "content": "Question:\nA circle of radius 5 is tangent to the sides AB and AC of triangle ABC at points D and E, respectively. If AB = 13, AC = 14, and the distance from A to the center of the circle is 8, find the length of BC.", "role": "user" } ]

[ { "content": "Question:\nIf x and y are positive integers such that x^2 - y^2 = 45 and x + y < 20, how many ordered pairs (x, y) satisfy these conditions?", "role": "user" } ]

[ { "content": "Question:\nHow many permutations of the digits 1 through 6 are there such that no two consecutive digits in the permutation are both odd?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of integers between 1 and 1000, inclusive, that are divisible by either 4 or 6 but not both. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all integers between 10 and 99, inclusive. How many numbers in S have the property that the sum of their digits is 10?", "role": "user" } ]

[ { "content": "Question:\nA right triangle has integer side lengths and a hypotenuse of length 13. Find the area of this triangle.", "role": "user" } ]

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

[ { "content": "Question:\nHow many four-digit numbers are there such that all four digits are distinct and the number is divisible by 5?", "role": "user" } ]

[ { "content": "Question:\nTwo fair six-sided dice are rolled. What is the probability that the product of the numbers rolled is a perfect square?", "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 xy?", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, angle C is a right angle, and the lengths of the legs are AC = 8 and BC = 15. A circle is inscribed in triangle ABC. What is the radius of the inscribed circle?", "role": "user" } ]

[ { "content": "Question:\nA five-digit number is formed using the digits 1, 2, 3, 4, and 5 exactly once. How many such numbers are divisible by 6?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all positive integers n such that 1 ≤ n ≤ 100 and the remainder when n is divided by 7 is equal to the remainder when n is divided by 11. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nLet a and b be positive integers such that a^2 + b^2 = 2020. What is the number of ordered pairs (a, b) that satisfy this equation?", "role": "user" } ]

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

[ { "content": "Question:\nLet $ABC$ be a triangle with $AB = 7$, $BC = 8$, and $CA = 9$. Find the length of the altitude from $A$ to side $BC$.", "role": "user" } ]

[ { "content": "Question:\nFor positive integers $n$, define $S(n)$ as the sum of the digits of $n$. Find the number of integers $n$ between $100$ and $999$ (inclusive) such that $S(n) = 12$.", "role": "user" } ]

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

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

[ { "content": "Question:\nLet n be the smallest positive integer such that 3^n ends with the digits 001. 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 different four-digit positive integers have digits that sum to 10?", "role": "user" } ]

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

[ { "content": "Question:\nLet x be a real number such that x^2 - 6x + 13 = 0. What is the value of x^4 + x^2 + 1?", "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^3 + y^3.", "role": "user" } ]

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

[ { "content": "Question:\nIn triangle ABC, the sides have lengths AB = 5, BC = 7, and CA = 8. What is the area of triangle ABC?", "role": "user" } ]

[ { "content": "Question:\nA bag contains 7 red balls and 5 blue balls. Two balls are drawn at random without replacement. What is the probability that both balls are red?", "role": "user" } ]

[ { "content": "Question:\nFind the smallest positive integer n such that the sum of the digits of n and the sum of the digits of 2n are both equal to 10.", "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 a positive integer. For how many values of x is the area of the triangle also an integer?", "role": "user" } ]

[ { "content": "Question:\nThe sum of three consecutive positive integers is equal to one-fourth the square of the smallest of the three. Find the value of the smallest integer.", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all three-digit integers whose digits sum to 9. Find the number of elements in S that are divisible by 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 x and y be real numbers such that x^2 + y^2 = 25 and x + y = 7. Find the value of xy.", "role": "user" } ]

[ { "content": "Question:\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:\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 positive integers n such that the sum of the digits of n is 12 and n < 100. Find the sum of all elements in S.", "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 a multiple of 5?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of positive integers less than 100 that are multiples of 6 or 8 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^3 + y^3.", "role": "user" } ]

[ { "content": "Question:\nA regular hexagon has area 54. What is the length of one 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 positive real numbers such that abc = 8 and a + b + c = 6. What is the minimum possible 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 multiples of 3 or 5. What is the sum of all the elements of S?", "role": "user" } ]

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

[ { "content": "Question:\nA function f satisfies f(x) + 2f(1 - x) = x^2 for all real x. Find f(2).", "role": "user" } ]

[ { "content": "Question:\nHow many four-digit positive integers are there such that the sum of the digits is 9?", "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) with a > b.", "role": "user" } ]

[ { "content": "Question:\nLet n be the smallest positive integer such that 5n is a perfect square and 8n is a perfect cube. What is the value of n?", "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:\nA positive integer is selected at random from the integers 1 to 1000, inclusive. What is the probability that its base-10 representation contains exactly two digits that are the digit 7?", "role": "user" } ]

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

[ { "content": "Question:\nLet f(x) = x^4 - 8x^2 + 16. Find the sum of all real values of x for which f(x) = 0.", "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^3 + y^3.", "role": "user" } ]

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

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

[ { "content": "Question:\nLet S be the set of 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:\nIf a, b, and c are integers such that 2^a + 2^b + 2^c = 56, where a > b > c > 0, find the value of a + b + c.", "role": "user" } ]

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

[ { "content": "Question:\nA right triangle has legs of integer length, and its perimeter is 30. What is the length of the hypotenuse?", "role": "user" } ]

[ { "content": "Question:\nIn how many ways can 5 identical balls be placed into 3 distinct boxes so that no box is empty?", "role": "user" } ]

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

[ { "content": "Question:\nLet n be the smallest positive integer such that 2^n ≡ 1 (mod 65). Find the value of n.", "role": "user" } ]

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

[ { "content": "Question:\nIn triangle ABC, AB = 7, AC = 9, and angle BAC = 60°. Find the area of triangle ABC in simplest form.", "role": "user" } ]

[ { "content": "Question:\nLet S be the sum of all three-digit numbers that leave a remainder of 1 when divided by 5. Find the value of S.", "role": "user" } ]

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

[ { "content": "Question:\nLet a and b be positive integers such that a^2 - b^2 = 77. How many ordered pairs (a, b) satisfy this equation?", "role": "user" } ]

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

[ { "content": "Question:\nHow many integers between 100 and 999 (inclusive) have digits that sum to exactly 19?", "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:\nEvaluate the sum \$ \\sum_{k=1}^{5} \\frac{1}{k(k+1)} \$ and 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. What is the sum of all possible values of x + y?", "role": "user" } ]

[ { "content": "Question:\nA triangle has side lengths 7, 24, and x, where x is an integer. What is the number of possible values of x that result in a triangle with integer area?", "role": "user" } ]

[ { "content": "Question:\nHow many integers n between 1 and 1000 inclusive are such that the sum of the digits of n is divisible by 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 = 20. How many elements of S are there?", "role": "user" } ]

[ { "content": "Question:\nLet f(x) = x^4 - 8x^2 + 16. What is the sum of all real values of x for which f(x) = 0?", "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:\nA rectangle has a perimeter of 60 units. If the length is twice the width, what is the area of the rectangle?", "role": "user" } ]

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

[ { "content": "Question:\nLet n be the smallest positive integer such that 7n is a perfect square and n is divisible by 10. Find n.", "role": "user" } ]

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

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

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

[ { "content": "Question:\nLet x and y be positive integers satisfying 3x + 4y = 100. Find the number of ordered pairs (x, y) that satisfy this equation.", "role": "user" } ]

[ { "content": "Question:\nEvaluate the sum of the squares of all positive integers less than 100 that are divisible by 7.", "role": "user" } ]

[ { "content": "Question:\nLet a, b, and c be real numbers such that a + b + c = 9 and ab + bc + ca = 26. Find the maximum possible value of c.", "role": "user" } ]

[ { "content": "Question:\nLet n be the smallest positive integer such that 3n is a perfect square and 4n is a perfect cube. Find the value of n.", "role": "user" } ]

[ { "content": "Question:\nA circle passes through the vertices A and B of a square ABCD and is tangent to the side CD at its midpoint. If the side length of the square is 6, 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 by 7. How many elements are in S?", "role": "user" } ]

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

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

[ { "content": "Question:\nLet S be the set of integers from 1 to 1000, inclusive. How many numbers in S are divisible by neither 4 nor 6?", "role": "user" } ]

[ { "content": "Question:\nThe sum of three consecutive positive integers is 99. What is the product of the smallest and largest of these integers?", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, AB = 13, AC = 15, and BC = 14. Let D be the point on BC such that AD bisects angle BAC. Find the length of BD.", "role": "user" } ]

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

[ { "content": "Question:\nLet x be a real number such that x + 1/x = 4. Compute x^3 + 1/x^3.", "role": "user" } ]