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 3n leaves a remainder of 2 when divided by 7. What is the value of n?", "role": "user" } ]
[ { "content": "Question:\nA triangle has side lengths 7, 24, and x, where x is a positive integer. If the area of the triangle is an integer, what is the smallest possible value of x?", "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:\nSuppose the quadratic equation x^2 - (2k+3)x + (k^2 + 2k) = 0 has two integer roots. Find the sum of all possible values of k.", "role": "user" } ]
[ { "content": "Question:\nIn how many ways can the letters in the word 'ALGORITHM' be arranged 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. How many ordered pairs (a, b) satisfy this equation?", "role": "user" } ]
[ { "content": "Question:\nA circle with radius 5 is tangent to both the x-axis and the y-axis. What is the area of the region inside the circle and above the line x + y = 5?", "role": "user" } ]
[ { "content": "Question:\nThere are 5 different books to be placed on a shelf. In how many ways can the books be arranged so that no two books originally next to each other remain adjacent?", "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:\nHow many positive integers n less than 1000 are divisible by neither 3 nor 5?", "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:\nHow many positive integers less than 1000 are divisible by neither 3 nor 5?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 8, AC = 6, and angle BAC = 60°. Find the area of triangle ABC.", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that the sum 1 + 2 + ... + n is divisible by 100. What is n?", "role": "user" } ]
[ { "content": "Question:\nA drawer contains 7 red socks and 9 blue socks. If two socks are drawn at random without replacement, what is the probability that both are the same color? Express your answer as a reduced fraction.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of integers between 1 and 1000 (inclusive) that are divisible by either 3 or 5 but not both. Find the number of elements in S.", "role": "user" } ]
[ { "content": "Question:\nA triangle has side lengths 7, 24, and 25. Find the area of the triangle.", "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:\nHow many positive integers less than 1000 are perfect squares or perfect cubes?", "role": "user" } ]
[ { "content": "Question:\nIn a regular hexagon, all diagonals are drawn. How many intersection points (inside the hexagon) are formed by these diagonals?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that when 2024 is divided by n, the remainder is 24. Find the value of n.", "role": "user" } ]
[ { "content": "Question:\nA triangle has side lengths 7, 8, and x, where x is a positive integer. If the area of the triangle is also an integer, find the sum of all possible values of x.", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit numbers are there such that the sum of their digits is exactly 10?", "role": "user" } ]
[ { "content": "Question:\nLet a and b be positive integers such that a^2 + b^2 = 2025. How many ordered pairs (a, b) satisfy this equation?", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, angle C is a right angle, and the length of altitude from C to hypotenuse AB is 6. If the lengths of sides AC and BC are both integers and AC > BC, find the number of possible ordered pairs (AC, BC).", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 6 and x^2 + y^2 = 20. What is the value of xy?", "role": "user" } ]
[ { "content": "Question:\nA rectangle has an area of 60 and a perimeter of 34. What is the length of its longer side?", "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 sum to 9. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nFind the smallest positive integer n such that the sum 1 + 2 + 3 + ... + n is a perfect square greater than 1.", "role": "user" } ]
[ { "content": "Question:\nLet x be a positive real number such that x + 1/x = 5. Compute the value of x^3 + 1/x^3.", "role": "user" } ]
[ { "content": "Question:\nA triangle has side lengths 7, 8, and 9. Find the area of the triangle.", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 2 nor 5?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of positive integers less than or equal to 1000 that are perfect squares. Find the sum of the elements of 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 x and y be real numbers such that x + y = 10 and x^2 + y^2 = 58. What is the value of xy?", "role": "user" } ]
[ { "content": "Question:\nA triangle has side lengths 5, 12, and 13. Let h denote the length of the altitude to the side of length 13. What is the value of h?", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 2 nor 5?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all integers n such that 1 ≤ n ≤ 100 and the greatest common divisor of n and 100 is 1. Find the sum of all elements in S.", "role": "user" } ]
[ { "content": "Question:\nSuppose a regular hexagon has area 54. What is the length of a side of the hexagon?", "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:\nIn how many ways can the letters in the word \"BALLOON\" be arranged so that no two L's are adjacent?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of integers n between 1 and 1000 inclusive such that n is divisible by 7 but not by 5. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nA right triangle has legs of integer length and a hypotenuse of length 25. What is the sum of all possible values of the area of such a triangle?", "role": "user" } ]
[ { "content": "Question:\nSuppose that a sequence {a_n} is defined by a_1 = 2 and a_{n+1} = 3a_n + 1 for n ≥ 1. Find the value of a_5.", "role": "user" } ]
[ { "content": "Question:\nLet the quadratic equation x^2 - (a+3)x + a = 0 have integer roots. Find the sum of all possible values 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 exactly 10?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of 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 ABCD be a square of side length 8. Points E and F are chosen on sides AB and BC, respectively, such that AE = BF = 3. Find the area of triangle AEF.", "role": "user" } ]
[ { "content": "Question:\nLet n be a positive integer such that 2^n + 2 is divisible by 7. Find the smallest such n greater than 1.", "role": "user" } ]
[ { "content": "Question:\nLet a, b, and c be positive integers such that a + b + c = 12 and a < b < c. How many ordered triples (a, b, c) satisfy these conditions?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of integers between 1 and 1000, inclusive, that are divisible by exactly two distinct primes. Find the number of elements in S.", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 7, AC = 9, and angle BAC = 60°. Find the length of BC.", "role": "user" } ]
[ { "content": "Question:\nLet x be a real number such that x^4 - 8x^2 + 15 = 0. Compute the sum of all possible values of x.", "role": "user" } ]
[ { "content": "Question:\nA fair six-sided die is rolled six times. What is the probability that exactly three different numbers appear among the outcomes?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 8 and x^2 + y^2 = 40. Find the value of x*y.", "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:\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 integers such that a + b + c = 15 and a < b < c. How many ordered triples (a, b, c) satisfy these conditions?", "role": "user" } ]
[ { "content": "Question:\nThe sum of the infinite geometric series 6 + 4 + 8/3 + ... is S. Find the value of 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 x*y.", "role": "user" } ]
[ { "content": "Question:\nIn how many ways can 7 distinct books be arranged on a shelf such that no two mathematics books are next to each other if there are 3 mathematics books among them?", "role": "user" } ]
[ { "content": "Question:\nA circle with center O passes through points A(2, 3) and B(4, 7). If the center O has integer coordinates and the circle passes through the origin (0, 0), find the sum of all possible x-coordinates of O.", "role": "user" } ]
[ { "content": "Question:\nLet triangle ABC have AB = 5, AC = 7, and angle BAC = 60°. Find the area of triangle ABC in simplest form.", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 2 nor 5?", "role": "user" } ]
[ { "content": "Question:\nLet f(x) = x^2 + 2x + 5. For how many integers x between 1 and 100 (inclusive) is f(x) divisible by 7?", "role": "user" } ]
[ { "content": "Question:\nA triangle has side lengths 8, 15, and x, where x is an integer. For how many values of x does the triangle have an area that is also an integer?", "role": "user" } ]
[ { "content": "Question:\nHow many four-digit positive integers have digits that sum to 25?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all positive integers less than 1000 that are divisible by neither 2 nor 5. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nSuppose a regular hexagon has area 72. What is the length of one of its sides, expressed in simplest radical form?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all positive integers less than 1000 that are divisible by 3 or 5 but not both. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x^2 + y^2 = 25 and x + y = 7. Find the value of xy.", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, the side lengths are all positive integers, and the perimeter is 17. If side AB = 7, how many possible values can AC take?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that the decimal representation of 2024n ends with the digits 888. Find n.", "role": "user" } ]
[ { "content": "Question:\nA regular hexagon has side length 2. A point P is chosen randomly inside the hexagon. What is the probability that the distance from P to the center is less than 1? Express your answer as a fraction in simplest form.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all positive integers n such that n^2 + 2n is divisible by 24. Find the sum of all n in S with 1 ≤ n ≤ 100.", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 13, AC = 15, and the area of triangle ABC is 84. Find the length of BC.", "role": "user" } ]
[ { "content": "Question:\nHow many positive integers less than 1000 are divisible by neither 2 nor 5?", "role": "user" } ]
[ { "content": "Question:\nLet f(x) = x^3 - 3x^2 + 2x. Find all real numbers x such that f(f(x)) = 0.", "role": "user" } ]
[ { "content": "Question:\nA fair six-sided die is rolled four times. What is the probability that exactly two different numbers appear among the four rolls?", "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 triangle has sides of integer length. Two of its sides have length 6 and 8, and the perimeter is 21. What is the length of the third side?", "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 by both. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nHow many ordered triples of positive integers (a, b, c) satisfy a + b + c = 10?", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that 2^n ≡ 1 (mod 101). Find n.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 8 and x^2 + y^2 = 34. Find the value of x^3 + y^3.", "role": "user" } ]
[ { "content": "Question:\nA fair coin is tossed six times. What is the probability that exactly three heads appear, and no two heads are consecutive? Express your answer as a common fraction.", "role": "user" } ]
[ { "content": "Question:\nIn triangle ABC, AB = 5, AC = 7, and angle BAC = 60°. Find the area of triangle ABC.", "role": "user" } ]
[ { "content": "Question:\nFind the sum of all positive integers n less than 100 such that n^2 + 2n + 2 is divisible by 5.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all integers between 1 and 1000 (inclusive) that are divisible by neither 2 nor 5. Find the number of elements in S.", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of positive integers less than 100 that are divisible by either 3 or 5 but not both. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nA triangle has sides of lengths 7, 8, and x, where x is an integer. Find the number of possible values of x.", "role": "user" } ]
[ { "content": "Question:\nFind the number of four-digit positive integers whose digits sum to 10.", "role": "user" } ]
[ { "content": "Question:\nLet n be the smallest positive integer such that 2^n ≡ 1 (mod 65). Find n.", "role": "user" } ]
[ { "content": "Question:\nIn parallelogram ABCD, the point E lies on side AB such that AE : EB = 1 : 2. If the area of ABCD is 60, find the area of triangle CED.", "role": "user" } ]
[ { "content": "Question:\nLet x and y be real numbers such that x + y = 10 and x^2 + y^2 = 58. What is the value of xy?", "role": "user" } ]
[ { "content": "Question:\nIn how many ways can the letters of the word \"MATH\" be arranged so that no two vowels are adjacent?", "role": "user" } ]
[ { "content": "Question:\nLet S be the set of all positive integers less than 1000 that are divisible by neither 2 nor 5. How many elements are in S?", "role": "user" } ]
[ { "content": "Question:\nTriangle ABC has sides AB = 13, BC = 14, and CA = 15. Find the length of the altitude from B to AC.", "role": "user" } ]
[ { "content": "Question:\nWhat is the sum of all integers n between 1 and 100 inclusive for which n^2 + 2n + 19 is a perfect square?", "role": "user" } ]

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

[ { "content": "Question:\nA triangle has side lengths 7, 24, and x, where x is a positive integer. If the area of the triangle is an integer, what is the smallest possible value of x?", "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:\nSuppose the quadratic equation x^2 - (2k+3)x + (k^2 + 2k) = 0 has two integer roots. Find the sum of all possible values of k.", "role": "user" } ]

[ { "content": "Question:\nIn how many ways can the letters in the word 'ALGORITHM' be arranged 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. How many ordered pairs (a, b) satisfy this equation?", "role": "user" } ]

[ { "content": "Question:\nA circle with radius 5 is tangent to both the x-axis and the y-axis. What is the area of the region inside the circle and above the line x + y = 5?", "role": "user" } ]

[ { "content": "Question:\nThere are 5 different books to be placed on a shelf. In how many ways can the books be arranged so that no two books originally next to each other remain adjacent?", "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:\nHow many positive integers n less than 1000 are divisible by neither 3 nor 5?", "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:\nHow many positive integers less than 1000 are divisible by neither 3 nor 5?", "role": "user" } ]

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

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

[ { "content": "Question:\nA drawer contains 7 red socks and 9 blue socks. If two socks are drawn at random without replacement, what is the probability that both are the same color? Express your answer as a reduced fraction.", "role": "user" } ]

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

[ { "content": "Question:\nA triangle has side lengths 7, 24, and 25. Find the area of the triangle.", "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:\nHow many positive integers less than 1000 are perfect squares or perfect cubes?", "role": "user" } ]

[ { "content": "Question:\nIn a regular hexagon, all diagonals are drawn. How many intersection points (inside the hexagon) are formed by these diagonals?", "role": "user" } ]

[ { "content": "Question:\nLet n be the smallest positive integer such that when 2024 is divided by n, the remainder is 24. Find the value of n.", "role": "user" } ]

[ { "content": "Question:\nA triangle has side lengths 7, 8, and x, where x is a positive integer. If the area of the triangle is also an integer, find the sum of all possible values of x.", "role": "user" } ]

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

[ { "content": "Question:\nLet a and b be positive integers such that a^2 + b^2 = 2025. How many ordered pairs (a, b) satisfy this equation?", "role": "user" } ]

[ { "content": "Question:\nIn triangle ABC, angle C is a right angle, and the length of altitude from C to hypotenuse AB is 6. If the lengths of sides AC and BC are both integers and AC > BC, find the number of possible ordered pairs (AC, BC).", "role": "user" } ]

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

[ { "content": "Question:\nA rectangle has an area of 60 and a perimeter of 34. What is the length of its longer side?", "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 sum to 9. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nFind the smallest positive integer n such that the sum 1 + 2 + 3 + ... + n is a perfect square greater than 1.", "role": "user" } ]

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

[ { "content": "Question:\nA triangle has side lengths 7, 8, and 9. Find the area of the triangle.", "role": "user" } ]

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

[ { "content": "Question:\nLet S be the set of positive integers less than or equal to 1000 that are perfect squares. Find the sum of the elements of 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 x and y be real numbers such that x + y = 10 and x^2 + y^2 = 58. What is the value of xy?", "role": "user" } ]

[ { "content": "Question:\nA triangle has side lengths 5, 12, and 13. Let h denote the length of the altitude to the side of length 13. What is the value of h?", "role": "user" } ]

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

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

[ { "content": "Question:\nSuppose a regular hexagon has area 54. What is the length of a side of the hexagon?", "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:\nIn how many ways can the letters in the word \"BALLOON\" be arranged so that no two L's are adjacent?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of integers n between 1 and 1000 inclusive such that n is divisible by 7 but not by 5. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nA right triangle has legs of integer length and a hypotenuse of length 25. What is the sum of all possible values of the area of such a triangle?", "role": "user" } ]

[ { "content": "Question:\nSuppose that a sequence {a_n} is defined by a_1 = 2 and a_{n+1} = 3a_n + 1 for n ≥ 1. Find the value of a_5.", "role": "user" } ]

[ { "content": "Question:\nLet the quadratic equation x^2 - (a+3)x + a = 0 have integer roots. Find the sum of all possible values 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 exactly 10?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of 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 ABCD be a square of side length 8. Points E and F are chosen on sides AB and BC, respectively, such that AE = BF = 3. Find the area of triangle AEF.", "role": "user" } ]

[ { "content": "Question:\nLet n be a positive integer such that 2^n + 2 is divisible by 7. Find the smallest such n greater than 1.", "role": "user" } ]

[ { "content": "Question:\nLet a, b, and c be positive integers such that a + b + c = 12 and a < b < c. How many ordered triples (a, b, c) satisfy these conditions?", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of integers between 1 and 1000, inclusive, that are divisible by exactly two distinct primes. Find the number of elements in S.", "role": "user" } ]

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

[ { "content": "Question:\nLet x be a real number such that x^4 - 8x^2 + 15 = 0. Compute the sum of all possible values of x.", "role": "user" } ]

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

[ { "content": "Question:\nLet x and y be real numbers such that x + y = 8 and x^2 + y^2 = 40. Find the value of x*y.", "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:\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 integers such that a + b + c = 15 and a < b < c. How many ordered triples (a, b, c) satisfy these conditions?", "role": "user" } ]

[ { "content": "Question:\nThe sum of the infinite geometric series 6 + 4 + 8/3 + ... is S. Find the value of 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 x*y.", "role": "user" } ]

[ { "content": "Question:\nIn how many ways can 7 distinct books be arranged on a shelf such that no two mathematics books are next to each other if there are 3 mathematics books among them?", "role": "user" } ]

[ { "content": "Question:\nA circle with center O passes through points A(2, 3) and B(4, 7). If the center O has integer coordinates and the circle passes through the origin (0, 0), find the sum of all possible x-coordinates of O.", "role": "user" } ]

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

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

[ { "content": "Question:\nLet f(x) = x^2 + 2x + 5. For how many integers x between 1 and 100 (inclusive) is f(x) divisible by 7?", "role": "user" } ]

[ { "content": "Question:\nA triangle has side lengths 8, 15, and x, where x is an integer. For how many values of x does the triangle have an area that is also an integer?", "role": "user" } ]

[ { "content": "Question:\nHow many four-digit positive integers have digits that sum to 25?", "role": "user" } ]

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

[ { "content": "Question:\nSuppose a regular hexagon has area 72. What is the length of one of its sides, expressed in simplest radical form?", "role": "user" } ]

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

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

[ { "content": "Question:\nIn triangle ABC, the side lengths are all positive integers, and the perimeter is 17. If side AB = 7, how many possible values can AC take?", "role": "user" } ]

[ { "content": "Question:\nLet n be the smallest positive integer such that the decimal representation of 2024n ends with the digits 888. Find n.", "role": "user" } ]

[ { "content": "Question:\nA regular hexagon has side length 2. A point P is chosen randomly inside the hexagon. What is the probability that the distance from P to the center is less than 1? Express your answer as a fraction in simplest form.", "role": "user" } ]

[ { "content": "Question:\nLet S be the set of all positive integers n such that n^2 + 2n is divisible by 24. Find the sum of all n in S with 1 ≤ n ≤ 100.", "role": "user" } ]

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

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

[ { "content": "Question:\nLet f(x) = x^3 - 3x^2 + 2x. Find all real numbers x such that f(f(x)) = 0.", "role": "user" } ]

[ { "content": "Question:\nA fair six-sided die is rolled four times. What is the probability that exactly two different numbers appear among the four rolls?", "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 triangle has sides of integer length. Two of its sides have length 6 and 8, and the perimeter is 21. What is the length of the third side?", "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 by both. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nHow many ordered triples of positive integers (a, b, c) satisfy a + b + c = 10?", "role": "user" } ]

[ { "content": "Question:\nLet n be the smallest positive integer such that 2^n ≡ 1 (mod 101). Find n.", "role": "user" } ]

[ { "content": "Question:\nLet x and y be real numbers such that x + y = 8 and x^2 + y^2 = 34. Find the value of x^3 + y^3.", "role": "user" } ]

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

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

[ { "content": "Question:\nFind the sum of all positive integers n less than 100 such that n^2 + 2n + 2 is divisible by 5.", "role": "user" } ]

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

[ { "content": "Question:\nLet S be the set of positive integers less than 100 that are divisible by either 3 or 5 but not both. How many elements are in S?", "role": "user" } ]

[ { "content": "Question:\nA triangle has sides of lengths 7, 8, and x, where x is an integer. Find the number of possible values of x.", "role": "user" } ]

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

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

[ { "content": "Question:\nIn parallelogram ABCD, the point E lies on side AB such that AE : EB = 1 : 2. If the area of ABCD is 60, find the area of triangle CED.", "role": "user" } ]

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

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

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

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

[ { "content": "Question:\nWhat is the sum of all integers n between 1 and 100 inclusive for which n^2 + 2n + 19 is a perfect square?", "role": "user" } ]