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[
{
"content": "Question: Let $a$ and $b$ be real numbers such that $a + b = 5$ and $a^2 + b^2 = 13$. Find the value of $ab$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the value of ab given that a + b = 5 a... |
[
{
"content": "Question: Let $x$ and $y$ be real numbers such that $x + y = 8$ and $x^2 + y^2 = 34$. Find the value of $xy$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the value of xy given that x + y = 8 a... |
[
{
"content": "Question: Let $x$ and $y$ be real numbers such that $x + y = 6$ and $x^2 + y^2 = 20$. Find the value of $xy$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, let me try to figure out how to find the value of xy giv... |
[
{
"content": "Question: Let $x$ and $y$ be real numbers such that $x + y = 5$ and $x^2 + y^2 = 13$. Find the value of $xy$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, let's see. I need to find the value of xy given that x +... |
[
{
"content": "Question: If $x$ and $y$ are real numbers such that $x^2 + y^2 = 25$ and $x + y = 7$, what is the value of $xy$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the value of $ xy $ given that $ x^... |
[
{
"content": "Question: Let $x$ and $y$ be real numbers such that $x+y=7$ and $x^2+y^2=37$. What is the value of $x^3+y^3$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the value of $ x^3 + y^3 $ given that ... |
[
{
"content": "Question: Let $x$ and $y$ be real numbers such that $x + y = 6$ and $xy = 5$. Compute $x^3 + y^3$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find x³ + y³ given that x + y = 6 and xy = 5. Hmm, let... |
[
{
"content": "Question: A circle with center $O$ passes through points $A$ and $B$. If $AB = 6$ and the measure of angle $AOB$ is $60^\\circ$, what is the radius of the circle?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so ... |
[
{
"content": "Question: There are $5$ red balls and $7$ blue balls in a bag. In how many ways can three balls be selected such that at least one red ball is chosen?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, let's see. I ne... |
[
{
"content": "Question: Let $x$ and $y$ be real numbers such that $x+y=4$ and $x^2+y^2=10$. Find the value of $x^3+y^3$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the value of $ x^3 + y^3 $ given that $ x... |
[
{
"content": "Question: Let $a$ and $b$ be real numbers such that $a^2 + b^2 = 10$ and $a + b = 4$. Find the value of $ab$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, let's see... I need to find the value of ab given that a... |
[
{
"content": "Question: Let $x$ and $y$ be real numbers such that $x + y = 7$ and $x^2 + y^2 = 37$. Find the value of $x^3 + y^3$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the value of $ x^3 + y^3 $ give... |
[
{
"content": "Question: The sum of the first $n$ positive integers is 2016. Find the value of $n$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the value of n where the sum of the first n positive integers e... |
[
{
"content": "Question: The sum of the first $n$ positive integers is $3003$. Find the value of $n$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the value of n where the sum of the first n positive integers... |
[
{
"content": "Question: If $x$ and $y$ are real numbers satisfying $x^2 + y^2 = 13$ and $x + y = 5$, what is the value of $xy$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the value of xy given that x² + y²... |
[
{
"content": "Question: Find the number of ordered triples of positive integers $(a, b, c)$ such that $a + b + c = 20$ and $a \\geq 5$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the number of ordered trip... |
[
{
"content": "Question: Let $x$ and $y$ be real numbers such that $x^2 + y^2 = 10$ and $x + y = 4$. Find the value of $x^3 + y^3$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the value of $ x^3 + y^3 $ give... |
[
{
"content": "Question: Let $x$ and $y$ be real numbers such that $x + y = 4$ and $x^2 + y^2 = 20$. Find the value of $xy$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the value of xy given that x + y = 4 a... |
[
{
"content": "Question: Let $a$ and $b$ be positive real numbers such that $a+b=6$ and $a^2+b^2=20$. Find the value of $ab$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the value of ab given that a and b ar... |
[
{
"content": "Question: A triangle has side lengths $5$, $12$, and $13$. What is the area of the circle inscribed in this triangle?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the area of the circle inscrib... |
[
{
"content": "Question: Let $f(x) = x^2 - 6x + 11$. If $f(a) = f(b)$ and $a \\neq b$, compute the value of $a + b$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the value of $ a + b $ given that $ f(a) = f(b... |
[
{
"content": "Question: Let $a$ and $b$ be positive integers such that $\\frac{a}{b} = \\frac{7}{13}$ and $\\gcd(a, b) = 6$. Find $a + b$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find positive integers a and... |
[
{
"content": "Question: Let $x$ be a real number such that $x^4 - 4x^3 + 6x^2 - 4x + 1 = 81$. Find the sum of all possible values of $x$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to solve this equation: $x^4 - 4... |
[
{
"content": "Question: Let $x$ be a real number such that $x + \\frac{1}{x} = 4$. Find the value of $x^5 + \\frac{1}{x^5}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the value of $ x^5 + \\frac{1}{x^5} $... |
[
{
"content": "Question: The arithmetic mean of five consecutive odd integers is $35$. What is the value of the largest of these five integers?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the largest of five... |
[
{
"content": "Question: For how many integers $n$ between $1$ and $100$, inclusive, is $\\gcd(n, 100) = 1$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to figure out how many integers between 1 and 100 inclusive ha... |
[
{
"content": "Question: In triangle $ABC$, the side lengths satisfy $AB = 13$, $BC = 14$, and $CA = 15$. Find the length of the altitude from $B$ to $AC$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the len... |
[
{
"content": "Question: Find the number of distinct ordered triples of positive integers $(a, b, c)$ such that $a + b + c = 10$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the number of distinct ordered tr... |
[
{
"content": "Question: The sum of the squares of three consecutive positive integers is $365$. What is the smallest of these integers?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find three consecutive positive... |
[
{
"content": "Question: Let $ABC$ be a triangle with $AB = 7$, $BC = 8$, and $CA = 9$. Compute the length of the altitude from $B$ to $AC$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the length of the alti... |
[
{
"content": "Question: In triangle $ABC$, $AB = AC = 5$, and $BC = 6$. Find the area of triangle $ABC$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the area of triangle ABC where AB and AC are both 5 units... |
[
{
"content": "Question: Triangle $ABC$ has side lengths $AB = 7$, $BC = 8$, and $CA = 9$. Find the length of the altitude from $B$ to side $AC$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find the length of the... |
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