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[
{
"content": "Question: Let $a$ and $b$ be real numbers such that $a^2 + b^2 = 25$ and $a + b = 7$. Compute 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 a² + b² ... |
[
{
"content": "Question: Let $x, y$ be real numbers such that $x + y = 4$ and $x^2 + y^2 = 10$. Compute 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 and... |
[
{
"content": "Question: A circle with center $O$ has radius $5$. Points $A$ and $B$ lie on the circle such that the length of chord $AB$ is $6$. Find the distance from $O$ to chord $AB$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think... |
[
{
"content": "Question: Let $a$ and $b$ be positive real numbers such that $a + b = 10$ and $a^2 + b^2 = 58$. What is 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 tha... |
[
{
"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 and x² + ... |
[
{
"content": "Question: The sum of three consecutive positive integers is equal to $111$. What is the largest of these integers?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, let me try to figure out this problem. The question... |
[
{
"content": "Question: Let $x$ and $y$ be real numbers such that $x + y = 6$ and $x^2 + y^2 = 20$. Find the value of $x^3 + y^3$.\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: If $a$ and $b$ are 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, so I need to find the value of ab given that a squared p... |
[
{
"content": "Question: If $x$ and $y$ are real numbers such that $x+y=6$ and $x^2+y^2=20$, find all possible values of $xy$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find all possible values of xy given that... |
[
{
"content": "Question: If $x$ and $y$ are real numbers such that $x + y = 6$ and $x^2 + y^2 = 20$, compute 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 $ g... |
[
{
"content": "Question: Let $x$ and $y$ be real numbers such that $x + y = 6$ and $x^2 + y^2 = 20$. 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 $ g... |
[
{
"content": "Question: Let $n$ be a positive integer such that $\\sum_{k=1}^{n} k^2 = 140$. 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 a positive integer n such that the sum of the ... |
[
{
"content": "Question: A triangle has side lengths $7$, $24$, and $25$. Find the area of the 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 a triangle with side lengths 7, 24, and 25. Hm... |
[
{
"content": "Question: Let $n$ be a positive integer such that $2^n + 2^{n+1} + 2^{n+2} = 224$. 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 2^n + 2^{n+1} + 2^{n+... |
[
{
"content": "Question: Let $f(x) = 2x^2 - 5x + 7$. If $f(a) = f(b)$ and $a \\neq b$, what is 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: The sum of the first $n$ positive integers is $210$. 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: Let $a$ and $b$ be real numbers such that $a^2 + b^2 = 10$ and $a + b = 4$. Find the value of $a^3 + b^3$.\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 $ a^3 + b^3 $ give... |
[
{
"content": "Question: The sum of three consecutive positive integers is $123$. What is the product of these three 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 integers ... |
[
{
"content": "Question: If $a$ and $b$ are real numbers satisfying $a^2 + b^2 = 20$ and $a + b = 6$, what is the value of $a^3 + b^3$?\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 $ a^3 + b^3 $ ... |
[
{
"content": "Question: Let $x$ and $y$ be real numbers such that $x + y = 4$ and $x^2 + y^2 = 20$. 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 $x^2 + y^2 = 25$ and $x + y = 7$. 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 digits of a two-digit number is 11. If the digits are reversed, the new number is 27 greater than the original number. What is the original number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think... |
[
{
"content": "Question: Let $S$ be the set of integers between $1$ and $1000$ inclusive that are not divisible by $2$ or $5$. How many elements are in $S$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to figure out ... |
[
{
"content": "Question: Let $ABC$ be a triangle with $AB = 8$, $BC = 15$, and $CA = 17$. 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 length of the altit... |
[
{
"content": "Question: The sum of the first $k$ positive integers is $3003$. Find the value of $k$.\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 k such that the sum of the first k positive inte... |
[
{
"content": "Question: The sum of five consecutive integers is $145$. What is 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 five consecutive integers that add up to 145, a... |
[
{
"content": "Question: Find the number of positive integers $n < 1000$ such that $n$ is divisible by $3$ but not by $5$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find how many positive integers less than 100... |
[
{
"content": "Question: A regular hexagon has area $54\\sqrt{3}$. What is the length of one of its sides?\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 one side of a regular hexagon when its are... |
[
{
"content": "Question: In triangle $ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. Find the area of the circle inscribed in 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 the circle in... |
[
{
"content": "Question: The sum of three consecutive positive integers is a perfect square. What is the smallest possible value of this sum?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, so I need to find three consecutive pos... |
[
{
"content": "Question: A sequence $(a_n)$ is defined by $a_1 = 2$ and $a_{n+1} = 3a_n + 4$ for $n \\geq 1$. Find the value of $a_5$.\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 a₅ in this sequ... |
[
{
"content": "Question: The sum of three consecutive positive integers is $111$. Find the product of these three integers.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "<think>Okay, let me try to figure out this problem. The question says ... |
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