instruction stringlengths 21 6.8k | input stringclasses 1
value | output stringlengths 815 71.5k | system stringclasses 1
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Given any positive integer \( N \), show that there is just one solution to \( m + \frac{1}{2}(m + n - 1)(m + n - 2) = N \) in positive integers. | <think>
Okay, so I need to show that for any positive integer N, there's exactly one pair of positive integers (m, n) that satisfies the equation m + ½(m + n - 1)(m + n - 2) = N. Hmm, let's start by trying to understand the structure of this equation.
First, let's rewrite the equation to make it a bit clearer. The mai... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1305597,
"category": "math",
"source": "NuminaMath-CoT"
} | |
Given a triangle $ABC$ with inradius $r$, circumradius $R$, and semiperimeter $s$, let $r_a$, $r_b$, and $r_c$ denote the exradii. Define $f(R, r, s) = \frac{R}{r} + 1 - \left(\frac{r_a + r_b + r_c}{s}\right)^2$. Calculate the minimum value of $f(R, r, s)$ over all possible triangles $ABC$. | <think>
Okay, so I have this problem here about a triangle ABC with inradius r, circumradius R, and semiperimeter s. The exradii are r_a, r_b, r_c. The function to minimize is f(R, r, s) = R/r + 1 - ( (r_a + r_b + r_c)/s )². Hmm, interesting. I need to find the minimum value of this function over all possible triangles... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1195915,
"category": "math",
"source": "OpenMathReasoning"
} | |
Find integers \( n \) and \( m \) such that \( 3n^2 + 3n + 7 = m^3 \). | <think>
Okay, so I need to find integers n and m such that 3n² + 3n + 7 equals m cubed. Hmm, let's see. Let me start by understanding the problem. The equation is 3n² + 3n + 7 = m³. Both n and m are integers, so I need to find integer solutions.
First, maybe I should try plugging in some small integer values for n an... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3267285,
"category": "math",
"source": "OpenMathReasoning"
} | |
A point moves along the parabola \( y = x^2 \). Its \( x \)-coordinate is increasing uniformly at 2 units per minute. At what rate is its distance from the point \( (1, -1) \) changing when it passes through \( (2, 4) \)? | <think>
Okay, so I have this problem here where a point is moving along the parabola y equals x squared. The x-coordinate is increasing at a constant rate of 2 units per minute. I need to find the rate at which the distance from the point (1, -1) is changing when the point passes through (2, 4). Hmm, let me think throu... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2099870,
"category": "math",
"source": "OpenMathReasoning"
} | |
Let \(ABC\) be an isosceles triangle with \(AB = AC\). Let \(Q\) be a point on \(BC\) such that \(BQ = 2QC\), and let \(T\) be a point on \(AQ\) such that \(\angle BTQ = \angle BAC\). If \(\angle BAC = \theta\), find the value of \(\angle CTQ\) in terms of \(\theta\). | <think>
Okay, so I have this geometry problem here. Let me read it again and try to visualize it. The problem says:
"Let ABC be an isosceles triangle with AB = AC. Let Q be a point on BC such that BQ = 2QC, and let T be a point on AQ such that angle BTQ = angle BAC. If angle BAC = θ, find the value of angle CTQ in ter... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3900537,
"category": "math",
"source": "OpenMathReasoning"
} | |
You are given a sequence of integers of length n and an integer m. Your task is to find the smallest integer x such that the sum of the floor division of each element in the sequence by x is at least m.
Note that floor division means dividing and taking the floor of the result (e.g., 7 // 3 = 2).
If no such x exists... | <think>
Okay, so I need to find the smallest integer x such that the sum of the floor divisions of each element in the sequence by x is at least m. Hmm. Let me think about how to approach this.
First, the problem is asking for the minimal x. Since x can range from 1 up to the maximum element in the array (because if x... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 2171521,
"category": "code",
"source": "OpenCodeReasoning"
} | |
Determine the value of \( n \) such that \(\frac{1! \cdot 2! \cdot 3! \cdots 100!}{n!}\) is a perfect square. | <think>
Okay, so I need to find the value of n such that when you divide the product of all factorials from 1! up to 100! by n!, the result is a perfect square. Hmm, let me think about how to approach this.
First, I know that a perfect square has all exponents even in its prime factorization. So, the problem boils dow... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1725764,
"category": "math",
"source": "OpenMathReasoning"
} | |
How many homomorphisms are there from \( \mathbb{Z}_n \) to \( \mathbb{Z}_n \), and what are they? | <think>
Okay, so I need to figure out how many homomorphisms there are from ℤ_n to ℤ_n and describe what they are. Hmm, homomorphisms between cyclic groups... I remember that cyclic groups have the property that their homomorphisms are determined by where the generator is sent. Let me think.
First, ℤ_n is a cyclic gro... | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3926347,
"category": "math",
"source": "OpenMathReasoning"
} | |
"Solve the following system in $\\mathbb{R}$:\n\\[\n\\sin(x+y) = \\frac{1}{2}\n\\]\n\\[\n\\cos(x-y) (...TRUNCATED) | "<think>\nAlright, let's see. I need to solve the system of equations involving sine and cosine. The(...TRUNCATED) | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 3915145,
"category": "math",
"source": "OpenMathReasoning"
} | |
"In triangle \\(ABC\\) with \\(ABC\\) not isosceles, the incircle touches the sides \\(BC\\), \\(CA\(...TRUNCATED) | "<think>\nOkay, so I need to find the ratio FG/GL in triangle ABC where the incircle touches the sid(...TRUNCATED) | Please reason step by step, and put your final answer within \boxed{}. | {
"index": 1020534,
"category": "math",
"source": "OpenMathReasoning"
} |
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