prompt stringlengths 24 2k | response stringlengths 4.32k 68.8k | task_name stringclasses 1
value | difficulty stringclasses 1
value | is_correct bool 1
class | shard int64 0 9 | pool_index int64 1 54.5k | problem_index int64 7 546k |
|---|---|---|---|---|---|---|---|
In a triangular pyramid \(SABC\) with edges \(SA = a\), \(SB = b\), \(SC = c\), \(AB = d\), \(BC = e\), and \(CA = f\), a sphere is exscribed, and the touchpoints of this sphere with the faces \(ABC\), \(SAB\), \(SBC\), and \(SCA\) are \(M\), \(N\), \(K\), and \(L\), respectively. How are the volumes of the pyramids \(... | <think>
Okay, let's tackle this problem step by step. It's about finding the relationship between the volumes of a triangular pyramid (tetrahedron) SABC and another tetrahedron MNKL formed by the touchpoints of an exscribed sphere. The problem has four parts depending on the type of pyramid: regular, regular pyramid, i... | math | true | 0 | 22,045 | 22,063 | |
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that
\[ \forall x, y \in \mathbb{R}, \quad f(x+xf(y))=xy+f(x). \] | <think>
Okay, so I need to find all real functions f that satisfy the equation f(x + x f(y)) = x y + f(x) for all real numbers x and y. Hmm, functional equations can be tricky, but let's see what approaches I can take here.
First, maybe I can try plugging in some specific values for x or y to simplify the equation and... | math | true | 0 | 51,468 | 51,515 | |
Find all pairs $(a,b) \in \mathbb{N}$ such that $a^3 + 2a + 1 = 2^b$. | <think>
Okay, let's see. I need to find all natural number pairs (a, b) such that a³ + 2a + 1 equals 2ᵇ. Hmm, starting off, maybe I can try plugging in some small natural numbers for a and see if the left side becomes a power of two. Let's start with a=1.
If a=1: 1³ + 2*1 + 1 = 1 + 2 + 1 = 4. Oh, that's 2². So (1, 2) ... | math | true | 0 | 38,493 | 38,525 | |
Find all positive integers \( x_1, x_2, \ldots, x_{2004} \) that satisfy:
1. \( x_i \geq 2 \) for \( i = 1, 2, \ldots, 2004 \)
2. \( \sum_{i=1}^{2004} x_i^2 + 4 = 2002 \cdot x_1 \cdot x_2 \cdots x_{2004} \) | <think>
Alright, let's try to tackle this problem. So, we have 2004 positive integers, each at least 2, and their squares sum up plus 4 equals 2002 times the product of all these numbers. That seems quite complex, but maybe there's a way to simplify it.
First, let me note the equation again:
x₁² + x₂² + ... + x₂₀₀₄² ... | math | true | 0 | 35,217 | 35,247 | |
Let \( A(0), A(1), A(2), \ldots, A(100) \) and \( B(1), B(2), \ldots, B(100) \) be two sequences of real numbers. For all \( n = 0, 1, \ldots, 99 \), one of the following conditions holds:
1) \( A(n+1) = \frac{A(n)}{2} \) and \( B(n+1) = \frac{1}{2} - A(n) \).
2) \( A(n+1) = 2A(n)^2 \) and \( B(n+1) = A(n) \).
Differe... | <think>
Okay, so I'm trying to solve this problem where there are two sequences, A and B, each with 101 and 100 terms respectively. The sequences are connected through some recursive conditions, and I need to find the maximum possible sum of the B terms from B(1) to B(100), given that A(100) is less than or equal to A(... | math | true | 0 | 4,098 | 4,104 | |
"Find a point \\(P\\) in the plane of triangle \\(A'B'C'\\) such that \\(PA' = a\\), \\(PB' = b\\), (...TRUNCATED) | "<think>\nOkay, so I have this geometry problem here. It says: Find a point P in the plane of triang(...TRUNCATED) | math | true | 0 | 22,094 | 22,113 | |
"Determine the integral \\(\\int_{C} f(z) \\,dz\\) where \\(f(z) = 1 + \\frac{2}{z}\\) and \\(C = \\(...TRUNCATED) | "<think>\nOkay, so I need to compute the integral of the function f(z) = 1 + 2/z along the contour C(...TRUNCATED) | math | true | 0 | 18,562 | 18,577 | |
"Given the equation \\( ax^2 + bx + c = x^3 + \\frac{1}{x} \\) with \\( a, b, c \\leq 0 \\) and the (...TRUNCATED) | "<think>\nOkay, let's see. I need to solve this equation \\( ax^2 + bx + c = x^3 + \\frac{1}{x} \\) (...TRUNCATED) | math | true | 0 | 3,411 | 3,415 | |
"In trapezoid \\(ABCD\\) with major base \\(AB\\) and minor base \\(CD\\), let \\(M\\) and \\(N\\) b(...TRUNCATED) | "<think>\nOkay, let's see. I need to find the area of trapezoid ABCD given that the area of triangle(...TRUNCATED) | math | true | 0 | 10,028 | 10,037 | |
Find the function $y=f(x)$ such that $y^2 \cdot y'' = c$, where $c$ is a constant. | "<think>\nOkay, so I need to find the function y = f(x) that satisfies the differential equation y²(...TRUNCATED) | math | true | 0 | 5,026 | 5,033 |
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