yashmoar/Qwen2.5-3B-SFT-cleaned-difficulty-weighted
3B • Updated • 2
Datasets:
instruction stringlengths 17 1.51k | input stringclasses 1
value | output stringlengths 749 67.8k | system stringclasses 1
value |
|---|---|---|---|
Prove that a graph with five vertices, each of which is connected by an edge to all the others, is not planar. | <think>
Okay, so I need to prove that a complete graph with five vertices, K5, is not planar. Hmm, planar graphs can be drawn on a plane without edges crossing, right? So I need to show that K5 doesn't meet the criteria for planarity.
First, maybe I should recall some theorems related to planar graphs. Oh, right, ther... | Please reason step by step, and put your final answer within \boxed{}. | |
Two players take turns modifying an integer starting from $2011^{2011}$. Each move consists of either subtracting an integer between $1$ and $2010$ inclusive, or dividing by $2011$ and rounding down to the nearest integer. The player who first obtains a non-positive integer wins. Which player has a winning strategy? | <think>
Okay, let's see. So we have this game where two players take turns modifying an integer that starts at 2011^2011. Each move can either subtract a number between 1 and 2010 inclusive, or divide by 2011 and round down. The goal is to be the one who first gets a non-positive integer. We need to figure out which pl... | Please reason step by step, and put your final answer within \boxed{}. | |
Is it true that if the sum of the distances from any internal point of a tetrahedron to its faces is constant, then the tetrahedron is regular? | <think>
Okay, so I need to figure out whether a tetrahedron must be regular if the sum of the distances from any interior point to its four faces is constant. Hmm, interesting. Let me start by recalling what a regular tetrahedron is—it's a tetrahedron where all four faces are equilateral triangles, and all edges are eq... | Please reason step by step, and put your final answer within \boxed{}. | |
Solve the equation ${25^x} - 3(2 - x){5^x} - 2x + 7 = 0$. | <think>
Okay, let's try to tackle this equation: 25^x - 3(2 - x)5^x - 2x + 7 = 0. Hmm, that looks a bit complicated. Let me see... So, 25^x is the same as (5^2)^x, which simplifies to 5^(2x). So maybe I can substitute 5^x with another variable to make it easier. Let's try that.
Let me set y = 5^x. Then 25^x becomes y^... | Please reason step by step, and put your final answer within \boxed{}. | |
Find the limit of \((\cos^2(a) \cdot \cos^{\frac{1}{n}}(a) + \sin^2(a) \cdot \sin^{\frac{1}{n}}(a))^n\) as \(n \to +\infty\) for \(a \in (0, \frac{\pi}{2})\). | <think>
Okay, so I need to find the limit as n approaches infinity of [cos²(a) * cos^(1/n)(a) + sin²(a) * sin^(1/n)(a)]^n, where a is between 0 and π/2. Hmm. Let me try to break this down step by step.
First, let's write the expression inside the brackets: cos²(a) * cos^(1/n)(a) + sin²(a) * sin^(1/n)(a). Let me simpli... | Please reason step by step, and put your final answer within \boxed{}. | |
Find all prime numbers \( p \) and \( q \) such that \( q(3q+5) = (2p^2 - 3p + 2)p \). | <think>
Okay, let's try to solve this problem: Find all prime numbers \( p \) and \( q \) such that \( q(3q+5) = (2p^2 - 3p + 2)p \).
First, I need to understand what the equation is saying. It's an equation involving two primes, p and q. Both sides of the equation must be equal, so I need to find primes p and q that... | Please reason step by step, and put your final answer within \boxed{}. | |
A window has a perimeter of 10 meters and is shaped like a rectangle with the top edge replaced by a semicircle. Find the dimensions of the rectangle that maximize the amount of light admitted through the window. | <think>
Okay, so I have this problem where I need to find the dimensions of a window that's a rectangle with a semicircle on top. The perimeter is 10 meters, and I need to maximize the amount of light, which I guess means maximizing the area. Let me try to work through this step by step.
First, let me visualize the wi... | Please reason step by step, and put your final answer within \boxed{}. | |
In the addition of three-digit numbers shown, the letters \(x\) and \(y\) represent different digits. The value of \(y - x\) is
(A) 3
(B) -5
(C) 7
(D) -7
(E) 2 | <think>
Okay, let me try to figure out this problem. So, we have an addition of two three-digit numbers, and the result is another three-digit number. The problem uses the letters x and y to represent different digits, and we need to find the value of y - x. The options are given as (A) 3, (B) -5, (C) 7, (D) -7, (E) 2.... | Please reason step by step, and put your final answer within \boxed{}. | |
Find all functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) such that for all positive integers \( n \), there exists a unique positive integer \( k \) satisfying \( f^k(n) \leq n + k + 1 \). | <think>
Okay, so I need to find all functions f from the natural numbers to themselves such that for every positive integer n, there's a unique positive integer k where the k-th iterate of f at n is less than or equal to n + k + 1. Hmm, let's unpack that.
First, let me make sure I understand the problem. The function ... | Please reason step by step, and put your final answer within \boxed{}. | |
Distinct prime numbers \( p, q, r \) satisfy the equation \( 2pqr + 50pq = 7pqr + 55pr = 8pqr + 12qr = A \) for some positive integer \( A \). Find \( A \). | <think>
Okay, let's try to tackle this problem. It says that distinct prime numbers p, q, r satisfy three equations that all equal some integer A. The equations are:
1. 2pqr + 50pq = A
2. 7pqr + 55pr = A
3. 8pqr + 12qr = A
Hmm, so all three expressions equal A. That means I can set them equal to each other. Maybe sta... | Please reason step by step, and put your final answer within \boxed{}. |
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AceReason 15K: cleaned, difficulty weighted (our final model)
15,000 rows from the cleaned pool, drawn at random with more weight on problems the base model often got wrong in a 4-try test. The cleaned pool is the 47,430 math rows of acereason11_100k left after removing code, solutions cut off at 16,384 tokens, solutions that give up, answers shown to be wrong, very repetitive solutions, overlap with the eval benchmarks, and repeated questions.
The rows come from redsgnaoh/acereason11_100k, a 100K subset of
NVIDIA's AceReason-1.1-SFT; see those pages for the license. The file
cleaned_difficulty_weighted.json is in LLaMA-Factory's format (instruction, input, output, system). Model trained on it:
yashmoar/Qwen2.5-3B-SFT-cleaned-difficulty-weighted. How we built it: GitHub.
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