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fykcluster_qwen3_gemma_k6

input large_stringlengths 19 4.75k	output large_stringlengths 4 56.1k	dataset_name stringclasses 5 values	global_index int64 1 250k	local_index int64 1 50k
Determine all positive integers $n > 1$ for which $n + D(n)$ is a power of $10$, where $D(n)$ denotes the largest integer divisor of $n$ satisfying $D(n) < n$.	<think> Okay, let's try to solve this problem where we need to find all positive integers n > 1 such that n + D(n) is a power of 10. D(n) is the largest divisor of n that's less than n, so that would be n divided by its smallest prime factor, right? Because the largest proper divisor of a number is n divided by its sma...	acereason_f50k	9,427	9,427
What rational number does the repeating decimal 0.9999... equal?	<think> Okay, so I need to figure out what rational number the repeating decimal 0.9999... equals. Hmm, I remember that repeating decimals can often be converted into fractions by using algebra. Let me try to recall the method. First, let's call the repeating decimal x. So, let x = 0.9999... The next step is to multip...	acereason_f50k	199	199
What is the probability that the product $a \times b \times c$ is a multiple of 24, where $a$, $b$, and $c$ are integers drawn from the set $S = \{1, 2, 3, \ldots, 2015, 2016\}$?	<think> Okay, so I need to find the probability that the product of three randomly chosen numbers a, b, c from the set S = {1, 2, 3, ..., 2015, 2016} is a multiple of 24. Hmm, probability is usually favorable outcomes over total outcomes. Total outcomes should be 2016^3 since each number is chosen independently. But ho...	acereason_f50k	39,489	39,489
Let $ a $ be a positive real number and $ n $ a positive integer. Define the power tower $ a\|n $ recursively with $ a\|1 = a $ and $ a\|(i+1) = a^{(a\|i)} $ for $ i = 1, 2, \ldots $. For each positive integer $ k $, let $ x_k $ denote the unique positive real number solution of the equation \( x\|k = 10\|(k+...	<think> Okay, let's see. I need to figure out whether x_42 is larger than x_43 or the other way around. The problem involves these power towers, right? Let me start by understanding the definitions. So, a power tower a\|n is defined recursively. For example, a\|1 is just a. Then a\|2 is a^(a\|1) which is a^a. Then a\|3 is ...	acereason_f50k	1,414	1,414
Let's call a natural number "useful" if it contains neither zeros nor repeated digits in its decimal notation, and the product of all its digits is divisible by their sum. Find the two largest consecutive (i.e., differing by 1) useful numbers.	<think> Okay, let's see. I need to find the two largest consecutive natural numbers that are "useful." A useful number is one that has no zeros, no repeated digits, and the product of its digits is divisible by their sum. Hmm. So first, I should probably figure out what makes a number useful, and then look for the larg...	acereason_f50k	18,471	18,471
Find the value of $\frac{x}{y}$ if $\frac{3}{\sqrt{y}} - \frac{1}{\sqrt{x}} = \frac{2}{\sqrt{x} + \sqrt{y}}$.	<think> Okay, let's try to solve this problem. We need to find the value of x/y given the equation 3/sqrt(y) - 1/sqrt(x) = 2/(sqrt(x) + sqrt(y)). Hmm, rational equations can be tricky, but maybe we can manipulate it to get a relationship between x and y. First, I notice that all the terms involve sqrt(x) or sqrt(y). M...	acereason_f50k	15,118	15,118
2014 students are lined up in a row. They follow these rules for counting: if a student calls out a single-digit number, the next student will call out the sum of that number and 8; if a student calls out a two-digit number, the next student will call out the units digit of that number plus 7. The first student starts ...	<think> Okay, let's try to figure out this problem. So, we have 2014 students lined up, and they're counting according to certain rules. The first student starts with 1. Then, depending on whether the number called is a single-digit or two-digit number, the next student does different things. If it's a single-digit num...	acereason_f50k	42,294	42,294
If $ n $ is a positive integer such that $ 2n $ has 28 positive divisors and $ 3n $ has 30 positive divisors, then how many positive divisors does $ 6n $ have?	<think> Okay, so I need to find the number of positive divisors of 6n, given that 2n has 28 divisors and 3n has 30 divisors. Hmm. Let me start by recalling that the number of divisors of a number is determined by its prime factorization. If a number N has a prime factorization of \( N = p_1^{a_1} p_2^{a_2} \dots p_k^{a...	acereason_f50k	17,523	17,523
Given that $a + b + c + d + e + f + g = 1$ and $a, b, c, d, e, f, g$ are non-negative, find the minimum value of $\max(a+b+c, b+c+d, c+d+e, d+e+f, e+f+g)$.	<think> Okay, let's see. I need to find the minimum value of the maximum of these five expressions: a+b+c, b+c+d, c+d+e, d+e+f, and e+f+g. All the variables a through g are non-negative and their sum is 1. Hmm, this feels like an optimization problem where I have to distribute the total sum of 1 among the variables suc...	acereason_f50k	24,712	24,712
Let $ m, n \in \mathbb{Z} $, and let $ k = (4n - 1)^3 - 4m^2 $. Suppose there is no prime of the form $ p = 4t + 1 $ dividing $ m $. Find the sum of all possible values of $ k $ for $ n = 1 $ and $ -10 \leq m \leq 10 $.	<think> Alright, let's tackle this problem step by step. So, we have integers m and n, and k is defined as (4n - 1)^3 minus 4m squared. The problem states that there are no primes of the form p = 4t + 1 dividing m. We need to find the sum of all possible values of k when n = 1 and m ranges from -10 to 10. First, let'...	acereason_f50k	37,213	37,213
If an integer $8$ is a solution of the equation $x^2 - ax + b = 0$ and the representation of $a$ in the base-$n$ number system is $18$, then what is the base-$n$ representation of $b$?	<think> Okay, let me try to solve this problem step by step. The question says that the integer 8 is a solution to the quadratic equation $x^2 - ax + b = 0$. Also, the number $a$ in base-$n$ is written as 18. We need to find the base-$n$ representation of $b$. First, since 8 is a solution to the equation, I ...	acereason_f50k	20,304	20,304
How can the expression $x(a-b)^2 + y(b-c)^2 + z(c-a)^2$ be converted to the form $m(a-b)^2 + n(a-c)(b-c)$?	<think> Okay, so I need to convert the expression $ x(a-b)^2 + y(b-c)^2 + z(c-a)^2 $ into the form $ m(a-b)^2 + n(a-c)(b-c) $. Hmm, let's see. Alright, let's start by expanding both expressions and then try to match the coefficients. First, let me write down the original expression and the target expression clear...	acereason_f50k	42,684	42,684
Find the remainder when $x^4 + 1$ is divided by $x^2 - 3x + 5.$	<think> Okay, so I need to find the remainder when $ x^4 + 1 $ is divided by $ x^2 - 3x + 5 $. Hmm, let's think about polynomial division. When you divide a polynomial by another polynomial, the remainder has a degree that's less than the divisor. Since the divisor here is quadratic (degree 2), the remainder should...	acereason_f50k	47,522	47,522
Find a trigonometric cosine expression that evaluates to $\sqrt{2+\sqrt{2+\sqrt{2}}}$.	<think> Okay, so I need to find a cosine expression that equals √(2 + √(2 + √2)). Hmm, this looks like it's related to multiple-angle formulas or maybe half-angle identities in trigonometry. Let me think. I remember that expressions with nested square roots like this often come from the cosine of angles that are fracti...	acereason_f50k	15,079	15,079

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fykcluster_qwen3_gemma_k6

This repository contains k-means cluster subsets for training. Each subset is a config (name=cluster_X), backed by Parquet.

Load examples

from datasets import load_dataset

# Single cluster
ds0 = load_dataset("{username}/fykcluster_qwen3_gemma_k6", name="cluster_0")

# Specific cluster
ds3 = load_dataset("{username}/fykcluster_qwen3_gemma_k6", name="cluster_3")

# Or with data_files directly (optional alternative):
ds = load_dataset("{username}/fykcluster_qwen3_gemma_k6", data_files={"train": "cluster_3/*.parquet"}, split="train")

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