input large_stringlengths 19 5.98k | output large_stringlengths 86 60.8k | kmeans_label int32 0 0 | dataset_id int16 0 4 | dataset_name stringclasses 5
values | global_index int64 0 250k | local_index int64 0 50k |
|---|---|---|---|---|---|---|
Let $\triangle ABC$ be an equilateral triangle with sides of length 6, and let $P$ be a point inside the triangle. Find the point $P$ such that the distances from $P$ to the three vertices $A$, $B$, and $C$ are in the ratio $1:2:3$. | <think> Okay, so I have this problem: there's an equilateral triangle ABC with each side length 6, and I need to find a point P inside the triangle such that the distances from P to the three vertices A, B, and C are in the ratio 1:2:3. Hmm, that seems challenging but interesting. Let me think about how to approach thi... | 0 | 2 | openthoughts_f50k | 115,445 | 15,445 |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Solve the equation \( e^{\sin(x - \frac{\pi}{4})} = \tan x \) in \( \mathbb{R} \). | <think>
Okay, so I need to solve the equation \( e^{\sin(x - \frac{\pi}{4})} = \tan x \) for real numbers x. Hmm, let's start by understanding what's going on here. The left side is an exponential function with a sine function inside its exponent, and the right side is the tangent of x. Both sides have different behavi... | 0 | 4 | nemotron_f50k | 226,827 | 26,827 |
In triangle \(ABC\), given that \(c > a\), \(c > b\), and \(a^2 + b^2 = 2Rc\), find the measure of angle \(C\). | <think>
Okay, let's see. I need to find the measure of angle C in triangle ABC where it's given that side c is longer than a and b, and also that a² + b² equals 2R times c. Hmm. Alright, let's start by recalling some basic triangle properties and theorems that might help here.
First, I know that in any triangle, the L... | 0 | 0 | acereason_f50k | 36,742 | 36,742 |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Solve the equation $64x^{6} -96x^{4} +36x^{2} -3=0$, giving the answer in trigonometric form. | <think>
Okay, so I need to solve the equation 64x⁶ - 96x⁴ + 36x² - 3 = 0. Hmm, that looks a bit complicated with all those powers. Let me see if I can simplify it somehow. Maybe substitution?
Right, since all the exponents are even, I can let y = x². Then the equation becomes 64y³ - 96y² + 36y - 3 = 0. That should mak... | 0 | 4 | nemotron_f50k | 234,618 | 34,618 |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find the equation of a circle with radius five that is tangent to both branches of \( y = |x| \). | <think>
Okay, let's see. I need to find the equation of a circle with radius 5 that's tangent to both branches of the graph y = |x|. Hmm, okay. So first, I remember that the graph of y = |x| is a V-shaped graph with two branches: one going up to the right and the other up to the left. The circle has to touch both of th... | 0 | 4 | nemotron_f50k | 226,068 | 26,068 |
Let \( x, y, z > 0 \) and \( x + y + z = 1 \). Find the maximum and minimum values of \( x^3 + 2y^2 + \frac{10}{3}z \). | <think>
Okay, so I have this problem here: given positive numbers x, y, z such that x + y + z = 1, I need to find the maximum and minimum values of the expression x³ + 2y² + (10/3)z. Alright, let's start thinking about how to approach this.
First, since there's a constraint x + y + z = 1, maybe I can use substitution... | 0 | 0 | acereason_f50k | 16,998 | 16,998 |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Given the function \( y = \frac{x^3 + 2^x}{x^2 + 3^x} \), determine the x-intercept. | <think>
Okay, so I need to find the x-intercept of the function y = (x³ + 2ˣ)/(x² + 3ˣ). Hmm, x-intercepts are where the function crosses the x-axis, which means y is zero there. So, I need to find all the points (x, 0) that satisfy this equation. That happens when the numerator is zero because a fraction is zero when ... | 0 | 4 | nemotron_f50k | 201,890 | 1,890 |
As a decimal, $\frac{9}{10}+\frac{8}{100}$ is:
(A) 1.098
(B) 0.98
(C) 0.098
(D) 0.0908
(E) 9.8 | <think>
Okay, so I need to figure out what 9/10 plus 8/100 is as a decimal. Let me start by understanding each fraction separately.
First, 9/10. Since the denominator is 10, that's a tenth place. So 9 divided by 10 is 0.9. Right? Because 10 goes into 9 zero times, add a decimal point, 10 goes into 90 nine times. So 0.... | 0 | 0 | acereason_f50k | 44,500 | 44,500 |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Given an integer \( n \) with \( n \geq 2 \), find the largest real number \( k_n \) such that for any positive numbers \( a_1, a_2, \ldots, a_n \) satisfying \( a_1 a_2 \cdots a_n = 1 \), the inequality \( a_1^{n-1} + a_2... | <think>
Okay, so I need to find the largest real number \( k_n \) such that for any positive numbers \( a_1, a_2, \ldots, a_n \) with \( a_1 a_2 \cdots a_n = 1 \), the inequality \( a_1^{n-1} + a_2^{n-1} + \cdots + a_n^{n-1} - n \geq k_n \left( \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n} - n \right) \) holds... | 0 | 4 | nemotron_f50k | 217,479 | 17,479 |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find all positive integers \( k \) that satisfy the following conditions:
1. The number \( k \) is odd.
2. For each positive integer \( n \), there exists a positive integer \( a_n \) such that \( a_n^2 + k \) is divisible... | <think>
Okay, so I need to find all positive integers k that are odd and satisfy the condition that for every positive integer n, there's some positive integer a_n such that a_n squared plus k is divisible by 2^n. Hmm, let's break this down.
First, since k is odd, that means k is congruent to 1 mod 2. So k can be writ... | 0 | 4 | nemotron_f50k | 217,373 | 17,373 |
End of preview. Expand in Data Studio
fykcluster_qwen25_qwen3_rank_only_k5
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_qwen25_qwen3_rank_only_k5", name="cluster_0")
# Specific cluster
ds3 = load_dataset("{username}/fykcluster_qwen25_qwen3_rank_only_k5", name="cluster_3")
# Or with data_files directly (optional alternative):
ds = load_dataset("{username}/fykcluster_qwen25_qwen3_rank_only_k5", data_files={"train": "cluster_3/*.parquet"}, split="train")
- Downloads last month
- 13