category stringclasses 2
values | source stringclasses 6
values | input stringlengths 17 21.5k | output stringlengths 3.96k 46.8k | output_token_count int64 970 8.97k | input_token_count int64 6 5.07k | total_count int64 3k 9k | id int64 0 139k | messages listlengths 2 2 |
|---|---|---|---|---|---|---|---|---|
math | OpenMathReasoning | For the given matrix \( T = \begin{pmatrix} 0.1 & 0.9 \\ 0.7 & 0.3 \end{pmatrix} \), verify that the row vector \( v = \begin{pmatrix} 0.4 & 0.6 \end{pmatrix} \) is the steady-state vector. | <think>
Okay, let me try to figure out how to verify that the vector v is the steady-state vector for the matrix T. Hmm, steady-state vector... I remember that a steady-state vector is a probability vector that doesn't change when multiplied by the transition matrix. So, basically, if we multiply v by T, we should get ... | 7,965 | 77 | 8,042 | 0 | [
{
"role": "user",
"content": "For the given matrix \\( T = \\begin{pmatrix} 0.1 & 0.9 \\\\ 0.7 & 0.3 \\end{pmatrix} \\), verify that the row vector \\( v = \\begin{pmatrix} 0.4 & 0.6 \\end{pmatrix} \\) is the steady-state vector."
},
{
"role": "assistant",
"content": "<think>\nOkay, let me try t... |
math | NuminaMath-CoT | Given \( n \geqslant 3 \) is a positive odd number, let \( z=\cos \frac{2 \pi}{n}+\mathrm{i} \sin \frac{2 \pi}{n} \). Consider the sets:
$$
\begin{array}{l}
A=\left\{z, z^{2}, \cdots, z^{n-1}\right\}, \\
B=\left\{1+z, 1+z+z^{2}, \cdots, 1+z+\cdots+z^{n-1}\right\}.
\end{array}
$$
Find \( A \cap B \). | <think>
Okay, let's see. I need to find the intersection of sets A and B, where A is the set of powers of z from z^1 to z^{n-1}, and B is the set of partial sums of z starting from 1+z up to 1+z+...+z^{n-1}. The number n is an odd integer greater than or equal to 3.
First, let me recall that z is a primitive nth root... | 6,196 | 136 | 6,332 | 1 | [
{
"role": "user",
"content": "Given \\( n \\geqslant 3 \\) is a positive odd number, let \\( z=\\cos \\frac{2 \\pi}{n}+\\mathrm{i} \\sin \\frac{2 \\pi}{n} \\). Consider the sets:\n$$\n\\begin{array}{l}\nA=\\left\\{z, z^{2}, \\cdots, z^{n-1}\\right\\}, \\\\\nB=\\left\\{1+z, 1+z+z^{2}, \\cdots, 1+z+\\cdots+z^... |
math | NuminaMath-CoT | Find the derivative of the specified order.
$$
y=\left(x^{2}+3\right) \ln (x-3), \quad y^{(IV)}=?
$$ | <think>
Okay, so I need to find the fourth derivative of the function y = (x² + 3) ln(x - 3). Hmm, derivatives can get tricky when you have to go up to the fourth order, especially with a product of functions like a polynomial and a logarithm. Let me break this down step by step.
First, the original function is y = (x... | 5,196 | 39 | 5,235 | 2 | [
{
"role": "user",
"content": "Find the derivative of the specified order.\n\n$$\ny=\\left(x^{2}+3\\right) \\ln (x-3), \\quad y^{(IV)}=?\n$$"
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to find the fourth derivative of the function y = (x² + 3) ln(x - 3). Hmm, derivatives can g... |
math | NuminaMath-CoT | In triangle \(ABC\), the angle bisector \(AP\) is drawn. It is known that \(BP = 16\), \(PC = 20\), and the circumcenter of triangle \(ABP\) lies on segment \(AC\). Find side \(AB\). | <think>
Okay, so I have this geometry problem here. Let me read it again carefully.
"In triangle ABC, the angle bisector AP is drawn. It is known that BP = 16, PC = 20, and the circumcenter of triangle ABP lies on segment AC. Find side AB."
Hmm. Let me visualize the triangle first. Triangle ABC, with angle bisector A... | 6,069 | 59 | 6,128 | 3 | [
{
"role": "user",
"content": "In triangle \\(ABC\\), the angle bisector \\(AP\\) is drawn. It is known that \\(BP = 16\\), \\(PC = 20\\), and the circumcenter of triangle \\(ABP\\) lies on segment \\(AC\\). Find side \\(AB\\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I have this ... |
math | NuminaMath-CoT | Let \( R \) be the reals. Define a sequence \( a_n \) by \( a_1 = \alpha \in R \), \( a_{n+1} = \cos a_n \). Show that \( a_n \) converges to a limit independent of \(\alpha\). | <think>
Okay, so I need to show that the sequence defined by a₁ = α, where α is a real number, and a_{n+1} = cos(a_n) converges to a limit that doesn't depend on α. Hmm, interesting. Let me think about this step by step.
First, I recall that for recursively defined sequences, especially ones involving functions like c... | 3,908 | 64 | 3,972 | 4 | [
{
"role": "user",
"content": "Let \\( R \\) be the reals. Define a sequence \\( a_n \\) by \\( a_1 = \\alpha \\in R \\), \\( a_{n+1} = \\cos a_n \\). Show that \\( a_n \\) converges to a limit independent of \\(\\alpha\\)."
},
{
"role": "assistant",
"content": "<think>\nOkay, so I need to show t... |
math | NuminaMath-CoT | A TV station aired $r$ commercials over $n$ days.
1. If at least $p$ commercials are aired each day $(n p \leqslant r)$, how many ways can the commercials be aired?
2. If at most $q$ commercials are aired each day $(n q \geqslant r)$, how many ways can the commercials be aired?
3. If the number of commercials aired e... | <think>
Okay, so I have this problem about a TV station airing commercials over several days, and I need to find the number of ways they can schedule these commercials under different constraints. There are three parts to the problem, each with a different condition: at least p commercials per day, at most q commercial... | 3,050 | 109 | 3,159 | 5 | [
{
"role": "user",
"content": "A TV station aired $r$ commercials over $n$ days. \n\n1. If at least $p$ commercials are aired each day $(n p \\leqslant r)$, how many ways can the commercials be aired?\n2. If at most $q$ commercials are aired each day $(n q \\geqslant r)$, how many ways can the commercials be... |
math | OpenMathReasoning | "For $x \\ge y$ and $(x, y) \\in \\mathbb{R^+}$, find the minimum value of $\\frac{(x+y)!(x-y)!}{2x!(...TRUNCATED) | "<think>\nOkay, let's try to figure out the minimum value of the expression \\(\\frac{(x+y)!(x-y)!}{(...TRUNCATED) | 7,338 | 45 | 7,383 | 6 | [{"role":"user","content":"For $x \\ge y$ and $(x, y) \\in \\mathbb{R^+}$, find the minimum value of(...TRUNCATED) |
code | MagicoderEvolInstruct | "Given a string, return the count of the number of words in the string, where a word is defined as a(...TRUNCATED) | "<think>\nOkay, I need to solve this problem where I have to count the number of words in a string. (...TRUNCATED) | 3,501 | 84 | 3,585 | 7 | [{"role":"user","content":"Given a string, return the count of the number of words in the string, wh(...TRUNCATED) |
math | OpenMathReasoning | Simplify the expression \(\frac{2\sqrt{x} - 3\sqrt{y}}{\sqrt{x} + \sqrt{y}}\). | "<think>\nOkay, so I need to simplify the expression (2√x - 3√y)/(√x + √y). Hmm, let's see. (...TRUNCATED) | 4,493 | 33 | 4,526 | 8 | [{"role":"user","content":"Simplify the expression \\(\\frac{2\\sqrt{x} - 3\\sqrt{y}}{\\sqrt{x} + \\(...TRUNCATED) |
math | NuminaMath-CoT | "For two perpendicular lines \\(a\\) and \\(b\\) in a plane, \\((a, b)\\) is called a \"perpendicula(...TRUNCATED) | "<think>\nOkay, let's see. The problem is about finding the maximum number of perpendicular pairs po(...TRUNCATED) | 5,234 | 75 | 5,309 | 9 | [{"role":"user","content":"For two perpendicular lines \\(a\\) and \\(b\\) in a plane, \\((a, b)\\) (...TRUNCATED) |
End of preview. Expand in Data Studio
No dataset card yet
- Downloads last month
- 63