problem_token_count int64 8 200 | generated_token_count int64 478 3.84k | dataset stringclasses 1
value | conversations listlengths 2 2 |
|---|---|---|---|
43 | 2,174 | DeepMath-4096 | [
{
"content": "Find the auxiliary equation for the ordinary differential equation with constant coefficients: \\((x^2D^2 + xD + 1)y = \\sin(2\\log x)\\sin(\\log x)\\).",
"role": "user"
},
{
"content": "Okay, so I need to find the auxiliary equation for the given ordinary differential equation (ODE) w... |
43 | 2,767 | DeepMath-4096 | [
{
"content": "Find the auxiliary equation for the ordinary differential equation with constant coefficients: \\((x^2D^2 + xD + 1)y = \\sin(2\\log x)\\sin(\\log x)\\).",
"role": "user"
},
{
"content": "Okay, so I need to find the auxiliary equation for this ordinary differential equation (ODE) with c... |
43 | 2,584 | DeepMath-4096 | [
{
"content": "Find the auxiliary equation for the ordinary differential equation with constant coefficients: \\((x^2D^2 + xD + 1)y = \\sin(2\\log x)\\sin(\\log x)\\).",
"role": "user"
},
{
"content": "Okay, so I need to find the auxiliary equation for the given ordinary differential equation (ODE) w... |
25 | 2,865 | DeepMath-4096 | [
{
"content": "Find the limit: \\[ \\lim_{x \\to \\infty} (x!)^{1/x} \\]",
"role": "user"
},
{
"content": "Okay, so I need to find the limit as x approaches infinity of (x!)^(1/x). Hmm, factorials can be tricky because they grow really fast, but maybe I can use some approximations or known formulas f... |
25 | 3,305 | DeepMath-4096 | [
{
"content": "Find the limit: \\[ \\lim_{x \\to \\infty} (x!)^{1/x} \\]",
"role": "user"
},
{
"content": "Okay, so I need to find the limit as x approaches infinity of (x!) raised to the power of 1/x. Hmm, that sounds a bit tricky, but maybe I can use some approximations or known formulas for factor... |
25 | 3,391 | DeepMath-4096 | [
{
"content": "Find the limit: \\[ \\lim_{x \\to \\infty} (x!)^{1/x} \\]",
"role": "user"
},
{
"content": "Alright, so I need to find the limit of (x!) raised to the power of 1/x as x approaches infinity. Hmm, okay. Let me think about how factorials grow. I remember that factorials grow really fast, ... |
49 | 2,518 | DeepMath-4096 | [
{
"content": "Find the expectation \\( \\mathbb{E}[X_t] \\) where \\( X_t = \\sin(B_t) \\) and \\( B_t \\) is a standard Brownian motion for \\( t \\geq 0 \\).",
"role": "user"
},
{
"content": "Okay, so I need to find the expectation of X_t, which is defined as sin(B_t), where B_t is a standard Brow... |
49 | 2,391 | DeepMath-4096 | [
{
"content": "Find the expectation \\( \\mathbb{E}[X_t] \\) where \\( X_t = \\sin(B_t) \\) and \\( B_t \\) is a standard Brownian motion for \\( t \\geq 0 \\).",
"role": "user"
},
{
"content": "Okay, so I need to find the expectation of \\( X_t = \\sin(B_t) \\), where \\( B_t \\) is a standard Brown... |
49 | 1,580 | DeepMath-4096 | [
{
"content": "Find the expectation \\( \\mathbb{E}[X_t] \\) where \\( X_t = \\sin(B_t) \\) and \\( B_t \\) is a standard Brownian motion for \\( t \\geq 0 \\).",
"role": "user"
},
{
"content": "Okay, so I need to find the expectation of X_t, which is defined as sin(B_t), where B_t is a standard Brow... |
41 | 2,314 | DeepMath-4096 | [
{
"content": "Determine the limit of \\( \\frac{\\pi(x)}{x} \\) as \\( x \\to \\infty \\), where \\( \\pi(x) \\) is the prime counting function.",
"role": "user"
},
{
"content": "Okay, so I need to figure out the limit of π(x)/x as x approaches infinity. π(x) is the prime counting function, which me... |
41 | 1,763 | DeepMath-4096 | [
{
"content": "Determine the limit of \\( \\frac{\\pi(x)}{x} \\) as \\( x \\to \\infty \\), where \\( \\pi(x) \\) is the prime counting function.",
"role": "user"
},
{
"content": "Okay, so I need to figure out the limit of π(x) divided by x as x approaches infinity. Hmm, π(x) is the prime counting fu... |
41 | 2,657 | DeepMath-4096 | [
{
"content": "Determine the limit of \\( \\frac{\\pi(x)}{x} \\) as \\( x \\to \\infty \\), where \\( \\pi(x) \\) is the prime counting function.",
"role": "user"
},
{
"content": "Okay, so I need to find the limit of π(x)/x as x approaches infinity, where π(x) is the prime counting function. Hmm, I r... |
78 | 1,746 | DeepMath-4096 | [
{
"content": "Consider a sequence $(r_n)$ resulting from infinite coin flips, where $R_n=1$ if $r_n$ is a head and $R_n=-1$ if $r_n$ is a tail. Determine if $P\\left(\\sum \\frac{R_n}{n} < \\infty\\right) = 1$. Provide a justification for your answer.",
"role": "user"
},
{
"content": "Okay, so I nee... |
78 | 2,023 | DeepMath-4096 | [
{
"content": "Consider a sequence $(r_n)$ resulting from infinite coin flips, where $R_n=1$ if $r_n$ is a head and $R_n=-1$ if $r_n$ is a tail. Determine if $P\\left(\\sum \\frac{R_n}{n} < \\infty\\right) = 1$. Provide a justification for your answer.",
"role": "user"
},
{
"content": "Okay, so I hav... |
78 | 2,297 | DeepMath-4096 | [
{
"content": "Consider a sequence $(r_n)$ resulting from infinite coin flips, where $R_n=1$ if $r_n$ is a head and $R_n=-1$ if $r_n$ is a tail. Determine if $P\\left(\\sum \\frac{R_n}{n} < \\infty\\right) = 1$. Provide a justification for your answer.",
"role": "user"
},
{
"content": "Alright, so I ... |
39 | 2,549 | DeepMath-4096 | [
{
"content": "Find parametric equations for a unit circle with a speed of \\( e^t \\), starting from \\( x=1 \\), \\( y=0 \\). Determine when the circle is completed.",
"role": "user"
},
{
"content": "Okay, so I need to find parametric equations for a unit circle that starts at (1,0) and has a speed... |
39 | 3,077 | DeepMath-4096 | [
{
"content": "Find parametric equations for a unit circle with a speed of \\( e^t \\), starting from \\( x=1 \\), \\( y=0 \\). Determine when the circle is completed.",
"role": "user"
},
{
"content": "Okay, let's tackle this problem step by step. So, we need to find parametric equations for a unit c... |
39 | 1,914 | DeepMath-4096 | [
{
"content": "Find parametric equations for a unit circle with a speed of \\( e^t \\), starting from \\( x=1 \\), \\( y=0 \\). Determine when the circle is completed.",
"role": "user"
},
{
"content": "Okay, so I need to find parametric equations for a unit circle that starts at (1, 0) and has a spee... |
53 | 1,637 | DeepMath-4096 | [
{
"content": "Using the generating function \\( \\sum_{n=0}^{\\infty}P_n(x)r^n=(1-2rx+r^2)^{-\\frac{1}{2}} \\), find the value of \\( P_n(1) \\).",
"role": "user"
},
{
"content": "Okay, so I need to find the value of \\( P_n(1) \\) using the generating function provided. The generating function is g... |
53 | 1,527 | DeepMath-4096 | [
{
"content": "Using the generating function \\( \\sum_{n=0}^{\\infty}P_n(x)r^n=(1-2rx+r^2)^{-\\frac{1}{2}} \\), find the value of \\( P_n(1) \\).",
"role": "user"
},
{
"content": "Alright, so I need to find the value of \\( P_n(1) \\) using the generating function given by \\( \\sum_{n=0}^{\\infty}P... |
53 | 1,523 | DeepMath-4096 | [
{
"content": "Using the generating function \\( \\sum_{n=0}^{\\infty}P_n(x)r^n=(1-2rx+r^2)^{-\\frac{1}{2}} \\), find the value of \\( P_n(1) \\).",
"role": "user"
},
{
"content": "Okay, so I need to find the value of \\( P_n(1) \\) using the generating function given by \\( \\sum_{n=0}^{\\infty}P_n(... |
51 | 2,615 | DeepMath-4096 | [
{
"content": "Evaluate the expression \\( \\log \\left| 1 + \\alpha + \\alpha^2 + \\alpha^3 - \\frac{1}{\\alpha} \\right| \\), where \\( \\alpha \\) is a fifth root of unity.",
"role": "user"
},
{
"content": "Okay, so I need to evaluate the expression \\( \\log \\left| 1 + \\alpha + \\alpha^2 + \\al... |
51 | 3,607 | DeepMath-4096 | [
{
"content": "Evaluate the expression \\( \\log \\left| 1 + \\alpha + \\alpha^2 + \\alpha^3 - \\frac{1}{\\alpha} \\right| \\), where \\( \\alpha \\) is a fifth root of unity.",
"role": "user"
},
{
"content": "Okay, so I need to evaluate the expression \\( \\log \\left| 1 + \\alpha + \\alpha^2 + \\al... |
130 | 2,255 | DeepMath-4096 | [
{
"content": "Find the kernel of the Vandermonde matrix \\( A \\) given by\n\\[\nA = \\begin{pmatrix} x_1^0 & x_1^1 & \\ldots & x_1^n \\\\ x_2^0 & x_2^1 & \\ldots & x_2^n \\\\ \\vdots & \\vdots & \\ldots & \\vdots \\\\ x_m^0 & x_m^1 & \\ldots & x_m^n \\end{pmatrix}\n\\]\nwhere \\( n < m - 1 \\) and the \\( x_i ... |
130 | 2,063 | DeepMath-4096 | [
{
"content": "Find the kernel of the Vandermonde matrix \\( A \\) given by\n\\[\nA = \\begin{pmatrix} x_1^0 & x_1^1 & \\ldots & x_1^n \\\\ x_2^0 & x_2^1 & \\ldots & x_2^n \\\\ \\vdots & \\vdots & \\ldots & \\vdots \\\\ x_m^0 & x_m^1 & \\ldots & x_m^n \\end{pmatrix}\n\\]\nwhere \\( n < m - 1 \\) and the \\( x_i ... |
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