messages listlengths 2 2 | enable_thinking bool 1
class | data_source stringclasses 1
value |
|---|---|---|
[
{
"content": "Let $f$ be a function such that $f$ and $f''$ are continuous on $[0, \\pi]$, $f(\\pi) = 1$, and \\( \\int^{\\pi}_0 (f(x) + f''(x)) \\sin(x) \\, dx = 2. \\) Find $f(0)$. \\( \\)\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "We... | false | DeepMath-103K |
[
{
"content": "Given real coefficient polynomials $P(x)$ and $Q(x)$ with odd degrees, and a function $f: \\mathbb{R} \\to \\mathbb{R}$, if for all $x, y \\in \\mathbb{R}$ the following equation holds:\n\\[ f(P(x) + Q(f(y)) + Q(y)P(f(x))) = Q(y) + P(f(x)) + P(x)Q(f(y)) \\]\nDetermine the function $f(x)$. Is $f(x)... | false | DeepMath-103K |
[
{
"content": "Evaluate \\( \\lim_{n \\to \\infty}\\sum_{k=2}^{n} \\frac{1}{\\sqrt[k]{n^k+n+1}+1} \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "We are to evaluate the limit:\n\n$$\n\\lim_{n \\to \\infty} \\sum_{k=2}^{n} \\frac{1}{\\sqr... | false | DeepMath-103K |
[
{
"content": "Compute the sixth derivative of \\( \\frac{\\cos{(5x^2)}-1}{x^2} \\) and evaluate it at zero. What is the result?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "We are asked to compute the sixth derivative of the function $ f(... | false | DeepMath-103K |
[
{
"content": "Determine whether the function \\( g(x) \\) defined by \\( g(x) = x^3 \\) if \\( x \\) is rational and \\( g(x) = 0 \\) if \\( x \\) is irrational is differentiable at any point \\( x \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"c... | false | DeepMath-103K |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\int_{0}^1 \\frac{nx^{n-1}}{1+x} \\, dx \\]\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "We are to evaluate the limit:\n\n$$\n\\lim_{n \\to \\infty} \\int_{0}^1 \\frac{n x^{n-1... | false | DeepMath-103K |
[
{
"content": "Calculate the fourth moment of Brownian motion at time \\( t \\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "We are asked to calculate the fourth moment of Brownian motion at time $ t $. Let me recall that Brownian motion ... | false | DeepMath-103K |
[
{
"content": "Evaluate the limit: $$ \\lim_{{M \\to +\\infty}} \\frac{1}{M} \\int_1^M M^{\\frac{1}{x}} \\, \\mathrm{d}x $$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "We are asked to evaluate the limit:\n\n$$\n\\lim_{M \\to +\\infty} \\f... | false | DeepMath-103K |
[
{
"content": "Are the fibers of a surjective polynomial submersion \\(\\mathbb{C}^n \\to \\mathbb{C}\\) all homeomorphic?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "We are asked whether the fibers of a surjective polynomial submersion f... | false | DeepMath-103K |
[
{
"content": "Determine whether the following statement is true or false: If a normed space $Y$ has the property that the identity operator can be approximated by finite range operators, then $Y$ must be separable.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
}... | false | DeepMath-103K |
[
{
"content": "Find the exact functional form of $g(y)$ for all $y > 0$ if $g(y) = \\lim_{x \\to \\infty} \\frac{f(xy)}{x^3}$ and $g(1) = 1$, where $f: \\mathbb{R} \\to \\mathbb{R}$ is a non-zero function such that the limit exists for all $y > 0$. Assume $f(x)$ is differentiable.\nPlease reason step by step, an... | false | DeepMath-103K |
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