prompt list | ground_truth string |
|---|---|
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0^+} x \\ln(\\sin(x)) \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Calculate the limit: $$\\lim_{n\\rightarrow \\infty} \\frac{1+\\sqrt{2}+\\sqrt[3]{3}+\\cdots+\\sqrt[n]{n}}{n}$$",
"role": "user"
}
] | 1 |
[
{
"content": "Calculate the sum of the series \\( \\sum\\limits_{k=0}^{\\infty}\\frac{k^4}{k!} \\).",
"role": "user"
}
] | 15e |
[
{
"content": "Consider the topological space $[0,\\infty]$ with the order topology. Define $F_n = (-\\infty, n]$. Determine the limit $\\lim_{n \\to \\infty}\\mu(F_n)$.",
"role": "user"
}
] | \infty |
[
{
"content": "Determine whether there exist distinct lambda terms $M$ and $B$ such that $M \\neq B$ and the normal form of $MB$ is the same as the normal form of $(MB)(MB)$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\frac{(1+x)^{1/x} - e + \\frac{1}{2}ex}{x^2} \\] using a method other than Taylor series expansion.",
"role": "user"
}
] | \dfrac{11e}{24} |
[
{
"content": "Find a solution to the congruence $x^5 \\equiv 2 \\pmod{221}$, given that $221 = 17 \\times 13$ and solutions exist modulo $13$ and $17$. Use the Chinese Remainder Theorem to determine the solution modulo $221$.",
"role": "user"
}
] | 32 |
[
{
"content": "Determine whether there exists an Artinian ring with exactly two prime ideals such that their product is non-zero, both prime ideals are non-zero, and their sum equals the entire ring.",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $8,043.30 |
[
{
"content": "Find the minimum value of the function \\( f(x, y)=6\\left(x^{2}+y^{2}\\right)(x+y)-4\\left(x^{2}+ xy+y^{2}\\right)-3(x+y)+5 \\) in the region \\( D=\\{(x, y) \\mid x>0, y>0\\} \\).",
"role": "user"
}
] | 2 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $1,666,666,666.67 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 906000 N |
[
{
"content": "Determine the number of distinct conjugacy classes in a finite group whose order is a multiple of 8, considering only the case where the order is even.",
"role": "user"
}
] | 5 |
[
{
"content": "Find the element in \\( \\mathbb{Z}/2550\\mathbb{Z} \\) that corresponds to \\((\\bar{14},\\bar{32})\\) under the Chinese Remainder Theorem map from \\( \\mathbb{Z}/50\\mathbb{Z} \\times \\mathbb{Z}/51\\mathbb{Z} \\).",
"role": "user"
}
] | 1664 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 724350.5 g |
[
{
"content": "Evaluate the limit: $$\\lim _{(x,y)\\to(2,1)}\\frac{\\arcsin(xy-2)}{\\arctan(3xy-6)}$$",
"role": "user"
}
] | \dfrac{1}{3} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1. Debit Notes Receivable $1,000, Credit Sales $857, Credit Discount on Notes Receivable $143
2. Debit Interest Receivable $68.56, Credit Interest Revenue $68.56
3. Debit Cash $1,000, Debit Discount on Notes Receivable $143, Credit Notes Receivable $1,000, Credit Interest Receivable $68.56, Credit Interest Revenue $7... |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $349,330.22 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.09e-20 J |
[
{
"content": "Find the value of the constant \\( C \\) in the equation \\( 1 = C\\int_{-\\infty}^\\infty\\int_{-\\infty}^\\infty e^{-(x^2+y^2)}dydx \\). The answer should involve \\( \\pi \\).",
"role": "user"
}
] | \dfrac{1}{\pi} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.125mi/h |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.65 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $9,000 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 18.3 s |
[
{
"content": "Solve for $x>0$ in the following arithmetic sequence: $1^2, x^2, 3^2, \\ldots$.",
"role": "user"
}
] | \sqrt{5} |
[
{
"content": "Let $u,v \\in \\mathcal{E}'(\\mathbb{R}^n)$ be two compactly supported distributions. Define the convolution $u*v$ by $u*v(\\phi) = u(Rv*\\phi)$, where $v*\\phi = v(\\tau_xR\\phi)$ for $\\phi \\in C^\\infty$, $\\tau_x\\phi(t) = \\phi(t-x)$, and $R\\phi(t) = \\phi(-t)$. Prove that the convolution i... | u * v = v * u |
[
{
"content": "Determine the order of the quotient ring \\( \\mathbb{Z}_7[i]/\\langle 2 \\rangle \\).",
"role": "user"
}
] | 1 |
[
{
"content": "How many integers are between $(10.2)^3$ and $(10.3)^3$?",
"role": "user"
}
] | 31 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Met - Ala - Phe - Leu - Ile |
[
{
"content": "Let $S$ be the disk $|z|<3$ in the complex plane, and let $f: S \\to \\mathbb{C}$ be an analytic function such that $f\\left(1+\\frac{\\sqrt{2}}{n}i\\right) = -\\frac{2}{n^2}$ for all natural numbers $n$. Determine the value of $f(\\sqrt{2})$. \\( \\boxed{} \\)",
"role": "user"
}
] | 3 - 2\sqrt{2} |
[
{
"content": "Evaluate the limit \\( \\lim_{h \\to 0} \\frac{\\int_{1-h}^{1+h} f(x) \\,dx}{h} \\) where \\( f \\) is a continuous function defined on \\( \\mathbb{R} \\).",
"role": "user"
}
] | 2f(1) |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2Cr(OH)_4^- + 3ClO^- + 2OH^- \to 2CrO_4^{2-} + 3Cl^- + 5H_2O |
[
{
"content": "What is the remainder when the sum of factorials from $1!$ to $45!$ is divided by $47$?",
"role": "user"
}
] | 18 |
[
{
"content": "For what value of $k$ does the line represented by the equation $-\\frac{1}{2}-2kx = 5y$ contain the point $\\left(\\frac{1}{4},-6\\right)$?",
"role": "user"
}
] | 59 |
[
{
"content": "In how many ways can a soccer game between two teams end with a total of 12 goals scored, such that the scores of the two teams never differ by more than 2 goals?",
"role": "user"
}
] | 972 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 14.6 A |
[
{
"content": "Determine the minimum sample size $n$ required such that the probability that the maximum of a random sample $X_1, \\cdots, X_n$ from a uniform distribution on the interval $[0, \\theta]$ is within 10% of $\\theta$ is at least 95%.",
"role": "user"
}
] | 29 |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty}\\int_{1}^{n}\\dfrac{1}{x^{n}}dx.$$",
"role": "user"
}
] | 0 |
[
{
"content": "Find the limit: \\[ \\lim_{x \\to \\infty} \\frac{x(x+1)^{x+1}}{(x+2)^{x+2}} \\]",
"role": "user"
}
] | \dfrac{1}{e} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Depreciation tax shield = Depreciation * Tax rate |
[
{
"content": "Show that the sequence of functions \\( f_n(x) = \\frac{x}{1+n^2x^2} \\) converges uniformly to 0 on \\( \\mathbb{R} \\). Use an \\( \\epsilon-n_0 \\) argument to demonstrate this convergence.",
"role": "user"
}
] | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | h, g, f, c, i, d, b, a, e, j |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 200.609 amu |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 64.3 g |
[
{
"content": "Determine the number of elements of order 15 in the symmetric group $S_{13}$. Provide your answer as a single integer.",
"role": "user"
}
] | 79567488 |
[
{
"content": "Given real numbers $A, B, C, D$ such that $A + B + C + D = \\pi$, determine the minimum value of the function $$f(A,B,C,D) = \\cos{A} + \\cos{B} + \\cos{C} + \\cos{D}.$$",
"role": "user"
}
] | -2\sqrt{2} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.23e5 J |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 42% |
[
{
"content": "Find the function $f(x)$ such that \\( \\sum_{n=0}^{\\infty} \\frac{n+1}{n!}x^n \\) is the Taylor series of $f(x)$. Provide your answer in terms of elementary functions.",
"role": "user"
}
] | (x + 1)e^x |
[
{
"content": "Find the value of the following expectation involving Brownian motion: \\[ E\\left[\\int_{0}^{T} W_t^{n} dW_t\\right] \\] where $n$ is a constant power of $W_t$.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the integral \\( \\int_{[0,1]^2} \\frac{1}{x+y} \\, d\\lambda_2(x,y) \\).",
"role": "user"
}
] | 2 \ln 2 |
[
{
"content": "Determine the radius of convergence of the power series \\( \\sum_{n=1}^{\\infty}{\\left(\\sum_{k=1}^{n}{\\frac{1}{k}}\\right)}z^n \\).",
"role": "user"
}
] | 1 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 21.6% |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.18e11 J |
[
{
"content": "A parallelogram is generated by the vectors $\\begin{pmatrix} 2 \\\\ 1\\\\ 1 \\end{pmatrix}$ and $\\begin{pmatrix} 1 \\\\ -1 \\\\ - 1 \\end{pmatrix}.$\n\n[asy]\nunitsize(0.4 cm);\n\npair A, B, C, D;\n\nA = (0,0);\nB = (7,2);\nC = (1,3);\nD = B + C;\n\ndraw(A--B,Arrow(6));\ndraw(A--C,Arrow(6));\ndr... | \frac{1}{3} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.70e+23 |
[
{
"content": "Let the Fibonacci sequence be defined by \\( F_1 = F_2 = 1 \\) and \\( F_n = F_{n-1} + F_{n-2} \\) for \\( n \\geq 3 \\). Let \\( k \\) be a positive integer. Suppose that for every positive integer \\( m \\), there exists a positive integer \\( n \\) such that \\( m \\mid F_n - k \\). Must \\( k ... | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 44.1 u |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 614.33J |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 16650.65 N |
[
{
"content": "Calculate the line integral of the vector field \\( \\vec{F} = \\left(x \\frac{e^{x^2+y^2} - e}{x^2+y^2}, y \\frac{e^{x^2+y^2} - e}{x^2+y^2}\\right) \\) along the curve \\( \\gamma \\) defined by \\( x = 4 \\cos t \\) and \\( y = 4 \\sin^2 t \\) for \\( t \\in [0, \\pi/2] \\).",
"role": "user"... | 0 |
[
{
"content": "Evaluate the limit: $$ \\lim_{x\\to \\infty} \\int_x^{x+1} \\frac{t^2+1}{t^2+20t+8}\\, dt $$",
"role": "user"
}
] | 1 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | -57.573 atm |
[
{
"content": "Find the analytic extension of the function $f(x) = |x|$ from the real numbers to the complex plane, excluding zero.",
"role": "user"
}
] | \sqrt{z^2} |
[
{
"content": "Calculate the surface integral \\(\\iint \\bar{N} \\cdot \\bar{F} \\, ds\\) where \\(\\bar{F} = 4x\\hat{i} + 3y\\hat{j} - 2z\\hat{k}\\) and \\(S\\) is the surface bounded by \\(x = 0, y = 0, z = 0\\), and \\(2x + 2y + z = 4\\). Given that \\(\\nabla\\cdot\\bar{F} = 5\\), determine the value of the... | \dfrac{40}{3} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 140 g |
[
{
"content": "Is there only one counterexample in the complete graph $K_5$ for the Ramsey number $R(3,3)$?",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | O2 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $67.01 |
[
{
"content": "Let $X$ be an infinite set and ${\\cal P}(X)$ its powerset. A subset ${\\cal F} \\subseteq {\\cal P}(X)$ is said to have the splitting property (SP) if there exists $A \\in {\\cal P}(X)$ such that for all $F \\in {\\cal F}$, we have $F \\cap A \\neq \\emptyset \\neq F \\cap (X \\setminus A)$. Let ... | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 47.5 |
[
{
"content": "Compute the gradient of the function \\( f(x) = \\|x\\|^p \\), where \\( f(x) = (x_1^2 + x_2^2 + \\ldots + x_N^2)^{p/2} \\).",
"role": "user"
}
] | p \|x\|^{p-2} x |
[
{
"content": "Evaluate the surface integral \\( \\iint_{S} \\text{curl}(\\vec{F}) \\cdot \\vec{dS} \\) for the vector field \\( \\vec{F} = -y\\vec{i} + x\\vec{j} + xyz\\vec{k} \\) over the part of the sphere \\( x^2 + y^2 + z^2 = 25 \\) that lies below the plane \\( z = 4 \\). The sphere is oriented positively ... | -18\pi |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 32745.39 N |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.153 J |
[
{
"content": "Let \\((X_n)\\) be a sequence of random variables on a shared probability space. Prove that if \\(\\mathbb{P}(\\lim_{n \\to \\infty} X_n = \\beta) = 1\\), then \\(\\lim_{n \\to \\infty} \\mathbb{P}(X_n > \\alpha) = 1\\) for all \\(\\alpha < \\beta\\).",
"role": "user"
}
] | 1 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 64.28 kg |
[
{
"content": "Let \\(A=[a_{ij}]\\in Mat_n(\\mathbb{R})\\) be the matrix defined by: \\[a_{ij}=\\begin{cases}n+1 & \\text{if } i=j\\\\ 1 & \\text{if } i\\not=j\\end{cases}\\] Calculate \\(\\det(A)\\).",
"role": "user"
}
] | 2n^n |
[
{
"content": "Find the number of functions $f : \\mathbb{R} \\to \\mathbb{R}$ such that\n\\[f(x + f(y)) = x + y\\]for all real numbers $x$ and $y.$",
"role": "user"
}
] | 1 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Cash flows from operating activities:
Cash receipts from customers $6,500
Cash paid to suppliers $(5,130)
Income taxes paid $(550)
Net cash provided by operating activities $820
Cash flows from investing activities:
Purchase of equipment $(110)
Proceeds on sale of investment $190
Net cash provided by investing activiti... |
[
{
"content": "Let $\\{A_n\\}_{n=1}^\\infty$ be a sequence of finite sets such that no two sets in the sequence are pairwise disjoint. Must there exist a finite set $F$ such that no two sets in the sequence $\\{ A_n \\cap F\\}_{n=1}^\\infty$ are pairwise disjoint?",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 10 ft/s |
[
{
"content": "Evaluate the integral \\( \\iiint_{K} (x^{2} - z^{2}) \\, dx \\, dy \\, dz \\), where \\( K \\) is the region defined by \\( x \\geq 0, \\: y \\geq 0, \\: z \\geq 0 \\), and \\( x+y+z \\leq 1 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 31 kJ |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 65.90 g |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 14.59% |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 270 nC |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 96.3 % |
[
{
"content": "Evaluate the integral \\( \\int \\int_E f(x,y) \\, dx \\, dy \\) where \\( f(x,y) = \\begin{cases} (x-1/2)^{-3} & \\text{if } 0 < y < |x-1/2| \\\\ 0 & \\text{else} \\end{cases} \\) and \\( E = [0,1] \\times [0,1] \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the inequality \\(2\\int_{0}^{y^{2}}c(t)\\,dt<3\\frac{1}{y}\\int_{0}^{y^{2}}\\sqrt{t}c(t)\\,dt\\) holds for any positive number \\(y\\), given that the function \\(c(t)\\) is smooth, strictly increasing, and positive on the domain \\([0, \\infty)\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Calculate the limit: \\[ \\lim_{n \\to \\infty} \\frac{1}{n}\\sqrt[n] {(n+1)(n+2)\\cdots(n+n)} \\]",
"role": "user"
}
] | \dfrac{4}{e} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 674.48 grams |
[
{
"content": "Determine whether the sum of the infinite series \\( \\sum\\limits_{x = -\\infty}^{\\infty}x \\) is equal to 0 or undefined.",
"role": "user"
}
] | undefined |
[
{
"content": "Consider the function \\( f(x) = e^{3x}(x^3 - 3x) \\). Find the sum of the x-coordinates at which \\( f'(x) = 2f''(x) \\).",
"role": "user"
}
] | -\dfrac{11}{5} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | f_o = 2f_c |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2166421.58 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 221700 |
[
{
"content": "Find the maximum value of the function \\( f(x) = \\frac{x^2 + x + 1}{x^2 - x + 1} \\) using the AM-GM inequality.",
"role": "user"
}
] | 3 |
[
{
"content": "Determine the value of \\(a\\) for which the equation \\( |x-1| + |x-2| + \\ldots + |x-99| = a \\) has a unique real solution.",
"role": "user"
}
] | 2450 |
[
{
"content": "What is the difference between the largest and smallest of the following numbers? \\[\n0.78 \\qquad 0.12 \\qquad 1.33 \\qquad 1.328\n\\]",
"role": "user"
}
] | 1.21 |
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