prompt list | ground_truth string |
|---|---|
[
{
"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... | 3.6 cm |
[
{
"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... | 7616 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... | $110,326 |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\rightarrow \\infty}\\int_0^{\\pi}{\\frac{\\sin nx}{x^2+1}}\\text{d}x$$",
"role": "user"
}
] | 0 |
[
{
"content": "Calculate the limit using the Squeeze theorem: $$\\lim_{n \\to \\infty} \\sqrt[n]{e^{2n}+(1+\\frac{2}{n})^{n^3}}$$",
"role": "user"
}
] | \infty |
[
{
"content": "Evaluate the integral \\( \\int_{\\mathbb{T}} z^2 \\, d\\mu \\), where \\( \\mathbb{T} = \\{z \\in \\mathbb{C} : |z| = 1\\} \\), \\( \\phi:[0,1] \\to \\mathbb{T} \\) is defined by \\( \\phi(t) = e^{2 \\pi i t} \\), \\( m \\) is the Lebesgue measure, and \\( \\mu(E) = m(\\phi^{-1}(E)) \\).",
"r... | 0 |
[
{
"content": "Find the derivative of the inverse function $f^{-1}(x)$ of $f(x) = e^x + x$ at the point $x = f(\\log(2))$. Express your answer as $\\frac{d}{dx}(f^{-1}(x))$ evaluated at this point.",
"role": "user"
}
] | \dfrac{1}{3} |
[
{
"content": "Suppose $S$ is a collection of subsets of the natural numbers $\\{1,2,3,\\ldots\\}$ such that:\n\n1. If $A \\in S$ and $B \\subset A$, then $B \\in S$ (i.e., $S$ is closed under taking subsets).\n2. If $A \\in S$ and $B \\in S$, then $A \\cup B \\in S$ (i.e., $S$ is closed under taking finite unio... | No |
[
{
"content": "Calculate the residue of the function \\( \\frac{(\\cos(z))^3}{(\\cosh(z))^7} \\) at \\( z=0 \\).",
"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... | 0.44 m/s |
[
{
"content": "A street has 20 houses on each side, for a total of 40 houses. The addresses on the south side of the street form an arithmetic sequence, as do the addresses on the north side of the street. On the south side, the addresses are 4, 10, 16, etc., and on the north side they are 3, 9, 15, etc. A sign ... | 84 |
[
{
"content": "Evaluate the partial derivative of \\( f(x,y) = (x^3+y^3)^{1/3} \\) with respect to \\( y \\) at the point \\( (0,0) \\) using the definition of the partial derivative.",
"role": "user"
}
] | 1 |
[
{
"content": "A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 3, 4, and 5. What is the area of the triangle?\n$\\mathrm{(A) \\ 6 } \\qquad \\mathrm{(B) \\frac{18}{\\pi^2} } \\qquad \\mathrm{(C) \\frac{9}{\\pi^2}(\\sqrt{3}-1) } \\qquad \\mathrm{(D) \... | \frac{9}{\pi^2}(\sqrt{3}+3) |
[
{
"content": "Is a matrix with one eigenvalue and rank less than the number of rows similar to a matrix with the eigenvalue on the diagonal and either 0s or 1s on the superdiagonal?",
"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... | $108,000 |
[
{
"content": "In how many ways can 11 fish, consisting of 9 black fish, 1 yellow fish, and 1 blue fish, be distributed to 4 distinct persons such that each person receives at least one fish?",
"role": "user"
}
] | 1776 |
[
{
"content": "Let F be a countable field and B an infinite set. Let \\((F^B)_0\\) be the set of all functions with finite support from F to B. Determine whether \\(|(F^B)_0| = |B|\\).",
"role": "user"
}
] | |(F^B)_0| = |B| |
[
{
"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... | 8.06e21 ions of K+ |
[
{
"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... | 2.3m/s |
[
{
"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... | 0.0029M |
[
{
"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... | 8.58% |
[
{
"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... | 36.2 mL |
[
{
"content": "Determine whether the function \\( f(\\epsilon) = \\lim_{n \\rightarrow \\infty}\\frac{\\sum_{i=1}^n\\left|\\{k\\le i^\\epsilon: k |i\\} \\right|}{\\sum_{i=1}^n\\left|\\{k |i\\} \\right|} \\) is well-defined for all \\( \\epsilon \\in [0,1] \\).",
"role": "user"
}
] | \epsilon |
[
{
"content": "Let \\( X \\) be a set of \\( n \\) points. Let \\( P \\) be a set of subsets of \\( X \\), such that if \\( A, B \\in P \\), then \\( X - A \\), \\( A \\cup B \\), and \\( A \\cap B \\) are also in \\( P \\). Determine the possible values for the number of elements in \\( P \\).",
"role": "us... | 2^k |
[
{
"content": "Calculate the determinant of the matrix expression $\\det(A^4 - A^3)$, where $A = \\begin{bmatrix} 1 & 1 & -1 \\\\ 0 & 1 & 1 \\\\ -1 & 0 & 2 \\end{bmatrix}$. Use determinant properties to find the solution efficiently.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit: \\( \\lim_{n\\to \\infty} \\{(3+2\\sqrt{2})^n\\} \\), where \\( \\{x\\} = x - \\lfloor x \\rfloor \\) denotes the fractional part of \\( x \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Determine the dimension of the ring \\( \\mathbb{K}[x,y,z]/(xy, xz, yz) \\), where \\( \\mathbb{K} \\) is a field.",
"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... | 46.7 kg |
[
{
"content": "How many four-digit numbers $N$ have the property that the three-digit number obtained by removing the leftmost digit is one ninth of $N$?",
"role": "user"
}
] | 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... | 4.45 pF |
[
{
"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... | 1.27 AuCi |
[
{
"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... | 115 J |
[
{
"content": "Evaluate the definite integral \\( \\int_0^\\pi\\frac{\\sin\\left(n+\\frac{1}{2}\\right)x}{\\sin \\frac{x}{2}}dx \\).",
"role": "user"
}
] | \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... | 6867.40 |
[
{
"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... | 3.43e+09 |
[
{
"content": "Find a polynomial with integer coefficients that has the primitive 8th roots of unity as its roots.",
"role": "user"
}
] | x^4 + 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... | 1.04e-5 M |
[
{
"content": "Determine the values of \\(\\alpha\\) for which the series \\[\\sum_{n=1}^\\infty {((-1)^n + \\alpha) \\left(\\frac 1{\\sqrt{n+1} + \\sqrt{n}} \\right) }\\] converges.",
"role": "user"
}
] | 0 |
[
{
"content": "Consider functions $f_1, f_2, \\ldots \\in L^p$ such that $\\|f_n - f\\|_p \\rightarrow 0$ as $n \\rightarrow \\infty$. Does there exist a subsequence $f_{n_i}$ that converges to $f$ almost everywhere?",
"role": "user"
}
] | Yes |
[
{
"content": "Find the derivative $g'(1)$ for the inverse function $g(x) = f^{-1}(x)$, where $f(x) = x^3 + e^x + \\ln(x + 1) + 3x^{4/3}$.",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Find the largest real number \\( z \\) such that \\( x + y + z = 5 \\) and \\( xy + yz + zx = 3 \\), where \\( x \\) and \\( y \\) are also real numbers.",
"role": "user"
}
] | \dfrac{13}{3} |
[
{
"content": "In a regular 2019-gon, numbers are placed at the vertices such that the sum of the numbers at any nine consecutive vertices is 300. Given that the number at the 19th vertex is 19 and the number at the 20th vertex is 20, determine the number at the 2019th vertex.",
"role": "user"
}
] | 61 |
[
{
"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... | $0.768 |
[
{
"content": "Evaluate the limit \\( \\lim_{n\\to\\infty}\\int_0^n\\frac{\\left(1-\\frac{x}{n}\\right)^n}{e^{x/2}}\\:dx \\) using the dominated convergence theorem.",
"role": "user"
}
] | \dfrac{2}{3} |
[
{
"content": "Let $X_{(1)}, X_{(2)}, X_{(3)}$ be the order statistics of three independent random variables $X_1, X_2, X_3$ uniformly distributed over $[0,1]$. Find the correlation coefficient between $X_{(1)}$ and $X_{(3)}$. Use the fact that $X_{(k)} \\sim \\text{Beta}(k, 4-k)$ and the formula for correlation... | \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... | Propyl formate, Isopropyl formate, Ethyl acetate, and Methyl propionate |
[
{
"content": "Is the boundary of an open ball in \\(\\mathbb{R}^{n}\\) homeomorphic to \\(\\mathbb{R}^{n-1}\\)?",
"role": "user"
}
] | No |
[
{
"content": "Determine the dimension of the cell to which the boundary of $e_n$ maps in a CW-complex.",
"role": "user"
}
] | n-1 |
[
{
"content": "Differentiate the expression \\( \\dfrac{\\partial\\text{RSS}}{\\partial\\beta} = -2X^{T}(y-X\\beta) \\) twice with respect to \\( \\beta \\).",
"role": "user"
}
] | 2X^{T}X |
[
{
"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... | red; green or black |
[
{
"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... | 6.2e-4 J |
[
{
"content": "Evaluate the limit or determine if it does not exist: $$ \\lim_{(x,y) \\to (0,0)} \\frac{x^2 y^2}{x^2 + y^4} $$",
"role": "user"
}
] | 0 |
[
{
"content": "Given the equation \\( x y z + y + z = 12 \\), determine the maximum value of the expression \\( \\log_{4} x + \\log_{2} y + \\log_{2} z \\).",
"role": "user"
}
] | 3 |
[
{
"content": "Ben rolls four fair 20-sided dice, and each of the dice has faces numbered from 1 to 20. What is the probability that exactly two of the dice show an even number?",
"role": "user"
}
] | \frac{3}{8} |
[
{
"content": "Find the smallest positive integer \\( n \\) such that for any selection of \\( n \\) integers, there exist at least two integers whose sum or difference is divisible by 2009.",
"role": "user"
}
] | 1006 |
[
{
"content": "Given the joint probability density function $f_{X,Y}(x,y) = \\frac{1}{2}x^3e^{-xy-x}$ for $x>0$ and $y>0$, find the covariance $\\text{Cov}(X,Y)$ using the formula $\\text{Cov}(X,Y) = E[XY] - E[X]E[Y]$. Provide your answer as a single numerical value.",
"role": "user"
}
] | -\dfrac{1}{2} |
[
{
"content": "Let $X \\sim N(0, 1)$. Calculate $E(X^4)$. Provide your answer as a single number.",
"role": "user"
}
] | 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... | 18.1 m/s |
[
{
"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... | 409.9 KJ/kg |
[
{
"content": "Let $F$ be the splitting field of $g(x)=x^3-2$ over $\\mathbb{Q}$. Find a basis for $F$ over $\\mathbb{Q}$ and determine the degree of the extension $[F:\\mathbb{Q}]$. \\( \\text{Provide your answer as a basis and the degree of the extension.} \\)",
"role": "user"
}
] | 6 |
[
{
"content": "Let \\( A \\) be a diagonal matrix with the characteristic polynomial \\( P(x) = (x-15)(x-14)^2(x-13)^3\\ldots(x-2)^{14}(x-1)^{15} \\). Determine the dimension of the set \\( V \\) of all \\( 120 \\times 120 \\) matrices that commute with \\( A \\).",
"role": "user"
}
] | 1240 |
[
{
"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... | arcsin(0.101 - sin(alpha)) |
[
{
"content": "Is the following application of Abel's convergence theorem valid? Given that $\\sum a_nx^n$ converges to $f(x)$ for $|x|<1$, and $\\sum (-1)^na_n$ converges, can we conclude that $\\lim_{x\\rightarrow -1}f(x)=\\sum (-1)^na_n$?",
"role": "user"
}
] | Yes |
[
{
"content": "Is every polynomial in \\(\\mathbb{C}[x]\\) solvable by radicals?",
"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... | 951.7 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... | $4,375 |
[
{
"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... | 835.9kg |
[
{
"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... | 0.01 |
[
{
"content": "Find the derivative of the expression \\( \\sum_{i=0}^{m} \\binom{m}{i+1} (i+1)x^{i}(1-x)^{m-i-1} \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Calculate the volume of the solid formed by revolving the curve \\( y = \\frac{\\sin x}{x} \\) around the y-axis from \\( x = \\frac{\\pi}{3} \\) to \\( x = \\pi \\).",
"role": "user"
}
] | 3\pi |
[
{
"content": "Using Rouche's Theorem, find the number of roots of the polynomial equation \\(z^4 - 6z^3 + 9z^2 - 24z + 20 = 0\\) inside the circle \\(|z| = 2\\).",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to ∞}\\int_{0}^{1} \\frac{x^n}{\\cos x} dx$$",
"role": "user"
}
] | 0 |
[
{
"content": "In an equilateral triangle \\( ABC \\) with side length \\( 2 \\sqrt{3} \\), two points \\( P \\) and \\( M \\) are in the same plane as \\( ABC \\). The point \\( P \\) satisfies \\( |\\overrightarrow{AP}| = 1 \\), and \\( \\overrightarrow{PM} = \\overrightarrow{MB} \\). Determine the minimum val... | \dfrac{5}{2} |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty} n \\int_1^e (\\ln x)^n \\, dx \\]",
"role": "user"
}
] | e |
[
{
"content": "In a simple, connected, non-bipartite graph with 100 vertices and 300 edges, calculate the stationary probability of a vertex with degree 3.",
"role": "user"
}
] | \dfrac{1}{200} |
[
{
"content": "In triangle $\\triangle ABC$, point $P$ is an interior point such that $Q$ and $R$ are the reflections of $P$ across sides $AB$ and $AC$, respectively. If points $Q, A,$ and $R$ are collinear, find the measure of $\\angle A$. \\(\\text{Express your answer in degrees.}\\)",
"role": "user"
}
] | 90 |
[
{
"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... | 0.02 kgm/s |
[
{
"content": "Determine whether a group can have two distinct subgroups of the same order whose intersection contains more elements than just the identity. Provide a justification for your answer.",
"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... | Cash Budget
| Quarter | Beginning Cash Balance | Total Cash Receipts | Total Cash Available | Total Cash Disbursements | Excess (Deficiency) | Financing (Borrowings, Repayments, Interest) | Ending Cash Balance |
|---|---|---|---|---|---|---|---|
| 1st | $42,000 | $290,000 | $332,000 | $337,000 | $(5,000) | $15,000 Bo... |
[
{
"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... | 38.88751 g |
[
{
"content": "Evaluate the integral \\( \\int_0^\\infty\\int_0^\\infty e^{-sxy}\\cos(x)\\,dx\\,dy \\) for \\( s>0 \\) using the double integration method.",
"role": "user"
}
] | \dfrac{\pi}{2s} |
[
{
"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... | 0.018 L |
[
{
"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... | 33 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... | $5,960 |
[
{
"content": "Determine whether the following statement is true: For a symmetric, square matrix $A$, the $L^2$ operator norm $||A||_2$ is less than or equal to the $L^\\infty$ operator norm $||A||_\\infty$. The $L^2$ operator norm is defined as $||A||_2 := \\sup_{||x|| =1}||Ax||$, and the $L^\\infty$ operator n... | True |
[
{
"content": "Determine whether there exists a constant \\( M \\) such that for all integers \\( c \\ge M \\), there is no prime number \\( p \\) for which \\( p^c \\) divides \\((p-1)!+1\\).",
"role": "user"
}
] | 3 |
[
{
"content": "Suppose random variables $Z_i$ are independently distributed according to $p(z)$, and given $Z_i$, the random variable $X_i$ is distributed according to $p(x|z)$. Does the sample mean $\\frac{\\sum_{i=1}^n X_i}{n}$ converge to the expected value $E(X)$ as $n$ approaches infinity?",
"role": "us... | Yes |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\left( \\frac{n^{1/3}}{2} \\arccos \\left(\\frac{1}{\\sqrt{1+\\frac{4}{(k(n)-1)^2}}\\sqrt{1+\\frac{8}{(k(n)-1)^2}}} \\right) \\right), \\] where \\( k(n) = \\frac{1}{12} (108n+12 \\sqrt{768+81n^2})^{1/3}-\\frac{4}{ (108n+12 \\sqrt{768+81n^2})^{1/3}} ... | 2\sqrt{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... | -0.73 M/s |
[
{
"content": "Compute the line integral \\( \\int_C \\vec{F} \\cdot d\\vec{s} \\) where \\( \\vec{F} = A_0(x\\hat{y} - y\\hat{x}) \\), along a circular path, counterclockwise, about the origin with radius 4.",
"role": "user"
}
] | 32\pi A_0 |
[
{
"content": "Determine the cardinality of the Julia set for a given complex parameter. Is it always countable, $2^{\\aleph_0}$, or does it depend on the parameter?",
"role": "user"
}
] | 2^{\aleph_0} |
[
{
"content": "Let \\( f:X\\rightarrow Y \\) be a continuous function. If \\( V\\cap f(X)\\neq \\emptyset \\) and \\( W\\cap f(X)\\neq \\emptyset \\) and \\( f(X)\\subseteq V\\cup W \\), is it true that \\( f^{-1}(V)\\cup f^{-1}(W)=X \\)?",
"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... | 170.24 s |
[
{
"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... | 4619 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... | 3.022 Diopters |
[
{
"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... | 31.30 m/s |
[
{
"content": "\"Modulo $m$ graph paper\" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x,y)$ where $0\\le x, y <m$. To graph a congruence on modulo $m$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. For example, a graph of $y\\equiv x^2\\pmod 5$ would ... | 32 |
[
{
"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... | If the time t is known, then the distance from Boston station, d = (440-10t)/3 can be evaluated. |
[
{
"content": "In a Dedekind domain, let $R$ be a commutative, Noetherian ring with $1$. Consider a non-prime ideal $I$ of $R$, and elements $a, b \\in R$ such that $a \\not\\in I$, $b \\not\\in I$, but $ab \\in I$. Let $\\mathcal{P}$ be the set of prime ideals appearing in the Dedekind factorization of $(I,a)$ ... | Yes |
[
{
"content": "Determine whether the function \\( h : \\mathbb{R} \\rightarrow \\mathbb{R} \\) defined by \\( h(x) = x \\) if \\( x \\in \\mathbb{Q} \\) and \\( h(x) = 0 \\) otherwise, is measurable.",
"role": "user"
}
] | Yes |
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