prompt list | ground_truth string |
|---|---|
[
{
"content": "Compute $\\tan 420^\\circ$.",
"role": "user"
}
] | \sqrt{3} |
[
{
"content": "Evaluate the limit \\( \\lim_{x\\to {0}}\\left(\\frac{1}{x\\arcsin x} - \\frac{1}{x^2}\\right) \\) without using L'Hôpital's rule.",
"role": "user"
}
] | -\dfrac{1}{6} |
[
{
"content": "In the binomial expansion of \\((1+\\sqrt{11})^{212}\\), determine the value of \\(k\\) for which the term \\(C_{212}^{k}(\\sqrt{11})^{k}\\) is maximized.",
"role": "user"
}
] | 163 |
[
{
"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... | 1.5*10^5 Pa |
[
{
"content": "Given a random variable $X$ with probability density function $f(x) = K|x|$ for $-1 < x < 2$, derive the probability density function of $Y = |X|$. Also, determine the value of the constant $K$.",
"role": "user"
}
] | \dfrac{2}{5} |
[
{
"content": "Find the value of \\( \\alpha \\) such that the following equation holds: \\[ \\sum _ { n = 1 } ^ { \\infty } \\tan ^ { -1 } \\left( \\frac { 2 } { n ^ { 2 } + n + 4 } \\right)= \\tan ^ { -1 } \\left( \\frac { \\alpha } { 5 } \\right) \\]",
"role": "user"
}
] | 10 |
[
{
"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... | $148,000 |
[
{
"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... | 5h/36 |
[
{
"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... | 108253.18 ft |
[
{
"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... | -8000 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... | 67.4 V |
[
{
"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... | 1.34 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... | 67.44 km heading 65.08 degrees north of east |
[
{
"content": "Calculate the size of the stabilizer of the permutation \\((1\\ 2\\ 3)(4\\ 5\\ 6)\\) in the symmetric group \\(S_7\\) under the action defined by \\(x \\cdot y = xyx^{-1}\\).",
"role": "user"
}
] | 18 |
[
{
"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... | The expected cash collection for month 3 is $119,000 and for month 4 is $100,000 |
[
{
"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... | 13.85 m/s |
[
{
"content": "Let $X$ be a completely regular space, and let $C = C(X,[0,1])$ be the collection of continuous functions from $X$ to $[0,1]$. Consider the map $\\phi: X \\to [0,1]^C$ defined by $\\phi(x) = (f(x))_{f \\in C}$. By Tychonoff's theorem, $[0,1]^C$ is compact in the product topology, and $\\overline{\... | No |
[
{
"content": "Determine whether the following statement is true or false: A function $f: U \\to \\mathbb{R}$, where $U \\subset \\mathbb{R}^n$ is open, is partially derivable if and only if it is derivable in the direction of the standard unit vectors $e_1, \\ldots, e_n$. Provide a justification for your answer... | A |
[
{
"content": "Determine if the function \\( h(x) = \\frac{g(f(x))-g(f(0))}{f(x)-f(0)} \\) has a well-defined limit at \\( x=0 \\), and if so, find its value, where \\( f(x)=x^2\\sin\\left(\\frac{1}{x}\\right) \\) for \\( x \\neq 0 \\), \\( f(0)=0 \\), and \\( g(x)=\\sqrt{x} \\) for \\( x \\geq 0 \\).",
"rol... | \infty |
[
{
"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... | 654.3N |
[
{
"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... | 62636 |
[
{
"content": "Calculate the surface integral \\( \\iint_S \\mathbf{F} \\cdot d\\mathbf{S} \\), where \\( S \\) is the boundary of the solid \\( E \\) enclosed by the paraboloids \\( z = x^2 + y^2 \\) and \\( z = 12 - 2x^2 - 2y^2 \\), and \\( \\mathbf{F}(x, y, z) = \\langle x^3 + y^2, 2yz + e^z, y^2 - z^2 \\rang... | 48\pi |
[
{
"content": "Determine the necessary condition on \\((k, p)\\) such that the integral \\(\\int_{\\mathbb{R}^d}\\frac{1}{(|x|^{2k}+1)^p}\\,dx\\) is finite. Is the condition \\(2kp > d\\) sufficient?",
"role": "user"
}
] | 2kp > d |
[
{
"content": "Evaluate the set \\( \\bigcap_{n \\in \\mathbb{N}}\\bigcup_{m\\in\\mathbb{N}}A_{m,n} \\), where \\( A_{m,n} \\) is the interval \\( \\left[n-\\frac{1}{m},n+m\\right] \\).",
"role": "user"
}
] | \emptyset |
[
{
"content": "Given the recurrence relation \\( a_0 = 2 \\), \\( a_1 = 16 \\), and \\( a_{n+1}^2 = a_n a_{n-1} \\), find the limit of the sequence \\( \\{a_n\\} \\) as \\( n \\to \\infty \\).",
"role": "user"
}
] | 8 |
[
{
"content": "Determine the maximum number of elements in a sigma-algebra generated by a collection of $N$ distinct subsets of a set $X$, given that the intersection of any two subsets in the collection is non-empty.",
"role": "user"
}
] | 2^{2^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... | -2 C |
[
{
"content": "Let $X$ and $Y$ be independent random variables, each uniformly distributed on the interval $[0,3]$. Calculate the expected value $E(Y | X < 1 \\cup Y < 1)$. Provide your answer as a single numerical value.",
"role": "user"
}
] | \dfrac{11}{10} |
[
{
"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... | 1022.69 m/s |
[
{
"content": "Find the integral part of the expression \\( \\Big\\lfloor \\underbrace{\\sqrt{2018+\\sqrt{2018+\\sqrt{...+\\sqrt{2018}}}}}_\\text{2018 roots} \\Big\\rfloor \\).",
"role": "user"
}
] | 45 |
[
{
"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... | 0.058 m |
[
{
"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... | 7.92 m/s |
[
{
"content": "Calculate \\( \\lim_{n \\to \\infty} \\cos\\left(\\frac{a}{n}\\right)^{n^2} \\) where \\( a \\in \\mathbb{R} \\setminus \\{0\\} \\).",
"role": "user"
}
] | e^{-\frac{a^2}{2}} |
[
{
"content": "Let \\( f(n) \\) be a function defined on the set of positive integers such that:\n1. For any positive integer \\( n \\), \\( f(f(n)) = 4n + 9 \\).\n2. For any non-negative integer \\( k \\), \\( f(2^k) = 2^{k+1} + 3 \\).\n\nFind the value of \\( f(1789) \\).",
"role": "user"
}
] | 3581 |
[
{
"content": "The larger circle has center $O$ and passes through $D$. The smaller circle has diameter $OD$. What percent of the larger circle's area is gray?\n\n[asy]import graph;\ndraw(Circle((0,0),30),black);\nfill(Circle((0,-15),15),gray(.6));\ndraw(Circle((0,-15),15),black);\ndraw((0,0)--(0,-30),black);\nl... | 25\% |
[
{
"content": "Suppose $N$ points $p_i$ are chosen with a constant probability distribution in the interval $[0,1]$ and $x_i > 0$. Given that $\\sum_{i=1}^N p_i = 1$, find the probability that the weighted geometric mean of $x_i$ is greater than or equal to the geometric mean of $x_i$. ",
"role": "user"
}
... | \dfrac{1}{2} |
[
{
"content": "Given the sum \\(\\sum_{r=1}^n t_r = n(n+1)(n+2)(n+3)\\), find the limit \\(\\lim_{n\\to\\infty} \\sum_{r=1}^n \\frac{1}{t_r}\\).",
"role": "user"
}
] | \dfrac{1}{16} |
[
{
"content": "Let $(X_n)_n$ be a sequence of random variables in $L^2$. Suppose there exists a constant $M$ such that for all $n$, $$P(X^2_n \\ge M) < P(X^2_n \\ge E(X^2_n)).$$ Determine whether it follows that $E(X_n^2) \\le M$ for all $n$.",
"role": "user"
}
] | E(X_n^2) \le M |
[
{
"content": "Evaluate the limit of the Riemann sum as \\( n \\) approaches infinity for the interval \\([0,1]\\):\n\n\\[ \\lim _{n\\to \\infty }\\sum _{i=1}^n\\left(\\frac{i^4}{n^5}+\\frac{i}{n^2}\\right) \\]",
"role": "user"
}
] | \dfrac{7}{10} |
[
{
"content": "Calculate the limit: $$\\lim_{n \\to \\infty}\\frac{1}{n}\\sum_{k=0}^{n}{\\frac{1}{1+\\frac{k^6}{n^3}}}$$",
"role": "user"
}
] | 0 |
[
{
"content": "Consider the function \\( f : \\mathbb{C} \\to \\mathbb{C} \\) with a single pole at \\( z = 0 \\) and analytic elsewhere. The function is even, meaning \\( f(z) = f(-z) \\) for all \\( z \\in \\mathbb{C} \\). Determine \\( \\text{Res}(f, 0) \\).",
"role": "user"
}
] | 0 |
[
{
"content": "A famous theorem states that given any five points in the plane, with no three on the same line, there is a unique conic section (ellipse, hyperbola, or parabola) which passes through all five points. The conic section passing through the five points \\[(-\\tfrac32, 1), \\; (0,0), \\;(0,2),\\; (3,... | \frac{4\sqrt3}{3} |
[
{
"content": "How many license plates consist of 2 letters followed by 2 digits, if one of the digits must be odd and the other must be even?",
"role": "user"
}
] | 33,\!800 |
[
{
"content": "Determine the order of the poles of the function $\\tan(z)$ at the points $z = (2n+1)\\frac{\\pi}{2}$.",
"role": "user"
}
] | 1 |
[
{
"content": "The quadratic $-6x^2+36x+216$ can be written in the form $a(x+b)^2+c$, where $a$, $b$, and $c$ are constants. What is $a+b+c$?",
"role": "user"
}
] | 261 |
[
{
"content": "Determine the value of $k$ such that the differential form \\((y^3 + k xy^4 - 2x) \\, dx + (3xy^2 + 20 x^2y^3) \\, dy\\) is exact.",
"role": "user"
}
] | 10 |
[
{
"content": "Calculate the expected value of the sixth power of a standard normal random variable \\( Y \\), using the Law of the Unconscious Statistician (LOTUS):\n\n\\[ \\text{E}(Y^6) = \\int_{-\\infty}^{\\infty} x^6 \\cdot \\frac{1}{\\sqrt{2 \\pi}} e^{-\\frac{1}{2}x^2} \\, dx \\]",
"role": "user"
}
] | 15 |
[
{
"content": "Is the closed upper hemisphere of the 2-sphere, denoted as $M$, diffeomorphic to a closed disk in $\\mathbb{R}^2$?",
"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... | 0.278 m^2 |
[
{
"content": "Determine whether for every polynomial \\( P \\) of degree \\(\\geq 1\\) with integer coefficients, there exist infinitely many prime numbers, each of which divides at least one value \\( P(n) \\) for some positive integer \\( n \\).",
"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
July August September
Beginning cash balance $ 9,600 $ 9,200 $ 9,200
Cash Receipts 25,200 33,200 41,200
Total cash available 34,800 42,400 50,400
Cash payments (29,800) (31,200) (33,200)
Interest on bank loans (0) (42) (22)
Preliminary cash balance 5000 11158 17178
Additional loan (loan repayment) 4200 (1958) (2242)
Ending cash balance 9,200 9,200 14936
Loan balance
Loan balance - Beginning of month 0 4200 2242
Additional loan (loan repayment) 4200 (1958) (2242)
Loan balance - End of month 4200 2242 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... | 538.4 nm |
[
{
"content": "Let $X$ be an infinite-dimensional vector space and $P$ be a subspace of $X$. If $P$ is isomorphic to $X$, must $P$ be equal to $X$? Answer with 'True' or 'False' and justify your answer.",
"role": "user"
}
] | False |
[
{
"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... | 812 km |
[
{
"content": "Determine the degree of the field extension $[\\Bbb Q(u,i):\\Bbb Q]$, where $u$ is a root of the polynomial $t^4-3t^2+4$. Provide your answer as a single integer.",
"role": "user"
}
] | 8 |
[
{
"content": "Determine all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) that satisfy the equation:\n\n\\[ f(x^2 - y^2) = (x + y)(f(x) - f(y)) \\]",
"role": "user"
}
] | f(x) = kx |
[
{
"content": "Find the sum of the relatively prime positive integers $m$ and $n$ such that the parabola $y = 4x^2$ is tangent to the parabola $x = y^2 + \\frac{m}{n}$. What is $m + n$?",
"role": "user"
}
] | 19 |
[
{
"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, 5) |
[
{
"content": "Given a sequence $\\{a_n\\}$ where all terms are positive and the sum of the first $n$ terms $S_n$ satisfies $6 S_n = a_n^2 + 3 a_n + 2$. If $a_2, a_4, a_9$ form a geometric sequence, determine the general formula for the sequence $\\{a_n\\}$.",
"role": "user"
}
] | 3n - 2 |
[
{
"content": "Find all pairs of prime numbers \\(p\\) and \\(q\\) such that \\(1 + \\frac{p^q - q^p}{p + q}\\) is a prime number.",
"role": "user"
}
] | (2, 5) |
[
{
"content": "Determine the coefficient of \\(x^{33}\\) in the expansion of \\((x+x^3)(1+5x^6)^{-13}(1-8x^9)^{-37}\\). Use the identity \\((1-x)^{-k} = \\binom{k+n-1}{k-1}x^n\\) to assist in your calculation.",
"role": "user"
}
] | 83019300 |
[
{
"content": "Let $G$ be a group and consider elements $a, b \\in G$. If the subgroup generated by $a$, denoted $\\langle a \\rangle$, is normal in the subgroup generated by $\\{a, b\\}$, $\\langle a, b \\rangle$, and both $a$ and $b$ have finite order, is $\\langle a, b \\rangle$ necessarily a finite group? An... | 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... | -$0.628 million |
[
{
"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... | 40.4 lb |
[
{
"content": "Determine whether the implication $p \\multimap p$ holds in linear logic.",
"role": "user"
}
] | p \multimap p |
[
{
"content": "Evaluate the contour integral \\( \\int_{c}(1-z)^2dz \\), where the contour \\( c \\) is defined by the parametric equations \\( x=3\\cos\\theta \\) and \\( y=3+3\\sin\\theta \\) for \\( \\theta \\) ranging from \\( 0 \\) to \\( \\pi/2 \\).",
"role": "user"
}
] | 51 - 69i |
[
{
"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 balance per book
Particulars Amount
Cash balance per books, July 31$7,784
Bank collection not recorded by company $1,166
Bank Service Charges($37)
Correct Cash Balance $8,913
Cash balance per bank
Particulars Amount
Cash balance per bank, July 31$7,763
Deposits In Transit$1,750
Outstanding Checks($600)
Correct Cash Balance$8,913 |
[
{
"content": "Evaluate the integral \\(\\int^{100}_{-100}x^2\\cos(3x)\\delta(x^2-16)dx\\) using the property of the delta function.",
"role": "user"
}
] | 4\cos(12) |
[
{
"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... | 8.373% |
[
{
"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... | 8.09% |
[
{
"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... | Office equipment
Debit: $6,900
Accumulated depreciation
Debit: $4,140
Loss on sale
Debit: $5,520
Welding machine
Credit: $12,420
Cash
Credit: $4,140 |
[
{
"content": "Find the sum of the series \\( \\sum \\limits_{n=1}^{\\infty} (-1)^{n}\\frac{n^2}{2^n} \\).",
"role": "user"
}
] | -\dfrac{2}{27} |
[
{
"content": "Determine the spectrum of a self-adjoint idempotent element in a unital $C^*$-algebra.",
"role": "user"
}
] | \{0, 1\} |
[
{
"content": "Determine whether the series \\( S_n = \\sum\\limits_{k=1}^n \\frac{1}{\\sqrt{n^2+k}} \\) converges as \\( n \\to \\infty \\), and if so, find the limit of \\( S_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... | 4.12e4 kJ |
[
{
"content": "Compute the value of \\( \\omega^{3n/2 + 1} + \\omega \\) where \\( \\omega \\) is an \\( n^{th} \\) root of unity and \\( n \\) is a multiple of 4.",
"role": "user"
}
] | 0 |
[
{
"content": "Let $V$ be an $n$-dimensional vector space over $F$, with basis $\\mathcal{B} = \\{\\mathbf{v_1, \\cdots, v_n}\\}$. Let $\\mathcal{B}^{*} = \\{\\phi_1, \\cdots, \\phi_n\\}$ be the dual basis for $V^{*}$. Let $\\psi : V \\to V$ and $A$ be the matrix representing $\\psi$ with respect to $\\mathcal{B... | B = A^\top |
[
{
"content": "Solve the equation $|y-6| + 2y = 9$ for $y$.",
"role": "user"
}
] | 3 |
[
{
"content": "Does there exist a metric $d$ on $\\mathbb{R}$ such that the function $f:(\\mathbb{R},d) \\to (\\mathbb{R},d)$ defined by $f(x) = -x$ is not continuous?",
"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... | $171.64 |
[
{
"content": "Corner $A$ of a rectangular piece of paper of width 8 inches is folded over so that it coincides with point $C$ on the opposite side. If $BC = 5$ inches, find the length in inches of fold $l$.\n\n[asy]\nsize(100);\n\ndraw( (0,0) -- (8,0) -- (8, 10) -- (0, 10) -- cycle);\ndot((8,10)); dot((0,6)); d... | 5\sqrt{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... | 86.2 g/mol |
[
{
"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... | methyl |
[
{
"content": "Evaluate the integral \\( \\int_{|z|=3} \\frac{e^{\\frac{1}{1-z}}}{z} \\, dz \\) using the residue theorem, considering the residue at infinity.",
"role": "user"
}
] | 2\pi i |
[
{
"content": "Consider a meromorphic function $g$ in a region containing the unit disk. Suppose $g$ has a simple pole at $z=2$ and no other poles in the region. If the Maclaurin series of $g$ is $\\sum_{n=0}^{\\infty}b_nz^n$, determine the value of $\\lim_{n\\to\\infty}b_n$. \\(\\text{Express your answer as a n... | 0 |
[
{
"content": "Find the radius of curvature of the plane curve $x^3 + y^3 = 12xy$ at the point $(0, 0)$. Provide your answer as a single numerical value.",
"role": "user"
}
] | 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... | 5,5-dibromo-2,2,3-trimethylhexane |
[
{
"content": "Let $T$ be an invertible linear operator on a finite-dimensional vector space over a finite field. Is it true that there exists a positive integer $n$ such that $T^n = I$, where $I$ is the identity operator?",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the integral \\( \\int_{-\\infty}^{\\infty} \\frac{x^2}{x^6+1} \\, dx \\) using the method of residues and a contour in the upper half-plane.",
"role": "user"
}
] | \dfrac{\pi}{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... | 180.81 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... | c. 22.08% |
[
{
"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... | 0.093kgm^2s^-1 |
[
{
"content": "Find the volume of the solid formed when the triangle with vertices at $(1, 0)$, $(2, 0)$, and $(2, 3)$ is rotated around the line $y = -1$.",
"role": "user"
}
] | 6\pi |
[
{
"content": "Consider a distribution \\( f(x) \\). How does the difference between the two largest samples drawn from this distribution scale with the sample size \\( n \\)? Assume that the difference scales as \\( O(n^{-\\alpha}) \\). Determine whether \\( \\alpha \\) is greater than or less than 1.",
"ro... | \alpha < 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... | 90.7 ft |
[
{
"content": "A circular garden is enlarged so that the new diameter is twice the old diameter. What is the ratio of the original area to the enlarged area? Express your answer as a common fraction.",
"role": "user"
}
] | \frac{1}{4} |
[
{
"content": "Determine whether the limit \\( \\lim_{n\\to\\infty}\\underset{n}{\\underbrace{\\cos(\\cos(...\\cos x))}} \\) exists. If it exists, find the limit.",
"role": "user"
}
] | 0.739085 |
[
{
"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... | 1720 N |
[
{
"content": "Find the real roots of the equation \\(e^{x-1}+x-2=0\\).",
"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... | 272.71 |
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