prompt list | ground_truth string |
|---|---|
[
{
"content": "Evaluate the integral \\( \\int_{-1}^1 \\frac{x}{x^6+\\mathrm{i}} \\, dx \\).",
"role": "user"
}
] | 0 |
[
{
"content": "The amount of a claim on a health insurance policy is exponentially distributed with a mean of $1000. If the premium for the policy is $1100, and 100 such policies are sold, calculate the approximate probability that the total claims will exceed the total premiums collected, assuming the claims ar... | 0.1587 |
[
{
"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... | 1.0026e-4 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... | 0.0402 |
[
{
"content": "Let $G$ be a connected compact finite-dimensional abelian group. Suppose $H$ is a disconnected subgroup of $G$ with Haar measure 0. Determine the dimension of $H$.",
"role": "user"
}
] | 0 |
[
{
"content": "Three points are placed independently and at random in a unit square. What is the expected value of the area of the triangle formed by these three points?",
"role": "user"
}
] | \dfrac{11}{144} |
[
{
"content": "Determine whether the function \\( G(x)=\\sum^\\infty_{n=1}\\frac{1}{n}\\mathbb{1}_{\\{\\tfrac{1}{n}\\}}(x) \\) is Riemann integrable on the interval \\([0,1]\\).",
"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... | 0.05625 m^3/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... | Year Depreciation Base $ (A) Depreciation % MARCS Annual Depreciation 1 12,000 20 2,400 2 12,000 32 3,840 3 12,000 19.2 2,304 4 12,000 11.52 1,382.40 5 12,000 11.52 1,382.40 6 12,000 5.76 691.20 |
[
{
"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. Add hydrochloric acid (HCl) to precipitate silver ions as silver chloride (AgCl). Decant the solution to remove the AgCl precipitate.
2. To the remaining solution, add 0.3M HCl and then hydrogen sulfide (H2S). This precipitates copper(II) ions as copper(II) sulfide (CuS) while nickel(II) ions remain in solution.
3. Decant the solution to separate the CuS precipitate from the nickel(II) ion solution. |
[
{
"content": "Determine the kernel of the linear transformation \\( l : P \\to P \\) defined by \\( p(x) \\mapsto x^3p(x) - xp'(x) \\), where \\( P \\) is the vector space of polynomials.",
"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... | 24.9 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... | Current Liabilities:
Accounts payable: $52,600
Interest payable: $38,700
Income tax payable: $26,900
Current portion of installment note: $30,000
Total Current Liabilities: $148,200
Long-term debt:
Bonds payable (9.4%, due in 2027): $300,000
Less: Discount on bonds payable: $12,600
Bonds payable (8.7%, due in 2023): $50,000
Notes payable (7.8%, due in 2025): $400,000
Add: Premium on notes payable: $6,100
Installment note payable (due 2021-2024 less current portion): $90,000
Zero coupon note payable (due in 2026): $31,900
Capital lease liability: $41,500
Deferred tax liability: $133,400
Total Long-term Debt: $940,400 |
[
{
"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... | Date AssetsLiabilitiesCommon StockRetained earningsRevenuesExpensesNet incomeCash flows
1/10 50,000 -50,000 NA
15/12-50,000-50,000 -50,000 FA |
[
{
"content": "Solve the initial value problem for the differential equation:\n\n$$y'' - y' - 2y = 12\\sin(t)\\delta(t - \\pi)$$\n\nwith the initial conditions:\n\n$$y(0) = 1, \\quad y'(0) = -1$$",
"role": "user"
}
] | e^{-t} |
[
{
"content": "Find the largest positive integer $k$ such that $\\phi(\\sigma(2^k)) = 2^k$. (Here, $\\phi(n)$ denotes the number of positive integers that are smaller than $n$ and relatively prime to $n$, and $\\sigma(n)$ denotes the sum of divisors of $n$). You are given the hint that $641|2^{32}+1$. \\(\\boxed... | 31 |
[
{
"content": "Solve the recurrence relation: \\( T(n) = T\\left(\\frac{n}{4}\\right) + T\\left(\\frac{n}{2}\\right) + n^2 \\).",
"role": "user"
}
] | \Theta(n^2) |
[
{
"content": "Calculate the limit: $$\\lim_{x\\to\\infty} x^2\\cos\\left(\\frac{3x+2}{x^2}-1\\right).$$ Determine its value without using L'Hôpital's rule.",
"role": "user"
}
] | \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... | 64 units |
[
{
"content": "Determine whether the sequence defined by $x_1 = 2$ and $x_{n+1} = 2 + \\frac{1}{x_n}$ for $n \\in \\mathbb{N}$ converges or diverges.",
"role": "user"
}
] | 1 + \sqrt{2} |
[
{
"content": "Calculate the line integral of the vector field \\( \\mathbf{v}(x, y) = y\\mathbf{i} + (-x)\\mathbf{j} \\) along the semicircle of radius 2 centered at the origin. The path \\( C \\) starts from the point \\((0, 2)\\), proceeds along the upper semicircle to \\((0, -2)\\), and then continues along ... | -4\pi |
[
{
"content": "Evaluate the limit: \\[ \\lim _{x \\to \\pi}\\left(\\frac{\\cos(x)}{\\cos(\\pi)}\\right)^{\\frac{1}{x-\\pi}} \\]",
"role": "user"
}
] | 1 |
[
{
"content": "A $100$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $100$ vertices equals 2009. The midpoints of the sides of $P_1$ form a second $100$-gon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third $100$-gon, $P_3$. Find the sum of the $x$-coordinat... | 2009 |
[
{
"content": "Evaluate the limit: $$ \\lim_{x\\to0}\\frac{\\cos x-\\cos (3x)}{\\sin (3x^2)-\\sin (x^2)} $$",
"role": "user"
}
] | 2 |
[
{
"content": "Determine the first 1963 digits after the decimal point in the decimal expansion of \\((\\sqrt{26} + 5)^{1963}\\).",
"role": "user"
}
] | 000\ldots000 |
[
{
"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... | 20 mph |
[
{
"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... | 1366.5 J |
[
{
"content": "If $a,b,c$ are positive integers less than $13$ such that \\begin{align*}\n2ab+bc+ca&\\equiv 0\\pmod{13}\\\\\nab+2bc+ca&\\equiv 6abc\\pmod{13}\\\\\nab+bc+2ca&\\equiv 8abc\\pmod {13}\n\\end{align*}then determine the remainder when $a+b+c$ is divided by $13$.",
"role": "user"
}
] | 5 |
[
{
"content": "Evaluate the limit as $x$ approaches $0^+$ for the expression $$\\frac{e^x+e^{-x}}{-\\cos x}\\cdot\\frac{\\sin^2x}{e^x-e^{-x}}.$$",
"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... | $24,000 Unfavorable |
[
{
"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... | 5.292 m/s^2 |
[
{
"content": "Find the smallest value of the parameter \\( a \\) such that the coefficient of \\( x^4 \\) in the expansion of \\( \\left(1-3x+ax^2\\right)^8 \\) is 70.",
"role": "user"
}
] | -50 |
[
{
"content": "Evaluate the integral \\( Q(B) = \\int_B \\cdots \\int dx_1 dx_2 \\cdots dx_n \\) for the set \\( B = \\{ (x_1, x_2, \\dots, x_n); 1\\leq x_1 \\leq x_2 \\leq \\dots \\leq x_n \\leq 2 \\} \\).",
"role": "user"
}
] | \dfrac{1}{n!} |
[
{
"content": "Determine the smallest constant \\(c_0 > 0\\) such that for every series \\(\\sum a_n\\) with \\(a_n \\geq 0\\), the convergence of \\(\\sum a_n\\) implies the convergence of \\(\\sum_{n=1}^\\infty \\frac{\\sqrt{a_n}}{n^c}\\) for all \\(c > c_0\\), and there exists a convergent series \\(\\sum a_n... | \dfrac{1}{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... | 136.1 grams |
[
{
"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... | $1.33 |
[
{
"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... | 1.125e4 sec |
[
{
"content": "In a three-valued fuzzy logic system with defined operators, can all functions from $Q^3$ to $Q$ be generated using only these operators, without using the truth-values as constants?",
"role": "user"
}
] | No |
[
{
"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 mL |
[
{
"content": "Calculate the conditional expectation \\( E\\left[\\left(\\sum_{i=1}^{\\tau} Y_{i}\\right)^2 \\mid \\tau\\right] \\), where \\( Y_{i} \\) are independent random variables taking values 2 and -2 with equal probability \\( \\frac{1}{2} \\), and \\( \\tau \\) is independent of \\( Y_{i} \\).",
"r... | 4\tau |
[
{
"content": "Let $V$ be a complex vector space with $\\dim V = 12$, and let $N \\in L(V)$ be a nilpotent operator. The Jordan blocks of $N$ have sizes $4, 4, 2, 2$. Find the value of $\\dim \\ker(N^4) - \\dim \\ker(N^2)$. \\( \\boxed{?} \\)",
"role": "user"
}
] | 4 |
[
{
"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... | 1.22e-3 m^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... | $1,742,376 |
[
{
"content": "Given that \\(x, y, z\\) are the roots of the cubic polynomial \\(t^3 - 2t^2 + \\frac{1}{2}t - 4 = 0\\), find the value of the cyclic sum:\n\\[\n\\sum_{\\mathrm{cyc}} \\frac{1}{xy + z - 1} = \\frac{1}{xy + z - 1} + \\frac{1}{yz + x - 1} + \\frac{1}{zx + y - 1}.\n\\]",
"role": "user"
}
] | -\dfrac{2}{9} |
[
{
"content": "Calculate the volume of the solid bounded by the surfaces defined by the equations \\( x=0 \\), \\( y^{2}+z^{2}=4 \\), and \\( x+z=4 \\).",
"role": "user"
}
] | 16\pi |
[
{
"content": "Calculate the limit: $$\\lim_{n \\to \\infty} \\frac{n^{\\ln(n)}}{2^n}$$",
"role": "user"
}
] | 0 |
[
{
"content": "Is 1024 the last power of 2 that can be doubled without carrying in base 10? If not, how many more such powers of 2 exist?",
"role": "user"
}
] | 0 |
[
{
"content": "Let $\\Omega \\subset \\mathbb{R}^N$ be an open and bounded domain, and consider a continuous operator $T: L^2(\\Omega) \\to L^2(\\Omega)$ with a nullspace consisting only of the zero function. Prove or disprove that there exists a constant $C > 0$ such that $\\|Tx\\|_{L^2(\\Omega)} \\geq C\\, \\|... | False |
[
{
"content": "Find the coefficient of $x^{60}$ in the expansion of the product $\\prod_{m=1}^{11} (x^m - m)$. ",
"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... | Points located approximately 11.6 miles from the South Pole meet this condition. |
[
{
"content": "Consider a sequence of functions $\\{f_n\\}$ on a closed interval $I \\subset \\mathbb{R}$. Assume each $f_n$ is bounded on $I$ by a constant $M \\in \\mathbb{R}$. If $\\{f_n\\}$ converges pointwise to a function $f$ on $I$, does it necessarily follow that $\\{f_n\\}$ converges uniformly to $f$ on... | No |
[
{
"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... | $1,370,000 |
[
{
"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... | circular |
[
{
"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... | 0.514 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... | Debit Cash $192,270; Debit Due from factor $13,260; Debit Loss on sale of receivables $20,000; Credit Accounts Receivable $221,000; Credit Recourse liability $4,530 |
[
{
"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... | 1.17m |
[
{
"content": "Let $S_n = \\sum_{i=1}^n X_i$ be a symmetric random walk. Compute the conditional expectation $E(X_i \\mid \\mathcal{G}_{n+1})$ for $1 \\leq i \\leq n$, where $\\mathcal{G}_{n} = \\sigma(S_{n},S_{n+1},\\dots)$. Assume $S_n$ is a martingale.",
"role": "user"
}
] | \dfrac{S_{n+1}}{n+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... | 8711 N |
[
{
"content": "Evaluate the limit of the sequence of definite integrals as \\( n \\) approaches infinity:\n\n\\[ I_n = \\int_0^1 \\frac{dx}{1 + \\frac{1}{\\displaystyle\\sum_{k=0}^{n-1} \\frac{1}{1+\\sqrt{x}}}} \\]\n\nFind \\( \\lim_{n \\to \\infty} I_n \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Find an integer x such that \\( x^3 \\equiv 2 \\pmod{59} \\).",
"role": "user"
}
] | 38 |
[
{
"content": "Given the recurrence relation \\(a_n = \\left(1 - \\frac{1}{n}\\right)a_{n-1} + \\frac{1}{n}a_{n-2}\\) with initial conditions \\(a_1, a_2 \\in \\mathbb{N}\\), determine \\(\\lim_{n \\to \\infty} a_n\\). Choose the correct answer from the following options:\n\n(A) \\(2(a_2-a_1) + a_1 e^{-1}\\)\n\n... | D |
[
{
"content": "Jill and Kelly are doing spring cleaning. Jill takes an exponentially distributed time with a mean of 30 minutes to clean the kitchen, and Kelly takes an exponentially distributed time with a mean of 40 minutes to clean the bathroom. The first one to finish will then rake leaves, which takes an ex... | \dfrac{408}{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... | -4.700e-6 C and 3.500e-6 C |
[
{
"content": "Compute the value of the polynomial \\( \\left(\\frac{2}{z}+\\frac{z}{2}\\right)^2+2 \\) where \\( z = -1 + \\sqrt{3}i \\).",
"role": "user"
}
] | 3 |
[
{
"content": "In an Eulerian graph, consider a vertex with degree $2k$. Determine the number of times an Eulerian circuit can pass through this vertex, given that the circuit does not start or end at this vertex.",
"role": "user"
}
] | k |
[
{
"content": "Is the unit ball in the space of trace-class operators $TC( \\mathcal{H} )$ compact in the trace norm topology?",
"role": "user"
}
] | No |
[
{
"content": "Determine the degree of the field extension $[\\mathbb{Q}(\\alpha) : \\mathbb{Q}]$, where $\\alpha = \\sqrt[m]{p} + \\sqrt[m+k]{p}$, $p$ is a prime number, and $m$ and $k$ are positive integers such that $\\gcd(m, m+k) = 1$. Provide your answer as an integer.",
"role": "user"
}
] | m(m + k) |
[
{
"content": "What is the ones digit of $7^{35}$ when written as an integer?",
"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... | +151 J/K |
[
{
"content": "Let $V$ be a vector space of dimension 6 over a field $F$. Determine whether there exists a linear operator $T \\in A(V)$ such that $\\text{Ker}(T) = \\text{Range}(T)$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Find the sum of all positive integers \\(k\\) for which the inequality \\(5x^2-2kx+1<0\\) has exactly one integral solution.",
"role": "user"
}
] | 9 |
[
{
"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... | $895,322.38 |
[
{
"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_2] = 0.14999 M
[I_2] = 0.04999 M
[HI] = 1.84e-4 M |
[
{
"content": "Determine whether a tangent to a cubic curve can intersect the curve orthogonally at another point. 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... | $v_{2f}=\frac{m_1v_{1i}}{m_2(\mathrm{sin}\theta_2\mathrm{cot}\theta_1+\mathrm{cos}\theta_2)}$ |
[
{
"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... | 15.44 |
[
{
"content": "Is it possible for an unbounded, non-negative function $f(x)$ to have roots $x_n$ such that $x_n \\to \\infty$ as $n \\to \\infty$? 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... | 184.7 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... | 1.88 mil |
[
{
"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... | -242.8 J/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... | work |
[
{
"content": "Evaluate the integral \\( \\iint \\dfrac {\\mathrm dx\\mathrm dy}{\\sqrt {x^{2}-3y^{2}}} \\) over the region defined by \\( x^{2}+y^{2}\\leq 4x-3 \\). Use polar coordinates to simplify the integration.",
"role": "user"
}
] | \dfrac{2\pi}{3} |
[
{
"content": "Given the differentiable functions $z$ and $w$ of $x$ and $y$ that satisfy the equations:\n\n$$xw^3 + yz^2 + z^3 = -1$$\n$$zw^3 - xz^3 + y^2w = 1$$\n\nFind the partial derivative of $z$ with respect to $x$, $\\frac{\\partial z}{\\partial x}$, at the point $(x, y, z, w) = (1, -1, -1, 1)$. \\( \\box... | -\dfrac{5}{4} |
[
{
"content": "Evaluate the expression \\[ \\frac{\\sum_{i=0}^{100} \\binom{k}{i} \\binom{M-k}{100-i} \\frac{k-i}{M-100}}{\\binom{M}{100}}. \\]",
"role": "user"
}
] | \dfrac{k}{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... | -$190 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\frac{\\sin^2\\left(\\frac{x}{2}\\right) - \\frac{x^2}{4}}{e^{x^2} + e^{-x^2} - 2} \\]",
"role": "user"
}
] | -\dfrac{1}{48} |
[
{
"content": "Let $f(x)$ be a rational non-constant polynomial such that $f(f(x)) = 3f(x)^4 - 1$. Determine the polynomial $f(x)$. \\( \\boxed{} \\)",
"role": "user"
}
] | 3x^4 - 1 |
[
{
"content": "Determine the Morley rank of the type $x = x$ in the theory $ACF_{0}$, where $ACF_{0}$ denotes the theory of algebraically closed fields of characteristic 0.",
"role": "user"
}
] | 1 |
[
{
"content": "What is the area in square inches of the pentagon shown?\n\n[asy]\ndraw((0,0)--(8,0)--(8,18)--(2.5,20)--(0,12)--cycle);\nlabel(\"8''\",(1.3,16),NW);\nlabel(\"6''\",(5.2,19),NE);\nlabel(\"18''\",(8,9),E);\nlabel(\"8''\",(4,0),S);\nlabel(\"12''\",(0,6),W);\ndraw((1,0)--(1,1)--(0,1));\ndraw((7,0)--(7... | 144 |
[
{
"content": "Let $M$ be a $4 \\times 4$ non-singular matrix and $A$ be a $4 \\times 4$ matrix with rank 2. Determine the dimension of the vector space $S$ composed of vectors $MAx$, where $x \\in \\mathbb{R}^4$. Specify whether the dimension refers to the range space or the null space of $S$. \\( \\text{Dimens... | 2 |
[
{
"content": "Let $U_1, U_2, U_3, U_4, U_5$ be independent random variables, each uniformly distributed on the interval (0, 1). Let $R$ be the distance between the minimum and maximum of these variables. Find the expected value $E(R)$. Express your answer as a single number.",
"role": "user"
}
] | \dfrac{2}{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... | $11,483.82 |
[
{
"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... | 35.075 MPa |
[
{
"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... | 0.273 atm |
[
{
"content": "Evaluate the limit: \\[ L = \\lim_{x \\to 0} \\frac{\\cos(\\sin x) - (1 + x^2)^{-\\frac{1}{2}}}{x^4} \\]",
"role": "user"
}
] | -\dfrac{1}{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... | -$38,534 |
[
{
"content": "Evaluate the Lebesgue integral \\( \\int_{0}^{\\infty} 10^{-2[x]} \\, dx \\), where \\([x]\\) denotes the integer part of \\( x \\).",
"role": "user"
}
] | \dfrac{100}{99} |
[
{
"content": "Determine the value of \\( a \\) such that the curves \\( ax^2+4xy+2y^2+x+y+5=0 \\) and \\( ax^2+6xy+5y^2+2x+3y+8=0 \\) intersect at four concyclic points. Choose the correct option: \\( a \\) = 4, \\( a \\) = -4, \\( a \\) = 6, \\( a \\) = -6.",
"role": "user"
}
] | -4 |
[
{
"content": "Find all real numbers \\(x\\) that satisfy the equation \\(|x^2 - \\lfloor x \\rfloor| = 1\\), where \\(\\lfloor x \\rfloor\\) is the greatest integer less than or equal to \\(x\\).",
"role": "user"
}
] | \sqrt{2} |
[
{
"content": "Calculate the integral \\( \\int_{0}^{2\\pi} |3+4e^{10ix}+5e^{100ix}|^{2}dx \\) using Parseval's identity.",
"role": "user"
}
] | 100\pi |
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