prompt list | ground_truth string |
|---|---|
[
{
"content": "Let $P=\\langle 7, 5 +\\sqrt{-10}\\rangle$ be an ideal in $\\mathbb{Z}[\\sqrt{-10}]$. Determine if $P \\cap \\mathbb{Z} = 7\\mathbb{Z}$.",
"role": "user"
}
] | P \cap \mathbb{Z} = 7\mathbb{Z} |
[
{
"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... | $16,076 |
[
{
"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... | 0.45 m per second |
[
{
"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... | $1.13 |
[
{
"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... | 26.59 mm |
[
{
"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... | 470.75 |
[
{
"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... | 39.90 g/mol |
[
{
"content": "Determine the tightest upper bound on the sum of the absolute values of a sequence with n elements, where each term can be 0, 1, or -x, and x is less than or equal to the sum of all preceding elements. Express this upper bound in terms of n, using big O notation for large n.",
"role": "user"
... | O(n) |
[
{
"content": "Determine the real number $a$ such that the integral \\( \\int_{-a}^a \\frac{x^4}{e^x+1} \\, dx \\) equals \\( -\\frac{32}{5} \\).",
"role": "user"
}
] | -2 |
[
{
"content": "Determine the smallest integer \\( k \\) such that for any set \\( S \\) of \\( k \\) integers, there exist two distinct integers \\( x \\) and \\( y \\) in \\( S \\) for which either \\( x+y \\) or \\( x-y \\) is divisible by 2007.",
"role": "user"
}
] | 1005 |
[
{
"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... | 588.6J |
[
{
"content": "Find \\( \\lim_{n \\to \\infty} x_n \\), where:\n\\[ x_n = \\sqrt{n^2+1} + \\sqrt{4n^2+2n} - \\sqrt[3]{27n^3+n^2}. \\]",
"role": "user"
}
] | \dfrac{25}{54} |
[
{
"content": "If Alex gives Bob a penny, Bob will have three times as many pennies as Alex has. If Bob gives Alex a penny, Bob will have twice as many pennies as Alex has. How many pennies does Bob currently have?",
"role": "user"
}
] | 17 |
[
{
"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... | 3.49 km |
[
{
"content": "Determine the number of connected components of the subset \\( \\mathcal{E} \\) of \\( GL_n(\\mathbb{R}) \\), where \\( \\mathcal{E} = \\{V \\in GL_n(\\mathbb{R}): v_1, \\dots, v_r \\text{ is a basis for } \\mathcal{S}\\} \\) and \\( \\mathcal{S} \\subset \\mathbb{R}^n \\) is a subspace with dimen... | 4 |
[
{
"content": "Non-negative real numbers $a_1, a_2, \\ldots, a_n$ satisfy $a_1 + a_2 + \\ldots + a_n = k$, where $k$ is a constant. Determine the maximum value of the expression $$a_1a_2 + a_2a_3 + \\ldots + a_{n-1}a_n + a_na_1.$$",
"role": "user"
}
] | \dfrac{k^2}{4} |
[
{
"content": "Find the remainder when \\(100!\\) is divided by \\(97^2\\).",
"role": "user"
}
] | 8827 |
[
{
"content": "Let $a$,$b$,$c$,$d$, and $e$ be five consecutive terms in an arithmetic sequence, and suppose that $a+b+c+d+e=30$. Which of the following can be found? $$\\text{(A)}\\ a \\qquad \\text{(B)}\\ b \\qquad \\text{(C)}\\ c \\qquad \\text{(D)}\\ d \\qquad \\text{(E)}\\ e $$Enter the letter of the corre... | \text{(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... | 1.3e-8 C |
[
{
"content": "Determine the largest perfect square that divides the expression \\(2014^3-2013^3+2012^3-2011^3+\\ldots+2^3-1^3\\). Choose from the following options: (a) \\(1^2\\), (b) \\(2^2\\), (c) \\(1007^2\\), (d) \\(2014^2\\).",
"role": "user"
}
] | 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... | 14.898% |
[
{
"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... | 0.125 L of 16 mol/L HCl |
[
{
"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... | 2.60e8 m/s |
[
{
"content": "In a hemispherical vase with a flat lid, there are four identical oranges, each touching the vase, and one grapefruit that touches all four oranges. Are the four points where the grapefruit touches the oranges necessarily coplanar? (Assume all fruits are perfect spheres.)",
"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... | 14519.781 N.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... | 15.1 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... | 0.775 g/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... | 140.86 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... | 14.41% |
[
{
"content": "Determine the values of \\( \\alpha \\in \\mathbb{R} \\) for which the function \\( \\frac{1}{1+\\lvert\\lvert x\\rvert\\rvert^{\\alpha}} \\) is integrable on \\( \\mathbb{R}^n \\setminus \\{0\\} \\), where \\( \\lvert\\lvert \\, .\\rvert\\rvert \\) is a norm on \\( \\mathbb{R}^n \\).",
"role"... | \alpha > n |
[
{
"content": "Find the domain of the function\n\\[f(x) = \\sqrt{1 - \\sqrt{2 - \\sqrt{3 - x}}}.\\]",
"role": "user"
}
] | [-1,2] |
[
{
"content": "Is a Lebesgue measurable subgroup of \\(\\mathbb{R}\\) necessarily a Borel subset of \\(\\mathbb{R}\\)?",
"role": "user"
}
] | Yes |
[
{
"content": "Compute the expected value \\( E[e^{(i-1)Z}] \\) where \\( Z \\) is a standard normal random variable and \\( i \\) is the imaginary unit.",
"role": "user"
}
] | e^{-i} |
[
{
"content": "Calculate the volume of the region that lies outside the cone defined by \\( \\varphi = \\frac{\\pi}{4} \\) and inside the sphere defined by \\( \\rho = 4\\cos(\\varphi) \\).",
"role": "user"
}
] | \dfrac{8\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... | 0.81 or 89.19 |
[
{
"content": "Evaluate the line integral \\( \\oint_C \\mathbf{G} \\cdot d\\mathbf{r} \\), where \\( C \\) is the boundary of the surface \\( T \\) defined by \\( z = 10x - 20y + 60 \\) over the rectangle \\( R = \\{ (x,y) \\mid -2 < x < 3, 0 < y < 2 \\} \\) and \\( \\mathbf{G}(x,y,z) = -y^2 \\mathbf{i} + xy \\... | -8970 |
[
{
"content": "Find all continuous functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) such that \\( f(x) = f(x^2 + C) \\) for all \\( x \\in \\mathbb{R} \\), where \\( C > 0 \\) is a constant.",
"role": "user"
}
] | f(x) = 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... | 2.42e-18 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... | 3.77% |
[
{
"content": "Use the predictor-corrector method with the following second-order multi-step methods to approximate \\( y(0.3) \\) for the initial value problem \\( y'(t) = 3ty, y(0) = -1 \\), using a step size of \\( h = 0.1 \\):\n\n1. Predictor: \\( \\omega_{i+1} = \\omega_i + \\frac{h}{2}(3f_i - f_{i-1}) \\)\... | -1.1284 |
[
{
"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... | 12.88 |
[
{
"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... | 165 |
[
{
"content": "Let \\( A = \\{1, 2, 3, \\ldots, 9\\} \\) and let \\( f: A \\rightarrow A \\) be a bijection such that \\( f^n = f \\) for the smallest positive integer \\( n \\), where \\( f^n \\) denotes the \\( n \\)-fold composition of \\( f \\) with itself. Determine the largest possible value of \\( n \\)."... | 21 |
[
{
"content": "Find the expected value of a random variable $Z$ with the following conditional distribution: given $X=x$, $Z$ follows a binomial distribution with parameters $n$ and $x$. The random variable $X$ is uniformly distributed over the interval $[0,1]$. Compute $E[Z]$. \\( \\text{(Express your answer in... | \dfrac{n}{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... | Be2+ < Mg2+ < Ca2+ < Ba2+ |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\rightarrow\\infty} \\frac{1}{n} \\sum_{k=0}^n \\left|\\cos\\left(\\frac{k\\pi}{2}(5-\\sqrt{3})\\right)\\right|$$",
"role": "user"
}
] | \dfrac{2}{\pi} |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\frac{\\left(\\int_{0}^x e^{t^2}dt\\right)^2 }{\\int_{0}^{x} e^{2t^2}dt} \\]",
"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... | 335.64 g Cl |
[
{
"content": "Compute \\( \\lim_{x\\to0} \\ln^x(x) \\).",
"role": "user"
}
] | 1 |
[
{
"content": "In a Poisson process with intensity \\( \\lambda > 0 \\), the arrival times are \\( W_1, W_2, \\ldots \\). Given that there are exactly 7 arrivals in the time interval \\((0, T]\\), find the expected value of the second arrival time \\( W_2 \\).",
"role": "user"
}
] | \dfrac{T}{4} |
[
{
"content": "Does there exist a metrizable separable Tychonoff space that is almost strongly zero-dimensional but not strongly zero-dimensional?",
"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... | 0.628 |
[
{
"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... | -4242.64 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... | 2.6 eV |
[
{
"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 speed of the canoeist is 32.71 mph during the first part and 27.71 mph during the second part of the trip. |
[
{
"content": "Evaluate the improper integral \\(\\int_{0}^{\\infty} \\frac{\\cos(2x) - 1}{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... | 2.5secs |
[
{
"content": "Does there exist an infinite \\(\\sigma\\)-algebra that contains only countably many members? Justify your answer.",
"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... | -1.79e-10 J |
[
{
"content": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^2 = s,$$where $r$ and $s$ are constants.\n\nWhat is $r+s$?",
"role": "user"
}
] | 11 |
[
{
"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... | 3.944 |
[
{
"content": "Calculate the flux of the vector field \\( \\mathbf{u} = x \\hat{\\imath} + y \\hat{\\jmath} + z \\hat{k} \\) through the surface of the ellipsoid \\( \\frac{x^2}{a^2} + \\frac{y^2}{b^2} + \\frac{z^2}{c^2} = 1 \\), where \\( a > b > c > 0 \\).",
"role": "user"
}
] | 4\pi abc |
[
{
"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... | T = \frac{2 v_0 sin \theta }{g} |
[
{
"content": "Given a model \\(X_t = X_{t-1} + Y_t\\) where \\(Y_t\\) are iid random variables with \\(\\mathbb{E}(Y_t) = 0\\) and \\(\\text{Var}(Y_t) = 4\\), evaluate \\(\\text{Var}(X_1 + X_2 + X_3 + \\ldots + X_8)\\).",
"role": "user"
}
] | 816 |
[
{
"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... | $44,965 |
[
{
"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... | 3.59 seconds |
[
{
"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... | 564.9 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... | 1.8e+2 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... | Bank Reconciliation
Kaycee Sisters Inc.
August 31, 20Y9
Cash balance according to bank statement: $22,040
Add deposit in transit, not recorded by bank: $3,450
Subtotal: $25,490
Deduct outstanding checks: $4,290
Adjusted balance: $21,200
Cash balance according to Kaycee Sisters Inc.: $21,140
Add error in recording check... |
[
{
"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... | 13.82 |
[
{
"content": "Evaluate the integral \\( \\int_0^{2\\pi} \\frac{\\cos(e^{ix})}{e^{2ix}} \\, dx \\) using Cauchy's integral formula.",
"role": "user"
}
] | -\pi |
[
{
"content": "Given that $x, y, z > 0$ and $x^2 + y^2 + z^2 = 1$, find the minimum value of the expression \\( \\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z} \\).",
"role": "user"
}
] | 3\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... | 375 mL |
[
{
"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... | 102.9 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... | 3.73e-5 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... | 1.92 V |
[
{
"content": "Find the limit of the sequence \\( \\lim_{n \\to \\infty}\\left\\{2n^2\\cos\\left(\\frac{1}{n}\\right)\\right\\} \\cos\\left(\\frac{1}{n}\\right) \\), where \\(\\left\\{x\\right\\}= x - \\left\\lfloor x\\right\\rfloor\\) is the fractional part.",
"role": "user"
}
] | 0 |
[
{
"content": "Is it possible to simulate from a joint distribution containing both continuous and discrete variables using Gibbs sampling?",
"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... | 77 |
[
{
"content": "Evaluate the complex integral \\( \\int_{|z| = 1}\\frac{z^2}{4e^z -z}\\,\\mathrm dz \\) using residue calculus.",
"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... | kg/m |
[
{
"content": "Evaluate the definite integral: $$\\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}}\\frac{x\\cos{x}}{1+\\sin^2{x}} dx$$",
"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... | M^3g^2/(6m^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... | 3.96 * sqrt(m / (k * A)) |
[
{
"content": "What is the total number of digits used when the first 2002 positive even integers are written?",
"role": "user"
}
] | 7456 |
[
{
"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... | 19.266 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... | 13.152 m/s |
[
{
"content": "Determine the degree of the field extension \\( \\mathbb{Q}(\\xi+\\xi^5+\\xi^8) / \\mathbb{Q} \\), where \\( \\xi = \\cos \\left(\\frac{2\\pi}{13}\\right) + i\\sin \\left(\\frac{2\\pi}{13}\\right) \\).",
"role": "user"
}
] | 12 |
[
{
"content": "Find the general solution to the recurrence relation \\(a_n=1+\\frac{1}{2}a_{n+1}+\\frac{1}{2}a_{n-1}\\) for \\(n=0,1,\\ldots,T\\) with boundary conditions \\(a_0=a_T=0\\).",
"role": "user"
}
] | a_n = n(T - 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... | 248 mL |
[
{
"content": "Let $X$ be a random variable in $L_{1}$. Given that $E[X] = 1$, does it necessarily follow that $E[X \\cdot 1_{A}] = P[A]$ for any event $A$?",
"role": "user"
}
] | No |
[
{
"content": "In a non-atomic measure space, determine if it is possible to split a measurable set into two disjoint parts of equal measure.",
"role": "user"
}
] | Yes |
[
{
"content": "A committee consists of 12 members. Each pair of members of the committee either mutually like each other or mutually dislike each other. Each member likes exactly 6 other members. Determine if it is necessarily possible to arrange all 12 members around a round table so that any two adjacent membe... | Yes |
[
{
"content": "Determine all real values of $a$ for which the system of equations \\[\\begin{cases} x + y + z = 0 \\\\ xy + yz + azx = 0 \\end{cases}\\] has exactly one solution.",
"role": "user"
}
] | (0, 4) |
[
{
"content": "Given real numbers $a, b, c$ such that $a^2 + b^2 + c^2 = 1$ and $a^3 + b^3 + c^3 = 1$, find the possible value(s) of $a + b + c$. Express your answer as a set of values.",
"role": "user"
}
] | 1 |
[
{
"content": "Let $m$ be the minimum value of $|z|$, where $z$ is a complex number such that \\( |z-3i | + |z-4 | = 5. \\) If $m$ can be written in the form $\\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, find the value of $a+b$. \\( \\boxed{} \\)",
"role": "user"
}
] | 17 |
[
{
"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... | 2.7e4 J |
[
{
"content": "Determine if the following representation of the derivative $f'(x)$ is correct for all real numbers $x$:\n$$f'(x)=\\sum_{k=0}^{2009}\\prod_{l=0, l \\neq k}^{2009}(x+l)$$\nwhere $f(x)=\\prod_{k=0}^{2009}(x+k)-c$.",
"role": "user"
}
] | Correct |
[
{
"content": "Determine whether the creation operator $c^*(f)$, where $f \\in H$ and $H$ is a Hilbert space representing single-particle states, is a bounded linear operator on the fermionic Fock space $\\mathfrak{F}$. If it is bounded, find its norm.",
"role": "user"
}
] | \|f\| |
[
{
"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... | 748.8 MN/C leftwards |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.