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... | 0.95 |
[
{
"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... | 3.33333333e-9 s |
[
{
"content": "Calculate the expected value of the integral of a pure birth process from time 0 to the first time it reaches state 4, given that the process starts at state 1 and has birth rates \\( \\lambda_k = k \\) for \\( k=1,2, \\dots \\).",
"role": "user"
}
] | 3 |
[
{
"content": "Determine whether a ring with finite non-zero characteristic must be finite. Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"content": "Let \\( n \\) be a positive integer such that \\( (n,8)=1 \\) and \\(-4\\) is a square modulo \\( n \\). Prove that \\( n \\) can be expressed in the form \\( n = x^2 + 4y^2 \\) for some integers \\( x \\) and \\( y \\).",
"role": "user"
}
] | n = x^2 + 4y^2 |
[
{
"content": "In an equilateral triangle $ABC$ with side length $a$, points $P, Q, R$ are located on sides $AB, BC, CA$, respectively, such that $AP = x$, $BQ = 2x$, and $CR = 3x$. Find the value of $x$ that maximizes the area of triangle $PQR$. Express your answer in terms of $a$. \\( \\boxed{} \\)",
"role... | 0 |
[
{
"content": "Evaluate the integral \\( \\frac{i}{4-\\pi} \\int_{|z|=4} \\frac{dz}{z \\cos z} \\) using the residue theorem, given that the function \\( f(z)=\\frac{1}{z \\cos z} \\) has simple poles at \\( z=0 \\) and \\( z=\\frac{(2n-1)\\pi}{2} \\) for \\( n\\in\\mathbb{Z} \\).",
"role": "user"
}
] | 2 |
[
{
"content": "Calculate the sum of all unique permutations of the digits of the number 1112334.",
"role": "user"
}
] | 999999900 |
[
{
"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... | 0.0268 |
[
{
"content": "Suppose $X_1, X_2, \\ldots, X_N$ are mutually independent and identically distributed random variables. Let $F_X$ denote the cumulative distribution function (cdf) of $X_1$. Consider the event $A$ that the largest $X_i$ is smaller than or equal to $t_1$, and the smallest $X_i$ is greater than $t_0... | True |
[
{
"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... | 52.11 |
[
{
"content": "Suppose $k$ is an algebraically closed field, and let $f \\in k(\\mathbb{A}^1)$ be a rational function over the affine line $\\mathbb{A}^1$. If $f^2 \\in k[\\mathbb{A}^1]$, does it follow that $f \\in k[\\mathbb{A}^1]$? Justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "In a class trial with 15 participants, each participant votes either \"guilty\", \"not guilty\", or \"abstain\". The voting stops as soon as at least half of the votes are \"guilty\". Determine the total number of possible voting sequences that can occur.",
"role": "user"
}
] | 13663745 |
[
{
"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... | 0.7% |
[
{
"content": "Find the limit of the sequence \\( a_n = (n^2 + 1)\\sin\\left(\\frac{1}{n^3+1}\\right) + (n^2 + 2)\\sin\\left(\\frac{1}{n^3+2}\\right) + \\dots + (n^2 + n)\\sin\\left(\\frac{1}{n^3+n}\\right) \\) as \\( n \\to \\infty \\).",
"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... | 2,064,516 km/h^2 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.35e10 C |
[
{
"content": "Calculate the surface integral of the vector field \\( \\vec{F}=y\\hat{i}-x\\hat{j} \\) over the surface defined by the part of the paraboloid \\( z=x^2+y^2 \\) that lies below the plane \\( z=1 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Given a curve \\( C \\) defined by a twice-differentiable function \\( y=f(x) \\) such that \\( f''(x) > 0 \\) for all \\( x \\in [0, a] \\), where \\( a \\) is a positive constant, find the value of \\( r \\in (0, a) \\) for which the area \\( A \\) of the region bounded by the \\( y \\)-axis, th... | \dfrac{a}{2} |
[
{
"content": "Let \\( \\Omega \\) be a smooth bounded subset of \\( \\mathbb{R}^{n} \\), and let \\( h \\) be an \\( L^{\\sigma_{\\alpha}} \\)-function with \\( h^{+} \\neq 0 \\). Given that \\( \\frac{1}{\\sigma_{\\alpha}} + \\frac{\\alpha}{p^*} = 1 \\), determine whether there exists a \\( C_{0}^{\\infty} \\)... | Yes |
[
{
"content": "In triangle \\( ABC \\), \\( \\angle ABC = 20^\\circ \\) and \\( AB = BC \\). Points \\( D \\) and \\( E \\) are on segments \\( AB \\) and \\( BC \\) respectively, such that \\( AD = BE = AC \\). Determine the measure of angle \\( \\angle BDE \\).",
"role": "user"
}
] | 20^\circ |
[
{
"content": "Let \\( n \\ge 2 \\) be an integer. Determine the largest integer \\( k \\ge 1 \\) such that for any \\( k \\) matrices \\( A_1, A_2, \\ldots, A_k \\in \\mathcal{M}_n(\\mathbb{C}) \\), if \\( I_n - A_1A_2\\cdots A_k \\) is invertible, then \\( I_n - A_{\\pi(1)}A_{\\pi(2)}\\cdots A_{\\pi(k)} \\) is... | 2 |
[
{
"content": "For functions mapping from positive reals to positive reals, if \\( f(f(x)) = f(x) \\) for all \\( x \\), does it necessarily imply that \\( f(x) = x \\) for all \\( x \\)? Provide a justification for 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... | -$2,000 |
[
{
"content": "Determine the range of values for the real number \\( t \\) such that the inequality \\( a x^{2} + t y^{2} \\geqslant (a x + t y)^{2} \\) holds for any real numbers \\( x \\) and \\( y \\), given that \\( a \\in (0,1) \\).",
"role": "user"
}
] | [0, 1 - a] |
[
{
"content": "Does there exist an infinite quadratically closed field extension of $\\mathbb{Z}_2$?",
"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... | $38,527.22 |
[
{
"content": "Minimize the function \\( f(x, y, z) = x^2 + y^2 + z^2 \\) subject to the constraints:\n\\[ x^2 + y^2 + z^2 - 5 = 0 \\]\n\\[ x^2 + y^2 + z^2 - 2x - 3 = 0 \\]",
"role": "user"
}
] | 5 |
[
{
"content": "Find the function $g(x)$ that is implicitly defined by the relation $\\sqrt{x^2 - y^2} + \\cos^{-1}\\left(\\frac{x}{y}\\right) = 0$. Show that $y = x$ is the only function that satisfies this relation.",
"role": "user"
}
] | 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... | 2 * arccos(0.5 * n_m) |
[
{
"content": "If an arc of $45^{\\circ}$ on circle $A$ has the same length as an arc of $30^{\\circ}$ on circle $B$, then what is the ratio of the area of circle $A$ to the area of circle $B$? Express your answer as a common fraction.",
"role": "user"
}
] | \frac{4}{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... | 0.141 |
[
{
"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... | $4,987.70 |
[
{
"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... | $1,200 |
[
{
"content": "Let $x_1, x_2, \\ldots, x_k$ be distinct elements of a group $G$ with $k \\geq 4$. If $x_i$ commutes with $x_{i+1}$ for $1 \\leq i < k$, and $x_k$ commutes with $x_1$, determine whether all $x_i$ must commute with each other.",
"role": "user"
}
] | No |
[
{
"content": "In an acute-angled triangle \\( \\triangle ABC \\), \\( AB = BC = 12 \\). Line segment \\( AN \\) is perpendicular to \\( BC \\). There is a point \\( M \\) on \\( BC \\) such that \\( M \\) is between \\( B \\) and \\( N \\), \\( AN = MN \\), and \\( \\angle BAM = \\angle NAC \\). Determine the l... | 6\sqrt{3} |
[
{
"content": "Given the equation for experience required for a level in an online game: \\[ \\frac{50}{3} (x^3-6x^2+17x-12) = 37800 \\] solve for the level \\( x \\).",
"role": "user"
}
] | 15 |
[
{
"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... | \Lambda = 1 and \Sigma = -1, 0, or 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... | n=(PV)/(RT) |
[
{
"content": "Given $Y_1 \\sim \\Gamma(2,1)$ and $Y_2|Y_1 \\sim U(0, Y_1)$, where $y_1 \\geq y_2 \\geq 0$, find the variance of $Y_2 - Y_1$. Use the following information:\n\n$$f_{Y_1}(y_1) = y_1 \\exp(-y_1)$$\n$$f_{Y_2|Y_1}(y_2) = \\frac{1}{y_1}$$",
"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... | PbSO4 > BaCO3 > AgCl > BaSO4 > CdS |
[
{
"content": "Find a function $g(x)$ such that $\\int_0^1 x^n e^{-x} \\, dx < \\int_0^1 g(x) \\, dx$ for all $n \\geq 0$, and $\\int_0^1 g(x) \\, dx = 1$. Provide the explicit form of $g(x)$. ",
"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... | 4.01*10^-7 m |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\rightarrow \\infty}\\frac{1}{n}\\sum _{k=1}^{\\lfloor\\frac{n}{2}\\rfloor} \\cos\\frac{k\\pi}{n} \\] where \\(\\lfloor\\frac{n}{2}\\rfloor\\) denotes the largest integer not exceeding \\(\\frac{n}{2}\\).",
"role": "user"
}
] | \dfrac{1}{\pi} |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to 0} \\frac{\\sqrt[5]{1+\\frac{x}{5}}-\\sqrt[6]{1+\\frac{x}{6}}}{1-\\sqrt{1-\\frac{x}{3}}}$$ without using L'Hospital's rule, power expansions, or derivatives.",
"role": "user"
}
] | \dfrac{11}{150} |
[
{
"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% |
[
{
"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... | s(t) = t^4/4 + t^2/2 - 3 |
[
{
"content": "You have a four-sided die numbered 1 to 4 and are playing a game. In the first round, you roll the die once. If you roll a 1, you lose the die. If you roll a 2, you keep the die. If you roll a 3 or 4, you gain an additional identical die. In the second round, you roll each die you have, applying t... | \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... | 370.71 K |
[
{
"content": "In a scalene triangle $ABC$, the angle bisector $AD$ intersects the circumcircle of the triangle at point $P$. Point $I$ is the incenter of the triangle $ABC$. Given that $ID = DP$, determine the ratio $AI: ID$. (20 points)",
"role": "user"
}
] | 2 |
[
{
"content": "Determine whether the statement \"The kernel of $A^3$ is equal to the kernel of $A$ if and only if the sum of the image of $A^2$ and the kernel of $A$ is equal to $\\mathbb{R}^n$\" is true or false.",
"role": "user"
}
] | True |
[
{
"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... | 29.21% |
[
{
"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... | 1.57e-15 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... | Cash flows from operating activities:
Net income $6,000
Depreciation expense $8,000
Increase in inventory $(16,000)
Decrease in accounts payable $(21,000)
Net cash provided by operating activities $(23,000)
Cash flows from investing activities:
Purchase of equipment $(15,000)
Net cash used by investing activities $(15,000)
Cash flows from financing activities:
Proceeds form Bonds issued $70,000
Proceeds from issuing stock $1,000
Dividends paid $(3,000)
Net cash provided by financing activities $68,000
Net increase in cash $30,000
Cash balance at January 1, 2012 $20,000
Cash balance at December 31, 2012 $50,000 |
[
{
"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... | $\theta = \cos ^{-1} \frac{d}{v.t}$ |
[
{
"content": "Determine the radius of convergence of the series \\( \\sum \\frac{h(n)}{n^k}r^n \\), where \\( h(n) \\) is a polynomial of degree \\( k \\).",
"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... | 15818 |
[
{
"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... | Total alkalinity (mg/L as CaCO3) = (V2 * N * 50 * 1000) / V1, where V2 is the volume of H2SO4 used in titration (mL), N is the normality of H2SO4 (M), and V1 is the volume of the water sample (mL). |
[
{
"content": "Let \\( f \\) be a holomorphic function on the unit disk \\( \\mathbb{D} \\) such that \\( |f(z)| < 1 \\) for all \\( |z| < 1 \\). Given that \\( f(0) = \\frac{1}{2} \\), find a bound for \\( |f'(0)| \\).",
"role": "user"
}
] | \dfrac{3}{4} |
[
{
"content": "Evaluate \\( \\frac{\\partial^6 f}{\\partial x^4 \\partial y^2}(0,0) \\) for the function \\( f(x,y) = \\frac{1}{1-x^2y} \\).",
"role": "user"
}
] | 48 |
[
{
"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... | Description Debit Credit
DR Salaries Expense $10,000
CR Accrued Salaries $7,085
CR Federal income taxes Payable 2,000
CR FICA taxes Payable = $10,000 x 0.765 765
CR Union Dues Payable 150
To record salaries incurred
DR Employer Payroll Taxes Expense $1,215
CR FICA taxes Payable $765
CR SUTA Payable 450
To record accrued employer payroll taxes |
[
{
"content": "Consider a differentiable function $f: [0,1] \\rightarrow [0,1]$ such that $f(f(x)) = x$ and $f(0) = 1$. Determine the value of the integral \\( \\int_{0}^{1} (x-f(x))^{2016} \\, dx \\).",
"role": "user"
}
] | \dfrac{1}{2017} |
[
{
"content": "In a sequence of 12 distinct letters, the letters 'N' and 'O' must not be adjacent, the letter 'I' must come before 'D', and there should be exactly three 'E's such that no two 'E's are next to each other. How many different arrangements of these letters are possible?",
"role": "user"
}
] | 16934400 |
[
{
"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... | 8.42% |
[
{
"content": "Evaluate the curvilinear integral \\( \\int_{\\gamma} x \\, ds \\) where the curve \\( \\gamma \\) is given in polar coordinates by \\( \\rho = 1 + \\cos(\\theta) \\) for \\( \\theta \\in [-\\pi, \\pi] \\).",
"role": "user"
}
] | \dfrac{32}{5} |
[
{
"content": "Given a sphere with center at the origin and radius 'r', calculate the average of the reciprocal of the distance between a point located at a distance 'c' from the sphere's center (where 'c' > 'r') and all points inside the sphere.",
"role": "user"
}
] | \dfrac{1}{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... | $36,868 |
[
{
"content": "Consider the curve $C$ defined by the intersection of the cylinder \\(\\frac{x^2}{25} + \\frac{y^2}{9} = 1\\) and the plane \\(3z = 4y\\). The line \\(L\\) is tangent to \\(C\\) at the point \\((0, -3, -4)\\). Find the x-coordinate of the point where \\(L\\) intersects the plane \\(2x - 3y - 4z = ... | 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) Q x b8, 2) Q-g2, 1) Q x b8, 2) Q-h8, and 1) Q x b8, 2) Q x h2 |
[
{
"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... | 32.16 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... | 3625.6 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... | Fe(OH)_3(s) + 3H_3O^+(aq) -> Fe^3+(aq) + 6H_2O(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... | 1020 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... | 92.16 nN |
[
{
"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... | $84 |
[
{
"content": "Is there a vector space $X$ with two norms $\\|\\cdot\\|_1$ and $\\|\\cdot\\|_2$ such that $(X,\\|\\cdot\\|_1)$ is separable and $(X,\\|\\cdot\\|_2)$ is not separable, considering the topology induced by each norm?",
"role": "user"
}
] | Yes |
[
{
"content": "Find the minimum value of \\( f(a, \\theta) = \\cos^3 \\theta + \\frac{4}{3a \\cos^2 \\theta - a^3} \\) given that \\( 0 < a < \\sqrt{3} \\cos \\theta \\) and \\( \\theta \\in \\left[-\\frac{\\pi}{4}, \\frac{\\pi}{3}\\right] \\).",
"role": "user"
}
] | 3 |
[
{
"content": "Let $X$ and $Y$ be independent random variables, each uniformly distributed on the interval $(0,1)$. Define the random variable $Z = \\frac{\\min\\{X,Y\\}}{\\max\\{X,Y\\}}$. Determine the distribution of $Z$ by calculating $\\mathbb{P}(Z \\le z)$ for $0 \\le z \\le 1$. Provide the cumulative distr... | 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... | (V_u - V_d) / (U - D) |
[
{
"content": "Find the minimum value of the expression \\[ \\frac{a^{10}+ b^{10}}{a^{7}+ b^{7}} +\\frac{b^{10}+ c^{10}}{b^{7}+ c^{7}} +\\frac{c^{10}+ a^{10}}{c^{7}+ a^{7}} \\] given that \\( a, b, c \\in (0, \\infty) \\) and \\( a+b+c=1 \\).",
"role": "user"
}
] | \dfrac{1}{9} |
[
{
"content": "Given that \\( f(x) \\in L^2 [-\\pi, \\pi] \\) and \\( \\int_{-\\pi} ^ {\\pi }f(x)e^{-inx}dx =0 \\) for all integers \\( n \\), determine the value of \\( \\int_{-\\pi} ^ {\\pi }|f(x) |^2 dx \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Given a configuration of points in $X \\in \\mathbb{R}^{n \\times 2}$ with zero column means, the variance of each axis $x$ and $y$ is given by:\n\\[ \\mathrm{var}(x) = \\frac{1}{n} x^T x \\quad \\text{and} \\quad \\mathrm{var}(y) = \\frac{1}{n} y^T y. \\]\nIf $\\mathrm{var}(x) > \\mathrm{var}(y)$... | 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... | 0.0372 kg |
[
{
"content": "Students arrive at a help center according to a Poisson process with rate \\( r \\). When there are \\( n \\geq 1 \\) students in the center, the first one to leave does so at a random \\( \\text{Exp}(2r) \\) time. Suppose that there are currently no students in the center. What is the expected ti... | \dfrac{4}{r} |
[
{
"content": "Evaluate the limit: \\[ \\lim_{a\\to 0}\\frac{\\delta_{a}+\\delta_{-a}-2\\delta}{a^{2}} \\] where \\( \\delta_{a} \\) is the Dirac distribution at the point \\( a \\).",
"role": "user"
}
] | \delta'' |
[
{
"content": "Let $M$ be a topological space and $f, g : M \\rightarrow \\mathbb{R}_+$ be two non-negative functions with $f(x_0) = g(x_0) = 0$ for some $x_0 \\in M$. Suppose $f(x) \\leq g(x)$ for each $x \\in M$, and $g$ is continuous at $x_0$. Is $f$ necessarily continuous at $x_0$?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the upper bound of the function $g(x)$ that satisfies the differential equation $g'(x) = \\frac{1}{x^{2} + (g(x))^{2}}$ with the initial condition $g(1) = 0$. Provide your answer as a single value.",
"role": "user"
}
] | 1 |
[
{
"content": "Consider an even cycle graph $C_{2n}$. Can contracting an edge in this graph increase its chromatic number by more than one?",
"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... | $9,732.70 |
[
{
"content": "Evaluate the limit \\( \\lim_{n\\rightarrow \\infty }\\int \\frac{n\\sin(x/n)}{x(1+x^2)}\\,dx \\) using the Lebesgue Dominated Convergence Theorem. Use the hint: \\( \\int \\frac{1}{1+x^2}\\,dx=\\pi \\).",
"role": "user"
}
] | \pi |
[
{
"content": "Determine whether a multivariate polynomial can be uniquely determined by its values on a finite set of points, similar to the univariate case. Answer with 'True' or 'False'.",
"role": "user"
}
] | False |
[
{
"content": "Let $\\Omega$ be the set of all permutations on the set \\( \\{1,2,\\ldots,n\\} \\). Define random variables \\( U_j \\) for \\( j=1,\\dots,n \\) such that \\( U_j(\\sigma)=1 \\) if \\( \\sigma(j)=j \\) and \\( U_j(\\sigma)=0 \\) otherwise. Find the expected value of \\( \\sum_{i=1}^{n}\\sum_{j=1}... | 2 |
[
{
"content": "Evaluate the limit \\( \\lim_{N \\to \\infty} \\sum_{n=1}^{N} \\frac{1}{(n+1)} \\sum_{i=1}^{n} \\frac{1}{i(n+1-i)} \\).",
"role": "user"
}
] | 2\zeta(3) |
[
{
"content": "Find the limit of the expression \\( \\left(2 \\sqrt{1 + \\frac{1}{n}}\\right)^n + \\left(4 + \\frac{4}{n}\\right)^{\\frac{n}{2}} \\) as \\( n \\to \\infty \\).",
"role": "user"
}
] | \infty |
[
{
"content": "Determine the weight of the Sorgenfrey line $S$. Recall that the weight of a topological space $X$ is defined as $\\text{Weight}(X) = \\min\\{|\\mathcal{B}| : \\mathcal{B} \\text{ is a base for } X\\}$. Provide your answer as a cardinal number.",
"role": "user"
}
] | \mathfrak{c} |
[
{
"content": "A driver and a passenger are in a car accident. Each of them independently has a probability of 0.3 of being hospitalized. When a hospitalization occurs, the loss is uniformly distributed on [0,1]. When two hospitalizations occur, the losses are independent. Calculate the expected number of people... | 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... | 1.46 M |
[
{
"content": "Determine the number of real-valued solutions to the equation \\( f(f(x)) = 25 \\) for \\( 0 \\le x \\le 2023 \\), where \\( f(x) = x\\{x\\} \\) and \\( \\{x\\} \\) is the fractional part of \\( x \\).",
"role": "user"
}
] | 1997001 |
[
{
"content": "In a cube with edge length \\( a \\), points \\( E, F, G, H \\) are the midpoints of edges \\( AB, A_{1}D_{1}, A_{1}B_{1}, BC \\) respectively. Calculate the ratio of the circumradius to the inradius of the tetrahedron \\( EFGH \\).",
"role": "user"
}
] | 2 + \sqrt{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... | $788 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.