prompt list | ground_truth string |
|---|---|
[
{
"content": "Determine whether there exists an independent family \\( \\mathcal{A} \\subseteq \\mathcal{P}(\\kappa) \\) such that the Boolean algebra generated by \\( \\mathcal{A} \\) and the subsets of \\( \\kappa \\) of size less than \\( \\kappa \\) is equal to \\( \\mathcal{P}(\\kappa) \\).",
"role": "... | Yes |
[
{
"content": "Find all positive integers \\( n \\) such that the sum \\( S_{n} = \\mathrm{C}_{n}^{1} f_{1} + \\mathrm{C}_{n}^{2} f_{2} + \\cdots + \\mathrm{C}_{n}^{n} f_{n} \\) is divisible by 8, where the sequence \\( \\{f_{n}\\} \\) is defined by \\[ f_{n} = \\frac{1}{\\sqrt{5}} \\left[ \\left(\\frac{1+\\sqrt... | n \equiv 0 \pmod{3} |
[
{
"content": "Evaluate the limit: $$ \\lim_{n\\to \\infty}\\left(\\frac{1}{n^4}+3^{\\frac{2}{2+n}}\\right)^{n} $$",
"role": "user"
}
] | 9 |
[
{
"content": "Evaluate the contour integral \\(\\oint_C \\left(\\frac{2x^3+2xy^2-2y}{ x^2+y^2} \\, dx+\\frac{2y^3+2x^2y+2x}{ x^2+y^2} \\, dy\\right)\\) where \\(C\\) is any simple closed curve encircling the origin.",
"role": "user"
}
] | 4\pi |
[
{
"content": "If $f(x)$ is a smooth function from $\\mathbb{R}$ to $\\mathbb{R}$ and $f(0)=0$, is it true that $f(x)/x$ is smooth? Justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Given that \\( m \\) and \\( n \\) are integers, the point \\( P(m, n^2) \\) is on the graph of the function \\[ f(x) = (x^2 - 4)(x^2 + 12x + 32) + 4. \\] Find the number of points \\( P \\) that satisfy these conditions.",
"role": "user"
}
] | 6 |
[
{
"content": "Determine the largest possible determinant of a matrix whose elements are either $\\pm1$ or $0$. Provide your answer as a single number or expression.",
"role": "user"
}
] | 16 |
[
{
"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... | 3.074 |
[
{
"content": "Given that \\(a=e^{\\frac{2\\pi i}{5}}\\), find the trace of the matrix \\(I+A+A^2\\), where \\(A=\\begin{bmatrix}1 & a & a^2 & a^3 & a^4 \\\\ 0 & a & a^2 & a^3 & a^4 \\\\ 0 & 0 & a^2 & a^3 & a^4 \\\\ 0 & 0 & 0 & a^3 & a^4 \\\\ 0 & 0 & 0 & 0 & a^4 \\end{bmatrix}\\).",
"role": "user"
}
] | 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... | 277.31 s |
[
{
"content": "Given a linearly recurrent sequence \\(a_k\\) such that \\(a_k \\geq 0\\) and \\(\\sum_{k=1}^{\\infty} a_k = 1\\), determine if there always exists a constant \\(x\\) and a constant \\(r < 1\\) such that \\(a_k \\leq x r^k\\) for all \\(k\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the integral \\( \\int_{|z|=2} \\frac{dz}{z^3(z^{10}-2)} \\) using the residue theorem. Consider the substitution \\( \\zeta = 1/z \\) to simplify the calculation.",
"role": "user"
}
] | 0 |
[
{
"content": "Calculate the area of the region defined by the inequality \\( \\sin^2(\\pi x) + \\sin^2(\\pi y) \\le 1 \\) for \\( x, y \\in [-1, 1] \\).",
"role": "user"
}
] | 2 |
[
{
"content": "Thirty people are seated around a circular table. Each person is either an idiot or a wise person. Each person is asked: \"Is the person to your right wise or an idiot?\" Wise people always give the correct answer, while idiots may answer correctly or incorrectly. Given that the number of idiots d... | 14 |
[
{
"content": "Determine the distance of the function \\( \\mathrm{id}_{[0,1]}(x) = x \\) from the closed subspace \\( \\left\\{ f \\in C[0,1] \\: : \\: \\int_0^{\\frac{1}{2}}f(t) \\, dt = f(0) \\right\\} \\) in the space \\( C[0,1] \\) with the supremum norm.",
"role": "user"
}
] | \dfrac{1}{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... | $0.3487382279 trillion |
[
{
"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... | 0.275m |
[
{
"content": "Prove that the sum of the n nth roots of any complex number is equal to zero.",
"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... | 1.44*10^9J |
[
{
"content": "Let G be an input graph to the max flow problem. Let (A, B) be a minimum capacity s−t cut in the graph. Suppose we add 1 to the capacity of every edge in the graph. Is it necessarily true that A is still a minimum cut? Answer yes or no.",
"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... | $893.52 |
[
{
"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... | Rate = k[C6H11OH] |
[
{
"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... | 1.69e-13 J |
[
{
"content": "Find the last two digits of the number \\(7^{7^{7^{10217}}}\\).",
"role": "user"
}
] | 43 |
[
{
"content": "Find the arc length of the horizontal curve on the cone given by $x^{2}+y^{2}=a^{2}z^{2}$ at height $z = h$.",
"role": "user"
}
] | 2\pi a h |
[
{
"content": "Determine algebraically the number of intersection points between the functions $10\\sin(x)$ and $y=x$ for $x \\geq 0$.",
"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... | +1.2 |
[
{
"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... | 0.55D |
[
{
"content": "Calculate the value of \\( \\displaystyle \\sum ^{2020}_{n=1} n^{2019} \\pmod{2019} \\).",
"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... | 77.23 |
[
{
"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... | 17.8 rev/min |
[
{
"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... | -$13,276 |
[
{
"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... | Del Ray Enterprises Inc.
Statement of Cash Flows (Indirect Approach) for the year ending December 31, 2016
Cash flows from operating activities:
Net income $332,000
Add (deduct) noncash effects on operating income
Depreciation expense $83,400
Decrease in accounts receivable $17,400
Increase in inventory ... |
[
{
"content": "Evaluate the integral \\( \\int_C \\frac{1}{z^4+1} \\, dz \\), where \\( C = \\{ z \\mid |z-1| = 2 \\} \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to 0}\\frac{\\sinh x - \\sin x}{x\\sin^2 x}$$.",
"role": "user"
}
] | \dfrac{1}{3} |
[
{
"content": "Determine the maximum number of non-attacking $2 \\times 2$ squares that can be placed on a $1000 \\times 1000$ board. Two $2 \\times 2$ squares are said to attack each other if they lie in the same pair of adjacent rows (or columns), and the cells between them are all empty. What is the largest n... | 250000 |
[
{
"content": "In an equilateral triangle ABC, a point P is located inside the triangle such that the distances from P to the vertices are PA = 5, PB = 7, and PC = 8. Determine the length of side AB.",
"role": "user"
}
] | \sqrt{129} |
[
{
"content": "Suppose $G$ is a Lie group and $X$ is a smooth vector field on $G$. Determine whether $X$ can be expressed as a linear combination of left-invariant vector fields with smooth functions as coefficients.",
"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... | 15015 |
[
{
"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... | 1473.8 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... | $448.11 |
[
{
"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... | 10 m/s^2 |
[
{
"content": "Calculate the norm of the operator $A: C[-\\pi/2,\\pi/2] \\to C[-\\pi/2,\\pi/2]$ defined by $Ax(t) = \\cos(t)x(t)$. Provide your answer as a single value.",
"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... | 0.3504 J |
[
{
"content": "Can you find an example of an unbounded linear operator on a Banach space whose rank is finite? Specifically, consider two Banach spaces $X$ and $Y$, and a linear map $T: X \\to Y$ such that $T(X) \\subseteq Y$ is of finite dimension. Is it possible for $T$ to be unbounded?",
"role": "user"
... | Yes |
[
{
"content": "Determine the minimum possible degree of a polynomial function $f(z)$ on the complex plane, given that $f(z) = \\delta$ whenever $z^3 = \\gamma$ and $z^3 = \\gamma$ has exactly 3 distinct solutions.",
"role": "user"
}
] | 3 |
[
{
"content": "Evaluate the integral \\( \\int_{\\frac{-\\pi}{2}}^{\\frac{\\pi}{2}} \\frac{\\cos^2 x}{1+a^x} \\, dx \\), where \\( a \\) is a constant.",
"role": "user"
}
] | \dfrac{\pi}{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... | 6a_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... | 10.1 |
[
{
"content": "Given that \\((x+\\sqrt{x^2+3})(y+\\sqrt{y^2+3})=3\\), find the value of \\(x+y\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Calculate the limit: $$\\lim \\limits_{n \\to \\infty}\\left(1-e^{\\left(\\frac{t+ \\log^2(n)}{t-\\log(n)}\\right)}\\right)^n$$",
"role": "user"
}
] | e^{-e^{-t}} |
[
{
"content": "Evaluate the integral: $$\\int_0^1\\frac{\\mathrm dx}{\\sqrt{1-x^\\frac{2}{3}}\\ \\sqrt[3]{2-x^\\frac{1}{2}}\\ \\sqrt[6]x}$$",
"role": "user"
}
] | \pi |
[
{
"content": "Consider a topology $T$ on $\\mathbb{R}^3$ that contains sets of the form $\\{(x,y,z) \\mid x^2 + y^2 + z^2 \\leq r\\}$, where $r \\in \\mathbb{R}^{\\geq 0}$. Determine if the union of all sets in $T$ for which $r \\in (0,1)$ belongs to $T$. Provide a justification for your answer.",
"role": "... | 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... | $1,086,378.57 |
[
{
"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 torque for the rod rotating about the center is =148.9Nm, and the torque for the rod rotating about one end is =595.7Nm |
[
{
"content": "Determine the smallest positive integer $n$ such that $5^n\\equiv n^5\\pmod 3$.",
"role": "user"
}
] | 4 |
[
{
"content": "Assume $A$ is a nontrivial submodule of a free $R$-module, where $R$ is a ring. Is it true that $A$ is also a free module?",
"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... | March 1st, 2014:
Investment on Bonds: $50000
Interest: $250
To Cash: $50250
March 16th, 2014:
Investment on Bonds: $84000
Interest: $175
To Cash: $84175
August 1st, 2014:
Cash: $1500
To Interest: $250
To Interest Revenue: $1250
August 31st, 2014:
Cash: $19900
Loss: $200
To Investment on Bonds: $20000
To Interest Rev... |
[
{
"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... | 2.14e6 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... | 10.3 |
[
{
"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... | 7.92 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... | attachment disorder |
[
{
"content": "Given that \\( z = x + iy \\in \\mathbb{C} \\) satisfies \\( \\arg \\left[\\frac{z-1}{z+1}\\right] = \\frac{\\pi}{4} \\), determine which of the following equations is true:\n\n(a) \\( x^2-y^2-2x-1 = 0 \\)\n\n(b) \\( x^2+y^2x-1 = 0 \\)\n\n(c) \\( x^2+y^2-2y-1 = 0 \\)",
"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... | 3549.32 cal |
[
{
"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... | 292 slits/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... | 19.5 |
[
{
"content": "Find an irreducible polynomial of degree 2 over a finite field of characteristic 2.",
"role": "user"
}
] | x^2 + x + 1 |
[
{
"content": "Determine the second homotopy group of the space obtained by removing a line and a point (where the line does not intersect the point) from \\( \\mathbb{R}^3 \\).",
"role": "user"
}
] | \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... | S = \frac{(\sqrt{21})u^2}{78.4} |
[
{
"content": "Evaluate the limit: \\[ \\lim _{n \\rightarrow \\infty} \\left[\\frac{1 \\cdot 1 !+2 \\cdot 2 !+\\cdots+n \\cdot n !}{(n+1) !}\\right]^{(n+1)!} \\]",
"role": "user"
}
] | \dfrac{1}{e} |
[
{
"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... | 5.71e13 C |
[
{
"content": "Find the smallest positive integer \\( x \\) such that:\n- \\( x \\equiv 1 \\pmod{7} \\)\n- \\( x \\equiv 1 \\pmod{9} \\)\n- \\( x \\equiv 3 \\pmod{64} \\)",
"role": "user"
}
] | 3907 |
[
{
"content": "Let $ABCDE$ be a pentagon inscribed in a circle such that $AB = CD = 3$, $BC = DE = 10$, and $AE= 14$. The sum of the lengths of all diagonals of $ABCDE$ is equal to $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ?\n$\\textbf{(A) }129\\qquad \\textbf{(B) }2... | 391 |
[
{
"content": "Evaluate the limit of the series: $$\\sum_{n=1}^\\infty (-1)^{n+1}\\frac{2n+1}{n(n+1)}.$$",
"role": "user"
}
] | 1 |
[
{
"content": "The largest value of $x$ that satisfies $\\sqrt{x+1}=x$ can be written as $\\dfrac{a+\\sqrt{b}}{c}$ where $c$ has no common factor with $a$ and $\\sqrt{b}$ and $b$ is not divisible by the square of any integer greater than 1. What is the value of $a+b+c$?",
"role": "user"
}
] | 8 |
[
{
"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... | magnesium |
[
{
"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 m/s^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... | 6.71e5 m/s |
[
{
"content": "Two planes \\(\\pi_1\\) and \\(\\pi_2\\) intersect in a line \\(l\\). The angle between planes \\(\\pi_1\\) and \\(\\pi_2\\) is \\(45^\\circ\\). Let \\(A\\) be a point on \\(l\\), and let \\(m\\) be the line in plane \\(\\pi_1\\) passing through \\(A\\) that is perpendicular to \\(l\\). Let \\(B\\... | 30^\circ |
[
{
"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... | 5578.0 |
[
{
"content": "Given the space $H$ of real homogeneous harmonic polynomials of degree $k$ in $n$ variables, equipped with the inner product \\( \\langle f,g\\rangle = \\int_{S^{n-1}} f(x) g(x) d\\sigma(x) \\), an orthonormal basis \\( \\{v_i\\}_i \\) exists. Define the function \\( Q: [-1,1] \\to \\mathbb{R} \\)... | Yes |
[
{
"content": "Determine whether there exists a non-abelian group for each integer $k > 1$ such that the group has a minimal generating set of size $k$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the supremum of the real number \\( M \\) such that the inequality \\( a^2 + b^2 + c^2 + 3abc \\geq M(ab+ bc + ca) \\) holds for all positive real numbers \\( a, b, c \\) satisfying \\( a+b+c=4 \\).",
"role": "user"
}
] | 2 |
[
{
"content": "Evaluate the double integral \\( \\iint_S \\frac{(x+y)^4}{(x-y)^5} \\,dA \\) where \\( S = \\{-1 \\leq x + y \\leq 1, 1 \\leq x - y \\leq 3\\} \\) using the change of variables \\( u = x + y \\) and \\( v = x - y \\).",
"role": "user"
}
] | \dfrac{4}{81} |
[
{
"content": "Evaluate the following limit as \\( x \\) approaches infinity:\n\n\\[\n\\lim_{x \\to \\infty} \\frac{(x \\log x + x)^5(1+2/x)^x}{(x+1/x)^5 (\\log x + 1/\\log x)^6}\n\\]",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the number of tuples \\((k_1, k_2, \\ldots, k_{26})\\) such that each \\(k_i\\) is either 0, 1, or 3, and the sum \\(k_1 + k_2 + \\ldots + k_{26} = 15\\).",
"role": "user"
}
] | 853423740 |
[
{
"content": "Let \\( f \\in \\mathcal{S(\\mathbb{R})} \\) be a positive function. Determine the point \\( \\xi \\) where the function \\( \\xi \\mapsto |\\hat{f}(\\xi)|^2 \\) attains its maximum value.",
"role": "user"
}
] | 0 |
[
{
"content": "What is the topological dimension of Sierpiński's Triangle?",
"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... | 185.74% |
[
{
"content": "Two lines have the same non-zero $y$-intercept. The first line has a slope of 10 and an $x$-intercept of $(s, 0)$. The second line has a slope of 6 and an $x$-intercept of $(t, 0)$. What is the ratio of $s $ to $t$? Express your answer as a common fraction.",
"role": "user"
}
] | \frac 35 |
[
{
"content": "Find the degree of the field extension \\( \\mathbb{Q} (\\sqrt{3}) \\cap \\mathbb{Q} (i) \\) over \\( \\mathbb{C} \\).",
"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... | 9000 |
[
{
"content": "Evaluate the complex line integral: \\[ I = \\int_{0}^{4+2i} z^* \\,\\mathrm{d}z \\] where \\( z^* = x - iy \\).",
"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... | 35,840 |
[
{
"content": "Evaluate the integral \\( \\int_0^{\\infty}\\frac{\\ln(x^2+1)}{x^2+1}dx \\).",
"role": "user"
}
] | \pi \ln 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... | 9.42e10 m/s |
[
{
"content": "Determine whether the series \\( \\Phi = \\sum_{k=1}^{\\infty} \\frac{1}{k^a} \\cdot \\frac{1}{(k+1)^b} \\) converges, where \\(a\\) and \\(b\\) are positive constants.",
"role": "user"
}
] | a + b > 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. January 1, 2020:
Investment in Towne Corporation Common Stock: $108,000
Cash: $108,000
2. December 30, 2020:
Cash: $1,920
Investment in Towne Corporation: $1,920
3. December 31, 2020:
Investment in Towne Corporation: $5,760
Income of Towne Corporation: $5,760 |
[
{
"content": "Derive a general formula for the number of subsets of an ordered set $S = \\{1, 2, 3, \\ldots, N\\}$ where adjacent elements can either be grouped together as one \"unit\" (indicated by a dash, e.g., $1-2$) or appear as separate elements. For example, for $N=2$, there are $5$ such subsets: $\\{\\}... | F_{2N+1} |
[
{
"content": "Determine the largest possible value of $a$ such that there exist real numbers $b, c > 1$ satisfying the equation:\n\\[a^{\\log_b c} \\cdot b^{\\log_c a} = 2023.\\]",
"role": "user"
}
] | \sqrt{2023} |
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