prompt list | ground_truth string |
|---|---|
[
{
"content": "Determine the cardinality of the quotient set \\( \\frac{\\mathbb{Z}}{I_{63} \\cap I_{84}} \\), where for any integer \\( n \\), \\( I_n \\) denotes the ideal \\( \\{m \\in \\mathbb{Z} \\mid \\exists r \\in \\mathbb{N} \\text{ such that } m^r \\in n \\mathbb{Z}\\} \\).",
"role": "user"
}
] | 42 |
[
{
"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... | 2100 N |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to0}\\left[1 + \\left(\\frac{\\log \\cos x}{\\log \\cos(x/2)}\\right)^2 \\right]^2$$",
"role": "user"
}
] | 289 |
[
{
"content": "Evaluate the limit of the function \\( \\exp\\left(\\frac{|x-2y|}{(x-2y)^2}\\right) \\) as \\((x,y)\\) approaches \\((2y, y)\\). Consider the behavior of the function when \\((x-2y)\\) is less than 0 and greater than 0.",
"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... | 32.78 degrees |
[
{
"content": "Evaluate the limit \\(\\lim_{\\epsilon\\to 0^+} \\frac{3}{4\\pi\\epsilon^3}\\oint_{S_\\epsilon} F \\cdot\\hat{N}dS\\) where \\(S_\\epsilon\\) is a sphere centered at \\((0,0,0)\\) with radius \\(\\epsilon\\), \\(F(x,y,z) = (yx^2, xyz, x^2y^2)\\) is a vector field, and \\(\\hat{N}\\) is the outward... | 0 |
[
{
"content": "If a series \\(\\sum_{n=1}^{\\infty} a_n\\) diverges to infinity, does the series \\(\\sum_{n=1}^{\\infty} \\frac{1}{a_n}\\) converge to zero? 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... | 471 nm |
[
{
"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... | $212,000 |
[
{
"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... | c*(NewConstant*t) + b |
[
{
"content": "Determine whether the following statement is true or false: A topological space $X$ is compact if there exists a basis $B$ for $X$ such that every open cover of $X$ which consists of elements of this basis has a finite subcover.",
"role": "user"
}
] | A |
[
{
"content": "Determine whether there exists a group with elements $a$ and $b$ such that the cyclic subgroups generated by $ab$ and $ba$ are equal ($\\langle ab \\rangle = \\langle ba \\rangle$), but the cyclic subgroups generated by $a$, $b$, and $ab$ are distinct.",
"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... | 30.09 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.0215 M^0.5 |
[
{
"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... | 97961.86 |
[
{
"content": "Determine whether every non-negative multivariate polynomial with $n$ variables on $\\mathbb{R}$ necessarily has an even degree. Justify 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... | $188,957 |
[
{
"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... | 22 m |
[
{
"content": "Calculate \\( \\frac{\\partial T}{\\partial x}\\left(\\frac{\\pi}{2}, - \\frac{\\pi}{2}\\right) \\) for the function \\( T(x,y) = \\int_{x-y}^{x+y} \\frac{\\sin(t)}{t} \\, dt \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Determine the number of solutions to the equation \\( \\ln(x) = \\frac{x^2}{2} - 1 \\).",
"role": "user"
}
] | 2 |
[
{
"content": "Determine the number of zeros of the function \\( f(z) = \\frac{1}{3}e^z - z \\) within the unit disc \\( |z| < 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... | 90 min |
[
{
"content": "Determine the degree of the field extension \\(\\mathbb{Q}(\\sqrt[3]{2} + \\sqrt{5}):\\mathbb{Q}\\).",
"role": "user"
}
] | 6 |
[
{
"content": "Let \\( f: \\mathbb{R} \\to \\mathbb{R} \\) be a differentiable function such that \\( f(2) = 6 \\) and \\( f'(2) = \\frac{1}{48} \\). Find \\( \\lim_{x \\to 2} \\int_{6}^{f(x)} \\frac{4t^3}{x-2} \\, dt \\).",
"role": "user"
}
] | 18 |
[
{
"content": "Evaluate the limit: \n\\[ \\lim_{n\\to\\infty}\\frac{\\sqrt[3]{n+1}-\\sqrt[3]{n+\\cos\\frac{3}{n}}}{\\sqrt[6]{n^2+\\sin\\frac{2}{n}}-\\sqrt[3]{n}} \\]",
"role": "user"
}
] | \dfrac{9}{2} |
[
{
"content": "In Chemistry class, Samantha finds that she can make a certain solution by mixing $.04$ liters of chemical A with $.02$ liters of water (this gives her $.06$ liters of the solution). She wants to make a total of $.48$ liters of this new solution. To do so, how many liters of water will she use?",... | 0.16 |
[
{
"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... | Years|Original cost of equipment|Depreciation|Accumulated depreciation|Net book value
---|---|---|---|---|
2016|$88,000|$2,933|$2,933|$85,067
2017|$88,000|$17,600|$20,533|$67,467
2018|$88,000|$17,600|$38,133|$49,867
2019|$88,000|$17,600|$55,733|$32,267
2020|$88,000|$17,600|$73,333|$14,667 |
[
{
"content": "In a scalene triangle with sides measuring 13, 14, and 15 units, two external angle bisectors are drawn from different vertices. The third vertex is then connected perpendicularly to these bisectors. Calculate the length of the segment joining the feet of these perpendiculars.",
"role": "user"... | 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... | 3.31 A |
[
{
"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... | 114.81 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... | P = 10, Q = 2 |
[
{
"content": "Determine the values of \\( p \\) for which the series \\( \\sum_{k=2}^\\infty \\frac{1}{k(\\ln(k))^p} \\) converges.",
"role": "user"
}
] | p > 1 |
[
{
"content": "Let \\( f : \\mathbb{R} \\to \\mathbb{R} \\) be a function satisfying the equation \\( f(x^2 + x + 3) + 2f(x^2 - 3x + 5) = 6x^2 - 10x + 17 \\) for all real numbers \\( x \\). Determine the value of \\( f(85) \\).",
"role": "user"
}
] | 167 |
[
{
"content": "Is the Lebesgue measure of a closed set with empty interior in \\(\\mathbb{R}^{n}\\) always zero?",
"role": "user"
}
] | No |
[
{
"content": "Calculate the conditional variance of a Poisson process given the sigma-algebra generated by the process at an earlier time. Specifically, find \\( \\text{Var}[(N_{t}-\\lambda t)\\mid F_{s}] \\), where \\( N_{t} \\) is a Poisson process, \\( F_{s} \\) is the sigma-algebra generated by \\( N_{s} \\... | \lambda (t - s) |
[
{
"content": "Determine whether there exists a set $\\mathfrak{S}(\\mathbb{R})$ that is strictly between the Borel sets $\\mathfrak{B}(\\mathbb{R})$ and the Lebesgue measurable sets $\\mathfrak{L}(\\mathbb{R})$ in terms of inclusion.",
"role": "user"
}
] | Yes |
[
{
"content": "Let $(a_n)_{n\\geq 1}$ and $(b_n)_{n\\geq 1}$ be positive real sequences such that \\(\\lim_{n\\to\\infty}\\frac{a_{n+1}-a_n}{n}=a\\in \\mathbb{R}_{>0}\\) and \\(\\lim_{n\\to\\infty}\\frac{b_{n+1}}{nb_n}=b\\in \\mathbb{R}_{>0}\\). Compute the limit:\n\\[\\lim_{n\\to\\infty}\\left(\\frac{a_{n+1}}{\... | \dfrac{a e}{2 b} |
[
{
"content": "Given that $X, Y, Z$ are independent random variables uniformly distributed over the interval $(0, 1)$, and that $0 < X < Y < Z < 1$, find the probability that $Z - X < \\frac{1}{3}$.",
"role": "user"
}
] | \dfrac{7}{27} |
[
{
"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... | 14.023% |
[
{
"content": "Find the coefficient of the $x^3$ term in the expansion of $(x^2 + x - 5)^7$. \\( \\text{(Use the multinomial theorem if necessary.)} \\)",
"role": "user"
}
] | -109375 |
[
{
"content": "Is it possible to construct functions \\( g \\) and \\( h \\) with periods 2 and \\( \\pi / 2 \\) respectively such that \\( g + h \\) is also a periodic function?",
"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... | 1122.39 J |
[
{
"content": "If a vector space $V$ has a proper subspace $W$ that is isomorphic to $V$, must the dimension of $V$ be infinite? Answer yes or no.",
"role": "user"
}
] | Yes |
[
{
"content": "Find all functions \\( f: \\mathbb{R}^+ \\rightarrow \\mathbb{R}^+ \\) satisfying the functional equation \\( f(1+xf(y)) = yf(x+y) \\) for all \\( x, y \\in \\mathbb{R}^+ \\).",
"role": "user"
}
] | \dfrac{1}{x} |
[
{
"content": "Evaluate the limit as \\( h \\to 0 \\) of the expression:\n\\[ \\frac{he^{-h}}{e^{-2h} + 3he^{-h} - 1} \\]",
"role": "user"
}
] | 1 |
[
{
"content": "When $f(x) = ax^3 - 6x^2 + bx - 5$ is divided by $x - 1,$ the remainder is $-5.$ When $f(x)$ is divided by $x + 2,$ the remainder is $-53.$ Find the ordered pair $(a,b).$",
"role": "user"
}
] | (2,4) |
[
{
"content": "Given a linear transformation $A$ on an $n$-dimensional vector space $V$, where there exists a vector $v \\in V$ such that $A^{n-1}v \\neq 0$ but $A^nv = 0$, find the eigenvalues of $A$.",
"role": "user"
}
] | 0 |
[
{
"content": "Compute the limit: \\[ \\lim_{x\\to 0} \\frac{\\sqrt{\\cos x} - \\sqrt[3]{\\cos x}}{\\sin^2x}. \\] Do not use l'Hospital's rule.",
"role": "user"
}
] | -\dfrac{1}{12} |
[
{
"content": "Determine the size of the conjugacy class of a 2+2+1 cycle in the alternating group A5.",
"role": "user"
}
] | 15 |
[
{
"content": "Let $S$ be a subset of $\\{1,2,3,\\ldots,1989\\}$ such that no two members of $S$ differ by $4$ or $7$. What is the largest number of elements $S$ can have?",
"role": "user"
}
] | 905 |
[
{
"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... | 0.34s |
[
{
"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... | 2.244e-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... | $1,260 Unfavorable |
[
{
"content": "Let \\(x_1, x_2, \\ldots, x_n\\) be the roots of the equation \\(x = \\tan x\\) such that \\(x_1 < x_2 < \\cdots < x_n\\). Find the value of \\(\\lim_{n \\to \\infty} (x_{n+1} - x_n)\\).",
"role": "user"
}
] | \pi |
[
{
"content": "A circle passes through the three vertices of an isosceles triangle that has two sides of length 3 and a base of length 2. What is the area of this circle? Express your answer in terms of $\\pi$.",
"role": "user"
}
] | \frac{81}{32}\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... | 0.226 gram |
[
{
"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... | {
"January": {
"motor": {
"quantity": 10000,
"cost_per_unit": 40,
"total_cost": 400000
},
"blade": {
"quantity": 42400,
"cost_per_unit": 3.5,
"total_cost": 148400
},
"total_cost": 548400
},
"February": {
"motor": {
"quantity": 12000,
"cost_per_unit": 40,
"total_cost": 480000
},
"blade": {
"quantity": 51600,
"cost_per_unit": 3.5,
"total_cost": 180600
},
"total_cost": 660600
},
"March": {
"motor": {
"quantity": 15000,
"cost_per_unit": 40,
"total_cost": 600000
},
"blade": {
"quantity": 55200,
"cost_per_unit": 3.5,
"total_cost": 193200
},
"total_cost": 793200
}
} |
[
{
"content": "Given $x = 2\\cos t - \\cos 2t$ and $y = 2\\sin t - \\sin 2t$, find the value of $\\frac{d^2y}{dx^2}$ at $t = \\frac{\\pi}{2}$. Provide your answer as a single value.",
"role": "user"
}
] | -\dfrac{3}{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... | 5.04 M |
[
{
"content": "Determine the radius of convergence of the function \\( f(z) = \\frac{\\sin(z)}{z} \\) for any point \\( z_0 \\in \\mathbb{C} \\setminus \\{0\\} \\).",
"role": "user"
}
] | \infty |
[
{
"content": "Find the remainder when $40!$ is divided by $5^{10}$. Express your answer as $40! \\mod 5^{10}$.",
"role": "user"
}
] | 7812500 |
[
{
"content": "Calculate the residue of \\( \\frac{z}{z^4+1} \\) at \\( z=e^{i\\pi/4} \\) without using the Taylor series formula.",
"role": "user"
}
] | -\dfrac{i}{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... | $1.50/unit |
[
{
"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 = IR |
[
{
"content": "Let $R$ be a commutative unital ring with no non-trivial idempotents. Is it true that the additive group $(R,+)$ is not isomorphic to the group of units $(R^\\times, \\cdot)$?",
"role": "user"
}
] | Yes |
[
{
"content": "Is there a group $G$ with a finite index subgroup $H$ such that for every subgroup $N$ of $H$ which is normal in $G$, the index $[G:N]$ is at least $n!$?",
"role": "user"
}
] | Yes |
[
{
"content": "Find the residue of the function \\( \\frac{z}{z^6+1} \\) at the point \\( z=i \\).",
"role": "user"
}
] | \dfrac{1}{6} |
[
{
"content": "Let $x, y, z \\in \\mathbb{C}$. Given that $x+y+z = 0$ and $x^2+y^2+z^2 = 0$, determine if $x^2y^2 + y^2z^2 + z^2x^2 = 0$ is true.",
"role": "user"
}
] | 0 |
[
{
"content": "Given that \\(|z-i| \\leq 2\\) and \\(z_0 = 5+3i\\), find the maximum value of \\(|iz + z_0|\\).",
"role": "user"
}
] | 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... | $Al_2(SO_4)_3 + 6H_2O \to Al_2(SO_4)_3 \cdot 6H_2O$
$Al_2(SO_4)_3 \cdot 6H_2O(aq) + 6NaOH(aq) \to 2Al(OH)_3(s) + 3Na_2SO_4(aq) + 6H_2O(l)$
$2Al(OH)_3(s) + 3H_2C_2O_4(aq) + 3K_2C_2O_4(aq) \to 2K_3[Al(C_2O_4)_3] \cdot 3H_2O(aq)$ |
[
{
"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... | pV=constant. Plot V against 1/p to get a straight line through the origin. |
[
{
"content": "Determine the number of elements in the quotient ring $\\mathbb{Z}[i]/\\langle 5+2i\\rangle$. Provide your answer as a single integer.",
"role": "user"
}
] | 29 |
[
{
"content": "Verify the Divergence Theorem for the vector field $\\mathbf{F} = 3x^2y^2\\mathbf{i} + y\\mathbf{j} - 6xy^2z\\mathbf{k}$ within the volume enclosed by the paraboloids $z = x^2 + y^2$ and $z = 2y$. Given that the divergence of $\\mathbf{F}$ is $\\operatorname{div}(\\mathbf{F}) = 1$, calculate the f... | \dfrac{\pi}{2} |
[
{
"content": "In a tetrahedron \\(A B C D\\), the side lengths are given as \\(B D = D C = C B = \\sqrt{2}\\), \\(A C = \\sqrt{3}\\), and \\(A B = A D = 1\\). Let \\(M\\) be the midpoint of \\(C D\\). Calculate the cosine of the angle between line \\(B M\\) and line \\(A C\\). Choose the correct answer:\n\n(A) ... | A |
[
{
"content": "Evaluate the contour integral \\( \\oint_C \\frac{\\cos z}{z^3} \\space dz \\) where the contour does not enclose any poles.",
"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... | 1.2e5 |
[
{
"content": "Determine whether the limit \\( \\lim \\limits_{n\\to\\infty} \\int_0^1 \\cos\\left(\\frac{1}{x}\\right) \\cos(nx)\\,dx \\) converges, and find the limit if it exists.",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the difference quotient \\( \\phi_h(t) = \\frac{\\phi(t+h) - \\phi(t)}{h} \\) converges uniformly to \\( \\phi'(t) \\) on \\( \\mathbb{R} \\) as \\( h \\to 0 \\), where \\( \\phi \\in C^\\infty_c(\\mathbb{R}) \\) is a smooth function with compact support.",
"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... | 11.6 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... | 1.39e-8 m |
[
{
"content": "Given that \\(a, b, c,\\) and \\(d\\) are positive integers such that \\(a + b + c + d = 63\\), find the maximum value of the expression \\(ab + bc + cd\\).",
"role": "user"
}
] | 991 |
[
{
"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... | $27,149,100 |
[
{
"content": "Evaluate the limit: $$ \\lim_{x\\to\\infty} \\frac { (1+2 \\sqrt[6]{x})^{2016} } {1+(2 + (3+4x^6)^7)^8} $$",
"role": "user"
}
] | 2^{1904} |
[
{
"content": "Determine the number of distinct Cayley tables that can be constructed for the set \\( \\{0, 1, \\ldots, n-1\\} \\).",
"role": "user"
}
] | n^{n^2} |
[
{
"content": "Find the sum of the infinite series: $$ \\sum_{k=0}^{\\infty}\\frac{(-1)^k}{2k+1} $$.",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "Calculate \\( \\lim_{n \\rightarrow \\infty} \\sum_{k=0}^{n-1} \\frac{1}{\\sqrt{n^{2}-k^{2}}} \\).",
"role": "user"
}
] | \dfrac{\pi}{2} |
[
{
"content": "What is the norm of the difference between the projection operators $P_n$ and $P_m$ on $l^2(\\mathbb{N})$ when $n = m$?",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the following limit exists: $$\\lim_{(x,y) \\to (0,0)} \\arctan\\left(\\frac {x^4+y^4}{x^2+y^2}\\right)$$",
"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... | 42.52 N/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... | E(r) = \frac{\rho r}{2\epsilon_0} |
[
{
"content": "Evaluate the double integral: $$ \\int_0^1 \\int_0^1 \\max(0,u_1+u_2-1) \\, du_1 \\, du_2 $$",
"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... | 11.74% |
[
{
"content": "Evaluate the expression \\( \\sec'' \\frac{\\pi}{4} + \\sec'' \\frac{3\\pi}{4} + \\sec'' \\frac{5\\pi}{4} + \\sec'' \\frac{7\\pi}{4} \\), where \\( \\sec'' \\) denotes the second derivative of the secant function.",
"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... | 2.46 M |
[
{
"content": "Find a single polynomial that generates the ideal \\( I = ((x^4 - 1)(x + 2), (x^2 - 4)(x + 1)) \\) in \\( \\mathbb{R}[x] \\).",
"role": "user"
}
] | x^2 + 3x + 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... | 1178 nm |
[
{
"content": "Consider the sets \\( S = \\{ x \\in \\mathbb{R} : x^6 - x^5 \\le 100 \\} \\) and \\( T = \\{ x^2 - 2x : x \\in (0, \\infty) \\} \\). Determine whether the intersection \\( S \\cap T \\) is:\n\n- Closed and bounded in \\( \\mathbb{R} \\)\n- Closed but not bounded in \\( \\mathbb{R} \\)\n- Bounded ... | A |
[
{
"content": "Given that the volume of tetrahedron \\(ABCD\\) is \\(V\\), point \\(E\\) is the midpoint of edge \\(AD\\), and point \\(F\\) lies on the extension of \\(AB\\) such that \\(BF = AB\\). The plane passing through points \\(C\\), \\(E\\), and \\(F\\) intersects \\(BD\\) at point \\(G\\). Determine th... | \dfrac{V}{3} |
[
{
"content": "Given the function \\( f(x) = x + 2x^3 + x^5 \\), evaluate \\( (f^{-1})'(4) \\).",
"role": "user"
}
] | \dfrac{1}{12} |
[
{
"content": "Let \\( f \\in L^1(\\mathbb{R}) \\) and \\( a > 0 \\). Prove that \\( \\lim_{a \\to 0} a \\cdot \\mu (\\{ x \\in \\mathbb{R} : |f(x)| > a \\}) = 0 \\).",
"role": "user"
}
] | 0 |
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