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... | f = \frac{d_i d_o}{d_o + d_i} |
[
{
"content": "Evaluate the determinant:\n\\[\n\\left|\n\\begin{array}{ccc} \n b^2c^2 & bc & b+c \\\\\n c^2a^2 & ca & c+a \\\\\n a^2b^2 & ab & a+b \n\\end{array}\n\\right|\n\\]",
"role": "user"
}
] | 0 |
[
{
"content": "Calculate the sum of the infinite series \\( \\sum\\limits_{k=1}^{\\infty} \\frac{k^2}{4^k} \\).",
"role": "user"
}
] | \dfrac{20}{27} |
[
{
"content": "Suppose $f(z)$ is an analytic function defined on a closed disc $D$ in the complex plane, where it is complex-differentiable inside $D$ and on the boundary of $D$ using the relative topology of the disc. Must the derivative $f'$ be continuous or bounded on the boundary of $D$?",
"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... | 443 m |
[
{
"content": "Does there exist a signature $L$ and an $L$-theory $T$ such that $T$ is not the theory of finitely many of its models? In other words, can you find a theory $T$ for which there is no finite set of models $\\{M_1,...,M_n\\}$ such that $Th(\\{M_1,...,M_n\\})=T$?",
"role": "user"
}
] | Yes |
[
{
"content": "If matrices $A$ and $B$ are invertible and $A + B$ is invertible, does it imply that $A^{-1} - B^{-1}$ is invertible? Answer 'yes' or 'no' and 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... | 908.7 |
[
{
"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... | 3231.4 J |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 9000 |
[
{
"content": "If \\( \\alpha , \\beta , \\gamma \\) are the roots of the equation \\( x^3 - x - 1 = 0 \\), evaluate the expression \\( \\frac{1+\\alpha}{1-\\alpha} + \\frac{1+\\beta}{1-\\beta} + \\frac{1+\\gamma}{1-\\gamma} \\).",
"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... | 1.27 atm |
[
{
"content": "Does Robinson's Arithmetic have infinitely many non-isomorphic models? Answer yes or no.",
"role": "user"
}
] | Yes |
[
{
"content": "Can a simple abelian variety have a Picard number of \\(n^2-1\\)?",
"role": "user"
}
] | Yes |
[
{
"content": "Given the transfer function \\( G(s) = \\frac{1}{(s+7)(s+12)} \\) and the input \\( x(t) = \\left(1 + \\frac{1}{t}\\right)^t \\), calculate the limit of the output \\( y(t) \\) as \\( t \\to \\infty \\).",
"role": "user"
}
] | \dfrac{e}{84} |
[
{
"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... | $3.40 |
[
{
"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... | 5.31% |
[
{
"content": "Evaluate the line integral \\( \\int_c xdx + ydy + zdz \\) where \\( c \\) is the intersection of the cylinder \\( x^2 + y^2 = 1 \\) and the sphere \\( x^2 + y^2 + z^2 = 9 \\), oriented clockwise when viewed from the positive z-axis.",
"role": "user"
}
] | 0 |
[
{
"content": "Let \\( T = \\{ 1, 2, 3, \\dots, 14, 15 \\} \\). A subset \\( S \\) of \\( T \\) is called *handy* if the sum of all the elements of \\( S \\) is a multiple of 5. For example, the empty set is handy (because its sum is 0) and \\( T \\) itself is handy (because its sum is 120). Compute the number o... | 6560 |
[
{
"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... | 173.21 |
[
{
"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... | Q = 200P - 10,000 |
[
{
"content": "Real numbers \\(a, b, c\\) satisfy the equations \\(a + \\frac{1}{b} = 9\\), \\(b + \\frac{1}{c} = 10\\), and \\(c + \\frac{1}{a} = 11\\). Determine the value of \\(abc + \\frac{1}{abc}\\).",
"role": "user"
}
] | 960 |
[
{
"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... | 5.00e-2 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... | 4.43 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} 2 \\int_0^{\\frac{\\pi}{2}} \\frac{\\sin((2n+1)x)}{\\sin x} \\, dx \\]",
"role": "user"
}
] | \pi |
[
{
"content": "Find the sum of the series \\( \\frac{1}{1!}+\\frac{1+2}{2!}+\\frac{1+2+3}{3!}+ \\ldots \\).",
"role": "user"
}
] | \dfrac{3e}{2} |
[
{
"content": "Determine whether there exist integer points \\((x, y)\\) on the cubic curve \\(y^2 = x^3 + 7x + 9\\) such that both \\(x\\) and \\(y\\) are odd numbers.",
"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... | 12.3% |
[
{
"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... | Br < Ga < Mn < Cr < Ca < K |
[
{
"content": "Let $b \\geq 2$ be an integer and $x \\in [0,1)$. Define $T_b(x) = bx \\mod 1$. It is known that $x$ is a normal number with respect to $b$ if and only if the sequence $(T_b^n(x))_{n \\in \\mathbb{N}}$ is uniformly distributed in $[0,1)$. If $x$ is normal with respect to $b$, then the orbit of $x$... | 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... | - 1.36 * 10^-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... | 2,219,500 |
[
{
"content": "How many integers $n$ satisfy the condition $100 < n < 200$ and the condition $n$ has the same remainder whether it is divided by $6$ or by $8$?",
"role": "user"
}
] | 25 |
[
{
"content": "Consider performing a Lagrange interpolation for the points $x_0$, $x_0 + \\varepsilon$, and $x_1$. As $\\varepsilon \\to 0$, what happens to the interpolation polynomial $P(x) = y_0L_0(x) + y_1L_1(x) + y_2L_2(x)$? Determine the degree of the resulting polynomial.",
"role": "user"
}
] | 2 |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty}\\frac{(n+1)(n+2)\\cdots(n+m)}{(n-m)!}x^n$$ where $x \\in (-1, 1)$ and $m$ is a positive integer.",
"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... | 128,000 units (Q1), 84,000 units (Q2), 112,000 units (Q3), and 153,000 units (Q4) |
[
{
"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... | $217,819,800 |
[
{
"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... | 7.44% |
[
{
"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... | 13.9e-4 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... | 7.95 *10^-13 F |
[
{
"content": "Determine the minimum number of points, \\( n \\), in a set \\( M \\) on a plane such that:\n1. There are 7 points in \\( M \\) that form the vertices of a convex heptagon.\n2. For any 5 points in \\( M \\) that form the vertices of a convex pentagon, the interior of this pentagon contains at leas... | 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... | 60.63 x 10^3 kJ |
[
{
"content": "Find the smallest positive integer $k$ such that $2^3$ divides $k$, $3^3$ divides $(k+1)$, and $7^3$ divides $(k+2)$. What is the value of $k$?",
"role": "user"
}
] | 40472 |
[
{
"content": "Find the positive integer solutions for $a$ and $b$ in the equation $a^2 - 4b^2 = 101$. Provide your answer as a set of ordered pairs $(a, b)$. ",
"role": "user"
}
] | (51, 25) |
[
{
"content": "Find the locus of the duals of lines that pass through the point (1, 1) and have variable slopes.",
"role": "user"
}
] | x + y + 1 = 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... | x = $c_1x_1+c_2x_2$ |
[
{
"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... | $111,445.82 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.98e+04 N/C |
[
{
"content": "Is the space $C_c^\\infty(\\Omega)$ dense in $W^{1,p}(\\Omega) \\cap C_0(\\Omega)$ for a bounded open set $\\Omega \\subset \\mathbb{R}^N$ and $1 \\leq p < \\infty$? Here, $C_0(\\Omega)$ denotes the space of continuous functions on $\\overline{\\Omega}$ that are zero on $\\partial\\Omega$, and $W^... | Yes |
[
{
"content": "Let $x=3^{a_1}5^{a_2}7^{a_3}$ and $y=3^{b_1}5^{b_2}7^{b_3}$ where $a_i,b_i$ are nonnegative integers such that $a_k + b_k \\le 5$ for all $k$. Find the sum $$\\sum \\gcd(x,y),$$ where the sum is over all possible $x,y$. Express your answer as a single number.",
"role": "user"
}
] | 1477773 |
[
{
"content": "Let $A$, $B$, $C$, and $D$ be points on a circle such that $AB = 11$ and $CD = 19.$ Point $P$ is on segment $AB$ with $AP = 6$, and $Q$ is on segment $CD$ with $CQ = 7$. The line through $P$ and $Q$ intersects the circle at $X$ and $Y$. If $PQ = 27$, find $XY$.",
"role": "user"
}
] | 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... | Mortgage payable (secured): $100,000
Administrative expenses payable: $16,000
Interest payable: $8,900
Mortgage payable (unsecured): $26,700
Accounts payable: $44,500
Salaries payable: $3,900 |
[
{
"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... | $22,900 |
[
{
"content": "Given a series \\(\\sum_{k=1}^\\infty p_k a_k\\) that converges to some \\(c<0\\), where \\(\\{a_k\\}_{k=1}^\\infty\\) is a known increasing sequence with \\(a < a_k < 0\\) for all \\(k\\), has a limit of 0, and a known sum \\(\\sum_{k=1}^\\infty a_k = A\\), and \\(\\{p_k\\}_{k=1}^\\infty\\) is an... | a_1 |
[
{
"content": "Is it possible to construct an uncountable subset of the unit square such that any line intersects at most a finite number of points in the subset?",
"role": "user"
}
] | Yes |
[
{
"content": "In triangle $\\triangle PQR$, the sides are $PQ = 16$, $QR = 30$, and $RP = 34$. The incircle of the triangle touches the sides at points $X, Y,$ and $Z$. The ratio of the area of $\\triangle XYZ$ to that of $\\triangle PQR$ is $\\frac{m}{n}$, where $m$ and $n$ are coprime positive integers. Find ... | 20 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.45 |
[
{
"content": "What is the smallest positive integer $a$ such that $a^{-1}$ is undefined $\\pmod{55}$ and $a^{-1}$ is also undefined $\\pmod{66}$?",
"role": "user"
}
] | 10 |
[
{
"content": "Find the sum function of the power series \\( \\sum\\limits _{n=1}^{\\infty}\\frac{x^{2n+1}}{n} \\).",
"role": "user"
}
] | -x \ln(1 - x^2) |
[
{
"content": "In square \\(ABCD\\) with side length 9, point \\(P\\) is on \\(AB\\) such that \\(AP: PB = 7:2\\). A quarter circle is drawn inside the square with center \\(C\\) and radius \\(CB\\). The tangent from \\(P\\) meets the circle at \\(E\\) and \\(AD\\) at \\(Q\\). Segments \\(CE\\) and \\(DB\\) inte... | \dfrac{85}{22} |
[
{
"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... | 0.375 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... | 18 ohms |
[
{
"content": "Given that $a, b, c > 0$ and $a + b + c = 3$, find the minimum value of the expression $$\\frac{a}{b^2+1} + \\frac{b}{c^2+1} + \\frac{c}{a^2+1}.$$",
"role": "user"
}
] | \dfrac{3}{2} |
[
{
"content": "Determine the number of conjugacy classes in the quaternion group $Q$ and find the size of each class.",
"role": "user"
}
] | 5 |
[
{
"content": "Let \\( \\{N(t), t \\geq 0\\} \\) be a Poisson process with rate \\( \\lambda \\). Define \\( T_1 \\) as the time of the first event, \\( T_n \\) as the elapsed time between the \\((n-1)\\)-th and the \\(n\\)-th event, and \\( S_n \\) as the time of the \\(n\\)-th event, for \\( n = 1, 2, \\ldots ... | \dfrac{s}{t} |
[
{
"content": "In a game, you start with two coins. Each round, you bet one coin. You have a 1/4 chance of losing the coin and a 3/4 chance of winning an additional coin. You continue playing until you either go bankrupt or play indefinitely. Define \\( f(i) \\) as the probability of going bankrupt with \\( i \\... | \dfrac{1}{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... | 1. Change in quantity demanded of the product produced
2. Changes in the production process
3. Technological changes |
[
{
"content": "Find the mass of the region $E$ bounded by the spheres $x^2+y^2+z^2 = 4z$ and $x^2+y^2+z^2 = 2z$, given that the density of the region is $\\rho(x,y,z) = (x^2+y^2+z^2)^{3/2}$.",
"role": "user"
}
] | 192\pi |
[
{
"content": "In a group of order 120, determine the number of elements of order 12, given that there is at least one Sylow 5-subgroup.",
"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... | 16627.27 |
[
{
"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... | The acidic hydrogens are the ones attached to the double-bonded carbon atoms and the aldehyde hydrogen. |
[
{
"content": "Find the minimum value of the function \\( f(a,b,c) = \\max\\{|a+1|,|2a+b|,|b+2c|,|c-2|\\} \\) for all \\( a, b, c \\in \\mathbb{R} \\).",
"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... | $22,443.45 |
[
{
"content": "Let $A$ be the $3 \\times 3$ matrix such that for any $v \\in \\mathbb{R}^3$, $Av$ gives the projection of $v$ onto the plane $x+y+z=0$. Determine $A^{15}$. Express your answer in terms of $A$. ",
"role": "user"
}
] | A |
[
{
"content": "Evaluate the limit \\( I = \\lim_{n \\to \\infty} \\sum_{k=0}^{n} \\frac{\\cos{\\sqrt{\\frac{k}{n}}}}{2^k} \\).",
"role": "user"
}
] | 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... | Pessimistic criterion
Texan equipment |
[
{
"content": "Find the limit: \\[ \\lim_{x\\to 0}\\frac{(1+\\sin 2x)^\\frac{1}{x}-e^2}{x} \\]",
"role": "user"
}
] | -2e^2 |
[
{
"content": "Find the Fourier expansion of the periodic function defined by:\n\\[ g(x) = \\begin{cases} 1 & \\text{for } x \\in [-\\pi, \\pi) \\setminus \\{0\\} \\\\ 0 & \\text{for } x = 0 \\end{cases} \\]\nwith the periodicity condition \\( g(x + 2\\pi) = g(x) \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Determine a sequence of the digits from 1 to 9, denoted as $a_1, a_2, \\ldots, a_9$, such that each two-digit number formed by consecutive digits ($a_ia_{i+1}$) is divisible by either 7 or 13. Express this sequence as a single number $10^{8}a_1 + 10^{7}a_2 + \\cdots + 10a_8 + a_9$. This problem is... | 784913526 |
[
{
"content": "A large parallelepiped is constructed from unit cubes, with each side of the parallelepiped being greater than 4 units. Two cubes are considered adjacent if they share a face, meaning a cube can have up to 6 neighbors. Given that there are 836 cubes with exactly 6 neighbors, determine the number o... | 144 |
[
{
"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... | [OH-] = [NH4+] = 4.2e-4 M, [NH3] = 0.0096M, [H3O+] = 2.4e-11M |
[
{
"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... | 13.59% |
[
{
"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... | 1.14e-5 m |
[
{
"content": "Evaluate the double integral \\( \\iint_\\Omega \\left(\\sqrt{\\frac xa}+\\sqrt{\\frac yb}\\right)\\,dxdy \\) where \\(\\Omega\\) is the region enclosed by the curve \\(\\sqrt{\\frac xa}+\\sqrt{\\frac yb} = 1\\) and the coordinate axes.",
"role": "user"
}
] | \dfrac{2ab}{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... | Red shift of Galaxies |
[
{
"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... | 3:1 |
[
{
"content": "Find the smallest possible value of \\( F = \\max_{1 \\leq x \\leq 3} |x^3 - ax^2 - bx - c| \\) as \\( a \\), \\( b \\), and \\( c \\) vary over all real numbers.",
"role": "user"
}
] | \dfrac{1}{4} |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\to \\infty} \\left(x\\sin \\frac{1}{x}\\right)^{1-x}$$ using Taylor expansions.",
"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... | 1.549*10^-27 kg |
[
{
"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... | 13.63 eV |
[
{
"content": "Let $E$ be an elliptic curve over the finite field $F_p$, where $p > 7$ is a prime number. If $E(F_p)$ has a point of order $p$, is it true that the number of points on $E(F_p)$, denoted as #$E(F_p)$, is equal to $p$?",
"role": "user"
}
] | Yes |
[
{
"content": "Given a group $G$ and an infinite sequence of independent and identically distributed (i.i.d.) random elements $\\{X_n\\}_{n=1}^{\\infty}$ in $G$, where for any subgroup $H \\leq G$,\n\\[P(X_1 \\in H) = \\begin{cases} \\frac{1}{[G:H]} & \\text{if $[G:H]$ is finite} \\\\ 0 & \\text{if $[G:H]$ is in... | No |
[
{
"content": "Given that \\( \\sin(\\theta_1) + \\sin(\\theta_2) + \\sin(\\theta_3) = 0 \\) and \\( \\cos(\\theta_1) + \\cos(\\theta_2) + \\cos(\\theta_3) = 0 \\), find the value of \\( \\cos^2(\\theta_1) + \\cos^2(\\theta_2) + \\cos^2(\\theta_3) \\).",
"role": "user"
}
] | \dfrac{3}{2} |
[
{
"content": "Determine whether the space $B^{s;1,\\theta}(\\Omega)$ is compactly embedded in $L^{r}(\\Omega)$ under the conditions $0<s<1$, $1\\leq \\theta \\leq \\frac{N}{N-s}$, $r<\\frac{N}{N-s}$, and $\\Omega$ is an open, bounded subset of $\\mathbb{R}^N$ with a smooth boundary.",
"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... | Consolidated Balance Sheet
Current Assets $466,000
Land 324,000
Building, net 816,000
Investment in Blue -
Total Assets $1,606,000
Liabilities ... |
[
{
"content": "Let $a_k > 0$ for $k \\in \\mathbb{N}$. Suppose that $\\sum_{n=1}^{\\infty} a_n^2 < \\infty$. Does this imply that $\\sup_{n \\in \\mathbb{N}} n a_n < \\infty$? Answer with 'Yes' or 'No' and justify your answer.",
"role": "user"
}
] | No |
[
{
"content": "Determine the sign of the map induced by the covering map \\( p: S^n \\to \\mathbb{R}P^n \\) on homology, specifically \\( p_{*}: H_n(S^n, R) \\to H_n(\\mathbb{R}P^n, R) \\), where \\( R \\) is an arbitrary ring. Given that \\( H_n(S^n, R) \\cong R \\) is generated by the class \\([\\sigma_n^{(1)}... | + |
[
{
"content": "Determine the number of Polish group topologies that can be placed on \\( \\text{PSL}_2(\\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... | 10.90% |
[
{
"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... | <img>/cimages/multimages/16/s8-lewis3327858767362130733.png</img> |
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