prompt list | ground_truth string |
|---|---|
[
{
"content": "Let $S = \\{c_1, \\dots, c_n\\}$ be a set of vectors in $\\mathbb{R}^M$. Determine the maximum cardinality of a subset $S' \\subset S$ such that $\\dim(\\text{span}(S')) < \\dim(\\text{span}(S))$. What is the largest size of such a subset?",
"role": "user"
}
] | n - 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... | 0.554 atom/cm |
[
{
"content": "Given positive integers \\( p \\) and \\( q \\), and \\( n = 2^{p} \\cdot 3^{q} \\), determine the number of positive divisors of \\( n^{2} \\) that are less than \\( n \\) and are not divisors of \\( n \\).",
"role": "user"
}
] | pq |
[
{
"content": "Given the equation \\( f(x+\\frac{y}{2})-f(x-\\frac{y}{2})=2x^2y+5y^2 \\), find the derivative \\( f'(3) \\). Assume \\( y \\) is a constant.",
"role": "user"
}
] | 18 |
[
{
"content": "Let $R$ be the ring of formal power series in $n$ indeterminates over $\\mathbb{C}$, and let $(I_{k})_{k\\in \\mathbb{N}}$ be a strictly decreasing chain of unmixed radical ideals, all having the same height $s$. Assume that $\\bigcap I_{n} = \\mathfrak{p}$ is prime and that $I_{1}$ is prime. Is i... | 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... | 1.65 cm |
[
{
"content": "Evaluate the limit: $$\\lim\\limits_{(x,y) \\to (0,0)} \\frac{x^3y}{(x^2+y^2)^\\frac 32}$$",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit \\( \\lim_{t \\to 0} \\frac{1}{t} \\int_{-\\infty}^{\\infty} g\\left(\\frac{x}{t}\\right) f(x) \\, dx \\), where \\( g: \\mathbb{R} \\to \\mathbb{R} \\) is a continuous function satisfying the following conditions:\n\n- \\( g(x) \\ge 0 \\) for all \\( x \\in \\mathbb{R} \\),\n- ... | f(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... | 1.31g |
[
{
"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... | 8.00 Nm -9.00 Nm |
[
{
"content": "Find the KKT point for the following optimization problem:\n\nMinimize \\( f(x_1, x_2) = (x_1 - 3)^2 + (x_2 - 2)^2 \\)\n\nSubject to:\n\n\\( x_1^2 + x_2^2 \\le 5 \\)\n\n\\( x_1 + 2x_2 = 4 \\)\n\n\\( x_1 \\ge 0, x_2 \\ge 0 \\)",
"role": "user"
}
] | (2, 1) |
[
{
"content": "Find the generating function (closed form) for the sequence: \\( \\binom{8}{1}, 2\\binom{8}{2}, 3\\binom{8}{3}, \\dots, 8\\binom{8}{8} \\).",
"role": "user"
}
] | 8x(1 + x)^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... | The average cost method is applied by calculating the average cost of all inventories and using it to determine the cost of goods sold and the ending inventory balance. Average cost is calculated by dividing the total cost of all inventories by the total number of units available. This method is suitable for companies ... |
[
{
"content": "Find a path with a maximum length of $2n+2$ that passes through $n^2$ points inside a unit square, where the path can be a broken line.",
"role": "user"
}
] | 2n + 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... | 8.09% |
[
{
"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... | 23.564 J/K |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 11.94 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... | $157,206.57 |
[
{
"content": "Determine the number of roots of the equation \\(z^4 - 8z + 10 = 0\\) that are located within the annulus defined by \\(1 < |z| < 3\\).",
"role": "user"
}
] | 4 |
[
{
"content": "Let $B$ be an $m \\times m$ real symmetric matrix with eigenvalues $b_1, \\ldots, b_m$. Given that $\\det(B) = -1$, what can be concluded about the product of the eigenvalues of $B$?",
"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... | 3.14% |
[
{
"content": "Given the function $f: \\mathbb{R} \\to \\mathbb{R}$ such that $f(f(x)) = x^2 - x + 1$ for all $x \\in \\mathbb{R}$, find the value of $f(0)$. \\( \\boxed{} \\)",
"role": "user"
}
] | 1 |
[
{
"content": "Let $\\phi: c_0 \\to \\mathbb{R}$ be a linear functional defined by $\\phi(x_n) = \\sum_{n=1}^\\infty (-2)^{1-n} x_n$, where $c_0$ is the space of sequences that converge to 0. Determine the norm of $\\phi$. \\( \\boxed{} \\)",
"role": "user"
}
] | 2 |
[
{
"content": "Let \\( f \\in \\mathcal{S}(\\mathbb{R}^3) \\) be a real function. Consider the integral \\[ I_f = \\int_{\\mathbb{R}^3} \\int_{\\mathbb{R}^3} \\frac{f(x)f(y)}{|x-y|^2} \\, dy \\, dx. \\] Determine whether it is true that \\( I_f \\geq 0 \\) for a generic function \\( f \\).",
"role": "user"
... | I_f \geq 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... | 3/5 |
[
{
"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... | $6,500 |
[
{
"content": "Consider all real functions \\( f \\) such that \\( f(f(x)) = 1 - x \\) for all \\( x \\in \\mathbb{R} \\). Define the sum \\( S_f \\) for each such function \\( f \\) as follows:\n\n\\[ S_f = f(-2017) + f(-2016) + \\cdots + f(-1) + f(0) + f(1) + \\cdots + f(2017) + f(2018). \\]\n\nDetermine the s... | 2018 |
[
{
"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... | 0.9808 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... | 1.342 * 10^-5 M |
[
{
"content": "Given a matrix $A$, if its column space is identical to its row space, does it necessarily imply that $A$ is a symmetric matrix, i.e., $A = A^T$? Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"content": "A batch of goods is transported by 17 trains from city $A$ to city $B$ at a uniform speed of $v$ kilometers per hour. The distance between city $A$ and city $B$ is 400 kilometers. For safety reasons, the distance between any two trains must be no less than $\\left(\\frac{v}{20}\\right)^2$ kilomete... | 8 |
[
{
"content": "Let $m$ be the largest real solution to the equation\n\\[\\dfrac{3}{x-3} + \\dfrac{5}{x-5} + \\dfrac{17}{x-17} + \\dfrac{19}{x-19} = x^2 - 11x - 4\\]There are positive integers $a, b,$ and $c$ such that $m = a + \\sqrt{b + \\sqrt{c}}$. Find $a+b+c$.",
"role": "user"
}
] | 263 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty}\\max_{x\\in[0,\\pi]}\\sum_{k=1}^n\\frac{\\cos(kx)}{k} \\]",
"role": "user"
}
] | \infty |
[
{
"content": "Evaluate the infinite product: \\[ \\prod_{x=1}^{\\infty} (2x)^{\\frac{1}{4x}} \\]",
"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... | variable |
[
{
"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... | h_n =(v_0^2 epsilon^(2n))/(2g) |
[
{
"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... | 444 Hz, 888 Hz, 1332 Hz |
[
{
"content": "Find the coefficient of \\( \\frac{1}{z} \\) in the Laurent series expansion of \\( \\log\\left(\\frac{z}{z-1}\\right) \\) for \\(|z| > 1\\).",
"role": "user"
}
] | 1 |
[
{
"content": "Consider a pair of random variables \\(X, Y\\) with a constant joint density on the parallelogram with vertices at \\((0, 0)\\), \\((2, 0)\\), \\((1, 1)\\), and \\((3, 1)\\). Calculate the probability \\(P(X + 2Y > 3)\\).",
"role": "user"
}
] | \dfrac{1}{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... | $19044 |
[
{
"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... | 0.003 Vm |
[
{
"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... | 0.216m |
[
{
"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... | 125 g O2 |
[
{
"content": "There are 20 cars in my building's parking lot. All of the cars are red or white. Also, all the cars are either 2-door or 4-door. 12 of them are red, 15 of them are 4-door, and 4 of them are 2-door and white. How many of the cars are 4-door and red?",
"role": "user"
}
] | 11 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 183 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... | 0.149 rad/s |
[
{
"content": "Determine the values of \\(a > 0\\) for which the series \\(\\sum_{n=1}^\\infty \\frac{a^{na}}{n^{2a}}\\) converges.",
"role": "user"
}
] | (0, 1] |
[
{
"content": "Consider two sequences \\(a_n\\) and \\(b_n\\) where \\(a_n, b_n > 0\\). If \\(a_n \\sim b_n\\) and both \\(\\sum a_n\\) and \\(\\sum b_n\\) diverge, determine whether it implies that \\(\\lim_{N\\rightarrow\\infty} \\frac{\\sum_{n\\leq N} a_n}{\\sum_{n\\leq N} b_n} = 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... | 4OH^- (aq) + 3MnCl_2 (aq) + 2KMnO_4 (aq) -> 5MnO_2 (s) + 6Cl^- (aq) + 2K^+ (aq) + 2H_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... | her genes |
[
{
"content": "Calculate the limit using definite integrals: \\[ \\lim_{n\\to\\infty}2n\\sum_{k=1}^n\\frac1{(n+2k)^2} \\]",
"role": "user"
}
] | \dfrac{2}{3} |
[
{
"content": "Evaluate the integral \\( \\int_C \\frac{\\log{z}}{(z-i)^2} \\, dz \\) along the contour \\( C: |z-i| = \\alpha \\), where \\( 0 < \\alpha < 1 \\). Use the Laurent series representation of \\( \\log{z} \\) on \\( |z-1|<1 \\) to solve this integral.",
"role": "user"
}
] | 2\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... | $8.40 |
[
{
"content": "Does the construction of the measure $P$ in the theorem for infinite (countable) product measures require the axiom of choice? 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... | 1024.28 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... | NH_2O_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... | Operating Income$30,067 |
[
{
"content": "Find the value of $a$ in the function $f(x) = ax^n$, where $a \\neq 0$, given that $f(xy) = f(x)f(y)$ for all real numbers $x$ and $y$.",
"role": "user"
}
] | 1 |
[
{
"content": "Let \\( \\triangle ABC \\) be a triangle in the \\( xy \\) plane, where \\( B \\) is at the origin \\( (0,0) \\). Extend \\( BC \\) to \\( D \\) such that \\( BC: CD = 1:1 \\), \\( CA \\) to \\( E \\) such that \\( CA: AE = 1:2 \\), and \\( AB \\) to \\( F \\) such that \\( AB: BF = 1:3 \\). Given... | 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... | density = (Molar mass * P) / (R * T) |
[
{
"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... | 5s |
[
{
"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... | 5.24% |
[
{
"content": "Find the limit points of the sequence \\( b_n = \\left(\\frac{\\sqrt{2}+2i}{4}\\right)^n \\).",
"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... | 741.48 m/s |
[
{
"content": "Let $p$ be a prime number and $A$ be a commutative ring with unity. Determine if it is possible for $A$ to have all its residue fields (i.e., $\\operatorname{Frac}(A/\\mathfrak{p})$ for all prime ideals $\\mathfrak{p}$) of characteristic $p$, while $A$ itself does not have characteristic $p$. Prov... | 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... | 59.2 m |
[
{
"content": "Evaluate \\( \\lim\\limits_{x \\to \\infty}\\frac{(x+a)^a-(x-a)^a}{x^{a-1}} \\) where \\(a\\) is a positive constant.",
"role": "user"
}
] | 2a^2 |
[
{
"content": "Determine the structure of all continuous functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) that satisfy the functional equation \\( f(x+y) = f(x) + f(y) + 2xy \\) for all real numbers \\( x \\) and \\( y \\), given that \\( f(0) = 0 \\) and \\( f(x+1) = f(x) + f(1) + 2x \\).",
"role": "user"
... | f(x) = x^2 + kx |
[
{
"content": "In the inscribed quadrilateral \\(ABCD\\) within a circle, the side lengths are given as \\(AB = 2\\), \\(BC = 6\\), and \\(CD = DA = 4\\). Calculate the area of quadrilateral \\(ABCD\\).",
"role": "user"
}
] | 8\sqrt{3} |
[
{
"content": "Find all surjective functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) such that for all \\( x, y \\in \\mathbb{R} \\), the following equation holds:\n\\[\nf(x+f(x)+2f(y))=f(2x)+f(2y)\n\\]",
"role": "user"
}
] | f(x) = x |
[
{
"content": "Determine the order of the normalizer of the subgroup \\( H = \\langle (1,2)(3,4),(1,3)(2,4) \\rangle \\) in the alternating group \\( A_5 \\).",
"role": "user"
}
] | 12 |
[
{
"content": "Let $f:X \\to Y$ be a morphism of schemes of finite type over a field $k$. Suppose $X$ is smooth, $f$ is a closed immersion, and the Zariski tangent space of $X$ at a $k$-point $x$ is isomorphic to the Zariski tangent space of $Y$ at $f(x)$. Does there exist an open neighborhood $U$ such that the ... | Yes |
[
{
"content": "The function $f(x)$ satisfies\n\\[f(x) - 2 f \\left( \\frac{1}{x} \\right) = 4^x\\]for all $x \\neq 0.$ Find $f(2).$",
"role": "user"
}
] | -\frac{20}{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... | In order to meet target net income, each outlet will need to perform $425,120 of oil changes and $106,280 of brake repairs |
[
{
"content": "Determine a non-constant function $f(n)$ that is asymptotically equivalent to the sum $$\\sum_{k=0}^{n}(p_{2k+1}-p_{2k}).$$",
"role": "user"
}
] | n \log n |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 4.22% |
[
{
"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... | 1.16 g |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Product D: $6,100
Product E: $7,679.8
Product F: ($3,400) |
[
{
"content": "For each positive integer $n$, a non-negative integer $f(n)$ is defined by the following rules:\n\n1. $f(a b) = f(a) + f(b)$ for all positive integers $a$ and $b$.\n2. $f(n) = 0$ if $n$ is a prime greater than 10.\n3. $f(1) < f(243) < f(2) < 11$.\n\nGiven that $f(2106) < 11$, find the value of $f(... | 31 |
[
{
"content": "A regular tetrahedron has both its base edge length and side edge length equal to $a$. Determine the minimum side length of a square wrapping paper needed to completely wrap the tetrahedron without cutting the paper, only folding it. Choose the correct option:\n\n(A) $\\frac{\\sqrt{2}+\\sqrt{6}}{2... | A |
[
{
"content": "The integers $G$ and $H$ are chosen such that\n\\[\\frac{G}{x+5}+\\frac{H}{x^2-4x}=\\frac{x^2-2x+10}{x^3+x^2-20x}\\]for all real values of $x$ except $-5$, $0$, and $4$. Find $H/G$.",
"role": "user"
}
] | 2 |
[
{
"content": "On a manifold with a metric tensor, let \\( \\Omega \\) be the volume form. Is it true that the Hodge star operator applied to \\( \\Omega \\) results in 1, i.e., \\( *\\Omega = 1 \\)?",
"role": "user"
}
] | 1 |
[
{
"content": "Suppose $f$ is a real function defined on $\\mathbb{R}$ which satisfies \\(\\lim_{h\\rightarrow 0}\\ [f(x+h)-f(x-h)]=0.\\) Does this imply that $f$ is continuous?",
"role": "user"
}
] | No |
[
{
"content": "Determine the points of differentiability of the function \\( f(z) = |z|^2 + i \\bar{z} + 1 \\). Choose the correct option:\n\n1) \\( i \\)\n2) \\( 1 \\)\n3) \\( -i \\)\n4) No point in \\( \\mathbb{C} \\)",
"role": "user"
}
] | C |
[
{
"content": "In NBG set theory, consider a category-theoretic product function \\( f: V \\times V \\to V \\) that satisfies the universal property for products. Determine whether it is true that for any such product function \\( f \\), there exists an ordered pair function \\( OP: V \\times V \\to V \\) such t... | True |
[
{
"content": "If a sequence $(a_n)$ satisfies $a_n > 0$ for all $n$ and $\\frac{a_{n+1}}{a_n} \\to 1$, does it follow that $(a_n)^{1/n} \\to 1$? Answer 'yes' or 'no' and 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... | 26{eq}^o{/eq}C |
[
{
"content": "Find the sum of the infinite series with the general term \\( \\frac{n^2}{2^n} \\).",
"role": "user"
}
] | 6 |
[
{
"content": "The variables $a$ and $b$ are inversely proportional. When the sum of $a$ and $b$ is 24, their difference is 6. What is $b$ when $a$ equals 5?",
"role": "user"
}
] | 27 |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to +\\infty} \\left(\\ln \\left(e^{2x} - e^x + 1\\right) - 2x\\right)$$",
"role": "user"
}
] | 0 |
[
{
"content": "Find the divergence of the vector field \\( \\nabla \\cdot (r^n \\vec{r}) \\), where \\( r \\) is the radial distance and \\( \\vec{r} \\) is the position vector.",
"role": "user"
}
] | (n + 3) r^n |
[
{
"content": "Compute the limit: \\[ \\lim \\limits_{x\\to \\infty} \\left((2x^3-2x^2+x)e^{1/x}-\\sqrt{x^6+3}\\right). \\] Use Taylor series to find the solution.",
"role": "user"
}
] | \infty |
[
{
"content": "Evaluate the limit \\( \\lim_{x\\to 0}\\sin^2\\left(\\frac{1}{x}\\right)\\sin^2 x \\).",
"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... | -7.73 cm |
[
{
"content": "The function $y=\\frac{x^3+8x^2+21x+18}{x+2}$ can be simplified into the function $y=Ax^2+Bx+C$, defined everywhere except at $x=D$. What is the sum of the values of $A$, $B$, $C$, and $D$?",
"role": "user"
}
] | 14 |
[
{
"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... | 171 N.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... | 35.488 g |
[
{
"content": "Given a double sequence \\(a_{n,m}\\) such that:\n\n- For all \\(n \\in \\mathbb{N}\\), \\(\\lim_m a_{n,m}\\) exists and \\((a_{n,m})_{m \\in \\mathbb{N}}\\) is non-decreasing.\n- For all \\(m \\in \\mathbb{N}\\), \\(\\lim_n a_{n,m} = l\\) exists and does not depend on \\(m\\).\n\nShow that \\(\\l... | l |
[
{
"content": "Simplify the expression: $$\\sqrt{\\dfrac{\\sqrt[4]{x^3}-8}{\\sqrt[4]{x}-2}+2\\sqrt[4]{x}}\\left(\\dfrac{\\sqrt[4]{x^3}+8}{\\sqrt[4]{x}+2}-\\sqrt{x}\\right)$$",
"role": "user"
}
] | 8 - 2\sqrt{x} |
[
{
"content": "Find all pairs of distinct positive integers \\((m, n)\\) such that the equation \\(m^n - n^m = m+n\\) holds.",
"role": "user"
}
] | (2, 5) |
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