prompt list | ground_truth string |
|---|---|
[
{
"content": "A whole number larger than 2 leaves a remainder of 2 when divided by each of the numbers 3, 4, 5, and 6. What is the smallest such number?",
"role": "user"
}
] | 62 |
[
{
"content": "Determine whether there exists a function \\( f: \\mathbb{C} \\to \\mathbb{C} \\), expressed as \\( f = u + iv \\), such that the partial derivatives \\( u_x, u_y, v_x, \\) and \\( v_y \\) exist and satisfy the Cauchy-Riemann equations at \\((0,0)\\), but \\( f \\) is not differentiable at \\((0,0... | Yes |
[
{
"content": "Determine the closed form of the generating function for the sequence \\( u_n = \\frac{n^2+3n+2}{(n+1)!} \\) for \\( n = 0, 1, 2, \\ldots \\).",
"role": "user"
}
] | (x + 2)e^x |
[
{
"content": "Consider a list of polynomials $P_a(k_1,\\ldots,k_n)$ for $a=1,\\ldots,n$ in real variables $k_1,\\ldots,k_n$ such that \\( \\sum_{a=1}^n k_a P_a(k_1,\\ldots,k_n) = 0 \\). Determine if there exists a list of polynomials $P_{ab}(k_1,\\ldots,k_n)$ for $a,b=1,\\ldots,n$ such that $P_{ab}+P_{ba}=0$ an... | 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... | 10.52% |
[
{
"content": "Find the limit of \\( \\frac{x^2 - \\log(1+x^2)}{x^2 \\sin^2 x} \\) as \\( x \\to 0 \\).",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $5.77 |
[
{
"content": "Compute the conditional expectation \\( E(e^{-\\Lambda}|X=1) \\) given that \\( X \\) has a probability mass function \\( P_{\\lambda}(X=x)=\\frac{\\lambda^{x}e^{-\\lambda}}{x!} \\) for \\( x=0,1,2,\\ldots \\) and \\( \\Lambda \\) is a random variable with probability density function \\( f(\\lamb... | \dfrac{4}{9} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.165/d |
[
{
"content": "How many cubic feet are in three cubic yards?",
"role": "user"
}
] | 81 |
[
{
"content": "What is the area enclosed by the region defined by the equation $x^2+y^2+12x+16y=0$?",
"role": "user"
}
] | 100\pi |
[
{
"content": "Consider a non-Hausdorff compactification $X$ of $\\mathbb{Z}$ with two points $\\infty_1$ and $\\infty_2$. A basis of neighborhoods for each $\\infty_i$ is formed by sets $U$ such that $\\infty_i \\in U$ and $U$ has a finite complement. Determine whether the space $X$ is first countable.",
"r... | 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... | $646,500 |
[
{
"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... | 1.791 ft |
[
{
"content": "Jo practices her tennis serve by serving until she achieves 50 successful serves. Each serve is successful with a probability of 0.4, independently of previous serves. What is the approximate probability that she will need more than 100 serves to accomplish her goal?",
"role": "user"
}
] | 0.974 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty}\\left(\\sqrt[n]{n+2}\\cdot\\sqrt[n^2+2n]{\\binom{n}{1}\\binom{n}{2}\\cdots\\binom{n}{n}}\\right) \\]",
"role": "user"
}
] | \sqrt{e} |
[
{
"content": "Find the arc length of the curve defined by the equation \\(24xy = x^4 + 48\\) from \\(x = 2\\) to \\(x = 4\\).",
"role": "user"
}
] | \dfrac{17}{6} |
[
{
"content": "If $0.8= \\frac{y}{186+x}$, find the smallest value of $x$, if $x$ and $y$ are positive integers.",
"role": "user"
}
] | x=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... | 88.85% |
[
{
"content": "Evaluate the limit \\( \\lim_{n\\to\\infty}\\sum_{k=1}^n \\frac{\\sin(k)}{nk} \\).",
"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... | 2.4e11 |
[
{
"content": "Find \\( \\varepsilon > 0 \\) such that the map \\( F \\colon NM \\cap (\\mathbb{R}^2 \\times B_{\\varepsilon}(0)) \\rightarrow M_{\\varepsilon} \\colon (p,w) \\mapsto p + w \\) is a diffeomorphism, where \\( M = \\{(x,x^2) : x \\in \\mathbb{R}\\} \\) and \\( M_{\\varepsilon} = \\bigcup_{p \\in M}... | \dfrac{1}{2} |
[
{
"content": "Find the radius of convergence for the series \\( \\sum_{n=1}^\\infty \\left (1+{1 \\over 2}+\\ldots+{1 \\over n} \\right )x^n \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Find the maximum value of the function \\( f(x,y,z) = x + 2y + 3z \\) using Lagrange Multipliers, subject to the constraints \\( x - y + z = 1 \\) and \\( x^2 + y^2 = 1 \\).",
"role": "user"
}
] | 3 + \sqrt{29} |
[
{
"content": "Solve the equation \\(2^{x+3} + 3^{x-5} = 2^{3x-7} + 3^{2x-10}\\).",
"role": "user"
}
] | 5 |
[
{
"content": "Let $G$ be a group and $H$ a subgroup with $[G:H]=n$. Is it true that $x^n \\in H$ for all $x \\in G$?",
"role": "user"
}
] | No |
[
{
"content": "If $f$ is a complex function with a constant derivative, prove that $f(z) = az + b$ for every $z \\in \\mathbb{C}$.",
"role": "user"
}
] | f(z) = az + b |
[
{
"content": "In an imaginary quadratic field $K$, is a prime element of the ring of integers of $K$ necessarily an integer?",
"role": "user"
}
] | No |
[
{
"content": "Determine the values of \\( a \\in \\mathbb{R} \\) for which the vectors \\( w_1 = v_1 + av_2 \\), \\( w_2 = v_2 + av_3 \\), and \\( w_3 = v_3 + av_1 \\) are linearly independent, given that \\( v_1, v_2, v_3 \\) are linearly independent vectors in a vector space \\( V \\).",
"role": "user"
... | a \neq -1 |
[
{
"content": "Evaluate the contour integral \\( \\int_\\gamma \\frac{\\cos^2 z}{z^2} \\, dz \\), where \\( \\gamma \\) is the unit circle in the complex plane, traversed in the anti-clockwise direction.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the integral of the function \\( f(x,y) = x^2 \\sin(y) \\) over the surface defined by \\( g(x,y) = 2x - 2y \\) on the domain \\([0,1] \\times [0,\\pi]\\).",
"role": "user"
}
] | 2 |
[
{
"content": "Evaluate the integral \\( \\int\\limits_{-\\infty}^{\\infty}\\frac{\\sin(mx)dx}{x^2+a^2} \\) using contour integration, where \\( m \\) and \\( a \\) are positive constants.",
"role": "user"
}
] | 0 |
[
{
"content": "Given two random variables $X$ and $Y$ with a joint distribution, where $E(X) = E(Y)$, $E(X^2) = E(Y^2)$, and $Cov(X, Y) = 2$, determine whether the statement $Var(X) < 2$ is true or false.",
"role": "user"
}
] | B |
[
{
"content": "An ant starts from vertex \\( A \\) of a rectangular prism \\( ABCD-A_1B_1C_1D_1 \\) and travels along the surface to reach vertex \\( C_1 \\) with the shortest path distance being 6. What is the maximum volume of the rectangular prism?",
"role": "user"
}
] | 12\sqrt{3} |
[
{
"content": "Find the supremum of the set \\( \\{2^n - 3^m : n, m \\in \\mathbb{N}\\} \\).",
"role": "user"
}
] | \infty |
[
{
"content": "Evaluate the line integral \\( \\oint \\mathbf{F} \\cdot \\mathbf{n} \\, ds \\) where \\( \\mathbf{F} = y \\mathbf{i} + x \\mathbf{j} \\) and the curve is a circle with radius one. Do not use Green's theorem in your solution.",
"role": "user"
}
] | 0 |
[
{
"content": "Find the value of $x$ such that $x^{x^{x^{\\dots}}}=2$.",
"role": "user"
}
] | \sqrt{2} |
[
{
"content": "Albert starts to make a list, in increasing order, of the positive integers that have a first digit of 1. He writes $1, 10, 11, 12, \\ldots$ but by the 1,000th digit he (finally) realizes that the list would contain an infinite number of elements. Find the three-digit number formed by the last thr... | 116 |
[
{
"content": "Given that $G$ is the centroid of triangle $\\Delta ABC$, with $GA = 2\\sqrt{3}$, $GB = 2\\sqrt{2}$, and $GC = 2$, calculate the area of $\\Delta ABC$.",
"role": "user"
}
] | 6\sqrt{2} |
[
{
"content": "A triangle has an area of \\( A \\). The lengths of the medians of this triangle are used to form a second triangle, then the medians of that one are used to form a third triangle, and so on. Calculate the sum of the areas of all the triangles in this sequence.",
"role": "user"
}
] | 4A |
[
{
"content": "Given that $W(t)$ is a standard Brownian motion, calculate the probability that $W(2) < 0$ given that $W(1) > 0$. Express your answer as a numerical value for $\\mathbb{P}[W(2)<0|W(1) > 0]$. Use the joint distribution of $(W(1), W(2)) \\sim N((0,0), \\begin{bmatrix} 1 & 1 \\\\ 1 & 2 \\end{bmatrix}... | 0.25 |
[
{
"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... | 3.492 |
[
{
"content": "Evaluate the integral \\( \\int_0^\\infty \\frac{t^2 - \\sin^2 t}{t^4} \\, dt \\).",
"role": "user"
}
] | \dfrac{\pi}{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... | 15.68 kg |
[
{
"content": "Let $X, X_1, X_2, \\ldots$ be IID random variables with $P(X=0) = P(X=1) = 1/2$. Let $N_1$ be the number of 0's and 1's until the first appearance of the pattern 10. Find $E[N_1]$. \\(\\text{Express your answer as a single number.}\\)",
"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... | 4670.88 |
[
{
"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... | 1095.07 |
[
{
"content": "Given a convergent sequence $\\{x_n\\}$ with limit $x$, if there is no constant subsequence in $\\{x_n\\}$, does the limit $x$ become a limit point of the range set of $\\{x_n\\}$? Answer yes or no and justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the line integral \\( \\int_D x\\ln(y) \\, dx \\) over the region \\( D \\) defined by \\( 1 \\leq x \\leq 2 \\) and \\( e^x \\leq y \\leq e^{x^2} \\) using Green's Theorem. The answer is given as \\( \\frac{-17}{12} \\). Verify this result.",
"role": "user"
}
] | -\dfrac{17}{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... | 9.39e23 molecules |
[
{
"content": "Is a matrix diagonalizable if its minimal polynomial is a product of distinct linear factors?",
"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... | 154.8 m, 31.1 degrees south of west |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{2\\pi} e^{\\cos(\\varphi)} \\cos(\\varphi - \\sin(\\varphi)) \\, d\\varphi \\) using the method of residues.",
"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... | 39000 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... | Cinerney Jeep Tours
Flexible Budget
For the Month Ended October 31
Actual guests 350
Actual jeeps 104
Revenue $35,350
Expenses:
Tour guide wages $12,480
Vehicle expenses $14,052
Administrative expenses $3,750
Total expense $30,282
Net operating income $5,068 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 788.1 m |
[
{
"content": "Determine whether the following statement is true or false: \"If a function $f$ is continuously differentiable on $\\mathbb{R}$ and $f(x) \\rightarrow 1$ and $f'(x) \\rightarrow b$ as $x \\rightarrow \\infty$, then the value of $b$ must be $1$.\" Provide a proof or a counterexample.",
"role": ... | B |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 6.875% |
[
{
"content": "Find the limit of the following expression as \\(n\\) approaches infinity, given that \\(f\\) is a continuous function on the interval \\([0,1]\\):\n\n\\[ \\lim_{n\\to \\infty} (n+1) \\int_0^1 x^n f(x) \\,dx \\]",
"role": "user"
}
] | f(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... | 3287 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... | 49.6% |
[
{
"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... | Rate of decomposition of NH4NO2 = -1 * Rate of formation of N2 |
[
{
"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... | 7.05e+22 molecules |
[
{
"content": "Given the joint probability density function:\n\\[\nf(x,y) =\n\\begin{cases}\ne^{-x-y}, & \\text{for } x>0 \\text{ and } y>0 \\\\\n0, & \\text{elsewhere}\n\\end{cases}\n\\]\nFind the probability \\( P(X+Y>3) \\).",
"role": "user"
}
] | 4e^{-3} |
[
{
"content": "Determine the minimum number of times two fair six-sided dice must be rolled to ensure a 95% probability that the proportion of times they show the same face is between \\( \\frac{5}{36} \\) and \\( \\frac{7}{36} \\) using the central limit theorem.",
"role": "user"
}
] | 692 |
[
{
"content": "Determine the region of the domain \\( D = \\{ ( x , y ) \\in \\mathbb{R}^2 \\mid x \\ge 1 , y \\ge 1 \\} \\) on which the function \\( f(x, y) = \\begin{cases} \\frac{2x^2 - xy - y^2}{x^2 - y^2} & x \\neq y \\\\ \\frac{3}{2} & x = y \\end{cases} \\) is continuous.",
"role": "user"
}
] | D |
[
{
"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... | 0.0469 M |
[
{
"content": "Determine the values of \\( z \\in \\mathbb{C} \\) for which \\( A_2(z) = 0 \\), where \\( e^{jz} = A_0(z) + j A_1(z) + j^2 A_2(z) \\) and:\n\\[\n\\begin{cases}\nA_0(z) = \\displaystyle \\sum_{n \\geq 0} \\frac{x^{3n}}{(3n)!} \\\\\nA_1(z) = \\displaystyle \\sum_{n \\geq 0} \\frac{x^{3n+1}}{(3n+1)!... | 0 |
[
{
"content": "In \\( \\triangle PQR \\), if \\( 3\\sin P + 4\\cos Q = 6 \\) and \\( 4\\sin Q + 3\\cos P = 1 \\), find the measure of angle \\( R \\).",
"role": "user"
}
] | 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... | 5.25 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... | 40670 V |
[
{
"content": "Suppose $f:[a,b]\\to\\mathbb{R}$ is a continuous function and there exists a sequence $(\\alpha_n)_{n=1}^{\\infty}$ with $\\alpha_n\\to 0$ such that for all $x\\in(a,b)$:\n\n$$\\lim_{n\\to\\infty}\\frac{f(x+\\alpha_n)+f(x-\\alpha_n)-2f(x)}{\\alpha_n^2}=0$$\n\nDoes this imply that $f$ is a linear f... | f(x) = ax + b |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 25.1% |
[
{
"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... | 9.38 |
[
{
"content": "In a finite metric space, if one singleton set is open, determine whether all singleton sets are open.",
"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... | 0 m/s |
[
{
"content": "Prove that the $\\mathbb{K}$-algebra $\\mathbb{K}[s, t]/\\langle s-t \\rangle$ is isomorphic to $\\mathbb{K}[t]$, where $\\mathbb{K}$ is a field.",
"role": "user"
}
] | \mathbb{K}[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... | 1.19e-2 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... | 5.98 * 10^-10 N/C, Southward |
[
{
"content": "Can Euler's equation be used to find extremals of the functional \\( \\int_0^1 [(y')^2 + 12 xy] \\, dx \\) for functions in the space \\( \\{f \\in C^1([0,1]) , f''\\in L^1([0,1]) : f(0)=0, f(1)=1\\} \\)?",
"role": "user"
}
] | y = x^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... | Lithium, Sodium, Magnesium, and Beryllium |
[
{
"content": "In a normed space $X$, is any ball around a point $x \\in X$ connected?",
"role": "user"
}
] | Yes |
[
{
"content": "Consider a continuous $2\\pi$-periodic function $f : \\mathbb{R} \\to \\mathbb{R}$ of bounded variation, with its Fourier series given by:\n\\[ f(\\theta) = \\frac{a_0}{2} + \\sum_{n = 1}^{\\infty} (a_n\\cos{n\\theta} + b_n\\sin{n\\theta}) \\]\nThe real part of the series can be expressed as:\n\\[... | Yes |
[
{
"content": "Evaluate the limit: $$\\lim_{n \\to \\infty} \\frac{\\sum_{k=1}^n\\lfloor kx\\rfloor}{n^2}$$",
"role": "user"
}
] | \dfrac{x}{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... | 0.027 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... | $4,936.33 |
[
{
"content": "Find the sum of the series \\(\\sum \\limits_{n=0}^{\\infty} (-1)^{n}\\frac{(n+1)^3}{3^n}.\\)",
"role": "user"
}
] | -\dfrac{9}{128} |
[
{
"content": "Evaluate the line integral \\( \\int_{\\Gamma} v \\cdot dr \\) for the vector field \\( v(x, y, z) = \\left(\\frac{-y}{x^2 + y^2}, \\frac{x}{x^2 + y^2}, 0\\right) \\) along the curve \\( \\Gamma \\) defined by \\( \\{(x, y, z) \\mid x^2 + y^2 = 1, z = 0\\} \\). Note that \\( \\text{Curl}(v) = 0 \\... | 2\pi |
[
{
"content": "Evaluate the limit: \\[ \\lim_{t \\to 0}\\frac{1}{t^{2}\\left(1+t^{2}\\right)}\\sin\\left(\\frac{1}{t}\\sin\\frac{1}{t}\\right)e^{-\\frac{1}{t^{2}}}. \\]",
"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... | Soil nutrient loss, Algae blooms, Pollution of water ways |
[
{
"content": "Evaluate the limit \\( \\lim_{x\\to\\infty}\\frac{e}{x}\\left(1-\\frac{1}{2 x}+\\frac{11}{24 x^2}+O\\left(\\frac{1}{x^3}\\right) \\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... | 1. Treat the mixture with NaHSO3 to precipitate benzophenone.
2. Filter the precipitate and treat it with acidified water to recover benzophenone.
3. Separate the remaining propanol and methanol mixture by fractional distillation, collecting methanol first. |
[
{
"content": "Let \\( X \\) and \\( Y \\) be independent random variables uniformly distributed on \\((0, a)\\). Find the expected value of the product \\( XY \\) using the moment generating function.",
"role": "user"
}
] | \dfrac{a^2}{4} |
[
{
"content": "Determine the number of integer pairs \\((x, n)\\) such that \\(x^2 + 615 = 2^n\\), where \\(x \\in \\mathbb{Z}\\) and \\(n \\in \\mathbb{N}\\).",
"role": "user"
}
] | 2 |
[
{
"content": "Find the limit of the sequence: \\[ \\lim_{n\\to \\infty} \\sqrt[n]{1^{2022}+2^{2022}+\\ldots+n^{2022}} \\]",
"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... | m(sqrt(2gh2)+sqrt(2gh1))/Dt |
[
{
"content": "Determine the value of the infinite product \\( \\prod_{n=1}^{\\infty}\\left(1 - \\frac{1}{3n + 2}\\right) \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the equation \\( x^{2}+y^{2}+z^{2}=2 x y z \\) has a solution in non-negative integers.",
"role": "user"
}
] | (0, 0, 0) |
[
{
"content": "Let \\( \\zeta_{2^{n+2}} \\) be a \\( 2^{n+2} \\)th root of unity, and let \\( \\overline{\\zeta}_{2^{n+2}} \\) be its complex conjugate. Show that the degree of the field extension \\([\\mathbb{Q}(\\zeta_{2^{n+2}}): \\mathbb{Q}(\\zeta_{2^{n+2}} + \\overline{\\zeta}_{2^{n+2}})] = 2\\).",
"role... | 2 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.04 M |
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