prompt list | ground_truth string |
|---|---|
[
{
"content": "For $0 < k < 6,$ the graphs of $\\frac{(x - k)^2}{9} + y^2 = 1$ and $\\frac{x^2}{9} + y^2 = 1$ intersect at $A$ and $C,$ and have $x$-intercepts at $B$ and $D$ respectively. Compute the value of $k$ for which $ABCD$ is a square.\n\n[asy]\nunitsize(1 cm);\n\npath ellone = xscale(3)*Circle((0,0),1)... | \frac{24}{5} |
[
{
"content": "In an acute-angled triangle \\(\\Delta ABC\\), let \\(H\\) be its orthocenter. Define \\(G_1, G_2,\\) and \\(G_3\\) as the centroids of triangles \\(\\Delta HBC, \\Delta HCA,\\) and \\(\\Delta HAB\\) respectively. If the area of triangle \\(\\Delta G_1G_2G_3\\) is 7 square units, find the area of ... | 63 |
[
{
"content": "Sergey wrote down numbers from 500 to 1499 in a row in some order. Under each number, except for the leftmost one, he wrote the greatest common divisor (GCD) of this number and its left neighbor, resulting in a second row of 999 numbers. He continued this process, creating subsequent rows by writi... | 10 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.64e13 m/s^2 |
[
{
"content": "Suppose that the sequences \\(\\{X_n\\}\\) and \\(\\{Y_n\\}\\) converge almost surely to constants \\(a\\) and \\(b\\), respectively. If \\(X_n\\) is bounded and \\(Y_n\\) is bounded away from zero, does this imply that \\(\\mathbb{E}\\left[\\frac{X_n}{Y_n}\\right]\\) converges to \\(\\frac{a}{b}\... | \dfrac{a}{b} |
[
{
"content": "Find all linear functionals \\( \\varphi: X \\to \\mathbb{R} \\) such that \\( |\\varphi(x)| \\leq \\|x\\|^2 \\) for all \\( x \\in X \\), where \\( X \\) is a normed space.",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the radius of convergence for the power series \\( \\sum_{n=1}^{\\infty} \\frac{(-1)^{n+1} (z-1)^{2n+1}}{(2n+1)!} \\).",
"role": "user"
}
] | \infty |
[
{
"content": "How many pairs of positive integers \\((x, y)\\) satisfy the conditions \\((x, y) = 5!\\) and \\([x, y] = 50\\)?",
"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... | 858.36 |
[
{
"content": "Given two random variables $X$ and $Y$ with a correlation of $\\rho = 1$, both having a mean of $100$ and a standard deviation of $40$, and two other random variables $U$ and $V$ with a correlation of $\\rho = 0.8$, both having a mean of $0$ and a standard deviation of $20$, compute the correlatio... | 0.96 |
[
{
"content": "Determine the set of real values of $x$ for which the series \\( \\sum_{n=1}^{\\infty}\\frac{n^{n^2}}{(n+x)^{n^2}} \\) converges.",
"role": "user"
}
] | (0, \infty) |
[
{
"content": "Determine if the region of interest on the XY plane, for finding the volume of the object bounded by the planes \\(z = y + 4\\), \\(x^2 + y^2 = 4\\), and \\(z = 0\\), is the entire circle defined by \\(x^2 + y^2 = 4\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\left(\\frac{(n+1)(n+2)\\cdots(2n)}{n^{2n}}\\right)^{1/n} \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the definite integral: \\[ \\int_0^\\infty \\frac{r^3}{(r^2+d^2)^3} \\, dr \\]",
"role": "user"
}
] | \dfrac{1}{4d^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... | 108.88 |
[
{
"content": "Given a differentiable function \\( f(x) \\) on the real line such that \\( \\lim_{x \\to \\infty}f(x)=1 \\) and \\( \\lim_{x \\to \\infty}f'(x)=\\alpha \\), determine the value of \\( \\alpha \\).",
"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... | $29,503.00 |
[
{
"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... | -0.033 m/s^2 |
[
{
"content": "Given a finite set $X$ and a subset $S \\subseteq X$, consider the equivalence relation $R$ on the power set $P(X)$ defined by: \\((A, B) \\in R \\iff A \\cup S = B \\cup S\\). Determine the size of the quotient set $P(X)/R$. Provide your answer as a single integer.",
"role": "user"
}
] | 2^{|X| - |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... | 3.001 cm/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... | 1007.59 L |
[
{
"content": "Can every nonempty open set in \\(\\mathbb{R}\\) be represented as a countable union of bounded open intervals of the form \\((a_k, b_k)\\), with \\(a_k, b_k \\in \\mathbb{R}\\) finite? Provide a proof or counterexample.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit \\( \\lim\\limits_{n\\rightarrow\\infty}\\sum\\limits_{k=2}^n\\frac{1-e^{-(x_k-x_{k-1})}}{1+e^{-(x_k-x_{k-1})}} \\) where \\( \\{x_1, \\ldots, x_n\\} \\subset [0,1] \\) is an ordered sequence satisfying \\( x_1 < x_2 < \\ldots < x_n \\) and forms a partition of \\([0,1]\\) with ... | \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... | 10972.8 centimeters |
[
{
"content": "Determine the codimension of the variety defined by the zero set of the determinants of all $m \\times m$ submatrices of a symbolic $m \\times n$ matrix, where $m \\leq n$. Express your answer in terms of $m$ and $n$.",
"role": "user"
}
] | n - m + 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... | +160e-12 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... | 8.85 |
[
{
"content": "Let \\( A = 4444^{4444} \\). Find the digital root of \\( A \\), denoted as \\( D \\).",
"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... | 13.01% |
[
{
"content": "Given an orthogonal $3 \\times 3$ matrix $R$ represented as:\n\n$$R = \\begin{bmatrix}\nr_{11}&r_{12}&r_{13} \\\\\nr_{21}&r_{22}&r_{23} \\\\\nr_{31}&r_{32}&r_{33} \\\\\n\\end{bmatrix}$$\n\nwhere the parameters $r_{11}, r_{21}, r_{31}, r_{12}, r_{22}, r_{32}$ are known, determine how many unique so... | 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... | 0.159 atm |
[
{
"content": "Determine the highest degree of the polynomial for which the quadrature formula \\( \\int _{-1}^1\\vert x \\vert f(x)dx \\approx \\frac{1}{2}[f(x_0)+f(x_1)] \\) is exact, where \\( x_0 \\) and \\( x_1 \\) are quadrature points.",
"role": "user"
}
] | 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... | -12 m |
[
{
"content": "Let $X$ be an $n$-dimensional Alexandrov space with curvature $\text{curvature} \\geq k$. Determine if $X$ satisfies a reverse doubling condition. Specifically, does there exist a constant $C > 1$ such that for any $x \\in X$ and $0 < r < \\mathrm{diam}(X)/2$, the inequality $\\mathrm{vol}(B_x(2r)... | Yes |
[
{
"content": "Evaluate the limit: $$\\lim_{n \\to \\infty} \\int_{0}^{1} f(nx+1) \\, dx$$ where $f(x)$ is a continuous, strictly increasing, and positive function on the interval $[0, \\infty)$, and $\\lim_{x \\to \\infty} f(x) = M$. Determine the value of this limit.",
"role": "user"
}
] | 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... | 78.2 g |
[
{
"content": "Assume the series \\( \\sum |f_n| \\) is uniformly Cauchy. Does this imply that the series \\( \\sum f_n \\) is also uniformly Cauchy? 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... | 0.93 m |
[
{
"content": "Find the infimum of the expression \\( \\inf\\limits_{k \\ge n} k^{2}\\chi_{[0,\\frac{1}{k}]} \\), where \\( \\chi \\) is the indicator 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... | 22.4 L |
[
{
"content": "In the context of linear regression, consider the matrix \\( M = I_n - X(X'X)^{-1}X' \\), where \\( M \\) is an \\( n \\times n \\) matrix. Is it true that \\( \\text{rank}(M) = n - k \\)?",
"role": "user"
}
] | Yes |
[
{
"content": "Given an $n$-dimensional vector space $V$ over a field $F$ with basis $\\mathcal{B} = \\{\\mathbf{v_1}, \\ldots, \\mathbf{v_n}\\}$, let $\\mathcal{B}^{*} = \\{\\phi_1, \\ldots, \\phi_n\\}$ be the dual basis for $V^{*}$. If $\\psi : V \\to V$ is a linear transformation and $A$ is the matrix represe... | B = A^\top |
[
{
"content": "Let \\( f \\) be a twice differentiable function such that \\( f''(x) = -f(x) \\) and \\( f'(x) = g(x) \\). If \\( h'(x) = f(x)^2 + g(x)^2 \\), \\( h(1) = 8 \\), and \\( h(0) = 2 \\), find \\( h(2) \\).",
"role": "user"
}
] | 14 |
[
{
"content": "Determine whether the closure of a countable union of disjoint open intervals can be an uncountable set.",
"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... | $24,247.52 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $686,363.45 |
[
{
"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-] = 0.039 M, pH = 12.6 |
[
{
"content": "In the product topology, determine whether the closure of the box of two points is equal to the box of the closures of the points. Provide a true or false answer.",
"role": "user"
}
] | True |
[
{
"content": "Determine the values of $s$ for which the integral \\[ \\int\\limits_{e_k}^{\\infty} \\frac{\\mathrm{d} x}{x \\cdot \\ln x \\cdot \\ln^{(2)} x \\cdot \\ln^{(3)} x \\cdots (\\ln^{(k)} x)^s } \\] converges, where $\\ln^{(k)}$ denotes the $k$-fold iterated logarithm.",
"role": "user"
}
] | s > 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... | 0.006426 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... | 3.75 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... | 8.996J |
[
{
"content": "Let \\( A B C D E F \\) be a regular hexagon with an area of 1. Define \\( M \\) as the midpoint of \\( D E \\). Let \\( X \\) be the intersection of lines \\( A C \\) and \\( B M \\), \\( Y \\) be the intersection of lines \\( B F \\) and \\( A M \\), and \\( Z \\) be the intersection of lines \\... | 0 |
[
{
"content": "In the context of Galois extensions of number fields, determine whether the statement is true or false: The ramification index of a prime ideal in the composite field is equal to the ramification index of its restriction to one of the subfields.",
"role": "user"
}
] | False |
[
{
"content": "Let $A$ be an invertible square matrix of dimension $n$, and $B$ be a square matrix of the same dimension such that $AB + BA = 0$. Determine the trace of the matrix $B$. \\( \\text{(Express your answer as a single number.)} \\)",
"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... | Preferred: 2015: $18,600, 2016: $28,000, 2017: $30,000, 2018: $30,000, Total: $106,600
Common: 2015: $0, 2016: $0, 2017: $248,000, 2018: $390,000, Total: $638,000 |
[
{
"content": "Evaluate the contour integral \\( \\int_C f(z) \\, dz \\) where \\( f(z) = 1 \\) for \\( y < 0 \\) and \\( f(z) = 4y \\) for \\( y > 0 \\). The contour \\( C \\) is the arc from \\( z = -1-i \\) to \\( z = 1+i \\) along the curve \\( y = x^3 \\).",
"role": "user"
}
] | 2 + 3i |
[
{
"content": "Given 984 people numbered from 1 to 984, select \\( n \\) people according to the following rules:\n\n- Person number 12 is always picked.\n- If person number \\( x \\) is picked and person number \\( y \\) is not picked, then if \\( x \\cdot y \\le 984 \\), person number \\( x \\cdot y \\) must b... | 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... | 57.69 V |
[
{
"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... | 814.94 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... | Alkylating chemicals, such as ethyl methanesulfonate and N-nitroso-N-methylurea, and radiation treatments like fast neutron bombardment, gamma rays, and X-rays can induce mutations. |
[
{
"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... | Marketing plan |
[
{
"content": "Determine the values of \\(\\alpha > 0\\) for which the fixed point iteration \\(x_{k+1} = e^{-\\alpha x_k}\\) converges, given that the initial value \\(x_0\\) is in the interval \\([0,1]\\).",
"role": "user"
}
] | (0, e) |
[
{
"content": "Given that the values of the quadratic polynomial \\(a x^{2} + b x + c\\) on the interval \\([-1,1]\\) do not exceed 1 in absolute value, find the maximum possible value of the sum \\(|a| + |b| + |c|\\).",
"role": "user"
}
] | 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... | {eq}F_{x} = 12t N{/eq} |
[
{
"content": "Find a real number \\( \\lambda \\) such that \\((A+B)^{-1} \\leq (A+\\lambda I)^{-1}\\), where \\(A\\) and \\(B\\) are real symmetric non-singular matrices and \\(A+B\\) is non-singular.",
"role": "user"
}
] | \lambda_{\min}(B) |
[
{
"content": "Let $p$ be a prime number, and consider the $p$-adic order function $\\ord_p: \\mathbb{C}_p \\rightarrow \\mathbb{Q}$, normalized such that $\\ord_p(p) = 1$. The roots of the equation $X^2 - 252X + 3^{11} = 0$ are denoted as $\\beta_1$ and $\\beta_2$, with the ordering $\\ord_3(\\beta_1) \\leq \\o... | 2 |
[
{
"content": "Determine whether the set of all functions $f(x)$ such that $f(q) = 0$ for all $q \\in \\mathbb{Q} \\cap [0,1]$ is a subring of $C[0,1]$.",
"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... | 15 rad/s |
[
{
"content": "Find the equation of a surface that contains circles with a curvature of \\( \\frac{1}{4} \\) and intersects planes parallel to the xy-plane in circles.",
"role": "user"
}
] | x^2 + y^2 = 16 |
[
{
"content": "Determine the slant asymptote of the function \\( f(x) = (x+2)\\left(1-\\frac{12x+40}{(x+2)^3}\\right)^{1/3} \\).",
"role": "user"
}
] | y = 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... | 2 atm |
[
{
"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... | 0.297 cm |
[
{
"content": "Find the largest integer \\( n \\leq 9,000,000 \\) such that the expression \\( \\sqrt{n+\\sqrt{n+\\sqrt{n+\\cdots}}} \\) is rational.",
"role": "user"
}
] | 8997000 |
[
{
"content": "Convert the double integral \\( I = \\int_1^2 \\int_0^\\sqrt{1-(1-x)^2} \\dfrac{x}{x^2+y^2} \\ \\mathrm dy\\ \\mathrm dx \\) from Cartesian coordinates to polar coordinates and evaluate the integral.",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Express the conjugate \\( \\bar{z} \\) in terms of \\( z \\) for the equation \\( z^2 = \\bar{z} \\).",
"role": "user"
}
] | \bar{z} = z^2 |
[
{
"content": "A line passes through the center of a regular $n$-gon inscribed in a unit circle. Calculate the sum of the squares of the distances from the vertices of the $n$-gon to this line.",
"role": "user"
}
] | \dfrac{n}{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... | 11352 ounces |
[
{
"content": "Calculate the volume of the solid region bounded by the paraboloid \\(z = 2x^2 + 2y^2\\) and the plane \\(z = 6\\) using a triple integral in elliptical coordinates.",
"role": "user"
}
] | 9\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... | 18.11 km |
[
{
"content": "Is a projection on a closed subspace of a Banach space always bounded?",
"role": "user"
}
] | No |
[
{
"content": "A person can be in one of three states: employed, unemployed, or early retirement. The transition rates per year are as follows: an employed person becomes unemployed with a rate of 0.2, an unemployed person becomes employed with a rate of 4, and an unemployed person retires early with a rate of 0... | 13048000 |
[
{
"content": "Find the degree of the extension field \\( \\mathbb{Q}(\\sqrt[3]{5},\\zeta) \\) over \\( \\mathbb{Q}(\\zeta) \\), where \\( \\zeta \\) is a third root of unity.",
"role": "user"
}
] | 3 |
[
{
"content": "Let $f: X \\rightarrow Y$ be an open, continuous map and let $X' \\subseteq X$ be a closed subspace such that the restriction $f|_{X'}: X' \\rightarrow f(X')$ is a bijection. Does this imply that $f|_{X'}$ is proper?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether a symmetric matrix with zero diagonals can be expressed as the commutator of a diagonal matrix with positive diagonals and a skew-symmetric matrix.",
"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... | $82,000 |
[
{
"content": "Evaluate the line integral \\( \\int_\\ell \\left(\\frac{-y}{\\sqrt{2x^2+2y^2}}\\right) \\, dx + \\left(\\frac{x}{\\sqrt{2x^2+2y^2}}\\right) \\, dy \\) over the curve \\( \\vec \\ell(t)=(\\sin(t), \\sin(t)\\cos(t)-1) \\) for \\( t \\in [0,2\\pi] \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Let \\( X = \\begin{pmatrix}\\cos\\frac{2\\pi}{5} & -\\sin\\frac{2\\pi}{5}\\\\\\sin\\frac{2\\pi}{5} & \\cos\\frac{2\\pi}{5}\\end{pmatrix} \\) and \\( Y = \\begin{pmatrix}1 & 0\\\\0 & -1\\end{pmatrix} \\). Determine the number of distinct matrices in the set \\( Z \\), which consists of all matrice... | 10 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.5 billion years |
[
{
"content": "What is the radius of the circle inscribed in triangle $ABC$ if $AB = 22, AC=12,$ and $BC=14$? Express your answer in simplest radical form.",
"role": "user"
}
] | \sqrt{10} |
[
{
"content": "The midpoint of a line segment is located at $(1, -2)$. If one of the endpoints is $(4, 5)$, what is the other endpoint? Express your answer as an ordered pair.",
"role": "user"
}
] | (-2,-9) |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\frac{1}{n}\\sqrt[n]{(n+1)^5+(n+2)^5+\\cdots+(3n)^5}. \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether there exists an element \\( b \\in \\mathbb{Z}_5[x]/(x+1)^2 \\) such that \\( b^2 = 0 \\). If such an element exists, explain why this is the case.",
"role": "user"
}
] | Yes |
[
{
"content": "Let $f$ be a non-negative continuous function on the interval $[0,1]$. Suppose that the sequence $f^{1/n}$ converges uniformly. Determine the number of zeros of $f$. Provide your answer as a single integer.",
"role": "user"
}
] | 0 |
[
{
"content": "Is it always possible to find a countable subset $M \\subseteq \\{\\beta \\mid \\beta < \\lambda\\}$ such that $\\lambda = \\sup M$ for a given limit ordinal $\\lambda$ defined as $\\lambda = \\sup\\{\\beta \\mid \\beta < \\lambda\\}$?",
"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... | {eq}\rm HCO_3^- (aq) + H^+ (aq) \rightarrow H_2CO_3 (aq){/eq} |
[
{
"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... | R(Q) = 90Q - 2Q^2 |
[
{
"content": "Evaluate the double integral \\( \\int\\int xy \\, dx\\,dy \\) over the square with corners \\((0,0), (1,1), (2,0),\\) and \\((1,−1)\\) using the change of variables \\(x = \\frac{u+v}{2}\\) and \\(y = \\frac{u-v}{2}\\). Compute the integral using the correct limits and Jacobian, and verify if the... | 0 |
[
{
"content": "Use Green's Theorem to transform the line integral \\( \\oint_L(y^4-x^2)dx+(x^4+y^2)dy \\) into a double integral, where \\( L: x^2-y^2=1 \\). Then, solve the double integral using polar coordinates.",
"role": "user"
}
] | 0 |
[
{
"content": "Calculate the integral \\( \\int_{[0,1]} (2 - \\omega) \\, d\\mu \\), where \\( \\mu = 2\\lambda_1 + 3\\delta_{1} \\) is a measure on \\([0,1]\\) with \\( \\lambda_1 \\) being the Lebesgue measure and \\( \\delta_{1} \\) the Dirac measure at 1.",
"role": "user"
}
] | 6 |
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