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... | 40,845.07 units |
[
{
"content": "Evaluate the integral \\( I = \\int_0^{\\pi/2} \\frac{1}{1+\\cot^{n}x} \\, dx \\) using the substitution \\( x = \\arccot(u) \\).",
"role": "user"
}
] | \dfrac{\pi}{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... | 188 N |
[
{
"content": "Four runners start at the same position on a circular track that is 1 km long, running in the same direction. Their speeds are \\( \\sqrt{5} \\), \\( e \\), 3, and \\( \\pi \\) meters per second. Determine the first time, in seconds, when the runner with speed \\( e \\) is at least 250 meters away... | \dfrac{250}{3 - e} |
[
{
"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... | 638 mL |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $90 |
[
{
"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... | \sqrt{3} < t \leq 5 |
[
{
"content": "Let \\( A \\) be the sum of the digits of the decimal number \\( 4444^{4144} \\), and let \\( B \\) be the sum of the digits of \\( A \\). Find the sum of the digits of \\( B \\).",
"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... | 4.9e2 V/m |
[
{
"content": "Calculate the volume of the larger region cut from the solid sphere defined by \\(x^2 + y^2 + z^2 = 4\\) by the plane \\(z = 1\\). Use double integrals in spherical coordinates to find the solution.",
"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... | 229 atm |
[
{
"content": "Determine the real function $f: \\mathbb{R} \\to \\mathbb{R}$ such that \\[ 3f(x) + 2 = 2f(\\left \\lfloor{x}\\right \\rfloor) + 2f(\\{x\\}) + 5x, \\] where $\\left \\lfloor{x}\\right \\rfloor$ is the floor function and $\\{x\\} = x - \\left \\lfloor{x}\\right \\rfloor$ is the fractional part of $... | 5x + 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... | 0.814T |
[
{
"content": "The expression \\(16^n + 4^n + 1\\) is equivalent to the expression \\(\\frac{2^{p(n)} - 1}{2^{q(n)} - 1}\\) for all positive integers \\(n > 1\\), where \\(p(n)\\) and \\(q(n)\\) are functions and \\(\\frac{p(n)}{q(n)}\\) is constant. Find \\(p(2006) - q(2006)\\).",
"role": "user"
}
] | 8024 |
[
{
"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... | 289.158 W |
[
{
"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... | 50 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... | 0.58 mol L^(-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... | The United States |
[
{
"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... | $x(t) = \frac{1}{S}\int_{t-\tau}^t u(t) dt + x(t-\tau)$ if the tank is not leaking. However, with leakage, $\dot{x}(t) = \frac{1}{S}u(t) - \frac{1}{S}\alpha\sqrt{2g x(t)}$ |
[
{
"content": "Determine the values of $p$ for which the series \\( \\sum_{n=2}^{\\infty} \\frac{1}{n(\\ln n)^p} \\) converges.",
"role": "user"
}
] | p > 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... | 124.45 pm |
[
{
"content": "Let \\( \\{ N(t) \\}_{t \\geq 0} \\) be a Poisson process with intensity \\( \\lambda = 3 \\). Compute the probability \\( P\\left[N(6) = 2 \\,|\\, N(8) = 4, N(3) = 1\\right] \\).",
"role": "user"
}
] | \dfrac{36}{125} |
[
{
"content": "Find the minimum length of segment \\(AB\\) where point \\(A\\) lies on the line \\(y=\\frac{12}{5} x-9\\) and point \\(B\\) lies on the parabola \\(y=x^{2}\\).",
"role": "user"
}
] | \dfrac{189}{65} |
[
{
"content": "Calculate the spectral radius of the Volterra operator \\( A \\) defined on \\( L^2[0,1] \\) by \\( A(f)(x) = \\int_0^x f(t) \\, dt \\). Use the formula \\( \\rho(A) = \\lim_{n\\rightarrow \\infty} \\|A^n\\|^{1/n} \\) and the fact that \\( \\|(x-s)^n\\|_\\infty = 1 \\) for \\( s \\in [0,x] \\), \\... | 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... | 0.39 |
[
{
"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... | 20 J |
[
{
"content": "In triangle \\(ABC\\), the area is 1. Point \\(A'\\) is on the extension of side \\(BC\\) beyond \\(C\\) such that \\(BC = CA'\\). Point \\(B'\\) is on the extension of side \\(CA\\) beyond \\(A\\) such that \\(CA = AB'\\). Point \\(C'\\) is on the extension of \\(AB\\) beyond \\(B\\) such that \\... | 7 |
[
{
"content": "Given two distributions \\( p \\) and \\( q \\) over the same finite set \\( X \\) with a total variation distance \\( TV(p,q) = \\epsilon \\), and a function \\( f: X \\to \\mathbb{R} \\) such that \\( 0 \\leq \\sum_{x \\in X} f(x) p(x) \\leq 1 \\) and \\( 0 \\leq \\sum_{x \\in X} f(x) q(x) \\leq... | 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... | N/A, $450, $225, $150, $112.50, $90, $75, $64.29, $56.25, $50 |
[
{
"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... | 3.4e9 A |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $10,400 |
[
{
"content": "The number of toy cars that Ray has is a multiple of $6$. When he loses two of them, the number of cars that he has left is a multiple of $n$. If $n$ is a positive even integer less than $10$, then how many possible values are there for $n$?",
"role": "user"
}
] | 3 |
[
{
"content": "Calculate the value of \\( \\left|\\frac{(\\pi+i)^{100}}{(\\pi-i)^{100}}\\right| \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the expression: \\[ \\tan\\left(\\sum_{r=1}^{\\infty} \\arctan\\left(\\frac{4}{4r^2 +3}\\right)\\right) \\]",
"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... | $\theta = \tan^{-1} \left(\dfrac{v^{2}}{gR}\right)$ |
[
{
"content": "Let $A$ and $B$ be symmetric $n \\times n$ matrices. Find the matrix $X$ such that the commutator $[A, B] = [B, X]$, where $[A, B] = AB - BA$.",
"role": "user"
}
] | -A |
[
{
"content": "Let $T_1, T_2: V \\rightarrow W$ be linear transformations, where $V$ and $W$ are vector spaces over a field $\\mathbb{F}$. Given that $\\dim(V) = n$, $\\ker(T_1) \\neq \\ker(T_2)$, and $\\dim(\\operatorname{Image}(T_1)) + \\dim(\\operatorname{Image}(T_2)) = \\dim(W)$, determine $\\dim(\\ker(T_1) ... | n - \dim(W) |
[
{
"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... | Sodium, potassium, chloride and calcium ions |
[
{
"content": "Let $a, b, c$ be positive real numbers, and let $\\Gamma$ denote the Gamma function. Evaluate the sum:\n\\[ \\sum_{n=0}^\\infty \\frac{a^n}{\\Gamma(b+nc)} \\]",
"role": "user"
}
] | E_{c,b}(a) |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Coefficient of variation = standard deviation / average return. For each decade, the coefficient of variation is:
1950-1959: 9.4
1960-1969: 4.2
1970-1979: 1.2
1980-1989: 1.2
1990-1999: 1.4
2000-2009: 1.2 |
[
{
"content": "Calculate the following limit superior:\n\\[ \\limsup_{n\\to\\infty}\\left(\\frac{A^n}{n}+\\frac{B^n}{n^2}\\right)^{\\frac{1}{n}} \\]\nwhere \\( A, B > 0 \\).",
"role": "user"
}
] | \max\{A, 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... | 225 N |
[
{
"content": "Let \\( f: \\mathbb{R} \\to \\mathbb{R} \\) be defined as \\( f(x) = \\begin{cases} x^2 - 2x, & x \\leq 0 \\\\ -x^2 - 2x, & x > 0 \\end{cases} \\). Find \\((f^{-1})'(3)\\).",
"role": "user"
}
] | -\dfrac{1}{4} |
[
{
"content": "Solve\n\\[\\arctan \\frac{1}{x} + \\arctan \\frac{1}{x^3} = \\frac{\\pi}{4}.\\]",
"role": "user"
}
] | \frac{1 + \sqrt{5}}{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... | $1,875 |
[
{
"content": "Evaluate the limit \\(\\lim_{{n \\to \\infty}} \\left(n - 1 - 2\\left(\\frac{\\Gamma\\left(\\frac{n}{2}\\right)}{\\Gamma\\left(\\frac{n - 1}{2}\\right)}\\right)^2\\right)\\) using Stirling's approximation.",
"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... | $33,950 |
[
{
"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... | 5.5e+03 kg/m^3 |
[
{
"content": "Two players, A and B, are playing a table tennis match. The probability that player A wins a game is \\(\\frac{2}{3}\\). The match ends when one player is ahead by two games. Calculate the expected total number of games played in the match.",
"role": "user"
}
] | \dfrac{18}{5} |
[
{
"content": "Determine whether there exists an isometry of \\(\\mathbb{R}^n\\) that cannot be expressed as a product of \\(n\\) reflections.",
"role": "user"
}
] | Yes |
[
{
"content": "Integrate the complex function \\( f(z) = xy \\) over the circle \\( C = [0, r] \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Calculate the integral \\( \\int{x^2-2y^2}dA \\) using the Jacobian matrix, where the substitutions are \\( x+y=u, x-y=v \\), with the limits \\( 1<u<3 \\) and \\( 0<v<2 \\), and the Jacobian is \\( \\frac{1}{2} \\).",
"role": "user"
}
] | \dfrac{19}{6} |
[
{
"content": "Determine whether the function \\( f(x) = \\frac{x}{2} + \\frac{1}{x} \\) is a contraction map on the set \\( X = \\{ x \\in \\mathbb{R} : x \\ge 1 \\} \\) with respect to the usual metric on \\( \\mathbb{R} \\). If it is a contraction map, find the smallest contraction constant \\( \\lambda \\)."... | \dfrac{1}{2} |
[
{
"content": "Calculate the limit of the function \\( f(x) = \\frac{e^{x^2}-\\ln(1+x^2)-1}{\\cos2x+2x\\sin x-1} \\) as \\( x \\) approaches 0.",
"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... | 177 mg/L |
[
{
"content": "Given a convex set \\( \\mathcal{S} \\) with respect to Synge's world function \\( \\sigma: \\mathcal{S}^2 \\rightarrow \\mathbb{R} \\), and a subset \\( \\mathcal{G}_P \\subset \\mathcal{S}^2 \\) consisting of point pairs \\((X, Y)\\) such that a point \\(P\\) lies on the unique straight line seg... | 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... | 40.4 kV |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 42 days |
[
{
"content": "Let $E$ be a vector bundle of rank $k$ over a smooth manifold $M$ of dimension $m$. Can $E$ be immersed in $\\mathbb{R}^{2m + k}$?",
"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... | 9.4608 x 10^25 m |
[
{
"content": "Evaluate the limit: \\[ \\lim _{ n\\rightarrow \\infty }{ \\frac { { 2 }^{ n }! }{ { 2 }^{ 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... | 1334.65 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... | 1.55% |
[
{
"content": "Consider an $n$-dimensional Brownian motion $B = (B^1, \times, B^n)$, where $B_t - B_s \\sim N(0, (t - s) E_n)$ for $s < t$. Are the components $B^k$ of $B$ independent?",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\frac{x^3 e^{\\frac{x^4}{4}} - \\sin^{\\frac{3}{2}}(x^2)}{x^7} \\]",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Evaluate the contour integral \\(\\oint_C \\cos(z) \\, dz\\) where \\(C\\) is the unit circle \\(|z| = 1\\) in the complex plane.",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the function \\( F(x) = \\int_{0}^x f(t) \\, dt \\), where \\( f(x) = \\begin{cases} \\cos{\\dfrac{1}{x}}, & x \\neq 0 \\\\ 0, & x = 0 \\end{cases} \\), is differentiable at \\( x = 0 \\).",
"role": "user"
}
] | Yes |
[
{
"content": "A company that sells widgets has to pay $\\$500$ in maintenance fees each day and then it pays each worker $\\$15$ per hour. Each worker makes 5 widgets per hour, which are sold at $\\$3.10$ each. What is the least number of workers the company has to hire in order to make a profit in an 8-hour wo... | 126 |
[
{
"content": "Calculate the limit \\( \\lim_{n\\to\\infty}\\int_{0}^{\\infty} \\frac{\\sqrt x}{1+ x^{2n}} dx \\).",
"role": "user"
}
] | \dfrac{2}{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... | XYZ |
[
{
"content": "Determine whether the following statement is true or false: An integrable sub-martingale $S_t$ with $\\mathbb{E}(S_t)$ being a constant is a martingale.",
"role": "user"
}
] | True |
[
{
"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... | $282,290 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 27 |
[
{
"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... | 5160 mg |
[
{
"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... | excess capacity |
[
{
"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... | 2.07e6 m/s |
[
{
"content": "Determine the value that \\(\\frac{B}{x}\\) approaches, where \\(B=\\sqrt[3]{x\\sqrt[5]{x^4\\sqrt[9]{x^{24}\\sqrt[17]{x^{240}\\ldots}}}}\\).",
"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... | 17.083 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... | $41,590 |
[
{
"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... | 145.96 m/s |
[
{
"content": "Suppose $l: k \\to K$ is a ring morphism and $\\bar k$, $\\bar K$ are the algebraic closures of $k$ and $K$, respectively. Does there exist a ring morphism $\\bar l: \\bar k \\to \\bar K$ such that the following diagram commutes?\n$$\n\\require{AMScd}\n\\begin{CD}\nk @>{l}>> K\n\\\\\n@VVV @VVV\n\\... | Yes |
[
{
"content": "Suppose $M$ is a square matrix with real coefficients such that \\( \\text{Rank}(M) > \\text{Rank}(M^2) \\). Determine whether $M$ must be nilpotent. Answer with 'True' or 'False'.",
"role": "user"
}
] | False |
[
{
"content": "Find the value of \\( x \\) such that the infinite power tower \\( x^{x^{x^{.^{.^.}}}} = 2 \\).",
"role": "user"
}
] | \sqrt{2} |
[
{
"content": "Determine the number of homomorphisms from the symmetric group $S_4$ to the group $\\mathbb{Z}/3\\mathbb{Z} \\times \\mathbb{Z}/6\\mathbb{Z}$.",
"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... | 8.17% |
[
{
"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... | 0.99 |
[
{
"content": "Let $\\mathbf{R}$ be the matrix for reflecting over the vector $\\begin{pmatrix} 1 \\\\ 5 \\end{pmatrix}.$ Find $\\det \\mathbf{R}.$",
"role": "user"
}
] | -1 |
[
{
"content": "Determine whether the following statement is true or false: If a function \\( g:[0,\\infty) \\to \\mathbb{R} \\) is continuous and \\( \\lim_{x\\to \\infty} g(x) = L \\) for some \\( L \\in \\mathbb{R} \\), then \\( g \\) is uniformly continuous on \\([0,\\infty)\\).",
"role": "user"
}
] | True |
[
{
"content": "Let $M$ be a compact, connected, oriented smooth Riemannian manifold with non-empty boundary. Let $f:M \\to M$ be a smooth orientation preserving isometric immersion. Determine whether $f$ is locally distance preserving, meaning that around every point $p \\in M$, there exists a neighborhood $U$ s... | Yes |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to \\infty} \\frac{1}{n^2} \\sum_{k=1}^n \\frac{f(b+\\frac{2k}{n^2}) -f(b)}{\\frac{2k}{n^2}}$$ where $f$ is a differentiable function.",
"role": "user"
}
] | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | -1.91e-4 J |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 22 m/s |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.1359 M |
[
{
"content": "Identify all closed sets in \\(\\mathbb{R}\\) that are subgroups of \\((\\mathbb{R}, +)\\) and have at least one limit point.",
"role": "user"
}
] | \mathbb{R} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | hydrogen gas |
[
{
"content": "Determine the number of ordered pairs \\((x, y)\\) of integers such that \\(x^2 \\equiv y^3 \\pmod{57}\\).",
"role": "user"
}
] | 57 |
[
{
"content": "Find the coefficient of \\(x^{12}\\) in the expansion of \\((1-x^6)^4 \\cdot (1-x)^{-4}\\).",
"role": "user"
}
] | 125 |
[
{
"content": "Find the positive integer $q \\leq 50$ such that the following equation holds: $$\\binom{98}{30} + 2\\binom{97}{30} + 3\\binom{96}{30} + \\dots + 68\\binom{31}{30} + 69\\binom{30}{30} = \\binom{100}{q}.$$",
"role": "user"
}
] | 32 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | P = 101.5 |
[
{
"content": "Determine whether there exists a group $G$ with a subgroup $H \\subseteq G$ of finite index and an element $g \\in G$ such that $H \\cap g^{-1} H g$ has infinite index in $G$. Provide a justification for your answer.",
"role": "user"
}
] | No |
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