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... | 7.11 degrees |
[
{
"content": "Given that \\( f(1)=1 \\) and \\( f(2)=2 \\), and the function satisfies the condition:\n\\[\nf(n+2) \\geq 2 f(n+1) - f(n) \\quad \\forall n \\in \\mathbf{N}^{+},\n\\]\nfind the function \\( f(n) \\).",
"role": "user"
}
] | n |
[
{
"content": "For the sequence \\(a_n = \\left(n + \\frac{1}{n}\\right)\\) where \\(n \\in \\mathbb{N}\\), determine the sum of the series \\(\\sum (-1)^{n+1} \\frac{a_{n+1}}{n!}\\). Use Stirling's approximation \\(n! \\approx \\sqrt{2\\pi n} \\left(\\frac{n}{e}\\right)^n\\) to help determine the answer. Choose... | 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... | 1. Covalent compounds are assumed to be in their standard state (s, l, or g) unless dissolved in water, as stated in the problem. 2. Charged ions in a reaction must be dissolved in water and written with (aq). |
[
{
"content": "Determine the order of an orthogonal matrix composed only of the elements $+\\frac{1}{4}$ and $-\\frac{1}{4}$. The matrix satisfies the condition $A A^{T} = I$, where $I$ is the identity matrix.",
"role": "user"
}
] | 16 |
[
{
"content": "For a periodic smooth function $f \\in C^\\infty$, is the set $\\{\\max(\\text{im}(f^{(n)})): n \\in \\mathbb{N}\\}$ always bounded?",
"role": "user"
}
] | No |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty} \\left( \\frac {c_n}{n} - \\frac{c_n}{p_n} - \\frac {c_n}{n^2} \\right) \\] where \\( c_n \\) is the nth composite number (excluding 1) and \\( p_n \\) is the nth prime number.",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the integral \\( \\int_0^\\frac{\\pi}{3}\\ln^n\\left(2\\sin\\frac{\\theta}{2}\\right)d\\theta \\) at \\( a=0 \\).",
"role": "user"
}
] | \dfrac{\pi}{3} |
[
{
"content": "Determine the limit of the sequence \\( x_n = \\frac{\\frac{2}{1} + \\frac{3^2}{2} + \\frac{4^3}{3^2} + \\ldots + \\frac{(n+1)^n}{n^{n-1}}}{n^2} \\).",
"role": "user"
}
] | \dfrac{e}{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... | 0.717 m/s^2 |
[
{
"content": "Determine whether the following statement is true: Given any field $F$, there exists a proper field extension $K$ of $F$. Provide a justification for your answer.",
"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... | 1.092e-3 Nm |
[
{
"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 maintenance cost allocated to the mixing department is $110,000.
The maintenance cost allocated to the bottling department is $90,000. |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to 1^{-}}(\\arccos (x))^{1-x} \\]",
"role": "user"
}
] | 1 |
[
{
"content": "Let \\( A \\in M_2 \\) be a matrix with characteristic polynomial \\( p(x) = x^2 - 3x - 5 \\). Find the values of \\( a \\) and \\( b \\) such that \\( -2A^2 + 4A - 3I_2 = aA + bI_2 \\). What is \\( a + b \\)?",
"role": "user"
}
] | -15 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 894 nm |
[
{
"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... | 1146.15 K |
[
{
"content": "Determine the largest possible size of a deck for which the following card-guessing puzzle is solvable: Alice is dealt five random cards from the deck, looks at them, and shows four of them to Bob in a chosen order. Using only these four cards and their order, Bob must determine the identity of th... | 124 |
[
{
"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... | 5.0 g H_2 |
[
{
"content": "Given that $X_1, X_2, \\ldots, X_{16}$ are observations from a normal distribution $N(\\mu, \\sigma^2)$, and the statistics are defined as:\n\\[ \\overline{X}=\\frac{1}{16}\\sum_{i=1}^{16}X_i \\]\n\\[ S^2=\\frac{1}{15}\\sum_{i=1}^{16}(X_i-\\overline{X})^2 \\]\nFind the conditional probability $P(\... | 0.5 |
[
{
"content": "If a sequence of functions $f_n$ converges to a function $f$ weakly in $L^p(U)$, where $1 \\leq p \\leq \\infty$, does it imply that $f_n$ converges to $f$ in the space of distributions $\\mathcal{D}^\\prime (U)$?",
"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... | 30.4 cm |
[
{
"content": "Find a single continuous function \\( f(x) \\) such that \\( f(0) = 1 \\), \\( \\int_{-\\infty}^{\\infty} f(x) \\, dx = 1 \\), and the function decreases on both sides of \\( x = 0 \\) at a rate faster than or equal to \\( e^{-2|x|} \\).",
"role": "user"
}
] | e^{-2|x|} |
[
{
"content": "If $g^{-1}(a,b)$ is not measurable for some $a,b \\in \\mathbb{R}$, can we conclude that the function $g(x)$ is not measurable?",
"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... | 2MnO_4^-(aq) + 5C_2O_4^{2-}(aq) + 16H^+(aq) -> 2Mn^{2+}(aq) + 8H_2O(l) + 10CO_2(g) |
[
{
"content": "Let $G$ be a non-solvable group of order 120 with a center $Z(G)$ such that $|Z(G)| = 2$. If $G/Z(G) \\cong A_5$, prove that $G \\cong SL(2,5)$. Provide a proof for this conclusion.",
"role": "user"
}
] | SL(2,5) |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to 0} \\frac{\\ln\\left(\\cos\\left(\\frac{x}{4}\\right)\\right)\\ln\\left(\\sin^2\\left(\\frac{x}{8}\\right)\\right)}{\\ln\\left(\\cos\\left(\\frac{x}{8}\\right)\\right)\\ln\\left(\\sin^2\\left(\\frac{x}{4}\\right)\\right)}$$",
"role": "user"
}
] | 4 |
[
{
"content": "A fair coin is tossed 100 times. Determine which of the following events has the greatest probability:\n\nA. The number of heads is exactly 50.\nB. The number of tails is at least 60.\nC. The number of heads is between 51 and 55, inclusive.\nD. The number of heads is at least 48 and the number of ... | 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... | 6.13e+10 km^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... | $82.89 |
[
{
"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... | 991.57 J |
[
{
"content": "Show that the expression \\(x = \\sqrt[3]{2+\\sqrt{5}} - \\sqrt[3]{-2+\\sqrt{5}}\\) simplifies to \\(x = 1\\) for the equation \\(x^3 + 3x - 4 = 0\\).",
"role": "user"
}
] | 1 |
[
{
"content": "Find the point at which the density function of the $\\chi^2(n)$ distribution reaches its maximum.",
"role": "user"
}
] | \max(n - 2, 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... | 3H_2A+2M(OH)_3->M_2A_3+6H_2O |
[
{
"content": "Simplify\n\\[\\frac{\\tan^3 75^\\circ + \\cot^3 75^\\circ}{\\tan 75^\\circ + \\cot 75^\\circ}.\\]",
"role": "user"
}
] | 13 |
[
{
"content": "Find the value of \\(s\\) that satisfies the equation: \\[2^s = 2s + 2\\]",
"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... | 8 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... | 7 mol |
[
{
"content": "Given the functions \\( f(x) = x^3 - ax \\) and \\( g(x) = 2x^2 - bx \\) where \\( a \\neq b \\), the curves intersect at two points, one of which is a tangent. Find the area enclosed between the curves from \\( x = 0 \\) to \\( x = 1 \\).",
"role": "user"
}
] | \dfrac{1}{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... | 15.2% |
[
{
"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... | 6502.31 N |
[
{
"content": "Determine the number of whole number terms in the expansion of \\((\\sqrt[11]{5} + \\sqrt[5]{11})^{2015}\\).",
"role": "user"
}
] | 37 |
[
{
"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... | 625 |
[
{
"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... | 351 N |
[
{
"content": "Evaluate the integral: $$\\int_{ - \\pi /2}^{\\pi /2} \\frac{1}{2007^{x} + 1} \\cdot \\frac{\\sin^{2008}x}{\\sin^{2008}x + \\cos^{2008}x} \\, dx.$$",
"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... | 481.98 J |
[
{
"content": "Let $x, y, z$ be positive real numbers such that $(1 + x + y + z)(1 + \\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z}) = 25$. Find the value of $x + y + z$.",
"role": "user"
}
] | 4 |
[
{
"content": "Consider the sum \\( S = 1 + 10 + 19 + 28 + 37 + \\ldots + 10^{2013} \\). Determine how many times the digit '5' appears in the number \\( S \\).",
"role": "user"
}
] | 4022 |
[
{
"content": "Calculate the work done against the force field \\( F(x,y) = -\\frac{(x,y)}{\\sqrt{x^2 + y^2}} \\) in moving a particle from \\((2a,0)\\) to the origin along the top half of the circle \\((x−a)^2+y^2=a^2\\).",
"role": "user"
}
] | -2a |
[
{
"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... | 0.098 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | q = m*sqrt(G/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... | 333K |
[
{
"content": "Determine whether the sequence \\( \\frac{n^n}{e^{n^2}} \\) converges to 0 as \\( n \\to \\infty \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Let $V$ be the vector space of polynomials of degree at most $99$ with real coefficients. Define a linear map $$T:V\\to\\mathbb{R}^{1000},\\quad T(p)=(p(1),p(2),\\ldots,p(1000)).$$ Find the dimension of the null space of $T$. \\( \\text{Provide your answer as a single integer.} \\)",
"role": "... | 0 |
[
{
"content": "What is the range of the function $y = \\frac{x^2 + 3x + 2}{x+1}$? (Express your answer using interval notation.)",
"role": "user"
}
] | (-\infty, 1)\cup(1, \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... | 72.04 degrees Celcius |
[
{
"content": "Line $L$ is the intersection of the planes $x + 2y + 3z = 2$ and $x - y + z = 3.$ A plane $P,$ different from both these planes, contains line $L,$ and has a distance of $\\frac{2}{\\sqrt{3}}$ from the point $(3,1,-1).$ Find the equation of plane $P.$ Enter your answer in the form\n\\[Ax + By +... | 5x - 11y + z - 17 = 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... | $971.36 |
[
{
"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... | 41558W |
[
{
"content": "Find the Laurent series expansion of the function \\( f(z) = z^2 \\cdot \\frac{e^{1/z}}{z-1} \\) around the point \\( z = 0 \\) and determine the residue of \\( f \\) at \\( z = 0 \\).",
"role": "user"
}
] | \frac{5}{2} - 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... | 2050.84 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 9.14 |
[
{
"content": "Find the maximum value of the expression \\( p^3 + q^3 + r^3 + 4pqr \\) given that \\( p, q, r \\) are non-negative real numbers such that \\( p + q + r = 3 \\) and each of \\( p, q, r \\) is less than or equal to 1.5.",
"role": "user"
}
] | \dfrac{243}{32} |
[
{
"content": "For positive real numbers $x$, $y$, and $z$ satisfying $x + 2y + 3z = 4$, determine the maximum value of $\\min\\{xy, xz, 2yz\\}$. Provide your answer as a single number.",
"role": "user"
}
] | \dfrac{32}{49} |
[
{
"content": "Given a flow network represented as a directed graph with a single source vertex \\(s\\) and a single sink vertex \\(t\\), and a total of \\(|V|\\) vertices, determine the number of possible cuts in the network.",
"role": "user"
}
] | 2^{|V| - 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... | 1.07 rad/s |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{+\\infty} \\frac{\\cos x \\sin \\sqrt{1+x^{2}}}{\\sqrt{1+x^{2}}} \\mathrm{~d} x \\).",
"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... | $320,001.29 |
[
{
"content": "Determine the number of primitive quartic polynomials over the field \\(\\mathbb{Z}_5\\).",
"role": "user"
}
] | 48 |
[
{
"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... | position (-7.14 cm) and height (0.86 cm) |
[
{
"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. Cash $150,000
Loss on Realization $50,000
Noncash Assets $200,000
2. Oslo, Capital $30,000
Decker, Capital $15,000
Silas, Capital $5,000
Loss on Realization $50,000
3. Liabilities $50,000
Cash $50,000
4. Oslo, Capital $45,000
Decker, Capital $85,000
Silas, Capital $20,000
Cash $150,000 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Q = 20 - 0.75P |
[
{
"content": "Circles $\\omega_1$, $\\omega_2$, and $\\omega_3$ each have radius $4$ and are placed in the plane so that each circle is externally tangent to the other two. Points $P_1$, $P_2$, and $P_3$ lie on $\\omega_1$, $\\omega_2$, and $\\omega_3$ respectively such that $P_1P_2=P_2P_3=P_3P_1$ and line $P_i... | 552 |
[
{
"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... | -7.16% |
[
{
"content": "Find the integer solution with the smallest sum for the equation \\(975a + 880b + 790c + 585d + 487e + 440f + 292g + 260h + 530i + 195j + 125k = 1002\\), subject to the constraints \\(0 \\leq a \\leq 27\\), \\(0 \\leq b \\leq 37\\), \\(0 \\leq c \\leq 14\\), \\(0 \\leq d \\leq 15\\), \\(0 \\leq e ... | 3 |
[
{
"content": "In triangle \\( \\triangle ABC \\), where \\( AB = AC = 5 \\) and \\( BC = 6 \\), the orthocenter \\( H \\) satisfies the vector equation \\( \\overrightarrow{AH} = m \\overrightarrow{AB} + n \\overrightarrow{BC} \\). Find the value of \\( m+n \\).",
"role": "user"
}
] | \dfrac{21}{32} |
[
{
"content": "Solve for $r$: $r = \\displaystyle\\frac{\\sqrt{5^2+12^2}}{\\sqrt{16+9}}$. Express as a common fraction.",
"role": "user"
}
] | \frac{13}{5} |
[
{
"content": "A fair six-sided die is tossed continuously, with two faces each bearing brown, black, and white colors. Let the random variable $Y$ denote the number of tosses required to observe all three colors. Specifically, $Y$ takes on the value $k$ if exactly two colors appear in the first $k-1$ tosses and... | \dfrac{11}{2} |
[
{
"content": "Given a probability distribution \\(f_X(x) = \\frac{e^{-x/\\lambda}x}{\\lambda^2}\\) for \\(x > 0\\) and \\(f_X(x) = 0\\) otherwise, where \\(\\lambda\\) is an unknown positive number, a sample of size 50 has a sum \\(\\sum_{i=1}^{50} x_i = 185.52\\). What is the maximum likelihood estimation of \... | 1.8552 |
[
{
"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... | 17.68 m |
[
{
"content": "What is the remainder when the polynomial \\( P(x) = (x+1)^{2015} + (x-1)^{2015} \\) is divided by \\( x^2 + 1 \\)?\n\nA) \\(-2^{2015}x\\)\nB) \\(-2^{1008}x\\)\nC) \\(2^{1008}x\\)\nD) \\(2^{2008}\\)\nE) \\(1\\)",
"role": "user"
}
] | B |
[
{
"content": "Let $U=\\left(0,\\frac{1}{2}\\right)\\times \\left(0,\\frac{1}{2}\\right)$ and $V=\\left(-\\frac{1}{2},0\\right)\\times \\left(-\\frac{1}{2},0\\right)$. Let $D$ be the open unit disk centered at the origin of $\\mathbb{R}^2$. Suppose $f$ is a real-valued continuous function on $D$ such that $f(U)=... | ABC |
[
{
"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... | $981.54 |
[
{
"content": "For any positive constant $c < 1$, determine whether there exists a positive integer $n$ that is coprime with 10, such that for all integers $k$, the number of zero digits in the decimal representation of $n^k$ is always greater than $c$ times the total number of digits in $n^k$. Provide a justifi... | Yes |
[
{
"content": "Evaluate the limit \\( \\lim_{x\\rightarrow 0}\\frac{\\sqrt[m]{1+P(x)}-1}{x} \\) where \\( P(x) = a_{1}x + a_{2}x^2+...+a_{n}x^n = \\sum_{i=1}^{n}a_{i}x^{i} \\).",
"role": "user"
}
] | \dfrac{a_1}{m} |
[
{
"content": "Evaluate the limit \\( \\lim_{a \\to -\\infty} b \\) for a point \\( P(a,b) \\) lying on the curve satisfying the differential equation:\n\\[\n2xy^2dx + 2x^2 y dy - \\tan(x^2y^2) dx = 0.\n\\]\nChoose the correct option:\n(a) 0\n(b) -1\n(c) 1\n(d) does not exist.",
"role": "user"
}
] | A |
[
{
"content": "A regular octagon $ABCDEFGH$ has an area of one square unit. What is the area of the rectangle $ABEF$?\n\n[asy]\npair A,B,C,D,I,F,G,H;\nA=(-1,1);\nB=(1,1);\nH=(-2.41,-0.41);\nC=(2.41,-0.41);\nG=(-2.41,-2.41);\nD=(2.41,-2.41);\nF=(-1,-3.82);\nI=(1,-3.82);\ndraw(A--B--C--D--I--F--G--H--cycle,linewi... | \frac{1}{2} |
[
{
"content": "Given a list of linearly independent vectors $\\{x_1,...,x_n\\}$ in a Banach space, and for each $1 \\leq i \\leq n$, a sequence of vectors $\\{y_m^{(i)}\\}_{m=1}^{\\infty}$ converging to $x_i$. Determine if there exists an $M$ such that for all $m \\geq M$, the set $\\{y_m^{(i)}:1 \\leq i \\leq n... | Yes |
[
{
"content": "Given two real square matrices $A$ and $B$ such that $A^TA = B^TB$, determine if there exists an orthogonal matrix $Q$ such that $A = QB$. Provide a justification for 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... | 344.83 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... | 5.56e-11 C |
[
{
"content": "If a group $G$ has only the trivial group and $G$ itself as its subgroups, can we be sure that $G$ has prime order?",
"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... | 2409/64 feet |
[
{
"content": "Determine the minimum number of weighings required to order 5 coins of different weights using a balance scale.",
"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... | 9.27 |
[
{
"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... | 6.1 x 10^-10 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... | 0.02 M |
[
{
"content": "In a $p$-group $G$, where $Z$ is the center of $G$ and $x^G$ is the conjugacy class of an element $x$, consider the equality $|\\langle x,Z\\rangle| = |Z| \\cup 2 \\cdot |x^G|$. Given that $|\\langle x,Z\\rangle|$, $|Z|$, and $|x^G|$ are all $p$-powers, determine the value of $p$.",
"role": "u... | 2 |
[
{
"content": "Determine the greatest natural number \\( x \\) such that \\( x \\) divides \\( 7^{y} + 12y - 1 \\) for every natural number \\( y \\).",
"role": "user"
}
] | 18 |
[
{
"content": "Find the volume of the region bounded by the plane \\( z = 1 - x - y \\) and the paraboloid \\( z = x^2 + y^2 \\).",
"role": "user"
}
] | \dfrac{9\pi}{8} |
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