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... | 117 Days |
[
{
"content": "Given a random variable \\(X\\) which is non-negative and integer-valued, with the probability generating function:\n\\[G_X(s) = e^{s-1}\\]\nFind the probability \\(P(X < 2)\\).",
"role": "user"
}
] | \dfrac{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... | 38.42 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... | 718.99 W |
[
{
"content": "Determine the weight function with respect to which the eigenfunctions of the differential equation \\( x^2y'' + xy' + \\lambda y = 0 \\) are orthogonal, given the boundary conditions \\( y(1) = 0 \\) and \\( y(b) = 0 \\) where \\( b > 1 \\). Verify the orthogonality of these eigenfunctions by dir... | \dfrac{1}{x} |
[
{
"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... | $163.68 |
[
{
"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... | 4.73% |
[
{
"content": "Let $G$ be a group of order 12. Does the group of automorphisms of $G$, denoted as $\\mathrm{Aut}(G)$, contain an element of order 5?",
"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... | $633,500 |
[
{
"content": "Evaluate the integral: $$\\int_{0}^{\\frac{\\pi}{2}} \\sqrt{\\frac{1+\\sin x}{2}} \\, dx$$",
"role": "user"
}
] | \sqrt{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... | 103.6 m/s |
[
{
"content": "Calculate the volume of the region defined by the inequality \\(|x| + |y| + |z| \\leq 1\\) using integration.",
"role": "user"
}
] | \dfrac{4}{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... | 15750 lbs |
[
{
"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... | 3119 meters |
[
{
"content": "Evaluate the expression \\( \\sum\\limits_{n=1}^{1000} \\gcd(3n, 3003) - \\sum\\limits_{n=0}^{999} \\gcd(3n+1, 3001) \\).",
"role": "user"
}
] | 16472 |
[
{
"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... | 0.75v |
[
{
"content": "Given that $f(x)$ and $g(x)$ are functions satisfying $f'(x) = g(x)$ and $g'(x) = -f(x)$, with initial conditions $f(2) = 4$ and $f'(2) = 4$, find the value of $f^2(24) + g^2(24)$. Provide your answer as a single number.",
"role": "user"
}
] | 32 |
[
{
"content": "Evaluate the limit \\( \\lim\\limits_{x \\to \\infty} \\left(1 + \\frac{\\pi}{2} - \\arctan(x)\\right)^x \\). Use the fact that \\( \\lim\\limits_{x \\to \\infty} \\arctan(x) = \\frac{\\pi}{2} \\) and consider techniques such as L'Hôpital's rule or the resemblance to \\( \\left(1 + \\frac{1}{x}\\r... | e |
[
{
"content": "Find the kernel of the map induced on homology by the inclusion map \\(i: \\mathbb{S}^1 \\to \\mathbb{S}^3\\), specifically \\(\\ker(i_*:H_0(\\mathbb{S}^1)\\to H_0(\\mathbb{S}^3))\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Given any number $c$, determine whether it is always possible to find functions $f$ and $g$ such that $\\lim_{x \\to a} f(x) = \\infty$, $\\lim_{x \\to a} g(x) = 0$, and $\\lim_{x \\to a}f(x)g(x) = c$ for some value $a$. 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... | $63,000 |
[
{
"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... | 11074 m/s |
[
{
"content": "Simplify $\\sqrt{64^3}$.",
"role": "user"
}
] | 512 |
[
{
"content": "Evaluate the integral \\( \\int_{|z|=1} \\overline{z}^2 \\sin z \\, \\mathrm{d}z \\).",
"role": "user"
}
] | 2\pi i |
[
{
"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... | 1084.8 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... | $10,400 |
[
{
"content": "Calculate the probability that $Z > X + Y$, where $X$, $Y$, and $Z$ are independent random variables uniformly distributed over the interval $(0,1)$. Express your answer as a single probability value.",
"role": "user"
}
] | \dfrac{1}{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... | 0.62026 and $5,830.45 |
[
{
"content": "Verify whether the function \\( f(t,x) = t^3 e^{-tx^2} \\) satisfies the Lipschitz condition on the domain \\( (t,x) \\in [0,1] \\times \\mathbb{R} \\).",
"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... | $148,727.69 |
[
{
"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... | $18,181.82 |
[
{
"content": "Given a function \\( f(x) = \\begin{cases} f_1(x) & \\text{if } x \\leq 0 \\\\ f_2(x) & \\text{if } x > 0 \\end{cases} \\) with the following properties:\n\n- \\( f \\) is continuous over its domain.\n- \\( f_1 \\) is differentiable over \\( (-\\infty, 0) \\).\n- \\( f_2 \\) is differentiable over... | L |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 5.45% |
[
{
"content": "Calculate the product of the elements $(1,1,1,0,1)$ and $(0,1,0,1,0)$ in the Galois Field $GF(2^5)$. What is the result?",
"role": "user"
}
] | (0,1,1,0,0) |
[
{
"content": "A fair die is tossed repeatedly until the last six outcomes form the pattern 131131. What is the expected number of rolls needed to achieve this pattern?",
"role": "user"
}
] | 46878 |
[
{
"content": "Let $O, A, B, C$ be points in space such that $\\angle AOB = 60^{\\circ}$, $\\angle BOC = 90^{\\circ}$, and $\\angle COA = 120^{\\circ}$. Determine $\\cos \\theta$, where $\\theta$ is the acute angle between the planes $AOB$ and $AOC$. \\( \\boxed{\\cos \\theta} \\)",
"role": "user"
}
] | \dfrac{1}{3} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.67e-14 N |
[
{
"content": "Determine the number of transitive $G$-sets of the symmetric group $S_3$ up to isomorphism.",
"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... | 3.74e+3 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... | 2.71 mm |
[
{
"content": "Find the value of \\( p \\) such that the function \\( f(x) = x^4 - 4x + p \\) has two identical real roots.",
"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... | 16.7 |
[
{
"content": "Let $A=\\{x^2:0<x<1\\}$ and $B=\\{x^3:1<x<2\\}$. Determine which of the following statements is true:\n1. There is a one-to-one, onto function from $A$ to $B$.\n2. There is no one-to-one, onto function from $A$ to $B$ taking rationals to irrationals.\n3. There is no one-to-one, onto function from ... | 1 |
[
{
"content": "Evaluate the limit as \\( x \\) approaches infinity:\n\n\\[\n\\lim_{x \\to \\infty} \\frac{1}{e^{x^2}} \\int_{x}^{x + \\frac{\\ln x}{x}} e^{t^2} dt\n\\]",
"role": "user"
}
] | \infty |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2017: $2.02
2018: $2.34
2019: $2.72
2020: $3.15
2021: $3.65 |
[
{
"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... | Number of Hours Worked | Output | Marginal Product | Fixed Cost | Variable Cost | Total Cost | Marginal Cost | Average Variable Cost | Average Total Cost
---|---|---|---|---|---|---|---|---
0 | 0 | - | 1,000 | 0 | 1,000 | - | - | -
50 | 400 | 8 | 1,000 | 250 | 1,250 | 0.625 | 0.625 | 3.125 |
[
{
"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... | $a_{1y} = g * \frac{m_2*sin \theta - m_1}{m_1+m_2}$ |
[
{
"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... | new f.p. = normal f.p. - DeltaT_f, where DeltaT_f = imK_f |
[
{
"content": "Let $X$ be a topological space and $K$ be a compact subset of $X$. Suppose a sequence $(x_n)$ in $K$ has the property that every convergent subsequence converges to the same point $p$. Does it follow that the sequence $(x_n)$ itself converges to $p$? Answer yes or no and justify your answer.",
... | 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... | t= \sqrt{\frac{2h}{g-\frac F m }} |
[
{
"content": "In a ring \\((R, +, \\times, 0, 1)\\) with unity, can a non-zero element \\(a \\in R\\) be both nilpotent and idempotent? Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"content": "Let $\\{a_i\\}_{i=0}^\\infty$ be a sequence of real numbers such that \\[\\sum_{n=1}^\\infty\\dfrac {x^n}{1-x^n}=a_0+a_1x+a_2x^2+a_3x^3+\\cdots\\] for all $|x|<1$. Find $a_{1000}$. \\(\\text{Proposed by David Altizio}\\)",
"role": "user"
}
] | 16 |
[
{
"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... | 19.5oC |
[
{
"content": "Is there a countable family of open subsets of \\( \\mathbb{R} \\) or \\( [0,1] \\) such that each rational number belongs to only finitely many of the open sets, while each irrational number belongs to infinitely many of the sets?",
"role": "user"
}
] | Yes |
[
{
"content": "In complex analysis, consider a domain $\\Omega \\subseteq \\mathbb{C}$ and a curve $\\gamma$ in $\\Omega$. If $\\gamma$ is homotopic to $0$ (i.e., $\\gamma \\sim 0$), does it imply that $\\gamma$ is homologous to $0$ (i.e., $\\gamma \\approx 0$)? Answer yes or no.",
"role": "user"
}
] | Yes |
[
{
"content": "For which primes $p$ does the series $\\sum_{k = 0}^{\\infty} \\left(\\frac{15}{7}\\right)^k$ converge in the $p$-adic numbers $\\mathbb{Z}_p$? Find the sum of the series in each case.",
"role": "user"
}
] | -\dfrac{7}{8} |
[
{
"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... | C = Ka/0.50 |
[
{
"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... | $85,000 |
[
{
"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... | 1.10 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... | 33 L |
[
{
"content": "If Greg rolls four fair six-sided dice, what is the probability that he rolls more 1's than 6's?",
"role": "user"
}
] | \dfrac{421}{1296} |
[
{
"content": "Find the largest integer that can be expressed as the sum of squares of two positive integers in three or more distinct ways.",
"role": "user"
}
] | 325 |
[
{
"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... | $915.82 million |
[
{
"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... | Debit Purchases 30,000
Debit Freight In 450
Credit Accounts Payable 30,450 |
[
{
"content": "During her birthday, her parents have decided to give Laura and her 2 younger brothers new cellphones. However, they are confused between the innumerable service providers. Assuming no child wants a provider that another sibling has, and that there are 20 service providers, in how many ways can t... | 6840 |
[
{
"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}{}_7^{14}{{
m{N}}^{3 - }} {/eq} |
[
{
"content": "Given that \\( x, y, z \\) are the side lengths of a triangle and \\( x + y + z = 1 \\), find the minimum value of the real number \\( \\lambda \\) such that the inequality \\( \\lambda(x y + y z + z x) \\geqslant 3(\\lambda + 1)x y z + 1 \\) always holds.",
"role": "user"
}
] | 5 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.5189 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... | 7293.2 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... | $0.00224 |
[
{
"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... | 41.01 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... | The tension in the left wire is 194.9 N and the tension in the right wire is 258.71 N. |
[
{
"content": "In a rectangular coordinate system, the line $3y = x$ intersects the line $2x + 5y = 11$ at point $A$. What is the sum of the coordinates of point $A$?",
"role": "user"
}
] | 4 |
[
{
"content": "An equilateral triangle has two vertices at $(0,5)$ and $(8,5)$. If the third vertex is in the first quadrant, what is the y-coordinate? Express your answer in simplest radical form. [asy]\ndraw((-1,0)--(11,0),Arrows);\ndraw((0,-1)--(0,12),Arrows);\nfor(int i=0;i<11;++i)\n{draw((i,-0.1)--(i,0.1));... | 5+4\sqrt{3} |
[
{
"content": "Consider $X$ as a set and let $\\mathcal{R}$ be the collection of all non-strict partial orders over $X$. Define a partial order $\\sqsubseteq$ on $\\mathcal{R}$ such that for $\\leqslant, \\trianglelefteq \\in \\mathcal{R}$, $\\leqslant \\sqsubseteq \\trianglelefteq$ if and only if for all $x, y ... | Yes |
[
{
"content": "Determine whether the following statement is true: If two random variables $X$ and $Y$ have the same probability distribution, then $E\\left(\\frac{X}{X+Y}\\right) = \\frac{1}{2}$. Provide a justification for your answer.",
"role": "user"
}
] | False |
[
{
"content": "Determine whether the statement is true or false: For a sequence of functions $g_n(x)$, if $\\sup_{x\\in[a,b]}|g_n(x)-g(x)|$ converges to $0$ for some function $g(x)$, then $g_n(x)$ converges uniformly to $g(x)$ on $[a,b]$. Provide a justification for your answer.",
"role": "user"
}
] | A |
[
{
"content": "How many of the 343 smallest positive integers written in base 7 use 4 or 5 (or both) as a digit?",
"role": "user"
}
] | 218 |
[
{
"content": "Rodney uses the following clues to try to guess a secret number:\n\\begin{tabular}{ c }\nIt is a two-digit integer.\\\\\nThe tens digit is odd.\\\\\nThe units digit is even.\\\\\nThe number is greater than 65.\n\\end{tabular}If Rodney guesses a number that has each of these properties, what is the... | \frac{1}{10} |
[
{
"content": "Find the smallest integer \\( n > 1 \\) such that the product of the factors of \\( n \\) is \\( n^{15} \\).",
"role": "user"
}
] | 720 |
[
{
"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... | 7.789% |
[
{
"content": "Determine the smallest integer \\( k \\) such that if the Fourier coefficients \\( a_n \\) and \\( b_n \\) of a function \\( f \\) satisfy \\( |a_n| \\leq Cn^{-k} \\) and \\( |b_n| \\leq Cn^{-k} \\) for some constant \\( C \\), then \\( f \\) is in \\( C^1(T) \\).",
"role": "user"
}
] | 3 |
[
{
"content": "Is simply connectedness a topological property? Specifically, if $f: X \\rightarrow Y$ is a homeomorphism and $X$ is a simply connected space, is $Y$ also simply connected?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the last two digits of \\(3^{3^{100}}\\).",
"role": "user"
}
] | 03 |
[
{
"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... | 360.16 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... | Teams |
[
{
"content": "Calculate the limit: $$\\lim\\limits_{t \\to 0} \\dfrac{\\sqrt[n]{(t+1)(2t+1)...(nt+1)}}{t+1}$$",
"role": "user"
}
] | 1 |
[
{
"content": "Given that the function \\( f \\) satisfies the equation \\( f(a + b) = f(a) + f(b) + ab \\) for all real numbers \\( a \\) and \\( b \\), and that \\( \\lim_{h \\to 0} \\frac{f(h)}{h} = 7 \\), determine \\( f'(x) \\).",
"role": "user"
}
] | x + 7 |
[
{
"content": "Does a homeomorphism between two topological spaces map clopen sets to clopen sets?",
"role": "user"
}
] | Yes |
[
{
"content": "Find the greatest common divisor (GCD) of the polynomials: \\(x^2-8x+15-9y^2+12xy-4x^2\\) and \\(4x^4-40y^2x^2-112xy^2-64y^2\\).",
"role": "user"
}
] | 1 |
[
{
"content": "Let $f$, $g$, and $h$ be polynomials such that $h(x) = f(x)\\cdot g(x)$. If the constant term of $f(x)$ is $-4$ and the constant term of $h(x)$ is 3, what is $g(0)$?",
"role": "user"
}
] | -\frac{3}{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... | 1.10 L |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | {a/2(i+j), a/2(j+k), a/2(i+k)} |
[
{
"content": "The numbers \\( a \\) and \\( b \\) are such that the polynomial \\( x^{4} + 3x^{3} + x^{2} + ax + b \\) is the square of some other polynomial. Determine the value of \\( b \\).",
"role": "user"
}
] | \dfrac{25}{64} |
[
{
"content": "Calculate the limit of the function \\( f(x,y)=\\frac{\\ln(1+x^2y^2)}{x^2} \\) as \\((x,y)\\) approaches \\((0,y_0)\\). Find an upper bound function that converges to \\(y_0^2\\) to confirm the limit.",
"role": "user"
}
] | y_0^2 |
[
{
"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... | {eq}{V_1} = \dfrac{Q}{{{C_1}}} {/eq} |
[
{
"content": "Let \\( a \\) be any odd positive integer and \\( n \\) be an integer greater than 5. Determine the smallest possible integer \\( N \\) such that \\( a^N \\equiv 1 \\pmod{2^n} \\).",
"role": "user"
}
] | 2^{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... | Left contraction: \(v = r\lcontr\Omega\) |
[
{
"content": "Given three independent exponential random variables with means 1, 2, and 3, calculate the expected value of the minimum of these variables, \\( E(\\min(X_1, X_2, X_3)) \\).",
"role": "user"
}
] | \dfrac{6}{11} |
[
{
"content": "Find the minimal polynomial of \\( \\alpha = 2^{\\frac{1}{4}} \\) over \\( \\mathbb{Q}(\\beta) \\), where \\( \\beta = e^{\\frac{2\\pi i}{5}} \\), and factor it into irreducibles in \\( \\mathbb{Q}(\\beta)[X] \\).",
"role": "user"
}
] | X^4 - 2 |
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