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... | 2.25 moles |
[
{
"content": "Suppose \\(\\alpha\\) is a real number within the range \\(0\\leq\\alpha\\leq\\pi\\), and the function \\(f(x) = \\cos x + \\cos(x+\\alpha) + \\cos(x+2\\alpha)\\) attains a constant value \\(c\\) for all \\(x \\in \\mathbb{R}\\). Determine the value of \\(\\lfloor c+\\alpha \\rfloor\\).",
"rol... | 2 |
[
{
"content": "Find the maximum value of the expression \\( \\left\\lvert \\frac{1-(-x^2)^{n+1}}{1+x^2} \\right\\rvert \\) for \\(-1 \\le x \\le 1\\).",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\rightarrow \\infty}\\; \\frac{1}{n}\\;\\sum_{k=1}^{\\Big\\lfloor\\frac{n}{2}\\Big\\rfloor} \\cos\\Big(\\frac{k\\pi}{n}\\Big) \\]",
"role": "user"
}
] | \dfrac{1}{\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... | 15 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... | 10e-9 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... | 247.6636 AUD |
[
{
"content": "In the diagram, $AB,$ $BC,$ $CD,$ $DE,$ $EF,$ $FG,$ $GH,$ and $HK$ all have length $4,$ and all angles are right angles, with the exception of the angles at $D$ and $F.$\n\n[asy]\ndraw((0,0)--(0,4)--(4,4)--(4,8)--(6.8284,5.1716)--(9.6569,8)--(9.6569,4)--(13.6569,4)--(13.6569,0)--cycle,black+linewi... | 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... | $2,319.50 |
[
{
"content": "Determine the conditions on the parameter \\(a \\in \\mathbb{C}\\) such that the field extension \\(\\mathbb{C}(X) / \\mathbb{C}(Y)\\), where \\(Y = X^3 + 3X^2 + aX\\), is a Galois extension.",
"role": "user"
}
] | 3 |
[
{
"content": "Determine whether the sequence of random variables $X_1, X_2, \\dots$ converges in distribution, where $$p(X_n = \\frac{1}{n}) = 1- \\frac{1}{n^2} ~\\text{and}~ p(X_n = n) = \\frac{1}{n^2}.$$",
"role": "user"
}
] | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Project A |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{\\frac{\\pi}{4}} \\frac{\\ln(1+\\tan x)}{\\cos^2 x} \\mathrm{d}x \\) using a trigonometric substitution.",
"role": "user"
}
] | 2 \ln 2 - 1 |
[
{
"content": "Consider a Markov chain where each day there is a 0.2 probability of rain in the morning. The probability of rain in the afternoon given it rained in the morning is 0.6, and the probability of rain in the afternoon given it did not rain in the morning is 0.3. John goes to the office in the morning... | \dfrac{27}{40} |
[
{
"content": "Evaluate the limit: $$\\lim \\limits_{x \\to \\infty} \\frac{1}{e^{x^2}}\\int \\limits_x^{x + \\frac{\\ln x}{x}} e^{t^2} dt.$$",
"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... | $29 |
[
{
"content": "The number $5\\,41G\\,507\\,2H6$ is divisible by $72.$ If $G$ and $H$ each represent a single digit, what is the sum of all distinct possible values of the product $GH?$ (Count each possible value of $GH$ only once, even if it results from multiple $G,$ $H$ pairs.)",
"role": "user"
}
] | 59 |
[
{
"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... | 1139.24 mmHg |
[
{
"content": "Determine whether the statement is true or false: Two atlases \\(\\mathcal{A}_1\\) and \\(\\mathcal{A}_2\\) are compatible if and only if their associated maximal atlases \\(\\mathcal{A}_1^+\\) and \\(\\mathcal{A}_2^+\\) are equal.",
"role": "user"
}
] | True |
[
{
"content": "Let $X$ be a real random variable with mean $\\mu > 0$ and variance $\\mu^2$. Determine whether there exists a non-trivial upper bound on the probability $\\Bbb P(X < 0)$, or provide a counterexample showing that this probability can be as high as possible.",
"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... | 2.076e10 N to the right |
[
{
"content": "A car is equipped with one spare tire. The lifetimes of the four tires at the beginning of a long-distance journey are independent identically distributed exponential random variables with a mean of 5000 miles. The spare tire has an exponential lifetime with a mean of 1000 miles. Compute the expec... | 1875 |
[
{
"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... | Nov 1: Dr Cash 26,000; Cr Common Stock 26,000
Nov 1: Dr Rent Expense 3,800; Cr Cash 3,800
Nov 6: Dr Equipment 12,710; Cr Accounts Payable 12,710
Nov 8: Dr Vehicles 30,400; Cr Cash 6,100; Cr Notes Payable 24,300
Nov 10: Dr Supplies 1,920; Cr Cash 1,920
Nov 12: Dr Cash 8,300; Cr Service Revenue 8,300
Nov 15: Dr Prepaid 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... | 2010 J |
[
{
"content": "Identify the algebra $H\\otimes_{\\mathbb{R}} \\mathbb{C}$ in terms of a familiar algebraic structure, where $H$ denotes the Hamiltonian quaternions, $\\mathbb{C}$ represents the complex numbers, and $\\mathbb{R}$ stands for the real numbers.",
"role": "user"
}
] | M_2(\mathbb{C}) |
[
{
"content": "How many digits does $700_{10}$ have when converted to base 5?",
"role": "user"
}
] | 5\text{ digits} |
[
{
"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 major factors include obsolescence, tax considerations, and the nature of the asset. |
[
{
"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... | $196,000 |
[
{
"content": "Evaluate the double integral \\( \\iint_D \\frac{\\sqrt{x+y}}{\\sqrt{x-2y}}~dA \\), where the region \\( D \\) is bounded by the lines \\( y = \\frac{x}{2} \\), \\( y = 0 \\), and \\( x+y = 1 \\). Using the change of variables \\( u = x+y \\) and \\( v = x-2y \\), find the Jacobian and express the... | \dfrac{1}{3} |
[
{
"content": "Given two real polynomials $p(x)$ and $q(x)$ of degrees $a$ and $b$ respectively, such that $p(x^3 - 1) = p(x)q(x)$ for every $x \\in \\mathbb{R}$, determine the number of distinct real roots of $p(x)$. Provide a justification for your answer.",
"role": "user"
}
] | 1 |
[
{
"content": "Determine the maximum product of positive numbers that sum to 10 using the method of Lagrange multipliers.",
"role": "user"
}
] | \dfrac{625}{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... | 2Cl^- + ClO_3^- + 2H^+ o Cl_2 + ClO_2^- + H_2O |
[
{
"content": "Determine the number of positive integers \\( n \\leq 1988 \\) for which the function \\( f(n) = n \\), given that \\( f(1) = 1 \\), \\( f(3) = 3 \\), \\( f(2n) = f(n) \\), \\( f(4n + 1) = 2f(2n + 1) - f(n) \\), and \\( f(4n + 3) = 3f(2n + 1) - 2f(n) \\) for all positive integers \\( n \\).",
... | 92 |
[
{
"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... | 5.92e-6 C |
[
{
"content": "A bank issues ATM cards to its customers, each with a unique 6-digit password. The password must not have three consecutive digits that are the same. What is the maximum number of unique ATM cards the bank can issue?",
"role": "user"
}
] | 963090 |
[
{
"content": "Given a sequence of independent identically distributed (i.i.d.) random variables $X_1, X_2, \\cdots$, each following a normal distribution with mean 1 and variance 1, and a Poisson random variable $N$ with mean 2, independent of the sequence $X_i$'s, find the variance of the sum $X_1 + X_2 + \\cd... | 5 |
[
{
"content": "A crew of 8 members needs to secure a safe such that any group of 5 or more members can open it, but any group of 4 members cannot. Each member receives an equal number of keys. Determine the minimum number of locks required to achieve this security condition.",
"role": "user"
}
] | 70 |
[
{
"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... | 0.18 A |
[
{
"content": "Given the total differential \\( dF = 2xy e^{xy^2} dy + y^2e^{xy^2} dx \\), find the function \\( F(x,y) \\).",
"role": "user"
}
] | e^{xy^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... | 0.041 cm |
[
{
"content": "Calculate the circulation of the vector field \\( \\vec{A} = \\hat{r} r \\cos(\\phi) + \\hat{z} z \\sin(\\phi) \\) around a circular sector defined by \\( 0 \\leq r \\leq 3 \\) and \\( 0^{\\circ} \\leq \\phi \\leq 90^{\\circ} \\). Assume the path of circulation is clockwise around the sector.",
... | -\dfrac{9}{2} |
[
{
"content": "Find an explicit expression for the function $f(n)$ defined recursively by $f(n) = 2f(n-1) + n - 1$ for $n > 1$ and $f(0) = 0$, using generating functions.",
"role": "user"
}
] | 2^n - n - 1 |
[
{
"content": "Consider an analytic function $h : \\Bbb C \\to \\Bbb C$ with the following properties:\n\n- $h(0) = 0$\n- $h(1/2) = 5$\n- $|h(z)| < 10$ for $|z| < 1$\n\nDetermine the maximum possible value of $|h'(1/4)|$. \\( \\boxed{} \\)",
"role": "user"
}
] | 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... | east |
[
{
"content": "Let $f(x,y)$ be a function analytic with respect to both $x$ and $y$. Assume $U$ is some open set in $\\mathbb{R}$. If $f: U \\times U \\rightarrow \\mathbb{R}$ is such that for a fixed $y$, the function $x \\mapsto f(x,y)$ is analytic, and for a fixed $x$, the function $y \\mapsto f(x,y)$ is anal... | Yes |
[
{
"content": "Let X and Y be independent binomial random variables with parameters n and p. Calculate the expected value of Y given that X + Y = m, using the hypergeometric distribution.",
"role": "user"
}
] | \dfrac{m}{2} |
[
{
"content": "Calculate the second mixed partial derivative \\( \\frac{\\partial^2 g}{\\partial x \\partial y} \\) of the function \\( g(x,y)= \\begin{cases} \\frac{x^2y(x^2-y^2)}{x^2+y^2} &\\text {if }(x,y) \\neq (0,0) \\\\ 0 & \\text{otherwise}\\end{cases} \\) at the point (0,0).",
"role": "user"
}
] | 0 |
[
{
"content": "Let $f: \\mathbb{R}^3 \\rightarrow \\mathbb{R}^3$ be defined by $f(x) = cx$ for some constant $c \\neq 0$, and let $S$ be a regular surface. Express the Gaussian curvature of $f(S)$ in terms of the Gaussian curvature of $S$. Provide your answer in terms of $c$ and the Gaussian curvature of $S$.",
... | \dfrac{K}{c^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... | 0.408 g |
[
{
"content": "Determine the value of the constant \\( \\Phi \\) such that the improper integral \\( \\int_0^\\infty \\left( \\frac{1}{\\sqrt{x^2+4}} - \\frac{\\Phi}{x+2} \\right) dx \\) converges.",
"role": "user"
}
] | 1 |
[
{
"content": "A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is $\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.",
"role": "user"
}
] | 187 |
[
{
"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... | 9.62% |
[
{
"content": "What is the probability of hitting a specific point on a dartboard with an uncountable number of points when a dart is thrown randomly at the board?",
"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... | $1,554,858.8 |
[
{
"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... | 40.83 m/s^2 |
[
{
"content": "Can the measure \\( \\mu \\) be chosen to be a finite positive measure such that \\( C[0,1]^{*} \\) is order isometric to \\( L_{1}(\\mu) \\)?",
"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... | 35 m/s |
[
{
"content": "Evaluate the infinite series \\(\\sum_{n=1}^{\\infty}\\frac{2n-1}{2^n}\\).",
"role": "user"
}
] | 3 |
[
{
"content": "Let $E_1, E_2$ be a smooth orthonormal frame of vector fields on a simply connected open domain $U \\subseteq \\mathbb{R}^2$. Let $E^i$ be the associated dual coframe of one-forms, and suppose that $dE^i = 0$. Are the vector fields $E_i$ constant on $U$?",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the expression \\( \\sqrt{2+\\sqrt{2+\\sqrt{2+...+\\sqrt{2(1+\\cos\\theta)}}}} \\) for \\( n \\) iterations, where \\( 0<\\theta <180 \\). Choose the correct option:\n\nA) \\( 2\\cos\\left(\\frac{\\theta}{2^n}\\right) \\) \nB) \\( 2\\cos\\left(\\frac{\\theta}{2^{n-1}}\\right) \\) \nC) \... | A |
[
{
"content": "Suppose E is a dense subset in a topological space X, and F is a proper finite subset of E. Determine whether the set E \\ F can be dense in X.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: $$\\lim_{y\\rightarrow 0}\\frac{2\\cos \\left(\\frac{\\tan y+\\sin y}{2}\\right)-2}{1+\\tan(\\tan y)\\tan(\\sin y)}$$ without using L'Hopital's rule or Maclaurin series.",
"role": "user"
}
] | 0 |
[
{
"content": "A certain regular tetrahedron has three of its vertices at the points $(0,1,2),$ $(4,2,1),$ and $(3,1,5).$ Find the coordinates of the fourth vertex, given that they are also all integers.",
"role": "user"
}
] | (3,-2,2) |
[
{
"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... | v = sqrt(3RT/M), where R = 8.314 J/(mol*K), T is temperature in Kelvin, and M is molar mass in kg/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... | $(8,778) |
[
{
"content": "An investor buys a stock worth $100. Each month, the stock has a 50% chance of increasing by $1.10 and a 50% chance of decreasing by $0.90, with changes being mutually independent. What is the probability that, after 100 months, the stock's value exceeds $91?",
"role": "user"
}
] | 0.9713 |
[
{
"content": "Determine whether it is always possible to find a countable open locally finite refinement of a given open cover in a paracompact second-countable space.",
"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... | $37,942.22 |
[
{
"content": "In a two-dimensional Cartesian coordinate system, there are two points $A(2, 0)$ and $B(2, 2)$ and a circle $c$ with radius $1$ centered at the origin $O(0, 0)$. If $P$ is a point on the circle $c$, determine the minimum value of the expression:\n\\[ f = 2\\sqrt{2}\\lvert{PA}\\rvert + \\lvert{PB}\... | 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... | 2.36e+05 J |
[
{
"content": "In the finite field \\( \\mathbb{F}_{p^n} \\), where \\( p \\) is a prime number and \\( n > 1 \\), does there exist a subgroup of the additive group \\( (\\mathbb{F}_{p^n}, +) \\) of order \\( p^k \\) for any integer \\( k \\) such that \\( 1 < k < n \\)?",
"role": "user"
}
] | Yes |
[
{
"content": "Find the minimum value of the function \\( \\sqrt{2x} - \\left\\lceil \\frac{\\sqrt{1+8x}}{2} \\right\\rceil \\) for positive integer values of \\( x \\).",
"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... | 5.714 m/s |
[
{
"content": "Let $k$ and $m$ be real numbers, and suppose that the roots of the equation \\[x^3 - 7x^2 + kx - m = 0\\]are three distinct positive integers. Compute $k + m.$",
"role": "user"
}
] | 22 |
[
{
"content": "Let $X$ and $Y$ be topological spaces, and let $f: X \\to Y$ be a continuous function. Suppose that for every $y \\in Y$, the preimage $f^{-1}(y)$ is compact, and $Y$ is a Hausdorff space. Determine whether it is true that if $Z \\subseteq Y$ is compact, then $f^{-1}(Z)$ is also compact.",
"ro... | False |
[
{
"content": "Determine the limit of \\( \\frac{\\hat\\sigma_n}{\\log(\\log(n))} \\) as \\( n \\to \\infty \\), where \\( \\hat\\sigma_n = \\sqrt{\\frac{1}{n}\\sum_{i=1}^n(X_i - \\bar X)^2} \\) and \\( X_i \\) is a sequence of iid random variables with finite variance.",
"role": "user"
}
] | 0 |
[
{
"content": "In a triangle, determine the ratio of the distance between a vertex and the orthocenter to the distance from the circumcenter to the side opposite that vertex.",
"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... | 204.5 m |
[
{
"content": "Calculate the limit of the expression as \\( n \\) approaches infinity: \n\\[ \\lim_{n\\to \\infty} e^{-n} \\sum_{i=0}^{n} \\frac{n^i}{i!} \\]",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Aaron samples a number from a uniform \\((0,1)\\) distribution. Brooke then repeatedly samples from the same distribution until she obtains a number higher than Aaron's. What is the expected number of samples Brooke needs to make?",
"role": "user"
}
] | \infty |
[
{
"content": "Find all natural numbers \\( n \\) such that the numbers \\( 1, 2, 3, \\ldots, 2n \\) can be partitioned into pairs in such a way that the product of the sums of the numbers in each pair is a perfect square.",
"role": "user"
}
] | n \geq 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... | 0.07268 J |
[
{
"content": "Given two regular languages $R_1$ and $R_2$, consider a language $L$ such that $R_1 \\subseteq L \\subseteq R_2$. Is it always the case that $L$ is also a regular language? Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"content": "Determine the splitting field $F$ of the polynomial $f(x)=x^4+16$ over the real numbers $\\mathbb{R}$ and find the degree of the extension $[F:\\mathbb{R}]$. Provide your answer as a pair $(F, [F:\\mathbb{R}])$. ",
"role": "user"
}
] | (\mathbb{C}, 2) |
[
{
"content": "Evaluate the limit \\( \\lim_{x \\rightarrow 0} \\int_{0}^{x^{3}} \\sin t^{3} \\, dt \\) without using Taylor series.",
"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... | $9.59 |
[
{
"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... | 0.00320 N |
[
{
"content": "Let $x, y, z > 0$ such that $x + y + z = xy + yz + zx$. Determine the minimum value of the expression:\n\\[ P = \\frac{x^3}{\\sqrt{2(y^4+1)}} + \\frac{y^3}{\\sqrt{2(z^4+1)}} + \\frac{z^3}{\\sqrt{2(x^4+1)}}. \\]",
"role": "user"
}
] | \dfrac{3}{2} |
[
{
"content": "Let \\( f \\) be an analytic function on the unit disk \\( D = \\{ z \\in \\mathbb{C} : |z| < 1 \\} \\) such that \\( |f(z)| \\le 1 \\) for all \\( z \\in D \\). Determine which of the following is not a possible value of \\( (e^f)''(0) \\):\n\n(a) \\( 2 \\)\n(b) \\( 6 \\)\n(c) \\( \\frac{7}{9} e^... | B |
[
{
"content": "Evaluate \\( \\lim_{n \\rightarrow \\infty} J_n \\), where \\( J_n = \\int_{0}^{1} \\frac{g(x)}{1+n^2x^2} \\, dx \\) and \\( g(x) \\) is an integrable function on \\([0,1]\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Given a quadratic function \\( f(x) \\), define a new function \\( F(x) \\) as follows:\n\n\\[ F(x) = \\int_0^x f(t) \\, dt \\]\n\nAssume that \\( a \\) is a positive number and \\( F(x) \\) has extreme values at \\( x = -2a \\) and \\( x = 2a \\). What is the local maximum value of the function \... | \dfrac{4a}{3} |
[
{
"content": "If $X$ and $Y$ are two spaces that are isometrically isomorphic, does the function $\\phi: X \\rightarrow Y$ have to be onto?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the set of integers \\( x \\) such that \\( \\sqrt[3]{\\frac{7x+2}{x+5}} \\) is an integer.",
"role": "user"
}
] | -38 |
[
{
"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... | 2.260 inches^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... | 6.80 |
[
{
"content": "Find the coordinates of point P on the graph of \\( y = ae^{-x} \\) for \\( a > 0 \\) such that the perimeter of the rectangle OAPB is minimized.",
"role": "user"
}
] | (\ln a, 1) |
[
{
"content": "Evaluate the integral \\( \\int \\frac{1}{\\sqrt{(x^2-4x)^3}} \\, dx \\) and determine which of the following solutions is correct:\n\n1. \\( \\frac{\\sqrt{(x-2)^2-4}}{x-2} - \\ln\\left| \\frac{x-2}{2}+\\frac{\\sqrt{(x-2)^2-4}}{2} \\right| + C \\)\n\n2. \\( \\frac{2-x}{\\sqrt{(x-2)^2-4}} + C \\)",... | 2 |
[
{
"content": "Evaluate the integral \\( \\oint_{|z-1|=1} \\frac{2z-1}{z^2 - 1} \\, dz \\) using the residue theorem, where the contour is the circle of radius 1 centered at 1, traversed anticlockwise.",
"role": "user"
}
] | \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... | 1N |
[
{
"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... | 28.125 g |
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