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... | 1.32e-5 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... | 1673.2 MeV |
[
{
"content": "Evaluate the limit: $$\\lim_{N\\to\\infty} \\frac{\\sum_{n=0}^N \\sin^4 (5n)}{N}$$",
"role": "user"
}
] | \dfrac{3}{8} |
[
{
"content": "Determine if there exists a distinct non-zero integer $k$ such that the equation $x^4 + y^4 = k$ has more than one solution in terms of distinct non-zero integer pairs $(x, y)$. If such a $k$ exists, provide an example.",
"role": "user"
}
] | 635318657 |
[
{
"content": "Solve the recurrence relation for the time complexity: \\[ T(n) = T(\\sqrt{n}) + n^2 \\]",
"role": "user"
}
] | O(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... | 125.10 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... | 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... | 68219N |
[
{
"content": "Evaluate the limit: $$\\lim_{(x,y)\\to(0,0)} x \\cdot \\ln{(x^2+2y^2)}$$",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the cardinality of the set \\( B \\), where \\( B \\) is defined as: \\[ B = \\{g \\in \\Bbb N \\to \\Bbb N \\mid \\exists T\\subseteq \\Bbb N, \\left.g\\right|_T = \\operatorname{Id}_T \\land g(\\Bbb N \\setminus T)\\subseteq \\Bbb N \\setminus T\\} \\]",
"role": "user"
}
] | 2^{\aleph_0} |
[
{
"content": "2 diagonals of a regular heptagon (a 7-sided polygon) are chosen. What is the probability that they intersect inside the heptagon?",
"role": "user"
}
] | \dfrac{5}{13} |
[
{
"content": "Determine whether the following statement is true: If the equation \\(a^2 - Db^2 = -1\\) has integer solutions, then the equation \\(x^2 - Dy^2 = D\\) also has integer solutions for the same discriminant \\(D\\). 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... | $1,284,021 |
[
{
"content": "Consider a series of i.i.d. random variables \\((X_n)_{n\\in\\mathbb{N}}\\). Is it possible that \\( \\limsup_{n\\to\\infty} \\frac{X_n}{n} = 0 \\quad a.s.\\) does not hold? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Show that the image of the unit disk $B_1(0)$ under the function $f(z) = 6z^6 - 2z^2 + 1$ contains the ball $B_4(1)$ using Rouché's theorem.",
"role": "user"
}
] | B_4(1) |
[
{
"content": "Determine the smallest positive integer that does not divide the expression \\(1 \\cdot 2 \\cdot 3 \\cdot 4 \\cdot 5 \\cdot 6 \\cdot 7 + 11 \\cdot 12 \\cdot 13 \\cdot 14 \\cdot 15 \\cdot 16 \\cdot 17\\).",
"role": "user"
}
] | 11 |
[
{
"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... | $3,589.79 |
[
{
"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 magnitude of the net force is 530.0e9 N and the direction is 203.5 degrees relative to the -x axis. |
[
{
"content": "Find the vector \\( w \\in W \\) that is closest to \\( (0,2,1,0) \\), where \\( W = \\text{span}\\left(\\{(1,-1,0,0),(1,2,0,-1),(1,0,0,1)\\}\\right) \\subseteq \\mathbb{R}^4 \\).",
"role": "user"
}
] | (0, 2, 0, 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... | 588.6 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... | 135.15 g |
[
{
"content": "Emma plays with her square unit tiles by arranging all of them into different shaped rectangular figures. (For example, a $5$ by $7$ rectangle would use $35$ tiles and would be considered the same rectangle as a $7$ by $5$ rectangle). Emma can form exactly ten different such rectangular figures th... | 240 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\sqrt[n^n]{(3n)! + n^n} \\]",
"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... | 37.59 |
[
{
"content": "Using Cauchy's integral formula, determine a tighter upper bound for \\(|f^{(4)}(0)|\\) for all functions \\(f\\) in the family \\(\\mathscr{F}\\) as defined in the original context.",
"role": "user"
}
] | 24 |
[
{
"content": "In the equation for the incomplete Gamma function given by:\n\n$$ \\Gamma(a,x)=\\frac{\\rho^{-x}x^{a}}{\\Gamma(1-a)} \\int_0^\\infty \\frac{e^{-t} t^{-a}}{x+t} dt $$\n\nfor \\(a < 1\\) and \\(x > 0\\), identify the correct value of \\(\\rho\\) if it is a typographical error.",
"role": "user"
... | 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... | 4.8e-26 kg |
[
{
"content": "Let $N \\subset 2^X$. Let $M$ be the $\\sigma$-algebra generated by $N$ and let $M'$ be the algebra generated by $N$. What is the relationship between $M$ and $M'$?",
"role": "user"
}
] | M' \subset 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... | 11.18 |
[
{
"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... | 2H{eq}_2\O_2{/eq}(l) {eq}\to\ 2H_2O{/eq}(l) + O{eq}_2{/eq}(g) |
[
{
"content": "Suppose $\\hat{\\theta}$ is an estimator derived from an independently and identically distributed (iid) random sample $X_1, ..., X_n$, with the property that $\\text{var}(\\hat{\\theta}) < \\infty$ for all $n$. If $\\hat{\\theta}$ is a consistent estimator, does it imply that $\\lim_{n\\to\\infty... | 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... | 1.36 m |
[
{
"content": "Let $f:\\mathbb{R}\\to\\mathbb{R}$ be a continuous function such that $f(i)=0$ for all integers $i$. Determine which of the following statements is always true:\n\nA) The image of $f$ is closed in $\\mathbb{R}$.\nB) The image of $f$ is open in $\\mathbb{R}$.\nC) $f$ is uniformly continuous.\nD) No... | D |
[
{
"content": "Calculate the volume of the solid in \\(\\mathbb{R}^3\\) defined by the inequality:\n\\[(2x + y + z)^2 + (x + 2y + z)^2 + (x + y + 2z)^2 \\leq 1.\\]\nAfter simplification, the inequality becomes:\n\\[6(x^2 + y^2 + z^2) + 10(xy + yz + xz) \\leq 1.\\]\nFind the volume of this solid.",
"role": "u... | \dfrac{\pi}{3} |
[
{
"content": "Let $X$ be a locally convex space over $\\mathbb{R}$, and let $D \\subset X$ be dense. Consider a Banach space $B$ over $\\mathbb{R}$ with a Schauder basis $\\{b_i\\}_{i=1}^{\\infty}$. Is the set \\( D^+ \\triangleq \\left\\{\\sum_{j=1}^n \\beta_j d_j\\otimes b_j: \\, d_j \\in D, \\, \\beta_j \\in... | Yes |
[
{
"content": "Solve the differential equation: \\( yy' + x = \\sqrt{x^2 + y^2} \\).",
"role": "user"
}
] | y^2 = 2Cx + 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... | The speed of the 10kg mass is 4.2 x 10^-5 m/s and the speed of the 20kg mass is 2.1 x 10^-5 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... | Date| Particulars| Debit($)| Credit($)
---|---|---|---
Sept. 6|Merchandise Inventory A/C|1,800|
|Cash A/C| |1,800
Sept. 9|Merchandise Inventory A/C|180|
|Cash A/C| |180
Sept. 10|Accounts Payable A/C|66|
|Merchandise Inventory A/C| |66
Sept. 12|Accounts Receivable A/C |858|
|Sales A/C| |858
|Cost of Goods Sold ... |
[
{
"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... | $1,301,000 |
[
{
"content": "Determine if the sequence \\(b_n\\) defined by \\(b_{n+1} = \\frac{5+4b_n}{4+3b_n}\\) is a Cauchy sequence, given that \\(-1 < b_0 < 1\\).",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 3.62 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... | 0.861 m/s |
[
{
"content": "Determine the smallest positive integer \\( n \\) such that for every real number \\( x \\) and for all integers \\( 1 \\le a \\le 100 \\), the inequality \\( \\left|\\sqrt{a}-x\\right| \\le \\dfrac{1}{10^n} \\) implies \\( \\left|a - x^2\\right| \\le \\dfrac{1}{10^{100}} \\).",
"role": "user"... | 102 |
[
{
"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... | 1.44 |
[
{
"content": "Our school's girls volleyball team has 14 players, including a set of 3 triplets: Alicia, Amanda, and Anna. In how many ways can we choose 6 starters if exactly two of the triplets are in the starting lineup?",
"role": "user"
}
] | 990 |
[
{
"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... | 39.15 degrees Celsius |
[
{
"content": "Given that $\\{x,y,z\\}\\subset\\Bbb{R}^+$ and the equations \\(x^2 + xy + y^2 = 3\\), \\(y^2 + yz + z^2 = 1\\), and \\(x^2 + xz + z^2 = 4\\) hold, find the value of $xy+yz+zx$.",
"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... | 48.3 m^3 |
[
{
"content": "Given the function \\( f(x) = \\lim\\limits_{n \\to \\infty} n(x^{1/n} - 1) \\), determine the value of \\( f(xy) \\). Choose the correct option:\n\nA) \\( f(x) + f(y) \\) \nB) \\( \\frac{f(x)}{f(y)} \\) \nC) \\( f(x) - f(y) \\) \nD) \\( f(x)f(y) \\)",
"role": "user"
}
] | A |
[
{
"content": "Given that $a^2 + b^2 + c^2 + d^2 = 4$, determine the range of the expression $a^3 + b^3 + c^3 + d^3$. Express your answer as an interval.",
"role": "user"
}
] | [-8, 8] |
[
{
"content": "Let $I = [t_0, t_1]$ be a bounded interval of $\\mathbb{R}$, and consider a function $f: I \\to \\mathbb{R}$ that is $C^k(I)$ for some $k \\ge 2$. If for every $C^k$ function $g: I \\to \\mathbb{R}$ with $g(t_0) = g(t_1) = 0$, the integral $\\int_{t_0}^{t_1} f(t)g(t) \\, dt = 0$, does it follow th... | 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... | 1.81 g/L |
[
{
"content": "Determine whether it is possible to construct a bounded function that is differentiable on $(0,\\infty)$, does not converge to a limit as $x$ approaches infinity, and has a derivative that approaches zero as $x$ approaches infinity.",
"role": "user"
}
] | Yes |
[
{
"content": "Find the minimum value of the function \\( f(x,y,z) = \\frac{x^2}{1+x} + \\frac{y^2}{1+y} + \\frac{z^2}{1+z} \\) given that \\( a, b, c, x, y, z \\) are positive real numbers satisfying the equations \\( bz+cy=a \\), \\( az+cx=b \\), and \\( ay+bx=c \\).",
"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... | 72.6 N |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\frac{(n^2)!}{(2n)!} \\]",
"role": "user"
}
] | \infty |
[
{
"content": "Find the intersection of the lines $9x-4y=30$ and $7x+y=11.$ Express your answer as an ordered pair $(x,y).$",
"role": "user"
}
] | (2,-3) |
[
{
"content": "Let $\\sigma$ and $\\tau$ be two permutations such that $\\sigma^a = \\tau^b = e$, where $a$ and $b$ are positive integers and $e$ is the identity permutation. If $\\langle H\\cap K \\rangle$ has order 2, where $H = \\langle \\sigma \\rangle$ and $K = \\langle \\tau \\rangle$, determine the order ... | \dfrac{ab}{2} |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\frac{x\\sin(\\sin x) - \\sin^2 x}{x^6} \\]",
"role": "user"
}
] | \dfrac{1}{18} |
[
{
"content": "Evaluate the limit \\( \\lim_{k \\to \\infty} \\int_{0}^{\\infty}k^{3/2}\\left(\\int_{0}^{x}\\frac{\\sin(t)}{t\\sqrt{t}}dt\\right)e^{-kx} dx \\) using the Laplace transform.",
"role": "user"
}
] | \sqrt{\pi} |
[
{
"content": "Calculate the expected value of \\( \\cos(aX) \\) where \\( X \\) is a standard normal random variable and \\( a \\) is a scalar. Express your answer in terms of \\( a \\).",
"role": "user"
}
] | e^{-a^2/2} |
[
{
"content": "Find the integer value of $x$ such that \\( \\sqrt{x+\\frac{1}{2}\\sqrt{2011}}-\\sqrt{x-\\frac{1}{2}\\sqrt{2011}} \\) is an integer.",
"role": "user"
}
] | 503 |
[
{
"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... | 14.6 g NaCH3COO |
[
{
"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... | 3-4 minutes |
[
{
"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... | 0.0735 |
[
{
"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... | $27,000 |
[
{
"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... | 23.04 |
[
{
"content": "Does the map \\( \\langle f, g \\rangle = \\int_0^1 \\left( \\left( f(x) - \\frac{d}{dx}f(x) \\right) \\left( g(x) - \\frac{d}{dx}g(x) \\right) \\right) dx \\) define an inner product on the set of all polynomial functions of order less than or equal to \\( n \\)?",
"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... | 5.8% |
[
{
"content": "Consider the sequence defined by \\(x_n=\\left(1-\\frac{1}{3}\\right)^2\\left(1-\\frac{1}{6}\\right)^2\\left(1-\\frac{1}{10}\\right)^2\\left(1-\\frac{1}{15}\\right)^2\\cdots\\left(1-\\frac{1}{\\frac{n(n+1)}{2}}\\right)^2\\) for \\(n>1\\). Determine \\(\\lim_{n\\to\\infty} x_n\\). Choose the correc... | 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... | -506.66 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... | 104094 |
[
{
"content": "Calculate the moment of inertia of the solid defined by the inequalities $0 \\leq x$, $0 \\leq y$, $0 \\leq z$, and $x + y + z \\leq 1$ about the axis given by the line $\\{(t,t,t)|t\\in \\mathbb{R}\\}$.",
"role": "user"
}
] | \dfrac{1}{60} |
[
{
"content": "Consider the function \\( f:[0,1] \\to \\mathbb{R} \\) defined by\n\\[\nf(x) =\n\\begin{cases}\nx^2 & \\text{if } x > 0, \\\\\n1 & \\text{if } x=0.\n\\end{cases}\n\\]\nThis function is convex. Determine the subdifferential of \\( f \\) at \\( x = 0 \\).",
"role": "user"
}
] | \emptyset |
[
{
"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. Extract the ether solution with dilute HCl to separate p-toluidine.
2. Extract the remaining ether solution with aqueous sodium bicarbonate to separate m-anisic acid.
3. Extract the remaining ether solution with aqueous 3 M sodium hydroxide to separate thymol.
4. The remaining ether solution contains indane. |
[
{
"content": "If \\( \\{\\ln(P_n(x))\\} \\) converges uniformly to the zero function, does it follow that \\( \\{P_n(x)\\} \\) converges uniformly to 1? Justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | l = 3h, where l is the length and h is the height of the inclined plane |
[
{
"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... | 43.43 m |
[
{
"content": "Given two distinct orthogonal matrices A and B, if there exists a sequence of applications of these matrices such that each matrix appears an equal number of times and the application of this sequence is equal to the application of the sequence's dual (where each matrix is replaced by the other), ... | No |
[
{
"content": "Determine whether every non-commutative C*-algebra contains a non-normal element.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit \\( \\lim_{x\\to \\infty}(x + \\ln(\\frac{\\pi}{2} - \\arctan(x))) \\) using L'Hôpital's rule.",
"role": "user"
}
] | \infty |
[
{
"content": "Determine the number of sets \\(\\mathcal{A}\\) such that \\(1 \\leq |\\mathcal{A}| \\leq 999\\) and \\(2^{|\\mathcal{A}|} = z|\\mathcal{A}|\\) for some positive integer \\(z\\).",
"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... | t_{1/2} = \frac{\ln 2}{\lambda} |
[
{
"content": "Determine the signature of the symmetric bilinear form on \\( \\mathbb{R}^n \\) defined by \\( e_i \\cdot e_j = 1 - \\delta_{ij} \\), where \\( e_1 \\cdot e_1 = e_2 \\cdot e_2 = e_3 \\cdot e_3 = \\cdots = 0 \\) and \\( e_1 \\cdot e_2 = e_1 \\cdot e_3 = e_2 \\cdot e_3 = \\cdots = 1 \\). The signatu... | (1, n-1, 0) |
[
{
"content": "If $(X,d_1)$ and $(X,d_2)$ are topologically equivalent metric spaces, does there exist positive constants $a$ and $b$ such that $a d_1(x,y) \\le d_2(x,y) \\le b d_1(x,y)$ for every $x,y$ in $X$?",
"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... | 0.4 mol/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... | $11,100 |
[
{
"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... | 13540.8 Joule |
[
{
"content": "A plane passes through the vertex \\( A \\) of a right circular cone, creating a cross-section of maximum area. This area is twice the area of a cross-section that passes through the cone's axis. Determine the angle at the vertex of the cone's axial cross-section.",
"role": "user"
}
] | 150^\circ |
[
{
"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... | $1,320.16 |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\rightarrow \\infty} {x (1-\\Phi(x))}$$ where $\\Phi$ is the standard normal cumulative distribution 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... | 15.072V |
[
{
"content": "Let \\( f(x) = \\sin x \\) for \\( x \\in [0, \\frac{\\pi}{2}] \\), \\( f(x) = 2 - \\sin(\\pi - x) \\) for \\( x \\in (\\frac{\\pi}{2}, \\pi] \\), and \\( f(x) = f(2\\pi - x) \\) for \\( x \\in (\\pi, 2\\pi] \\). If the area enclosed by \\( y = f(x) \\) and the x-axis is \\( a\\pi + b \\), find \\... | 4 |
[
{
"content": "Determine whether there exists a characteristically simple group that is not a direct product of simple groups. A characteristically simple group is defined as one without non-trivial proper characteristic subgroups.",
"role": "user"
}
] | Yes |
[
{
"content": "Find the quadratic variation of the process \\( Y_t = B_{2t} \\), where \\( (B_t)_{t \\geq 0} \\) is a standard Brownian motion.",
"role": "user"
}
] | 2t |
[
{
"content": "Given an abelian group $G$ that is a cubic over $\\mathbb{C}$, determine the general pattern for the order of the subgroup $G_{n} = \\{x \\in G \\mid \\exists n \\in \\mathbb{N}: nx = 0\\}$. For example, $G_{2}$ is isomorphic to $\\mathbb{Z}_{2} \\oplus \\mathbb{Z}_{2}$, and $G_{3}$ is isomorphic ... | n^2 |
[
{
"content": "Determine whether the following conjecture is true or false:\n\\[ \\frac{P(s)^2}{\\zeta(s)} \\sim 2 \\sum_{r=3}^\\infty \\frac{G(2r)}{(2r)^{2s}} \\]\nas \\(s \\nearrow 1\\). In other words, evaluate:\n\\[ \\lim_{s \\to 1} \\frac{\\frac{P(s)^2}{\\zeta(s)} }{ \\sum_{r=3}^\\infty \\frac{G(2r)}{(2r)^{... | False |
[
{
"content": "Let \\( a > 1 \\) and \\( f, g: [-a,a] \\rightarrow \\mathbb{R} \\) be twice differentiable functions such that:\n- \\( f(x) = 0 \\) only for \\( x = -a, 0, a \\)\n- \\( f'(x) = 0 = g(x) \\) only for \\( x = -1, 0, 1 \\)\n- \\( g'(x) = 0 \\) only for \\( x = -c, c \\) where \\( 0 < c < 1 < a \\)\n... | f' = g |
[
{
"content": "Consider a sequence $(x_n)$ of real numbers defined recursively by $x_{n+1} = f(x_n)$ for all natural numbers $n$, where $f: \\mathbb{R} \\to \\mathbb{R}$. Suppose the sequence $(x_n)$ converges to a real number $x$. Is it necessarily true that $f(x) = x$? Justify your answer.",
"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... | 0.579% |
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