prompt list | ground_truth string |
|---|---|
[
{
"content": "In an isosceles triangle $ABC$ with $AB = BC = 8$, a height $BD$ is drawn to the base $AC$. In triangle $BCD$, a median $DE$ is drawn. A circle is inscribed in triangle $BDE$, touching side $BE$ at point $K$ and side $DE$ at point $M$. If segment $KM$ is equal to 2, find the measure of angle $A$ i... | 30 |
[
{
"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... | 0.1 N/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... | {eq}\begin{align*}\rm 2NO &\rightleftharpoons N_2O_2\\ N_2O_2 + H_2 &\to N_2O + H_2O\\ N_2O + H_2 &\to N_2 + H_2O \end{align*} {/eq} |
[
{
"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... | 0.138 kg |
[
{
"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... | 23.80% |
[
{
"content": "Let $\\mathbb{D}$ denote the open unit disk of the complex plane and let $f : \\mathbb{D} \\rightarrow \\mathbb{C}$ be an analytic function. Assume that $f$ has non-tangential limits on a dense set on $\\partial\\mathbb{D}$ such that the values can be extended to a continuous function on $\\partia... | 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... | 259.1 M/s |
[
{
"content": "Calculate the total curvature of the curve given by the parameterization \\( \\exp(t) \\cdot (\\cos(t), \\sin(t)) \\), where \\( t \\in [0, 2\\pi] \\).",
"role": "user"
}
] | 2\pi |
[
{
"content": "Let $X$ be a Hausdorff topological space and $G$ be a group acting on $X$ by homeomorphisms. Suppose that for every point $x \\in X$, there exists a neighborhood $U$ of $x$ such that $g \\cdot U \\cap U = \\emptyset$ for all but finitely many $g \\in G$. Prove or disprove: For every pair of points... | True |
[
{
"content": "Let \\( \\mathcal{H} = \\mathbb{C}^2 \\) and \\( \\mathcal{M}_1 = \\mathbb{C}|0\\rangle \\). Given \\( |\\psi\\rangle = \\alpha |0\\rangle + \\beta|1\\rangle \\), show that the probability \\( Pr(\\mathcal{M}_1) = |\\alpha|^2 \\).",
"role": "user"
}
] | |\alpha|^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... | 0.00330 m/s or 3.30 mm/s |
[
{
"content": "Given a prime $p$ such that $3$ does not divide $p-1$, determine the order of the elliptic curve over $\\mathbb{F}_p$ defined by $E(\\mathbb{F}_p) = \\{ (x,y) \\in \\mathbb{F}_p^2 \\mid y^2 = x^3 + 7 \\}$. Use Hasse's theorem to guide your calculation.",
"role": "user"
}
] | p + 1 |
[
{
"content": "Person A chooses 60 numbers randomly from the set \\( \\{1, 2, \\ldots, 100\\} \\). Person B chooses 30 numbers randomly from the set \\( \\{1, 2, \\ldots, 50\\} \\). What is the expected value of the number of numbers chosen by person A that are not chosen by person B?",
"role": "user"
}
] | 42 |
[
{
"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... | $16.4 |
[
{
"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... | 1.08s |
[
{
"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... | 5.96 |
[
{
"content": "Three circles $S$, $S_1$, and $S_2$ have radii 5, 3, and 2, respectively. Circle $S_1$ and $S_2$ touch externally, and both $S_1$ and $S_2$ touch circle $S$ internally. Find the radius of circle $S_3$, which externally touches $S_1$ and $S_2$ while internally touching circle $S$. Provide your answ... | \dfrac{30}{19} |
[
{
"content": "Given that $T \\sim t_7$ (with $n=7$ degrees of freedom), solve the following probability equation:\n\n$$P(-t < T) + P(0 < T) + P(t < T) + P(2t < T) = 1.75.$$ \n\nBreak down the problem by considering the overlapping areas:\n\n$$P(-t < T < 0) + 2P(0 < T < t) + 3P(t < T < 2t) + 4P(2t < T) = 1.75.$$... | 0.356 |
[
{
"content": "Let \\( A \\) be the sum of the digits of the number \\( 16^{16} \\), and \\( B \\) be the sum of the digits of the number \\( A \\). Determine the sum of the digits of the number \\( B \\) without directly calculating \\( 16^{16} \\).",
"role": "user"
}
] | 7 |
[
{
"content": "In triangle \\(ABC\\), where \\(AB = AC = 1\\), points \\(D\\) and \\(E\\) are the midpoints of \\(AB\\) and \\(AC\\) respectively. A point \\(P\\) lies on line segment \\(DE\\). The extensions of lines \\(BP\\) and \\(CP\\) intersect sides \\(AC\\) and \\(AB\\) at points \\(G\\) and \\(F\\) respe... | 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... | 320 g SO_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... | (11/26 l, 0) |
[
{
"content": "What is a necessary condition for a matrix to be invertible?",
"role": "user"
}
] | \det(A) \neq 0 |
[
{
"content": "Find the maximum value of k such that \\( \\frac{1001 \\times 1002 \\times \\ldots \\times 1985 \\times 1986}{11^k} \\) is an integer.",
"role": "user"
}
] | 99 |
[
{
"content": "Evaluate the limit \\( \\lim_{(x,y)\\to(0,0)}\\frac{x^3-xy^3}{x^2+y^2} \\) and show that it equals zero.",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the following statement is true: \\[ \\sum_{x \\in \\mathbb{Z}^n} e^{-2x^Tx} = \\left(\\sum_{y=-\\infty}^{\\infty} e^{-2y^2}\\right)^n \\]",
"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... | 1.76e-9 N/m |
[
{
"content": "Let \\((R, \\mathfrak{m})\\) be a Noetherian local ring containing a field. Does the \\(\\mathfrak{m}\\)-adic completion \\(\\widehat{R}\\) of \\(R\\) also contain a field? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Let $a$ and $b$ be real numbers. The function $h(x)=ax+b$ satisfies $h(1)=5$ and $h(-1)=1$. What is $h(6)$?",
"role": "user"
}
] | 15 |
[
{
"content": "Find the limit of \\( n!^{1/n^2} \\) as \\( n \\to \\infty \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Let $X$ be a normal Hausdorff space. Let $A_1$, $A_2$, and $A_3$ be closed subsets of $X$ which are pairwise disjoint. Which of the following statements is true regarding the existence of a continuous real-valued function $f$ on $X$ such that $f(x) = a_i$ if $x$ belongs to $A_i$, $i=1,2,3$?\n\n(A)... | 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... | 0.61 atm |
[
{
"content": "Compute the limit: $$\\lim_{x \\to 0^+} \\frac{\\sqrt{e^x-1}-e^x\\sqrt{x}}{x}.$$",
"role": "user"
}
] | 0 |
[
{
"content": "How many distinct garlands can be formed by arranging all $6n$ flowers of one kind and $3$ flowers of another kind in a circular fashion, considering that the garland is considered the same if it can be rotated or reflected to match another arrangement?",
"role": "user"
}
] | 3n^2 + 3n + 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... | Date|Particulars|Debit ($)|Credit ($)
---|---|---|---
Jan. 5|Cash account|2,400,000|
|Common Stock capital||300,000
|Additional paid in capital-Common||2,100,000
Jan. 5|Cash Account|330,000|
|Preferred Stock capital||300,000
|Additional paid in capital-Preferred||30,000
Feb 12|Equipment|310,000|
|Common Stock capital||... |
[
{
"content": "Minimize the determinant of the matrix \\( A = (VV^T + \\lambda I)^{-1} \\), where \\( V \\) is an \\( m \\times n \\) matrix and \\( \\lambda \\) is a very small positive scalar.",
"role": "user"
}
] | 0 |
[
{
"content": "Let $q(x)$ be a differentiable function on $\\mathbb{R}$ such that $\\lim\\limits_{x \\to \\infty} q'(x) = 0$. Determine $\\lim\\limits_{x \\to \\infty} [q(x+1) - q(x)]$. \\( \\text{Provide your answer in a simplified form.} \\)",
"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... | 0.51 g I2 |
[
{
"content": "Evaluate the limit: $$\\lim\\limits_{x \\to 1^-} \\left[\\ln\\left(\\left(\\frac{x-1}{x+1}\\right)^2\\right) \\cdot x - 2 \\cdot \\ln|x^2 - 1|\\right]$$",
"role": "user"
}
] | -4 \ln 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... | Cash balance per bank: $8,243
Add: Deposits in transit: $1,990
Less: Outstanding checks: $599
Adjusted bank balance: $9,634
Cash balance per books: $8,264
Add: Electronic funds transfer: $1,406
Less: Bank service charge: $36
Adjusted book balance: $9,634 |
[
{
"content": "Determine the largest subset of the set $\\{A, B, \\cdots, Z\\}$, where each letter represents a mathematician, such that no two mathematicians in your subset have birthdates within 20 years of each other. Calculate your score for the subset using the formula $\\max (3(k-3), 0)$, where $k$ is the ... | 69 |
[
{
"content": "Determine the number of ordered pairs \\((a, b)\\) of positive integers such that both \\(a\\) and \\(b\\) divide \\(20^{19}\\), but their product \\(ab\\) does not divide \\(20^{19}\\).",
"role": "user"
}
] | 444600 |
[
{
"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... | 135,000 |
[
{
"content": "Find the volume of the region bounded by the surfaces \\( z = x^2 + y^2 \\) and \\( z = 1 - x^2 - y^2 \\).",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "Find the Lebesgue measure of the set of numbers in the interval (0,1) that contain 100 consecutive 4's in their decimal expansion.",
"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... | Copper sulfate > 2-pentanol > ethane > methane |
[
{
"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... | Anuria |
[
{
"content": "For which values of $n$ can a graph $G$ with $n$ vertices have more edges but fewer components than another graph $G'$ with the same number of vertices?",
"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... | T_1+T_2=147 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... | 3.77e-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... | $25,701.75 |
[
{
"content": "Determine the number of ordered pairs \\((X,Y)\\), where \\(X\\) and \\(Y\\) are \\(n \\times n\\) matrices, such that the commutator \\(XY - YX = I_n\\) holds. Choose the correct option: \\(0\\), \\(1\\), \\(n\\), \\(\\infty\\).",
"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... | inward |
[
{
"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... | 74.168 m |
[
{
"content": "Determine whether the following statement is true or false: If a group \\( G \\) has a normal subgroup \\( N \\) such that both \\( N \\) and \\( G/N \\) are abelian, then \\( G \\) must also be abelian.",
"role": "user"
}
] | False |
[
{
"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... | 121 years |
[
{
"content": "Determine the number of times the number 3 appears in all compositions of a natural number \\( n \\geq 4 \\).",
"role": "user"
}
] | n \cdot 2^{n - 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... | 0.19 N/m |
[
{
"content": "Calculate the order of 3 modulo 257.",
"role": "user"
}
] | 256 |
[
{
"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... | 4.85e-5 degrees |
[
{
"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... | 67500 N |
[
{
"content": "Find the value of \\( n \\) such that the area under the curve \\( y = x^n \\) from \\( x = 1 \\) to \\( x = a \\) is one third of the area under the curve from \\( y = 1 \\) to \\( y = a^n \\).",
"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... | 3.06 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... | 73.5 N |
[
{
"content": "Find the limit of \\( \\left(\\frac{nx}{1+nx}\\right)^{n} \\) as \\( n \\to \\infty \\) for \\( x > 0 \\).",
"role": "user"
}
] | e^{-1/x} |
[
{
"content": "Arrange the numbers $0, 1, 2, 3, 4, 5, 6, 7$ on the vertices of a cube such that the sum of the numbers on each edge is a prime number. Determine the maximum possible sum of the numbers on one face of the cube.",
"role": "user"
}
] | 18 |
[
{
"content": "Evaluate the limit \\( \\lim_{x \\to \\frac{\\pi}{2}^-} \\left(\\csc\\left(-x + \\frac{\\pi}{2}\\right)\\right)^{\\sin(x)} \\).",
"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... | 4810.02 A/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... | 756.257 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... | 20,086 km/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... | 7.34 |
[
{
"content": "Given a regular pentagon, each vertex is assigned a real number. You can repeatedly choose two adjacent vertices and replace the numbers at these vertices with their arithmetic mean. If the initial sum of the numbers at all five vertices is zero, is it always possible to make all five numbers zero... | Yes |
[
{
"content": "The polynomial \\( P(x) = x^4 + ax^3 + bx^2 + cx + d \\) satisfies \\( P(k) = 11k \\) for \\( k = 1, 2, 3, 4 \\). Find the value of \\( c \\).",
"role": "user"
}
] | -39 |
[
{
"content": "Determine whether there exists an infinite topological space $X$ such that there are only countably many continuous functions from $X$ to itself.",
"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... | 11.19 ft |
[
{
"content": "Given three unit vectors \\(\\mathbf{a}\\), \\(\\mathbf{b}\\), and \\(\\mathbf{c}\\), where any two vectors make an angle of \\(\\arccos\\left(\\frac{3}{5}\\right)\\) with each other, and \\(\\mathbf{a} \\times \\mathbf{b} = p\\mathbf{a} + q\\mathbf{b} + r\\mathbf{c}\\). Find the value of \\(55q^2... | 9 |
[
{
"content": "Let $M$ be a manifold with boundary $\\partial M$. Form the manifold $M'$ by attaching a half-infinite cylinder $\\partial M\\times[0,\\infty)$ to $M$ along its boundary. In other words, \\[M'=M\\cup_{\\partial M}\\partial M\\times[0,\\infty),\\] where we identify $\\partial M\\sim\\partial M\\tim... | Yes |
[
{
"content": "Compute the work done against the force field \\( F(x, y) = -\\frac{(x, y)}{\\sqrt{x^2 + y^2}} \\) when a particle is moved from the point \\((2a, 0)\\) to the origin along the upper half of the circle with the equation \\((x - a)^2 + y^2 = a^2\\). Use the parametrization \\(x = a + a\\cos(t)\\), ... | -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... | 1.7e-5 J |
[
{
"content": "Evaluate the integral: $$\\int_{-\\infty}^{+\\infty}e^{-\\frac{1}{x^{2}}}\\, dx$$",
"role": "user"
}
] | \infty |
[
{
"content": "Compute the integral \\( \\int_{\\gamma} e^z \\, dz \\), where \\( \\gamma \\) is the semicircular arc defined by \\( \\gamma = 3e^{i\\theta} \\) for \\( 0 \\leq \\theta \\leq \\pi \\).",
"role": "user"
}
] | e^{-3} - e^{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... | 40.95% |
[
{
"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... | 1.5 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... | 0.170 m |
[
{
"content": "Find the maximum value of the function \\( f(x) = \\sin x + \\sqrt{3} \\cos x - 2 \\sin 3x \\).",
"role": "user"
}
] | \dfrac{16\sqrt{3}}{9} |
[
{
"content": "Can every real, symmetric matrix be orthogonally diagonalized as \\( A = QDQ^T \\), where \\( Q \\) is an orthogonal matrix?",
"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... | Selling price * 140.33 |
[
{
"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... | 11.57% |
[
{
"content": "Given two discrete random variables $X$ and $Y$, if their probability generating functions $E(t^X)$ and $E(t^Y)$ are equal for all $t$ in the interval $(0,1)$, can we conclude that $X$ and $Y$ have the same distribution? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the integral \\(\\int_0^{1/2}\\int_0^{\\sqrt{1-x^2}}xy\\sqrt{x^2+y^2}\\,dy\\,dx\\) using polar coordinates.",
"role": "user"
}
] | \dfrac{19}{480} |
[
{
"content": "Let $G$ be a finite group with no element of order $p^2$ for any prime $p$, and suppose that $|G| \\neq p$ for each prime $p$. Does there always exist an automorphism $\\phi$ of order 2 such that for at least one subgroup $H$ of $G$, we have $\\phi(H) \\neq H$? Provide a justification for your ans... | No |
[
{
"content": "Calculate the value of the following sum for $a_{1}, \\ldots, a_{n} \\in \\mathbb{R}$ pairwise distinct:\n\\[ \\sum_{i=1}^{n} \\prod_{j \\neq i} \\frac{x-a_{j}}{a_{i}-a_{j}}. \\]",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 3} \\frac{\\sqrt{19-x} - 2\\sqrt[4]{13+x}}{\\sqrt[3]{11-x} - x + 1} \\]",
"role": "user"
}
] | \dfrac{9}{52} |
[
{
"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... | 48.308 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... | 0.4962 m/s |
[
{
"content": "Let $G:[0,\\infty)\\to \\mathbb{R}^n$ be a continuous and differentiable function such that $G(0) = 0\\in\\mathbb{R}^n$. Suppose there exists a constant $c>0$ such that for all $t>0$, the inequality $$||DG(t)||\\leq c||G(t)||$$ holds, where $DG(t)$ is the derivative of $G$ and $||\\cdot||$ denotes... | G(t) \equiv 0 |
[
{
"content": "Evaluate \\( \\lim_{R \\to \\infty} \\int_{C_{R}} \\frac{e^{iz^{2}}}{1+z^{4}} \\, \\mathrm{d}z \\), where \\( C_{R} \\) is the portion of the circle \\( |z|= R \\) in the first quadrant.",
"role": "user"
}
] | 0 |
[
{
"content": "Given that X and Y are independently distributed Poisson random variables with means 1 and 3 respectively, find the conditional probability P(X ≤ Y | X + Y = 1).",
"role": "user"
}
] | \dfrac{3}{4} |
[
{
"content": "Use the Neyman-Pearson lemma to find the most powerful statistical test for testing the hypothesis \\( H_0 : \\theta = \\theta_1 \\) versus \\( H_1 : \\theta = \\theta_2 \\) at a significance level of \\( \\alpha = 0.05 \\). Given the following probability distributions for a discrete random varia... | \{1\} |
[
{
"content": "Let $f: X \\to X \\times X$ be the diagonal inclusion map defined by $f(x) = (x, x)$, where $X$ is a manifold in $\\mathbb{R}^m$. Show that the derivative of $f$ at a point $x$, denoted $df_x$, maps a tangent vector $v \\in T_x(X)$ to $(v, v) \\in T_{(x,x)}(X \\times X)$. Verify that $df_x(v) = (v... | df_x(v) = (v, v) |
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