prompt list | ground_truth string |
|---|---|
[
{
"content": "Evaluate the integral \\( \\int_{0}^{1}\\int_{0}^{1-x}\\int_{0}^{2-x} xyz\\, dz\\, dy\\, dx \\).",
"role": "user"
}
] | \dfrac{13}{240} |
[
{
"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... | 45.98 kJ/mol |
[
{
"content": "Solve the differential equation \\( y(t)[y''(t)+2\\lambda y'(t)]=(y'(t))^2 \\) with the initial condition \\( y(0)=0 \\). Consider the case where \\( \\lambda \\neq 0 \\).",
"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... | $45 |
[
{
"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. The element with the lowest electronegativity (excluding hydrogen) is typically the central atom.
2. If there's a tie in electronegativity, the element with the least number of occurrences in the molecule becomes the central atom.
3. A priority order can also be used: C > Si > N > P > S > O |
[
{
"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... | $76.5 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... | 130.67 J |
[
{
"content": "Find the minimum arc length of the curve defined by \\( \\alpha(t) = \\left(t, t\\sin\\left(\\frac{\\pi}{t}\\right)\\right) \\) for \\( t \\neq 0 \\) and \\( \\alpha(0) = (0,0) \\), between \\( t = \\frac{1}{n+2} \\) and \\( t = \\frac{1}{n+1} \\).",
"role": "user"
}
] | 0 |
[
{
"content": "A website offers one of three quizzes, A, B, or C, each day. If the majority of visitors pass the quiz, the next day's quiz is chosen randomly among A, B, and C with equal probability. If the majority fail, quiz A is chosen with certainty. The probabilities of passing each quiz are: \\( P(A) = \\f... | \dfrac{5}{7} |
[
{
"content": "Let $X$ be a compact metric space and $\\mathcal{M}(X)$ be the set of probability measures on $X$ equipped with the topology of weak convergence. Consider $A\\subseteq X$ to be any measurable set in $X$. Is the map $P \\mapsto P(A)$ from $\\mathcal{M}(X)$ to $[0,1]$ measurable?",
"role": "user... | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | -4.41% |
[
{
"content": "Let \\((X_n)\\) be a sequence of random variables such that \\(X_n \\to +\\infty\\) almost surely. Prove that for any \\(M > 0\\), \\(P(X_n < M \\text{ i.o.}) = 0\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Given a set of pairwise distinct positive integers \\( \\{a_1, a_2, \\ldots, a_7\\} \\), if the sequence \\( a_1, 2a_2, \\ldots, 7a_7 \\) forms an arithmetic series in this order, determine the smallest positive value of \\( |a_7 - a_1| \\).",
"role": "user"
}
] | 360 |
[
{
"content": "Find the number of integer solutions to the congruence $x^3 \\equiv 1 \\pmod{31}$ within the interval $[1, 100]$. Provide your answer as a single integer.",
"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... | 0.00367 mol |
[
{
"content": "Consider a Lorentzian manifold $(M,g)$ and a spacelike curve $\\gamma:[0,1]\\rightarrow M$ connecting two distinct points $A$ and $B$. Determine whether the infimum of the integral \\( \\int_0^1\\sqrt{g(\\gamma'(t),\\gamma'(t))}dt \\) over all such spacelike curves with $\\gamma(0)=A$ and $\\gamma... | 0 |
[
{
"content": "Calculate the probability $P(X > Y + Z, X > Y)$, where $X, Y,$ and $Z$ are independent and identically distributed exponential random variables with rate parameter 1. The joint probability density function is $e^{-(x + y + z)}$. Determine the correct limits for the variables in the triple integral... | \dfrac{1}{4} |
[
{
"content": "Given two sets of basis vectors \\( V = \\{v_1, \\ldots, v_n\\} \\) and \\( U = \\{u_1, \\ldots, u_n\\} \\), each spanning \\( \\mathbb{R}^n \\), is it possible to select a subset \\( T \\) such that \\( T = \\{v_1, \\ldots, v_m\\} \\cup \\{u_1, \\ldots, u_o\\} \\) with \\( m + o = n \\) and \\( T... | 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... | 0.625 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... | 3.854e8 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... | 6 g |
[
{
"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... | 1419000 |
[
{
"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... | 1.216 g |
[
{
"content": "Find the multiplicative inverse of the polynomial $2x+5$ in the field $\\frac{\\Bbb F_7[x]}{\\langle x^2+2 \\rangle}$.",
"role": "user"
}
] | x + 1 |
[
{
"content": "Determine whether there exists a 10-digit integer such that more than 13,000 permutations of its digits are 10-digit multiples of 7. 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... | 32 N |
[
{
"content": "A sector with acute central angle $\\theta$ is cut from a circle of radius 6. The radius of the circle circumscribed about the sector is\n$\\textbf{(A)}\\ 3\\cos\\theta \\qquad \\textbf{(B)}\\ 3\\sec\\theta \\qquad \\textbf{(C)}\\ 3 \\cos \\frac12 \\theta \\qquad \\textbf{(D)}\\ 3 \\sec \\frac12 \... | 3 \sec \frac{1}{2} \theta |
[
{
"content": "Evaluate the integral \\( \\int_0^a \\frac{(a-x)^{n-1}}{(a+x)^{n+1}} \\, dx \\).",
"role": "user"
}
] | \dfrac{1}{2 a n} |
[
{
"content": "Find a generator of the maximal ideal of the local ring of rational functions on the algebraic curve $C: Y^2 = X + X^3$ at the point $P = (0,0)$. This local ring is a discrete valuation ring and thus a principal ideal domain.",
"role": "user"
}
] | Y |
[
{
"content": "Consider a random variable with an expected value \\( \\mathbb{E}[X] < 0 \\) and a parameter \\( \\theta \\neq 0 \\). Given that the expectation \\( \\mathbb{E}[e^{\\theta X}] = 1 \\), determine whether it is true that \\( \\theta > 0 \\).",
"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... | Year 1: $59,200
Year 2: $44,400
Year 3: $33,300
Year 4: $24,975
Year 5: $6,200
Year 6: $6,200
Year 7: $6,200
Year 8: $6,200 |
[
{
"content": "Determine the limit of the sequence \\(\\{b_n\\}\\), where \\(b_n\\) is the unique positive real root of the equation \\(x^{2n+1} + x^{2n} + \\cdots + x - 1 = 0\\).",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "In a topological space $X$, determine whether it is possible for a compact subset $Y$ to not be closed.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether the cardinality of the set of sequences of functions from \\( \\mathbb{R} \\) to \\( \\mathbb{R} \\) is \\( \\beth_1 \\), where \\( \\beth_0 \\) is the cardinality of the natural numbers, \\( \\beth_1 \\) is the cardinality of the continuum, and \\( \\beth_2 \\) is the cardinalit... | \beth_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... | 3.40 s |
[
{
"content": "Determine the number of distinct real roots of the polynomial equation \\(x^7 + 20x^6 - x^5 - 576 = 0\\).",
"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... | 2.50 m |
[
{
"content": "Calculate the volume of the solid obtained by rotating the region bounded by the curve \\( y = e^{-x} \\sin x \\) for \\( x \\ge 0 \\) about the y-axis.",
"role": "user"
}
] | \pi |
[
{
"content": "Determine whether the following statement is true for all functions $f(x)$ that have a series expansion at $x=0$: \\[ \\frac{d O(1)}{d x} = O(1) \\] Here, $O$ denotes the big-O notation, and the expansion is around the point $x=0$.",
"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... | 2.562e-4 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... | 3.1 pC |
[
{
"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}h = \frac{v^2c^2}{2g} - \mu\cos\alpha{/eq} |
[
{
"content": "Given a quadratic polynomial \\(ax^2 + bx + c\\) with positive coefficients \\(a, b, c\\) in arithmetic progression, and integer roots \\(\\alpha\\) and \\(\\beta\\), find the value of \\(\\alpha + \\beta + \\alpha \\beta\\).",
"role": "user"
}
] | 7 |
[
{
"content": "Find the greatest value of \\( n \\) such that the equation \\( 2\\tan^2x - 5\\sec x = 1 \\) has exactly 7 distinct solutions for \\( x \\in [0, \\frac{n\\pi}{2}] \\), where \\( n \\in \\mathbb{N} \\).",
"role": "user"
}
] | 15 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 17.15 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... | 0.285 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... | 0.58 A |
[
{
"content": "Determine the maximum number of digits a natural number can have such that all the digits are different and the number is divisible by each of its digits.",
"role": "user"
}
] | 7 |
[
{
"content": "In a Cartesian plane defined over a prime finite field $P = 79$, consider the points $Q = (16, 56)$ and $R = (67, 11)$. Determine how many times a line with the equation $y = ax + b \\pmod{P}$, passing through the collinear points $Q$ and $R$, intersects the $y$-axis at $y = P$. Provide your answe... | 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... | 35.9 g |
[
{
"content": "Evaluate the sum \\( \\sum_{n=1}^{\\infty}\\frac{(-1)^{n-1}}{n^{\\alpha}} \\) and determine its limit as \\( \\alpha \\) approaches \\( 0^+ \\) by grouping successive terms.",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Find the largest integer \\( n \\) such that \\(\\frac{(n-2)^{2}(n+1)}{2n-1}\\) is an integer.",
"role": "user"
}
] | 14 |
[
{
"content": "Determine the multiplicity of the zero of the function \\( f(z) = (z^2+1)^3(1+e^{\\pi z})^2 \\) at the point \\( z = i \\).",
"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... | 7.35% |
[
{
"content": "Given that $a$, $b$, and $c$ are the side lengths of $\\triangle ABC$ such that $a^{2} + b^{2} = 2018 c^{2}$, find the value of $(\\cot A + \\cot B) \\tan C$. Express your answer as a single number.",
"role": "user"
}
] | \dfrac{2}{2017} |
[
{
"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... | 4.038e-5 |
[
{
"content": "Determine the constant $p$ such that the geometric mean of any positive number $N_0$ multiplied by successive random numbers between $0$ and $p$ will, on average, neither diverge to infinity nor converge to zero.",
"role": "user"
}
] | e |
[
{
"content": "Determine whether the statement $E \\circ E \\subseteq E$ holds for every entourage $E$ in every uniform space. Answer with 'True' or 'False'.",
"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... | 16.9 mmoles |
[
{
"content": "Determine the number of bidders Jane should pay David to find in order to maximize the difference between the expected value of the second-highest bid from a uniform distribution of $n$ bidders and the cost $10n$. Each bidder values the item uniformly between $[500, 1000)$. The highest bidder wins... | 9 |
[
{
"content": "John has five children. What is the probability that at least half of them are girls? (We can assume a boy is equally likely to be born as is a girl, and vice-versa.)",
"role": "user"
}
] | \frac{1}{2} |
[
{
"content": "Evaluate the limit of the function \\( g(x,y)=\\frac{x^4-x^2y^2}{x^2+y^2} \\) as \\((x,y)\\) approaches \\((0,0)\\) using a method other than polar coordinates.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the integral \\( \\displaystyle \\int_C \\frac{1}{e^z-1} \\, dz \\) where \\( C \\) is a circle with center \\( 2i \\) and radius \\( 4 \\).",
"role": "user"
}
] | 2\pi i |
[
{
"content": "Let $X$ and $Y$ be metric spaces. Consider a convergent sequence $\\{x_n\\}$ in $X$ with distinct terms and limit $x$, and a convergent sequence $\\{y_n\\}$ in $Y$ with limit $y$. Define the function $f: \\{x\\} \\cup \\{x_n : n \\in \\mathbb{N}\\} \\to \\{y\\} \\cup \\{y_n : n \\in \\mathbb{N}\\}... | Yes |
[
{
"content": "Determine the natural number \\( n \\) such that the equation \\( [\\log_2 1] + [\\log_2 2] + [\\log_2 3] + \\dotsb + [\\log_2 n] = 1538 \\) holds, where \\([\\cdot]\\) denotes the greatest integer function.",
"role": "user"
}
] | 255 |
[
{
"content": "Determine whether every subring of a finite commutative ring that contains the identity is also an ideal.",
"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... | -2Q |
[
{
"content": "Find the surface area of the part of the sphere defined by the equation \\(x^2 + y^2 + z^2 = 4\\) that lies above the plane \\(z = 1\\).",
"role": "user"
}
] | 4\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... | 23.58 mV |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\frac{e^{-\\frac{1}{x^2}}}{x^2}. \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Given that \\( g(x) = \\frac{P(x)}{x^2-4} \\), \\( \\lim_{x\\rightarrow\\infty}g(x)=5 \\), and \\( \\lim_{x\\rightarrow2}g(x)=\\frac{9}{4} \\), find the polynomial \\( P(x) \\).",
"role": "user"
}
] | 5x^2 - 11x + 2 |
[
{
"content": "If \\( f(x) = (2x-3 \\pi)^5 + \\frac{4}{3}x + \\cos(x) \\) and \\( g \\) is the inverse function of \\( f \\), find the value of \\( g'(2 \\pi) \\).",
"role": "user"
}
] | \dfrac{3}{7} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 90.0 N |
[
{
"content": "Determine whether there exists a field that is not algebraically closed but has the property that every polynomial of degree less than or equal to \\( n \\) has a root.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the number of subgroups of index 2 in the group \\((C_2)^3 \\times C_3\\).",
"role": "user"
}
] | 7 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 70 m/s |
[
{
"content": "A person wishes to visit 6 different cities, each exactly twice, such that no city is visited consecutively. In how many distinct ways can this be done?",
"role": "user"
}
] | 2631600 |
[
{
"content": "Is it always true that a random walk with non-negative step sizes, starting at a positive value, will never reach zero?",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\( f:[0,1] \\rightarrow \\mathbb{R} \\) be a continuous function such that \\( \\int_0^1 f(x) \\, dx = 0 \\). Compute the limit:\n\\[ \\lim _{t\\to 0} \\frac{1}{t} \\int_0^1 (f(x+t)-f(x))x \\, dx \\]",
"role": "user"
}
] | f(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... | 11.6 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... | 2.6 m/s^2 |
[
{
"content": "Does the result of diagonalizing a matrix change if the eigenvectors used as columns of the matrix U are multiplied by different non-zero constants?",
"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... | 2.63 |
[
{
"content": "For \\( n = 2^k \\) where \\( k \\ge 1 \\), find a formula for the number of silver matrices of size \\( n \\). A silver matrix of size \\( n \\) is an \\( n \\times n \\) matrix where the \\( i \\)-th row and \\( i \\)-th column contain all the numbers in \\( \\{1, 2, \\ldots, 2n-1\\} \\).",
... | (2n - 1)! |
[
{
"content": "Find all quadruples of positive integers \\((a, b, c, d)\\) such that \\(a^3 = b^2\\), \\(c^3 = d^2\\), and \\(c - a = 7\\).",
"role": "user"
}
] | (9, 27, 16, 64) |
[
{
"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 |
[
{
"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... | 266.67% |
[
{
"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... | 5.70 |
[
{
"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... | 2.9e-4 |
[
{
"content": "Determine if an $n \\times n$ matrix with $m$ non-zero elements in each row and column can be decomposed into a sum of $m$ permutation matrices using bipartite graph decomposition into perfect matchings.",
"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... | -2.9012% |
[
{
"content": "If A and B are \\(n \\times n\\) matrices such that \\(AB = BA\\) and A has \\(n\\) distinct eigenvalues, can we conclude that the eigenvectors of A are also eigenvectors of B? Answer with 'Yes' or 'No' and justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the integral: \\[ \\int_0^\\pi \\cos(a \\sin(x)) e^{a \\cos(x)} \\, dx \\] where \\( a \\neq 0 \\) and \\( a \\in \\mathbb{R} \\).",
"role": "user"
}
] | \pi |
[
{
"content": "Let $(U_n)_n$ be an arbitrary sequence of open convex subsets of the unit disk $D(0,1) \\subseteq \\mathbb{R}^2$ such that $\\sum_{n=0}^\\infty \\lambda(U_n) = \\infty$, where $\\lambda$ is the Lebesgue measure. Does there exist a sequence $(q_n)_n$ in $\\mathbb{R}^2$ such that $D(0,1) \\subseteq ... | Yes |
[
{
"content": "Let $n$ be a positive integer and let $V$ be the vector space of all polynomial functions over the real numbers with degree at most $n$, i.e., functions of the form $f(x) = c_0 + c_1x + \\dots + c_nx^n$. Let $D$ be the differentiation operator on $V$. Find a basis for the null space of the transpo... | \{x^n\} |
[
{
"content": "The circumferences of Lebanese cedar cones are modeled as Gamma-distributed random variables with a mean of 5.40 cm and a variance of 3.24 cm². A botanist is interested in collecting cones with a circumference greater than 6 cm. Calculate the expected number of cones she needs to inspect to find h... | 9.01 |
[
{
"content": "Find the maximum value of the function \\( f(x) = (2\\sin A + 3\\cos A + 4)^2 \\times (6-2\\sin A-3\\cos A)^3 \\) as \\( A \\) ranges over all real numbers.",
"role": "user"
}
] | 3456 |
[
{
"content": "Determine the value of the parameter \\( a \\) for which the magnitude \\( |x + y| \\), where \\( (x, y) \\) is a solution of the system of equations \\( 4x^2 + 4y^2 = -a^2 + 16a - 32 \\) and \\( 2xy = a \\), reaches its maximum value.",
"role": "user"
}
] | 8 |
[
{
"content": "What is the remainder when $3x^7-x^6-7x^5+2x^3+4x^2-11$ is divided by $2x-4$?",
"role": "user"
}
] | 117 |
[
{
"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... | 36.123 degree Celsius |
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