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... | 0.2592 N |
[
{
"content": "Find all odd prime numbers $p$ such that both $2p-1$ and $\\frac{p-1}{2}$ are also prime.",
"role": "user"
}
] | 7 |
[
{
"content": "Determine whether the implication \"If \\( \\Phi \\) is an isomorphism, then \\( E \\) has finite dimension\" is correct, where \\( E \\) is a \\( K \\)-vector space, \\( E^{**} \\) is its bidual, and \\( \\Phi: E \\longrightarrow E^{**} \\) is defined by \\( \\Phi(x)=\\left<f,x\\right> \\) for \\... | 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... | 4.089 m^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... | 41.94 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... | 0.011 kg |
[
{
"content": "Evaluate the integral \\( \\int_0^{\\infty} \\frac{x^4}{e^x} \\, dx \\).",
"role": "user"
}
] | 24 |
[
{
"content": "Determine the minimum value of the sum \\( S=|x_2-x_1|+|x_3-x_2|+\\ldots+|x_{100}-x_{99}|+|x_1-x_{100}| \\), where \\( \\{x_1,x_2,\\ldots,x_{100}\\} \\) is a permutation of the integers \\( \\{1,2,\\ldots,100\\} \\).",
"role": "user"
}
] | 198 |
[
{
"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... | 294.12 atm |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 310 ft |
[
{
"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... | 0.0391 M |
[
{
"content": "Find the minimum value of \\( \\int_0^1 (1+x^2)f^2(x) \\, dx \\) where \\( f \\) is a continuous function on \\([0,1]\\) such that \\( \\int_0^1 f(x) \\, dx = 1 \\).",
"role": "user"
}
] | \dfrac{4}{\pi} |
[
{
"content": "Compute the integral \\( \\int_{\\gamma} \\frac{f(z)}{z^3}dz \\), where \\( f(z) = az^3 + bz^2 + cz + d \\) and \\( \\gamma : [0, 4\\pi] \\to \\mathbb{C} \\), \\( \\gamma(t) = e^{it} \\). Use the Cauchy integral formula to find the result, considering the interval \\([0, 4\\pi]\\).",
"role": "... | 4\pi i b |
[
{
"content": "Given the constraints $0 \\le x \\le 3$ and $0 \\le y \\le 3$, find the value of $x+y$ such that the equation \\( y^2 + 4 + 3\\cos(xy + y) = 2y \\) has at least one real solution.",
"role": "user"
}
] | \pi |
[
{
"content": "Suppose we have a countable set of intervals that covers the rational numbers. If each interval is extended by a factor of $1/(2^n)$, where $n$ is the index of the interval, does this new set of intervals cover all real numbers? Answer with 'Yes' or 'No'.",
"role": "user"
}
] | No |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty} \\frac{\\sqrt[n]{n!}}{n}$$",
"role": "user"
}
] | \dfrac{1}{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... | $2,520,000 |
[
{
"content": "Let $\\alpha$ be a root of the irreducible polynomial $X^3 + X^2 + 1$ over the finite field $\\mathbb{F}_2$. Consider the field extension $K = \\mathbb{F}_2(\\alpha)$, which has 8 elements. Suppose $f \\in K[X]$ is an irreducible polynomial of degree 4, and let $\\beta$ be a root of $f$. Let $L$ b... | (4096, 3) |
[
{
"content": "In the given figure, $AB = BC$ and $AC = BD$. Find the angle $x$.",
"role": "user"
}
] | 30^\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... | $46,500 |
[
{
"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... | 276 |
[
{
"content": "Let $f(n)$ be the sum of the positive integer divisors of $n$. For how many values of $n$, where $1 \\le n \\le 25$, is $f(n)$ prime?",
"role": "user"
}
] | 5 |
[
{
"content": "Suppose that $f$ is a function and $f^{-1}$ is the inverse of $f$. If $f(3)=4$, $f(5)=1$, and $f(2)=5$, evaluate $f^{-1}\\left(f^{-1}(5)+f^{-1}(4)\\right)$.",
"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... | 0.307 m |
[
{
"content": "Determine the Lebesgue measure of the set \\( S = \\left\\{ \\frac{1}{2} \\right\\} \\cup \\left\\{ \\frac{1}{4}, \\frac{3}{4} \\right\\} \\cup \\left\\{ \\frac{1}{8}, \\frac{3}{8}, \\frac{5}{8}, \\frac{7}{8} \\right\\} \\cup \\cdots \\). The set \\( S \\) is constructed by iteratively removing mi... | 0 |
[
{
"content": "Find the limit of the sequence of polynomials \\( \\{f_n\\} \\) defined recursively by \\( f_0(x) = 0 \\) and \\( f_{n+1}(x) = f_n(x) + \\frac{x^2 - f_n^2(x)}{2} \\) for \\( n = 0, 1, 2, \\ldots \\), in the supremum norm on the interval \\([-1, 1]\\).",
"role": "user"
}
] | |x| |
[
{
"content": "Find the complete solution set of the equation \\(\\sin^{-1}\\left(\\frac{x + \\sqrt{1 - x^2}}{\\sqrt{2}}\\right) = \\frac{\\pi}{4} + \\sin^{-1}x\\). Choose the correct option:\n\n(A) \\([-1,0]\\)\n(B) \\([0,1]\\)\n(C) \\([-1,\\frac{1}{\\sqrt{2}}]\\)\n(D) \\([\\frac{1}{\\sqrt{2}},1]\\)",
"role... | 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... | $67,000,000 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1. Required rate of return (r): Increases
2. Dividends (D1): Decreases |
[
{
"content": "Given the functional equation \\(4f(x^2 + y^2) = (f(x) + f(y))^2\\), determine the value of \\(f(3)\\) if \\(f: \\mathbb{N} \\rightarrow \\mathbb{N}\\) satisfies \\(f(0) = 0\\) and \\(f(1) = 4\\).",
"role": "user"
}
] | 36 |
[
{
"content": "Find all positive integer solutions \\((x, y)\\) to the Diophantine equation \\[24x^4 + 1 = y^2.\\]",
"role": "user"
}
] | (1, 5) |
[
{
"content": "Let $n > 1$ be a non-square odd integer. Determine whether there necessarily exists a prime $p$ such that $p + n$ is a perfect square.",
"role": "user"
}
] | Yes |
[
{
"content": "Given a symmetric, doubling, and strictly monotone increasing function $F: [0,1] \\to [0,1]$ with the following properties:\n1. $F(1) = 1$, $F(\\frac{1}{2}) = \\frac{1}{2}$, $F(0) = 0$\n2. For all $n \\in \\mathbb{Z}$, if $(2^n \\cdot x) \\in [0,1]$, then $F(\\frac{1}{2^n} \\cdot x) = \\frac{1}{2^... | F(x) = x |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | quasi-experimental design |
[
{
"content": "Let \\(X\\) be a random variable with probability density function (PDF) given by: \\[f(x)=\\begin{cases}4xe^{-2x} & 0\\le x<\\infty,\\\\ 0 & \\text{otherwise}\\end{cases}\\] Calculate \\(\\mathrm{Var}(X^2)\\) using the Gamma function.",
"role": "user"
}
] | \dfrac{21}{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... | 2.23e+3 |
[
{
"content": "What is $w + 2 - 3w - 4 + 5w + 6 - 7w - 8$?",
"role": "user"
}
] | -4w - 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... | $67.83 |
[
{
"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... | $166,421.70 |
[
{
"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... | 208.88 N/m |
[
{
"content": "For natural numbers \\( a \\) and \\( b \\), consider the expression \\( a \\wedge \\{n+1\\} + b \\wedge \\{n+1\\} \\). This expression is divisible by \\( a \\wedge n + b \\wedge n \\) for an infinite set of distinct natural numbers \\( n \\). Is it necessary that \\( a = b \\)?",
"role": "us... | 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... | 12500 |
[
{
"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... | -4.91 eV |
[
{
"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... | 3.636e-33 m |
[
{
"content": "The parabolas defined by the equations $y=2x^2-4x+4$ and $y=-x^2-2x+4$ intersect at points $(a,b)$ and $(c,d)$, where $c\\ge a$. What is $c-a$? Express your answer as a common fraction.",
"role": "user"
}
] | \frac{2}{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... | 1.17 |
[
{
"content": "Given the group \\( G = C_{25} \\times C_{45} \\times C_{48} \\times C_{150} \\), where \\( C_n \\) denotes a cyclic group of order \\( n \\), determine the number of elements in \\( G \\) that have an order of 5.",
"role": "user"
}
] | 124 |
[
{
"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... | 1071.17 |
[
{
"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... | 2,000,000 |
[
{
"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... | 50 g 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... | (AB/(A+B)) * T |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $895.64 |
[
{
"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... | 1/2 atm |
[
{
"content": "With all angles measured in degrees, the product $\\prod_{k=1}^{45} \\csc^2(2k-1)^\\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$.",
"role": "user"
}
] | 91 |
[
{
"content": "Given the force of interest \\( F(t) = \\frac{0.02 + 0.01t}{1 + 0.02t + 0.005t^2} \\), where \\( t \\) is the number of years starting from March 31, 2001, calculate the accumulated value of two investments: $100 invested on March 31, 2002, and $200 invested on March 31, 2006. Find the total accum... | 413.41 |
[
{
"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}\frac{\lambda}{\lambda_c}=\frac{1}{\sqrt{ \frac{2E_K}{E_o} +\left( \frac{E_k}{E_o}\right)^2}}{/eq} |
[
{
"content": "If $AAA_4$ can be expressed as $33_b$, where $A$ is a digit in base 4 and $b$ is a base greater than 5, what is the smallest possible sum $A+b$?",
"role": "user"
}
] | 7 |
[
{
"content": "Determine the minimum value of the function \\( f(x) = \\frac{\\sin{(\\pi x)}}{x(1-x)} \\) for \\( x \\in (0,1) \\).",
"role": "user"
}
] | \pi |
[
{
"content": "Determine the set of all values of \\( p \\) such that for every \\( q > 0 \\), the inequality \\( \\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q} > 2p^2q \\) holds. Express your answer in interval notation in decimal form.",
"role": "user"
}
] | [0, 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... | 4.094 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... | 121.6 nm |
[
{
"content": "If \\(\\dfrac{1}{100}\\sum_{n=1}^{199}\\dfrac{3^{100}}{3^n+3^{100}}=\\dfrac{a}{b}\\), where \\(a\\) and \\(b\\) are relatively prime, find \\(|a-b|\\).",
"role": "user"
}
] | 1 |
[
{
"content": "Find the minimal polynomial of $\\sqrt[6]{3}$ over $\\mathbb{Q}(\\sqrt{3})$. Provide your answer as a polynomial with rational coefficients.",
"role": "user"
}
] | x^3 - \sqrt{3} |
[
{
"content": "What is the area, in square units, of the interior region formed by the lines $y = 2x - 4, y = -3x +16$ and the $y$-axis?",
"role": "user"
}
] | 40 |
[
{
"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... | 512/81W |
[
{
"content": "Determine the Lebesgue measure of the set \\( \\left\\{ \\frac{x}{y} : x, y \\in I_\\alpha^* \\right\\} \\), where \\( I_\\alpha \\subset [0,1] \\) is an \\( \\alpha \\)-Cantor set of Lebesgue measure \\( \\alpha \\) and \\( I_\\alpha^* = I_\\alpha \\setminus \\{0\\} \\).",
"role": "user"
}
... | \infty |
[
{
"content": "Find \\( \\lim_{r \\to \\infty} \\frac{f(r)}{\\pi r^2} \\), where \\( f(r) \\) is the number of integral points inside a circle of radius \\( r \\) centered at the origin.",
"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... | 0.65 m/s |
[
{
"content": "Evaluate the limit:\n\\[ \\lim_{n \\to \\infty} \\frac{\\frac{\\eta(1-\\frac{1}{p_n})}{\\zeta(1-\\frac{1}{p_n})}}{\\frac{\\eta(1-\\frac{1}{n})}{\\zeta(1-\\frac{1}{n})}} \\]\nwhere \\( p_n \\) is the sequence of prime numbers. Assume the known relationship between the Dirichlet eta function and the... | 0 |
[
{
"content": "A square and isosceles triangle of equal height are side-by-side, as shown, with both bases on the $x$-axis. The lower right vertex of the square and the lower left vertex of the triangle are at $(10, 0)$. The side of the square and the base of the triangle on the $x$-axis each equal $10$ units. A... | 20 \text{ sq units} |
[
{
"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... | 167.25N |
[
{
"content": "Determine the possible values of the determinant $|A|$ for a square matrix $A$ such that $A = A^2$. Provide your answer as a set of values.",
"role": "user"
}
] | \{0, 1\} |
[
{
"content": "Convert the double integral \\( I = \\int_0^1 \\int_0^{2-2x} \\frac{y+2x}{x^2+y^2} \\, dy \\, dx \\) into polar coordinates and evaluate the integral. Determine the value of \\( I \\).",
"role": "user"
}
] | \pi |
[
{
"content": "Let $f:[0,1]\\to [0,\\infty)$ be a function such that \\(\\int_0^1 \\left|\\frac{d^r}{d x^r} f(x)\\right|^2 d x<\\infty\\) for some \\(1\\leq r<\\infty, r\\in\\mathbb{N}\\). Does it follow that \\(\\int_0^1 \\left|\\frac{d^{r-1}}{d x^{r-1}} f(x)\\right|^2 d x<\\infty\\)? Provide a justification fo... | 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... | $2,500 |
[
{
"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... | 550 |
[
{
"content": "Given \\( x \\in \\mathbf{R} \\), find the sum of the coefficients of the real parts in the expanded form of \\( (1+ix)^{50} \\).",
"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... | 2.97e4 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... | 782 km |
[
{
"content": "Let $A$ and $B$ be $n \\times n$ matrices. If for every $x \\in \\mathbb{R}^n$, $Ax = 0$ implies $Bx = 0$, which of the following statements is true?\\n\\n(a) There exists an $n \\times n$ matrix $C$ such that $A = BC$.\\n(b) There exists an $n \\times n$ matrix $C$ such that $A = CB$.\\n(c) There... | 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 m/s |
[
{
"content": "Each of the quadratic polynomials $P(x)$, $Q(x)$, and $P(x) + Q(x)$ with real coefficients has a repeated root. Is it guaranteed that these roots are the same?",
"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... | 0.617 |
[
{
"content": "Let $x$, $y$, $z$, $w$ be positive real numbers such that $x^2 + y^2 + z^2 + w^2 = 4$. Find the maximum possible value of $x^3 + y^3 + z^3 + w^3$.",
"role": "user"
}
] | 8 |
[
{
"content": "Through the focus \\( F \\) of the parabola \\( y^2 = 8(x+2) \\), a line with an inclination angle of \\( 60^\\circ \\) is drawn. If this line intersects the parabola at points \\( A \\) and \\( B \\), and the perpendicular bisector of chord \\( AB \\) intersects the \\( x \\)-axis at point \\( P ... | A |
[
{
"content": "Evaluate the limit: $$\\lim _{x \\to 0} |x|^{\\frac{1}{x^2}}$$ without using series expansions or L'Hôpital's rule.",
"role": "user"
}
] | 0 |
[
{
"content": "Let $P \\in \\mathbb{R}^{n \\times n}$ be an orthogonal matrix. Determine whether there exists a vector $x \\in \\mathbb{R}^n$ with no zero entries such that $Px$ also has no zero entries.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty}\\frac{4^n-C(2n+1,n)}{2^n}. \\]",
"role": "user"
}
] | \infty |
[
{
"content": "Evaluate the infinite series: $$ \\sum_{k=0}^{\\infty} \\frac{2^{k+7}}{5^{2^{k}} + 1} $$",
"role": "user"
}
] | 32 |
[
{
"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... | Property, Plant, & Equipment
Land $230,000,000
Plant Assets $925,000,000
Machinery & Equipment $2,020,000,000
Buildings $940,000,000
Less: Accumulated Depreciation ($2,110,000,000)
Total Property, Plant & Equipment: $1,005,000,000
Intangibles
Goodwill $210,000,000
Patents and Trademarks $500,000,000
Less: Accumulated Amortization ($59,000,000)
Total Intangibles: $651,000,000 |
[
{
"content": "Calculate the surface integral \\( \\iint_S \\vec{F} \\cdot \\vec{n} \\, dA \\) for the vector field \\( \\vec{F} = [x^2, z, -y]^T \\) over the unit sphere \\( S = \\{(x,y,z) \\in \\mathbb{R}^3 \\mid x^2 + y^2 + z^2 = 1\\} \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Suppose $(x_n)_n$ is a sequence in $\\mathbb{C}$ such that for every sequence $(y_n)_n$ in $\\mathbb{C}$ where the series $\\sum_n y_n$ converges, the series $\\sum_n (x_n y_n)$ also converges. Is the sequence $(x_n)_n$ necessarily bounded?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine all integers \\( n > 3 \\) for which there exist \\( n \\) points \\( A_1, A_2, \\ldots, A_n \\) on a plane and real numbers \\( r_1, r_2, \\ldots, r_n \\) such that:\n1. No three points \\( A_1, A_2, \\ldots, A_n \\) are collinear.\n2. For every set of three points \\( \\{A_i, A_j, A_k\... | 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... | 1,553.575KJ |
[
{
"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... | 3.9e2 km |
[
{
"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... | 1.9 s |
[
{
"content": "Let $A$ be a set with $\\text{card}(A) = a$. Determine the cardinal number of the set of countably infinite subsets of $A$. Express your answer in terms of $a$. \\( \\text{Hint: Consider the set of injective functions from } \\mathbb{N} \\rightarrow A. \\)",
"role": "user"
}
] | a^{\aleph_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... | R(smaller) = 1.089 ohm, R(larger) = 4.941 ohm |
[
{
"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... | 5.62e-21 J |
[
{
"content": "Determine the number of solutions for the congruence equation \\(2x^2 + 7x - 5 \\equiv 0 \\pmod{1099}\\).",
"role": "user"
}
] | 0 |
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