prompt list | ground_truth string |
|---|---|
[
{
"content": "Determine whether the converse of Dirichlet's test for convergence is true. Specifically, if a sequence of complex numbers $\\lambda_n$ yields an unbounded partial sum $\\sum_{k=1}^n \\lambda_k$ as $n$ approaches infinity, does there exist a sequence $x_n$ converging to zero such that $\\sum \\lam... | Yes |
[
{
"content": "If $X$ and $Y$ are two Banach spaces and $A : X \\rightarrow Y$ is a linear injective open operator, is $A$ necessarily bounded?",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\( A = \\begin{bmatrix}2 & -1\\\\-1 & 2\\end{bmatrix} \\). Define the linear transformation \\( T \\) as \\( T(X) = AX - XA \\), where \\( X \\) is a \\( 2 \\times 2 \\) matrix. Find the dimension of the null space of \\( T \\).",
"role": "user"
}
] | 2 |
[
{
"content": "Calculate the limit: $$\\lim_{x\\to\\infty} x^2\\operatorname{cos}\\left(\\frac{3x+2}{x^2}-1\\right)$$ without using L'Hôpital's rule.",
"role": "user"
}
] | \infty |
[
{
"content": "Find all integer values of $x$ such that $54x^3 + 1$ is a perfect cube.",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether there exist two distinct quaternions that represent the same rotation matrix $R_{3\\times3}$. 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... | 4.72 |
[
{
"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... | $4,889.27 |
[
{
"content": "Find the limit of the sequence \\( A_n = \\frac{(n!)^{1/n}}{n} \\) as \\( n \\to \\infty \\), given that \\( \\int_{0}^{1}\\ln{x}\\,dx = -1 \\).",
"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... | $30,144 |
[
{
"content": "Evaluate the double integral \\( \\int \\int_{R} \\frac{1}{(x^2+y^2)^2} \\, dA \\) where \\( R \\) is the region enclosed by the circle \\( x^2+y^2 = 2 \\) and the line \\( x=1 \\), using polar coordinates.",
"role": "user"
}
] | \dfrac{1}{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... | 8.24% |
[
{
"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": "Determine if there are any integer solutions for the equation \\(5^x - 2^y = 3\\).",
"role": "user"
}
] | (1, 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... | 29.16%N, 8.39%H, 12.5%C, 49.95%O |
[
{
"content": "Tim wants to invest some money in a bank which compounds quarterly with an annual interest rate of $7\\%$. To the nearest dollar, how much money should he invest if he wants a total of $\\$60,\\!000$ at the end of $5$ years?",
"role": "user"
}
] | \$42409 |
[
{
"content": "In triangle \\( \\triangle ABC \\), point \\( D \\) is the midpoint of \\( BC \\) such that \\( DA = DB = DC \\). The angle bisectors \\( BE \\) and \\( CF \\) of \\( \\angle ABC \\) and \\( \\angle ACB \\), respectively, are drawn. Let \\( EG \\) and \\( FH \\) be perpendiculars from \\( E \\) an... | 45^\circ |
[
{
"content": "Determine whether there exists a random variable $Z$ such that the sequence of random variables $(X_n)_{n \\in \\mathbb{N}}$, defined by \\(P(X_n = 2^n) = \\frac{1}{2^n}\\) and \\(P(X_n = 0) = 1-\\frac{1}{2^n}\\) for all \\(n \\in \\mathbb{N}\\), converges to $Z$ in distribution, in probability, a... | 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... | 9.30e-5 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... | 5212.05 N |
[
{
"content": "Evaluate the integral \\( W = \\int_{{\\textstyle\\frac{\\pi}{2}} + i \\log(2)}^{{\\textstyle\\frac{\\pi}{2}} + i \\log (5)} \\frac{\\mathrm{d}z}{\\cos(x) \\cosh(y) + i \\sin(x) \\sinh(y)} \\).",
"role": "user"
}
] | \ln 2 |
[
{
"content": "Let $f(x) \\in F[x]$ be a polynomial over a field $F$. Let $K$ be the splitting field of $f(x)$ over $F$, and let $G(K/F)$ be the Galois group of $K$ over $F$. The roots of $f(x)$ are $A = \\{\\alpha_1, \\alpha_2, \\ldots, \\alpha_n\\}$. Is it true that for all $\\sigma \\in G(K/F)$, $\\sigma$ is ... | Yes |
[
{
"content": "Let $H$ be a Hilbert space, and let $H_n$ be an increasing sequence of finite-dimensional subspaces such that $\\bigcup _{n=1}^{\\infty} H_n$ is dense in $H$. Suppose $P_n$ is the projection from $H$ to $H_n$ and $T:H\\to H$ is a bounded linear operator such that $||TP_n-P_nT||\\leq \\frac{1}{2^{n... | T = S + K |
[
{
"content": "Suppose \\( f \\) is an entire function such that \\( f(\\sqrt{n}) = 0 \\) for \\( n = 1, 2, 3, \\ldots \\), and there exists a positive constant \\( \\alpha \\) such that \\( |f(z)| < \\exp(|z|^\\alpha) \\) for all sufficiently large \\( |z| \\). Determine the values of \\( \\alpha \\) for which ... | \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... | 28 |
[
{
"content": "Let $f$ be a complex-valued bounded function on a locally compact group $G$. The function $f$ is called right uniformly continuous if for every $\\varepsilon > 0$, there exists an $e$-neighborhood $U$ such that for all $x, x' \\in G$, if $x(x')^{-1} \\in U$, then $|f(x) - f(x')| < \\varepsilon$. I... | Yes |
[
{
"content": "Evaluate the limit of the integral \\( \\int_x^{3x}\\frac{1}{\\arctan t}dt \\) as \\( x \\rightarrow +\\infty \\).",
"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... | 66 g |
[
{
"content": "What is the maximum number of days that 289 students can be divided into 17 groups of 17, such that no two students are in the same group more than once?",
"role": "user"
}
] | 18 |
[
{
"content": "Three cards are dealt at random from a standard deck (13 clubs, 13 diamonds, 13 hearts, and 13 spades) of 52 cards. What is the probability that all three cards are $\\heartsuit$s?",
"role": "user"
}
] | \frac{11}{850} |
[
{
"content": "Find the residue of the function \\( f(z) = \\frac{\\cos(z^2)}{z^7} \\) at \\( z = 0 \\) by determining its Laurent series expansion in the region \\( 0 < |z| < \\infty \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the definite integral: $$\\int_0^{3\\pi} \\frac{1}{\\sin^4 x + \\cos^4 x} \\, dx$$",
"role": "user"
}
] | 3\pi\sqrt{2} |
[
{
"content": "Find the sum to infinity of the series:\n\n$$\\frac{1}{2} + \\frac{1}{4}+ \\frac{2}{8} + \\frac{3}{16} + \\frac{5}{32} +\\frac{8}{64} + \\frac{13}{128} +\\frac{21}{256} +\\frac{34}{512}+ \\frac{55}{1024} + \\cdots$$",
"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... | 4.37e21 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... | 10,417 units |
[
{
"content": "Compute the line integral \\( \\int_{C}(x^2+y)dx + (z+x)dy + (x+2y)dz \\) where \\( C \\) is the intersection of the cylinder \\( x^2+y^2=4 \\) and the plane \\( x+y=z \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Let $n > 1$ be an odd number. Let $A$ be an $n \\times n$ matrix defined as follows:\n\\[\na_{i, j} = \\begin{cases} 1, & \\text{for}\\ i - j \\equiv \\pm 2 \\pmod n\\\\ 2, & \\text{for}\\ i = j\\\\ 0, & \\text{otherwise}.\\end{cases}\n\\]\nCalculate the determinant of matrix $A$. \\( \\boxed{?} ... | 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... | 5*10^6 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... | $6.892 |
[
{
"content": "Calculate the integral of \\( f(z) = \\frac{z^2}{e^{\\frac{1}{(z+i)^2}}} \\) over the contour \\(|z|=2\\).",
"role": "user"
}
] | -4\pi |
[
{
"content": "Is it true that if a function $f:[0,1]\\to\\mathbb{R}$ is in $L^1(0,1)$ and satisfies \\( \\int_0^t f(s) \\, ds = 0 \\) for all \\( t \\in [0,1] \\), then \\( f(t) = 0 \\) almost everywhere?",
"role": "user"
}
] | Yes |
[
{
"content": "In a convex quadrilateral, let points P, Q, R, and S be the midpoints of the sides AB, BC, CD, and DA, respectively. The segments PR and QS intersect and divide each other into three parts. If the ratio of the length of the middle part to the length of the entire segment is the same for both segme... | false |
[
{
"content": "Evaluate the integral \\( \\int \\int (x-y)e^{x^2-y^2} \\, dx \\, dy \\) over the region \\( R \\), which is bounded by the lines \\( x+y=1 \\) and \\( x+y=3 \\) and the curves \\( x^2 - y^2 =1 \\) and \\( x^2 - y^2 = -1 \\), using the change of variables \\( u=x-y \\) and \\( v=x+y \\). Find the ... | \dfrac{2}{3e} |
[
{
"content": "What is the 1000th digit to the right of the decimal point when $\\frac{4}{15}$ is expressed as a decimal?",
"role": "user"
}
] | 6 |
[
{
"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... | 0.107e-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... | 2.58e-6 M |
[
{
"content": "Compute the result of the operation $1*2*3*\\ldots*25$ in the monoid $(\\mathbb{N},*)$, where $x*y = xy + x + y$, and find this result modulo 29.",
"role": "user"
}
] | 13 |
[
{
"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... | Particulars Amount ($) % ASSETS Current assets Cash and cash equivalent $14968 53% Accounts receivable $450 2% Inventories $1836 6% Total current assets $17254 61% Property & equipment, net $6060 21% Long-term investments $5186 18% Total assets $28500 100% LIABILITIES Current liabilities Accounts payable $866 3% Accrued liabilities $5304 19% Total current liabilities $6170 22% Long-term liabilities $2430 8% Total liabilities $8600 30% Total stockholders' equity $19900 70% Total liabilities & stockholders' equity $28500 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... | 1.284 kg |
[
{
"content": "Find the order of the multiplicative group of the field \\( \\mathbb{Z}_3[x] / (x^4 + x^2 + 2) \\).",
"role": "user"
}
] | 80 |
[
{
"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... | 133.5 N |
[
{
"content": "The least common multiple of two numbers is 3720, and their greatest common divisor is 8. Given that one of the numbers is 120, what is the other number?",
"role": "user"
}
] | 248 |
[
{
"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... | Easter Bummy Corporation\nBalance Sheet\nDecember 31, 2012\n\nCurrent Assets:\nCash 35,000\nInventory 102,000\nPrepaid Advertising 5,000\nSupplies 1,860\nTotal Current Assets 143,860\n\nLong-Term Assets:\nLand 137,320\nBuildings 80,400\nAccumulated Depr. - Bldg 15,000\nNet Book Value of Buildings 65,400\nEquipment 50,000\nLess: Accumulated Depreciation - Equip 10,000\nNet Book Value of Equipment 40,000\nTotal Long-Term Assets 242,720\n\nTotal Assets 386,580\n\nCurrent Liabilities:\nNotes Payable (due in 6 months) 19,400\nInterest payable 600\nTaxes payable 3,000\nSalaries and Wages payable 900\nTotal Current Liabilities 23,900\n\nLong-Term Liabilities:\nBonds Payable 78,000\n\nTotal Liabilities 101,900\n\nStockholders' Equity:\nCommon Stock 60,000\nRetained Earnings 224,680\nTotal Stockholders' Equity 284,680\n\nTotal Liabilities and Stockholders' Equity 386,580 |
[
{
"content": "A park contains two adjacent equilateral triangles formed by five straight paths, each of length \\(a\\). Additionally, six shorter paths, each of length \\(b\\), connect the centers of these triangles to their vertices. Determine the shortest route a watering machine must take to travel along all... | 6a + 8b |
[
{
"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... | 32.88 |
[
{
"content": "If the digits represented by $A$ and $B$ satisfy the following subtraction problem $$ \\begin{array}{c@{}c@{\\;}c@{}c@{}c@{}c} & & & & A & B_4\\\\ &- & & & B & A_4\\\\ \\cline{2-6} & & & & & 3_4\\\\ \\end{array} $$what is the nonnegative difference of the digits represented by $A$ and $B$?",
"... | 1 |
[
{
"content": "Let $B$ be an operator on a Hilbert space $H$. Suppose there exists a sequence $(y_{n})$ of unit vectors such that $ \\langle x_{n} , y_{n} \\rangle=0 $ for all $n$ and $$ \\lim_{n}\\vert \\langle By_{n},y_{n} \\rangle\\vert= w(B).$$ Does it follow that $$ \\lim_{n}\\vert \\langle (A+B)y_{n},y_{n}... | No |
[
{
"content": "Consider the equation $f(x)=0$, where $f:\\mathbb{R} \\to \\mathbb{R}$ is a function composed using the operations $+, -, \\times, \\div, \\sqrt{},$ and integer constants $n \\in \\mathbb{Z}$. If $f$ has only finitely many solutions, are all the solutions necessarily algebraic?",
"role": "user... | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Thr-Thr-Arg-Val |
[
{
"content": "Find the measure in degrees of the angle \\( \\delta \\) given by:\n\\[\n\\delta = \\arccos \\left( \\left( \\sin 3271^{\\circ} + \\sin 3272^{\\circ} + \\cdots + \\sin 6871^{\\circ} \\right)^{\\cos 3240^{\\circ}} + \\cos 3241^{\\circ} + \\cdots + \\cos 6840^{\\circ} \\right)\n\\]",
"role": "us... | 59 |
[
{
"content": "Evaluate the limit using L'Hospital's rule: $$\\lim_{m \\rightarrow \\infty} \\frac{\\W\\left( \\tfrac{1}{m} e^{\\tfrac{1}{m}}\\right)}{\\tfrac{1}{m}}$$ where $\\W$ denotes the Lambert W function.",
"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... | 8.5 C |
[
{
"content": "Determine whether the expression \\(2^n5^{2n+1}+3^{2n+1}7^{n+1}\\) is divisible by 13 for all integers \\(n\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether a continuous, nonnegative, and nonconstant function $y = f(x)$ can have a nonzero derivative at a point $x_0$ where $y(x_0) = 0$. Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"content": "Evaluate the limit \\( \\lim_{n \\rightarrow \\infty} \\frac{\\ln(n!)}{n^n} \\) using the squeeze theorem.",
"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... | 18.94 kW/K |
[
{
"content": "Determine whether the polynomial $xy$ can be constructed solely from the function $s(x) = x^2$ and simpler operations like addition or negation, without using any higher-order operations like division or multiplication, within the set $P$ of integer polynomials defined as follows:\n\n1. $x$ and $y... | xy \notin P |
[
{
"content": "For how many integer values of $n$ between 1 and 120 inclusive does the decimal representation of $\\frac{n}{120}$ terminate?",
"role": "user"
}
] | 40 |
[
{
"content": "Suppose $PABCD$ is a right square pyramid with apex $P$ and base $ABCD$. If $PBD$ is an equilateral triangle with side length 6, then what is the volume of $PABCD$?",
"role": "user"
}
] | 18\sqrt{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... | $4,461.40 |
[
{
"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... | 0.60 |
[
{
"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... | 0.201 |
[
{
"content": "Determine whether there exists a constant \\( C \\) such that for the sigmoid function \\( \\sigma(x) = \\frac{1}{1+e^{-x}} \\), the inequality \\( \\left| \\frac{\\sigma^{(n)}(x)}{n!} \\right| \\leq C \\) holds for any \\( x \\in [-1/2, 1/2] \\) and any \\( n \\geq 1 \\).",
"role": "user"
}... | Yes |
[
{
"content": "How many ordered triples of rational numbers \\((e, r, t)\\) are there such that the polynomial \\(g(x) = x^3 + ex^2 + rx + t\\) has roots \\(e, r,\\) and \\(t\\)?",
"role": "user"
}
] | 3 |
[
{
"content": "Find the smallest constant $K$ such that for all positive real numbers $x$, the inequality \\( \\sqrt{(x^2 + ax)} - \\sqrt{(x^2 + bx)} < K \\) holds, given that $a$ and $b$ are positive real numbers with $a - b = 10$. What is the value of $K$?",
"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... | 35.9 N |
[
{
"content": "Consider the function \\( f(x) = \\sin^{49}x \\cos^{50}x \\). Determine if there exist any pairs of distinct values \\( x, x' \\) in the interval \\( (0, \\pi/2) \\) such that \\( f(x) = f(x') \\).",
"role": "user"
}
] | Yes |
[
{
"content": "Given that \\( f(x) \\) is a polynomial of degree 3, and its remainders are \\( 2x-5 \\) and \\( -3x+4 \\) when divided by \\( x^2 - 1 \\) and \\( x^2 - 4 \\) respectively, find the value of \\( f(-3) \\).",
"role": "user"
}
] | 53 |
[
{
"content": "Suppose we have a sequence of functions $f_n$ defined on a set $S$ that converges uniformly to a function $f$. If $x_n$ is any sequence in the interval, does $f_n(x_n) - f(x_n)$ converge to zero?",
"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... | 1.22 |
[
{
"content": "When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $abc = 180$.",
"role": "user"
}
] | \frac1{72} |
[
{
"content": "Suppose that $g(x)=f^{-1}(x)$. If $g(-15)=0$, $g(0)=3$, $g(3)=9$ and $g(9)=20$, what is $f(f(9))$?",
"role": "user"
}
] | 0 |
[
{
"content": "Let \\( g : \\mathbb{R}^{3} \\to \\mathbb{R}^{3} \\) be a function defined by \\( g(x,y,z) = (3y+4z, 2x-3z, x+3y) \\). Define the set \\( S = \\{(x,y,z) \\in \\mathbb{R}^{3} : 0 \\leq x \\leq 1, 0 \\leq y \\leq 1, 0 \\leq z \\leq 1 \\} \\).\n\nIf \\[ \\int\\int\\int_{g(S)} (2x+y-2z) \\, dx \\, dy ... | 75 |
[
{
"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... | [H2] = 0 M, [Br2] = 0.01 M, [HBr] = 0.02 M |
[
{
"content": "Let $\\ell_1$ and $\\ell_2$ be two lines passing through the point $M_0 = (1,1,0)$ that lie on the hyperboloid $x^2 + y^2 - 3z^2 = 2$. Calculate the cosine of the angle between the two lines.",
"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... | 1.1367kJ/kg |
[
{
"content": "Given a matrix $A$ with a null trace, what is the minimum number of linear measurements required to determine $A$? A linear measurement is defined as knowing the quantities $A\\vec{x}_i \\cdot \\vec{h}_j$ for given vectors $\\vec{x}_i$ and $\\vec{h}_j$.",
"role": "user"
}
] | n^2 - 1 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\sum_{k=n}^{2n} \\frac{1}{n+\\sqrt{k}} \\]",
"role": "user"
}
] | 1 |
[
{
"content": "What is the greatest number of points of intersection that can occur when $2$ different circles and $2$ different straight lines are drawn on the same piece of paper?",
"role": "user"
}
] | 11 |
[
{
"content": "Determine whether the polar decomposition theorem is valid for non-invertible matrices.",
"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... | 355081 |
[
{
"content": "Determine the number of distinct circular permutations that can be formed by selecting 6 elements, with repetition allowed, from the set \\( A = \\{a, b, c\\} \\).",
"role": "user"
}
] | 130 |
[
{
"content": "Evaluate the limit \\( \\lim _{x\\to e}\\left(\\frac{1-\\log _e\\left(x\\right)}{x-e}\\right) \\) without using l'Hôpital's Rule.",
"role": "user"
}
] | -\dfrac{1}{e} |
[
{
"content": "Find the 6th derivative of the function \\( f(x) = \\frac{\\cos(5x^2) - 1}{x^2} \\) at \\( x = 0 \\).",
"role": "user"
}
] | 18750 |
[
{
"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... | 0.746 kW/K |
[
{
"content": "In a $T_1$-space, if every countably infinite subset of a non-compact set $A$ has an accumulation point in $A$, does this imply that the space is countably compact?",
"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... | 1876.135 N/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... | 126500 W |
[
{
"content": "Evaluate the definite integral: $$\\lim_{x \\to +\\infty} \\sinh^{-1}(C \\sinh (ax)) - \\lim_{x \\to -\\infty} \\sinh^{-1}(C \\sinh (ax))$$ where $C$ and $a$ are real positive constants.",
"role": "user"
}
] | 2 \ln C |
[
{
"content": "Four dogs start at the corners of a square $ABCD$ with side length $30$ meters, running anti-clockwise. Dog $A$ chases dog $B$, dog $B$ chases dog $C$, dog $C$ chases dog $D$, and dog $D$ chases dog $A$. Each dog runs at a constant speed of $7$ meters per second. The pursuit stops when a dog reach... | \dfrac{30}{7} |
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