prompt list | ground_truth string |
|---|---|
[
{
"content": "Given \\( y = \\int_0^x f(t)\\sin[k(x-t)]dt \\), calculate \\( \\frac{d^2y}{dx^2} + k^2y \\).",
"role": "user"
}
] | k f(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... | 3930 |
[
{
"content": "Determine whether the following statement is true for a polynomial $f(x)$ over the finite field $GF[q]$, where $q = p^m$ and $p$ is a prime number: $$f^{p^m}(X) = f(X^{p^m}).$$",
"role": "user"
}
] | True |
[
{
"content": "In a country with 110 cities, each pair of cities may or may not have a road connecting them. A driver starts in a city with exactly one road leading out of it. He travels along this road to a second city, which has exactly two roads leading out of it. Continuing this pattern, he travels to a thir... | 55 |
[
{
"content": "Let \\(f(x)\\) and \\(g(x)\\) be two integrable functions on \\([1,3]\\) satisfying\n\n\\[\\left( \\int_1^3 f(x) g(x) \\, dx \\right)^2 = \\left( \\int_1^3 f^2(x) \\, dx \\right) \\left( \\int_1^3 g^2(x) \\, dx \\right)\\]\n\nGiven that \\(f(1)=2\\) and \\(g(1)=4\\), find the value of \\(\\frac{g(... | 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... | 7.51e-2 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... | Ca(OH)_2 + H_2 |
[
{
"content": "Determine the number of 10-digit numbers that can be formed using the digits from the set \\(\\{1, 2, 3, 4\\}\\) such that no two adjacent digits differ by 1. Use a minimal number of recurrence relations to find your answer.",
"role": "user"
}
] | 21892 |
[
{
"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 Flow from Operating Activities:
Net income: $280,000
Depreciation: $180,000
Increase in accounts receivables: ($15,000)
Increase in inventories: ($96,000)
Decrease in prepaid insurance: $1,500
Decrease in accounts payable: ($331,500)
Decrease in salaries payable: ($30,000)
Net cash used in operating activities: ($461,000)
Cash Flow from Investing Activities:
Purchase of equipment: ($1,123,500)
Sale of equipment: $18,000
Net cash used in investing activities: ($1,105,500)
Cash Flow from Financing Activities:
Issue of note payable: $1,500,000
Dividend paid: ($90,000)
Net cash flow from financing activities: $1,410,000
Net cash increase/(Decrease) in cash: ($156,500)
Add: Beginning cash and cash equivalent balance: $675,000
Ending cash and cash equivalents balance: $518,500 |
[
{
"content": "Consider a fair die that is rolled repeatedly. Let $X_n$ represent the sum of the first $n$ rolls. Determine the limit \\( \\lim_{{n\\rightarrow \\infty}}P(X_n \\text{ is a multiple of 13}) \\).",
"role": "user"
}
] | \dfrac{1}{13} |
[
{
"content": "Determine the dimension of the space of \\( n \\times n \\) matrices with non-zero trace, given that the dimension of the space of all \\( n \\times n \\) matrices is \\( n^2 \\) and the dimension of the space of \\( n \\times n \\) matrices with trace zero is \\( n^2 - 1 \\).",
"role": "user"... | n^2 |
[
{
"content": "Evaluate the line integral \\( \\oint_C \\mathbf{F} \\cdot d\\mathbf{r} \\) using Stokes' Theorem, where \\( S \\) is the surface defined by the graph of \\( f(x,y) = 4x - 8y + 30 \\) over the rectangle \\( R = \\{ (x,y) \\mid -2 < x < 3, 0 < y < 2 \\} \\). The vector field is given by \\( \\mathb... | 160 |
[
{
"content": "Determine the radius of convergence of the power series: $$\\sum_{k = 0}^{\\infty}\\frac{ k^{3 k - 2} \\log^{8} k}{\\left((k+1)!\\right)^2} \\,x^k$$",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the algebraic structure \\((N,+)\\), where \\(N\\) is the set of natural numbers defined by the Peano axioms, is an ordered commutative monoid.",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\( \\mathcal{F} \\) be the set of continuous functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) such that \\( e^{f(x)}+f(x) \\geq x+1 \\) for all \\( x \\in \\mathbb{R} \\). For \\( f \\in \\mathcal{F}, \\) define \\( I(f)=\\int_0^e f(x) \\, dx \\). Determine \\( \\min_{f \\in \\mathcal{F}} I(... | \dfrac{3}{2} |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\to 0} \\frac{\\sin x -\\arctan x}{x^2\\log(1+x)}$$",
"role": "user"
}
] | \dfrac{1}{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... | cal/g degree C |
[
{
"content": "In a game with two players, each player selects a number between 0 and b (inclusive). The payoff for each player is determined by the function \\( Z = u(x_1 + x_2) + v(2b - x_1 - x_2) \\), where \\( u \\) and \\( v \\) are strictly concave functions. Derive the slope of the best response function ... | -1 |
[
{
"content": "Evaluate the limit \\( \\lim\\limits_{n\\to\\infty}\\left(\\frac{1}{n+1}+\\frac{1}{n+2}+\\cdots+\\frac{1}{2n}\\right) \\) using the concept of Riemann sums and the integral \\( \\int_1^2\\frac{1}{x}dx \\).",
"role": "user"
}
] | \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... | P = MC |
[
{
"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... | 17.592 m/s |
[
{
"content": "Find all values of \\(a\\) such that the domain of \\(f(x) = \\sqrt{\\frac{2a+x}{a-x}}\\) includes the range of \\(g(x) = \\frac{1}{x^2 + 2x + 4a - 2}\\). The domain of \\(f(x)\\) is \\(x \\in [-2a, a)\\) for \\(a > 0\\) or \\(x \\in (a, -2a]\\) for \\(a < 0\\).",
"role": "user"
}
] | (1, \infty) |
[
{
"content": "Determine the range of the positive integer $w$ such that there exist $a, b, c$ with $\\pi \\le a < b < c \\le 2\\pi$ satisfying the equation $\\sin(aw) + \\sin(bw) + \\sin(cw) = 3$. Provide your answer as an interval or set of values for $w$.",
"role": "user"
}
] | \{6, 7, 8, \ldots\} |
[
{
"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... | 2.7e-6 M |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to 0} {x\\sqrt{y^2-(y-x)^2}\\over (\\sqrt{8xy-4x^2}-\\sqrt{8xy})^3}$$",
"role": "user"
}
] | -\infty |
[
{
"content": "Evaluate the limit \\( \\lim_{z \\rightarrow 0} z\\sin\\left(\\frac{1}{z}\\right) \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the quartic polynomial \\( P(x) = x^4 - 6x^3 + 9x^2 - 3x \\) can take on rational square values for infinitely many rational values of \\( x \\).",
"role": "user"
}
] | Yes |
[
{
"content": "Given that $a + b + c = 0$, find the value of the expression \\[ \\left(\\frac{b-c}{a}+\\frac{c-a}{b}+\\frac{a-b}{c}\\right)\\left(\\frac{a}{b-c}+\\frac{b}{c-a}+\\frac{c}{a-b}\\right). \\]",
"role": "user"
}
] | 9 |
[
{
"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... | 0.3333 |
[
{
"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... | $160,000 |
[
{
"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... | 7.90 years |
[
{
"content": "Consider the open unit disc $D$ in the complex plane. Let $g:D \\rightarrow D$ be a holomorphic function with $g(0) = 1/2$. If $h(z)$ is an automorphism of $D$ that maps 1/2 to 0, determine the maximum possible value of $|g(h^{-1}(1/3))|$. Provide your answer as a single numerical value.",
"ro... | \dfrac{17}{19} |
[
{
"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... | {"P1": 2, "P2": 1, "XA1": 10, "XA2": 5, "XB1": 10, "XB2": 10} |
[
{
"content": "Evaluate the integral \\( \\int_0^\\infty x^2 e^{-2x} \\, dx \\) using the Gamma function.",
"role": "user"
}
] | \dfrac{1}{4} |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 1} \\frac{\\sin^a{\\pi x}}{(1-x)^a} , \\quad a \\in \\mathbb{R} \\]",
"role": "user"
}
] | \pi^a |
[
{
"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... | 47.8 kg |
[
{
"content": "Let $G$ be a finite group of order $n$, and let $k$ be a positive integer. Suppose $G = \\{x^k : x \\in G\\}$. Must it be true that the greatest common divisor of $n$ and $k$ is 1?",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.308 |
[
{
"content": "Simplify the expression: \\[ a^{\\frac{2}{\\log_{b} a}+1} \\cdot b - 2 a^{\\log_{a} b+1} \\cdot b^{\\log_{b} a+1} + a b^{\\frac{2}{\\log_{a} b}+1} \\]",
"role": "user"
}
] | a b (a - b)^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... | 90.6 N/m |
[
{
"content": "Evaluate the improper integral: \\[ \\int_0^\\pi \\frac{\\sin x}{\\sqrt{|\\cos x|}} \\, dx \\]",
"role": "user"
}
] | 4 |
[
{
"content": "Determine all possible values of the positive constant \\( k \\) such that the series \\( \\sum_{n=1}^{\\infty}\\ln\\left(1+\\frac{1}{n^k}\\right) \\) is convergent.",
"role": "user"
}
] | k > 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... | $12,927,200.52 |
[
{
"content": "Determine the cardinality of the set of all subsets of the integers \\( \\mathbb{Z} \\) that have exactly 3 elements.",
"role": "user"
}
] | \aleph_0 |
[
{
"content": "Determine the limit of the sequence \\((a_n)\\) as \\(n\\) approaches infinity, where \\(a_n = \\frac{\\sqrt[3n]{4} - 6\\sqrt[3n]{2} + 9}{\\sqrt[2n]{9} - 4\\sqrt[2n]{3} + 4}\\).",
"role": "user"
}
] | 4 |
[
{
"content": "Does the triple $(j_\\ast A_X^\\ast,j_\\ast F^\\bullet,\\tau^{\\leq\\bullet} j_\\ast A_X^\\ast)$ induce the canonical mixed Hodge structure on $H^\\ast(X)\\simeq H^\\ast(\\bar{X},j_\\ast A_X^\\ast)$?",
"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... | Bacteria can use pumps to regulate electrolytes and water. Gram-positive bacteria have rigid cell walls to maintain cell shape. Eukaryotic cells like amoeba use vacuoles to expel excess water and wastes. |
[
{
"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... | $852,930.00 |
[
{
"content": "Calculate the probability $P(X_1 < X_2 < X_3)$ where $X_1, X_2, X_3$ are independent and identically distributed random variables following an exponential distribution with rate $\\beta$. Express your answer as a single probability value.",
"role": "user"
}
] | \dfrac{1}{6} |
[
{
"content": "In a cyclic quadrilateral $PQRS$, the sides are given as $PQ = RS = x$, $QR = 2x$, and $PS = 1$. Determine the value of $x$.",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Suppose $a_n > 0$, $\\lim\\limits_{n\\to\\infty} a_n b_n = -1$, and $\\lim\\limits_{n\\to\\infty} a_n = 0$. Determine if $\\lim\\limits_{n\\to\\infty} b_n = -\\infty$.",
"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... | 1.08e-11 m |
[
{
"content": "True or false? In any inner product space $V$, if $u, v \\in V$ satisfy $\\|u\\| = 5$, $\\|v\\| = 3$, and $|\\langle u, v \\rangle| = 16$, then the Cauchy-Schwarz inequality is violated.",
"role": "user"
}
] | A |
[
{
"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... | 236 m/s |
[
{
"content": "Determine the value of \\( c \\) that leads to the fastest convergence of the sequence \\( D_n(c) = \\frac {4^n}{\\binom {2n}{n} \\sqrt{4n + c}} \\) to \\( \\frac{\\sqrt{\\pi}}{2} \\) as \\( n \\to \\infty \\).",
"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... | 337.56 g |
[
{
"content": "Is every open set in $\\mathbb{R}$ an $F_{\\sigma}$ set?",
"role": "user"
}
] | Yes |
[
{
"content": "What is the sum of the digits of the decimal representation of $2^{2005} \\times 5^{2007} \\times 3$?",
"role": "user"
}
] | 12 |
[
{
"content": "Determine if a simple group $G$ of order $3420 = 2^2 \\cdot 3^2 \\cdot 5 \\cdot 19$ has an element of order $855 = 3 \\cdot 5 \\cdot 19$. Use Sylow's Theorems to justify your answer.",
"role": "user"
}
] | No |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 9.3 s |
[
{
"content": "Determine the radius of convergence of the power series \\( \\sum_{n=0}^\\infty a_n x^n \\), where the coefficients \\( a_n \\) are defined by the recurrence relation:\n\\[\n\\begin{aligned}\na_{n+2} &= \\frac{n(n+1) a_{n+1} - a_n}{(n+2)(n+1)}, \\quad n \\ge 1, \\\\\na_2 &= -\\frac{a_0}{2}\n\\end{... | 1 |
[
{
"content": "Given a linearly independent set of vectors $\\{v_1, v_2, \\ldots, v_k\\}$ with $\\|v_i\\| \\leq 1$ for all $1 \\leq i \\leq k$, determine if it is true that for any real numbers $\\alpha_1, \\alpha_2, \\ldots, \\alpha_k$ satisfying $\\|\\sum_{i=1}^k \\alpha_i v_i\\| = 1$, there exists a constant ... | True |
[
{
"content": "In triangle \\( \\triangle ABC \\), let \\( I \\) be the midpoint of side \\( [AB] \\). Consider point \\( J \\) on segment \\( [BC] \\) such that \\( BJ = 2JC \\) and \\( \\angle AIC = \\angle BAJ \\). Find the measure of angle \\( \\angle BAC \\).",
"role": "user"
}
] | 90^\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... | 418 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... | Attitude |
[
{
"content": "Let \\(f:\\mathbb{R} \\to \\mathbb{R}\\) be a differentiable function such that \\(f(x)=f(x/2)+\\dfrac{x}{2}f'(x)\\). Find the value of \\(f''(16)-f''(8)+2f''(4)\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Let $\\Omega = S^1$ be the unit circle in $\\mathbb{C}$, and let $T : \\Omega \\to \\Omega$ be the multiplication by $e^{i\\alpha}$. If $\\alpha \\notin \\pi\\mathbb{Q}$, is the set $T^{\\mathbb{N}}x$ dense in $\\Omega$ for every $x \\in \\Omega$? Provide a justification for your answer.",
"ro... | Yes |
[
{
"content": "Given that $z \\in \\mathbb{C}$ satisfies the equation $\\frac{\\sqrt{2}}{z+i}=-|z|$, find the value of $2z-\\bar{z}$.",
"role": "user"
}
] | -1 - 3i |
[
{
"content": "Evaluate the integral \\( \\int_0^{2\\pi}\\frac{\\cos(2\\theta)}{5+4\\cos(\\theta)}\\, d\\theta \\) using the residue theorem.",
"role": "user"
}
] | \dfrac{\pi}{6} |
[
{
"content": "Evaluate the double integral \\( \\iint_D \\sqrt{x} \\ dx \\ dy \\) where \\( D = \\{(x,y) \\mid x^2+y^2 < x\\} \\) using the change of variables to polar coordinates.",
"role": "user"
}
] | \dfrac{8}{15} |
[
{
"content": "Find the values of $k$ such that the quadratic form \\( Q(x_1,x_2) = (4+k)x_1^2 + 2kx_1x_2 + 2x_2^2 \\) is positive semi-definite.",
"role": "user"
}
] | [-2, 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... | 2.66V_o |
[
{
"content": "Determine the points of differentiability of the function \\( f:(0,2)\\rightarrow \\mathbb{R} \\) defined by \\( f(x)=x^2 \\) if \\( x\\in (0,2)\\cap \\mathbb{Q} \\) and \\( f(x)=2x-1 \\) if \\( x\\in (0,2)- \\mathbb{Q} \\). Evaluate the derivative of \\( f \\) at those points.",
"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... | 3.1 K |
[
{
"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 m/s, west |
[
{
"content": "Let $X_1, X_2, X_3, X_4, X_5$ be a random sample from a standard normal distribution $N(0,1)$. Define $T = \\sum_{i=1}^5 (X_i - \\bar{X})^2$, where $\\bar{X}$ is the sample mean. Calculate $E(T^2 \\bar{X}^2)$. Provide your answer as a single value.",
"role": "user"
}
] | \dfrac{24}{5} |
[
{
"content": "Derive the Taylor expansion of \\( f(x) = \\arctan{\\frac{1-x}{1+x}} \\) and determine its radius of convergence.",
"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... | prospective |
[
{
"content": "Determine the dimension of the tangent space at a point where multiple circles intersect in \\(\\mathbb{R}^2\\).",
"role": "user"
}
] | 2 |
[
{
"content": "Calculate the limit: \\[ \\lim_{n \\to \\infty} \\frac{1}{n^5}(1^4 + 2^4 + \\cdots + n^4). \\]",
"role": "user"
}
] | \dfrac{1}{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... | 0.469 meters |
[
{
"content": "Evaluate the integral \\( \\int_0^1 \\sqrt{1+x^4} \\hspace{1mm} dx \\) using series expansion, and provide the answer correct to two decimal places.",
"role": "user"
}
] | 1.09 |
[
{
"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... | 1.89e8 m/s |
[
{
"content": "Evaluate the limit \\( \\lim _{n\\to \\infty }\\left(\\frac{\\sqrt[n]{n!}}{n}\\right) \\) using Cauchy-d’Alembert's criterion or Stirling's approximation.",
"role": "user"
}
] | \dfrac{1}{e} |
[
{
"content": "Let \\( P(x) \\) and \\( Q(x) \\) be distinct polynomials of degree 2020 with non-zero coefficients. Suppose they have \\( r \\) common real roots (counting multiplicity) and \\( s \\) common coefficients. Determine the maximum possible value of \\( r + s \\).",
"role": "user"
}
] | 2020 |
[
{
"content": "Determine the rank of the matrix $I + A^T A$, where $A$ is an $m \\times n$ matrix with non-negative eigenvalues.",
"role": "user"
}
] | 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... | linear |
[
{
"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... | 23.1 mm |
[
{
"content": "Evaluate whether \\( \\lim_{x\\to\\infty}x\\log(f(x))=\\infty \\) given that \\( f(x)=1+\\frac{\\log x}{x}+o\\left(\\frac{\\log x}{x}\\right), x\\to\\infty. \\)",
"role": "user"
}
] | \infty |
[
{
"content": "In a circle, there are 103 numbers. Among any five consecutive numbers, there are at least two positive numbers. What is the minimum number of positive numbers that can be present among these 103 numbers?",
"role": "user"
}
] | 42 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\int_{-\\infty}^\\infty \\frac{1}{1+x^2e^{nx^2}}\\,dx. \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Find the ones digit of the largest power of $2$ that divides into $(2^4)!$.",
"role": "user"
}
] | 8 |
[
{
"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... | {
"Finished inventory": 510814,
"Labor rate variance": 39720,
"Labor efficiency variance": 26230,
"Wage payable": 444864
} |
[
{
"content": "Find the limit of the function \\( f(x) = x\\left(\\left(1+\\frac{1}{x}\\right)^{x}-e\\right) \\) as \\( x \\to \\infty \\).",
"role": "user"
}
] | -\dfrac{e}{2} |
[
{
"content": "Let $z_1, z_2, \\ldots, z_n$ be complex numbers such that $|z_1| = |z_2| = \\ldots = |z_n| = 1$. Determine whether the following inequality holds: \\[ \\limsup_{k \\in \\mathbb{N}} \\sqrt[k]{|z_1^k + z_2^k + \\cdots + z_n^k|} \\ge 1. \\]",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $174,890 |
[
{
"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... | 67542.85 |
[
{
"content": "The length of the segment between the points $(2a, a-4)$ and $(4, -1)$ is $2\\sqrt{10}$ units. What is the product of all possible values for $a$?",
"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... | 500 nm |
[
{
"content": "Solve for all values of $x$ that satisfy the equation: \\[ 2^{\\vert x - 1 \\vert} - 2^{x+1} = \\vert 2^{x+1} - 1\\vert + 1. \\]",
"role": "user"
}
] | -\dfrac{1}{2} |
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