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... | 105.35 |
[
{
"content": "Consider a continuous function \\(f(x):[0, 1] \\rightarrow \\mathbb{R}\\) that attains a maximum at exactly 3 points. Determine whether the function \\(g(f(x)) = f^3(x) - f^2(x) + 5f(x) - 14\\) also attains a maximum at exactly 3 points.",
"role": "user"
}
] | Yes |
[
{
"content": "Find a function \\( f(x) \\) that satisfies the integral equation: \\[ f(x) = \\int_0^x f(s) \\, \\mathrm{d}s \\]",
"role": "user"
}
] | 0 |
[
{
"content": "If $\\log (xy^3)= 1$ and $\\log (x^2y) = 1$, what is $\\log (xy)$?",
"role": "user"
}
] | \frac{3}{5} |
[
{
"content": "Determine the number of solutions to the equation \\( \\underbrace{\\cos(\\cos(\\ldots(\\cos x)\\ldots))}_{n\\text{ times }\\cos}=x \\) in the interval \\((0, \\frac{\\pi}{2})\\), where \\(n\\) is a positive integer.",
"role": "user"
}
] | 1 |
[
{
"content": "The polynomials\n\\[x^3 + 5x^2 + px + q = 0\\]and\n\\[x^3 + x^2 + px + r = 0\\]have exactly two roots in common, so each cubic has a root they do not have in common. Find the sum of the two roots they do not have in common.",
"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... | 1.39 m |
[
{
"content": "Calculate the conditional expectation \\( \\mathbb{E}(X^2Y + XY^2 | X - Y) \\) for independent random variables \\( X \\) and \\( Y \\) that follow \\( \\mathcal{N}(0,1) \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine if the equilibrium states of the following system of differential equations are only possible when $S=0$: \\[ \\begin{aligned} \\frac{dI}{dt} &= -bkIS \\\\ \\frac{dS}{dt} &= bkIS - akS(S+R) \\\\ \\frac{dR}{dt} &= akS(S+R) \\end{aligned} \\] given that \\( \\frac{dI}{dt} = \\frac{dS}{dt} ... | Yes |
[
{
"content": "Find the smallest possible value of \\( |2^m - 181^n| \\), where \\( m \\) and \\( n \\) are positive integers.",
"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... | 0.09 mol |
[
{
"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... | 4.04 |
[
{
"content": "Determine the number of monic irreducible quadratic polynomials over the finite field $\\Bbb{F}_{11}$.",
"role": "user"
}
] | 55 |
[
{
"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... | 9.36 L |
[
{
"content": "Find the formula of a function \\( g: \\mathbb{R} \\to \\mathbb{R} \\) that satisfies the following conditions for all \\( x, y \\in \\mathbb{R} \\): \\( g(x) \\geq x \\) and \\( g(x+y) \\geq g(x) + g(y) \\).",
"role": "user"
}
] | g(x) = x |
[
{
"content": "Determine the cardinality of the set \\(E(a)\\), where \\(E(a) = \\{(r_1, r_2, \\ldots, r_n, \\ldots) \\mid r_n \\in \\mathbb{Q} \\cap [0,1], r_n \\leq r_{n+1}, \\forall n \\geq 1, \\sum_{n=1}^\\infty \\frac{r_n}{2^n} = a\\}\\) for any \\(a \\in (0,1)\\). What is \\(|E(a)|\\)?",
"role": "user"... | 2^{\aleph_0} |
[
{
"content": "Determine whether a bounded sequence of real numbers can have infinitely many limit points. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\( k \\) be the product of every third positive integer from \\( 2 \\) to \\( 2006 \\), that is \\( k = 2\\cdot 5\\cdot 8\\cdot 11 \\cdots 2006 \\). Find the number of zeros at the right end of the decimal representation of \\( k \\).",
"role": "user"
}
] | 168 |
[
{
"content": "In an equilateral triangle \\( ABC \\) with side length 1, points \\( D \\) and \\( E \\) are located on sides \\( AB \\) and \\( AC \\) respectively. When the triangle is folded along the line segment \\( DE \\), vertex \\( A \\) lands exactly on side \\( BC \\). Determine the minimum length of \... | 2\sqrt{3} - 3 |
[
{
"content": "Find the sum of the 11th powers of all the roots of the polynomial equation \\(x^5 + 5x + 1 = 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... | 327.37 W |
[
{
"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... | $115.6 per machine hour |
[
{
"content": "Find the value of the real number \\(a\\) such that the quadratic equation \\(a(1+\\mathrm{i}) x^{2}+\\left(1+a^{2} \\mathrm{i}\\right) x+a^{2}+\\mathrm{i}=0\\) has real roots.",
"role": "user"
}
] | -1 |
[
{
"content": "Determine the greatest value of the constant \\( C \\) such that the inequality \\( |\\ln(x) - \\ln(y)| \\geq C|x - y| \\) holds for all \\( x, y \\in (0, 1] \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Determine the value of $c \\in \\mathbb{R}$ such that the series \\( s_n = \\sum_{t=1}^{n} tc^t \\) converges.",
"role": "user"
}
] | (-1, 1) |
[
{
"content": "Determine whether the following system of congruences has a solution:\n\\[ x \\equiv 9 \\pmod{11} \\]\n\\[ x \\equiv 6 \\pmod{13} \\]\n\\[ x \\equiv 6 \\pmod{12} \\]\n\\[ x \\equiv 9 \\pmod{15} \\]",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether the method described in the given text can be used to find all subgroups of a finite Abelian group $A$ that are isomorphic to a given subgroup $B$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether there exist six six-digit numbers, each consisting of the digits 1 to 6 without repetition, such that any three-digit number formed by the digits 1 to 6 without repetition can be obtained by deleting three digits from one of these six-digit numbers.",
"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... | 9650 |
[
{
"content": "Consider the one-sided limits $\\lim_{x\\to c^-} \\{ \\ln x \\}$ and $\\lim_{x\\to c^+} \\{ \\ln x \\}$, where $\\{ \\cdot \\}$ denotes the fractional part function. If both limits exist finitely but are not equal, determine the possible nature of the value $c$. Choose the correct option:\n\n(a) $... | D |
[
{
"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... | 35.37 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... | 0.15 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... | 2CCl_4 -> 2C + 4Cl_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... | 0.577 |
[
{
"content": "Let $M$ and $N$ be $n$-dimensional manifolds. Suppose $D_1, D_2, \\dots, D_k$ are $n$-dimensional manifolds embedded in $M$ such that $\\bigcup_{i=1}^k D_i = M$ and each $D_i$ is homeomorphic to $N$. Determine the minimum value of $k$. For example, if $M$ is a 2-dimensional torus and each $D_i$ is... | 3 |
[
{
"content": "Evaluate the integral: \\[ \\int_{-\\pi}^{\\pi} \\frac{x^3 \\sin x}{1+e^{x \\cos x}} \\, dx \\]",
"role": "user"
}
] | \pi^3 - 6\pi |
[
{
"content": "Find the derivative of the function \\( f(x) = x^t Ax \\) with respect to the column vector \\( x \\), where \\( x \\) is a column vector and \\( A \\) is a symmetric square matrix. Simplify your answer to \\( 2Ax \\).",
"role": "user"
}
] | 2Ax |
[
{
"content": "Find the symmetric matrix associated with the quadratic form \\( Q(X) = \\text{trace}(X^2) \\), where \\( X \\) is an \\( n \\times n \\) real matrix.",
"role": "user"
}
] | 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... | 5.50e-5 m |
[
{
"content": "Find all positive integer solutions \\((x, y)\\) to the equation \\(2x^2 - 1 = y^{15}\\).",
"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... | {eq}r(t)= < \frac{1}{9}e^{-3t}+At-\frac{1}{3}t+\frac{-1}{4}, G(t)- F(0)t- 2t+4-G(0), - \sin t + 5 t-2>{/eq} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.05c |
[
{
"content": "Let $a,$ $b,$ and $c$ be the roots of $x^3 - 7x^2 + 5x + 2 = 0.$ Find\n\\[\\frac{a}{bc + 1} + \\frac{b}{ac + 1} + \\frac{c}{ab + 1}.\\]",
"role": "user"
}
] | \frac{15}{2} |
[
{
"content": "Determine the number of distinct terms in the expansion of \\(\\left(x+\\frac{1}{x}+\\frac{1}{x^2}+x^2\\right)^{15}\\) with respect to powers of \\(x\\).",
"role": "user"
}
] | 61 |
[
{
"content": "Determine the dimension of the vector space \\( \\mathbb{C}[x,y]/I \\) over \\( \\mathbb{C} \\), where \\( I \\) is the ideal generated by the polynomials \\( y^2 - x^2 \\) and \\( y^2 + x^2 \\) in \\( \\mathbb{C}[x,y] \\).",
"role": "user"
}
] | 4 |
[
{
"content": "Determine the eigenvalues of the matrix: $$ A = \\begin{pmatrix} 1&1&1 \\\\ 1&w&w^2 \\\\ 1&w^2&w^4 \\end{pmatrix} \\begin{pmatrix} 1&0&0 \\\\ 0&-1&0 \\\\ 0&0&0 \\end{pmatrix} \\begin{pmatrix} 1&1&1 \\\\ 1&1/w&1/w^2 \\\\ 1&1/w^2&1/w^4 \\end{pmatrix} $$ where $w$ is a non-real cube root of unity. Ch... | D |
[
{
"content": "Let $X$ be a set of points in $\\mathbb{R}^2$, and define $g(X) = \\arg \\min_{y \\in \\mathbb{R}^2} \\sum_{x_i \\in X} \\| x_i - y \\|_2$ as the geometric median of $X$. If $X$ and $X'$ are neighboring sets, differing by only one element (denoted $X \\sim X'$), compute $\\sup_{X':X \\sim X'} \\| ... | \infty |
[
{
"content": "Evaluate the contour integral \\( \\oint_C \\frac{z^2+1}{\\cos^3(z)}dz \\) using the Residue Theorem, where \\( C \\) is the unit circle.",
"role": "user"
}
] | 0 |
[
{
"content": "As $n$ ranges over the positive integers, what is the sum of all possible values of the greatest common divisor of $3n+4$ and $n$?",
"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... | 8.62 m/sec |
[
{
"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... | 8.96e-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... | 0.232 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... | <img>/cimages/multimages/16/tyui4662380048177947040.png</img> |
[
{
"content": "In the Cartesian coordinate system, point \\(A\\) lies on the circle \\(x^{2} + y^{2} = 1\\), and point \\(B\\) has coordinates \\((3,0)\\). If point \\(C\\) forms an equilateral triangle \\(\\triangle ABC\\), find the maximum value of \\(|OC|\\).",
"role": "user"
}
] | 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... | 0.088 N |
[
{
"content": "Determine whether every group of even order has a normal subgroup of index 2.",
"role": "user"
}
] | No |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\to \\infty} \\frac{\\ln(3^x - 2x^2)}{x}$$",
"role": "user"
}
] | \ln 3 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | +3.37e+02 kJ |
[
{
"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... | 5.55e3 m/s |
[
{
"content": "Find the maximum value of the function \\( f(x,y,z) = xy^2z \\) subject to the constraint \\( x^2 + y^2 + z^2 = 36 \\) using Lagrange's method.",
"role": "user"
}
] | 162 |
[
{
"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... | 13.42% |
[
{
"content": "Determine the maximum number of negative eigenvalues that the matrix $AB + BA$ can have, where $A$ and $B$ are symmetric positive definite matrices in $\\mathbb{R}^{n \\times n}$. Can this number exceed $\\lfloor n/2 \\rfloor$?",
"role": "user"
}
] | \lfloor n/2 \rfloor |
[
{
"content": "Use the Chinese Remainder Theorem to find the remainder of \\(1030^{989}\\) when divided by 3003.",
"role": "user"
}
] | 1933 |
[
{
"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... | Brent Bosch:
Cash (Debit) \$60,000
Brent Bosch Capital (Credit) \$60,000
Carol Colaw:
Land (Debit) \$20,000
Building (Debit) \$80,000
Carol Colaw Capital (Credit) \$100,000
Debbie Dole:
Cash (Debit) \$11,000
Accounts Receivable (Debit) \$34,000
Equipment (Debit) \$21,000
Allowance for Doubtful Debts (Credit) \$4,000
... |
[
{
"content": "Determine the maximum repulsion degree that a circular function \\( f: \\mathbb{N} \\rightarrow \\mathbb{N} \\) can have, given that for every \\( p \\in \\mathbb{N} \\), there exists \\( n \\in \\mathbb{N} \\), \\( n \\leq p \\), such that \\( f^n(p) = p \\), and \\( f \\) has a repulsion degree ... | \dfrac{1}{2} |
[
{
"content": "Determine whether there exists a function $f(x)$ such that both $\\lim_{x\\to+\\infty}x\\left[f(x)\\right]^5=0$ and $\\lim_{x\\to+\\infty}x\\left[f(x)\\right]^3=+\\infty$ are satisfied simultaneously.",
"role": "user"
}
] | Yes |
[
{
"content": "Does the space $H(\\operatorname{div};\\Omega)$ have a Schauder basis, where $\\Omega$ is an open bounded subset of $\\mathbb{R}^n$ for $n \\in \\{2,3\\}$ and $H(\\operatorname{div};\\Omega) = \\{v \\in L^2(\\Omega) : \\operatorname{div} v \\in L^2(\\Omega)\\}$?",
"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... | 6-hydroxyhexanoic acid |
[
{
"content": "A chord \\( AB \\) of fixed length slides its ends along a circle of radius \\( R \\). A point \\( C \\) on this chord, which is at distances \\( a \\) and \\( b \\) from the ends \\( A \\) and \\( B \\) of the chord respectively, traces another circle as the chord makes a full rotation. Compute t... | \pi ab |
[
{
"content": "Let $f(x)$ be a twice differentiable function. If there exists a point $c$ such that $f''(c) = 0$, does it imply that there are no two values of $a,b \\in \\mathbb{R}$ for which $(b-a)f'(c) = f(b)-f(a)$? Answer with 'Yes' or 'No' and 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... | 1.47e6 N/C downwards |
[
{
"content": "Evaluate the integral: \\[ I = \\int_{-2}^2 4(4-x^2)^{3/2} \\, dx \\]",
"role": "user"
}
] | 24\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... | $176 |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\rightarrow -\\infty}{e^{\\frac {1}{2-x}}\\cdot\\frac{x^2+2x-1}{x-2}}-x$$",
"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... | {
"Date": "Jan. 1, 2016",
"Account Titles and Explanation": "Bonds Payable",
"Debit": 105000000,
"Credit": "",
"Remarks": "To reverse the liability account previously booked"
},
{
"Date": "Jan. 1, 2016",
"Account Titles and Explanation": "Loss on Retirement of Bonds",
"Debit": 2100000,
"Credit": "",
"Remarks... |
[
{
"content": "Given the differential equation \\( \\frac{dy}{dx} = 1 - y^2 \\), find the value of \\( y(\\ln(2)) \\) if the function \\( y \\) is symmetric about the point \\((\\ln(4), 0)\\).",
"role": "user"
}
] | -\dfrac{3}{5} |
[
{
"content": "Determine the number of groups of non-negative integers \\(b_1, b_2, \\ldots, b_m\\) such that the bitwise XOR of all the elements in the group equals a given integer \\(k\\), where \\(0 \\leq b_i \\leq 2^{20}-1\\).",
"role": "user"
}
] | 2^{20(m-1)} |
[
{
"content": "In an arithmetic progression, the sum of the first thirteen terms is 50% of the sum of the last thirteen terms. Additionally, the sum of all terms, excluding the first three, is to the sum of all terms, excluding the last three, as 6 is to 5. Determine the total number of terms in this arithmetic ... | 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... | 0.02943 N |
[
{
"content": "Calculate \\( \\lim\\limits_{x\\to\\infty}x\\left(\\int_0^x te^{-2t}\\,dt-\\frac{1}{4}\\right) \\).",
"role": "user"
}
] | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.901 g |
[
{
"content": "Does the convergence of \\(\\sum \\|f_n\\|_{\\infty}\\) imply the uniform convergence of \\(\\sum f_n\\), where \\(f_n\\) are continuous functions on a closed interval \\([a,b]\\)?",
"role": "user"
}
] | Yes |
[
{
"content": "Alex needs to catch a train. The train arrives randomly some time between 1:00 and 2:00, waits for 10 minutes, and then leaves. If Alex also arrives randomly between 1:00 and 2:00, what is the probability that the train will be there when Alex arrives?",
"role": "user"
}
] | \frac{11}{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... | 3 m/s |
[
{
"content": "Determine whether the conjunction of the following three conditions is sufficient for a set $S$ of integers to be the set of all integers $n$ for which a fixed group $G$ is an $n$-abelian group:\n\n1. $1 \\in S$.\n2. If $m,n \\in S$, then $mn \\in S$.\n3. If $n \\in S$, then $1-n \\in S$. Provide ... | Yes |
[
{
"content": "Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point $(0,0)$ and the directrix lines have the form $y=ax+b$ with $a$ and $b$ integers such that $a\\in \\{-2,-1,0,1,2\\}$ and $b\\in \\{-3,-2,-1,1,2,3\\}$. No three of these parabolas have a common point. How man... | 810 |
[
{
"content": "Let S be a set with N elements, and let \\( A_1, A_2, \\ldots, A_{101} \\) be 101 (possibly not disjoint) subsets of S with the following properties:\n\n1. Each element of S belongs to at least one subset \\( A_i \\) for \\( i \\in \\{1, 2, \\ldots, 101\\} \\).\n2. Each subset \\( A_i \\) contains... | 90900 |
[
{
"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... | 4.93 Years |
[
{
"content": "Given a linearly independent vector set $\\{u_1, u_2, u_3, u_4\\}$, find the value of $a$ such that the vector set $\\{u_1+au_2, u_2+au_3, u_3+au_4, u_4+au_1\\}$ remains linearly independent.",
"role": "user"
}
] | a \neq \pm 1 |
[
{
"content": "Determine the range of $xy$ given the conditions: $x^2 - xy + y^2 = 9$ and $|x^2 - y^2| < 9$, where $x, y \\in \\mathbb{R}^+$. Find the range of $xy$. \\( \\text{(Express your answer as an interval.)} \\)",
"role": "user"
}
] | (6, 9] |
[
{
"content": "Let \\((W_t)_t\\) be a Brownian motion with respect to \\((F_t)_t\\). Compute the expectation \\(E[(W_t^2 - t)(W_s^2 - s)]\\) for \\(0 < s < t\\).",
"role": "user"
}
] | 2s^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... | 95355 |
[
{
"content": "For the function \\( f: \\mathbb{R} \\to \\mathbb{R} \\) defined by \\( f(x) = x^r \\cdot \\sin(1/x) \\) for all \\( x \\neq 0 \\) and \\( f(0) = 0 \\), determine the values of \\( r > 0 \\) where \\( f'(0) \\) exists, given that \\( r \\) is rational.",
"role": "user"
}
] | r > 1 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty}\\left(\\sum_{k=2n+1}^{4n}\\frac{1}{k} - \\sum_{k=n+1}^{2n}\\frac{1}{k}\\right). \\]",
"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... | 1.1552e5 N |
[
{
"content": "In triangle $\\Delta ABC$, points $P$ and $Q$ are located on sides $AB$ and $AC$, respectively. Given that the ratio $|AP|:|PB|=1:5$ and $|AQ|:|QC|=2:1$, lines $BQ$ and $CP$ intersect at point $F$. Using vector methods, find the ratio $|CF|:|FP|$.",
"role": "user"
}
] | 3:5 |
[
{
"content": "Calculate the limit: $$\\lim _{x\\rightarrow 0^{+}} e^{x\\log{\\sin x}}.$$",
"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... | 97.45 kPa |
[
{
"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... | 100 mL |
[
{
"content": "Calculate the line integral \\( \\int_C (x-y) \\, dz \\) using Stokes' theorem, where \\( C \\) is the curve formed by the intersection of the paraboloid \\( z = x^2 + y^2 \\) and the plane \\( z - 2x - 2y = 1 \\). Orient \\( C \\) counterclockwise as viewed from the point \\( (0,0,10) \\).",
... | 12\pi |
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