prompt list | ground_truth string |
|---|---|
[
{
"content": "Determine whether the following statement is true or false: If a language $L$ is context-sensitive, then every grammar $G$ that generates $L$ is also context-sensitive.",
"role": "user"
}
] | B |
[
{
"content": "Let $f(r) = \\sum_{j=2}^{2008} \\frac{1}{j^r} = \\frac{1}{2^r}+ \\frac{1}{3^r}+ \\dots + \\frac{1}{2008^r}$. Find $\\sum_{k=2}^{\\infty} f(k)$.",
"role": "user"
}
] | \frac{2007}{2008} |
[
{
"content": "If $X$ is a topological space and $Y$ is a subspace of $X$, determine whether any subset $Z$ of $Y$ that is connected in the topology of $X$ is also connected in the subspace topology of $Y$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether a group in which every element is its own inverse must necessarily be finite. Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"content": "If the direct sum $\\oplus_{i\\in I}P_{i}$ of a family of $R$-modules is projective, does it follow that each $P_{i}$ is a projective module for all $i \\in I$?",
"role": "user"
}
] | Yes |
[
{
"content": "Given a set of $n$ points uniformly distributed in a hypersphere $H$ of radius $r$ in $\\mathbb{R}^d$, consider one of the chosen points located at $x \\in H$. Calculate the probability that $x$ is the closest chosen point to another point located at $y \\in H$. Provide your answer as a probabilit... | \dfrac{1}{n} |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to \\infty} \\frac{\\sum_{k=1}^{n}\\left(1+\\frac{k}{n}\\right)^{k}}{n}$$",
"role": "user"
}
] | \infty |
[
{
"content": "For a certain value of \\(c\\), the limit \\(\\lim_{x \\to \\infty}\\left[\\left(x^{5} + 7x^{4} + 2\\right)^c - x\\right]\\) is finite and non-zero. Let \\(l\\) be this limit. Calculate \\(l - 2c\\).",
"role": "user"
}
] | 1 |
[
{
"content": "Determine whether the following limit is equal to zero:\n\\[ \\lim_{R \\to \\infty} \\int_S \\frac{jRe^{j\\theta}}{1+R^4e^{j4\\theta}}d\\theta \\]\nwhere \\( S \\) is a semicircle. Does the presence of \\( j \\) in the differential make any difference to the limit?",
"role": "user"
}
] | 0 |
[
{
"content": "Does the convergence of $E(|X_n|) \\to 0$ and $E(X_n^2) \\to 1$ for a sequence of random variables $X_n$ imply that $X_n$ does not converge in square mean?",
"role": "user"
}
] | Yes |
[
{
"content": "Is the set of discontinuity points, which are all jumps, of a càdlàg function necessarily countable?",
"role": "user"
}
] | Yes |
[
{
"content": "Calculate the Lebesgue integral \\( \\int_0^1 f(x) \\, dx \\) for the function \\( f: [0,1] \\to \\mathbb{R} \\) defined as follows: \\( f(x) = 0 \\) if \\( x \\) is rational, and \\( f(x) = 9^n \\) if \\( x \\) is irrational, where \\( n \\) is the number of zeros immediately after the decimal po... | 9 |
[
{
"content": "Given the parabola \\(y^2 = 3x\\), two perpendicular lines are drawn from the origin, intersecting the parabola at points \\(P\\) and \\(Q\\). Find the equation, in Cartesian form, of the locus of the centroids of all triangles \\(OPQ\\), where \\(O\\) is the origin.",
"role": "user"
}
] | y^2 = x - 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... | 0.141 g |
[
{
"content": "Find the value of the series \\( \\sum_{k=1}^{\\infty} \\frac{k^2}{2^k} \\).",
"role": "user"
}
] | 6 |
[
{
"content": "In a tetrahedron \\(ABCD\\), a plane passes through the intersection points of the medians of the faces \\(ABC\\), \\(ABD\\), and \\(BCD\\). Determine the ratio in which this plane divides the edge \\(BD\\).",
"role": "user"
}
] | 2:1 |
[
{
"content": "In right triangle $\\triangle ABC$ with right angle at $A$, the angle bisectors $BD$ and $CE$ divide the triangle such that $AE = 8$ and $AD = 9$. Find the perimeter of $\\triangle ABC$. Use the angle bisector theorem and the Pythagorean theorem to assist in your calculations.",
"role": "user"... | 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... | 414.96 m |
[
{
"content": "Each interior angle of a polygon measures 170 degrees. How many sides does the polygon have?",
"role": "user"
}
] | 36 |
[
{
"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... | $282,113.87 |
[
{
"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... | Coefficient of static friction = Force of Static friction / Normal force |
[
{
"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... | 1.75 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... | 83 million |
[
{
"content": "Given the system of equations:\n\\[\\begin{align*}\nx + y + z &= 1\\\\\nx^2 + y^2 + z^2 &= 2\\\\\nx^3 + y^3 + z^3 &= 3\n\\end{align*}\\]\nDetermine the value of \\(x^5 + y^5 + z^5\\).",
"role": "user"
}
] | 6 |
[
{
"content": "Determine the number of positive integer triples \\((x, y, z)\\) that satisfy the equation \\(x^2 + 2021y^2 = z^2\\).",
"role": "user"
}
] | \infty |
[
{
"content": "Suppose $(T_n)$ is a sequence of unitary and linear bounded operators on a Hilbert space $H$, and $T_n$ converges to $T$ in the norm topology of $B(H,H)$. Prove that $T$ is also linear, bounded, and unitary.",
"role": "user"
}
] | 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... | 14.9 N |
[
{
"content": "Given the definition of the less-than relation in the construction of real numbers, where \\( a < b \\) means that \\( a \\) is contained in \\( b \\) and \\( a \\neq b \\), does this imply that all negative numbers are contained in any positive number?",
"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... | 1.9e-2 T, along negative y direction |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 36.67% |
[
{
"content": "Evaluate the integral \\(\\int_{-1}^{1}x^{2}\\delta(x^3)dx\\), where \\(\\delta(x)\\) is the Dirac delta function.",
"role": "user"
}
] | \dfrac{1}{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... | Joules |
[
{
"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... | 321 m/s |
[
{
"content": "Is it possible to find three consecutive identical digits in the decimal expansion of an irrational number?",
"role": "user"
}
] | Yes |
[
{
"content": "Given that \\(|ax^2+bx+c|\\leq 2\\) for all \\(x\\in[-1,1]\\), find the maximum value of \\(\\left|cx^2+2bx+4a\\right|\\) for all \\(x\\in [-2,2].\\)",
"role": "user"
}
] | 16 |
[
{
"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... | $6.57 |
[
{
"content": "Find the smallest positive integer \\(a\\) such that there exists a positive odd number \\(n\\) for which \\(2001 \\mid (55^n + a \\cdot 32^n)\\).",
"role": "user"
}
] | 436 |
[
{
"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... | 200 N |
[
{
"content": "Calculate the sum of the series \\( \\sum_{x=1}^{\\infty} \\frac{2^{-x}}{x} \\).",
"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... | 69.000% |
[
{
"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... | 19.18% |
[
{
"content": "Is every function in an $L^p$ space with $p<\\infty$ essentially bounded? Provide a justification for 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... | 39612.82 J |
[
{
"content": "Determine the condition on the real number $p$ such that the integral \\( \\int_0^\\infty \\frac{x^2 + 1 - \\cos x}{(x^2 + 1)x^p} \\, dx \\) converges.",
"role": "user"
}
] | 1 < p < 3 |
[
{
"content": "Calculate the distance \\( \\operatorname{dist}(f,B) \\) where \\( f(x) = x^2 \\) and \\( B = \\{ bx : b \\ge 0 \\} \\) in the metric space \\( (C[0,1], d_{\\sup}) \\), with \\( d_{\\sup}(f,g) = \\sup_{x \\in [0,1]}|f(x)-g(x)| \\).",
"role": "user"
}
] | 3 - 2\sqrt{2} |
[
{
"content": "Determine the number of Sylow 2-subgroups in the symmetric group $S_5$.",
"role": "user"
}
] | 15 |
[
{
"content": "Find the limit: \\[ \\lim_{x \\to \\infty}\\left(\\sum_{n=1}^{\\infty}\\left(\\frac{x}{n}\\right)^{n}\\right)^{\\frac{1}{x}}. \\]",
"role": "user"
}
] | e^{\frac{1}{e}} |
[
{
"content": "Given the cost function \\( C(x) = 125 \\sqrt{x^2 + 5^2} + 100|k-x| \\), where the cost of underwater cable is \\( \\alpha \\) per kilometer and the cost on land is \\( \\beta \\) per kilometer with \\( \\alpha > \\beta \\), determine the largest value of \\( k > 0 \\) such that the most efficient... | \dfrac{20}{3} |
[
{
"content": "A set \\( \\Phi \\) consists of a finite number of points on the plane. The distance between any two points in \\( \\Phi \\) is at least \\( \\sqrt{2} \\). A regular triangle with side length 3 can cover all points of \\( \\Phi \\). What is the greatest number of points that \\( \\Phi \\) can cont... | 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... | 6.1 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... | The highest energy d orbital in dubnium is 6d, which contains 3 electrons. The orbital diagram is shown above. |
[
{
"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... | -91.59e-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... | passing score |
[
{
"content": "Determine the dimension of the ring \\( k[x,y]/(x^2-y) \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Find the derivative of the inverse function of \\( f(x) = x - \\tan x \\) at the point \\( (\\pi/4) - 1 \\). Assume the domain of \\( f(x) \\) is \\((-\\pi/2, \\pi/2)\\) to ensure it is one-to-one.",
"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... | 0.60 |
[
{
"content": "Find all non-zero functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) such that \\( f(x) f(y) = f(x-y) \\) for all real numbers \\( x \\) and \\( y \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the double integral \\( \\iint_{Q} (x+y)^{2013} \\, dx \\, dy \\), where \\( Q \\) is the square \\([-1,1]^2 \\subset \\mathbb{R}^2\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Let \\((a, b)\\) denote the greatest common divisor of \\(a\\) and \\(b\\). Find the smallest integer \\(n > 2021\\) such that \\((63, n+120) = 21\\) and \\((n+63, 120) = 60\\). What is the sum of the digits of this smallest \\(n\\)?",
"role": "user"
}
] | 21 |
[
{
"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... | $\int_0^{20}(m_0-\frac{2x}{d})\cdot gdx$ |
[
{
"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... | Sales revenue: $357,000
Cost of goods sold: $132,000
Gross Margin: $225,000
Fixed costs: $43,000
Selling, general, and administrative expenses: $27,100
Depreciation: $45,600
Total Operating Expenses: $115,700
EBIT: $109,300
Interest expense: $17,700
Taxable income: $91,600
Taxes (40%): $36,640
Net income: $54,960 |
[
{
"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... | 1.334e-3 C |
[
{
"content": "Evaluate the integral \\( \\int_{-\\infty} ^{\\infty} \\frac{x \\sin{3x} }{x^4 +1}dx \\) and express it in the form \\( \\pi^a e^{\\frac{-b}{\\sqrt{c}}}\\sin \\big({\\frac {d}{\\sqrt{e}}}\\big) \\), where \\(a, b, c, d, e\\) are positive integers and \\(c\\) and \\(e\\) are square-free numbers. Fi... | 11 |
[
{
"content": "Let \\( f: \\mathbb{R} \\rightarrow \\mathbb{C} \\) be a differentiable function. Is it true that for all \\( a, b \\in \\mathbb{R} \\), there exists some \\( c \\in (a, b) \\) such that \\[ \\left| \\frac{f(b) - f(a)}{b - a} \\right| \\leq |f'(c)|? \\]",
"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... | 0.3 |
[
{
"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... | Neither |
[
{
"content": "Find a sequence $b_n$ such that $a_n = \\frac{(n!)^n}{n^{n^2}} < b_n$ for all $n$, and $\\lim_{n \\to \\infty} b_n = 0$.",
"role": "user"
}
] | 2 e^{-n^2/2} |
[
{
"content": "In a rectangular parallelepiped $ABCDEFGH$, the edge lengths are $AB = 30$, $AD = 32$, and $AA_1 = 20$. Point $E$ is the midpoint of edge $A_1B_1$, and point $F$ is the midpoint of edge $B_1C_1$. Calculate the distance between the lines $AE$ and $BF$. Express your answer in simplest form.",
"r... | \dfrac{96}{5} |
[
{
"content": "Let $U$ be an unbounded open subset of $\\mathbb{R}$. Does there exist a number $a$ such that $U$ contains infinitely many integer multiples of $a$?",
"role": "user"
}
] | Yes |
[
{
"content": "Suppose $f:[0, \\infty) \\rightarrow \\mathbb{R}$ is a continuous function such that $f(x) \\neq 0$ for all $x > 0$. If the equation \\[ (f(x))^2 = 2 \\int_0^x f(t) \\, dt \\] holds for all $x > 0$, determine whether it is true that $f(x) = x$ for all $x \\geq 0$. Provide a justification for your ... | 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... | 0.986 |
[
{
"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... | 2.34e-7 m |
[
{
"content": "Find all possible values of $m$ such that the equations $\\sin^4x+\\cos^4x=m$ and $\\sin^6x+\\cos^6x=m$ have common real solutions, where $m$ is a non-negative real number.",
"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... | 5.82e-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... | {eq}\frac{\epsilon_0 V^2}{d^2}{/eq} |
[
{
"content": "Calculate the limit: $$\\lim_{n \\to \\infty} \\left(\\frac{n}{n+3}\\right)^\\sqrt{n(1+n)}$$",
"role": "user"
}
] | e^{-3} |
[
{
"content": "Evaluate the integral \\(\\int_{-\\pi/4}^{\\pi/4}(\\cos(t)+\\sqrt{1+t^2}\\cos^3(t)\\sin^3(t))dt\\). Consider the parity of the integrand and any suitable substitution to find the result.",
"role": "user"
}
] | \sqrt{2} |
[
{
"content": "Find the larger of the two positive integers \\(n \\ge 3\\) for which the expression \\(\\frac{\\log 2 \\times \\log 3 \\times \\log 4 \\times \\cdots \\times \\log(n-1)}{10^n}\\) takes its smallest value.",
"role": "user"
}
] | 22027 |
[
{
"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... | d. 9.00% |
[
{
"content": "Determine the number of real roots of the equation \\(x^8 - x^5 + x^2 - x + 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... | 7.92 g/cm^3 |
[
{
"content": "Find all functions \\( f: \\mathbb{N} \\to \\mathbb{N} \\) such that \\( f(f(m) + f(n)) = m + n \\) for all \\( m, n \\in \\mathbb{N} \\).",
"role": "user"
}
] | f(n) = n |
[
{
"content": "Let \\( a, b, c \\) be non-zero complex numbers such that \\(\\frac{a}{b} = \\frac{b}{c} = \\frac{c}{a}\\). Determine the value of \\(\\frac{a+b-c}{a-b+c}\\). Choose from the following options:\n\n(A) 1\n\n(B) \\(\\pm \\omega\\)\n\n(C) \\(1, \\omega, \\omega^{2}\\)\n\n(D) \\(1, -\\omega, -\\omega^... | C |
[
{
"content": "Given an arbitrary polygon $p$, consider its midpoint polygon $p'$. By iterating this process to form $p''$, $p'''$, and so on, determine if there is always a convex polygon in this sequence.",
"role": "user"
}
] | Yes |
[
{
"content": "Triangle $ABC$ is a right triangle with $AC = 7,$ $BC = 24,$ and right angle at $C.$ Point $M$ is the midpoint of $AB,$ and $D$ is on the same side of line $AB$ as $C$ so that $AD = BD = 15.$ Given that the area of triangle $CDM$ may be expressed as $\\frac {m\\sqrt {n}}{p},$ where $m,$ $n,$ and $... | 578 |
[
{
"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... | $29,000 |
[
{
"content": "Evaluate the integral \\( \\int_0^{2\\pi} \\sum_{n=1}^{\\infty} \\frac{\\sin(nx)}{n} \\, dx \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the integral \\( \\int^{\\pi}_0\\int^{\\pi}_x \\frac{\\sin(y)} {y} \\, dy \\, dx \\).",
"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... | cancer |
[
{
"content": "Use Green's Theorem to compute the value of the line integral \\( \\int_{\\gamma} y\\,dx + x^2\\,dy \\), where \\( \\gamma \\) is the circle given by \\( g(t) = (\\cos t, \\sin t), 0 \\leq t \\leq 2\\pi \\).",
"role": "user"
}
] | -\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... | 3s |
[
{
"content": "Given the integer function \\( f(z) = u(x,y) + iv(x, y) \\) where \\( u(x, y) = e^{-x}\\sin(y -1) + y \\), find the value of \\( f'(i) \\).",
"role": "user"
}
] | -2i |
[
{
"content": "Find the closed-form solution for the sequence \\(a_n\\) defined by the recursive relation \\(a_n = a_{n-1} + 2n - 1\\) with the initial condition \\(a_0 = 0\\) for \\(n \\ge 1\\), using generating functions.",
"role": "user"
}
] | n^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... | 10 degrees |
[
{
"content": "Let $f: \\mathbb{R} \\to \\mathbb{R}$ be a convex function. Prove that there exists a linear function $g(y) = ay + b$ such that $f(y) \\geq g(y)$ for all $y \\in \\mathbb{R}$ and $f(x) = g(x)$ for some $x \\in \\mathbb{R}$. Consider the case where $f$ is not differentiable.",
"role": "user"
... | g(y) = ay + b |
[
{
"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... | 4.652 kg |
[
{
"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... | Genetic Data |
[
{
"content": "Find the sum of the series \\(\\sum_{n=1}^{\\infty} \\frac{b_{n}}{5^{n-1}}\\) given that the sequence \\(b_n\\) satisfies the recurrence relation \\(b_{n+2}=2b_{n+1}-b_{n}\\) with initial conditions \\(b_{1}=-1\\) and \\(b_{2}=2\\).",
"role": "user"
}
] | -\dfrac{5}{16} |
[
{
"content": "Determine whether the following equality holds for three random variables $X, Y$, and $Z$ that are not necessarily independent: $$ P(X \\leq Y \\leq Z) = P(X \\leq Y)P(Y \\leq Z). $$",
"role": "user"
}
] | No |
[
{
"content": "Evaluate the limit \\( \\lim_{r \\to 0^+} \\int_{\\gamma} \\frac{\\sin(z)}{z}dz \\), where \\( \\gamma(t)=re^{it}, t \\in [0,\\pi] \\).",
"role": "user"
}
] | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.