prompt list | ground_truth string |
|---|---|
[
{
"content": "Determine if the function \\( f(x) = \\sin^2\\left(x \\sin \\frac{1}{x}\\right) \\) is right differentiable at \\( x = 0 \\). If it is, find the right derivative of \\( f \\) at \\( x = 0 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Let \\( P = (3^1 + 1)(3^2 + 1)(3^3 + 1) \\ldots (3^{2020} + 1) \\). Determine the largest integer \\( n \\) such that \\( 2^n \\) divides \\( P \\).",
"role": "user"
}
] | 3030 |
[
{
"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... | 10.99 x 10^27N |
[
{
"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... | $9,525,000 |
[
{
"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... | 2.36 |
[
{
"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... | 9.70 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... | 0.025% |
[
{
"content": "Let \\( f(x) = x \\sin(x^N) \\). Determine the smallest integer \\( k \\) such that the \\( k \\)-th derivative of \\( f(x) \\) evaluated at 0, \\( f^{(k)}(0) \\), is not equal to zero. Express your answer in terms of \\( N \\).",
"role": "user"
}
] | N + 1 |
[
{
"content": "Let $a, b \\in \\mathbb{R}$ with $a < b$, and let $f \\in C^{1}([a, b])$. Prove that \\( \\lim \\limits_{n \\rightarrow \\infty} \\int \\limits_{a}^{b} f(x) \\sin (n x) \\, dx = 0. \\)",
"role": "user"
}
] | 0 |
[
{
"content": "In an arithmetic progression \\(a_{1}, a_{2}, \\ldots\\), consisting of natural numbers, it is given that for any \\(n\\), the product \\(a_{n} a_{n+31}\\) is divisible by 2005. Determine whether all terms of the progression must be divisible by 2005.",
"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... | 2.75 eV |
[
{
"content": "Determine the limit of the probability, as \\( n \\) approaches infinity, of selecting a number between \\( 1 \\) and \\( 10^n \\) (excluding numbers with zeros) such that the product of its digits is less than the square root of the number.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit \\( \\lim_{x \\to 0} \\frac{\\int_0^x e^t\\cos t\\, dt}{x} \\) without using L'Hospital's rule.",
"role": "user"
}
] | 1 |
[
{
"content": "For a given natural number $n$, determine the greatest possible value of $k$ such that there exist subsets $A, B, C, D$ of the set of the first $n$ natural numbers with the property that any two elements of the set are in at least one of the subsets, and at least one of $|A|,|B|,|C|,|D|$ is greate... | \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... | 9.09% |
[
{
"content": "Find the least positive integer $M$ such that $M^{49} \\equiv 21 \\pmod{209}$. \\( \\text{(Note: 209 = 11 * 19)} \\)",
"role": "user"
}
] | 109 |
[
{
"content": "Evaluate the limit using the Riemann sum approach: \\[ \\lim_{n \\rightarrow +\\infty} \\sum_{k=1}^{n}\\frac{(3k-2n)^{5}}{n^{6}}. \\]",
"role": "user"
}
] | -\dfrac{7}{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... | $716,072.53 |
[
{
"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... | 90.46 |
[
{
"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... | 328.56 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... | $20,800 |
[
{
"content": "Determine the number of 6-tuples \\((a, b, c, d, e, f)\\) of natural numbers such that \\(a > b > c > d > e > f\\) and \\(a+f = b+e = c+d = 30\\).",
"role": "user"
}
] | 364 |
[
{
"content": "Let $f(x)$ be a quartic polynomial with distinct real roots. If we substitute $x = by$ and divide by $b^4$, where $b$ is a nonzero real number, does the number of distinct real roots of the resulting polynomial depend on $b$? Answer with 'Yes' or 'No'.",
"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... | 6.65 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... | 6.38g |
[
{
"content": "Let $f(x+y)=f(x)f(y)$ for all $x, y \\in \\Bbb R$. Suppose that $f(3)=3$. Find the value of $f'(3)$. \\( \\text{(Express your answer as a number.)} \\)",
"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... | 20.45 |
[
{
"content": "In triangle $ABC$, $3 \\sin A + 4 \\cos B = 6$ and $4 \\sin B + 3 \\cos A = 1$. Find all possible values of $\\angle C,$ in degrees. Enter all the possible values, separated by commas.",
"role": "user"
}
] | 30^\circ |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.816 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... | The supply curve shifts to the right from S1 to S2. The new equilibrium point is E1, where the price is P1 and the quantity is Q1. Before the price adjusts, there will be a temporary surplus at the old equilibrium price P*. |
[
{
"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... | 69.3 J/m^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... | $6,809.12 |
[
{
"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... | 13.8% |
[
{
"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... | -535.08 J |
[
{
"content": "Calculate the surface integral \\( \\iint_Y z \\, dS \\) where \\( Y \\) is the part of the sphere \\( x^2 + y^2 + z^2 = 4 \\) where \\( z \\ge 0 \\).",
"role": "user"
}
] | 8\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... | 722.37 J |
[
{
"content": "The sides of the base of a triangular pyramid are 3, 4 and 5 feet and the altitude is 6 feet. What is the number of cubic feet in the volume of the pyramid?",
"role": "user"
}
] | 12 |
[
{
"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... | $6,501.06 |
[
{
"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... | 0.11kgm^2s^-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... | 30*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... | Computational biology, biostatistics, and bioinformatics |
[
{
"content": "Find the greatest common divisor of $5616$ and $11609$.",
"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... | 16854.24 MPa |
[
{
"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... | 10.51% |
[
{
"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... | 0.679 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... | 7.11% |
[
{
"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... | 15.791 m/s |
[
{
"content": "Given that \\( \\sum_{n=1}^{\\infty} \\frac{1}{n^2} = \\frac{\\pi^2}{6} \\), find the value of \\( \\lambda \\) such that \\( \\int_{0}^{1} \\frac{\\ln x}{1-x^2} \\, dx = -\\frac{\\pi^2}{\\lambda} \\).",
"role": "user"
}
] | 8 |
[
{
"content": "Find all polynomials \\( P(a) \\in \\mathbb{R}[a] \\) such that for all \\( x, y, z \\in \\mathbb{R} \\setminus \\{0\\} \\), the following equation holds:\n\\[ P(x + y + z) = \\frac{P(y)P(z)}{x^2} + \\frac{P(z)P(x)}{y^2} + \\frac{P(x)P(y)}{z^2}. \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the integral \\( \\iint_D \\left[3-\\frac{1}{2}\\left( \\frac{x^2}{a^2}+\\frac{y^2}{b^2}\\right)\\right] \\, dx \\, dy \\) where \\( D \\) is the region defined by \\( \\frac{x^2}{a^2}+\\frac{y^2}{b^2} \\le 4 \\).",
"role": "user"
}
] | 8\pi ab |
[
{
"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... | 3.083 |
[
{
"content": "Determine whether the following statement is true or false: Let $T:H \\rightarrow H$ be a continuous linear operator, where $H$ is a pre-Hilbert space. Then $H=\\ker(T) \\oplus\\ker(T)^{\\perp}$, where $\\ker(T)^{\\perp}$ is isomorphic to $H/\\ker(T)$. Furthermore, $\\ker(T)^{\\perp}$ is isomorphi... | False |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty} n\\int_{0}^{\\pi/4} \\sin^n(x) \\,{\\rm d}x$$",
"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... | 4.32e-3 |
[
{
"content": "Consider a closed irreducible algebraic set $V \\subseteq \\mathbb{A}^n_k$ in the affine space over a closed field $k$. Let $X$ be the topological space of all closed irreducible subsets of $V$, equipped with the Zariski topology, and $f:V \\to X$ be a continuous map such that $f(x) = \\overline{\... | True |
[
{
"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... | Static Budget(1)
Sales-Volume Variance(absolute value of 1-2)
Flexible Budget(2)
Flexible Budget Variance(absolute value of 2-3)
Actual Results(3)
Revenues$4,000,000
$212,000 unfavorable
$3,788,000
$75,760 favorable
$3,863,760
Variable Costs$2,280,000
$120,840 favorable
$2,159,160
$37,880 unfavorable
2,197,040
Fixed Costs$850,000
-0- n/a
$850,000
$20,000 unfavorable
$870,000
Income$870,000
$91,160 unfavorable
$778,840
$17,880 favorable
$796,720 |
[
{
"content": "Consider a non-abelian group $G$ and its automorphism group $\\Aut(G)$. Does there exist an element $x \\in G$ such that the centralizer of $x$ in $\\Aut(G)$, denoted by $C_{\\Aut(G)}(x)=\\{\\alpha \\in \\Aut(G) \\mid \\alpha(x)=x\\}$, is equal to the identity automorphism $I$ alone?",
"role":... | 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... | 13.7 rad/s^2 |
[
{
"content": "Determine the constant \\(c\\) such that the function \\(f(x,y)\\) defined by\n\\[f(x,y)=\\begin{cases}cxy,&-1\\leq x\\leq 0\\wedge 0\\leq y\\leq 1\\wedge y-x\\leq 1,\\\\cxy,&0\\leq x\\leq 1\\wedge -1\\leq y\\leq 0\\wedge x-y\\leq 1,\\\\0,&\\text{else}.\\end{cases}\\]\nis a probability density fun... | -12 |
[
{
"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*10^8 m |
[
{
"content": "Let $X$ be a Gaussian random variable with mean $0$ and variance $\\sigma^2$. Define $Y = \\max(X, 0)$. Determine the median of $Y$. Provide a clear and concise explanation of your reasoning.",
"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... | 4.747 |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{6}\\int_{-\\sqrt{36 - x^2}}^{\\sqrt{36 - x^2}}\\int_{-\\sqrt{36 - x^2 - z^2}}^{\\sqrt{36 - x^2 - z^2}} \\dfrac{1}{\\sqrt{x^2 + y^2 + z^2}} \\ dy \\, dz \\, dx \\) by changing to spherical coordinates.",
"role": "user"
}
] | 36\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... | Spectroscope |
[
{
"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... | $34,534 |
[
{
"content": "Given two positive-valued functions $f, g: [0,1]\\to(0,1)$, a pair $(f,g)$ is defined as good if, for every real number $r$, the set $Z_r := \\{x ~\\big|~ {g(x) / f(x)} = r \\}$ has a measure of zero in $\\mathbb{R}$. For example, if $f(x)=1/2$ and $g(x)=(x+1)/2$, the pair $(f,g)$ is good. Convers... | 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... | $420,800 |
[
{
"content": "In any category \\( C \\), are the hom-sets between objects \\( X \\) and \\( Y \\) always Abelian groups, regardless of whether \\( C \\) is an Abelian category? Answer yes or no.",
"role": "user"
}
] | No |
[
{
"content": "Determine whether a non-biconnected outerplanar graph is necessarily a tree. Answer 'yes' or 'no'.",
"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... | 236.66% |
[
{
"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... | 467J |
[
{
"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. Pure time preference: Individuals prefer present purchasing power to future purchasing power, even if future rates are potentially better. They tend to enjoy current benefits and costs rather than uncertain future ones.
2. High discount rates: Using constant high discount rates leads to low present values of future amounts, contrary to the belief that future generations will be wealthier.
3. Low discount rates: While some economists argue that low discount rates promote investment by allowing cheap bank loans, low discount rates shift environmental degradation costs to future generations, reducing incentives for environmentally favorable long-term projects and potentially leading to environmentally damaging investments. |
[
{
"content": "Determine whether there exist right ideals in the matrix ring \\(M_n(k)\\) that are not of the form of zeroing one or more rows at a time. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: $$ \\lim_{x\\to\\infty} \\biggl(\\frac {e^x-x^2 }{e^x-5x}\\biggr)^{\\log x} $$",
"role": "user"
}
] | 1 |
[
{
"content": "Determine the number of real solutions to the equation \\(8^x = 6x\\).",
"role": "user"
}
] | 2 |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to 1}\\left(\\frac{2-x}{3-x}\\right)^{(2-\\sqrt{x})/(2-x)}$$",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Let $L \\in GL_n(\\mathbb{Q})$ be a linear invertible map on $\\mathbb{Q}^n$. If $L^{-1} = L + L^2$, prove that $3 \\mid n$. ",
"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... | \mathbf{r}(t) =\left( \frac{3}{10}t^5+2t+ 3\right)\mathbf{i} + \left(e^t(t-2)+6t + 5\right)\mathbf{j} + \left(t - \frac{\sin(4t)}{2} -2 \right)\mathbf{k} |
[
{
"content": "Compute the limit: \\[ \\lim_{x \\to 0}{\\frac{\\int_x^{x+1}f(t)dt-\\int_0^{1}f(t)dt}{x}} \\] where \\( f \\) is continuous on \\( \\mathbb{R} \\) and satisfies \\( f(0) + 1 = f(1) \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Calculate the covariance \\( \\text{Cov}(U,V) \\) where \\( U = \\frac{1}{2}(X+Y) \\) and \\( V = \\frac{1}{2}(X-Y) \\), given that \\( X \\) and \\( Y \\) are independent random variables with moment generating functions \\( M_{X}(t) = (1-t)^{-1}, \\ t<1 \\) and \\( M_{Y}(t) = (1-2t)^{-1}, \\ t< ... | -\dfrac{3}{4} |
[
{
"content": "If $(f_n)$ converges uniformly to $f$ on $[\\delta,1]$ for all $\\delta>0$, does it converge uniformly to $f$ on $(0,1]$? Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"content": "A cylinder has a height of 6 and a radius of 3, with the centers of its bases at (0,0,0) and (0,0,6). Determine the volume of the smallest convex polyhedron that encloses all integer lattice points inside the cylinder.",
"role": "user"
}
] | 144 |
[
{
"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... | 196.6 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... | 0.91 km |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2 balls |
[
{
"content": "Find the coefficient of \\(x^{29}\\) in the expansion of \\((1 + x^{5} + x^{7} + x^{9})^{16}\\).",
"role": "user"
}
] | 65520 |
[
{
"content": "Determine whether there exists a probability density function (PDF) \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) such that \\( f \\in L^1(\\mathbb{R}) \\) and has countably infinite points \\( \\{x_n\\}_{n=1}^{\\infty} \\) with \\( f(x_n) = +\\infty \\), excluding the Dirac Comb. Assume the sup... | Yes |
[
{
"content": "Find the quadratic variation of the stochastic process $Y(t)$ over the interval $[-1, 2]$, where $Y(t)$ is defined as follows:\n\n$$Y(t) = \\begin{cases} 0 & -1 \\le t \\le 0 \\\\\\ Z(t) - tZ(1) & 0 < t < 1 \\\\\\ 1 - t & 1 \\le t \\le 2 \\end{cases}$$\n\nand $Z(t)$ is a standard Brownian motion."... | 1 |
[
{
"content": "Determine the radius of convergence of the series: $$1+ \\frac{a.b}{1.c}z + \\frac{a(a+1)b(b+1)}{1.2...c(c+1)}z^2+.... $$",
"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... | 1.71 m/s |
[
{
"content": "Use the divergence theorem to compute the flux of the vector field \\( \\vec{F}(x, y, z) = z\\vec{i} + x\\vec{j} + y\\vec{k} \\) through the surface of the solid bounded by the paraboloid \\( z = 4 - x^2 - y^2 \\) and the plane \\( z = 0 \\). What is the flux through the paraboloid surface?",
... | 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... | 392.4 J |
[
{
"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... | 635.5 nm |
[
{
"content": "Let $f$ be a continuous, real-valued function on $[0,1]$ such that there exists a constant $K > 0$ for which \\( |f(x)| \\le K \\int_{0}^{x} |f(t)| \\, dt \\) for all \\( x \\in [0,1] \\). Does it follow that $f(x) = 0$ for all $x \\in [0,1]$?",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{1}\\frac{x}{x^2+(1-x)^2}\\,dx \\) using the substitution \\(x = \\frac{1}{2}(1+\\tan \\theta)\\).",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "Let $h(x)$ be a bounded continuous function. Define $L_h(\\lambda)$ as the Laplace transform of $h(x)$. Let $Y_n$ be a sequence of independent random variables with an exponential distribution of rate $\\frac{1}{z}$. Derive the formula:\n\n$$ h(z)=\\lim_{n \\to \\infty}(-1)^{n-1}\\frac{\\left(\\fr... | h(z) |
[
{
"content": "In an infinite coin flipping game, the first player to achieve the sequence $HHT$ wins. If the last three tosses are $HTH$, the second player wins. The sequence $TT$ causes the game to restart. Determine which sequence, $HHT$ or $HTH$, is more likely to occur first.",
"role": "user"
}
] | HHT |
[
{
"content": "In the figure, vertex $B$ of $\\triangle ABC$ is at the center of a unit circle, with points $A$ and $C$ on the circumference. The angle $\\angle ABC = 2\\alpha$ where $0 < \\alpha < \\frac{\\pi}{3}$. The triangle is rotated counterclockwise within the circle as follows: \n\n1. Rotate around point... | D |
[
{
"content": "Let \\( F_{1} \\) and \\( F_{2} \\) be the foci of the ellipse \\(\\frac{x^{2}}{9} + \\frac{y^{2}}{4} = 1\\). \\( P \\) is a point on the ellipse such that \\(\\left|P F_{1}\\right| : \\left|P F_{2}\\right| = 2 : 1\\). Find the area of triangle \\(\\triangle P F_{1} F_{2}\\).",
"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... | The equilibrium price of good x is 70.68 and equilibrium price of good y is 21.7. The equilibrium quantity of good x is 50.68 units and equilibrium quantity of good y is 28.4 units. |
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