prompt list | ground_truth string |
|---|---|
[
{
"content": "Determine whether the polynomial \\(x^4 - 10x + 16\\) is irreducible over \\(\\mathbb{Q}[x]\\). If it is reducible, factor it into irreducible polynomials over \\(\\mathbb{Q}[x]\\).",
"role": "user"
}
] | x^4 - 10x + 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... | Genotypes: GgBb, Ggbb, ggBb, ggbb
Genotypic ratio: 1:1:1:1
Phenotypes: grey coat, brown eyes; grey coat, blue eyes; black coat, brown eyes; black coat, blue eyes
Phenotypic ratio: 1:1:1:1 |
[
{
"content": "Compute the integral \\( \\int_{|z| = 2} \\frac{dz}{z^2 - 1} \\) without using winding numbers.",
"role": "user"
}
] | 0 |
[
{
"content": "The inhabitants of the island of Jumble use the standard Kobish alphabet ($20$ letters, A through T). Each word in their language is $4$ letters or less, and for some reason, they insist that all words contain the letter A at least once. How many words are possible?",
"role": "user"
}
] | 30860 |
[
{
"content": "Evaluate the triple integral \\(I=\\iiint\\limits_\\Omega (x+y+z)^2 \\, dv\\), where \\(\\Omega\\) is the region defined by \\(x^2+y^2+z^2 \\le 1\\).",
"role": "user"
}
] | \dfrac{4\pi}{5} |
[
{
"content": "Calculate the line integral of the vector field \\( G(x,y,z) = (x-y)i + (y-z)j + (z-x)k \\) along the circular arc in \\( \\mathbb{R}^3 \\) on a plane parallel to the xy-plane, centered at the origin with radius \\( b \\). The arc follows the path from \\((b,0,c)\\) to \\((0,b,c)\\) to \\((-b,0,c)... | \pi b^2 |
[
{
"content": "Evaluate the integral \\( \\int_{-L}^L e^{i\\frac{(m-n)\\pi}{L}x} \\, dx \\), where \\( 0 < L \\in \\mathbb{R} \\) and \\( m, n \\in \\mathbb{N} \\).",
"role": "user"
}
] | 2L \delta_{mn} |
[
{
"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... | Nobelium |
[
{
"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... | 0.105 g |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 30.946 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... | 2.752 M |
[
{
"content": "A lamp is positioned at the point \\((-2, h)\\). A light ray from the lamp touches the circle defined by \\(x^2 + y^2 = 1\\) at a point in the first quadrant and then stops when it hits the x-axis at the point \\((1.25, 0)\\). Determine the height \\(h\\) of the lamp.",
"role": "user"
}
] | \dfrac{13}{3} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 4.67e+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... | $231.60 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | The unit product cost for Rascon is $28.60 and for Parcel is $29.80. |
[
{
"content": "Let $x_1< x_2 < x_3$ be the three real roots of the equation $\\sqrt{2014} x^3 - 4029x^2 + 2 = 0$. Find $x_2(x_1+x_3)$.",
"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... | 3.456 N in the positive z-axis |
[
{
"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... | 130.8*10^6 V |
[
{
"content": "Given the volume \\( V = xyz \\) and the constraint on surface area \\( S(x,y,z) = xy + yz + zx - 5 = 0 \\), confirm that \\( \\lim_{x \\to \\infty} V = 0 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Does there exist a prime number $p$ that does not divide $2^n - 1$ for any natural number $n$?",
"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... | $542,814.99 |
[
{
"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... | 0.44 mol |
[
{
"content": "In a regular tetrahedron \\( ABCD \\) with side length \\( \\sqrt{2} \\), the points \\( P, Q, \\) and \\( R \\) are defined by \\( \\overrightarrow{AP} = \\frac{1}{2} \\overrightarrow{AB} \\), \\( \\overrightarrow{AQ} = \\frac{1}{3} \\overrightarrow{AC} \\), and \\( \\overrightarrow{AR} = \\frac{... | \dfrac{1}{36} |
[
{
"content": "Consider a function $u \\in H^1(U)$ such that $Du = 0$ on an open and connected set $U$. If $Tu = 0$ on the boundary of $U$, determine whether $u = 0$ throughout $U$.",
"role": "user"
}
] | u = 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... | Marginal Utility of Leisure: 3.6, 3.2, 2.8, 1.6, 0.8, 1.0, 0.8, 0.6. Marginal Utility of Income: 0.78, 0.53, 0.40, 0.20, 0.13, 0.13, 0.11, 0.09 |
[
{
"content": "Find all solutions for $x$ that satisfy the equation:\n\\[ \\frac{1}{2(x-1)} = \\int_0^\\infty \\frac{\\sin (x\\arctan y)}{(1+y^2)^{x/2}(e^{y\\pi}+1)}\\,dy \\]",
"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... | $60,225.22 |
[
{
"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... | 0.72 m |
[
{
"content": "Find the equation of the plane that passes through the point \\((2,2,1)\\) and forms a tetrahedron with the smallest volume in the first octant, along with the coordinate planes.",
"role": "user"
}
] | x + y + 2z = 6 |
[
{
"content": "Determine the minimum number of colors required to paint all the vertices, edges, and diagonals of a convex $n$-gon such that:\n\n1. Any two segments emerging from the same vertex must be different colors.\n2. The color of any vertex must be different from the color of any segment emerging from it... | n |
[
{
"content": "Evaluate the integral \\( \\oint_C \\frac{1}{z+2} \\, dz \\) where \\( C \\) is the circle \\( \\left\\vert z-1 \\right\\vert=3 \\).",
"role": "user"
}
] | \pi i |
[
{
"content": "Determine the number of ways the sum \\(1 + 2 + \\cdots + 2007\\) can be expressed as a sum of consecutive positive integers.",
"role": "user"
}
] | 12 |
[
{
"content": "Find all positive integer solutions \\((x, y)\\) to the equation \\(7^x - 3^y = 4\\).",
"role": "user"
}
] | (1, 1) |
[
{
"content": "Evaluate the limit \\( \\lim_{n \\to \\infty} I_{n} \\), where \\( I_{n} = \\int_{0}^{1} x^n e^{\\sqrt{x}} \\, dx \\) and \\( n = 0, 1, 2, \\ldots \\).",
"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... | 1160 torr |
[
{
"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... | 813 L-atm/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... | 28.4 liters |
[
{
"content": "Given that the functions \\( f(x) \\) and \\( g(x) \\) are defined for all real numbers and satisfy the equation \\( f(x-y) = f(x) g(y) - g(x) f(y) \\), with the condition \\( f(-2) = f(1) \\neq 0 \\), find the value of \\( g(1) + g(-1) \\).",
"role": "user"
}
] | -1 |
[
{
"content": "Evaluate the integral \\( \\int_C \\cos \\left( \\cos \\frac{1}{z} \\right) dz \\) where \\( C \\) is a closed curve enclosing the origin in the positive sense.",
"role": "user"
}
] | 0 |
[
{
"content": "Find the values of \\(x\\) such that two of the numbers \\(\\log _{\\sqrt{2 x-3}}(x+1)\\), \\(\\log _{2 x^{2}-3 x+5}(2 x-3)^{2}\\), \\(\\log _{x+1}\\left(2 x^{2}-3 x+5\\right)\\) are equal and the third is smaller by 1.",
"role": "user"
}
] | 4 |
[
{
"content": "Describe the ring obtained by adding an inverse of the element \\((2, 0)\\) to the ring \\(\\Bbb{R \\times R}\\).",
"role": "user"
}
] | \mathbb{R} |
[
{
"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... | 1.44e+05 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... | 21.71 kW |
[
{
"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... | nitrogen oxide |
[
{
"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 flows from operating activities:
Cash receipts from customers: 327042
Cash paid to suppliers and employees: -232413
Cash generated from operations: 94629
Interest paid: -11345
Income taxes paid: -8466
Net cash generated (used) from operating activities: 74818
Cash flows from investing activities:
Net change in property, plant and equipment: -48628
Gains from sale of asset: 1928
(Acquisition) Liquidation of portfolio investments: -17002
Net cash generated (used) in investing activities: -63702
Cash flows from financing activities:
Proceeds (Payments) from short-term borrowings: -1740
Proceeds (Payments) from long-term payable: -6732
Dividends: -5917
Net cash generated (used) in financing activities: -14389
Net increase in cash and cash equivalents: -3273
Cash and cash equivalents at beginning of period: 9101
Cash and cash equivalents at end of period: 5828 |
[
{
"content": "Determine whether there exists a real number \\(\\alpha\\) such that for all \\(x > -1\\), the inequality \\(\\ln(1+x) \\leq x - \\frac{x^2}{2} + \\alpha x^3\\) holds. If such an \\(\\alpha\\) exists, find its value.",
"role": "user"
}
] | \dfrac{1}{3} |
[
{
"content": "Let $(R, \\mathfrak m)$ be a complete local normal domain of dimension $2$ with an algebraically closed residue field $R/\\mathfrak m$ of characteristic $0$. Assume that Spec$(R)$ has a rational singularity, and let $\\pi: X \\to \\text{Spec}(R)$ be the minimal resolution of singularities with the... | 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... | $875.38 |
[
{
"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... | $76.74 million |
[
{
"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... | 0.568 kmph |
[
{
"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... | 477626.6355 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... | Model 94: 2,875 units
Model 81: 8,625 units |
[
{
"content": "Given $a$, $b$, and $c$ as fixed positive integers, there are $a+b+c$ ducks sitting in a circle, each choosing either rock, paper, or scissors. Specifically, $a$ ducks choose rock, $b$ ducks choose paper, and $c$ ducks choose scissors. A move is defined as follows:\n\n- A duck picking rock sitting... | ab + bc + ca |
[
{
"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... | 4129.15 |
[
{
"content": "Consider the group $G = SL_2(F_3)$, where $F_3$ is the finite field of order 3. The order of $G$ is 24. Determine the number of Sylow-3 subgroups of $G$ and verify whether it is congruent to 1 modulo 3. If not, identify the mistake in the reasoning that leads to a contradiction with the Sylow Conj... | 4 |
[
{
"content": "Is the product space \\( \\prod\\limits_{n \\in \\mathbb{N}} A_n \\) compact when each \\( A_n = \\{0, 1\\} \\) is given the discrete topology? Justify your answer using Tychonoff's theorem.",
"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... | MnSO4*H2O + H2C2O4 + 2H2O -> MnC2O4*3H2O + H2SO4 |
[
{
"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... | $902.62 |
[
{
"content": "Suppose $x$ is a non-unit in a non-commutative ring $R$ with unity $1$. Determine whether $x$ is contained in some maximal left ideal of $R$. Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"content": "Let $f: \\mathbb{D} \\rightarrow \\mathbb{C}$ be a univalent function such that $f(0) = 0$ and $f'(0) = 1$. Determine the minimum area of the image $f(\\mathbb{D})$.",
"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... | -13200 kg*m/s |
[
{
"content": "Find the probability distribution function (PDF) for the random variable \\( Z = |X - Y| \\), where \\( X \\) and \\( Y \\) are independent random variables uniformly distributed between 0 and 1.",
"role": "user"
}
] | 2(1 - z) |
[
{
"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... | nature |
[
{
"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... | 145,122 L |
[
{
"content": "Consider a (3,3) mixed tensor $A^{ijk}_{mnp}= u^i_m v^j_n w^k_p$. Perform tensor contraction to obtain a (2,2) mixed tensor $B^{ij}_{mn}$ by setting $k=p$. Now, perform a different contraction on $A^{ijk}_{knp}$ by setting $k=m$. Determine whether this results in the same (2,2) tensor $B^{ij}_{mn}... | C^{ij}_{np} |
[
{
"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... | 476563.56 J |
[
{
"content": "Circles centered at $A$ and $B$ each have radius 2, as shown. Point $O$ is the midpoint of $\\overline{AB}$, and $OA=2\\sqrt{2}$. Segments $OC$ and $OD$ are tangent to the circles centered at $A$ and $B$, respectively, and $\\overline{EF}$ is a common tangent. What is the area of the shaded regi... | 8\sqrt{2}-4-\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... | 43.69 m/s |
[
{
"content": "Compute the limit: $$ \\lim_{n\\to\\infty}\\frac{\\dfrac{n}{1}+\\dfrac{n-1}{2}+\\dots+\\dfrac{2}{n-1}+\\dfrac{1}{n}}{\\ln(n!)} $$",
"role": "user"
}
] | 1 |
[
{
"content": "Find the radical of the ideal \\((4,x)\\) in the ring \\(\\mathbb{Z}[x]\\). Justify your steps.",
"role": "user"
}
] | (2, x) |
[
{
"content": "How many 3rd order determinants, formed using 0 and 1 as elements, have a determinant value of 0? The possible determinant values are -2, -1, 0, 1, and 2, with the number of determinants with value 2 equal to those with value -2, and the number with value 1 equal to those with value -1.",
"rol... | 338 |
[
{
"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... | 4.4 m |
[
{
"content": "Simplify the expression \\((5-\\sqrt{3}) \\sqrt{7+\\frac{5\\sqrt{3}}{2}}\\) to obtain 11.",
"role": "user"
}
] | 11 |
[
{
"content": "Let \\(f\\) be a non-decreasing and continuous function on \\([0,1]\\) such that \\(\\int_0^1 f(x) \\, dx = 2 \\int_0^1 x f(x) \\, dx\\). Given that \\(f(1) = 10.5\\), find the value of \\(f(0) + f(0.5)\\).",
"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... | c(w_1,w_2,w_3, y) = min(y^2[w_1+w_2], w_3y^{1/3}) |
[
{
"content": "Let $(M,d)$ be a compact metric space and $f: M \\to M$ be a bijective function such that $d(f(x),f(y)) \\leq d(x,y)$ for all $x, y \\in M$. Is $f$ an isometry, i.e., does $d(f(x),f(y)) = d(x,y)$ for all $x, y \\in M$?",
"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... | 7230 kg.m/s^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... | $y(r)=\frac{1}{r}\left(c_1e^{\sqrt{-a}\:r}+c_2e^{-\sqrt{-a}\:r} \right)$ |
[
{
"content": "Find the limit of the sequence as \\( n \\to \\infty \\):\n\n\\[\n\\lim _{n \\rightarrow \\infty}\\left(\\frac{3 n^{2}-5 n}{3 n^{2}-5 n+7}\\right)^{n+1}\n\\]",
"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... | 100 km/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... | 0.288mol |
[
{
"content": "Consider the group homomorphism \\(\\phi:M_2(\\mathbb{R}) \\to \\mathbb{R}\\) given by \\(\\phi(A) = \\text{trace}(A)\\). The kernel of \\(\\phi\\) is isomorphic to which of the following groups?\n\n(a) \\(M_2(\\mathbb{R})/ \\{A \\in M_2(\\mathbb{R}):\\phi(A)=0\\}\\)\n(b) \\(\\mathbb{R}^2\\)\n(c) ... | C |
[
{
"content": "What is the least possible degree of a polynomial with real coefficients that has the roots \\(2\\omega, 2+3\\omega, 2+3\\omega^2, 2-\\omega-\\omega^2\\)?",
"role": "user"
}
] | 5 |
[
{
"content": "Given the joint probability density function \\( f_{xy}(x,y) = 8xy \\) for \\( 0 < y < x < 1 \\), and the marginal densities \\( f_x(x) = 4x^3 \\) for \\( 0 < x < 1 \\) and \\( f_y(y) = 4y(1-y^2) \\) for \\( 0 < y < 1 \\), find the conditional probability \\( P(Y \\leq 1/2 \\ | \\ X > 1/2) \\).",
... | \dfrac{2}{5} |
[
{
"content": "Determine whether the following statement is true: \\( \\lim_{n\\to \\infty} r(n,k)=\\infty \\) for all \\( k \\in \\mathbb{N} \\), where \\( r(n,k)=\\dfrac {p_n \\cdots p_{n+k}}{p^k_{n+k+1}} \\) and \\( p_n \\) denotes the \\( n \\)-th prime number.",
"role": "user"
}
] | 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... | 4.98*10^(-3) mol |
[
{
"content": "Find the area enclosed by the equation: \\[ \\left\\lfloor\\frac{\\left|3x + 4y\\right|}{5}\\right\\rfloor + \\left\\lfloor\\frac{\\left|4x - 3y\\right|}{5}\\right\\rfloor = 3 \\] where \\(\\lfloor\\cdot\\rfloor\\) represents the greatest-integer function.",
"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... | (4H-pyran-4-ylidine)oxonium |
[
{
"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... | 91.3J |
[
{
"content": "Given that the angle bisector of the lines represented by the equation \\(a(x-1)^2+2h(x-1)(y-2)+b(y-2)^2=0\\) is \\(x+2y-5=0\\), and the ratio \\(\\frac{b-a}{h}\\) can be expressed as \\(\\frac{p}{q}\\), where \\(p\\) and \\(q\\) are coprime integers, find the value of \\(p+q\\).",
"role": "us... | 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... | 0.0319 rad/s |
[
{
"content": "Determine whether the following statement is true or false: For a stochastic matrix (where each row sums to 1), if an eigenvalue is 0, then the corresponding eigenfunction is identically 0.",
"role": "user"
}
] | B |
[
{
"content": "Find the solution to the partial differential equation \\( \\frac{\\partial}{\\partial t} f(x,y,t) = x \\frac{\\partial f}{\\partial y} - y \\frac{\\partial f}{\\partial x} \\) with the initial condition \\( f(x,y,0) = x^2 + y^2 \\).",
"role": "user"
}
] | x^2 + y^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... | 133.5 N |
[
{
"content": "Let \\( f \\in \\mathcal{C}^1(\\mathbb{R}^3,\\mathbb{R}) \\) be a function such that:\n\\[ f(0,1,1)=0, \\quad \\frac{\\partial f}{\\partial x}(0,1,1)=1, \\quad \\frac{\\partial f}{\\partial y}(0,1,1)=2, \\quad \\frac{\\partial f}{\\partial z}(0,1,1)=3. \\]\nDetermine the limit:\n\\[ \\lim_{t \\to ... | 3 |
[
{
"content": "Triangle $ABC$ has side lengths $AB=7, BC=8,$ and $CA=9.$ Circle $\\omega_1$ passes through $B$ and is tangent to line $AC$ at $A.$ Circle $\\omega_2$ passes through $C$ and is tangent to line $AB$ at $A.$ Let $K$ be the intersection of circles $\\omega_1$ and $\\omega_2$ not equal to $A.$ Then $A... | 11 |
[
{
"content": "For each pair of real numbers \\((x, y)\\) with \\(0 \\leq x \\leq y \\leq 1\\), consider the set\n\\[ A = \\{ x y, x y - x - y + 1, x + y - 2 x y \\}. \\]\n\nLet the maximum value of the elements in set \\(A\\) be \\(M(x, y)\\). Determine the minimum value of \\(M(x, y)\\).",
"role": "user"
... | \dfrac{4}{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... | (- 8.94 i^ + 55.6 j^ + 32.4 k^) * 10^-10 T |
[
{
"content": "If $f(x)=\\frac{x+4}{x^2+ax+b}$, and $f(x)$ has two vertical asymptotes at $x=1$ and $x=-2$, find the sum of $a$ and $b$.",
"role": "user"
}
] | -1 |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\to 0}\\frac{x-\\sin(x)\\cos(x)}{\\sin(x)-\\sin(x)\\cos(x)}.$$",
"role": "user"
}
] | \dfrac{4}{3} |
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