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... | 2.706 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... | $482,991.34 |
[
{
"content": "Verify the correctness of the following integral setup after changing the order of integration in polar coordinates:\n\nEvaluate the integral \\( \\int\\int_D \\sqrt{x^2+y^2} \\, dx \\, dy \\) where \\( D = \\{(x,y): x \\leq x^2+y^2 \\leq 2x\\} \\).\n\nAfter applying the change of variables \\( x ... | Yes |
[
{
"content": "Evaluate the limit \\( \\lim_{(x,y) \\to (0,0)} \\frac{\\sin(xy^2)}{x^2+y^2} \\).",
"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... | 6.105% |
[
{
"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... | Mg : |
[
{
"content": "Evaluate the integral of the function \\( f(x,y,z) = z \\) over the region defined by the constraints \\( y^2 + z^2 = 9 \\), \\( x = 0 \\), \\( z = 0 \\), and \\( y = 3x \\). Set up and compute the integral:\n\\[\n\\int_0^1 \\int_{3x}^3 \\int_0^{\\sqrt{9-y^2}} z \\, dz \\, dy \\, dx\n\\]",
"ro... | \dfrac{27}{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... | 11 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... | $170,000 |
[
{
"content": "Evaluate the expression: \\( \\sum_{d|n}\\frac{1}{d}\\sum_{d'|d}d'\\mu(d') \\)",
"role": "user"
}
] | \dfrac{1}{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... | 99 km/hr |
[
{
"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... | $22,000 |
[
{
"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... | 16.016e12 C |
[
{
"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... | 0.95 m |
[
{
"content": "Let $G$ be a group, $H$ a normal subgroup of index $n$, and $K$ a subgroup of $G$ not contained in $H$. If $n$ is a power of a prime, determine whether $G = HK$.",
"role": "user"
}
] | No |
[
{
"content": "In preparation for the World Cup, the coach of country $F$ evaluates players $A_{1}, A_{2}, \\ldots, A_{7}$ in 3 training matches, each lasting 90 minutes. Only one player is on the field at any time. Players $A_{1}, A_{2}, A_{3},$ and $A_{4}$ each spend a total time on the field that is divisible... | 208806 |
[
{
"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... | 494 km/h |
[
{
"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... | 6.7 cm behind the mirror |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 23.64m |
[
{
"content": "Find all functions $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ that depend on a real number $\\alpha$ and satisfy the equation:\n\n$$\nf(f(x+y) f(x-y))=x^{2}+\\alpha y f(y)\n$$\n\nfor all $x, y \\in \\mathbb{R}$. Provide a complete characterization of such functions.",
"role": "user"
}
] | f(x) = x |
[
{
"content": "Determine the smallest positive integer $n$ such that each cell in an $n \\times n$ grid can be filled with one of the numbers $-1, 0, 1$, and the sums of the numbers in each row and each column are all different.",
"role": "user"
}
] | 2 |
[
{
"content": "What is the product of the coordinates of the midpoint of a line segment with endpoints at $(2,3)$ and $(-6,5)$?",
"role": "user"
}
] | -8 |
[
{
"content": "In the right triangular prism $ABC-A_{1}B_{1}C_{1}$, where $\triangle ABC$ is a right triangle with $\theta ACB = 90^\text{°}$, $BC = CC_1 = 2$, and $AC = 4\\sqrt{2}$, point $P$ is located on the line segment $BC_1$. Determine the minimum value of the sum $CP + PA_1$. Provide your answer as a sing... | 2\sqrt{13} |
[
{
"content": "Let \\( G = \\left\\{ \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\mid a, b, c, d \\in \\mathbb{Z}, ad - bc = 1 \\right\\} \\) be the group of 2x2 integer matrices with determinant 1. Let \\( H = \\left\\{ \\begin{pmatrix} 1 & n \\\\ 0 & 1 \\end{pmatrix} \\mid n \\in \\mathbb{Z} \\right\\} \... | \infty |
[
{
"content": "Evaluate the contour integral \\( \\int_{c} f(z) \\, dz \\) where \\( c \\) is the square with vertices at \\( i, 2-i, 2+i, \\) and \\( -i \\), and \\( f(z) = \\frac{z^2 - iz}{z+1} \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Let $a$ and $b$ be nonnegative real numbers such that\n\\[\\sin (ax + b) = \\sin 29x\\]for all integers $x.$ Find the smallest possible value of $a.$",
"role": "user"
}
] | 10 \pi - 29 |
[
{
"content": "Find the sum of all possible values of $f(2)$ such that $f(x)f(y) - f(xy) = \\frac{y}{x} + \\frac{x}{y}$ for every positive real numbers $x, y$. \\(\\textbf{(A)}\\ \\frac{5}{2} \\qquad\\textbf{(B)}\\ -\\frac{5}{4} \\qquad\\textbf{(C)}\\ \\frac{5}{4} \\qquad\\textbf{(D)}\\ \\frac{3}{2} \\qquad\\tex... | A |
[
{
"content": "Does a continuous function \\( f: ]a,b[ \\rightarrow ]0,\\infty[ \\) ensure that a maximal solution of the differential equation \\( x' = f(x) \\), defined on \\( ]\\alpha,\\omega[ \\), is a homeomorphism between \\( ]\\alpha,\\omega[ \\) and \\( ]a,b[ \\)?",
"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/(1*10^11) |
[
{
"content": "Compute the value of the double integral \\( \\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty \\exp(-x^2 y^2) \\; dx \\, dy \\).",
"role": "user"
}
] | \infty |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 11.31% |
[
{
"content": "Given an $n_1 \\times n_1$ matrix $E$ and a $p_1 \\times n_1$ matrix $C$, it is known that the rank of the matrix \\( \\begin{bmatrix} I_{n_1} - E \\\\ C \\end{bmatrix} \\) is equal to the rank of $C$. Prove that the rank of the matrix \\( \\begin{bmatrix} E \\\\ C \\end{bmatrix} \\) is $n_1$. ",
... | n_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... | 6.642e17 m |
[
{
"content": "Does the construction of the fundamental group of a connected space require the axiom of choice? Answer yes or no.",
"role": "user"
}
] | No |
[
{
"content": "Evaluate the line integral of the vector field \\( \\vec{F}(x, y) = \\langle x^2, y^2 \\rangle \\) along the boundary of a triangle with vertices at \\((0,0)\\), \\((1,0)\\), and \\((0,1)\\), following a counter-clockwise direction.",
"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... | 3.0 m/s |
[
{
"content": "Find all invertible matrices $A$ over a field $\\mathbb{F}$ such that $A^2 = A$. Provide your answer in terms of matrix properties.",
"role": "user"
}
] | I |
[
{
"content": "Determine the value of \\( \\alpha \\in \\mathbb{R} \\) such that the wave equation\n\\[\n\\begin{align*}\n & u_{tt}-c^2u_{xx}=0 & \\text{in } Q \\\\\n & u(t,0)=0 & t \\ge 0 \\\\\n & u_t(t,ct)=\\cos t - \\alpha & t \\ge 0\n\\end{align*}\n\\]\nhas a solution in \\( C^2(Q) \\cap C^1(\\overline{Q}) \... | 1 |
[
{
"content": "Determine whether the order of a Sylow $p$-subgroup of a group $G$ with $p^nq^m$ elements, where $p < q$ are primes and the order of $[p] \\in \\mathbb{Z}_q^{\\times}$ is larger than $n$, guarantees the existence of a normal Sylow $p$-subgroup.",
"role": "user"
}
] | No |
[
{
"content": "Determine all positive integers $n$ such that when magpies at each vertex of a regular $n$-gon fly away and return to the vertices (not necessarily their original ones), there must exist three magpies such that the triangles formed by their initial and final vertices are all either acute, right, o... | n \geq 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... | 3H_2O + Se +2Cd^{2+} -> 2Cd + SeO_3^{2-} + 6H^+ |
[
{
"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... | 110 g |
[
{
"content": "Calculate the residue of the function \\( f(z) = \\frac{e^z + 1}{\\sin^2(iz)} \\) at the point \\( z = i\\pi \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Determine the number of points at which the function \\( f(x) = \\lim_{n\\to\\infty}\\left(4^n+x^{2n}+\\frac{1}{x^{2n}}\\right)^{\\frac{1}{n}} \\) is non-derivable.",
"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... | 2.78e-5 kg/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... | [B] = 6.9e-4 M
[C] = 3.21e-4 M |
[
{
"content": "Determine the necessary and sufficient condition on the exponent $d$ such that $g^d$ is a primitive root modulo $29$, given that $g$ is a primitive root modulo $29$.",
"role": "user"
}
] | \gcd(d, 28) = 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... | 3.3 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... | 7.13 V |
[
{
"content": "The probability of snow for each of the next three days is $\\frac{2}{3}$. What is the probability that it will snow at least once during those three days? Express your answer as a common fraction.",
"role": "user"
}
] | \dfrac{26}{27} |
[
{
"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: $969,642.20
Expenses:
Sales commission expense: $106,660.64
Rent expense: $25,600
Advertising expense: $116,357.06
Salaries expense: $209,090
Payroll tax expense: $24,154.92
Depreciation: $26,000
Total expenses: $507,862.62
Income before taxes: $461,779.58
Income tax expense: $138,533.87
Net Income: $323,245.71 |
[
{
"content": "Determine whether the number of non-isomorphic posets of cardinality $n$ overwhelms the number of non-isomorphic groups of cardinality $n$. A function $f:\\mathbb{Z}_{\\geq 1}\\to\\mathbb{Z}_{\\geq 1}$ overwhelms $g:\\mathbb{Z}_{\\geq 1}\\to\\mathbb{Z}_{\\geq 1}$ if for any $k\\in \\mathbb{Z}_{\\g... | 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... | 1.6e5 m/s |
[
{
"content": "Determine the difference in the number of solutions to the equation \\(z^2 + 1 = 0\\) in the tessarine number system compared to the split-complex number system.",
"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... | $92.71 |
[
{
"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... | -1.1 rev/min^2 |
[
{
"content": "Determine the Galois group of the polynomial \\(x^4 - 4x^2 - 11\\) over \\(\\mathbb{Q}\\).",
"role": "user"
}
] | D_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... | 0.55 m |
[
{
"content": "Find an ideal \\( I \\) of \\( \\mathbb{Z}[\\sqrt{3}] \\) generated by an integer prime such that \\( \\mathbb{Z}[\\sqrt{3}]/I \\) is not an integral domain.",
"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... | 4650 |
[
{
"content": "Let $C$ be a smooth projective plane curve, and let $P$ be a point on $C$. The tangent line to $C$ at $P$ is denoted by $\\ell$. The dual curve to $C$ is $C^*$, in the dual plane $(\\mathbb{P}^2)^*$. The line $P^*$ in $(\\mathbb{P}^2)^*$ parameterizes the lines passing through the point $P$. Given... | Yes |
[
{
"content": "What is the 100th derivative of the function \\( y=\\ln(2x-x^2) \\) evaluated at \\( x=1 \\)?\n\nChoose the correct answer from the options below:\n\n(a) \\( 2 \\times 99! \\)\n\n(b) \\( -2 \\times 99! \\)\n\n(c) \\( 2 \\times 101! \\)\n\n(d) \\( -2 \\times 101! \\)",
"role": "user"
}
] | 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... | 10\sqrt{3} 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... | Rugen Company\nBalance Sheet\nJuly 31, 2013\n\nAssets:\nCash $8,200\nAccounts receivable 10,780\nEquipment 38,000\nLess: Accumulated depreciation 6,000\nTotal Assets$50,980\n\nLiabilities\nAccounts payable $5,100\nSalaries and wages payable 2,980\nNotes payable (long-term) 3,000\nTotal Liabilities 11,080\n\nShareholders' Equity:\nCommon stock 20,000\nRetained earnings 19,900 (July 31, 2013)\nTotal Shareholders' Equity39,900\n\nTotal Liabilities and Stockholders' Equity$50,980 |
[
{
"content": "Determine the smallest natural number that cannot be expressed as \\(\\frac{2^{a} - 2^{b}}{2^{c} - 2^{d}}\\), where \\(a, b, c,\\) and \\(d\\) are natural numbers.",
"role": "user"
}
] | 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... | 35.44 min |
[
{
"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... | 29.6N upward |
[
{
"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... | decrease |
[
{
"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... | 154 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... | 1.17 g |
[
{
"content": "Determine the maximum number of faces of an n-sided pyramid that can be perpendicular to its base.",
"role": "user"
}
] | 2 |
[
{
"content": "Find the probability density function of the random variable $U = Y_1 + Y_2$, where $(Y_1, Y_2)$ is uniformly distributed over the region defined by $0 \\leq y_1 \\leq 1$, $0 \\leq y_2 \\leq 1$, and $0 \\leq y_1 + y_2 \\leq 1$. Express your answer in terms of $u$. \\[ \\int_0^1\\int_0^{u-y_1}\\,dy... | 2u |
[
{
"content": "Calculate the integral \\( \\int_1^2 \\frac{1}{x^2} \\, dx \\) using Darboux sums with the partition \\( 2^{\\frac{i}{n}} \\) for \\( 0 \\leq i \\leq n \\).",
"role": "user"
}
] | \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... | episodic |
[
{
"content": "Given the quadratic function \\( f(x) = ax^2 + bx + c \\), where it is known that \\( |f(x)| \\le 100 \\) for all \\( |x| \\le 1 \\), determine an upper bound for \\( |a| + |b| + |c| \\).",
"role": "user"
}
] | 300 |
[
{
"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... | 5.5 s |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to 0^+} \\frac{e^{-1/x}}{x^k}, \\text{ where } k \\in \\mathbb{N} \\cup \\{0\\}. \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Find the coefficient of \\(x^{50}\\) in the expansion of \\((\\sum_{n=1}^{\\infty}x^n)^3\\).",
"role": "user"
}
] | 1176 |
[
{
"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... | 10.93 Btu/lb |
[
{
"content": "Determine the number of orbits of the action of $GL_3(\\mathbb{R})$ on $\\mathbb{R}^3$ by left multiplication.",
"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... | 8196.16 |
[
{
"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... | 748.40 |
[
{
"content": "Find all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) such that for all real numbers \\( x \\) and \\( y \\), the following equation holds:\n\\[ f(x f(x-y))-f(f(y) f(x-y))=(x-y)^{2}. \\]",
"role": "user"
}
] | f(x) = x |
[
{
"content": "Find the sum of the infinite series \\( \\sum_{n=1}^{\\infty} \\frac{n^2}{2^n} \\).",
"role": "user"
}
] | 6 |
[
{
"content": "Determine if there exists a $3 \\times 3$ invertible matrix $B$ such that $B^2 = I$ and $\\det(B) = -1$. Justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\( \\rho: G \\to GL(V) \\) be a finite-dimensional representation of a group \\( G \\). Show that if the tensor product representation \\( \\rho \\otimes \\rho: G \\to GL(V \\otimes V) \\) is irreducible, then the vector space \\( V \\) is one-dimensional.",
"role": "user"
}
] | 1 |
[
{
"content": "Find the value of \\( x \\) such that the matrix \\( H = \\begin{bmatrix} 2 & 2 & 1 \\\\ -4 & -1 & 4 \\\\ 4 & 5 & x \\end{bmatrix} \\) has at least one eigenvalue equal to 2.",
"role": "user"
}
] | -1 |
[
{
"content": "Determine the value of \\( p \\) in a subgame perfect equilibrium if player I commits to a mixed strategy \\((1-p, p)\\) for strategies \\(A\\) and \\(B\\), and player II knows the probability \\((1-p, p)\\) applied to both strategies.",
"role": "user"
}
] | \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... | 799 ml |
[
{
"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... | 2.97 m/s |
[
{
"content": "Let \\( f: \\mathbb{C} \\to \\mathbb{C} \\) be a meromorphic function analytic at \\( 0 \\) satisfying \\( f\\left(\\frac{1}{n}\\right) = \\frac{n}{2n+1} \\) for all \\( n \\ge 1 \\). Determine if \\( f(2) = \\frac{1}{4} \\).",
"role": "user"
}
] | \dfrac{1}{4} |
[
{
"content": "Calculate the dual norm \\(\\|T\\|_{E^*}\\) of the linear functional \\(T: V \\rightarrow \\mathbb{R}\\), where \\(V\\) is the vector space of continuous functions \\(v:[0,1] \\rightarrow \\mathbb{R}\\) with \\(v(0) = 0\\), equipped with the supremum norm \\(\\|v\\| = \\sup_{x \\in [0,1]} |v(x)|\\... | 1 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\left( \\frac{1}{n+1} + \\frac{1}{n+2} + \\cdots + \\frac{1}{kn} \\right) \\] where \\( k > 1 \\) is an integer.",
"role": "user"
}
] | \ln 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... | 455.2 g |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\frac{\\ln(\\cos(4x))}{\\ln(\\cos(3x))} \\]",
"role": "user"
}
] | \dfrac{16}{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... | 126.67 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... | 1.96 |
[
{
"content": "Verify that \\( \\int \\sec x\\ dx = \\frac{1}{2} \\ln \\left|\\frac{1+\\sin x}{1-\\sin x}\\right| + C \\) by differentiating the right-hand side and showing it equals \\( \\sec x \\).",
"role": "user"
}
] | \sec 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... | 258714.5 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... | 0.7976 m |
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