prompt list | ground_truth string |
|---|---|
[
{
"content": "Find the minimal distance between the origin and the intersection of the surfaces given by the equations \\(x^2 = 2yz\\) and \\(x^2 + 3y^2 + 2z^2 = 30\\).",
"role": "user"
}
] | \sqrt{10} |
[
{
"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... | 38.24 N |
[
{
"content": "Let $G$ be a group of order $p^n$, where $p$ is a prime number, and let $H$ be a subgroup of $G$ with order $p^{n-2}$ that is not normal in $G$. Determine whether the index $[G:N_G(H)]$ is equal to $p$.",
"role": "user"
}
] | p |
[
{
"content": "Let \\( Q \\in \\mathbb{Z}[x] \\) be a polynomial that defines an injective function from \\( \\mathbb{Z} \\) to \\( \\mathbb{Z} \\). Determine whether there exists a prime \\( p \\) such that \\( Q \\) defines an injective function from \\( \\mathbb{Z}/p\\mathbb{Z} \\) to \\( \\mathbb{Z}/p\\mathb... | Yes |
[
{
"content": "Given a function $f$ such that $f^2$ is Riemann integrable, determine whether $f$ is always Riemann integrable. 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... | $314,994.42 |
[
{
"content": "Given that $y_1 = \\sin(t^2)$ is a solution to the differential equation $t y'' - y' + 4t^3 y = 0$, use the method of reduction of order to find a second linearly independent solution.",
"role": "user"
}
] | \cos(t^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... | 0.12 Nm |
[
{
"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... | 8.73 |
[
{
"content": "Determine the radius of convergence for the power series solution of the differential equation \\( u''+(x+1)u'-u=0 \\) with initial conditions \\( u(-1)=2 \\) and \\( u'(-1)=0 \\).",
"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... | 8.00 cm |
[
{
"content": "Evaluate the integral \\( \\int_{\\Lambda}\\left(\\sin\\frac{1}{z}+\\sin^2\\frac{1}{z}\\right)dz \\), where \\( \\Lambda \\) is a circle centered at zero in the complex plane excluding zero. Choose the correct value from the following options:\n\n(a) \\( 0 \\) \n(b) \\( \\pi i \\) \n(c) \\( 2\\p... | 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... | 16.40% |
[
{
"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... | \frac{1}{(\sqrt{t^{2}+1})^3} ~\rm{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... | 20,700 direct labor hours |
[
{
"content": "Suppose $M$ is a complex manifold and $\\Omega$ is a bounded pseudoconvex domain in $M$. Let $u:M\\setminus\\Omega\\to\\mathbb{R}$ be a pluriharmonic function. Determine whether $u$ can be extended as a pluriharmonic function to the entire manifold $M$. Assume $\\dim_{\\mathbb{C}}(M)\\geq 2$. Prov... | Yes |
[
{
"content": "Determine the conditions under which the identity \\(A^2 - B^2 = (A+B)(A-B)\\) holds true for matrices \\(A\\) and \\(B\\).",
"role": "user"
}
] | AB = BA |
[
{
"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... | A nonpolar solvent such as hexane or carbon tetrachloride |
[
{
"content": "Let $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ be a differentiable function such that $f'(x) = [f(x)]^2$ for all $x \\in \\mathbb{R}$ and $f(0) = 0$. Prove that $f(x) = 0$ for all $x \\in \\mathbb{R}$.",
"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... | 7.27% |
[
{
"content": "Find the limit of the integral \\( \\lim_{n\\rightarrow \\infty} \\int_0^\\infty n f(x) e^{-n|x|} \\, dx \\), where \\( f: [0,\\infty) \\rightarrow \\mathbb{R} \\) is Lebesgue integrable and \\( \\lim_{x\\rightarrow 0} f(x) = 2016 \\), using the Dominated Convergence Theorem.",
"role": "user"
... | 2016 |
[
{
"content": "Let \\(a_n\\) and \\(b_n\\) be two sequences such that \\(a_n \\xrightarrow{n\\to\\infty} a > 1\\) and \\(b_n \\xrightarrow{n\\to\\infty} b > 1\\). Find \\(\\lim_{n\\to\\infty}\\sqrt[n]{a_n^n+b_n^n+2018}\\).",
"role": "user"
}
] | \max\{a, 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... | $15,168 |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty}\\left(\\frac{1}{n}+\\frac{1}{2^n-1}\\sum_{k=1}^{n} \\frac{n \\choose k}{k}\\right)$$",
"role": "user"
}
] | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 93 nm |
[
{
"content": "Find the quadratic factor \\( f(x) = x^2 + bx + c \\), where \\( b, c \\in \\mathbb{R} \\), that is common to both quartic polynomials \\( x^4 + 6x^2 + 25 \\) and \\( 3x^4 + 4x^2 + 28x + 5 \\). Determine \\( f(x) \\).",
"role": "user"
}
] | x^2 - 2x + 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... | The marginal tax rate is 35%, and the average tax rate is 35%. |
[
{
"content": "Given the following system of equations, where \\(q, r, s,\\) and \\(t\\) are different constant values, find the sum \\(q + r + s + t\\):\n\\[\n\\frac{1}{qa+1} + \\frac{1}{qb+1} + \\frac{1}{qc+1} + \\frac{1}{qd+1} = 1\n\\]\n\\[\n\\frac{1}{ra+1} + \\frac{1}{rb+1} + \\frac{1}{rc+1} + \\frac{1}{rd+1... | 0 |
[
{
"content": "Determine whether the statement \"If a finite abelian group $G$ has multiple elements of order 3, then the product of all elements in $G$ is the identity element\" is true or false.",
"role": "user"
}
] | False |
[
{
"content": "Given an $n \\times n$ matrix $B$, define $e^B$ by \\[ e^B = \\sum_{j=0}^{\\infty} \\frac{B^j}{j!} \\] Let $p$ be the characteristic polynomial of $B$. Determine the matrix $e^{p(B)}$. Provide your answer in terms of $B$ and its properties.",
"role": "user"
}
] | I |
[
{
"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... | 1.57 |
[
{
"content": "Each outcome on the spinner below has equal probability. If you spin the spinner three times and form a three-digit number from the three outcomes, such that the first outcome is the hundreds digit, the second outcome is the tens digit and the third outcome is the units digit, what is the probabil... | \frac{2}{9} |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{\\pi}\\left\\{\\int_{y^2}^{\\pi}\\frac{y\\sin x}{x} \\, dx\\right\\}dy \\).",
"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% |
[
{
"content": "Determine the smallest integer \\( n \\) such that the inequality \\[ \\sum_{i=n}^{100} \\binom{100}{i} \\left(\\frac{80}{100}\\right)^i \\left(\\frac{20}{100}\\right)^{100-i} \\geq 0.9 \\] holds.",
"role": "user"
}
] | 75 |
[
{
"content": "Evaluate the integral \\( \\int_\\gamma \\frac{1}{1+z^2} \\, dz \\) using Cauchy's Theorem, where \\( \\gamma \\) is the circular contour defined by \\( t \\mapsto 2+e^{it} \\) for \\( 0 < t < 2\\pi \\).",
"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... | 88.6 degrees |
[
{
"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... | 1.62 m/s^2 |
[
{
"content": "Find the value of \\( p \\) such that the limit \\( \\lim_{x \\to \\infty} \\left( x^p \\left( \\sqrt[3]{x+1} + \\sqrt[3]{x-1} - 2\\sqrt[3]{x} \\right) \\right) \\) is a finite, non-zero number.",
"role": "user"
}
] | \dfrac{5}{3} |
[
{
"content": "Find all continuous functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) that satisfy the equation \\( f(xy) = f\\left(\\frac{x^2 + y^2}{2}\\right) + (x-y)^2 \\) for all real numbers \\( x \\) and \\( y \\).",
"role": "user"
}
] | f(x) = -2x + C |
[
{
"content": "Suppose $b > 1$ is a real number such that $\\log_5 (\\log_5 b + \\log_b 125) = 2$. Find the value of $\\log_5 \\left(b^{\\log_5 b}\\right) + \\log_b \\left(125^{\\log_b 125}\\right)$. \\(\\boxed{\\text{Your answer here}}\\)",
"role": "user"
}
] | 619 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 327 degree Celsius, Liquid phase |
[
{
"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... | (4.88, 0.52) |
[
{
"content": "Evaluate whether the following limit holds true:\n\\[ \\lim\\limits_{n\\to\\infty} \\int_{-\\infty}^{+\\infty}f(x)\\sin(nx)dx = 0 \\]\nwhen \\( \\int_{-\\infty}^{+\\infty}f(x)dx \\) is convergent but not absolutely convergent.",
"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... | Barium |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\sum_{r=1}^n \\frac{r^4}{4r^2-1} \\]",
"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... | -7.33 cm/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... | Rate of Inflation = (Current CPI-U - Previous CPI-U) / Previous CPI-U * 100 |
[
{
"content": "Evaluate the line integral of the conservative vector field \\(G(x,y,z) = (y^2+z^2)i + (2xy+z)j + (2xz+y)k\\) along the parametric curve \\(r(t) = (t^2, t^3, t)\\) from \\(t=0\\) to \\(t=1\\).",
"role": "user"
}
] | 3 |
[
{
"content": "Determine whether the sequence of operators \\( A_{n}x(t) = t^{n}(1-t)x(t) \\) converges in the operator norm on \\( C[0, 1] \\).",
"role": "user"
}
] | Yes |
[
{
"content": "For a real symmetric matrix with repeated eigenvalues, is it always possible to find an orthogonal matrix \\( P \\) such that the eigen decomposition satisfies a given equation?",
"role": "user"
}
] | Yes |
[
{
"content": "Provide an example of a linear operator $T \\in L(V)$ on the vector space $V$ of infinite sequences of finite length, such that:\n\n1. $T$ admits an adjoint, i.e., there exists $T^*$ such that $\\langle Ta, b \\rangle = \\langle a, T^*b \\rangle$ for all $a, b \\in V$.\n2. $T^*T$ is the identity o... | 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... | 26.4 kJ |
[
{
"content": "Calculate the limit: $$\\lim_{x \\to 0^+}\\frac{\\tan x \\cdot \\sqrt {\\tan x}-\\sin x \\cdot \\sqrt{\\sin x}}{x^3 \\sqrt{x}}$$",
"role": "user"
}
] | \dfrac{3}{4} |
[
{
"content": "Calculate the sum of the infinite series: \\[ \\sum_{j=0}^{\\infty} (j+1) \\left(\\frac{1}{1.05}\\right)^{j+1} \\]",
"role": "user"
}
] | 420 |
[
{
"content": "Let $A$ be a full triangulated subcategory of $B$, and let $u: A \\rightarrow B$ be the corresponding embedding. Consider a triangulated functor $f: B \\rightarrow A$ satisfying the following conditions:\n\n1. $f \\circ u = \\text{id}$\n2. For any $b \\in B$, if $f(b) = 0$, then $b = 0$.\n\nDeterm... | K_0(A) = K_0(B) |
[
{
"content": "Let $f$ be a continuous function such that $f(f(x)) = 1 + x$. Determine the value of $f'(1)$. ",
"role": "user"
}
] | 1 |
[
{
"content": "The area of one lateral face of a right pyramid with an equilateral triangular base is 75 square meters. If the slant height is 30 meters, what is the length of the side of its base, in meters?",
"role": "user"
}
] | 5 |
[
{
"content": "Let $f:(G,*) \\rightarrow (H,\\circ)$ be a homomorphism between groups $(G,*)$ and $(H,\\circ)$. If $a \\in G$ and $f(a)$ has an inverse in $H$, does it follow that $a$ has an inverse in $G$? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "In the figure, $ABCD$ and $BEFG$ are squares, and $BCE$ is an equilateral triangle. What is the number of degrees in angle $GCE$?\n\n[asy]\n\ndraw(rotate(32)*shift((-.48,.85))*unitsquare); draw(unitsquare);\ndraw( (-.85, .46) -- (0,0));\nlabel(\"$C$\", (-.85, .46), SW); label(\"$E$\", (0, 0), S); ... | 45^\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... | $85,215 |
[
{
"content": "For a linear transformation $T$ on an $n$-dimensional vector space $V$, if there exists a vector $\\mathbf{v}$ such that the set $\\{\\mathbf{v}, T(\\mathbf{v}), T^2(\\mathbf{v}), \\ldots, T^{n-1}(\\mathbf{v})\\}$ is linearly independent, does it imply that each eigenvalue of $T$ has a unique eige... | 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... | 206 months or 17.18 years |
[
{
"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... | n=PV/RT |
[
{
"content": "Determine the value of \\( y_0 \\) such that for the initial value problem (IVP) given by \\( y'+\\frac{2x^2-4xy-y^2}{3x^2}=0, x>0, y(1)=y_0 \\), the condition \\( \\lim_{x\\to 0}y'(x) \\neq 1 \\) is satisfied, while \\( \\lim_{x\\to 0}y'(x) = 1 \\) for all other values of \\( y_0 \\). What is thi... | -2 |
[
{
"content": "Suppose \\(x_1, x_2, x_3, x_4,\\) and \\(x_5\\) are the roots of the equation \\(x^5 - 2x^4 + x^3 + 1 = 0\\). Define a matrix \\(A\\) with elements \\(a_{ij} = 1 + x_i x_j\\) when \\(i = j\\), and \\(a_{ij} = x_i x_j\\) otherwise, where \\(1 \\leq i, j \\leq 5\\). Compute the determinant of matrix... | 3 |
[
{
"content": "Piravena must make a trip from $A$ to $B$, then from $B$ to $C$, then from $C$ to $A$. Each of these three parts of the trip is made entirely by bus or entirely by airplane. The cities form a right-angled triangle as shown, with $C$ a distance of 3000 km from $A$ and with $B$ a distance of 3250 ... | 7500 |
[
{
"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 N |
[
{
"content": "Let $p$ be a prime number. Determine the order of the group $G$, which consists of all $2 \\times 2$ matrices over $\\mathbb{Z}_p$ with determinant $1$, under matrix multiplication.",
"role": "user"
}
] | p(p^2 - 1) |
[
{
"content": "Calculate the expected distance between two points chosen randomly and independently on the circumference of a circle with radius 1.",
"role": "user"
}
] | \dfrac{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... | 81 J/s.K |
[
{
"content": "Given that $a > 0$, if $f(g(a)) = 8$, where $f(x) = x^2 + 8$ and $g(x) = x^2 - 4$, what is the value of $a$?",
"role": "user"
}
] | 2 |
[
{
"content": "Find the greatest common divisor (gcd) of the numbers $2020^{1830} + 2$ and $2020^{1830} - 2$. Provide your answer as a single integer.",
"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... | 52.1% |
[
{
"content": "Calculate the area under the curve \\( y = \\sqrt{x} \\) on the interval \\([0, 1]\\) using Darboux sums.",
"role": "user"
}
] | \dfrac{2}{3} |
[
{
"content": "A $24 \\times 8$ table is filled such that each row contains a permutation of the numbers $1, 2, \\ldots, 8$. In each column, the numbers are multiplied together. Determine the minimum possible sum of all the column products.",
"role": "user"
}
] | 8 \times (8!)^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... | $40,457.42 |
[
{
"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... | 2230.84 ft^3/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... | 1867 kg/m^3 |
[
{
"content": "Determine the number of distinct terms in the expansion of \\(\\left( x + \\frac{1}{x} + x^2 + \\frac{1}{x^2} \\right)^{20}\\).",
"role": "user"
}
] | 81 |
[
{
"content": "Determine the largest natural number \\( m \\) such that the polynomial \\( P(x) = x^5 - 3x^4 + 5x^3 - 7x^2 + 6x - 2 \\) is divisible by \\((x-1)^m\\).",
"role": "user"
}
] | 3 |
[
{
"content": "Consider a homogeneous Markov chain $(X_n)_{n \\in \\mathbb{N}}$ with the following transition probabilities: the only missing arrow is from state 0 to state 0 with $\\mathbb{P}(X_1=0 \\mid X_0=0) = 1/5$. Compute $h_0^6$, the probability of starting from state 0 and eventually hitting state 6. Giv... | \dfrac{1}{4} |
[
{
"content": "Find a 4-tuple \\((a, b, n, k) \\in \\mathbb{N}^4\\) with \\(a, b, n, k > 1\\) and \\(a \\neq b\\) such that the reverse of \\(a^n\\) is equal to \\(b^k\\).",
"role": "user"
}
] | (12, 21, 2, 2) |
[
{
"content": "Let $K = \\mathbb{Q}(\\sqrt[3]{3}, \\eta)$ where $\\eta = (e^{\\frac{\\pi}{3}i})^2$. Determine the degree of the field extension $[K: \\mathbb{Q}(\\sqrt[3]{3})]$. \\(\\eta\\) is a primitive sixth root of unity.",
"role": "user"
}
] | 2 |
[
{
"content": "Let \\( g(t) = t \\) for all \\( t \\in [-1,1] \\) and \\( f(t) = t^{1/n} \\), where \\( n \\) is a large odd integer. Consider the operator \\( T: C[-1,1] \\rightarrow \\mathbb{R} \\), defined by \\( T_g(f) = \\int_{-1}^1 g(t)f(t)dt \\). Given the norm \\( \\|f\\| = \\max_{t \\in [-1,1]} |f(t)| \... | 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... | 1.066e+3 g |
[
{
"content": "Evaluate the infinite series: \\( \\sum_{k=1}^{\\infty} \\frac{(2k + 1)^2}{2^k} \\)",
"role": "user"
}
] | 33 |
[
{
"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... | 852 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... | 107 dB |
[
{
"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... | 5.69e-2 m |
[
{
"content": "Given a sequence \\( \\{f_n\\} \\) of conformal, one-to-one maps from the unit disk \\( \\mathbb{D} \\) onto the right half-plane \\( \\mathbb{A} = \\{ z \\in \\mathbb{C} : \\mathfrak{R} z > 0 \\} \\), suppose \\( \\{f_n\\} \\) converges to \\( f \\) uniformly on compact subsets of \\( \\mathbb{D}... | 0 |
[
{
"content": "Given the function \\( f(x) = 1 + \\int_a^x{(x-t)^n f(t)\\,dt} \\) with the condition \\( f(a) = 1 \\), find a formula for \\( f^{(n+1)}(a) \\).",
"role": "user"
}
] | 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... | 74.1 mL |
[
{
"content": "Let \\( f: \\mathbb{C} \\rightarrow \\mathbb{C} \\) be a differentiable function such that \\( |f(z)-z|^{4/3} \\le 1 + 3|f(z)-z|^{6/5} \\). Find \\( f'(-i) \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Determine the Lebesgue measure of the set $B = \\{(x,y) \\in [0,1] \\times [0,1] : \\forall n (\\{x_n, y_n\\} \\subset \\{1,5\\}) \\}$, where $x = 0.x_1x_2 \\ldots$ and $y = 0.y_1y_2 \\ldots$ are the decimal representations of $x$ and $y$, respectively.",
"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... | 750 seconds or 12.5 minutes |
[
{
"content": "Let $P$ be a point in coordinate space, where all the coordinates of $P$ are positive. The line between the origin and $P$ is drawn. The angle between this line and the $x$-, $y$-, and $z$-axis are $\\alpha,$ $\\beta,$ and $\\gamma,$ respectively. If $\\cos \\alpha = \\frac{1}{3}$ and $\\cos \\... | \frac{\sqrt{191}}{15} |
[
{
"content": "Compute the following limit without using L'Hopital's rule: $$\\lim _{x\\to 0}\\left(\\frac{xe^x-2+2\\cos x-x}{\\left(\\sin x\\right)^2 \\tan\\left(2x\\right)+xe^{-\\frac{1}{\\left|x\\right|}}}\\right)$$",
"role": "user"
}
] | \dfrac{1}{4} |
[
{
"content": "Find the smallest positive real number \\( C \\) such that for any positive integer \\( k \\), the following inequality holds:\n\\[ S = \\sum_{i=k-\\lfloor \\frac{k-1}{2} \\rfloor+1}^k \\frac{1}{i} + \\sum_{i=k-\\lceil \\frac{k-1}{2} \\rceil+1}^k \\frac{1}{i} < C. \\]",
"role": "user"
}
] | 2\ln 2 |
[
{
"content": "Calculate the limit: \\[ \\lim_{x \\to 0}((-2^x+1)^{\\sin(x)}) \\]",
"role": "user"
}
] | 1 |
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