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... | 28 m |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} n \\cos x \\cos(\\cos x) \\cdots \\underbrace{\\cos(\\cos(\\cdots(\\cos x)))}_{n \\text{ times of } \\cos}. \\]",
"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... | $148,300 |
[
{
"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... | 880.6 N |
[
{
"content": "If a group \\( G \\) does not contain an element of order 2, can we conclude that for all elements \\( a \\) in \\( G \\), \\( a^2 = e \\) where \\( e \\) is the identity element?",
"role": "user"
}
] | No |
[
{
"content": "Given a continuous function \\( f: (0, \\infty) \\to \\mathbb{R} \\) such that for all \\( x > 0 \\), \\( y > 0 \\), and any \\( x' \\) in \\((0, \\infty)\\), the following holds:\n\\[ \\int_{x}^{xy} f(u) \\, du = \\int_{x'}^{x'y} f(u) \\, du. \\]\nIf \\( f(17) = 17 \\), calculate \\( A(x) = \\int... | 289 \ln x |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | -0.8 m/s |
[
{
"content": "Determine the number of ordered integer pairs \\((a, b)\\) such that the equation \\(x^{2} + ax + b = 167y\\) has integer solutions \\((x, y)\\), where \\(1 \\leq a, b \\leq 2004\\).",
"role": "user"
}
] | 2020032 |
[
{
"content": "Calculate the expected value of the following limit:\n\\[ \\lim_{N \\to \\infty} \\left( \\prod_{i = 1}^{N} X_{i} \\right)^{\\dfrac{1}{N}} \\]\nwhere each random variable $X_i$ for $i = 1, \\ldots, N$ is an independently and identically distributed random integer from 1 to 9.",
"role": "user"
... | 4.147 |
[
{
"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... | 9.0 J |
[
{
"content": "Evaluate the integral \\( \\int_T x^2 \\, dV \\), where \\( T \\) is the tetrahedron bounded by the planes \\( x=0 \\), \\( y=0 \\), \\( z=0 \\), and \\( 4x+2y+z=4 \\). Set up the region of integration correctly for the triple integral.",
"role": "user"
}
] | \dfrac{2}{15} |
[
{
"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... | 1660.67 kJ |
[
{
"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... | $129,94,670 |
[
{
"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... | $25,882 |
[
{
"content": "Identify the possible topological forms of a compact surface $Y$ such that for any compact surface $X$, the connected sum $X\\#Y$ is homeomorphic to $X$.",
"role": "user"
}
] | S^2 |
[
{
"content": "Given two primes $p$ and $q$, determine an upper bound for the smallest natural number $K$ such that the set of natural number multiples $pN + qN$ spans all natural numbers greater than or equal to $K$. Provide your answer as an expression or a specific value for $K$.",
"role": "user"
}
] | pq - p - q + 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... | 36J |
[
{
"content": "Suppose $\\mu_n$ for $n\\in \\mathbb{N}$ and $\\mu$ are probability measures on $\\mathbb{C}^d$ with uniformly bounded support. If \\(\\int_{\\mathbb{C}^d} f(z_1, \\ldots, z_d) \\, d\\mu_n \\to \\int_{\\mathbb{C}^d} f(z_1, \\ldots, z_d) \\, d\\mu \\) for any polynomial $f$ in $z_1, \\ldots, z_d, \... | Yes |
[
{
"content": "Determine whether all continuous functions \\(g: (\\mathbb{R}, \\tau_{ST}) \\to (\\mathbb{N}, \\tau_{c})\\) are constant functions, where \\(\\tau_{ST}\\) is the standard topology on \\(\\mathbb{R}\\) and \\(\\tau_{c}\\) is the cofinite topology on \\(\\mathbb{N}\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Are two finite positive measures \\(\\mu\\) and \\(\\nu\\) on \\(\\mathbb{R}^N\\) with the Borel \\(\\sigma\\)-algebra equal if their integrals over all bounded continuous functions \\(f\\) are equal?",
"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... | 25.98 km/s |
[
{
"content": "Determine the Lebesgue measure of the subset $A$ of $[0,1]$, where $A$ consists of all numbers that do not contain the digit $4$ in their decimal expansion.",
"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... | 0.4988 |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{2\\pi}\\frac{1}{1+\\cos^2(\\theta)}d\\theta \\) using contour integration techniques.",
"role": "user"
}
] | \pi \sqrt{2} |
[
{
"content": "Given the probability density functions \\( f_x(x) = \\begin{cases} \\frac{x^2}{9}, & 0 < x < 3, \\\\ 0, & \\text{otherwise}, \\end{cases} \\) and \\( f_{y \\mid x}(y) = \\begin{cases} \\frac{4y^3}{x^4}, & 0 < y < x, \\\\ 0, & \\text{otherwise}, \\end{cases} \\) find the probability \\( P(Y < 1) \... | \dfrac{1}{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... | Cu + NO_3^- + 4H_3O^+ + e^- \to Cu^{2+} + NO +6H_2O |
[
{
"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... | 0.0456 L |
[
{
"content": "Given the real numbers \\( x \\) and \\( y \\) satisfy the equations:\n\\[ 2^x + 4x + 12 = \\log_2{(y-1)^3} + 3y + 12 = 0 \\]\nfind the value of \\( x + y \\).",
"role": "user"
}
] | -2 |
[
{
"content": "Evaluate the integral \\( \\int_{\\gamma} |z-1| |dz| \\), where \\( \\gamma(t) = e^{it} \\) for \\( t \\in [0, 2\\pi] \\).",
"role": "user"
}
] | 8 |
[
{
"content": "Determine whether the space of infinitely differentiable functions, $C^{\\infty}(\\mathbb{R}^n)$, is dense in the Sobolev space $H^{k}(\\mathbb{R}^n)$ for $n \\geq 1$ and $k \\geq 0$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Calculate the curvature of the space curve given by \\( r(t) = \\langle \\sin 2t, \\cos 2t, 4t \\rangle \\) at the point \\( P(0,1,0) \\).",
"role": "user"
}
] | \dfrac{1}{5} |
[
{
"content": "Find the fixed point of the operator \\( T(f) = \\int_0^x \\frac{f(y)}{2} + \\frac{yf(y)}{2}\\, dy \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Find the value of $x$ such that $[809x] = [214]$ in $\\mathbb{Z}_{5124}$, where $0 \\le x < 5124$. Use the inverse $[5105]$ of $[809]$ in $\\mathbb{Z}_{5124}$ to help solve the problem.",
"role": "user"
}
] | 1058 |
[
{
"content": "Evaluate the limit: $$ \\lim_{{(x,y)\\to (0,0)}}\\frac{\\sin(x^2+9y^2+|x|+|3y|)}{|x|+|3y|} $$",
"role": "user"
}
] | 1 |
[
{
"content": "Determine whether every Riemann integrable function can be approximated arbitrarily closely by a linear combination of monotone functions. Provide a justification for your answer.",
"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... | x = 11, y = 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... | 36.6% |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 9:51 |
[
{
"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... | 1.70 * 10^-5 C^{-1} |
[
{
"content": "Determine the cardinality of the \\(\\sigma\\)-algebra generated by the sets \\(X_{i}:=\\{(x_1,x_2,...,x_n):x_i > \\frac{1}{2}\\}\\) for \\(i=1,2,...,n\\), where \\(X=[0,1]^{n}\\).",
"role": "user"
}
] | 2^{2^n} |
[
{
"content": "Evaluate the double integral \\( \\iint_{\\Omega} (x+y^2) \\, dx \\, dy \\) where \\( \\Omega \\) is the region bounded by the lines \\( x+y=0 \\), \\( x+y=1 \\), \\( 2x-y=0 \\), and \\( 2x-y=3 \\).",
"role": "user"
}
] | \dfrac{22}{27} |
[
{
"content": "Compute the limit: \\[ \\lim_{x \\to 0} \\left(\\frac{x^5 e^{-1/x^2} + x/2 - \\sin(x/2)}{x^3}\\right) \\]",
"role": "user"
}
] | \dfrac{1}{48} |
[
{
"content": "In the diagram, $K$, $O$ and $M$ are the centers of the three semi-circles. Also, $OC = 32$ and $CB = 36$.\n\n[asy]\npair A, K, O, C, M, B, X, Y, Z;\nO=(0,0);\nC=(32,0);\nM=(50,0);\nB=(68,0);\nA=(-68,0);\nK=(A+C)/2;\nX=(0,68);\nY=(-18,50);\nZ=(50,18);\npath nom, bigc, middlec, smallc;\nnom=A--B--(... | 1250\pi |
[
{
"content": "Find the first three digits of \\( \\frac{1}{x} \\) in \\( \\mathbb{Z}/7^3\\mathbb{Z} \\), where \\( x = 1+2\\cdot7+4\\cdot7^2 +1\\cdot7^3 +\\ldots \\in \\mathbb{Z}_7 \\).",
"role": "user"
}
] | 156 |
[
{
"content": "Consider a full column-rank square matrix with entries of either $0$ or $1$. Is it always possible to rearrange the columns of this matrix such that the diagonal of the resulting matrix consists entirely of $1$'s?",
"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... | Project A |
[
{
"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... | 0.50 |
[
{
"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... | 3402.619 |
[
{
"content": "How many different counting numbers will each leave a remainder of 5 when divided into 47?",
"role": "user"
}
] | 5 |
[
{
"content": "Is it accurate to apply the fundamental theorem of calculus to conclude that the integral $$\\int_{S_1^+(0)}\\left(1 - \\frac{1}{z^2}\\right)e^{z + \\frac{1}{z}}dz$$ is zero, given that the function has an antiderivative in $\\mathbb{C}\\setminus\\{0\\}$ and the path of integration lies within thi... | 0 |
[
{
"content": "Find the maximum value of $k$ such that there exists at least one point on the line $y = kx - 2$ which can be the center of a circle with radius 1 that intersects the circle given by $x^2 + y^2 - 8x + 15 = 0$.",
"role": "user"
}
] | \dfrac{4}{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... | 20296.8J |
[
{
"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... | 0.23 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... | $3800 billion |
[
{
"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... | 1.05 |
[
{
"content": "Find the remainder when \\((1+x)^{2010}\\) is divided by \\(1+x+x^{2}\\).",
"role": "user"
}
] | 1 |
[
{
"content": "Consider a regular octagon. How many triangles can be formed whose vertices are the vertices of the octagon?",
"role": "user"
}
] | 56 |
[
{
"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... | 600 ft |
[
{
"content": "What is the smallest two-digit prime that is the sum of three (not necessarily distinct) positive perfect cubes?",
"role": "user"
}
] | 17 |
[
{
"content": "Determine the computational complexity of finding the largest multiplicity of eigenvalues for a Hermitian matrix using the specified method.",
"role": "user"
}
] | O(n^3) |
[
{
"content": "Evaluate the line integral \\(\\int_C \\mathbf{F} \\cdot d\\mathbf{r}\\) for \\(\\mathbf{F} = x\\,\\mathbf{i} + 3y\\,\\mathbf{j} + 3x^2\\,\\mathbf{k}\\) where \\(C\\) is the circle \\(x^2 + y^2 = 16\\) lying in the plane \\(z = 3\\), oriented clockwise.",
"role": "user"
}
] | 0 |
[
{
"content": "A quadrilateral in the plane has vertices $(1, 3)$, $(1, 1)$, $(2, 1)$ and $(2006, 2007)$. How many square units is the area of the quadrilateral?",
"role": "user"
}
] | 3008 |
[
{
"content": "Determine whether the following proposition is true or false: Let \\( f:\\overline{D(0,1)}\\longrightarrow \\mathbb{C} \\) be a holomorphic function on \\( D(0,1) \\) and continuous on \\( \\overline{D(0,1)} \\) such that there is a sequence \\( z_n\\in D(0,1) \\) with \\( z_n\\longrightarrow a \\... | True |
[
{
"content": "Calculate the residue of the function \\( g(z) = \\frac{1}{z^2 - \\cosh(z)} \\) at \\( z = 0 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Calculate the value of \\( K \\) in the following expression:\n\\[\nK = \\lim_{(x,y) \\to (0,0)} \\frac{xy}{\\sqrt{x^2+y^2}}\\cdot \\frac{1-\\cos(x^2+y^2)}{(x^2+y^2)^2}\n\\]",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\frac{\\sqrt[n]{e} + 2\\sqrt[n]{e^2} + \\cdots + n\\sqrt[n]{e^n}}{n^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... | 1.01 GW |
[
{
"content": "Let $a$ and $t$ be positive real numbers. Consider the curve $C$ given by $y = ax^3$. A tangent line $\\\\ell\\\\$ is drawn to the curve at the point $P(t, at^3)$. This tangent line intersects the curve $C$ again at a point $Q$. The $x$-coordinate of $Q$ is given by $-Et$. Determine the value of $... | 2 |
[
{
"content": "Find all real values of \\(a\\) such that the quadratic equation \\(x^2 + (a+i)x - 5i = 0\\) has at least one real solution.",
"role": "user"
}
] | -5 |
[
{
"content": "Determine the range of values of $p > 0$ for which the improper integral \\( \\int_0^{\\infty} x^p e^{-x^8 \\sin^2 x} \\, dx \\) converges.",
"role": "user"
}
] | (0, 3) |
[
{
"content": "Given the differential equation \\( y'' = y' + y \\) with initial conditions \\( y(0) = 4 \\) and \\( y'(0) = 7 \\), find the value of \\( y^{(5)}(0) \\), the fifth derivative of \\( y \\) evaluated at zero.",
"role": "user"
}
] | 47 |
[
{
"content": "Identify the only multiplicative functional on the algebra $M_n(\\mathbb{C})$ for $n > 1$.",
"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... | 1.53e+23 CH_3Cl molecules |
[
{
"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... | 0.23 A |
[
{
"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... | 10758.4 |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to 1} \\frac{(1-x)(1-x^2)\\cdots(1-x^{2n})}{[(1-x)(1-x^2)\\cdots(1-x^n)]^2}$$",
"role": "user"
}
] | \dbinom{2n}{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... | 0.3 Hz |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\int_0^x \\frac{\\cos(t^3)}{t+x} \\, dt \\]",
"role": "user"
}
] | \ln 2 |
[
{
"content": "Determine the radius of convergence for the power series \\( \\sum_{n = 0} ^ {\\infty} \\frac {1}{\\pi (n)} x^n \\), where \\( \\pi(n) \\) is the number of prime numbers less than or equal to \\( 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... | 300,000 |
[
{
"content": "Consider the sequence \\((x_n)\\) defined by \\(x_0 = 1\\), \\(x_n = 2x_{n−1} + 1\\) if \\(n\\) is odd, and \\(x_n = 3x_{n−1} + 2\\) if \\(n\\) is even. Determine the value of \\(\\lim_{n\\to \\infty} \\frac{\\log(x_n)}{n}\\). Choose the correct option:\n\n(a) The limit does not exist.\n(b) \\(\\l... | 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... | 0.0225 J |
[
{
"content": "Determine whether a bijection exists between the set $|3^{\\mathbb N}|$ and the power set $\\mathcal P(\\mathbb N)$, given that a bijection exists between $2^{\\mathbb N}$ and $\\mathcal P(\\mathbb N)$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Does there exist a function that is continuous at all points except for a countable dense subset and has a finite supremum?",
"role": "user"
}
] | Yes |
[
{
"content": "Let $m$ be the smallest positive integer such that $m^2 + (m+1)^2 + \\cdots + (m+10)^2$ is the square of a positive integer $n$. Find the value of $m+n$. \\( \\text{(Express your answer as a single integer.)} \\)",
"role": "user"
}
] | 95 |
[
{
"content": "Determine whether the following statement is true or false: For a curve \\( \\gamma \\) that is continuously differentiable on \\([a, b)\\), the length of the curve can be expressed as \\( \\Lambda(\\gamma) = \\lim_{d \\to b} \\int_a^d |\\gamma'(t)| \\, dt \\).",
"role": "user"
}
] | A |
[
{
"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 rad/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... | $1,336.01 |
[
{
"content": "A number $P$ is randomly chosen from the uniform distribution on the interval [0,1]. A coin with probability $P$ of landing heads is flipped $n$ times. Let $X$ be the number of heads obtained. Compute the probability $P(X=k)$ for a given integer $k$.",
"role": "user"
}
] | \dfrac{1}{n+1} |
[
{
"content": "In the right parallelepiped $ABCDA^{\\prime}B^{\\prime}C^{\\prime}D^{\\prime}$, with $AB=12\\sqrt{3}$ cm and $AA^{\\prime}=18$ cm, consider the points $P$ on $AA^{\\prime}$ and $N$ on $A^{\\prime}B^{\\prime}$ such that $A^{\\prime}N=3B^{\\prime}N$. Find the length of the line segment $AP$ such tha... | \dfrac{27}{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... | $1.88 |
[
{
"content": "Evaluate the definite integral: \\[ \\int_{0}^{2} x^3 \\sqrt{2x - x^2} \\, dx \\]",
"role": "user"
}
] | \dfrac{7\pi}{8} |
[
{
"content": "Let \\( T: C[0,1] \\rightarrow C[0,1] \\) be defined by \\( T(f(x))= \\int_{0}^{1} \\sin(x+y)f(y)dy \\). Determine the dimension of the range space of \\( T \\).",
"role": "user"
}
] | 2 |
[
{
"content": "Find the multiplicative inverse of \\( \\alpha = x^2 + \\langle 3 + x^3 \\rangle \\) in the quotient ring \\( \\frac{\\mathbb{Z}_7[x]}{\\langle 3+x^3\\rangle} \\).",
"role": "user"
}
] | 2x |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{\\infty} \\frac{x^2}{(x^2 + a^2)^2} \\, dx \\).",
"role": "user"
}
] | \dfrac{\pi}{4a} |
[
{
"content": "Find the surface area of the cone defined by $x^2 + y^2 = z^2$ that lies above the $xy$-plane and beneath the cylinder $y^2 + z^2 = 16$. Provide your answer in terms of exact values or expressions.",
"role": "user"
}
] | 16\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... | 14.82 % |
[
{
"content": "In the triangular prism \\( S-ABC \\), the three lateral edges are pairwise perpendicular. Let \\( O \\) be a point inside the base triangle \\( \\triangle ABC \\). Determine the minimum value of \\( W = \\tan \\angle OSA \\cdot \\tan \\angle OSB \\cdot \\tan \\angle OSC \\).",
"role": "user"
... | 2\sqrt{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... | 21% |
[
{
"content": "Given an infinite set \\(P\\), is there an ordering relation \\(\\leq\\) such that the poset \\((P, \\leq)\\) is surjectively rigid, meaning the only order-preserving surjective map \\(f: P \\to P\\) is the identity \\(\\text{id}_P\\)?",
"role": "user"
}
] | Yes |
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