messages listlengths 2 2 | deepmath_final_answer stringlengths 0 305 | solution_column stringclasses 1
value | solution_clipped bool 2
classes |
|---|---|---|---|
[
{
"role": "user",
"content": "Let $\\mathcal{H}^{\\oplus n}$ denote the n-dimensional Hilbert space. Define a linear operator $A = [a_{i,j}]: \\mathcal{H}^{\\oplus n} \\to \\mathcal{H}^{\\oplus n}$ by\n\\[\nA\\begin{bmatrix}\nh_{1}\\\\\n\\vdots\\\\\nh_{n}\n\\end{bmatrix}=\n \\begin{bmatrix}\n\\sum_{k=1}^{n}... | Yes | r1_solution_1 | false |
[
{
"role": "user",
"content": "Evaluate the integral \\( \\int_{0}^{+\\infty} \\frac{M}{a}x \\big(1-\\exp(-\\frac{x}{a})\\big)^{M-1}\\exp(-\\frac{x}{a})dx \\)."
},
{
"role": "assistant",
"content": "Okay, so I have this integral to solve: the integral from 0 to infinity of (M/a) times x times (1 ... | a \sum_{k=1}^{M} \frac{1}{k} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Evaluate the contour integral \\( \\int_{\\gamma}{\\frac{e^{z^{2}}}{z-1}dz} \\), where \\( \\gamma \\) is the rectangle with vertices at \\( (0, -1), (3, -1), (3, 1), (0, 1) \\)."
},
{
"role": "assistant",
"content": "Alright, let's tackle this contour integral prob... | 2\pi i e | r1_solution_1 | false |
[
{
"role": "user",
"content": "Let $y \\in \\mathbb{R}^n \\setminus \\{0_n\\}$ and let $X \\subset \\mathbb{R}^n$ be a compact polytope. Determine whether there exists $(x_0, \\lambda_0) \\in X \\times \\mathbb{R}_+$ such that the projection of $\\lambda y$ onto $X$ is $x_0$ for all $\\lambda \\geq \\lambda_... | Yes | r1_solution_1 | true |
[
{
"role": "user",
"content": "Compute the expected value of the piecewise function \\( u(x,y) \\) given two independent continuous random variables \\( x \\) and \\( y \\) with uniform distribution in the range \\([0,b]\\). The joint PDF is \\( f(x,y)=\\frac{1}{b^2} \\). The piecewise function is defined as... | \dfrac{5b}{12} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Determine the values of $p$ for which the series \\( \\sum_{n=2}^{\\infty} \\frac{\\sin(\\frac{\\pi}{n})}{n^p} \\) converges."
},
{
"role": "assistant",
"content": "Okay, so I need to figure out for which values of p the series ∑_{n=2}^∞ sin(π/n)/n^p converges. Hmm,... | p > 0 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Calculate the symmetric-decreasing rearrangement of the function \\( f(x) = x \\) on the interval \\([0, 10]\\), with \\( f(x) = 0 \\) elsewhere."
},
{
"role": "assistant",
"content": "Okay, so I need to find the symmetric-decreasing rearrangement of the function f(... | 10 - 2|x - 5| | r1_solution_1 | false |
[
{
"role": "user",
"content": "Consider a square matrix with orthogonal, non-zero rows that are not orthonormal. Must each row have exactly one non-zero element for the columns of the matrix to be orthogonal as well? Answer yes or no."
},
{
"role": "assistant",
"content": "Okay, so I need to figu... | No | r1_solution_1 | false |
[
{
"role": "user",
"content": "Evaluate the limit: \\[ \\lim_{x\\to\\infty} x\\left(\\sqrt{x^2+2x}-2\\sqrt{x^2+x}+x\\right) \\]"
},
{
"role": "assistant",
"content": "Okay, so I have this limit to evaluate: the limit as x approaches infinity of x times (sqrt(x² + 2x) minus 2 times sqrt(x² + x) pl... | -\dfrac{1}{4} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Calculate the value of \\(13^{498} \\mod 997\\)."
},
{
"role": "assistant",
"content": "Okay, so I need to calculate 13 raised to the 498th power modulo 997. Hmm, 997 is a prime number, right? Let me check that first. Well, 997 is a well-known prime number; it's the... | 1 | r1_solution_1 | true |
[
{
"role": "user",
"content": "Determine the values of \\( q \\ge 1 \\) for which the function \\( f(x) = |x-a|^q \\) is strictly convex, where \\( a \\in \\mathbb{R} \\) is fixed."
},
{
"role": "assistant",
"content": "Alright, so I need to figure out for which values of \\( q \\ge 1 \\) the fun... | q > 1 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Find the value of \\( n \\) such that \\( \\sum_{r=0}^{n}(2r+1)\\binom{n}{r}=2^{n+4} \\)."
},
{
"role": "assistant",
"content": "Okay, so I need to find the value of \\( n \\) such that the sum \\( \\sum_{r=0}^{n}(2r+1)\\binom{n}{r} = 2^{n+4} \\). Hmm, let me think ... | 15 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Evaluate the limit \\( \\lim_{h \\to 0} \\frac{f(a-h^2)-f(a)}{h} \\) where \\( f \\) is a function differentiable at \\( a \\)."
},
{
"role": "assistant",
"content": "Okay, so I need to find the limit as h approaches 0 of [f(a - h²) - f(a)] divided by h. Hmm, let me... | 0 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Find the value of the convergent series: \\[ \\sum_{n=0}^\\infty 3^{n-1}\\sin^3\\left(\\frac{\\pi}{3^{n+1}}\\right) \\]"
},
{
"role": "assistant",
"content": "Okay, so I need to find the value of the convergent series: \n\n\\[ \\sum_{n=0}^\\infty 3^{n-1}\\sin^3\\lef... | \dfrac{\pi}{12} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Find the value of the fractional part of \\( \\frac{3^{1001}}{82} \\)."
},
{
"role": "assistant",
"content": "Okay, so I need to find the fractional part of \\( \\frac{3^{1001}}{82} \\). Hmm, fractional part means the part after the decimal point, right? So if I hav... | \dfrac{3}{82} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Evaluate the integral: \\[ \\int_{-\\infty}^\\infty e^{-x^{-2}} \\, dx \\]"
},
{
"role": "assistant",
"content": "Alright, so I have this integral to evaluate: the integral from negative infinity to positive infinity of e^{-x^{-2}} dx. Hmm, that looks a bit tricky. ... | \infty | r1_solution_1 | false |
[
{
"role": "user",
"content": "Determine whether there exists a continuous onto function from the interval $[0,1)$ to the interval $(0,1)$. Provide a justification for your answer."
},
{
"role": "assistant",
"content": "Okay, so I need to figure out if there's a continuous onto function from [0,1... | Yes | r1_solution_1 | false |
[
{
"role": "user",
"content": "Determine whether the sequence defined by \\(a_{n+2} = \\sqrt{a_n} + \\sqrt{a_{n+1}}\\) with initial conditions \\(a_1 > 0\\) and \\(a_2 > 0\\) converges or diverges. If it converges, find the limit."
},
{
"role": "assistant",
"content": "Okay, so I need to figure o... | 4 | r1_solution_1 | true |
[
{
"role": "user",
"content": "Determine the type of conic section formed by the intersection of the plane \\(2x + y + z - 2 = 0\\) and the cone \\(x^2 + y^2 = z^2\\)."
},
{
"role": "assistant",
"content": "Okay, so I need to figure out what kind of conic section is formed when the plane \\(2x + ... | hyperbola | r1_solution_1 | false |
[
{
"role": "user",
"content": "Evaluate the double integral \\( \\int_{B} \\int \\sin(y + x) \\, dB \\) over the triangular region \\( B \\) with vertices \\((0, 0)\\), \\((\\pi, 0)\\), and \\((\\pi/2, \\pi/2)\\)."
},
{
"role": "assistant",
"content": "Alright, so I need to evaluate this double i... | \dfrac{\pi}{2} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Find the residue of the function \\( e^{\\left(\\frac{1}{z^2}\\right)} \\) at the point \\( z = 0 \\)."
},
{
"role": "assistant",
"content": "Okay, so I need to find the residue of the function \\( e^{\\left(\\frac{1}{z^2}\\right)} \\) at \\( z = 0 \\). Hmm, residue... | 0 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Evaluate the ordinal expression $1^\\omega$ using the definition of ordinal exponentiation for limit ordinals."
},
{
"role": "assistant",
"content": "Okay, so I need to evaluate the ordinal expression \\(1^{\\omega}\\) using the definition of ordinal exponentiation ... | 1 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Determine the limiting distribution of the sequence \\( U_n = \\sqrt{\\frac{1}{n}\\sum\\limits_{i=1}^n Y_i^2} \\), where \\( Y_1, Y_2, \\ldots \\) are independent and identically distributed Poisson random variables with mean 1."
},
{
"role": "assistant",
"content":... | \sqrt{2} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Evaluate the limit of the sequence \\( S_n = \\int_{0}^{1} \\frac{nx^{n-1}}{1+x}dx \\) as \\( n \\to \\infty \\). Choose the correct answer from the following options:\n\n- \\(0\\)\n- \\(\\frac{1}{2}\\)\n- \\(1\\)\n- \\(+\\infty\\)"
},
{
"role": "assistant",
"conten... | B | r1_solution_1 | false |
[
{
"role": "user",
"content": "Determine the Krull dimension of the ring \\( A = \\mathbb{Q}[\\pi,\\sqrt{11},x,y]/I \\), where \\( I = \\langle x^2-y^3+xy+3 \\rangle \\)."
},
{
"role": "assistant",
"content": "Okay, so I need to figure out the Krull dimension of the ring A, which is the quotient ... | 2 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Evaluate the integral \\( \\int\\limits_{|x|=2}\\frac{x}{\\cos (x)}\\mathrm{dx} \\) using complex analysis techniques."
},
{
"role": "assistant",
"content": "Okay, so I need to evaluate the integral \\( \\int\\limits_{|x|=2}\\frac{x}{\\cos (x)}dx \\) using complex a... | -2\pi^2 i | r1_solution_1 | false |
[
{
"role": "user",
"content": "Evaluate the complex integral \\( \\int_{|z|=2} \\frac{z^3}{z^2-2z+1} \\, dz \\)."
},
{
"role": "assistant",
"content": "Okay, so I need to evaluate this complex integral: the integral of z cubed divided by (z squared minus 2z plus 1) around the circle where the abs... | 6\pi i | r1_solution_1 | false |
[
{
"role": "user",
"content": "Given that \\( X_1, X_2, \\ldots, X_n \\sim N(\\mu, 1) \\), consider the estimator \\( \\hat{g} = \\overline{X}^2 \\) for the mean squared \\( g(\\mu) = \\mu^2 \\). Calculate \\( E_{\\mu}(\\overline{X}^2) - \\mu^2 \\)."
},
{
"role": "assistant",
"content": "Okay, so... | \dfrac{1}{n} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Compute the integral of \\( z^2 + z \\overline{z} \\) over the contour \\( C = \\{ z \\mid |z| = 1 \\} \\), where \\( \\overline{z} \\) is the complex conjugate of \\( z \\)."
},
{
"role": "assistant",
"content": "Okay, so I need to compute the integral of \\( z^2 +... | 0 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Solve the equation \\(2^{-3x^3+5x^2-x}=\\frac{x^2+1}{x}\\) for all possible values of \\(x\\)."
},
{
"role": "assistant",
"content": "Okay, so I need to solve the equation \\(2^{-3x^3 + 5x^2 - x} = \\frac{x^2 + 1}{x}\\) for all possible values of \\(x\\). Hmm, let's... | 1 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Determine whether the following statement is true or false: For a polynomial \\( f \\in \\mathbb{Z}[x] \\), is it true that \\( \\deg(\\gcd_{\\mathbb{Z}_q}(f, x^p - 1)) \\geq \\deg(\\gcd_{\\mathbb{Q}}(f, x^p - 1)) \\)? Here, \\( \\mathbb{Z}_q = \\mathbb{Z}/q\\mathbb{Z} \\) for ... | True | r1_solution_1 | false |
[
{
"role": "user",
"content": "Find the exponential generating function for the sequence \\( t_n \\), where \\( t_n \\) represents the number of ways a teacher can divide \\( n \\) students into groups, assigning one student as president and another as vice president within each group."
},
{
"role": ... | e^{x^2 e^x} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Let $R$ be an integral domain with a total ordering. Determine if the field of fractions $F$ of $R$ can be given a total ordering that is naturally induced by the ordering of $R$."
},
{
"role": "assistant",
"content": "Alright, so I need to figure out if the field o... | Yes | r1_solution_1 | false |
[
{
"role": "user",
"content": "Let \\( \\alpha \\) be a root of the polynomial \\( f(x) = x^2 - x + 2 \\) over the field \\( \\mathbb{F}_5 \\). Consider the field extension \\( \\mathbb{F} = \\mathbb{F}_5(\\alpha) \\). Determine the order of \\( \\alpha \\) in the multiplicative group \\( \\mathbb{F}^* \\)."... | 24 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Let $f$ be a continuous function on the interval $[0,1]$. Define the sequence $S(n) = \\int_0^1 x^n n f(x) \\, dx$. Determine the limit of $S(n)$ as $n$ approaches infinity."
},
{
"role": "assistant",
"content": "Okay, so I have this problem here: given a continuous... | f(1) | r1_solution_1 | false |
[
{
"role": "user",
"content": "Compute the distance between the unilateral shift operator $S$ and the set of compact operators in a Hilbert space, denoted as $d(S, \\mathbb{K}(\\mathcal{H})) = \\inf\\{\\|S-K\\| : K \\in \\mathbb{K}(\\mathcal{H})\\}$, where $S \\in \\mathbb{B}(\\mathcal{H})$ is a unilateral s... | 1 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Find the quotient group \\( \\frac{\\mathbb{Z}\\times\\mathbb{Z}\\times\\mathbb{Z}}{\\langle(1,1,1),(1,3,2)\\rangle} \\)."
},
{
"role": "assistant",
"content": "Okay, so I need to find the quotient group \\(\\frac{\\mathbb{Z} \\times \\mathbb{Z} \\times \\mathbb{Z}}... | \mathbb{Z} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Compute the Fourier transform of the function \\( f(t) = 1 - t^2 \\) for \\(|t| < 1\\) and \\(f(t) = 0\\) elsewhere. Use the result to evaluate the integral:\n\\[\n\\int_{-\\infty}^{\\infty} \\frac{\\sin t - t \\cos t}{t^3} \\, dt.\n\\]"
},
{
"role": "assistant",
"c... | \dfrac{\pi}{2} | r1_solution_1 | true |
[
{
"role": "user",
"content": "Let $X$ be a smooth projective curve of genus $g > 2$ over an algebraically closed field $k$. Determine whether there exists a line bundle $L$ on $X$ of degree $(g-1)$ such that $H^0(X, L) = 0$. Provide a justification for your answer."
},
{
"role": "assistant",
"co... | Yes | r1_solution_1 | false |
[
{
"role": "user",
"content": "What is the value of $1^i$?"
},
{
"role": "assistant",
"content": "Okay, so I need to figure out what the value of \\(1^i\\) is. Hmm, complex numbers can be tricky, especially when dealing with exponents. Let me start by recalling some basic concepts. \n\nFirst, I k... | 1 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Evaluate the double integral \\( \\int\\int_R \\frac{1}{x} \\, dx \\, dy \\) over the region \\( R \\), which is a circular disc in \\( \\mathbb{R}^2 \\) with radius \\( a \\) and center \\( (a,0) \\). Use polar coordinates to express the integral and determine the appropriate ... | 2\pi a | r1_solution_1 | false |
[
{
"role": "user",
"content": "An array $\\mathbb{T}$ has elements $T_{ijkl}$ where $i,j,k,l=1,2,3,4$. It is given that\n$$T_{ijkl}=T_{jikl}=T_{ijlk}=-T_{klij}$$\nfor all values of $i,j,k,l$. Determine the number of independent components in this array."
},
{
"role": "assistant",
"content": "Okay... | 45 | r1_solution_1 | true |
[
{
"role": "user",
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty}\\left(\\dfrac{(2n)!}{n^n\\cdot n!}\\right)^{1/n}$$"
},
{
"role": "assistant",
"content": "Okay, so I need to find the limit as n approaches infinity of the expression ((2n)! / (n^n * n!))^(1/n). Hmm, let me think about how ... | \dfrac{4}{e} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Let $S$ be the $n$-simplex defined as \\( S = \\{ x \\in \\mathbb{R}_+^n \\mid \\sum_{i=1}^n x_i = 1 \\} \\), and let $E$ be a linear subspace of codimension 2 in $\\mathbb{R}^n$ such that $E \\cap \\mathring{S} \\neq \\varnothing$. For a face $F$ of $E \\cap S$, there exists a... | Yes | r1_solution_1 | false |
[
{
"role": "user",
"content": "Find the best approximation of the polynomial \\( t^3 + t^2 + t + 1 \\) using polynomials from the subspace \\( M \\) of \\( P_4 \\), where \\( M = \\{ p \\in P_4 : \\deg{p} \\leq 2, p(t) = p(-t) \\} \\). The scalar product is defined as: \\( \\langle q, p \\rangle = \\int_{-1}... | t^2 + 1 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Consider the functions \\( f_1(x) = \\psi(x)\\cos(\\phi(x)) \\) and \\( f_2(x) = \\psi(x)\\sin(\\phi(x)) \\) defined on \\((0, \\infty)\\), where \\( \\psi: [0, \\infty) \\to [0, \\infty) \\) is a smooth, strictly increasing function with \\( \\psi(0) = 0 \\) and \\( \\psi'(0) ... | Yes | r1_solution_1 | true |
[
{
"role": "user",
"content": "A huge pie is divided among $N$ guests. The first guest receives $\\frac{1}{N}$ of the pie. Guest number $k$ receives $\\frac{k}{N}$ of what remains, for all $1 \\leq k \\leq N$. A guest is considered fortunate if their share of the pie is strictly greater than the average shar... | 0 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Consider a random walk on a finite state space $0, 1, \\\\ldots, N$ with absorbing barriers at states $0$ and $N$. At each state $1, \\\\ldots, N-1$, the probability of moving to the adjacent states is $\\\\frac{1}{2}$ each. At states $0$ and $N$, the process is absorbed, meani... | 2 | r1_solution_1 | true |
[
{
"role": "user",
"content": "Find the maximum value of the product $xyz$ given that $x, y, z$ are integers and $x + y + z = 3n$, where $n$ is a constant integer."
},
{
"role": "assistant",
"content": "Okay, so I need to find the maximum value of the product xyz, given that x, y, z are integers ... | n^3 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Find all polynomials \\(P(x)\\) such that for all real numbers \\(x, y, z\\) satisfying \\(xy + yz + zx = 0\\), the following equation holds:\n\\[ P\\left((x - y)^2\\right) + P\\left((y - z)^2\\right) + P\\left((z - x)^2\\right) = 18P\\left(\\left(\\frac{(x + y + z)^2}{3}\\righ... | 0 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Evaluate the double integral \\( \\int_{0}^1\\int_0^{\\sqrt{2y-y^2}}\\ dxdy \\) using polar coordinates and find its value."
},
{
"role": "assistant",
"content": "Okay, so I need to evaluate this double integral: the integral from y = 0 to 1, and for each y, x goes ... | \dfrac{\pi}{4} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Compute \\( \\lim_{n \\to \\infty} \\int_{0}^{\\pi/3} \\frac{1}{1+\\tan^n(x)}\\,dx \\)."
},
{
"role": "assistant",
"content": "Okay, so I need to compute the limit as n approaches infinity of the integral from 0 to π/3 of 1/(1 + tan^n(x)) dx. Hmm. Alright, let me st... | \dfrac{\pi}{4} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Find the minimal polynomial of $\\zeta_9 + \\zeta_9^{-1}$ over $\\mathbb{Q}$, given that the degree of $\\mathbb{Q}(\\zeta_9 + \\zeta_9^{-1})$ over $\\mathbb{Q}$ is 3."
},
{
"role": "assistant",
"content": "Okay, so I need to find the minimal polynomial of ζ₉ + ζ₉⁻¹... | x^3 - 3x + 1 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Suppose we have a unit square ABCD. Arbitrarily pick a point E within the interior of the unit square. Let the line through E parallel to AB intersect AD at F, and let the line through E parallel to BC intersect DC at G. What is the expected value of the area of the rectangle E... | \dfrac{1}{4} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Evaluate the integral \\( I = \\int_{-r}^r \\int_{-\\sqrt{r^2-x^2}}^{\\sqrt{r^2-x^2}} \\sqrt{1 - \\frac{x^2 + y^2}{x^2 + y^2 - r^2}} \\, dy \\, dx \\) using an appropriate substitution and the identity \\( \\frac{1}{\\sqrt{1 - x^2}} = \\frac{d}{dx} \\arcsin(x) \\)."
},
{
... | 2\pi r^2 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Let $u: M \\rightarrow N$ be an $A$-module homomorphism, and let $N'$ be the image of $u$. Suppose $a$ is an ideal of $A$, and the induced map $\\bar{u}: M/aM \\rightarrow N/aN$ is surjective. Prove that the module $N/N'/a(N/N')$ is the zero module."
},
{
"role": "assis... | 0 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Determine the upper bound of the magnitude of \\(|e^{\\sin(z)}|\\) over the line segment from \\(z = 0\\) to \\(z = i\\)."
},
{
"role": "assistant",
"content": "Okay, so I need to figure out the upper bound for the magnitude of |e^{\\sin(z)| along the line segment f... | 1 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Given $n$ independent Gaussian random variables $x_i \\sim N(0,1)$ and constants $c_i \\geq 0$, let the set $K$ contain $k$ indices corresponding to the smallest $c_i$. What is the probability that $\\sum_{i \\in K} c_ix_i \\leq 0$?"
},
{
"role": "assistant",
"conte... | \dfrac{1}{2} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Determine the degree of the map \\( \\bar{r} : \\mathbb{CP}^n \\rightarrow \\mathbb{CP}^n \\) induced by \\( r(z_0, z_1, \\ldots, z_n) = (-z_0, z_1, \\ldots, z_n) \\) on \\( \\mathbb{C}^{n+1} \\). Consider the cases for even and odd \\( n \\)."
},
{
"role": "assistant",... | 1 | r1_solution_1 | true |
[
{
"role": "user",
"content": "Calculate \\( \\lim_{n\\to\\infty}\\left( \\frac{4n^2+5n-6}{4n^2+3n-10}\\right)^{3-4n} \\)."
},
{
"role": "assistant",
"content": "Okay, so I need to find the limit as n approaches infinity of this expression: ((4n² + 5n -6)/(4n² +3n -10))^(3-4n). Hmm, that looks a ... | e^{-2} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Let $(M, d)$ be a metric space, $K$ a compact set, and $G$ an open set such that $K \\subset G$. Is it true that there exists an $\\epsilon > 0$ such that $K \\subset K_\\epsilon \\subset G$? Here, $K_\\epsilon = \\{x \\in M : d(x, K) \\leq \\epsilon\\}$ and $d(x, K) = \\inf \\... | Yes | r1_solution_1 | false |
[
{
"role": "user",
"content": "Find the value of \\(i^i\\), where \\(i\\) is the imaginary unit."
},
{
"role": "assistant",
"content": "Okay, so I need to find the value of \\(i^i\\). Hmm, I remember that \\(i\\) is the imaginary unit, defined by \\(i^2 = -1\\). But raising \\(i\\) to the power o... | e^{-\pi/2} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Find the limit of the sequence: \\( \\lim_{n \\to \\infty} (\\sqrt[3]{1-n^3} + n) \\)"
},
{
"role": "assistant",
"content": "Okay, so I need to find the limit as n approaches infinity of the sequence given by the expression cube root of (1 - n^3) plus n. Hmm, let me... | 0 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Determine whether the following statement is true or false: In any compact metric space \\( X \\), every sequence \\( \\{x_n\\} \\subset X \\) has an accumulation point in \\( X \\), but not necessarily a limit in \\( X \\)."
},
{
"role": "assistant",
"content": "Ok... | A | r1_solution_1 | false |
[
{
"role": "user",
"content": "Compute the integral \\( \\int_{-\\infty}^{+\\infty} 2^{-4^t}(1-2^{-4^t})\\,dt \\)."
},
{
"role": "assistant",
"content": "Okay, so I need to compute the integral \\( \\int_{-\\infty}^{+\\infty} 2^{-4^t}(1 - 2^{-4^t})\\,dt \\). Hmm, let me start by understanding the... | \dfrac{1}{2} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Evaluate the limit \\( \\lim_{n\\to\\infty} \\sum\\limits_{i=1}^{n^2} \\frac{e^{i/n}}{ne^n} \\) by recognizing it as a Riemann sum of an integral."
},
{
"role": "assistant",
"content": "Okay, so I need to evaluate the limit \\( \\lim_{n\\to\\infty} \\sum\\limits_{i=... | 1 | r1_solution_1 | true |
[
{
"role": "user",
"content": "Let $D \\in \\mathbb{R}^{n,n}$ be a diagonal matrix with non-zero diagonal entries $d_{ii}$. Let $\\tilde{D} \\in \\mathbb{R}^{n,n}$ be another diagonal matrix with the same diagonal entries as $D$, but possibly permuted. Does there exist a unitary matrix $U$ such that $UDU^H =... | Yes | r1_solution_1 | false |
[
{
"role": "user",
"content": "Evaluate the limit: \\[ \\lim_{x\\rightarrow 0^+} \\frac{\\ln(x)}{\\ln(\\sin x)} \\] without using l'Hôpital's rule."
},
{
"role": "assistant",
"content": "Okay, so I need to evaluate the limit as x approaches 0 from the positive side of ln(x) divided by ln(sin x), ... | 1 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Express the conjugate \\( \\bar{z} \\) in terms of \\( z \\) for the equation \\( z^2 = \\bar{z} \\)."
},
{
"role": "assistant",
"content": "Okay, so I need to find the conjugate of z, which is \\(\\bar{z}\\), in terms of z for the equation \\(z^2 = \\bar{z}\\). Hmm... | \bar{z} = z^2 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Consider a hash function \\( H(x) \\rightarrow \\{ 0,1 \\}^{160} \\) that maps inputs to 160-bit strings. What is the minimum number of attempts required to find a collision, i.e., two different inputs \\( x_1 \\) and \\( x_2 \\) such that \\( H(x_1) = H(x_2) \\)?"
},
{
... | 2^{80} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Calculate the expected stopping time $E[S]$ for the first occurrence of the sequence HTH in a series of fair coin tosses using the optional stopping theorem."
},
{
"role": "assistant",
"content": "Okay, so I need to calculate the expected stopping time E[S] for the ... | 10 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Is every countably compact subset of a Hausdorff space closed?"
},
{
"role": "assistant",
"content": "Okay, so the question is: Is every countably compact subset of a Hausdorff space closed? Hmm, interesting. Let me try to work through this step by step. First, I ne... | No | r1_solution_1 | true |
[
{
"role": "user",
"content": "Maximize the function \\( f(x,y,z) = xy + z^2 \\) subject to the constraints \\( 2x - y = 0 \\) and \\( x + z = 0 \\)."
},
{
"role": "assistant",
"content": "Okay, so I need to maximize the function f(x, y, z) = xy + z², given the constraints 2x - y = 0 and x + z = ... | \infty | r1_solution_1 | false |
[
{
"role": "user",
"content": "Evaluate the limit: \\[ \\lim_{(x,y) \\to (0,0)} \\frac{3\\tan(x^3+y^3)-(x^3+y^3)}{(x^3+y^3)^3} \\]"
},
{
"role": "assistant",
"content": "Alright, so I need to evaluate the limit:\n\n\\[\n\\lim_{(x,y) \\to (0,0)} \\frac{3\\tan(x^3+y^3)-(x^3+y^3)}{(x^3+y^3)^3}\n\\]\... | \infty | r1_solution_1 | false |
[
{
"role": "user",
"content": "Find the norm of the bounded compact operator \\( A \\) defined on the space \\( C[0,1] \\) by:\n\\[ Ax(t) = 2x(0) - tx(1), \\quad t \\in [0,1] \\]\nWhat is \\( \\|A\\| \\)?"
},
{
"role": "assistant",
"content": "Okay, so I need to find the norm of this operator A d... | 3 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Evaluate the limit: \\[ \\lim_{(x,y) \\to (0,2)} \\left(1+x \\right)^{y/x} \\]"
},
{
"role": "assistant",
"content": "Okay, so I need to evaluate the limit as (x, y) approaches (0, 2) of (1 + x)^(y/x). Hmm, let's see. I remember that when dealing with limits involvi... | e^2 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Determine whether there exists a star-Lindelöf space that is not DCCC."
},
{
"role": "assistant",
"content": "Okay, so I need to figure out if there's a star-Lindelöf space that isn't DCCC. Hmm, let me start by recalling what these terms mean. \n\nFirst, a space is ... | Yes | r1_solution_1 | false |
[
{
"role": "user",
"content": "A gun is located at the origin of an infinite number line and starts shooting bullets along the positive x-axis at a rate of one bullet per second. Each bullet's velocity is randomly chosen from a uniform distribution between 0 and 1 m/s. If two bullets collide, they explode an... | 0 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Let \\( X \\sim \\mathcal{N}(0,1) \\) be a standard normal random variable. Compute \\( \\mathbb{E}[\\cos(X)] \\)."
},
{
"role": "assistant",
"content": "Okay, so I need to compute the expectation of cos(X) where X is a standard normal random variable. Hmm, let's se... | e^{-1/2} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Determine whether the following implication is true: If \\( u(x,t) \\in W^{1,1}([0,T],L^2(\\Omega^d)) \\), then \\( u \\in L^{\\infty}([0,T],L^2(\\Omega^d)) \\). Here, \\( \\Omega \\subset \\mathbf{R}^d \\) (\\(d=2,3\\)) is a domain with a smooth boundary \\( \\Gamma \\)."
},... | True | r1_solution_1 | false |
[
{
"role": "user",
"content": "Use Green's Theorem to compute the value of the line integral \\( \\int_{\\gamma} y\\,dx + x^2\\,dy \\), where \\( \\gamma \\) is the circle given by \\( g(t) = (\\cos t, \\sin t), 0 \\leq t \\leq 2\\pi \\)."
},
{
"role": "assistant",
"content": "Okay, so I need to ... | -\pi | r1_solution_1 | false |
[
{
"role": "user",
"content": "Calculate the limit: \\[ \\lim_{n\\to \\infty}\\frac{(-1)^n\\cdot 6^n-5^{1+n}}{5^n-(-1)^{n+1}\\cdot 6^{n+1}} \\]"
},
{
"role": "assistant",
"content": "Alright, so I need to calculate the limit: \n\n\\[\n\\lim_{n\\to \\infty}\\frac{(-1)^n\\cdot 6^n - 5^{1+n}}{5^n - ... | \dfrac{1}{6} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Find the natural number \\( n \\) such that the volume of the solid formed by revolving the function \\( f(x) = \\cos(n \\arccos(x)) \\) around the x-axis over the interval \\([-1, 1]\\) is \\( \\frac{14\\pi}{15} \\)."
},
{
"role": "assistant",
"content": "Okay, so ... | 2 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Find the sum of all natural numbers $n$ such that the function $g(x) = 100|e^{x+1} - 1| - \\sum_{k=1}^n |e^{x^k + 1} - 1|$ is differentiable over the entire real line $\\mathbb{R}$. Here, $n \\in \\mathbb{N}$. Provide your answer as a single number."
},
{
"role": "assis... | 39 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Find the value of $a$ such that \\( \\lim_{x \\to 0^+} \\ln(x\\ln a)\\ln\\bigg(\\frac{\\ln(ax)}{\\ln(\\frac{x}{a})}\\bigg)=6 \\)."
},
{
"role": "assistant",
"content": "Okay, so I need to find the value of \\( a \\) such that the limit as \\( x \\) approaches 0 from... | e^3 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Given a distribution with probability density function \\( f(x) = \\frac{2x}{49} \\), find the probability that the 5th smallest observation from a sample exceeds 0.01."
},
{
"role": "assistant",
"content": "Okay, so I need to find the probability that the 5th small... | 1 | r1_solution_1 | true |
[
{
"role": "user",
"content": "Calculate the conditional expectation \\( \\mathbb{E}(X \\mid XY) \\) where \\( X \\) and \\( Y \\) are independent standard normal random variables."
},
{
"role": "assistant",
"content": "Alright, so I need to find the conditional expectation \\( \\mathbb{E}(X \\mi... | 0 | r1_solution_1 | true |
[
{
"role": "user",
"content": "What is the smallest possible value of the correlation \\( \\rho \\) in an \\( n \\times n \\) correlation matrix where the correlation between any pair of two random variables is \\( \\rho \\)?"
},
{
"role": "assistant",
"content": "Okay, so I need to figure out th... | -\dfrac{1}{n-1} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Given the probability density function \\( f(x;\\lambda) = 1 - \\frac{2}{3}\\lambda + \\lambda\\sqrt{x} \\) for \\( 0 \\le x \\le 1 \\) and 0 otherwise, find the maximum likelihood estimate of the parameter \\( \\lambda \\) based on two independent observations \\( x_1 = \\frac... | -3 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Let \\( f(x,y): \\mathbb{R}^2 \\rightarrow \\mathbb{R} \\) be a smooth (\\( {\\cal C}^\\infty \\)) function such that \\( f(0,y)=0 \\) for all \\( y \\in \\mathbb{R} \\) and \\( f(x,y)>0 \\) for \\( x \\neq 0 \\). Determine whether there exists \\( y_0 \\in \\mathbb{R} \\), \\(... | Yes | r1_solution_1 | true |
[
{
"role": "user",
"content": "There are 100 people in a queue waiting to enter a hall with exactly 100 seats numbered from 1 to 100. The first person in the queue enters the hall and chooses any seat to sit in. Each subsequent person, from the 2nd to the 100th, will sit in their corresponding seat number if... | 2^{98} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Evaluate the line integral \\( \\int_C \\vec{F} \\cdot d\\vec{r} \\) using Stokes' theorem, where \\( \\vec{F} = (xz, xy, y^2) \\). The curve \\( C \\) is the boundary of the surface of the cylinder \\( z = 4-x^2 \\), bounded by the planes \\( x=2 \\) and \\( y=3 \\) in the fir... | 45 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Let $A$ and $B$ be $\nC$-algebras, which are also integral domains, and suppose there is an injective ring homomorphism $f: A \\to B$. Assume that $f$ is a finite morphism, meaning it induces a finite $A$-module structure on $B$. Let $M$ be a finitely generated $A$-module, and ... | No | r1_solution_1 | false |
[
{
"role": "user",
"content": "Given a Fourier sine series $A\\sin x + B\\sin 2x + C\\sin 3x + \\cdots$ that represents the function $x$ on the interval $[0, \\pi]$ and $[-\\pi, 0]$, determine the sum of the series at $x = \\pi$. Assume the series represents a periodic \"sawtooth function\" with period $2\\p... | 0 | r1_solution_1 | false |
[
{
"role": "user",
"content": "Find the probability density function of the random variable $Z = XY$, where $X$ and $Y$ are independent and identically distributed random variables with density $f(x) = 3x^2$ for $x \\in (0, 1)$. Use the product distribution formula to determine $f_Z(z)$. Note that $X, Y > 0$... | -9 z^2 \ln(z) | r1_solution_1 | false |
[
{
"role": "user",
"content": "Evaluate the limit: $$\\lim_{x\\to0} \\frac{(1+x)^{1/x}-e}{x}$$"
},
{
"role": "assistant",
"content": "Okay, so I have this limit to evaluate: the limit as x approaches 0 of [(1+x)^{1/x} - e] divided by x. Hmm, let's see. I remember that (1+x)^{1/x} is a well-known ... | -\dfrac{e}{2} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Let \\( f(x) = x^3 + x \\) and \\( g(x) = x^3 - x \\) for all \\( x \\in \\mathbb{R} \\). Find the derivative of the composition function \\( g \\circ f^{-1} \\) at the point \\( x = 2 \\)."
},
{
"role": "assistant",
"content": "Okay, so I need to find the derivativ... | \dfrac{1}{2} | r1_solution_1 | false |
[
{
"role": "user",
"content": "Evaluate the integral \\( \\int^a_0{\\cfrac{dx}{x + \\sqrt{a^2 - x^2}}} \\)."
},
{
"role": "assistant",
"content": "Alright, so I need to evaluate this integral: the integral from 0 to a of 1 over (x plus the square root of (a squared minus x squared)) dx. Hmm, okay... | \dfrac{\pi}{4} | r1_solution_1 | false |
[
{
"role": "user",
"content": "In triangle $ABC$ inscribed in a circle, the perpendicular bisector of $AB$ intersects $BC$ at $M$ and the extension of $AC$ at $N$. Given that $O$ is the center of the circle and $OH = OM = MN$ where $H$ is the midpoint of $AB$, calculate the measure of angle $OCB$."
},
{
... | 18.5^\circ | r1_solution_1 | false |
[
{
"role": "user",
"content": "Let \\( Q(b) \\) denote the right-tail probability of a standard Gaussian random variable. Suppose that \\( X \\) and \\( Y \\) are jointly-distributed Gaussian random variables with \\( X \\sim N(3, 8) \\) and \\( Y \\sim N(10, 5) \\). If the correlation coefficient between \\... | -7.453 | r1_solution_1 | false |
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