prompt list | ground_truth string |
|---|---|
[
{
"content": "Jimmy owns a cube-shaped container that measures $10$ inches on each side. He fills this container with water until it is half full. Then he throws ten giant ice cubes that measure $2$ inches on each side into the container. In inches cubed, how much of the container is unoccupied by ice or water?... | 420 |
[
{
"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... | 10.5 |
[
{
"content": "Determine the order of convergence of the fixed point iteration for the function \\( f(x) = \\sin x^{100} \\) as it converges to the fixed point 0. Justify your answer.",
"role": "user"
}
] | 100 |
[
{
"content": "Find the minimum value of $xy+yz+xz$ given that $x, y, z$ are real numbers satisfying $x^2+y^2+z^2=1$.",
"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... | 5.79e5 m/s |
[
{
"content": "Let \\( \\varphi: \\mathbb{R}^n \\rightarrow \\mathbb{R}^{n+k} \\) be a global parametrization for the differentiable manifold \\( M = \\varphi(\\mathbb{R}^n) \\subseteq \\mathbb{R}^{n+k} \\). The inner product \\( \\langle , \\rangle_{\\varphi(0)} \\) in \\( T_{\\varphi(0)}M \\) is such that \\( ... | g_{ij} = \delta_{ij} |
[
{
"content": "Let \\((R, m)\\) be a commutative regular local Noetherian ring. Determine whether it is true that for every prime \\(p \\in \\text{Spec}(R)\\), the factor ring \\(R/p\\) has a maximal Cohen-Macaulay \\(R/p\\)-module.",
"role": "user"
}
] | True |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $783.50 |
[
{
"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... | increase by $17,500 |
[
{
"content": "Consider 1-dimensional CW-complexes $A$ and $B$, both having a fundamental group isomorphic to $\\mathbb{Z}$. Does the inclusion map $A \\hookrightarrow B$ induce an isomorphism on their fundamental groups?",
"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 for potassium and +6 for chromium |
[
{
"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... | 3.245 s |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\to \\infty} \\frac{\\frac{1}{2}\\pi-\\text{tan}^{-1}x}{\\ln\\left(1+\\frac{1}{x^2}\\right)}$$",
"role": "user"
}
] | \infty |
[
{
"content": "In the ring $R = F[x,y]/(x^2 - y^2)$, where $F$ is a field, determine if there exist any elements that are both nilpotent and idempotent.",
"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... | Y = (100/X^0.33)^1/0.67 |
[
{
"content": "Evaluate the integral \\( \\int_{-a}^a \\frac{1}{1+e^{h(x)}} \\, dx \\), given that \\( h(x) \\) is an odd function.",
"role": "user"
}
] | a |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\rightarrow \\infty}\\frac{n!n^{x}}{x(x+1)...(x+n)} \\] given that \\( \\text{Re}(x) > 0 \\).",
"role": "user"
}
] | \Gamma(x) |
[
{
"content": "Let \\( M = \\{(x, y) : |xy| = 1, x > 0 \\} \\) and \\( N = \\{(x, y) : \\operatorname{arctg} x + \\operatorname{arcctg} y = \\pi \\} \\). Determine which of the following is true:\n\n(A) \\( M \\cup N = \\{(x, y) : |xy| = 1 \\} \\);\n\n(B) \\( M \\cup N = M \\);\n\n(C) \\( M \\cup N = N \\);\n\n(... | 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... | Increase in Cash=-+Increase in Capital |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | -1.44e-8 C |
[
{
"content": "Find an invariant of the recurrence relation \\( S_{n+2} = 3S_{n+1} - S_n \\) with initial conditions \\( S_1 = S_2 = 1 \\), in the form \\( S^2_{n+1} + aS_nS_{n+1} + bS^2_n \\).",
"role": "user"
}
] | S_{n+1}^2 - 3S_nS_{n+1} + S_n^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... | November: $158,000, December: $249,000 |
[
{
"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... | 23.75 cents |
[
{
"content": "Evaluate the limit: $$ \\lim_{x\\to0^-} \\frac{\\ln(\\cot(-3x))}{\\ln(\\cot(-2x))} $$",
"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... | 3000 A |
[
{
"content": "Find the limit of \\( y(x) \\) as \\( x \\to \\infty \\), where \\( y(x) \\) is defined by the integral equation:\n\\[ y(x) = 7 + \\int_0^x 4\\frac{(y(t))^2}{1+t^2}dt \\]",
"role": "user"
}
] | \infty |
[
{
"content": "Determine the range of integer values of $a$ for which the inequality \\( \\frac{x^2+a^2}{a(4+x)} \\ge 1 \\) holds for all \\( x \\in (-1, 1) \\).",
"role": "user"
}
] | [5, \infty) |
[
{
"content": "Determine the equation whose roots are independent of \\(\\lambda\\) for the following expression:\n\n$$(x-3)(x^3 + 3x^2 + 2x + 1) + \\lambda(4x^3 + \\beta x + \\beta) = 0$$",
"role": "user"
}
] | x^4 - 7x^2 - 5x - 3 = 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. Analyze source documents.
2. Journalize transactions.
3. Post journal entries to the ledger.
4. Prepare a trial balance. |
[
{
"content": "Let $X_1, X_2, \\ldots, X_n$ be independent and identically distributed random variables, each following a Bernoulli distribution with parameter $\\theta$, where $\\theta \\in (0,1)$. Define $Y = \\sum_{k=1}^n X_k$. Find the conditional expectation $E(X_k \\mid Y)$. \\(k = 1, 2, \\ldots, n\\)",
... | \dfrac{Y}{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... | 2.62 J |
[
{
"content": "Determine the norm \\( \\left\\lVert f\\right\\rVert \\) of the function \\( f: l^2 \\to \\mathbb{R} \\) defined by \\( f(x_1, x_2, x_3, \\dots) = \\sum_{n=1}^\\infty \\frac{x_n}{2^\\frac{n}{2}} \\) for all \\( x = (x_1, x_2, \\dots) \\in l^2 \\), where \\( l^2 \\) denotes the space of square-summ... | 1 |
[
{
"content": "Find the Fourier Transform of the distribution \\( f(x) = \\frac{2\\pi\\delta(x)}{e^{ix}+1} \\).",
"role": "user"
}
] | \pi |
[
{
"content": "Does there exist a sequence \\((x_n)\\) of real numbers such that for every \\(d\\) in \\((0,1]\\) and every \\(S \\subseteq \\mathbb{N}\\) of natural density \\(d\\), the subsequence \\((x_i\\, : \\, i \\in S)\\) is divergent?",
"role": "user"
}
] | Yes |
[
{
"content": "Find the equation of a plane that contains the intersection line of the planes \\(x + 2y + 3z = 4\\) and \\(2x + y + z = 2\\), and is perpendicular to the vector \\((0, 1, 0)\\).",
"role": "user"
}
] | 3x - z = 0 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\frac{x - \\sin(\\sin(\\cdots(\\sin x)\\cdots))}{x^3} \\] where the sine function is applied \\( n \\) times.",
"role": "user"
}
] | \dfrac{n}{6} |
[
{
"content": "Given a circle with radius 1, an arc subtends an angle of \\(x\\) radians at the center \\(O\\). Points \\(A\\) and \\(B\\) are the endpoints of the arc, and \\(C\\) is the intersection point of the tangents at \\(A\\) and \\(B\\). Let \\(T(x)\\) be the area of triangle \\(ABC\\). Find \\(\\lim_{x... | \dfrac{1}{8} |
[
{
"content": "Determine if there exists a sufficiently large value of \\( x \\) such that the inequality \\( e^{5x} > 555555e^{4.999x} + 3450000e^{3x} + 10000e^{2x} + 999999x^{500} \\) holds true.",
"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... | V=iR |
[
{
"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... | 3.03e-4 J |
[
{
"content": "What is the minimum number of times you need to roll a six-sided die so that the probability of rolling each number at least once is greater than 0.5?",
"role": "user"
}
] | 13 |
[
{
"content": "Evaluate the integral \\( \\int_{-1}^{1} x^2 P_{2n-1}(x) \\, dx \\), where \\( P_{2n-1}(x) \\) is the Legendre polynomial of degree \\( 2n-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... | 1.44 g |
[
{
"content": "In a right triangle $ABC$ with $\\angle A = 90^\\circ$, the inradius is $r$. A cevian $AD$ is drawn such that the inradii of triangles $ABD$ and $ADC$ are both $r_1$. Given that $AD = 2$, calculate the value of $\\frac{1}{r_1} - \\frac{1}{r}$.",
"role": "user"
}
] | 0.5 |
[
{
"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... | From most to least soluble: Ethanol, Acetic acid, Oxygen gas, Hexanes |
[
{
"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) Having a lazy afternoon. - (F) Working an 8 hour shift.
(B) Eating out. - (D) Cooking dinner.
(C) Catching up with an old friend. - (E) Reading your favourite book. |
[
{
"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... | $26,630 |
[
{
"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... | 2.7 years |
[
{
"content": "What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 420 = 0$ has integral solutions?",
"role": "user"
}
] | 130 |
[
{
"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... | 1s^22s^22p^63s^23p^63d^{10}4s^24p^64d^{10}5s^25p^6 |
[
{
"content": "Find the limit of the expression as \\(n\\) approaches infinity: \n\\[ \\lim_{n\\to \\infty} \\left[n - \\frac{n}{e}\\left(1 + \\frac{1}{n}\\right)^n\\right] \\]",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Two triangular pyramids \\( MABC \\) and \\( NABC \\) share a common base \\( ABC \\) and have no other common points. All vertices of both pyramids lie on the same sphere. Given that the lengths of all edges except \\( MA \\) and \\( MB \\) are \\( \\sqrt{3} \\), and \\( MA = MB \\), find the len... | 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... | $3.58 per share |
[
{
"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... | activity |
[
{
"content": "Let $N$ be a positive integer that is a multiple of $5$. In a sequence, one red ball and $N$ green balls are randomly arranged. Define $P(N)$ as the probability that at least $\\frac{3}{5}$ of the green balls are located on the same side of the red ball. Given $P(5) = 1$ and $P(N)$ approaches $\\f... | 12 |
[
{
"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... | -0.340 m/s |
[
{
"content": "Evaluate the complex expression \\( y = i\\log(x-i) + i^2\\pi + i^3\\log(x+i) + i^4(2\\arctan x) \\) for \\( x > 0 \\), and rewrite it in the form \\( y = 0 \\).",
"role": "user"
}
] | y = 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... | {eq}dQ = \left( \dfrac{\partial U}{\partial P }\right)_VdP + \left[ \left( \dfrac{\partial U}{\partial V }\right)_P + P \right]dV{/eq} |
[
{
"content": "Calculate the flux integral \\( \\int_S (\\nabla \\times \\vec{F}) \\cdot \\vec{n} \\, dS \\) directly, where \\( \\vec{F} = (y,z,x^2y^2) \\) and \\( S \\) is the surface defined by \\( z = x^2 + y^2 \\) with \\( 0 \\leq z \\leq 4 \\), oriented downwards.",
"role": "user"
}
] | 4\pi |
[
{
"content": "Given the function \\( y = f(x) \\) with the property that its derivative's domain is \\( \\mathbb{R} \\) and \\( xf'(x) = e^x - 1 \\) for all \\( x \\in \\mathbb{R} \\), and knowing that \\( f(1) = 0 \\), determine the value of \\( \\int_0^1 xf(x) \\, \\mathrm{d}x \\).",
"role": "user"
}
] | -\dfrac{1}{4} |
[
{
"content": "Determine the number \\( x \\) such that \\( 579^{65} \\equiv x \\pmod{679} \\) and \\( x < 676 \\). Given the pattern: \\( 579^2 \\equiv 494 \\pmod{679} \\), \\( 494^2 \\equiv 275 \\pmod{679} \\), \\( 275^2 \\equiv 256 \\pmod{679} \\), \\( 256^2 \\equiv 352 \\pmod{679} \\), \\( 352^2 \\equiv 326 ... | 108 |
[
{
"content": "Determine whether there exists a polynomial $g(x)$ such that for all distinct positive integers $m$ and $n$, the greatest common divisor of $g(m)$ and $g(n)$ is 1.",
"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... | January: $13,530
February: $20,130 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty}\\sum_{k=1}^n\\frac{k}{n}\\sin\\left(\\frac{2\\pi k}{n}\\right)\\sin\\left(\\frac{2\\pi k-\\pi}{n}\\right)\\sin\\left(\\frac{\\pi}{n}\\right) \\]",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "Determine the value of \\( n \\) for which the polynomial \\( x^{2n} + 1 + (x+1)^{2n} \\) is not divisible by \\( x^2 + x + 1 \\). Choose from the following options:\n\n\\( \\text{(A)} \\ 17 \\qquad \\text{(B)} \\ 20 \\qquad \\text{(C)} \\ 21 \\qquad \\text{(D)} \\ 64 \\qquad \\text{(E)} \\ 65 \\)... | 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... | -$1,961.20 |
[
{
"content": "A circular cone has a vertex at point \\( I \\), a base with radius 1, and a slant height of 4. Point \\( A \\) is located on the circumference of the base, and point \\( R \\) is on the line segment \\( IA \\) such that \\( IR = 3 \\). Shahid draws the shortest possible path starting at \\( R \\)... | \dfrac{12}{5} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.7906 |
[
{
"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... | 7495.89 |
[
{
"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... | 3.35e-6 C/m^2 |
[
{
"content": "Let $f: \\Bbb{Z} \\times \\Bbb{Z} \\to G$ be an epimorphism with $\\ker f$ generated by $\\langle (3,0),(0,5) \\rangle$. Determine the type of abelian group $G$.",
"role": "user"
}
] | \mathbb{Z}_{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... | 54.72 gm |
[
{
"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... | 9.66% |
[
{
"content": "Find the exact value of the infinite nested radical expression: \\[ R = \\sqrt{-3+2\\sqrt{-5+3\\sqrt{-7+4\\sqrt{-9+\\dots}}}} \\]",
"role": "user"
}
] | 1 |
[
{
"content": "Let \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) be a continuous function such that \\(|f(x)-f(y)| \\geq \\frac{1}{2}|x-y|\\) for all \\(x, y \\in \\mathbb{R}\\). Is \\(f\\) bijective?",
"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... | -55.45KJ |
[
{
"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... | 3.97 m/s |
[
{
"content": "Let $n \\geqslant 11$ be a positive integer. Define set $A$ as the set of sums of any 10 consecutive positive integers not greater than $n$, and set $B$ as the set of sums of any 11 consecutive positive integers not greater than $n$. If the number of elements in $A \\cap B$ is 181, find the maximu... | (2001, 2011) |
[
{
"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... | 8.59 g |
[
{
"content": "Evaluate the surface integral \\( \\iint_S \\mathbf{r} \\cdot \\mathbf{n} \\, dS \\) over the surface of a sphere with radius \\( a \\) centered at the origin, where the surface is given by the equation \\( x^2 + y^2 + z^2 = a^2 \\). Express the normal vector \\( \\mathbf{n} \\) as \\( \\hat{\\mat... | 4\pi a^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... | 532 mm Hg |
[
{
"content": "Does a map $T:V\\to W$ between vector spaces $V$ and $W$ that preserves addition, i.e., $T(v_1 + v_2)=T(v_1)+T(v_2)$ for all $v_1,v_2\\in V$, necessarily make $T$ a linear map? Answer yes or no and justify your answer.",
"role": "user"
}
] | No |
[
{
"content": "Given an element $\\alpha$ in a field of characteristic $2$ with its minimal polynomial $f(x) = x^9 + x + 1$ over $GF(2)$, determine the order of $\\alpha$. \\(\\text{(Express your answer as a single integer.)}\\)",
"role": "user"
}
] | 73 |
[
{
"content": "Determine whether there exists a sequence of real numbers \\( a_k \\) such that \\( e^{\\lambda-\\lambda^2} = \\sum \\limits_{k=0}^\\infty \\frac{\\lambda^k}{k!}a_k \\) for all \\( \\lambda \\in \\mathbb{R} \\).",
"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... | 73.3 m |
[
{
"content": "Let \\( P(x) \\) be a polynomial in \\( \\mathbb{R}[x] \\) with a leading coefficient of 1. Determine the leading coefficient of the polynomial \\( P(2x+1) - 2P(x) \\).",
"role": "user"
}
] | 2^n - 2 |
[
{
"content": "Evaluate the limit \\( \\lim_{n\\to\\infty} \\frac{S_2(n)S_4(n)-S_3(n)S_5(n)}{S_8(n)} \\), where \\( S_k(n)=\\sum_{r=1}^n r^k \\).",
"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... | 4650 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... | 2.17 g/cm^3 |
[
{
"content": "Given a sequence of non-negative, identically distributed random variables $X_i$ with expectation $\\mu = \\mathbb{E}[X_i]$, determine the expected stopping time $\\tau$ for the sum $\\sum_{i=1}^k X_i$ to first reach or exceed a threshold $t$. Express $\\mathbb{E}[\\tau]$ in terms of $t$ and $\\mu... | \dfrac{t}{\mu} |
[
{
"content": "In a forest consisting of 6 trees with a total of 27 vertices, what is the minimum number of simple paths? Calculate the minimum number of simple paths for this configuration.",
"role": "user"
}
] | 48 |
[
{
"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... | $600,000 |
[
{
"content": "Determine the upper bound for numbers such that the sum of the fifth powers of their digits equals the number itself.",
"role": "user"
}
] | 354294 |
[
{
"content": "In a compact and Hausdorff uniform space \\((X, \\mathcal{U})\\), where \\(X\\) is totally disconnected, and given \\(E \\in \\mathcal{U}\\), determine if there exists a \\(D \\in \\mathcal{U}\\) such that every \\(D\\)-chain \\(\\{x_i\\}_{i\\in\\mathbb{Z}}\\) satisfies \\((x_i, x_j) \\in E\\) for... | Yes |
[
{
"content": "Simplify $1-(1+(1-(1+(1-x))))$.",
"role": "user"
}
] | 1-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... | 0.447 |
[
{
"content": "In a game with 10 piles of 10 stones each, two players, Varya and Miron, take turns. On each turn, a player must divide a pile with more than one stone into several equal piles. The player who cannot make a move because all piles contain exactly one stone loses. If both players play optimally, who... | Miron |
[
{
"content": "Evaluate the limit: $$\\lim\\limits_{n\\to\\infty}\\sum\\limits_{r=0}^\\infty \\left(\\frac{r}{n}\\right)^n$$",
"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... | 3200 m |
[
{
"content": "Given the interval \\(\\mathcal{I} = \\{[a,b) \\mid a,b \\in \\mathbb{R}, a < b\\}\\), define the length function \\(\\ell(I) = b-a\\) for \\(I \\in \\mathcal{I}\\). Deduce that the length of an open interval \\((a,b)\\) is \\(b-a\\) using the properties of the length function.",
"role": "user... | b - a |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.