prompt list | ground_truth string |
|---|---|
[
{
"content": "Use continued fractions to find a rational number that approximates \\(\\sqrt{11}\\) to within \\(10^{-4}\\).",
"role": "user"
}
] | \dfrac{199}{60} |
[
{
"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... | Cash
125,000
Goodwill
155,000
Sergio, Capital
125,000
Tiger, Capital
77,500
Phil, Capital
62,000
Ernie, Capital
15,500 |
[
{
"content": "Determine the power of the group of languages A, where A consists of languages L such that L = L* and L is defined over the alphabet Σ = {1}.",
"role": "user"
}
] | 2^{\aleph_0} |
[
{
"content": "Suppose $K$ and $F$ are subfields of some field $L$, and $\\sigma$ is an automorphism of $L$ which fixes $K$ and $F$. Does $\\sigma$ necessarily fix the composite field $KF$? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "If \\(p\\) is a prime number and \\(G\\) is a non-abelian group of order \\(p^3\\), determine the number of elements in the center of the group \\(G\\), denoted as \\(Z(G)\\).",
"role": "user"
}
] | p |
[
{
"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... | $31.93 |
[
{
"content": "Let \\( f(x) = \\left(\\int_{0}^{x} e^{-t^2} dt\\right)^2 \\) and \\( g(x) = \\int_{0}^{1} \\frac{e^{-x^2(1+t^2)}}{1+t^2} dt \\). Find the value of \\( f'(\\sqrt{\\pi}) + g'(\\sqrt{\\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... | The final velocity of the first block is 7.1 m/s and the final velocity of the second block is 5.29 m/s |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | K+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... | 6.75 mg |
[
{
"content": "Calculate the surface area generated by revolving the polar curve \\( r = \\sqrt{\\cos 2\\theta} \\) around the y-axis, with the limits of integration from \\( \\theta = 0 \\) to \\( \\theta = \\frac{\\pi}{4} \\).",
"role": "user"
}
] | \pi \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... | 8.5% |
[
{
"content": "Compute the limit: $$\\lim_{n\\to\\infty}\\frac{1+\\frac1{\\sqrt[3]{2}}+\\ldots+\\frac1{\\sqrt[3]{n}}}{\\sqrt[3]{n^2}}$$",
"role": "user"
}
] | \dfrac{3}{2} |
[
{
"content": "Determine the number of planes that are tangent to both spheres \\((x+2)^2+y^2+z^2=1\\) and \\((x-2)^2+y^2+z^2=1\\), and also pass through the point \\((1, 0, 0)\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the minimum value of \\( k \\) such that the inequality \\( \\int_{0}^{1}f^2(x)\\,dx \\leq k\\left(\\int_{0}^{1}f(x)\\,dx\\right)^2 \\) holds for all integrable functions \\( f(x) \\) satisfying \\( 1 \\leq f(x) \\leq 2 \\) for \\( x \\in (0,1) \\).",
"role": "user"
}
] | \dfrac{9}{8} |
[
{
"content": "Determine the limit of the sequence \\( S(n) = \\sum_{k=0}^{2n}{\\frac{k}{k+n^2}} \\) as \\( n \\to \\infty \\).",
"role": "user"
}
] | 2 |
[
{
"content": "Given the sequence defined by $u_0=2$ and $u_{n+1} = \\frac{2}{3}(u_n)-n-\\frac{8}{3}$, and another sequence $v_n=u_n + x\\cdot n -1$, determine the possible values of $x$ such that $v_n$ forms a geometric sequence.",
"role": "user"
}
] | 3 |
[
{
"content": "For a holomorphic function $f$, define $h(z)=\\sin(f(z^2))$. Calculate the residue of $h(z)$ at $z=0$ in terms of $f(0)$ and $f'(0)$. Express your answer as a single complex number.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the integral \\( \\int_{-n}^{n} (-1)^{\\lfloor x \\rfloor} \\, dx \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Find the remainder \\( S(x) \\) when a polynomial \\( f(x) \\) is divided by \\((x^2+4)(x^2+6)\\). Given that when \\( f(x) \\) is divided by \\( x^2+4 \\), the remainder is \\( 2x+3 \\), and when divided by \\( x^2+6 \\), the remainder is \\( 6x-1 \\), calculate \\( S(4) \\).",
"role": "user"... | -109 |
[
{
"content": "Factor the expression \\(\\dfrac{2^{3x}+3^{3x}}{2^{2x}\\cdot 3^{x}+3^{2x}\\cdot{2^{x}}}\\) and verify that it equals \\(\\dfrac{7}{2\\cdot 3}\\).",
"role": "user"
}
] | \dfrac{7}{6} |
[
{
"content": "Determine whether a Calabi-Yau n-manifold can be homeomorphic to a non-Calabi-Yau complex manifold. Provide a yes or no answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether there exists an $L^1$ function $f(x)$ such that the limit \\( \\lim_{\\omega \\to \\infty} \\int_{\\mathbb{R}} f(x) \\cdot e^{-\\imath \\, x \\omega} \\, dx \\neq 0 \\).",
"role": "user"
}
] | No |
[
{
"content": "Determine the limit of the sequence \\( b_n = \\frac{n^{2023}}{1.002^n} \\) as \\( n \\to \\infty \\) without using logarithms.",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the group presentation \\( \\langle a, b \\mid ab = ba = 1 \\rangle \\) is equivalent to the group \\( \\langle a \\mid \\rangle = \\mathbb{Z} \\). Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\(ABCD\\) be a square and \\(P\\) be a point on the shorter arc \\(AB\\) of the circumcircle of the square. Determine the range of values that the expression \\(\\frac{AP+BP}{CP+DP}\\) can take.",
"role": "user"
}
] | \sqrt{2} - 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... | $3,110 |
[
{
"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... | 174*10^3 kJ |
[
{
"content": "If $n$ is a positive integer, then let $f(n)$ be the sum of the digits of $\\frac{1}{5^{{}^n}}$ that are to the right of the decimal point. What is the smallest positive integer $n$ such that $f(n) > 10$?",
"role": "user"
}
] | 7 |
[
{
"content": "Given complex numbers \\( z \\) and \\( \\omega \\) such that:\n1. \\( z + \\omega + 3 = 0 \\);\n2. \\( |z|, 2, |\\omega| \\) form an arithmetic sequence.\nFind the maximum value of \\( \\cos (\\arg z - \\arg \\omega) \\).",
"role": "user"
}
] | \dfrac{1}{8} |
[
{
"content": "Let \\(V\\) be a vector bundle over an algebraic curve \\(C\\), and assume that \\(V \\cong V \\otimes L\\) for some line bundle \\(L\\). If \\(V\\) is simple, does this imply that \\(L \\cong \\mathcal{O}_C\\)? Answer yes or no and justify 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... | Net book value * Double declining rate where Net book value = Cost - Accumulated depreciation and Double declining rate = (1 / Useful life) * 2 |
[
{
"content": "Evaluate the limit: $$\\lim _{x\\to0}\\dfrac{\\displaystyle \\int _0^{x^2}(\\sin t)^{\\frac{3}{2}} dt}{\\displaystyle\\int ^x_0t\\left(t-\\sin t\\right)dt}$$",
"role": "user"
}
] | 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... | -1.20e-38 eV |
[
{
"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... | 18.38 V |
[
{
"content": "Find the inverse of the coset $2x + 3 + I$ in the quotient ring $\\mathbb{Z}_5[x]/\\langle x^2 + x + 2 \\rangle$. Express your answer as a coset in the form $ax + b + I$ where $a, b \\in \\mathbb{Z}_5$. \\(\\boxed{\\text{Your answer here}}\\)",
"role": "user"
}
] | 3x + 1 + I |
[
{
"content": "Given the initial value problem:\n\n\\[\n\\begin{cases}\ny'_1 =y_1+y_2+1 \\\\\ny'_2= -y_1+y_2+1\n\\end{cases}\n\\]\nwith initial conditions:\n\\[\ny_1(0)=1, \\quad y_2(0)=-1\n\\]\nfind the value of \\( y_1(\\pi) \\).",
"role": "user"
}
] | -e^{\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... | 31360 |
[
{
"content": "Determine the range of values for the real number \\( m \\) such that for an odd function \\( f(x) \\) that is decreasing in its domain \\((-1,1)\\), the condition \\( f(1-m) + f\\left(1-m^{2}\\right) < 0 \\) is satisfied.",
"role": "user"
}
] | (0, 1) |
[
{
"content": "Given that $a, b, c > 0$ and $a + b + c = 2(ab + bc + ca)$, find the minimal value of the expression:\n\\[ P = \\sqrt{\\frac{1}{ab} + \\frac{1}{bc} + 1} + \\sqrt{\\frac{1}{bc} + \\frac{1}{ca} + 1} + \\sqrt{\\frac{1}{ca} + \\frac{1}{ab} + 1}. \\]",
"role": "user"
}
] | 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... | 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... | 0.00326 s |
[
{
"content": "Is it always possible to find a complex matrix $S$ such that $e^S = T$ for any invertible matrix $T$ in the space of linear transformations on $\\mathbb{C}^n$?",
"role": "user"
}
] | Yes |
[
{
"content": "Find the minimum value of the function \\( y(x) = x^x \\) using calculus.",
"role": "user"
}
] | e^{-1/e} |
[
{
"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... | 2.45 times that on Earth |
[
{
"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... | 5.12 Hz, 10.2 Hz |
[
{
"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. Cash $33,000
Truck $28,000
Unearned Profit Sale-Leaseback $5,000
(To Record Sale)
2. Leased Truck $33,000
Leased Liability $33,000
(To Record Lease Back)
3. Depreciation Expenses $6,600
Accumulated Depreciation $6,600
(To Record Depreciation)
4. Unearned Profit Sale-Leaseback $1,000
Depreciation Expenses $1,000
(To ... |
[
{
"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... | 12.5m |
[
{
"content": "Use Stokes' theorem to evaluate the integral \\( I = \\int\\limits_C \\textbf{F} \\centerdot \\textbf{ds} \\) where \\( \\textbf{F} = 3zx\\textbf{i} + 3xy\\textbf{j} + yz\\textbf{k} \\) and \\( C \\) is the path consisting of the three edges of the triangle \\( \\Delta ABC \\) formed by the portio... | \dfrac{7}{6} |
[
{
"content": "Let $p \\geq 23$ be a prime number such that the decimal expansion (base 10) of $\\frac{1}{p}$ is periodic with a period of $p-1$. Consider the multiplicative group of integers modulo $p$, denoted as $(\\mathbb{Z}/p\\mathbb{Z})^*$. Determine the order of the element $10$ in this group.",
"role... | p-1 |
[
{
"content": "Let $f \\in L^1(\\mathbb{R}^d)$ be a uniformly continuous function, where $d > 1$. Determine whether the following limit holds: $$\\lim_{R\\to\\infty} \\int_{|x|=R} |f(x)| \\ dS(x) = 0,$$ where $dS$ is the surface measure on the sphere of radius $R$. Provide a proof or a counterexample.",
"rol... | 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. Separate the unknown solution into two vials.
2. Add 1 M sulfuric acid to one vial.
3. Add 1 M hydrochloric acid to the other vial.
If no precipitate forms in either vial, the cation is sodium (Na+).
If a precipitate forms in both vials, the cation is silver (Ag+).
If a precipitate forms only in the vial with sulfu... |
[
{
"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... | 3.165 g |
[
{
"content": "Given the integral equation \\( y(x)= f(x) + \\int_{0}^x \\sin(x-t)y(t) dt \\), determine the correct expression for \\( y(x) \\) from the following options:\n\n1. \\( y(x) = f(x) + \\int_{0}^x (x-t)f(t) dt \\)\n2. \\( y(x) = f(x) - \\int_{0}^x (x-t)f(t) dt \\)\n3. \\( y(x) = f(x) - \\int_{0}^x \\... | 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... | 1/3 |
[
{
"content": "Can Vitya determine the first digit of a 9-digit number, which is a permutation of 123456789, in no more than 4 attempts by selecting any 9-digit number and receiving feedback on positional matches?",
"role": "user"
}
] | Yes |
[
{
"content": "Given a convergent series \\(\\sum_{n=1}^{\\infty} a_n\\) with positive terms, determine whether there exists a convergent series \\(\\sum_{n=1}^{\\infty} b_n\\) with positive terms such that \\(\\lim_{n \\to \\infty} \\frac{b_n}{a_n} = +\\infty\\).",
"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... | 10 mV |
[
{
"content": "Given that each potato has an equally-likely probability of dying, living, or living and producing a clone, what is the probability that an infinite number of potatoes will survive indefinitely?",
"role": "user"
}
] | 0 |
[
{
"content": "An isosceles right triangle has a leg length of 36 units. Starting from the right angle vertex, an infinite series of equilateral triangles is drawn consecutively on one of the legs. Each equilateral triangle is inscribed such that their third vertices always lie on the hypotenuse, and the opposit... | 324 |
[
{
"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... | 10.42 m/s |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to 0^+}\\left[\\big(\\sin(x)\\big)^{\\frac{1}{x}}+\\Big(\\frac{1}{x}\\Big)^{\\sin(x)}\\right]$$",
"role": "user"
}
] | 1 |
[
{
"content": "In triangle $ABC$, where $AB=9$, $BC=10$, and $CA=11$, the orthocenter is denoted as $H$. Point $D$ is located on line segment $\\overline{BC}$ such that $AH=HD$. Find the length of $AD$.",
"role": "user"
}
] | \sqrt{102} |
[
{
"content": "Find all functions \\( g : \\mathbb{Q} \\to \\mathbb{Q} \\) that satisfy the equation \\( g \\left( \\frac{x+y}{2} \\right) = \\frac{g(x) + g(y)}{2} \\) for all \\( x, y \\in \\mathbb{Q} \\).",
"role": "user"
}
] | g(x) = ax + b |
[
{
"content": "It takes 24 minutes for Jana to walk one mile. At that rate, how far in miles will she walk in 10 minutes? Express your answer as a decimal to the nearest tenth.",
"role": "user"
}
] | 0.4\mbox{ miles} |
[
{
"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... | 2.352 m/s |
[
{
"content": "Evaluate the integral \\( \\int_0^\\pi \\left[\\frac{\\sin(nx)-\\sin((n-1)x)}{\\sin(x/2)}\\right]^2 dx \\) analytically for \\( n \\in \\mathbb{N} \\).",
"role": "user"
}
] | 2\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... | 63710 |
[
{
"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... | 17 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 22.9 s |
[
{
"content": "The first term of an arithmetic sequence is $2$. The sum of the third and the sixth terms is $25$. What is the fourth term?",
"role": "user"
}
] | 11 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Department D: $344,000
Department E: $368,000
Department K: $177,960 |
[
{
"content": "Determine the radius of convergence of the Laurent series for the function \\( f(z) = \\frac{\\sin z}{(z-\\pi)^2} \\) centered at \\( z = \\pi \\).",
"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... | 9.90 |
[
{
"content": "If $G$ is a non-abelian group of order 27, what is the order of its commutator subgroup $G'$?",
"role": "user"
}
] | 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... | ```cpp
#include <iostream>
#include <limits.h>
#include <vector>
using namespace std;
int isPrime(int N) {
for (int i = 2; i <= N / 2; i++) {
if (N % i == 0)
return 0;
}
return 1;
}
int countPrimes(vector<int> vec) {
int count = 0;
for (auto i = vec.begin(); i != vec.end(); ++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... | Bent geometry with bond angles of 98 degrees |
[
{
"content": "Evaluate the integral \\( \\int_{-\\infty}^{\\infty} |x| e^{-x^2} \\, dx \\).",
"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... | 29921 |
[
{
"content": "Find the power series solution to the Laguerre differential equation:\n\\[ xy'' + (1-x)y' - y = 0 \\]\nwith the initial condition \\( P(0) = 1 \\).",
"role": "user"
}
] | e^x |
[
{
"content": "Given a transitive set $\\{x\\}$, determine the nature of $x$.",
"role": "user"
}
] | \emptyset |
[
{
"content": "Using Burnside's lemma, determine the number of distinct necklaces that can be formed with 12 equally spaced stones, using 4 red, 5 green, and 3 blue beads. Consider the group of symmetries given by the dihedral group \\(D_{24}\\), which includes rotations and reflections.",
"role": "user"
}... | 1170 |
[
{
"content": "In triangle \\(ABC\\), angle \\(A\\) is \\(60^{\\circ}\\). The angle bisector \\(AM\\), median \\(BN\\), and altitude \\(CL\\) intersect at a single point. Determine the measures of the other two angles of the triangle.",
"role": "user"
}
] | 60^\circ |
[
{
"content": "Compute the limit: $$\\lim\\limits_{n \\to \\infty}n\\int_{0}^{1}(\\cos x-\\sin x)^n dx$$ using elementary calculus methods.",
"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... | $1,207.12 |
[
{
"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... | 2.986*10^7 J |
[
{
"content": "Determine if the sequence \\( \\left\\{ \\left(\\frac{\\ln (n+2)}{\\ln (n+1)}\\right)^{n+1} \\right\\} \\) is convergent. If it converges, find its limit.",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the limit \\(\\lim_{\\substack{x\\to1\\\\ y\\to-1}}{\\frac{\\arcsin{(x^5+y^5)}}{x^5+\\tan{(x^4y+xy^4)}+\\log{(1+x^3y^2+x^2y^3)}+y^5}}\\) using Taylor series expansions for \\(\\arcsin{(x^5+y^5)}\\), \\(\\tan{(x^4y+xy^4)}\\), and \\(\\log{(1+x^3y^2+x^2y^3)}\\).",
"role": "user"
}
] | \dfrac{5}{3} |
[
{
"content": "Given the function \\( f(x) = 4\\arcsin(\\sqrt{x}) + 2\\arcsin(\\sqrt{1-x}) \\) with its derivative \\( f'(x) = \\frac{1}{\\sqrt{x-x^2}} \\), find the maximum value of \\( f^{-1}(x) \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Let \\( f(x) \\in C[0,1] \\) be a decreasing function and \\( \\lambda \\in (0,1) \\). Define \\( I_1 = \\int_{0}^{\\lambda} f(x) \\, dx \\) and \\( I_2 = \\lambda \\int_{0}^{1} f(x) \\, dx \\). Find the numerical relationship between \\( I_1 \\) and \\( I_2 \\).",
"role": "user"
}
] | I_1 \geq I_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... | externalities |
[
{
"content": "Find the smallest prime number \\( p \\) such that \\( p \\), \\( p+1 \\), and \\( p-1 \\) each have at least three distinct prime factors.",
"role": "user"
}
] | 131 |
[
{
"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 = nRT/P. Suppose 1 mol of an ideal gas is at standard temperature and pressure. We can calculate its volume V = (1 mol * 0.0821 L atm/K mol * 273K) / 1 atm = 22.4 L |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Lead acetate |
[
{
"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... | 2.26e+3 L |
[
{
"content": "In a semicircle with diameter $AB$ and center $S$, points $C$ and $D$ are located such that point $C$ is on arc $AD$. Given that angle $CSD$ is 120°, let $E$ be the intersection of lines $AC$ and $BD$, and $F$ be the intersection of lines $AD$ and $BC$. Find the ratio $\\frac{EF}{AB}$.",
"role... | \sqrt{3} |
[
{
"content": "Given a sequence \\(\\delta_n\\) of positive numbers decreasing to 0 and a fixed real number \\(t > 0\\), determine whether there exists a sequence of integers \\(N_n\\) such that \\(N_n \\delta_n \\rightarrow t\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Find the radius of convergence of the power series \\( \\sum\\limits_{n=2}^\\infty a_nx^n \\) given that \\( \\lim\\limits_{n \\to \\infty} \\frac{a_n}{n^3} = 1 \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the limit \\( \\lim_{k \\rightarrow \\infty} \\int_0^{\\pi} e^{x(\\sin(x))^k} \\, dx \\).",
"role": "user"
}
] | \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... | 1891 reps |
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