prompt list | ground_truth string |
|---|---|
[
{
"content": "Find the sum of all positive integers less than \\(2n\\) that are relatively prime to \\(n\\), where \\(n\\) is a positive integer.",
"role": "user"
}
] | 2n\phi(n) |
[
{
"content": "In a linear programming problem, consider a variable $x_k$ without a non-negativity constraint, expressed as $x_k = u_k - v_k$ where $u_k \\ge 0$ and $v_k \\ge 0$. Can a basic feasible solution exist where both $u_k$ and $v_k$ are basic variables? Answer yes or no.",
"role": "user"
}
] | No |
[
{
"content": "In an equilateral triangle with side length \\( a \\), points \\( X \\) and \\( Y \\) are located on sides \\( AB \\) and \\( BC \\) respectively, such that \\( |AX| = \\frac{a}{3} \\) and \\( |BY| = \\frac{a}{3} \\). Let \\( T \\) be the intersection of lines \\( AY \\) and \\( CX \\). Determine ... | 90^\circ |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 25cm |
[
{
"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 Hour |
[
{
"content": "Does a function $u$ in $\\mathbb{R}^n$ belonging to the Sobolev space $H^1(\\Omega)$ guarantee the existence of the classical first-order derivative almost everywhere in $\\Omega$?",
"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... | $21,76,000 |
[
{
"content": "Let $U = \\text{Spec}(A)$ and $V = \\text{Spec}(B)$ be affine schemes, and let $X$ be a separated scheme. Suppose there exist morphisms $U \\to X$ and $V \\to X$. Is the natural map \\( A \\otimes_{\\Gamma(X, \\mathcal{O}_X)} B \\to \\Gamma(U \\times_X V, \\mathcal{O}_{U \\times_X V}) \\) surjecti... | 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... | $24,637.09 |
[
{
"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... | (625, 1250) |
[
{
"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=3.5 cm |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 15.56 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... | 15.65 years |
[
{
"content": "Let the function $f(x)$ be thrice differentiable and satisfy $f(f(x)) = 1 - x$ for all $x \\in [0, 1]$. Given that $f''\\left(\\frac{4}{5}\\right) = 0$, find the value of \\( \\int_0^1 f(x) \\, dx \\).",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Compute the value of \\(5^{456789} \\mod 125000\\).",
"role": "user"
}
] | 78125 |
[
{
"content": "Let $(E,\\mathcal{A},\\mu)$ be a finite measure space. If a sequence of integrable functions converges in measure to a function $f$, does it imply that $f$ is integrable? Answer yes or no.",
"role": "user"
}
] | No |
[
{
"content": "Determine the Galois group of the splitting field of the polynomial \\( f(x) = x^3 - 11 \\) over \\( \\mathbb{Q} \\).",
"role": "user"
}
] | S_3 |
[
{
"content": "Evaluate the limit \\( \\lim_{N \\to \\infty} \\sum_{k=1}^N \\frac{k^3}{N^4} \\) using the concept of Riemann sums.",
"role": "user"
}
] | \dfrac{1}{4} |
[
{
"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... | 0.219 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... | $32,000 |
[
{
"content": "Is there a subset $\\mathfrak{F}$ of $\\omega^\\omega$, with cardinality $\\aleph_1$, such that no subset $\\mathfrak{G}$ of $\\mathfrak{F}$ with cardinality less than $\\aleph_1$ can generate $\\mathfrak{F}$, where generation means every function in $\\mathfrak{F}$ is a finite composition of func... | Yes |
[
{
"content": "If $f_n \\xrightarrow{wk\\star} f$, does it follow that $\\int_{\\mathbb{R}^d} f_n g \\to \\int_{\\mathbb{R}^d} f g$ for all $g \\in L^q(\\mathbb{R}^d)$, where $1/p + 1/q = 1$?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether an odd number $n > 1$ can divide $2^n - 1$. Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"content": "Find the maximum value of \\( \\frac{x^2+y^2+z^2}{xy+xz+yz} \\) where \\( x, y, z \\in [1, 2] \\).",
"role": "user"
}
] | \dfrac{6}{5} |
[
{
"content": "Determine the possible values of $n$ for which an automorphism $\\sigma$ of $\\mathbb{Q}(\\omega)$ can map $\\omega$ to $\\omega^n$, where $\\omega = e^{\\frac{2\\pi i}{10}}$. Provide your answer as a set of integers.",
"role": "user"
}
] | \{1, 3, 7, 9\} |
[
{
"content": "Find the function $f(z) = u(x, y) + iv(x, y)$, where $z = x + iy$, given that $if = -v + iu$ and $\\Re[(1+i)f(z)] = e^x (\\cos y - \\sin y)$. What is $f(z)$?",
"role": "user"
}
] | e^z |
[
{
"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... | 2.1e-9 M |
[
{
"content": "Evaluate the definite integral: $$\\int_{\\sqrt[3]{4}}^{\\sqrt[3]{3+e}}x^2 \\ln(x^3-3)\\,dx.$$",
"role": "user"
}
] | \dfrac{1}{3} |
[
{
"content": "Find the remainder when the polynomial $f(x)$, which satisfies the conditions $f(x) \\equiv 5 \\pmod{(x-1)}$, $f(x) \\equiv 3 \\pmod{(x+1)}$, and $f(x) \\equiv 5 \\pmod{(x+2)}$, is divided by $x^3 + 2x^2 - x - 2$. The polynomial $x^3 + 2x^2 - x - 2$ can be factored as $(x-1)(x+1)(x+2)$. Determine ... | x^2 + x + 3 |
[
{
"content": "Malmer Pebane's apartment uses a six-digit access code, with leading zeros allowed. He noticed that his fingers leave smudges that reveal which digits were pressed. He decided to change his access code to provide the largest number of possible combinations for a burglar to try when the digits are ... | 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... | 1112.781 Nm |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 50.94 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty}\\left(\\dfrac{n^2+5n+3}{n^2+n+2}\\right)^n \\]",
"role": "user"
}
] | e^4 |
[
{
"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... | #r = (\sqrt{R^2+4H^2})/2# |
[
{
"content": "In trapezoid \\(ABCD\\), the bases \\(AB\\) and \\(CD\\) are 101 and 20, respectively. The diagonals of the trapezoid are mutually perpendicular. Calculate the scalar product of the vectors \\(\\overrightarrow{AD}\\) and \\(\\overrightarrow{BC}\\).",
"role": "user"
}
] | 2020 |
[
{
"content": "Let $A$ be a finite-dimensional commutative algebra over $\\mathbb{R}$ without zero divisors. Determine the degree of the minimal polynomial of an element $a \\in A \\setminus \\mathbb{R}$.",
"role": "user"
}
] | 2 |
[
{
"content": "Suppose that there are two congruent triangles $\\triangle ABC$ and $\\triangle ACD$ such that $AB = AC = AD,$ as shown in the following diagram. If $\\angle BAC = 20^\\circ,$ then what is $\\angle BDC$? [asy]\npair pA, pB, pC, pD;\npA = (0, 0);\npB = pA + dir(240);\npC = pA + dir(260);\npD = pA +... | 10^\circ |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 48.37 V |
[
{
"content": "In how many ways can 8 distinguishable people, consisting of 2 Americans, 2 Canadians, 2 Mexicans, and 2 Jamaicans, be seated around a round table such that at least two pairs of countrymen are seated together? Consider rotations as identical.",
"role": "user"
}
] | 1632 |
[
{
"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... | $58,000 |
[
{
"content": "What is $0.4\\overline8 + 0.\\overline{37}$? Express your answer as a common fraction in lowest terms.",
"role": "user"
}
] | \frac{427}{495} |
[
{
"content": "Determine whether the following statement is true or false: If $(f^2)'$ and $f'$ are in $L^1(0,\\infty)$, then the limit of $f^2(x)$ as $x$ approaches infinity exists, and if $f$ is in $L^2$, then the limit of $f(x)$ as $x$ approaches infinity is 0.",
"role": "user"
}
] | True |
[
{
"content": "Evaluate the limit: $$\\lim\\limits_{n \\to \\infty} \\sum\\limits_{k=n}^{3n} \\frac{1}{k}$$",
"role": "user"
}
] | \ln 3 |
[
{
"content": "Given a random variable \\(X\\) such that \\(-1 \\leq X \\leq \\frac{1}{2}\\) and \\(\\mathbb{E}[X] = 0\\), find the maximal value of \\(\\mathbb{E}[X^2]\\).",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Evaluate the limit \\( \\lim_{(x,y)\\to 0} \\frac{|x^{3}+y^{3}|}{x^{2}+y^{2}} \\).",
"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... | 76.38 |
[
{
"content": "Evaluate the double integral \\(\\iint\\limits_{x^2+y^2\\ge k}\\frac{\\exp(-(x^2+y^2)/2)}{2\\pi}dxdy\\).",
"role": "user"
}
] | e^{-k/2} |
[
{
"content": "Evaluate the integral \\( \\int_0^{2\\pi} \\frac{\\sin^2x}{5+4\\cos x} \\ dx \\) using complex integration techniques.",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "In a real vector space, consider $L = \\sum_{i} t_i L_i$, where the $L_i$'s are linear functionals. If for all $x$ such that $L_i x \\geq 0$ for every $i$, it follows that $Lx \\geq 0$, can the coefficients $t_i$ be redefined to ensure $t_i \\geq 0$ for every $i$? Provide a proof or a reference to... | Yes |
[
{
"content": "Find the variance of the expression $${\\bf a}^* {\\bf N}$$ where \\({\\bf a} = \\frac{1}{\\sqrt{N}}[1, e^{jA}, e^{j2A},\\cdots, e^{j(N-1)A}]^T\\) and \\({\\bf N}\\) is a vector of i.i.d \\(\\mathcal{N}(0,1)\\) random variables. Simplify the expression for the variance.",
"role": "user"
}
] | 1 |
[
{
"content": "Calculate $f(f(f(2^{1388})))$, where $f(x)$ is defined as the sum of the digits of $x$. Provide a simple and logical method to find the result.",
"role": "user"
}
] | 4 |
[
{
"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... | -2.859e-6 C/m^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... | 102 N/C, 155 degrees counterclockwise from the positive x-axis |
[
{
"content": "Consider the sequence of operators $\\{T_n\\}$ defined as $T_n:\\ell^2\\to \\ell^2$, where $T_nx = (\\underbrace{0,0,...,0}_n,a_{n+1},a_{n+2},a_{n+3}...)$ for $x=(a_{1},a_{2},a_{3}...)\\in\\ell^2$. Show that the norm of this operator is $\\|T_n\\|=1$.",
"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... | NH3 |
[
{
"content": "Find the limit of the sequence \\(\\{u_n\\}\\) defined by \\(u_n = \\frac{1 + 2^2 + 3^3 + \\cdots + n^n}{n^n}\\) as \\(n\\) approaches infinity.",
"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... | 5792.016 kg |
[
{
"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... | 0.090 M |
[
{
"content": "Can a real-valued square matrix \\(\\textbf{A} \\in \\mathbb{R}^{n \\times n}\\) have no real eigenvalues when \\(n\\) is odd?",
"role": "user"
}
] | No |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 10 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... | 17755.86 |
[
{
"content": "Determine the minimal polynomial of $b = \\bar{y}$ over $\\mathbb{Q}$, where $L = \\mathbb{Q}[y]/(y^2 + 3)$ and $L \\simeq \\mathbb{Q}(\\sqrt{-3})$. Given that $[L:\\mathbb{Q}]=2$, find the minimal polynomial of $b$.",
"role": "user"
}
] | x^2 + 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... | The electronic transitions in a hydrogen atom that result in the emission of visible light (400 nm - 700 nm) are those in the Balmer series where nf = 2 and ni = 3, 4, 5, 6, or 7. Specifically, the wavelengths of the visible light emitted are 656.3 nm (ni=3), 486.2 nm (ni=4), 434.1 nm (ni=5), 410.2 nm (ni=6), and 397.0... |
[
{
"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... | 0.73 |
[
{
"content": "Is it possible to construct a sequence of real numbers such that the set of its limit points is exactly the set of natural numbers $\\mathbb{N}$? Answer with 'Yes' or 'No' and justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "If $f(x)$ is a differentiable function such that \\( f(x) = \\left(1 + \\frac{x^3}{3}\\right) + \\int_0^x e^{-t} f(x-t) \\, dt \\), find the value of \\( \\int_0^1 f(x) \\, dx \\).",
"role": "user"
}
] | \dfrac{8}{5} |
[
{
"content": "Determine whether the space of real-valued continuous functions on $[0, \\infty)$, denoted by $C([0, \\infty))$, is metrizable.",
"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... | 2.0e-2 m |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\int_{\\frac{1}{n+1}}^{\\frac{1}{n}} \\frac{\\sin(x)}{x^3} \\, dx. \\]",
"role": "user"
}
] | 1 |
[
{
"content": "Determine whether the sequence \\( \\frac{f(n)}{2^n} \\), where \\( f(n) \\) is the \\( n^{th} \\) Fibonacci number, converges as \\( n \\to \\infty \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Let $G$ be a finite non-nilpotent group. Determine whether there exists a non-normal maximal subgroup $M$ such that the index $|G:M|$ is a prime power.",
"role": "user"
}
] | No |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.7 atm |
[
{
"content": "Find a non-zero function $f(x)$ such that $\\dfrac{d}{dx}f(x)=f(x-1)$. If no such function exists, explain why.",
"role": "user"
}
] | e^{W(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... | -26.4 m/s^2 |
[
{
"content": "In a finite group $G$, determine the lower bound for the proportion of irreducible rational-valued characters that take the values $0$ or $\\pm 1$. Provide your answer as a fraction or decimal.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the integral: \\[ I = \\int_{0}^{\\infty} J_{0}(x) \\, dx \\] where \\( J_{0}(x) \\) is the Bessel function of order zero.",
"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... | +23.08e-9 C |
[
{
"content": "Can the real numbers on the interval $[0,1]$ be well-ordered?",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the integral by changing the order of integration: \\[ \\int^{\\infty}_{0} \\int^{x}_{0} x e^{-x^2/y} \\, dy \\, dx \\]",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Find the position of the term \\( \\frac{2020}{2187} \\) in the sequence: \\( \\frac{2}{3}, \\frac{2}{9}, \\frac{4}{9}, \\frac{6}{9}, \\frac{8}{9}, \\frac{2}{27}, \\frac{4}{27}, \\ldots, \\frac{26}{27}, \\ldots, \\frac{2}{3^{n}}, \\frac{4}{3^{n}}, \\ldots, \\frac{3^{n}-1}{3^{n}}, \\ldots \\).",
... | 1553 |
[
{
"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... | 1.365 mm |
[
{
"content": "Determine the values of \\(a\\) and \\(b\\) such that the polynomial equation \\(x^4 - 4x^3 + ax^2 + bx + 1 = 0\\) has four real roots. Choose from the following options:\n\n- \\((4, -6)\\)\n- \\((6, -4)\\)",
"role": "user"
}
] | (6, -4) |
[
{
"content": "Determine the number of non-equivalent formulas that can be constructed using the propositions \\( p_1, p_2, \\ldots, p_n \\).",
"role": "user"
}
] | 2^{2^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... | 319.62 g |
[
{
"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... | 13.09% |
[
{
"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... | 50 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... | $3,437,500 |
[
{
"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... | KOH or LiOH |
[
{
"content": "Given the expressions \\(a = \\lg z + \\lg \\left[ \\frac{x}{yz} + 1 \\right]\\), \\(b = \\lg \\frac{1}{x} + \\lg (xyz + 1)\\), and \\(c = \\lg y + \\lg \\left[ \\frac{1}{xyz} + 1 \\right]\\), determine the minimum value of the largest number among \\(a\\), \\(b\\), and \\(c\\).",
"role": "use... | \lg 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... | 17.58% |
[
{
"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... | 3.66 m/s^2 |
[
{
"content": "Evaluate the convergent sequence \\( \\sum_{n=1}^{\\infty} \\frac{n-1}{n!} \\) without using power series or Taylor expansions.",
"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-(d/16)^2)^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... | 15.69e-6 C and positive |
[
{
"content": "Find the values of \\(\\alpha\\) for which the series \\(\\sum_{k=0}^{\\infty}(-1)^{k}\\frac{x^{2k}}{(2k)!^{\\alpha}}\\) converges for all real numbers \\(x\\).",
"role": "user"
}
] | \alpha > 0 |
[
{
"content": "Evaluate the limit using L'Hopital's Rule: $$\\lim_{x \\to 0} \\frac{\\ln(1+\\ln(1+x))}{\\ln(1+\\ln(1+\\frac{1}{x}))}$$",
"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... | 29.39 L |
[
{
"content": "Given a quadratic polynomial $f(x)$ with a positive leading coefficient and a function $g(x) = 1 - \\frac{2}{x-5}$ defined for $x < 5$, define $h(t)$ as the minimum value of $f(g(x))$ for $t \\leq x \\leq t+2$. If $h(-1) = 7$, $h(t) = 6$ for $1 \\leq t < 3$, and $h(t)= f(g(t+2))$ for $t < 1$, find... | 42 |
[
{
"content": "Find the integer part of the sum \\( \\sum^{9999}_{k=1}\\frac{1}{\\sqrt[4]{k}} \\).",
"role": "user"
}
] | 1332 |
[
{
"content": "In an isosceles triangle \\(ABC\\) with \\(AB = AC\\) and \\(\\angle ABC = 53^\\circ\\), point \\(K\\) is such that \\(C\\) is the midpoint of segment \\(AK\\). Point \\(M\\) is chosen such that \\(B\\) and \\(M\\) are on the same side of line \\(AC\\), \\(KM = AB\\), and the angle \\(\\angle MAK\... | 44 |
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