prompt list | ground_truth string |
|---|---|
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 417.478 V |
[
{
"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... | 2Ag^+ + 2OH^- -> 2AgOH(s) |
[
{
"content": "On a $23 \\times 23$ grid board, there are four pieces: one white piece in the bottom-left corner, one white piece in the top-right corner, one black piece in the top-left corner, and one black piece in the bottom-right corner. The pieces move alternately, starting with the white pieces. Each move... | 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... | 199 g |
[
{
"content": "Determine whether there exists an arc $\\overset{\\frown}{AB}$ in the unit circle $\\mathbb{S}^{1}$ that connects two disjoint compact sets $F_{1}$ and $F_{2}$, such that $A \\in F_{1}$, $B \\in F_{2}$, and the open arc $\\overset{\\frown}{AB}$ contains no points from $F_{1} \\cup F_{2}$. Answer w... | Yes |
[
{
"content": "Determine the order of the group generated by elements $x$ and $y$ satisfying the relations $x^3y = xy^4 = 1$.",
"role": "user"
}
] | 11 |
[
{
"content": "Let $P$ be an orthogonal projection onto a subspace $E$ of an inner product space $V$. If the dimensions of $V$ and $E$ are $n$ and $m$ respectively, determine the dimensions of the eigenspaces of $P$ corresponding to the eigenvalues $0$ and $1$. Provide your answer as a pair of dimensions.",
... | (m, n - m) |
[
{
"content": "Determine all positive integers \\( n \\) such that for any sequence \\( a_1, a_2, \\ldots, a_n \\) with each \\( a_i \\in (0, 1] \\), the inequality \\( \\sum_{i=1}^{n} \\frac{\\sqrt{1-a_{i}}}{a_{i}} \\leq \\frac{1}{\\prod_{i=1}^{n} a_{i}} \\) holds.",
"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... | 0.133 g H_2 |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\to 0} \\frac{\\ln\\left[\\dfrac{(1-3x)(1+x)^3}{(1+3x)(1-x)^3}\\right]}{x^3}$$ without using L'Hôpital's rule, Taylor series, or differentiation (or integration).",
"role": "user"
}
] | -16 |
[
{
"content": "Find the function \\( f: \\mathbb{R} \\to \\mathbb{R} \\) that satisfies the equation \\( f(-y^2) + f(x^2 + y) - y = (x + y)f(x - y) \\) for all real numbers \\( x \\) and \\( y \\).",
"role": "user"
}
] | f(x) = x |
[
{
"content": "Evaluate the integral \\( \\frac{3}{4\\pi}\\int\\int\\int\\limits_{\\mathbb{R}^3} e^{-\\sqrt{x^2+y^2+z^2}^3} \\, dV \\).",
"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... | E = \frac{2k\lambda}{a} |
[
{
"content": "Let $X$ be a compact metric space, and let $\\{f_k\\}$ be a sequence of continuous functions on $X$ that converges uniformly to a function $f$. If $\\int_X f_k = 1$ for all $k$, does it follow that $\\int_X f = 1$?",
"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... | 5 N with 53.13 degrees with East direction |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\rightarrow 0}\\frac{(1+2x)^\\frac{1}{x}-(1+x)^\\frac{2}{x}}{x}$$",
"role": "user"
}
] | -e^2 |
[
{
"content": "For real numbers \\(x_1, x_2, x_3, \\dots, x_{333} \\in [-1, \\infty)\\), let \\(S_k = \\sum_{i = 1}^{333} x_i^k\\) for each \\(k\\). Given that \\(S_2 = 777\\), find the minimum possible value of \\(S_3\\).",
"role": "user"
}
] | 999 |
[
{
"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... | $724,188 |
[
{
"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... | 32.6 m/s |
[
{
"content": "Evaluate the limit \\( \\lim_{{n\\to\\infty}}\\frac{\\int_{0}^{1}\\left(x^2-x-2\\right)^n\\,dx}{\\int_{0}^{1}\\left(4x^2-2x-2\\right)^n\\,dx} \\).",
"role": "user"
}
] | 2 |
[
{
"content": "A certain organism begins as two cells. Each cell splits and becomes two cells at the end of three days. At the end of another three days, every cell of the organism splits and becomes two cells. This process lasts for a total of 15 days, and no cells die during this time. How many cells are there... | 64 |
[
{
"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... | 1.2 kg*m^2/s |
[
{
"content": "Assume we have two continuous integrable functions $f, g \\in L^1(\\mathbb{R})$ such that $f(x_0) \\leq g(x_0)$ for some $x_0 \\in \\mathbb{R}$. Define the Gaussian function with variance $\\sigma > 0$ as $p_\\sigma(x) = \\frac{1}{\\sqrt{2 \\pi} \\sigma} e^{-\\frac{x^2}{2\\sigma^2}}$. Can we concl... | 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... | 132.629 Hz |
[
{
"content": "Given the equations $x^2 y + y^2 z + z^2 x = 2186$ and $xy^2 + yz^2 + zx^2 = 2188$, where $x, y, z$ are integers, find the value of $x^2 + y^2 + z^2$. Express your answer as an integer.",
"role": "user"
}
] | 245 |
[
{
"content": "Let $(s_n)$ be a sequence such that $s_1, s_2, \\ldots, s_{100}$ are positive and $s_n < 0$ for all $n > 100$. Assume $\\lim_{n \\to \\infty} s_n = 0$. Determine $\\lim_{n \\to \\infty} \\frac{1}{s_n}$.",
"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... | 135.41 g |
[
{
"content": "Determine whether the limit \\( \\lim_{x\\to \\infty} \\left[\\frac{p(x)}{e^x}\\right] \\) always converges to 0, where \\( p(x) \\) is a polynomial.",
"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... | 8.31e-4 V |
[
{
"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... | $1,167 |
[
{
"content": "Find a number that can be expressed as a series of 1s and 0s in any base from 2 to 9 and also in base 64.",
"role": "user"
}
] | 1 |
[
{
"content": "For any prime $p$, does there exist a constant $C$ such that if $x \\ge C$, then at most one integer among $x+1, x+2, \\dots, x+p$ has its greatest prime factor less than or equal to $p$?",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the integral \\( \\int (\\log x + 1)x^x \\, dx \\).",
"role": "user"
}
] | x^x + C |
[
{
"content": "In the diagram, $P$ is on $RS$ so that $QP$ bisects $\\angle SQR$. Also, $PQ=PR$, $\\angle RSQ=2y^\\circ$, and $\\angle RPQ=3y^\\circ$. What is the measure, in degrees, of $\\angle RPQ$? [asy]\n// C14\nimport olympiad;\nsize(7cm);\n\nreal x = 50; real y = 20;\n\npair q = (1, 0);\npair r = (0, 0)... | 108 |
[
{
"content": "Consider a continuous function $f: \\mathbb{R}^2 \\rightarrow \\mathbb{R}$ with the property that $f(x) = 0$ for only finitely many points $x \\in \\mathbb{R}^2$. Which of the following statements is true?\n\n1. $f(x) \\leq 0$ for all $x$ or $f(x) \\geq 0$ for all $x$\n2. The function $f$ is one-t... | 1 |
[
{
"content": "Determine the condition under which the integral \\( \\int \\frac{ax^2+bx+c}{x^3(x-1)^2} \\, dx \\) represents a rational function.",
"role": "user"
}
] | a + 2b + 3c = 0 |
[
{
"content": "Calculate the flow of the vector field \\( F(x,y,z) = (xz, yz, -z^2) \\) out of the surface \\( S = \\{(x,y,z) \\in \\mathbb{R}^3 : z^2 = 1 + x^2 + y^2, 2 \\leq z \\leq 3\\} \\) using the definition of surface integral.",
"role": "user"
}
] | -60\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... | 1580 m/s |
[
{
"content": "Evaluate the integral \\( \\int_{-\\pi/2}^{\\pi/2} \\frac{1}{1+e^{\\sin x}} \\, dx \\).",
"role": "user"
}
] | \dfrac{\pi}{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... | 0.13 g |
[
{
"content": "Calculate the residues of the function \\( f(z) = e^z \\csc^2(z) \\) at the poles \\( z = (2n+1)\\pi \\), where \\( n \\) is an integer.",
"role": "user"
}
] | e^{(2n+1)\pi} |
[
{
"content": "Determine the largest real value of \\( r \\) such that the ranges of the functions \\( f(x) = 2x^2 - 3x + r \\) and \\( f(f(x)) \\) are identical.",
"role": "user"
}
] | \dfrac{15}{8} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 82 N |
[
{
"content": "Find the maximum value of the function \\( f(x) = \\int_0^1 e^{|t-x|} \\, dt \\) for \\( 0 \\leq x \\leq 1 \\).",
"role": "user"
}
] | e - 1 |
[
{
"content": "Does there exist a non-separable topological space $(X, \\tau)$ such that the sigma algebra generated by $\\tau$ is equal to the sigma algebra generated by a minimal topological base $\\tau_0$ of $\\tau$?",
"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... | 15.96 m/s |
[
{
"content": "Find the center of the circle on which the roots of the equation \\((z-i)^n = \\sqrt{21} + 2i\\) lie, where \\(z\\) is a complex number and \\(i = \\sqrt{-1}\\).",
"role": "user"
}
] | i |
[
{
"content": "Evaluate the sum \\( \\sum_{k=0}^{49}(-1)^k\\binom{99}{2k} \\). Choose the correct answer from the following options: \\( \\textbf{(A)}\\ -2^{50} \\qquad\\textbf{(B)}\\ -2^{49} \\qquad\\textbf{(C)}\\ 0 \\qquad\\textbf{(D)}\\ 2^{49} \\qquad\\textbf{(E)}\\ 2^{50} \\)",
"role": "user"
}
] | B |
[
{
"content": "Determine whether the following statement is true or false: Every finite-dimensional vector space is the dual space of another vector space.",
"role": "user"
}
] | A |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $18,513,000 |
[
{
"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... | A = Q/sqrt(2 * permittivity of free space * u) |
[
{
"content": "Let G be a cyclic group of order n, where n is odd. Determine the infimum of the average of the sum of the element orders in G.",
"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... | 45.47 |
[
{
"content": "Find the number of different homomorphisms \\( \\varphi : \\mathbb{Z}_{18} \\to \\mathbb{Z}_{24} \\) such that \\(|\\text{Im}\\varphi| = 6\\).",
"role": "user"
}
] | 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... | 3.613e+24 atoms |
[
{
"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... | malate dehydrogenase (MDH) |
[
{
"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}\frac{\rho R}{18\epsilon_0}{/eq} |
[
{
"content": "Find the value of the expression \\( \\cos\\left(\\frac{\\pi}{7}\\right) + \\cos\\left(\\frac{3\\pi}{7}\\right) + \\cos\\left(\\frac{5\\pi}{7}\\right) \\) without using trigonometric tables.",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Determine the radius of convergence of the power series \\( \\sum_{k=0}^{\\infty} a_k x^k \\), given that the series \\( \\sum_{k=0}^\\infty (-1)^k a_k \\) converges and \\( \\sum_{k=0}^\\infty a_k \\) diverges.",
"role": "user"
}
] | 1 |
[
{
"content": "Suppose that $F(z)$ is an analytic function with $f(z)$ as its derivative. Consider the complex line integral \\( \\int_\\gamma f(z)\\ dz \\). Is this integral equal to zero if $\\gamma$ is a closed curve? Justify your answer.",
"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... | 5.23 mm |
[
{
"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... | 51618.8 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... | Myocardial scar |
[
{
"content": "In a plane, 36 points are marked, with no three points being collinear. Some pairs of these points are connected by segments such that no more than 3 segments originate from any point. Determine the maximum number of distinct closed 4-segment broken lines that can be formed, where the vertices are... | 54 |
[
{
"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... | 45000 N/C, away from the charge |
[
{
"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... | 2.81 N |
[
{
"content": "Determine the degree of the differential equation: $$(y''')^\\frac{4}{3} + (y')^\\frac{1}{5} + 4=0$$.",
"role": "user"
}
] | 20 |
[
{
"content": "Evaluate the limit: $$\\lim\\limits_{n\\to\\infty}\\prod_{n=1}^{n}\\left(1+\\left(\\frac{1}{2}\\right)^{2^{n}}\\right)$$",
"role": "user"
}
] | \dfrac{4}{3} |
[
{
"content": "Given a set of identical regular pentagons, each with natural numbers from 1 to 5 written at its vertices. The pentagons can be rotated and flipped. They are stacked vertex to vertex such that the sums of numbers at each of the five vertices are equal. Determine the number of pentagons in this sta... | 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... | 119.114e5 N |
[
{
"content": "Determine the number of roots of the polynomial \\( g(x) = x^3 + 2x + 1 \\) in the finite field \\( \\mathbb{F}_{3^2} \\) using Proposition 11.1.5 from David Cox's \"Galois Theory.\"",
"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... | 80\pi / 3 km/min |
[
{
"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... | 29.8 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... | 7.1 mT |
[
{
"content": "Given a positive real number $x$, a sequence $\\{a_n(x)\\}$ is defined as follows: \\(a_1(x)=x\\) and \\(a_n(x)=x^{a_{n-1}(x)}\\) recursively for all \\(n\\geq 2\\). Determine the largest value of $x$ for which \\(\\lim_{n\\to\\infty}a_n(x)\\) exists.",
"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... | $158,400 |
[
{
"content": "Consider two independent real random variables \\(X_1, X_2\\) and a real function \\(g:\\mathbb{R}^2\\to\\mathbb{R}\\) such that \\(\\lim_{x\\to-\\infty}g(x, x_2) = -\\infty\\) for any \\(x_2\\in\\mathbb{R}\\). Prove or disprove that for any \\(d\\in\\mathbb{R}\\), the following holds: \\[ \\lim_{... | 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... | 61.92 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... | 1.5g |
[
{
"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... | 31.2 W |
[
{
"content": "Determine if there are any other solutions to the differential equation \\( y = 2\\sqrt{x} - \\frac{1}{2} \\frac{(y')^2}{y''} \\) besides \\( y = 4\\sqrt{x} \\).",
"role": "user"
}
] | y = 4\sqrt{x} |
[
{
"content": "Find the largest binomial coefficient in the expansion of \\(\\left(n+\\frac{1}{n}\\right)^{n}\\) given that the product of the fourth term from the beginning and the fourth term from the end equals 14400.",
"role": "user"
}
] | 252 |
[
{
"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... | 1.89e+27 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... | 4.9 seconds |
[
{
"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... | dollars |
[
{
"content": "Calculate the limit: $$\\lim_{n\\rightarrow\\infty}\\sqrt[n]{\\prod_{i=1}^n\\frac{1}{\\cos\\frac{1}{i}}}$$",
"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... | $341,928.51 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2MR^2 |
[
{
"content": "Suppose $E_1$ and $E_2$ are two elliptic curves over $\\mathbb{Q}$ such that there exists a $\\mathbb{Q}$-isogeny from $E_1$ to $E_2$. If the Main Conjecture for elliptic curves is true for one of these curves, is it also true for the other? Provide a justification for your answer.",
"role": "... | Yes |
[
{
"content": "Consider a connected and bipartite graph $G = (V_1 \\cup V_2, E)$ with the following properties:\n1. The vertex sets $V_1$ and $V_2$ have equal size: $|V_1| = |V_2| = N$, where $N$ is a positive integer.\n2. For every vertex $v \\in V_1$ and $v' \\in V_2$, the degree satisfies $2 \\leq \\deg(v), \... | 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... | $4,800 |
[
{
"content": "Determine whether there exists a sequence of non-negative numbers $a_n$ and $b_n$, for $n \\ge 1$, such that the following limit holds: \\[ \\lim_{x\\to 0^+} \\dfrac{\\sum_{x>a_n} b_n}{x} = 1. \\]",
"role": "user"
}
] | Yes |
[
{
"content": "Find the largest possible value of $f(4)$ if the continuously differentiable function $f : \\mathbb{R} \\rightarrow \\mathbb{R}$ satisfies $f(0) = 0$ and $f(x)f'(x) \\leq 2$ for all $x \\in \\mathbb{R}$. The function is defined on the entire real line.",
"role": "user"
}
] | 4 |
[
{
"content": "In a cube, the edges originating from a single vertex form a corner tetrahedron. Calculate the volume of the cube that is occupied by the points lying within the common regions of at least two corner tetrahedra.",
"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... | no effect |
[
{
"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... | 2801.88 N |
[
{
"content": "Calculate the volume of the region bounded by the surfaces \\( z = x^2 + y^2 \\) and \\( z = 2x \\).",
"role": "user"
}
] | \dfrac{\pi}{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... | 34.33 mL |
[
{
"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... | 214m |
[
{
"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... | 7.49 |
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