prompt list | ground_truth string |
|---|---|
[
{
"content": "Evaluate the integral \\( \\int_{0}^{2} x \\, d\\alpha(x) \\), where \\( \\alpha(x) = \\begin{cases} x & ,0 < x \\leq 1 \\\\ 2 + x & , 1 < x \\leq 2 \\end{cases} \\).",
"role": "user"
}
] | 4 |
[
{
"content": "Calculate the limit of the sequence \\(a_n = n\\left(\\frac{1}{n^2 + 1} + \\frac{1}{n^2 + 3} + \\dots + \\frac{1}{n^2 + 2n + 1}\\right)\\).",
"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... | 4-ethyl-2,2-dimethylhexane |
[
{
"content": "Compute the limit: \\[ \\lim_{(x,y)\\to (0,0)} \\frac{\\sin(xy)-xy}{x^2y} \\] using Taylor's theorem.",
"role": "user"
}
] | 0 |
[
{
"content": "Real numbers \\(x\\) and \\(y\\) satisfy the following equations:\n\\[ x^2 + y^2 = 2023 \\]\n\\[ (x-2)(y-2) = 3 \\]\nDetermine the largest possible value of \\(|x-y|\\).",
"role": "user"
}
] | 13\sqrt{13} |
[
{
"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... | 0.04 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... | 1.1275*10^-22J |
[
{
"content": "Given an $n$-dimensional vector space $V$ with basis $\\{e_i\\}$, consider the Young diagram $\\lambda$ for the partition $1+1+\\ldots+1=n$ and the corresponding Young symmetrizer $c_{\\lambda}$. Let $\\hat{c}_{\\lambda}$ be the scalar multiple of $c_{\\lambda}$ such that $\\hat{c}_{\\lambda}$ is ... | 1 |
[
{
"content": "Find the area enclosed by the polar curve \\( r = \\sin(3\\theta) \\).",
"role": "user"
}
] | \dfrac{\pi}{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... | 0.9 mol Al2O3 |
[
{
"content": "Determine whether the following statement is true or false: If $E \\subset \\mathbb{R}^n$ is a Lebesgue measurable set with finite measure, then $E$ is bounded.",
"role": "user"
}
] | 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... | 0.125 m \hat{i} + 0.125 m \hat{j} |
[
{
"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... | Take 192.5 mL of 1 normal solution in standard flask and dilute it up to 550 mL |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\to 0}\\frac{\\cos(x) - 8x\\sin(x/2) - \\cos(3x)}{x^4}$$",
"role": "user"
}
] | -\dfrac{19}{6} |
[
{
"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... | 22,475 |
[
{
"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... | 0.35 meters |
[
{
"content": "Suppose that a function $f$ has a pole of order $m$ at $z = 0$, and \\( \\frac{1}{|z|^{3/2}} \\le |f(z)| \\le \\frac{1}{|z|^{5/2}} \\) for all $z$ satisfying $0 < |z| \\le 1$. Determine the order $m$ of the pole at $z = 0$.",
"role": "user"
}
] | 2 |
[
{
"content": "Suppose $H$ is a Hilbert space, and $T$ is a continuous operator that satisfies $$(Tx,x)\\leq k(x,x),$$ where $k<0$ and $x\\in H$. Is $T$ necessarily injective?",
"role": "user"
}
] | Yes |
[
{
"content": "Triangle $ABC$ has sides of length $5$, $6$, and $7$, with vertices $A$, $B$, and $C$ located on the positive $x$-axis, $y$-axis, and $z$-axis, respectively. Let $O$ be the origin. Calculate the volume of tetrahedron $OABC$. Express your answer in simplest form.",
"role": "user"
}
] | \sqrt{95} |
[
{
"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... | $840,952 |
[
{
"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... | 11.30 |
[
{
"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... | $49,113.06 |
[
{
"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... | 2.52 kg/s |
[
{
"content": "Compute the line integral of the vector field \\( v_1(x_1, x_2, x_3) = \\begin{pmatrix} x_1 x_2 \\\\ -x_1^2 \\\\ x_3 \\end{pmatrix} \\) along the curve \\( \\gamma(t) = (\\cos(t), \\sin(t), t) \\) for \\( t \\in [0, 2\\pi] \\).",
"role": "user"
}
] | 2\pi^2 |
[
{
"content": "Joanie takes a $\\$6,\\!000$ loan to pay for her car. The annual interest rate on the loan is $12\\%$. She makes no payments for 4 years, but has to pay back all the money she owes at the end of 4 years. How much more money will she owe if the interest compounds quarterly than if the interest co... | \$187.12 |
[
{
"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... | 2.55 |
[
{
"content": "Find the minimal polynomial over \\( \\mathbb{Q} \\) for the number \\( \\sqrt{9-4\\sqrt{2}} \\), given that the minimal polynomial of \\( 9-4\\sqrt{2} \\) is \\( x^2-18x+49 \\).",
"role": "user"
}
] | x^2 + 2x - 7 |
[
{
"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... | 9.508 |
[
{
"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... | 4.05e7 J |
[
{
"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}K_c=[O_2]^{25}{/eq} |
[
{
"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... | 12.4% |
[
{
"content": "Evaluate the integral \\( \\iint_D |x-y| \\, dx \\, dy \\) where \\( D = \\{(x, y) \\in \\mathbb{R}^2 : x \\ge 0, 0 \\le y \\le 3-2x \\} \\).",
"role": "user"
}
] | \dfrac{15}{8} |
[
{
"content": "Suppose $f: \\mathbf{R} \\to \\mathbf{R}$ is a continuous function, and for any $a \\in \\mathbf{R}$, there exist real numbers $c_1$ and $c_2$ with $(c_1, \\infty) \\cap \\mathbf{R} \\neq \\emptyset$ and $(-\\infty, c_2) \\cap \\mathbf{R} \\neq \\emptyset$ such that $f(t) > f(a)$ for all $t \\in (... | Yes |
[
{
"content": "Triangle ABC has sides AB = 15, BC = 13, and AC = 14. Let O be the orthocenter of this triangle, and let the reflections of the orthocenter across sides AB, BC, and AC be points D, E, and F, respectively. Calculate the area of hexagon ADBECF.",
"role": "user"
}
] | 168 |
[
{
"content": "Determine the type of undirected, 2-connected n-vertex graph $G$ for which the sum of the shortest paths between all pairs of vertices, $S(G) = \\sum_{x,y \\in V[G]} d_G(x,y)$, is maximized.",
"role": "user"
}
] | C_n |
[
{
"content": "Determine whether there exist positive integers $m$ and $n$ such that the ratio of any two distinct perfect squares is arbitrarily close to $1$. Provide a justification for 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... | 85.6 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... | o-chloromethyltoluene, m-chloromethyltoluene, and p-chloromethyltoluene |
[
{
"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... | 37.5 g |
[
{
"content": "Calculate the mass of the region in the first quadrant where the tile lies between $1 < xy < 3$ and $1 < x^2 - y^2 < 3$, given the density function $\\sigma(x, y) = (x^2 + y^2)(x^4 + y^4)$.",
"role": "user"
}
] | 26 |
[
{
"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... | 2.53 rad/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... | n_m= (sin i)/(sin r) |
[
{
"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... | Histones |
[
{
"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 recursive formula for the resistance at step n+1 is given by:
R_(n+1) = R + 1 / (1/R + 1/R_n)
where R is the resistance at each step. Alternatively, if R_n = R * a_n, then the terms for a_n are defined by:
a_1 = 2
a_(n+1) = 1 + [a_n / (1 + a_n)] |
[
{
"content": "Evaluate the limit in terms of $a$ and $b$: $$ \\lim_{s\\rightarrow 0^{+}} \\frac{\\Gamma(b-as)}{\\Gamma(s)\\Gamma(1-s)} $$",
"role": "user"
}
] | 0 |
[
{
"content": "Find the value of \\( \\lambda \\) such that \\( \\frac{3- \\tan^2 \\left(\\frac{\\pi}{7}\\right)}{1 - \\tan^2 \\left(\\frac{\\pi}{7}\\right)} = \\lambda \\cos\\left(\\frac{\\pi}{7}\\right) \\).",
"role": "user"
}
] | 4 |
[
{
"content": "Evaluate the limit using the Mean Value Theorem: $$\\lim_{h \\to 0} \\frac{\\int_0^{h} \\cos(x^2)dx}{h}$$",
"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... | -12e6 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... | 0.189 N |
[
{
"content": "Let \\( D^n \\subseteq \\mathbb{R}^n \\) be the closed \\( n \\)-disk, and let \\( p \\in \\text{int}(D^n) \\). Does there exist a homeomorphism \\( h: D^n \\to D^n \\) such that \\( h(p) = 0 \\) and \\( h(\\partial D^n) = \\partial D^n \\)?",
"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. Tide profile variation
2. Tidal current velocities |
[
{
"content": "Suppose that $s = a + bi$ is a complex root of the function $f(s) = 6^s + 2^s - 3^s$. Is the complex conjugate $\\overline{s} = a - bi$ also a root of the function?",
"role": "user"
}
] | Yes |
[
{
"content": "Define the sequence \\((a_p)_{p\\ge0}\\) as follows: \\[ a_p = \\frac{\\binom{p}{0}}{2\\cdot 4} - \\frac{\\binom{p}{1}}{3\\cdot 5} + \\frac{\\binom{p}{2}}{4\\cdot 6} - \\ldots + (-1)^p \\cdot \\frac{\\binom{p}{p}}{(p+2)(p+4)}. \\]\nFind \\( \\lim_{n\\to\\infty}(a_0+a_1+\\ldots+a_n) \\).",
"rol... | \dfrac{1}{3} |
[
{
"content": "For the function \\( f(z) = 2x^3 + 3iy^2 \\), find \\( f'(x + ix^2) \\).",
"role": "user"
}
] | 6x^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... | 6.95 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... | force = if not accelerating, (0, 0); otherwise, (-sin(current rotation), cos(current rotation)) |
[
{
"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... | 0.856 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... | 1.28e-7 T |
[
{
"content": "Find all prime numbers \\( p \\) such that \\( 7^p - p - 16 \\) is a perfect square.",
"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... | 12.36% |
[
{
"content": "Let $X$ and $Y$ be independent random variables with the same moment generating function $M(t)=E(e^{tX})$ for $-\\infty<t<\\infty$. Find the value of $E\\left(\\dfrac{e^{2tX}}{e^{tY}}\\right)$. \\(\\text{Express your answer in terms of } M(t).\\)",
"role": "user"
}
] | M(2t)M(-t) |
[
{
"content": "Determine the time complexity of the conjugate gradient method when solving the system of linear equations $Ax = b$, where $A$ is a sparse, symmetric, and positive definite matrix.",
"role": "user"
}
] | O(n^2) |
[
{
"content": "Verify the inverse Fourier transform of the function \\( F(k) = \\frac{\\sqrt{\\pi}}{\\sqrt{a}}e^{\\frac{k^2}{4a}} \\) to confirm that it corresponds to the original function \\( f(t) = e^{-at^2} \\), where \\( a > 0 \\).",
"role": "user"
}
] | e^{-at^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... | 3.67e4 kJ/mol |
[
{
"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... | 62.96 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... | a, b and c (tied), d |
[
{
"content": "Evaluate the integral \\( \\int_{-\\infty}^{+\\infty} \\frac{x^4 - 5x^2 + 3}{x^6+1} \\mathrm{d}x \\) using the residue theorem.",
"role": "user"
}
] | \pi |
[
{
"content": "Given a metric on \\(\\mathbb{R}^2\\) in polar coordinates of the form \\(dr^2 + f(r,\\theta)d\\theta^2\\), and a curve \\(\\gamma(t) = (t, \\theta_0)\\) where \\(\\theta_0\\) is a constant, find the covariant derivative \\(\\nabla_{\\gamma'(t)}\\gamma'(t)\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Find the minimum value of \\((2+x)(2+y)\\) subject to the constraint \\(x^2+xy+y^2=12\\), where \\(x\\) and \\(y\\) are real numbers.",
"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... | $538,000 |
[
{
"content": "Determine the number of conjugacy classes of the group \\( G = \\langle s,t \\mid s^8=t^2=1, st=ts^3 \\rangle \\).",
"role": "user"
}
] | 7 |
[
{
"content": "Determine the minimum number of equally spaced subintervals, N, required such that the difference between the Upper and Lower Riemann sums for the integral \\( \\int_{1}^{17} \\frac{1}{x^3} \\, dx \\) is less than \\( \\frac{1}{100} \\).",
"role": "user"
}
] | 1600 |
[
{
"content": "Determine whether the following conjecture is true or false: For every undirected graph $G=(V,E)$, there exist three pairwise disjoint sets of vertices $V_1,V_2,V_3$ such that for every $i\\in\\{1,2,3\\}$ and for every cycle $C$ in $G$, $C\\cap V_i\\neq \\emptyset$. Each $V_i$ must intersect all c... | False |
[
{
"content": "Determine all prime numbers that can be expressed in the form $n^5 - 1$ for some integer $n$. Provide a complete solution.",
"role": "user"
}
] | 31 |
[
{
"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... | $8,645.95 |
[
{
"content": "Find the closed form of the infinite product: \\[ \\prod_{k=0}^\\infty \\left(1+\\frac{1}{2^{2^k}}\\right) \\]",
"role": "user"
}
] | 2 |
[
{
"content": "Let \\( g(\\theta) = \\tan \\theta \\) be a strictly increasing continuous function mapping \\([0, b]\\) onto \\([0, \\tan b]\\), where \\( b \\in (0, \\pi/2) \\). Verify the equation \\( \\int_0^b \\tan \\theta \\, d\\theta + \\int_0^{\\tan b} \\tan^{-1} y \\, dy = b \\tan b \\).",
"role": "u... | b \tan b |
[
{
"content": "Can any polynomial with coefficients in \\(\\mathbb{C}\\) be expressed as a product of linear factors with coefficients in \\(\\mathbb{C}\\)?",
"role": "user"
}
] | Yes |
[
{
"content": "Circles \\(\\omega_{1}\\) and \\(\\omega_{2}\\) intersect at points \\(A\\) and \\(B\\). A circle centered at point \\(O\\) encompasses circles \\(\\omega_{1}\\) and \\(\\omega_{2}\\), touching them at points \\(C\\) and \\(D\\) respectively. Given that points \\(A, C\\), and \\(D\\) are collinear... | 90^\circ |
[
{
"content": "Let \\(f(x) = x^3 + 3x + 2\\). Determine the number of points \\(c\\) such that \\(f'(c) \\neq \\frac{f(b) - f(a)}{b - a}\\) for any two values of \\(a\\) and \\(b\\), where \\(a, b, c \\in \\mathbb{R}\\).",
"role": "user"
}
] | 1 |
[
{
"content": "Let $A_1 A_2 \\dotsb A_{11}$ be a regular $11$-gon inscribed in a circle of radius $2$. Let $P$ be a point such that the distance from $P$ to the center of the circle is $3$. Find the value of $PA_1^2 + PA_2^2 + \\dots + PA_{11}^2$. Express your answer as an integer.",
"role": "user"
}
] | 143 |
[
{
"content": "Evaluate the integral \\( \\int_{\\gamma_2} f(z) \\, dz \\) where \\( f(z) = \\frac{z^9}{z^{10} -1} \\) and \\( \\gamma_2 \\) is a circle of radius 2, traced in the positive direction. Determine if the following steps to evaluate the integral using residues on the Riemann sphere are correct:\n\n1.... | 2\pi 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... | $469,000 |
[
{
"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... | -500 mm |
[
{
"content": "Let \\( n \\ge 3 \\) be a natural number. Find the infimum of the expression \\( \\sum_{i=1}^n\\left( \\frac{1}{x_i} -x_i \\right) \\) subject to the constraint \\( 1 = P(x_1, x_2, \\ldots, x_n) \\), where \\( P(x_1, x_2, \\ldots, x_n) := \\sum_{i=1}^n \\frac{1}{x_i+n-1} \\) and \\( x_1, x_2, \\ld... | 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... | $29.73 |
[
{
"content": "Determine the degree of the splitting field of the polynomial \\((x^2+x+1)(x^3+x+1)\\) over \\(\\mathbb{Q}\\).",
"role": "user"
}
] | 12 |
[
{
"content": "In a Hausdorff space $T$ that is also a topological ring with identity, if a sequence $(a_n)$ converges to zero, does the sequence $(ra_n)$, where $r$ is a fixed element of $T$, also converge to zero?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the maximum area of a quadrilateral with vertices \\((0,0), (x, 0), (0, y), (x,y)\\), where the last vertex lies on the line \\(\\frac{x}{a} + \\frac{y}{b} = 1\\). Use a modified Lagrange multiplier method to ensure a solution exists.",
"role": "user"
}
] | \dfrac{ab}{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... | 431.8 /mm |
[
{
"content": "Given $n$ integers $a_1 = 1, a_2, \\ldots, a_n$ such that $a_i \\le a_{i+1} \\le 2a_i$ for $i = 1, 2, \\ldots, n - 1$ and the sum of all integers is even, determine if it is possible to partition them into two groups with equal sums.",
"role": "user"
}
] | Yes |
[
{
"content": "Solve the logarithmic inequality for $x$:\n\n$$\\log_x(x^3+1)\\cdot\\log_{x+1}(x)>2$$",
"role": "user"
}
] | (2, \infty) |
[
{
"content": "Consider a group $G$ and a subset $A$ of $G$. Suppose there is a group action $\\cdot: G \\times G \\to G$. Determine if it is possible for a non-identity element $g \\in G$ to exist such that the set $g \\cdot A = \\{g \\cdot a \\mid a \\in A\\}$ is a strict subset of $A$. Provide a justification... | 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... | 752750 m |
[
{
"content": "Suppose $A$ is a doubly stochastic matrix, meaning each row and column sum to 1 and all entries are nonnegative. If $A = BC$ where $B$ is idempotent and $C$ is unitary, does it follow that $B$ and $C$ are also doubly stochastic matrices?",
"role": "user"
}
] | Yes |
[
{
"content": "Calculate the sum of the series: \\[ \\sum_{n=0}^{\\infty}\\frac{x^{2n+2}-(-x)^{2n+2}}{(2n+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... | 2.98% |
[
{
"content": "The numbers 1, 3, 6, 10, $\\ldots$, are called triangular numbers, as shown geometrically here. What is the $20^{\\text{th}}$ triangular number?\n\n[asy]\n\ndot((0,0));\nlabel(\"1\",(0,-1.5));\n\ndot((3,0));\ndot((4,0));\ndot((3,1));\nlabel(\"3\",(3.5,-1.5));\n\ndot((7,0));\ndot((8,0));\ndot((9,0... | 210 |
[
{
"content": "Let $G$ be the group of all $2\\times 2$ matrices with integer entries, and let $H$ be the subgroup of all $2\\times 2$ matrices with even integer entries. Determine the index of $H$ in $G$. \\( \\boxed{} \\)",
"role": "user"
}
] | 16 |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to 0} \\frac{1}{2x}\\int_{-x}^{x} \\cos(t^{2})dt$$",
"role": "user"
}
] | 1 |
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