prompt list | ground_truth string |
|---|---|
[
{
"content": "Does the convergence of a sequence \\((a_n)\\) to 0 guarantee the convergence of the series \\(\\sum_{n=1}^{\\infty} \\frac{a_n}{n}\\)? Provide a justification for your answer.",
"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... | 2.41e+29 kg*m/s |
[
{
"content": "Find the limit of the sequence: \\[ \\lim_{n\\rightarrow \\infty}\\frac{\\frac{n}{1}+\\frac{n-1}{2}+\\frac{n-3}{3}+\\cdots+\\frac{2}{n-1}+\\frac{1}{n}}{\\ln(n!)} \\]",
"role": "user"
}
] | 1 |
[
{
"content": "Given the equations:\n\\[ ax + by = 7 \\]\n\\[ ax^2 + by^2 = 49 \\]\n\\[ ax^3 + by^3 = 133 \\]\n\\[ ax^4 + by^4 = 406 \\]\nfind the value of \\(2014(x+y-xy) - 100(a+b)\\).",
"role": "user"
}
] | 5956 |
[
{
"content": "Let \\( x_n \\) be the smallest natural number such that \\( 1 + \\frac{1}{2} + \\cdots + \\frac{1}{x_n} \\geq n \\). Determine \\( \\lim_{n \\to \\infty} \\frac{x_{n+1}}{x_n} \\).",
"role": "user"
}
] | e |
[
{
"content": "The quartic polynomial \\( P(x) \\) satisfies \\( P(1)=0 \\) and attains its maximum value of 3 at both \\( x=2 \\) and \\( x=3 \\). Find the value of \\( P(5) \\).",
"role": "user"
}
] | -24 |
[
{
"content": "Determine whether a countable collection of open intervals in \\(\\mathbb{R}\\) can be rearranged to satisfy the following conditions: (i) The intervals are disjoint. (ii) There exists a positive minimum distance between all intervals. (iii) The total measure of the intervals remains finite and un... | 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... | 0.427 m |
[
{
"content": "Consider a function $g: \\Bbb Z\\setminus\\{0\\}\\to\\Bbb Z_2$ that is onto and satisfies the equation $g(xy)=g(x)g(y)$ for all $x,y \\in \\Bbb{Z}\\setminus 0$. Determine whether there exists a set of primes $A$ such that $g(n)=0$ if and only if some $p\\in A$ divides $n$. 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... | 350 N |
[
{
"content": "Calculate the limit of the sequence: \\[ \\lim_{n \\to \\infty} \\frac{\\left(2\\,\\sqrt[\\Large n]{\\, n\\,}\\, - \\,\\sqrt[\\Large n]{\\, 2\\,}\\right)^{n}}{n^2} \\]",
"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... | 1760 |
[
{
"content": "Is the expectation value map Lipschitz with respect to the 1-Wasserstein metric on the space of Borel measures on the interval $[0,1]$? Specifically, can a constant $C$ be found such that for every pair of measures $p$ and $q$, the inequality $|E_p - E_q| \\le C \\cdot d_W(p, q)$ holds, where $E_p... | 1 |
[
{
"content": "Evaluate the flux of the vector field \\(\\mathbf{G}(x,y,z) = (2xy, 3z^2, -y)\\) through the surface \\(S\\) defined by \\(z = 1 - x^2 - y^2\\) for \\(z > 0\\). Assume the surface is oriented in the direction of the positive \\(z\\)-axis.",
"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... | 68 s |
[
{
"content": "For which complex numbers \\( w \\) does the equation \\( \\exp(z) = w \\) have solutions?",
"role": "user"
}
] | \mathbb{C} \setminus \{0\} |
[
{
"content": "In an abelian category \\( \\mathcal{A} \\) that contains all colimits, let \\( \\mathcal{P} \\) be a full subcategory of generating objects, which are assumed to be projective. Is it true that for any object \\( X \\in \\mathcal{A} \\), there exists an epimorphism \\( \\bigoplus_{i \\in I} P_i \\... | Yes |
[
{
"content": "In triangle \\( \\triangle ABC \\), where \\( AC > AB \\), point \\( P \\) is the intersection of the perpendicular bisector of \\( BC \\) and the internal angle bisector of \\( \\angle A \\). Draw \\( PX \\perp AB \\), intersecting the extension of \\( AB \\) at point \\( X \\). Draw \\( PY \\per... | 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... | 4.96e5 m/s |
[
{
"content": "An ellipse is defined parametrically by\n\\[(x,y) = \\left( \\frac{2 (\\sin t - 1)}{2 - \\cos t}, \\frac{3 (\\cos t - 5)}{2 - \\cos t} \\right).\\]Then the equation of the ellipse can be written in the form\n\\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,\\]where $A,$ $B,$ $C,$ $D,$ $E,$ and $F$ are integ... | 1381 |
[
{
"content": "Find the sum of the series \\( \\sum_{x=5}^{\\infty} \\frac{C}{(x+1)(x+2)} \\), where \\( C \\) is a constant.",
"role": "user"
}
] | \dfrac{C}{6} |
[
{
"content": "Calculate the limit: \\[ \\lim_{x \\to 1} \\big(4^x - 3^x\\big)^{\\frac{1}{x - 1}} \\] without using L'Hospital's rule or Taylor series.",
"role": "user"
}
] | \dfrac{256}{27} |
[
{
"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... | 0.149 |
[
{
"content": "Determine if there is a particular solution to the equation $x = \\log(x) + c$ without using numerical methods.",
"role": "user"
}
] | c \geq 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... | ions and water molecules |
[
{
"content": "Evaluate the triple integral \\(\\int_{-a}^a\\int_{-\\sqrt{a^2-x^2}}^{\\sqrt{a^2-x^2}}\\int_{-\\sqrt{a^2-x^2-y^2}}^{\\sqrt{a^2-x^2-y^2}}(xy^2z^2)\\,dz\\,dy\\,dx\\) without using integration.",
"role": "user"
}
] | 0 |
[
{
"content": "The polynomial $4x^4 - ax^3 + bx^2 - cx + 5,$ where $a,$ $b,$ and $c$ are real coefficients, has four positive real roots $r_1,$ $r_2,$ $r_3,$ $r_4,$ such that\n\\[\\frac{r_1}{2} + \\frac{r_2}{4} + \\frac{r_3}{5} + \\frac{r_4}{8} = 1.\\]Find $a.$",
"role": "user"
}
] | 19 |
[
{
"content": "Let $X$ be a smooth projective toric variety over $\\mathbb{C}$, acted upon by the compact torus $T=(S^1)^n$. The $T$-equivariant cohomology $H^*_T(X)$ is an algebra over the ring $H^*_T(pt)$. Consider the ideal $H^{>0}_T(pt)\\cdot H^*_T(X)$. Is the quotient algebra $H^*_T(X)/H^{>0}_T(pt)\\cdot H^... | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 85 km |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 3.77e-6 H |
[
{
"content": "$P(x)$ is a polynomial such that when $P(x)$ is divided by $x-13$ the remainder is $19$, and when $P(x)$ is divided by $x+17$ the remainder is $-11$. What is the remainder when $P(x)$ is divided by $(x-13)(x+17)$?",
"role": "user"
}
] | x+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... | $16,109 |
[
{
"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... | 0.27 m |
[
{
"content": "Suppose $f$ is a real function defined on $E$, which satisfies $\\lim_{h\\rightarrow 0}[f(x+h)-f(x-h)]=0$ for every $x\\in \\mathbb{R}^1$. Does this imply that $f$ is continuous?",
"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... | 194 cm Hg |
[
{
"content": "Use Green's Theorem to evaluate the line integral \\( \\int_C (y + e^\\sqrt{x})\\,dx + (2x + \\cos y^2)\\,dy \\), where \\( C \\) is the boundary of the region enclosed by the parabolas \\( y = x^2 \\) and \\( x = y^2 \\). Determine the appropriate limits of integration for the double integral.",
... | \dfrac{1}{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... | 3Mg(OH)_2(aq) + 2H_3PO_4(aq) o Mg_3(PO_4)_2(s)+6H_2O(l) |
[
{
"content": "Determine the number of path components in the set $A_n \\setminus \\{0\\}$, where $A_n = \\{z \\in \\mathbb{C} \\mid z^n \\text{ is a non-negative real number}\\}$.",
"role": "user"
}
] | 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... | 4.915 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... | 326 m |
[
{
"content": "Evaluate the definite integral \\( \\int_0^\\infty e^{-ax} x^4 \\, \\text{dx} \\) for \\( a > 0 \\) and verify that it equals \\( 24a^{-5} \\) using a method other than applying integration by parts four times.",
"role": "user"
}
] | 24a^{-5} |
[
{
"content": "Evaluate whether the expression \\( \\sum_{k=0}^n \\binom{n}{k}(-1)^k(k+1) \\) equals zero. Justify your answer.",
"role": "user"
}
] | 0 |
[
{
"content": "Given a hyperbola with foci $F_{1}$ and $F_{2}$, and vertices $M$ and $N$, consider triangle $\\triangle P F_{1} F_{2}$ where vertex $P$ lies on the hyperbola. Determine the location of the contact point of the incircle of $\\triangle P F_{1} F_{2}$ with side $F_{1} F_{2}$. Choose from the followi... | B |
[
{
"content": "Determine whether there exists a sequence of positive real numbers $a_n$ such that $a_n \\to \\infty$ as $n \\to \\infty$, yet the series $\\sum_{n=1}^{\\infty} \\frac{a_n}{2^n}$ converges.",
"role": "user"
}
] | Yes |
[
{
"content": "Given the function \\( f(x) = \\frac{1}{2}\\int_{0}^{x}(x-t)^2g(t) \\, dt \\) where \\( g(x) \\) is a function such that \\( g(1) = 6 \\), find \\( f'''(1) \\).",
"role": "user"
}
] | 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... | 8.5 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... | 363.63 N/C |
[
{
"content": "Determine the values of $p$ for which the function \\[ f(x,y) = \\begin{cases} \\frac{(x+y)^{p}}{x^2-y^2} & \\text{if $(x,y) \\neq (0,0)$} \\\\ 0 & \\text{if $(x,y)= (0,0)$} \\end{cases} \\] is differentiable at the point $(0,0)$. Provide your answer as a set of values for $p$.",
"role": "user... | p > 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... | Strength |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} x^2\\left(1+2+3+\\dots+\\left\\lfloor\\frac{1}{|x|}\\right\\rfloor\\right) \\] where \\( \\left\\lfloor x \\right\\rfloor \\) denotes the floor function of \\( x \\).",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "If there are 6 matrices in \\( M_{6,6}(\\Bbb C) \\) such that each satisfies \\( A^2 = 0 \\), does this imply that at least two of them are similar? Provide a true or false answer.",
"role": "user"
}
] | True |
[
{
"content": "Prove that \\( \\operatorname{exp}\\left(\\sum_{n=1}^\\infty (-1)^{n-1}\\frac{z^n}{n}\\right)=z+1 \\) for all \\( z\\in\\mathbb{C} \\) where \\( |z|<1 \\).",
"role": "user"
}
] | z + 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... | $56,000 |
[
{
"content": "Determine the set of real numbers \\(x\\) for which the series \\(\\sum_{k=0}^{\\infty} \\frac{a_k b_k x^k}{k!}\\) converges, given that \\((b_n)\\) is a sequence converging to \\(b \\in \\mathbb{R}\\).",
"role": "user"
}
] | \mathbb{R} |
[
{
"content": "Find the volume of the region in \\( \\mathbb{R}^3 \\) described by the inequalities \\( x^2 + y^2 + z^2 \\le 1 \\) and \\( |y| \\le x^2 + y^2 \\).",
"role": "user"
}
] | \dfrac{16}{9} |
[
{
"content": "Let \\( f : \\mathbb{R} \\to \\mathbb{R} \\) be defined by \\( f(x) = x^r \\sin(1/x) \\) for \\( x \\neq 0 \\), and \\( f(0) = 0 \\), where \\( r > 0 \\) is a rational number. Determine the values of \\( r \\) for which \\( f'(0) \\) exists.",
"role": "user"
}
] | r > 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... | 0.0024 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... | 5.02 m |
[
{
"content": "Consider a finite group $G$ and the complex group algebra $\\mathbb{C}(G)$, consisting of formal sums \\( \\sum_{g \\in G} a_g g \\) where \\( a_g \\in \\mathbb{C} \\). The algebra structure is defined by:\n\\[ \\sum_g a_g g + \\sum_g b_g g = \\sum_g (a_g + b_g) g, \\]\n\\[ \\left( \\sum_g a_g g \... | 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... | 296 |
[
{
"content": "Determine the number of zeros of the function \\( F(x) = \\int_0^x e^t(t^2 - 3t - 5)dt \\) within the interval \\((0, 4)\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Calculate the covariance of the random variables $X$ and $Y$ given their joint density function: $$ f_{X,Y}(x,y) = \\begin{cases} 2-x-y,& 0<x,y<1\\\\ 0,&\\text{otherwise}. \\end{cases} $$",
"role": "user"
}
] | -\dfrac{1}{144} |
[
{
"content": "Evaluate the integral: $$\\int^1_0 \\left (\\sum^\\infty_{n=0}\\frac{\\left\\lfloor 2^nx\\right\\rfloor}{3^n} \\right )^2{\\rm d}x$$",
"role": "user"
}
] | \dfrac{27}{32} |
[
{
"content": "Determine whether it is possible to define a topology on a countable set that is homeomorphic to $\\mathbb{Q}^2$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Find the tangent direction at the point \\((1,1,1)\\) where the ellipsoid \\(2x^2 + y^2 + z^2 = 4\\) intersects the plane \\(x - y + 2z = 2\\).",
"role": "user"
}
] | \langle 1, -1, -1 \rangle |
[
{
"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... | 46890 |
[
{
"content": "Consider a set of closed segments on a real line, where each pair of segments intersects. Determine whether there exists a point that is contained in all of these segments.",
"role": "user"
}
] | Yes |
[
{
"content": "Calculate the limit: $$\\lim_{x\\to 0^{-}}(1-2x)^{-\\frac{1}{4x^{2}}}$$",
"role": "user"
}
] | 0 |
[
{
"content": "For \\( n \\in \\mathbb{N} \\), consider the functions \\( g_n(x) = x^{1/n} \\) and \\( f_n(a) = \\int_0^1 (1 - x^a)^n \\, dx \\). If the series \\( \\sum_{a=2}^\\infty \\left( \\lim_{n \\to \\infty} \\frac{g_n(f_n(a))}{a!} \\right) \\) converges to a constant \\( \\alpha \\), find the value of \\... | e - 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... | 1 > 2 > 4 > 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... | 13 m/s |
[
{
"content": "Using generating functions, find the number of distinct ways to roll five dice, each with three sides labeled $1$ and three sides labeled $2$, such that the sum of the numbers rolled is $9$. The generating function for a single die is $3x+3x^2$. Determine the coefficient of $x^9$ in the expansion ... | 1215 |
[
{
"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 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... | 19.3% |
[
{
"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... | 4.40 atm |
[
{
"content": "Determine the highest power of 50 that divides the expression \\(49^{{50}^{51}} + 51^{{50}^{49}} + 50^{{49}^{51}}\\).",
"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... | 13.59% |
[
{
"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... | 193.24 g |
[
{
"content": "Find all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) that satisfy the functional equation \\( f(x + y) = x + f(y) \\) for all \\( x, y \\in \\mathbb{R} \\). What is the general form of such functions?",
"role": "user"
}
] | f(x) = x + c |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.353 Am^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... | 94.188 g/mol |
[
{
"content": "Let \\((1+\\sqrt{2})^{2012} = a + b\\sqrt{2}\\), where \\(a\\) and \\(b\\) are integers. Determine the greatest common divisor of \\(b\\) and \\(81\\). Choose the correct answer from the following options: \\(\\text{(A) }1\\), \\(\\text{(B) }3\\), \\(\\text{(C) }9\\), \\(\\text{(D) }27\\), \\(\\te... | 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... | 0.10 cm/s^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... | 15384.6156003L |
[
{
"content": "Compute the integral \\( \\int_{|z|=1} \\text{Re}(z)e^{z^2} \\, dz \\) using Cauchy's theorem by finding a holomorphic function \\( f \\) such that \\( f(z) = \\text{Re}(z) \\) on the unit circle \\( |z|=1 \\).",
"role": "user"
}
] | \pi i |
[
{
"content": "Evaluate the integral \\( \\iint_D xy\\ dxdy \\), where \\( D \\) is the region bounded by the curves \\( y=x^2 \\), \\( y=2x^2 \\), \\( x=y^2 \\), and \\( x=3y^2 \\).",
"role": "user"
}
] | \dfrac{1}{18} |
[
{
"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 years |
[
{
"content": "Consider the matrix \\( A \\) given by:\n\\[ \\begin{bmatrix} a & b & 1 \\\\ c & d & 1 \\\\ 1 & -1 & 0 \\\\ \\end{bmatrix} \\]\nwhere \\( a, b, c, d \\) are distinct real numbers such that \\( a + b = c + d \\). Determine an eigenvalue of the matrix from the following options:\n\n(1) \\( a + c \\)... | B |
[
{
"content": "An object weighing 16 pounds is placed beneath the surface of a lake. The buoyancy of the object is 30 pounds, causing it to rise. The resistance of the water in pounds is numerically equal to the square of the velocity in feet per second. If the object surfaces in 5 seconds, what is the velocity ... | \sqrt{14} |
[
{
"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... | 9840.44 |
[
{
"content": "Evaluate the complex integral: $$\\displaystyle\\oint_{|z|=1}\\sin\\left (\\displaystyle\\frac{1}{z}\\right ) dz$$",
"role": "user"
}
] | 2\pi i |
[
{
"content": "In a regular tetrahedron \\( PQRS \\), each vertex and each edge has a number written on it, totaling 10 numbers. These numbers are \\( 1, 2, 3, 4, 5, 6, 7, 8, 9, \\) and \\( 11 \\), each used exactly once. The number on each edge is the sum of the numbers on its connecting vertices. If the number... | 5 |
[
{
"content": "Determine whether the inequality \\( \\sum_{n\\leq [x]}e^{-\\sqrt{\\frac{\\log x}{r}}}\\ll \\frac{x}{e^{c \\sqrt{\\log x}}} \\) holds for some constant \\( c>0 \\), where \\( n \\) is a square-free positive integer and \\( x>1 \\).",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 673N |
[
{
"content": "Determine the injective dimension of a module $M$ given that $\\text{id}(M) \\leq i$ if and only if $\\text{Ext}^{i+1}(N,M)=0$ for all modules $N$. What is the injective dimension of $M$?",
"role": "user"
}
] | i |
[
{
"content": "The quadratic polynomial \\( f(x) = ax^2 + bx + c \\) has exactly one root. Additionally, the quadratic polynomial \\( 2f(2x - 3) - f(3x + 1) \\) also has exactly one root. Determine the root of the polynomial \\( f(x) \\).",
"role": "user"
}
] | -11 |
[
{
"content": "Determine if the Dominated Convergence Theorem can be applied to evaluate the limit \\(\\lim_{n\\to \\infty}\\int_0^1 \\frac{\\sin(nx)}{1+x^2}\\ dx\\), where \\(n\\) is a positive integer.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the line integral \\(\\int_C x^2 \\, dx + (x+y) \\, dy\\) where \\(C\\) is the path of the right triangle with vertices \\((0,0), (4,0), (0,10)\\), starting from the origin, going to \\((4,0)\\), then to \\((0,10)\\), and finally back to the origin.",
"role": "user"
}
] | 20 |
[
{
"content": "Assume \\(f \\in C^{\\infty}(\\mathbb{R}^2)\\) satisfies \\(f(\\sin 2t, \\sin t)=0\\) for all \\(t\\in \\mathbb{R}\\). Evaluate the limit \\[ \\lim_{h \\to 0}\\frac{f(2h,h)}{h}. \\]",
"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... | At the time of devaluation, there will be an increase in demand for cheaper export-oriented products, leading to increased output. Consumers shifting from foreign to domestic products will also increase domestic output. However, it will be challenging to increase domestic production immediately. Due to excess demand, i... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.