prompt list | ground_truth string |
|---|---|
[
{
"content": "Evaluate the double integral \\( \\int_{-1}^1\\int_{-1}^1 h(x,y) \\, dx \\, dy \\) where the function \\( h(x,y) \\) is defined as follows: \\[ h(x,y) = \\begin{cases} x^2 + y^2 & \\text{if } x = y \\\\ 1 & \\text{if } x \\neq y \\end{cases} \\]",
"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... | 2.8m/s |
[
{
"content": "A right circular cone contains two spheres. The radius of the larger sphere is twice the radius of the smaller sphere. Each sphere is tangent to the other sphere and to the lateral surface of the cone. The larger sphere is also tangent to the cone's circular base. Calculate the fraction of the con... | \dfrac{7}{16} |
[
{
"content": "Points \\( A, B, C, D \\) lie on a circle with diameter 1. Point \\( X \\) is on the diameter \\( AD \\), and it is given that \\( BX = CX \\) and \\( 3 \\angle BAC = \\angle BXC = 36^\\circ \\). Determine the length of \\( AX \\).\n\n(A) \\(\\cos 6^\\circ \\cos 12^\\circ \\sec 18^\\circ \\).\n\n(... | B |
[
{
"content": "Consider two stochastic matrices A and B with different dominant eigenvectors. Is it possible for the largest eigenvalues of both A and B to be equal to 1?",
"role": "user"
}
] | Yes |
[
{
"content": "Let $p$ be a positive integer. Determine whether the following statement is true: If $2p+1$ divides $\\left(\\binom{2p}{p}+(-1)^{p-1}\\right)$, then $2p+1$ is a prime number.",
"role": "user"
}
] | True |
[
{
"content": "Determine all functions \\( f: \\mathbb{Q}_+ \\to \\mathbb{Q}_+ \\) that satisfy the following conditions: \\( f(x+1) = f(x) + 1 \\) for every \\( x \\in \\mathbb{Q}_+ \\) and \\( f(x^2) = f(x)^2 \\) for every \\( x \\in \\mathbb{Q}_+ \\).",
"role": "user"
}
] | f(x) = x |
[
{
"content": "Let \\( z_1 \\) and \\( z_2 \\) be two complex numbers that satisfy the following conditions:\n\\[ \\frac{z_2}{\\bar{z_1}} = \\frac{3}{8} \\left(\\cos(75^{\\circ}) + i\\sin(75^{\\circ})\\right) \\]\n\\[ z_1 z_2^2 = \\frac{1}{3} \\left(\\cos(120^{\\circ}) + i\\sin(120^{\\circ})\\right) \\]\nDetermi... | C |
[
{
"content": "Consider a Markov chain with parameters $\\alpha, \\beta \\in (0,1)$ where the variables $X_0, X_1, \\dots$ are confined to the interval $[0,1]$. At each step, $X_n$ transforms into:\n\n1. $\\frac{\\alpha X_n}{\\alpha X_n + \\beta (1-X_n)}$ with probability $\\alpha X_n + \\beta (1-X_n)$\n2. $\\fr... | X_1 |
[
{
"content": "Given the probability mass function $f_X(x)=0.1\\delta(x)+0.9u(x)e^{-x}$, find the probability $P(X=0)$. Express your answer as a number.",
"role": "user"
}
] | 0.1 |
[
{
"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... | 5.1205e5 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... | 0.1406 |
[
{
"content": "Determine the number of integer solutions to the equation \\(a + b + c + d = 30\\) where \\(0 \\leq a, b, c, d \\leq 9\\).",
"role": "user"
}
] | 84 |
[
{
"content": "Determine the conditions on the parameter $p$ such that the curve $y = x^p\\cos\\left(\\frac{\\pi}{x}\\right)$, for $0 < x \\leqslant 1$, has a finite length.",
"role": "user"
}
] | p > 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... | -0.3 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... | 100,000 W |
[
{
"content": "Determine whether the following statement is true or false: If a point is a source for a one-dimensional smooth map $f(x)$, meaning $x$ satisfies $|f'(x)| > 1$, then this point is sensitive to initial conditions. Specifically, for every $\\epsilon > 0$, there exists an $\\bar{x}$ and a $d > 0$ suc... | True |
[
{
"content": "Does there exist a poset whose quasiwidth strictly exceeds its width?",
"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... | 4167 nm |
[
{
"content": "Determine the number of positive integer solutions \\((a, b, c)\\) that satisfy the equation:\n\\[ \\frac{a^2+b^2-c^2}{ab}+\\frac{c^2+b^2-a^2}{cb}+\\frac{a^2+c^2-b^2}{ac}=2+\\frac{15}{abc} \\]",
"role": "user"
}
] | 9 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.11 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... | $189,889.88 |
[
{
"content": "Corn costs 99 cents per pound, and beans cost 45 cents per pound. If Shea buys 24 total pounds of corn and beans, and it costs $\\$18.09$, how many pounds of corn did Shea buy? Express your answer as a decimal to the nearest tenth.",
"role": "user"
}
] | 13.5\text{ pounds} |
[
{
"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... | $2,051.25 F |
[
{
"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... | 1.92 m/s |
[
{
"content": "Find all functions \\( f(x) \\) that satisfy the differential equation:\n\\[(2f(x)xy)'_x - (f(x)(2y^2+3x))'_y = 0.\\]",
"role": "user"
}
] | f(x) = Cx |
[
{
"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... | 1.03e-26 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... | 10 years |
[
{
"content": "Determine whether the following statement is true or false: If $g \\in L_1(X)$, then for every $\\epsilon>0$, there exists a set $E$ with $m(E)<\\infty$, $\\sup_E(|g|)<\\infty$, and $\\int_{E^c} |g| < \\epsilon$. Provide a justification for your answer.",
"role": "user"
}
] | True |
[
{
"content": "Determine the positive value of $x$ that minimizes the integral \\( \\int_0^{\\sqrt{x}} \\cos \\left(\\frac{\\pi t}{t+1}\\right) dt \\).",
"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... | -14.51% |
[
{
"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... | 52 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... | 1.26 * 10^16 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... | 66,500 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty}{\\frac{\\sum_{k=1}^n k^7}{n^8}} \\]",
"role": "user"
}
] | \dfrac{1}{8} |
[
{
"content": "Compute the value of the series \\( \\sum_{n=1}^{\\infty}\\frac{(-1)^n (n+1)}{n!} \\).",
"role": "user"
}
] | -1 |
[
{
"content": "Find the coefficient of \\(t^{17}\\) in the expansion of the generating function \\((t^2 + t^3 + t^4 + t^5)(t^3 + t^4 + t^5 + t^6)(t^4 + t^5 + t^6 + t^7)\\). This coefficient represents the number of integer solutions to the equation \\(x_1 + x_2 + x_3 = 17\\) where \\(2 \\leq x_1 \\leq 5\\), \\(3... | 3 |
[
{
"content": "Determine the remainder when the polynomial $P(x)$ is divided by $(x-3)$, given that $P(x^{3})=ax^{6}+(b-2)x^{5}+(a+3)x^{4}+bx^{3}+3$ for some constants $a$ and $b$ in $\\mathbb{R}$. The remainder is known to be $-18$. Find the value of $P(3)$. \\( \\text{(Hint: Use the Remainder Theorem)} \\)",
... | -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... | 38g |
[
{
"content": "Three friends each tell the truth with a probability of \\( \\frac{2}{3} \\) and lie with a probability of \\( \\frac{1}{3} \\). You ask them independently if it is raining in your city, and they all say yes. Assuming no prior probability of rain, what is the probability that it is actually rainin... | \dfrac{8}{9} |
[
{
"content": "Suppose $G$ is an affine algebraic group acting on $\\mathbb{C}\\mathbb{P}^2$. Is the action of $G$ on $\\mathbb{C}\\mathbb{P}^2$ equivalent to the action of a closed subgroup of $\\mathbb{P}\\mathbb{G}\\mathbb{L}(3,\\mathbb{C})$ on $\\mathbb{C}\\mathbb{P}^2$?",
"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... | $3,423 |
[
{
"content": "In a triangle, the angles satisfy the conditions $\\cos A + \\cos B + \\cos C = 0$ and $\\sin A + \\sin B + \\sin C = 0$. Determine the value of $\\sin^3A + \\sin^3B + \\sin^3C$. Express your answer as a single number.",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the radius of convergence of the power series \\(\\sum_{n=0}^{\\infty} a_n z^n\\), where \\(a_n = \\frac{f^{(n)}(0)}{n!}\\) and \\(f(z) = \\frac{\\sin z}{(z-1-i)^2}\\).",
"role": "user"
}
] | \sqrt{2} |
[
{
"content": "Kevin needs to compute the product \\(9 \\times 99 \\times 999 \\times \\cdots \\times 999999999\\). Kevin takes exactly \\(ab\\) seconds to multiply an \\(a\\)-digit integer by a \\(b\\)-digit integer. What is the minimum number of seconds necessary for Kevin to evaluate the expression by perform... | 870 |
[
{
"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... | 0.99 Btu |
[
{
"content": "Find the rational number $x \\in \\mathbb{Q}$ such that $1 + \\frac{2}{\\lfloor x \\rfloor} + \\frac{3}{x - \\lfloor x \\rfloor} = 4x$. Determine the value of $x$.",
"role": "user"
}
] | \dfrac{7}{4} |
[
{
"content": "Consider a uniformly continuous function $f: X \\rightarrow Y$ between metric spaces $X$ and $Y$. For a given $\\varepsilon > 0$, define $\\delta(\\varepsilon)$ as the supremum of all $\\delta > 0$ such that $d_X(x,y) < \\delta$ implies $d_Y(f(x),f(y)) < \\varepsilon$. Let $g: \\mathbb{R}^+ \\righ... | Yes |
[
{
"content": "What is the minimum number of points required to color an entire 3D space black using a device that colors all points at an irrational distance from each given point, assuming the initial color of the space is white?",
"role": "user"
}
] | 4 |
[
{
"content": "Determine the cardinality of the set of all functions from the set of natural numbers \\( \\mathbb{N} \\) to the set of real numbers \\( \\mathbb{R} \\).",
"role": "user"
}
] | 2^{\aleph_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... | 2.24e-3 m^3 |
[
{
"content": "Let \\(X_t\\) be a Brownian motion and \\(a < 0\\). Define the stopping time \\(T_x = \\inf\\{t \\geq 0: X_t + at \\geq x\\}\\). If \\(Y = \\sup_{t \\geq 0}(X_t + at) = x\\), determine whether \\(T_x < \\infty\\).",
"role": "user"
}
] | T_x < \infty |
[
{
"content": "Given the system of equations for positive numbers \\(x, y, z\\):\n\n\\[ \n\\left\\{\n\\begin{array}{l}\nx^{2} + x y + y^{2} = 27 \\\\\ny^{2} + y z + z^{2} = 25 \\\\\nz^{2} + x z + x^{2} = 52\n\\end{array}\n\\right.\n\\]\n\nFind the value of the expression \\(x y + y z + x z\\).",
"role": "use... | 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... | 35.6 L |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to0}\\int_{0}^{x}\\frac{\\cos(t^3)}{t+x}dt.$$",
"role": "user"
}
] | \ln 2 |
[
{
"content": "Calculate the integral \\( S = \\int_0^{5p} |\\sin x + \\cos x| \\, dx \\), where \\( p \\) is the period of the function \\( f(x) = |\\sin x + \\cos x| \\). Find the value of \\( S \\).",
"role": "user"
}
] | 10\sqrt{2} |
[
{
"content": "Consider i.i.d. random variables $X_1, X_2, \\ldots, X_n$ distributed according to a Weibull distribution with shape parameter $0 < \\epsilon < 1$, such that $\\mathbf{Pr}[X_i \\geq t] = e^{-\\Theta(t^{\\epsilon})}$. Define the random variable $S_n = X_1 + X_2 + \\ldots + X_n$. As $n$ tends to inf... | \epsilon |
[
{
"content": "Determine whether the following statement is true or false: If $X$ and $Y$ are independent random variables, then $\\mathbb{E}(f(X)\\mid X\\le Y)=\\mathbb{E}(f(X))$. Provide a justification for your answer.",
"role": "user"
}
] | False |
[
{
"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... | 10983.69 |
[
{
"content": "Evaluate the integral \\( \\int_{\\mathbb{T}^2} xyz \\, dw\\wedge dy \\), where \\( \\mathbb{T}^2\\subset\\mathbb{R}^4 \\) is the 2-torus defined by the equations \\( w^2+x^2=y^2+z^2=1 \\).",
"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... | 13.88 m |
[
{
"content": "Calculate \\(\\lim_{n\\to\\infty} \\frac{\\sum_{i=1}^n ia_i}{n^2+n}\\) given that \\(a_n \\to a\\).",
"role": "user"
}
] | \dfrac{a}{2} |
[
{
"content": "Let $X$ and $Y$ be manifolds of dimensions $k$ and $l$ respectively, with $k > l$. Given a smooth map $F : X \\to Y$ and a regular value $y \\in Y$, determine if there exists a smooth map $G : X \\to \\mathbb{R}^l$ such that $F^{-1}(y) = G^{-1}(0)$, with $G$ defined on the entire manifold $X$. Pro... | 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... | 1.447 |
[
{
"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... | 5085 |
[
{
"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... | 3.41 |
[
{
"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... | xc = m/pc if pc < 2pw; xc = 0 if pc > 2pw; xc = m/2pc if pc = 2pw |
[
{
"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... | 46.8 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... | -1.44e-2 V |
[
{
"content": "Given that the radius of convergence of the power series \\(\\sum_{n=0}^{\\infty} a_n x^n\\) is 6, determine the radius of convergence of the series \\(\\sum_{n=0}^{\\infty} a_n x^{\\frac{n}{2}}\\).",
"role": "user"
}
] | 36 |
[
{
"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... | 0.62 nC |
[
{
"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... | 5.04 kJ |
[
{
"content": "Find the smallest integer $n$ such that every 2-coloring of the edges of the complete graph $K_n$ contains a monochromatic cycle of length 4, $C_4$.",
"role": "user"
}
] | 6 |
[
{
"content": "Determine whether the following statement is true or false: If a function $f(x)$ is continuous on the interval $[a, \\infty)$ and the limit $\\lim_{{x\\to\\infty}} \\int_{x}^{2x} f(t)dt$ equals 0, then the improper integral $\\int_{a}^{\\infty} f(t)dt$ converges. Use Cauchy's test for integral con... | False |
[
{
"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 m/s |
[
{
"content": "Calculate the limit: \\[ \\lim_{x\\to\\infty}\\frac{\\gamma(x,-c\\sqrt{x}+x)}{\\Gamma(x,c\\sqrt{x}+x)} \\] where \\( \\gamma(a,b) \\) and \\( \\Gamma(a,b) \\) are the lower and upper incomplete gamma functions, respectively.",
"role": "user"
}
] | 1 |
[
{
"content": "Find the radius of the circle with equation $x^2 - 4x + y^2 - 6y - 36 = 0$.",
"role": "user"
}
] | 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... | $8,454.32 |
[
{
"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... | 10.04 |
[
{
"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... | $109,475.64 |
[
{
"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... | 6927.415 kW |
[
{
"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... | 3.1e-12 J |
[
{
"content": "Determine the number of group homomorphisms \\( \\phi: S_5 \\to C_5 \\), where \\( S_5 \\) is the symmetric group on 5 elements and \\( C_5 \\) is the cyclic group of order 5.",
"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... | 0.0281 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... | w_(rev) = -nRTln(V_2/V_1) |
[
{
"content": "In a first countable topological space $X$, consider a sequence $(x_n)$ and let $y$ be an accumulation point of the set $I = \\{x_n : n \\geq 1\\}$. Is it true that $y$ is also a limit point of the sequence $(x_n)$, meaning every neighborhood of $y$ contains infinitely many points from $I$? Provid... | Yes |
[
{
"content": "Differentiate the expression \\( \\frac{\\partial(x^T Bx)}{\\partial x} \\) where \\( x \\) is a column vector and \\( B \\) is a square matrix.",
"role": "user"
}
] | (B + B^T)x |
[
{
"content": "Find the limit: \\[ \\lim_{n \\to \\infty} \\sum_{k=n}^{\\infty} e^{-k^2} \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Given a series \\( S(x) = \\sum_{n=1}^\\infty s_n(x) \\) where each real-valued term \\( s_n(x) \\) converges to \\( s_n \\) as \\( x \\) approaches infinity, and the series \\( S=\\sum_{n=1}^\\infty s_n \\) is convergent, does it imply that \\( S(x) \\) converges to \\( S \\) as \\( x \\) approac... | No |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{2}\\left(x^{2}+x-1\\right) e^{x / 2} \\, d x \\).",
"role": "user"
}
] | 6e - 10 |
[
{
"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... | 11.00 |
[
{
"content": "Find the general formula for the number of subwords of the form \\((ba)^m\\) in a word of the form \\((ba)^n\\).",
"role": "user"
}
] | n - m + 1 |
[
{
"content": "Given three composite numbers \\( A, B, C \\) that are pairwise coprime and satisfy \\( A \\times B \\times C = 11011 \\times 28 \\), find the maximum value of \\( A + B + C \\).",
"role": "user"
}
] | 1626 |
[
{
"content": "Estimate the integral of the function \\( f(z) \\) over a circular contour of radius 3 using the ML inequality, given that the function has \\( |z^2| - |i| \\) in the denominator.",
"role": "user"
}
] | \dfrac{3\pi}{4} |
[
{
"content": "Consider a decreasing sequence of compact and connected sets \\(S_1 \\supset S_2 \\supset S_3 \\supset \\dots\\) in a metric space. Is the intersection \\(S = \\bigcap_n S_n\\) also connected?",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\rightarrow \\infty }\\sum_{1\\leqslant k\\leqslant n}\\frac{1}{k\\ln (n+k)} \\]",
"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... | 14500 kg/m^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... | 12.88% |
[
{
"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... | 6.3e-5 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... | $670 |
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