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... | 0.65 kg |
[
{
"content": "A non-square rectangle has integer dimensions. The number of square units in its area is numerically equal to the number of units in its perimeter. What is the number of units in the perimeter of this rectangle?",
"role": "user"
}
] | 18 |
[
{
"content": "Calculate the product of the trigonometric expression: \\( \\cot 1^\\circ \\cdot \\cot 2^\\circ \\cdot \\cot 3^\\circ \\cdots \\cot 89^\\circ. \\)",
"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... | 161572 m/s^2 |
[
{
"content": "Evaluate the limit: $$\\lim \\limits_{x \\to +\\infty}\\sin(x)\\left(\\ln\\left(\\sqrt{x}+1\\right)-\\ln\\left(\\sqrt{x+1}\\right)\\right)$$",
"role": "user"
}
] | 0 |
[
{
"content": "Calculate \\( \\lim_{n\\rightarrow \\infty} \\frac{1}{n}\\sum_{k=1}^{n}k\\sin\\left(\\frac{a}{k}\\right) \\).",
"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... | 15.76MW |
[
{
"content": "Evaluate the limit: $$ \\lim_{x \\rightarrow \\infty} \\frac{x^{\\ln(x)}}{(\\ln(x))^x} $$",
"role": "user"
}
] | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $67.30 |
[
{
"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... | 12.6 m/s |
[
{
"content": "Let $X_1, X_2, \\ldots, X_n$ be independent and identically distributed random variables with probability density function $f(x) = e^{-x}$ for $x > 0$. Define $X_n = \\max(X_1, X_2, \\ldots, X_n)$. Find \\( \\lim_{n \\to \\infty} P(X_n < 2 + \\log_e n) \\).",
"role": "user"
}
] | e^{-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... | Ba^2+, Ca^2+, and Sr^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... | 50 s |
[
{
"content": "A convex polygon has both an inscribed circle and a circumscribed circle, and the centers of these two circles coincide. Does this imply that the polygon is regular?",
"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... | 0.655 moles |
[
{
"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... | 8.3287% |
[
{
"content": "Arrange the numbers $0, 1, 2, 3, 4, 5, 6, 7$ on the vertices of a cube such that the sum of the numbers at the ends of each edge is a prime number. What is the maximum sum of the four numbers on one face of the cube?",
"role": "user"
}
] | 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... | $149,659 |
[
{
"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... | 14.1 |
[
{
"content": "Determine the positive integer $n$ such that the product of the real number solutions of the equation $x^{\\log_2(x^3)-n} = 13$ is equal to 32.",
"role": "user"
}
] | 15 |
[
{
"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... | 66.48 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... | 486.19m |
[
{
"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... | 1,931.4 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... | $4,800 |
[
{
"content": "Determine the order type of the set \\( \\omega \\times n \\) with the lexicographic order, considering the first position first.",
"role": "user"
}
] | \omega |
[
{
"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... | 13.8 |
[
{
"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... | 13 mL |
[
{
"content": "Determine the limit of the sequence \\( \\{a_n\\} \\) defined by \\( a_{n+1} = \\frac{a_n}{1 + \\frac{1}{n} + a_n + a_n^3} \\) for all positive integers \\( n \\), given that \\( a_1 > 0 \\).",
"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... | 11.44 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... | A contribution margin income statement is similar to a traditional income statement and produces the same operating and net profit.
Sales ($9.51 per unit, 81,000 units) = $770,310 |
[
{
"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... | $6,293.70, reject |
[
{
"content": "Find the largest natural number \\( n \\ge k \\) such that any selection of \\( k \\) numbers from a set of \\( n \\) natural numbers results in a composite sum.",
"role": "user"
}
] | 2k - 1 |
[
{
"content": "Find the length of the normal chord of the parabola \\(y^2 = 4x\\) that subtends a right angle at the vertex.",
"role": "user"
}
] | 6\sqrt{3} |
[
{
"content": "In triangle $ABC$ with an obtuse angle at $C$, a point $M$ is marked on side $BC$. A point $D$ is chosen such that triangle $BCD$ is acute, with points $A$ and $D$ on opposite sides of line $BC$. Circles $\\omega_{B}$ and $\\omega_{C}$ are circumscribed around triangles $BMD$ and $CMD$, respective... | 1 |
[
{
"content": "Let \\( f \\) be the real-valued function defined on the interval \\((-1,1)\\) such that \\( e^{-x}f(x)=2+ \\int_0^x\\sqrt{t^4+1}dt \\) for all \\( x\\in(-1,1) \\). Let \\( f^{-1} \\) be the inverse function of \\( f \\). Find the value of \\((f^{-1})'(2)\\).",
"role": "user"
}
] | \dfrac{1}{3} |
[
{
"content": "Evaluate the limit using Taylor series:\n\\[ \\lim_{x \\to 0} \\frac{(\\ln(1+x) - x + \\frac{x^2}{2})^4}{(\\cos(x) - 1 + \\frac{x^2}{2})^3} \\]",
"role": "user"
}
] | \dfrac{512}{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... | 685.82 K |
[
{
"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... | 10.08% |
[
{
"content": "Evaluate the limit: $$\\lim\\limits_{x\\to 0}\\int\\limits_0^1 \\frac{1}{x}\\left(e^{-x^2(1+t)}(1+t)^{-1}-(1+t)^{-1}\\right)dt.$$",
"role": "user"
}
] | 0 |
[
{
"content": "Compute \\( a(x)b(x) + c(x) \\) in \\( \\mathrm{GF}(2^4) \\) using the irreducible generator polynomial \\( x^4 + x + 1 \\). Given:\n\n\\[ a(x) = x^3 + x^2 + 1, \\]\n\\[ b(x) = x^2 + x + 1, \\]\n\\[ c(x) = x^3 + x^2 + x + 1 \\]\n\nFind the result of \\( a(x)b(x) + c(x) \\) modulo \\( x^4 + x + 1 \... | x^3 + x |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1156 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... | -2386 J |
[
{
"content": "Determine whether the inequality \\(|\\int_a^b f'(x) \\, dx| \\leq TV(f, [a, b])\\) holds for a function \\(f\\) of bounded variation on the interval \\([a, b]\\), where \\(TV(f, [a, b])\\) denotes the total variation of \\(f\\) on \\([a, b]\\).",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.53e-8 g |
[
{
"content": "In a $2n \\times 2n$ table, each cell is colored with one of $2n^2$ colors, with each color used exactly twice. Jana starts in one cell, and a chocolate bar is in another cell. Jana can move by either walking to an adjacent cell or teleporting to the other cell of the same color. She must alternat... | 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... | 5.22 |
[
{
"content": "Given positive numbers $a<1$ and $\text{finite } C>0$, consider a real nonnegative, bounded, and nonincreasing function $f(i)$ that satisfies $f(i) \\leq Ca^i$ for all positive integers $i$. Determine whether there exist finite positive numbers $C_0$ and $b$ such that $f(i) \\leq C_0i^{-b}$ for an... | 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... | 119.04 J |
[
{
"content": "Calculate the line integral \\( \\int_C \\mathbf{F} \\cdot d\\mathbf{r} \\) where \\( \\mathbf{F} = (e^y + ye^x, xe^y + e^x) \\) and \\( C \\) is the path from \\((a,0)\\) to \\((-a,0)\\) over the upper half of the ellipse \\( b^2x^2 + a^2y^2 = a^2b^2 \\). The answer is known to be \\(-2a\\).",
... | -2a |
[
{
"content": "Can a finitely generated subgroup of infinite order be constructed within a group of homeomorphisms that fix the boundary pointwise of a closed unit disc?",
"role": "user"
}
] | Yes |
[
{
"content": "Let $f$ be a function defined on $[0,1]$. If for any $c>0$, $f$ is Riemann integrable on $[c,1]$, and if $\\lim_{c\\to0^+}\\int_c^1|f(x)|dx=0$, does it follow that $f$ is Riemann integrable on $[0,1]$? Answer yes or no.",
"role": "user"
}
] | No |
[
{
"content": "Evaluate the limit: \\[ y = \\lim_{n \\to \\infty} \\left( \\sin \\frac{\\pi}{2n} \\times \\sin \\frac{2\\pi}{2n} \\times \\cdots \\times \\sin \\frac{(n-1)\\pi}{2n} \\right) \\] Find the value of \\( y \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the number of independent components of an antisymmetric rank five tensor on \\(\\mathbb{R}^5\\).",
"role": "user"
}
] | 1 |
[
{
"content": "In an additive category, let $A$ and $C$ be two objects. If $A \\oplus C \\cong A$, is it true that $C = 0$?",
"role": "user"
}
] | No |
[
{
"content": "Evaluate the limit of the sequence as \\( n \\) approaches infinity: \n\\[ \\lim_{n\\to\\infty}\\frac{1+\\sqrt[n]{2}+\\sqrt[n]{3}+\\cdots+\\sqrt[n]{n}}{n} \\]",
"role": "user"
}
] | 1 |
[
{
"content": "Calculate \\(73^{29} \\mod 2047\\) without directly multiplying all the intermediate results together.",
"role": "user"
}
] | 1894 |
[
{
"content": "Determine the radius of convergence of the series:\n\\[ 1 + \\frac{ab}{1c}z + \\frac{a(a+1)b(b+1)}{1 \\cdot 2 \\cdot c(c+1)}z^2 + \\cdots \\]",
"role": "user"
}
] | 1 |
[
{
"content": "True or False: The span of the functions \\(\\{1, x, x^2, \\dots\\}\\) contains all trigonometric functions.",
"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... | 426 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... | 5.9 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... | 37.2916 months |
[
{
"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... | 1.07e11 Nm |
[
{
"content": "Evaluate the limit: \\[ L = \\lim_{n\\to\\infty} \\left(n-\\sum_{k=1}^n e^{\\frac{k}{n^2}}\\right) \\]",
"role": "user"
}
] | -\dfrac{1}{2} |
[
{
"content": "Find the number of integer values of $y$ such that the expression $\\frac{9^n + 35n - 4}{y}$ is an integer for every positive integer $n$. Provide your answer as a single integer.",
"role": "user"
}
] | 2 |
[
{
"content": "Given that $m$ divides $n$ ($m \\mid n$), determine if there exists a subgroup of the dihedral group $D_n$ that is isomorphic to the dihedral group $D_m$.",
"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... | 4.135 nm |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 671.25 |
[
{
"content": "Find the spectral radius of the compact operator $T:C[0,1]\\to C[0,1]$ defined by $Tf(x)= \\int_{0}^{1-x} f(y)\\,dy$. Use the formula $r(T)=\\lim_{n \\to \\infty} ||T^n||^{\\frac{1}{n}}$ to determine the spectral radius.",
"role": "user"
}
] | \dfrac{2}{\pi} |
[
{
"content": "Let $G$ be a finite group and $f: G \\rightarrow \\mathbb{Z}_n$ be a group epimorphism. If $n$ has more than one prime factor, determine whether $G$ must have a normal subgroup of order $n$. Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"content": "Let $S$ be a graded ring generated by finitely many elements of $S_1$ as an $S_0$-algebra, and let $M$ be a graded $S$-module. For an element $m \\in M$, if $m = 0$ in $M_f$ for all generators $f \\in S_1$, does it follow that $m = 0$ in $M$?",
"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... | Pb(NO_3)_2(aq) + 2KI(aq) rarr PbI_2(s) + 2KNO_3(aq) |
[
{
"content": "Determine the number of distinct patterns that can be created by filling a 5x5 grid using 3 colors, considering symmetry. You do not have to use all three colors in each pattern.",
"role": "user"
}
] | 105918450471 |
[
{
"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... | 210.272 degrees |
[
{
"content": "Determine whether there exists a continuous function \\( g: \\mathbb{R} \\to \\mathbb{R}^{\\mathbb{N}} \\) such that the range of \\( g \\) is not eventually constant in any coordinate.",
"role": "user"
}
] | Yes |
[
{
"content": "Given that $a$ and $b$ are positive integers where $b$ is a multiple of $a$, and \\[ \\log\\left(\\left(\\frac{b}{a}\\right)^{\\frac{b}{3}}\\right) + \\log\\left(\\left(\\sqrt[3]{\\frac{a}{b}}\\right)^{9a}\\right) = 1, \\] find the value of $a^2 + b^2$. ",
"role": "user"
}
] | 909 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | x(t)=(m/k) * [log(cos(sqrt(gk*sin(theta))*t/m - arctan(sqrt(k)*vo/sqrt(g*sin(theta))))) - log(1/sqrt((k*vo^2)/(g*sin(theta)) + 1))] and v(t)=sqrt(g*sin(theta)/k) * tan(arctan(sqrt(k/(g*sin(theta)))*vo) - (sqrt(k*g*sin(theta))/m)*t) |
[
{
"content": "Let $f: \\mathbb{R} \\to \\mathbb{R}$ be a continuous function such that $f(\\mathbb{Q}) \\subset \\mathbb{N}$. Determine whether $f$ is (1) unbounded, (2) constant, or (3) non-constant 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... | $22,172.50 |
[
{
"content": "Consider a function \\( f(z) \\) that is holomorphic in some neighborhood of \\( z = 0 \\) and expands at zero into a power series \\( \\sum_{n=0}^{\\infty}a_nz^n \\) with convergence radius \\( R \\). Given a natural number \\( k \\), determine the holomorphy and convergence radius of the series ... | R |
[
{
"content": "Given two positive continuous random variables X and Y that are approximately normally distributed with E(X) = 50, sd(X) = 6 and E(Y) = 30, sd(Y) = 4, calculate Pr(X/Y > 2). Choose the correct answer from the following options:\n\n0.023\n0.159\n0.238\n0.460\n0.841",
"role": "user"
}
] | 0.159 |
[
{
"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... | 2.54 atm |
[
{
"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... | 6.53% |
[
{
"content": "Find the value of the limit \\( \\lim_{x\\to\\infty}e^{-\\left((\\log(\\frac{x+a}{\\sigma}))^\\nu-(\\log(\\frac{x}{\\sigma}))^\\nu\\right)} \\), where \\(\\nu\\) and \\(\\sigma\\) are positive constants.",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty}\\frac{\\lfloor x\\cdot 10^n\\rfloor}{10^n} \\] where \\( x \\in \\mathbb{R} \\setminus \\mathbb{Q} \\).",
"role": "user"
}
] | x |
[
{
"content": "Let $R$ be a ring (not necessarily with 1) containing a non-zero element $a$ such that $a^4 + a = 2a^3$. Determine whether $R$ must have at least one non-zero idempotent element.",
"role": "user"
}
] | Yes |
[
{
"content": "A biased coin lands on heads with an unknown probability $p$, where the cumulative distribution function of $p$ is given by $P(p \\leq x) = x^4$. If you flip the coin 5 times and it lands on heads each time, calculate the probability that the next flip will also result in heads.",
"role": "use... | \dfrac{9}{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... | 1.425 m |
[
{
"content": "Given the joint density function \\( f(x, y)=\\begin{cases} \\dfrac{x^2+y^2}{50} & 0<x<2, 1<y<4, \\\\\\ 0 & \\text{elsewhere.} \\end{cases}\\), calculate the probability \\( P(X+Y>4) \\).",
"role": "user"
}
] | \dfrac{8}{15} |
[
{
"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... | 1.51 L |
[
{
"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... | 0.784 m^2 |
[
{
"content": "Determine the radius of convergence for the complex power series \\(\\sum_{n=1}^{\\infty}\\frac{(-1)^{n+1}}{n}z^{n(n+1)}\\) using the Hadamard formula.",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the improper integral \\( \\int \\limits_{0}^{\\infty}\\left(\\frac{1}{\\sqrt{x^2+4}}-\\frac{k}{x+2}\\right)\\text{d}x \\) where \\( k \\) is chosen such that the integral converges. Determine the value of the integral from the following options: \\( \\ln 2 \\), \\( \\ln 3 \\), \\( \\ln 4... | \ln 2 |
[
{
"content": "Does the Central Limit Theorem fail to provide a good approximation for the Cauchy distribution due to its undefined mean and variance?",
"role": "user"
}
] | Yes |
[
{
"content": "Given the numbers from 1 to n, is it possible to repeatedly erase two numbers and add their difference to the list such that eventually all numbers in the list are odd?",
"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... | 115.5Nm for the rod rotating about the center, and 461.8Nm for the rod rotating about one end |
[
{
"content": "Evaluate the double integral \\( \\int \\int_D (1-y) \\, dx \\, dy \\) over the region \\( D \\) defined by the inequalities:\n\n1. \\( x^2 + y^2 \\leq 2y \\)\n2. \\( y \\leq x^2 \\)\n3. \\( x \\geq 0 \\)\n\nDetermine the appropriate domain for \\( x \\) and \\( y \\) and compute the integral.",
... | \dfrac{1}{15} |
[
{
"content": "Given a continuous polynomial function $f(x): \\mathbb{R} \\to \\mathbb{R}$ that satisfies the differential equation $f(2x) = f'(x) f''(x)$, find the value of $f(3)$. \\( \\boxed{} \\)",
"role": "user"
}
] | 12 |
[
{
"content": "Calculate the surface integral \\( \\int \\mathbf{E} \\cdot d\\mathbf{S} \\) of the vector field \\( \\mathbf{E} = (x+y^2)\\mathbf{i} + (y^3+z^3)\\mathbf{j} + (x+z^4)\\mathbf{k} \\) over the entire surface of a cube defined by the region \\( 0 < x < 2, -1 < y < 1, 0 < z < 2 \\). The outward normal... | 80 |
[
{
"content": "Calculate the winding number of the ellipse \\( \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 \\) at the origin using the formula:\n\\[ \\omega(\\gamma,0) = \\frac{1}{2i\\pi}\\oint_{\\gamma}\\frac{1}{z}dz \\]\nwhere the ellipse is parametrized by \\( \\gamma(t) = a\\cos(t) + ib\\sin(t) \\) for \\( t \\i... | 1 |
[
{
"content": "Show that the space $X+_fY$, where $X=[0,1]$, $Y=[2,3]$, $A=\\{0,1\\}$, and $f\\colon A\\to Y$ is defined by $f(0)=2$, $f(1)=3$, is homeomorphic to the circle $\\mathbb{S}^1$. Use the known result that if $X$ is any space, $A$ is a subset of $X$, and $p\\notin X$, the space $X+_f\\{p\\}$ resulting... | \mathbb{S}^1 |
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