prompt list | ground_truth string |
|---|---|
[
{
"content": "Find the sum of the roots of the equation:\n\\[(x − 1) + (x − 2)^2 + (x − 3)^3 + \\ldots + (x − 10)^{10} = 0.\\]",
"role": "user"
}
] | 99 |
[
{
"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... | 196 N |
[
{
"content": "According to Tao's definition of positive real numbers, determine if a Cauchy sequence with a finite number of negative terms can have a positive real number as its limit.",
"role": "user"
}
] | Yes |
[
{
"content": "Find the number of integer solutions to the equation \\(x_1 + x_2 + x_3 + x_4 = 30\\) subject to the constraints: \\(1 \\leq x_1 \\leq 5\\), \\(4 \\leq x_2 \\leq 9\\), \\(6 \\leq x_3 \\leq 11\\), and \\(10 \\leq x_4 \\leq 15\\).",
"role": "user"
}
] | 125 |
[
{
"content": "Let $r$ be the remainder when $1342$ is divided by $13$.\n\nDetermine the smallest positive integer that has these two properties:\n\n$\\bullet~$ It is a multiple of $1342$.\n\n$\\bullet~$ Its remainder upon being divided by $13$ is smaller than $r$.",
"role": "user"
}
] | 6710 |
[
{
"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... | 0.957 g |
[
{
"content": "Consider a population of people where each person's height is independently and identically distributed (IID). You randomly choose a person and observe that their height is $X$. Let $N$ be the number of additional samples needed to randomly select a person whose height is greater than $X$. Calcula... | \infty |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 8.88*10^-3 H |
[
{
"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... | 0.534I_0 |
[
{
"content": "Let \\(U, V, W\\) be subspaces of \\(\\mathbb{R}^{n}\\) with dimensions \\(u, v, w\\) respectively. Given that \\(U \\cap V = U \\cap W = V \\cap W = \\{0\\}\\) and \\(U + V + W = \\mathbb{R}^{n}\\), find the minimum possible value of \\(u + v + w\\).",
"role": "user"
}
] | n |
[
{
"content": "Find all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) such that for all real numbers \\( x \\) and \\( y \\), the equation \\( f(x^2 + f(y)) = y + f(x)^2 \\) holds.",
"role": "user"
}
] | f(x) = x |
[
{
"content": "Find the limit of the sequence: \\( \\lim_{n \\to \\infty} (\\sqrt[3]{1-n^3} + n) \\)",
"role": "user"
}
] | 0 |
[
{
"content": "Let $\\mathcal{H}$ be a separable Hilbert space and $U \\subset \\mathcal{H}$ a closed subspace. Is it possible to choose an orthonormal basis $\\{e_{i}\\}_{i=1}^{\\infty}$ of $\\mathcal{H}$ such that there exists a subsequence of the $e_{i}$'s that span $U$?",
"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... | {{I}_{new}}={{I}_{0}}+\left( \sum{M} \right){{R}^{2}} |
[
{
"content": "Evaluate the double integral \\( \\iint_E x\\, dx\\, dy \\) where \\( E = \\{ (x,y) \\mid 0 \\le x, 0 \\le y \\le 1, 1 \\le x^2 + y^2 \\le 4 \\} \\).",
"role": "user"
}
] | \dfrac{3}{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... | $20,532 |
[
{
"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... | 1,153,700 |
[
{
"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... | 3.45 mL |
[
{
"content": "If ZFC has a model, does it have a countable model whose Turing degree is bounded by $\\emptyset^{(n)}$ for some finite $n$?",
"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... | $14.90 million |
[
{
"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 + \sqrt{5} and 4 - \sqrt{5} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | -3870 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... | 138.24 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... | 0.7286 M |
[
{
"content": "Evaluate the series representation of \\( \\sin(i) \\) given by:\n\\[ \\sin(i) = i \\sum_{j=0}^\\infty \\frac{1}{(2j+1)!} \\]\nWhat does this series converge to?",
"role": "user"
}
] | i \sinh(1) |
[
{
"content": "Find the residue of the function \\( e^{-e^{-1/z}} \\) at \\( z=0 \\) by expressing it as a Laurent series.",
"role": "user"
}
] | \dfrac{1}{e} |
[
{
"content": "For any finite alphabet $A$ with $|A| \\geq 2$ and any natural number $k$, does there always exist a thin base of order $k$?",
"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... | 0.358 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... | 2.88e6 V |
[
{
"content": "Determine the number of ways to express \\(2^{2012}\\) as the sum of four (not necessarily distinct) positive squares.",
"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... | Year 1: $1.3750
Year 2: $1.5125
Year 3: $1.6638
Year 4: $1.7636
Year 5: $1.8694 |
[
{
"content": "Let $X$ and $Y$ be independently uniformly distributed random variables with densities $f(x) = 1$ for $0 < x < 1$ and $f(y) = 1$ for $0 < y < 1$. Define $V = XY$. Determine the density function of $V$.",
"role": "user"
}
] | -\ln v |
[
{
"content": "Calculate the sum of the series \\( \\sum_{n=2}^{\\infty}\\frac{(n-1)^{2}}{n!} \\).",
"role": "user"
}
] | e - 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... | 21.0 m/s^2 |
[
{
"content": "Using the central limit theorem, determine the minimum number of 100W bulbs needed to ensure that an advertising board is lit for at least 20,000 hours with a probability of at least 0.9. Assume that the life of each bulb follows an exponential distribution with an average life of 600 hours.",
... | 42 |
[
{
"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... | 3.24e-4 kg/m^2 h |
[
{
"content": "Determine the number of ways to select three subsets \\(A\\), \\(B\\), and \\(C\\) from the set \\(\\{1, 2, \\ldots, n\\}\\) such that \\(A \\subseteq C\\), \\(B \\subseteq C\\), and \\(A \\cap B \\neq \\emptyset\\).",
"role": "user"
}
] | 5^n - 4^n |
[
{
"content": "Find the limit of the sequence \\(b_n = \\Psi(2n + 1) - \\Psi(n + 1)\\), where \\(\\Psi(z)\\) is the digamma function.",
"role": "user"
}
] | \ln 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... | 35.2 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... | 8.29e+22 atoms |
[
{
"content": "Calculate the volume of the solid bounded by the sphere $x^2 + y^2 + z^2 = 4$ and the plane $y = 1$, within the region $-\n\\sqrt{3} \\leq x \\leq \\sqrt{3}$, $1 \\leq y \\leq 2$, and $-\\sqrt{3} \\leq z \\leq \\sqrt{3}$. Verify the integration boundaries.",
"role": "user"
}
] | \dfrac{5\pi}{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... | 8.97 cm |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0, y \\to 0} \\left(1 + x^2y^2\\right)^{-1/(x^2+y^2)}. \\]",
"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... | 644.2 s |
[
{
"content": "Determine whether for any odd number $x \\in 2\\mathbb{N} + 1$, there exist numbers $m, n \\in \\mathbb{N} \\cup \\{0\\}$ such that $2^n + 1 = 3^m x$. Answer with 'True' or 'False'.",
"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... | $294 Unfavorable |
[
{
"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 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... | y = 0.32 m |
[
{
"content": "Determine the largest integer \\( n \\) such that the function \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\), defined by \\[ f(x) = \\begin{cases} \\frac{e^x - 1}{x}, & x \\neq 0 \\\\ 1, & x = 0 \\end{cases} \\] is \\( n \\)-times differentiable.",
"role": "user"
}
] | \infty |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | H2C=CH2 |
[
{
"content": "Determine the least ordinal \\( \\beta \\) such that it is provable in \\( \\mathsf{ZFC} \\) that \\( 2^{\\aleph_0} \\leq \\aleph_{\\aleph_\\beta} \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Let $M$ be a Riemannian manifold such that for all $x \\in M$ and large $R > 0$, \\( \\text{Vol}(B(x, R)) \\leq \\exp(R) \\). Let $p \\in M$ and $d$ be the geodesic distance. Suppose that for large enough $r$ and all $x, y \\in \\partial B_{r}(p)$, \\( d(x, y) \\leq c \\log r \\). Is it true that ... | 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... | -16 kg.m/s |
[
{
"content": "Determine the limit of the sequence \\( \\left\\{ \\frac{(n!)^3}{(3n)!} \\right\\} \\).",
"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... | 0.00132 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... | I = S + T - G |
[
{
"content": "Determine whether the following statement is true or false: \"If $R$ is a field, then $R[x]$ is an Euclidean Domain (ED).\"",
"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... | $1,943,588.91 |
[
{
"content": "Let \\(a\\) be a root of the polynomial \\(X^4 + 2X + 1 = 0\\). Express \\(\\frac{a+1}{a^2 - 2a + 2}\\) as a polynomial in \\(a\\) with rational coefficients.",
"role": "user"
}
] | a^3 + a^2 |
[
{
"content": "Let \\( f: \\mathbb{R} \\to \\mathbb{R} \\) be a continuous function such that \\( f(\\mathbb{Q}) \\subseteq \\mathbb{N} \\). Choose the correct statement from the following options:\n\n(a) \\( f(\\mathbb{R}) = \\mathbb{N} \\)\n(b) \\( f(\\mathbb{R}) \\subseteq \\mathbb{N} \\) but \\( f \\) need n... | D |
[
{
"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... | 57.55 KJ |
[
{
"content": "Calculate the limit: $$\\lim_{n \\to \\infty} \\left(\\cos \\left(\\frac{a}{n}\\right)\\right)^{n^2}$$ for $n \\in \\mathbb{N}$ and $a \\in \\mathbb{R} \\setminus \\{0\\}$.",
"role": "user"
}
] | e^{-\frac{a^2}{2}} |
[
{
"content": "You are fighting a dragon with three heads. Each time you swing at the dragon, you have a 20% chance of hitting off two heads, a 60% chance of hitting off one head, and a 20% chance of missing altogether. If you knock off one head, the head grows back immediately before the next iteration. If you ... | \dfrac{2}{3} |
[
{
"content": "Given a Poisson process $\\lbrace N(t)\\rbrace_{t\\geq 0}$ with intensity $\\lambda = 3$, calculate the conditional probability:\n\n$$P\\left[N(6) = 2 \\,|\\, N(8) = 4, N(3) = 1\\right]$$",
"role": "user"
}
] | \dfrac{36}{125} |
[
{
"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... | 1066.8 |
[
{
"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... | 4.666 |
[
{
"content": "If a matrix \\( B \\) has rank \\( m-2 \\), where \\( m > 2 \\), what is the rank of its adjoint matrix, \\( \\text{adj}(B) \\)?",
"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... | $260 |
[
{
"content": "Calculate the value of the power series \\( f(x)=\\sum _{n=1}^{\\infty }\\frac{\\left(-1\\right)^{n+1}x^{2n+1}}{2n-1} \\) at \\( x=\\frac{\\sqrt{3}}{3} \\).",
"role": "user"
}
] | \dfrac{\pi}{18} |
[
{
"content": "Evaluate the integral \\(\\int_{|z|=2}\\frac{dz}{z^4(z^{12}-1)}\\) using the residue theorem.",
"role": "user"
}
] | 0 |
[
{
"content": "Given real numbers \\(b\\) and \\(c\\), and the function \\(f(x) = x^2 + bx + c\\), the equation \\(f(f(x)) = 0\\) has exactly three different real roots. Determine the maximum value of the sum of the roots of \\(f(x)\\).",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Find the function $u(x, y)$ such that $f(z) = u(x, y) + j(8xy + 4y)$ is analytic for $z = x + jy$ and satisfies $f(0) = 0$. Then express $f(z)$ as a function of $z$.",
"role": "user"
}
] | 4z^2 + 4z |
[
{
"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... | 0.2492 |
[
{
"content": "Evaluate the limit \\( \\lim \\limits_{n\\to \\infty} \\{ (2+\\sqrt{3})^n \\} \\), where \\(\\{\\cdot\\}\\) denotes the fractional part. Given that \\([(2+\\sqrt{3})^n]\\) is an odd number for all \\(n\\), prove the value of the limit.",
"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... | $19,306 |
[
{
"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... | 1s |
[
{
"content": "Find the equation of the circle that is transformed into a straight line by the function \\( f(z) = \\frac{z-1}{iz+i} \\) and passes through the point \\( 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... | 392.3 kJ |
[
{
"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... | 4.47 |
[
{
"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... | -66.7 nC |
[
{
"content": "In a rectangular parallelepiped ABCDA'B'C'D', where ABCD is the lower face and A, B, C, and D are directly below A', B', C', and D', respectively, the parallelepiped is divided into eight parts by three planes parallel to its faces. Given that the volume of the part containing vertex A is 40, the ... | 2015 |
[
{
"content": "The midpoint of the line segment between $(x,y)$ and $(2,4)$ is $(-7,0)$. Find $(x,y)$.",
"role": "user"
}
] | (-16,-4) |
[
{
"content": "Let $n = 2m$, where $m$ is odd. Determine the number of elements of order 2 in the group $D_n/Z(D_n)$. Provide your answer as a single integer.",
"role": "user"
}
] | m |
[
{
"content": "Consider a number $0 < x < 1$ where the $n$th digit after the decimal point has a $\\frac{1}{n}$ probability of being $2$, instead of the default $1$. Would such a number be considered irrational, given that the probability of a digit being $2$ approaches $0$ as $n$ goes to infinity?",
"role":... | 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... | 30.76% |
[
{
"content": "Compute the remainder when $1761^{4784}$ is divided by 35, given that $1761^{4784} \\equiv 1 \\pmod{5}$. First, find $1761^{4784} \\pmod{7}$, and then use the Chinese Remainder Theorem to determine the final result $R_{35}(1761^{4784})$. Express your answer as a single integer in the range 0 to 34... | 16 |
[
{
"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... | 25.4 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... | Canada |
[
{
"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... | 1.57 |
[
{
"content": "Determine the number of distinct real values of $x$ that satisfy the equation $17^x + 9^{x^2} = 23^x + 3^{x^2}$.",
"role": "user"
}
] | 2 |
[
{
"content": "Determine the rank of an $n \\times n$ matrix $A$, where $n \\ge 3$, with entries $A_{ij} = \\alpha i + \\beta j + \\gamma$, given that $\\alpha, \\beta$, and $\\gamma$ are three arbitrary positive numbers.",
"role": "user"
}
] | 2 |
[
{
"content": "Find the asymptotic equivalent of the sequence \\((u_n)_{n \\geq 1}\\) defined by \\(u_1 > 0\\) and \\(u_{n+1} = \\frac{u_n}{n} + \\frac{1}{n^2}\\) as \\(n \\to +\\infty\\).",
"role": "user"
}
] | \dfrac{1}{n^2} |
[
{
"content": "Find all prime numbers $p$ such that the polynomial $x^3 - 120x + p$ is reducible over the rational numbers.",
"role": "user"
}
] | 11 |
[
{
"content": "What is the probability that all increments of a Brownian motion are positive over an infinite time horizon?",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit: $$\\lim _{x\\to \\infty }\\frac{\\left(x!\\right)^3\\left(3\\left(x+1\\right)\\right)!}{\\left(\\left(x+1\\right)!\\right)^3\\left(3x\\right)!}$$",
"role": "user"
}
] | 27 |
[
{
"content": "Simplify the expression \\( \\sum_{n=-\\infty}^\\infty h^n(J_{n+1}(x) - J_{n-1}(x) + 2\\frac{\\mathrm{d}J_n}{\\mathrm{d}x}(x)) = 0 \\) by eliminating the summation signs and the variable \\( 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... | 3.80 * 10^4 rpm |
[
{
"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... | 1.40 m/s |
[
{
"content": "Here are two functions: $$\\begin{array}{ccc}\nf(x) & = & 3x^2-2x+ 4\\\\\ng(x) & = & x^2-kx-6\n\\end{array}$$ If $f(10) - g(10) = 10,$ what is the value of $k?$",
"role": "user"
}
] | -18 |
[
{
"content": "Evaluate the limit \\( \\lim_{x\\to\\infty}(1-\\frac{\\ln x}{2x})^x \\).",
"role": "user"
}
] | 0 |
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