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... | 0.0253 M |
[
{
"content": "Determine whether the following statement is true or false: If \\(f:[0,1]\\to \\mathbb{R}\\) is a continuous function such that \\(\\int_0^1 f(x)dx=\\int_0^1 xf(x)dx\\), then there exists \\(c\\in(0,1)\\) such that \\(\\int_0^c f(x)dx=0\\).",
"role": "user"
}
] | True |
[
{
"content": "Given a function $f(n)$ defined for every positive integer $n$, with the properties that $f(2) = 2$, $f(mn) = f(m)f(n)$, and $f(m) > f(n)$ if $m > n$, find the value of $f(1998)$. \n\nExpress your answer as a whole number.",
"role": "user"
}
] | 1998 |
[
{
"content": "It is a beautiful day at the beach and ten beach volleyball players have shown up at the volleyball courts. Each two-person volleyball team should consist of a setter and a spiker. Five of the players prefer to be a spiker, four of the players prefer to be a setter, and one player is fine either w... | 29 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to \\pi/4} \\frac{x}{4x-\\pi} \\int_{\\pi/4}^x \\frac{\\tan^2\\theta}{\\theta^2} \\, d\\theta \\]",
"role": "user"
}
] | \dfrac{1}{\pi} |
[
{
"content": "Find the residue of the function \\( f(z)=\\frac{\\cot(z)\\coth(z)}{z^3} \\) at \\( z=0 \\).",
"role": "user"
}
] | -\dfrac{7}{45} |
[
{
"content": "Suppose \\(f : [0,1] \\to \\mathbb{R}\\) is a bounded function such that \\(f\\) is Riemann integrable on \\([a,1]\\) for every \\(a \\in (0,1)\\). Is \\(f\\) Riemann integrable on \\([0, 1]\\)? Justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether the following statement is true or false: Let $x^n - a$ be a polynomial over $\\mathbb{Q}$, where $a$ is a positive rational number. If the constant term of any factor of $x^n - a$ in $\\mathbb{Q}[x]$ has an irrational absolute value, does it imply that $x^n - a$ is irreducible o... | True |
[
{
"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... | 10.272 |
[
{
"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. Meeting customer demand
2. Maintaining production continuity
3. Taking advantage of quantity discounts
4. Hedging against price increases
5. Taking advantage of seasonal demand
6. Reducing lead time
7. Supporting promotional activities |
[
{
"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... | 0.44 m |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\sum_{j=0}^{n}{{j+n-1} \\choose j}\\frac{1}{2^{j+n}} \\]",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Determine the number of binary $n \\times n$ matrices such that every contiguous $2 \\times 2$ submatrix contains exactly two ones and two zeroes. Here, $a_{ij} \\in \\{0,1\\}$ for $i,j \\in \\{1,2,\\dots,n\\}$ are the matrix entries, and the condition is given by:\n\\[ a_{i,j} + a_{i,j+1} + a_{i+... | 2^{n+1} - 2 |
[
{
"content": "Find the integer value of \\(x\\) such that \\(7^{30} \\equiv x \\pmod{100}\\).",
"role": "user"
}
] | 49 |
[
{
"content": "Determine the value of $k$ for which the limit \\[ \\lim_{x\\rightarrow0} \\frac{(e^{x^2}-x^2-1)(\\cos(x)-1)}{x^k} \\] has a finite, non-zero value. Use the Taylor Series expansions of $e^x$ and $\\cos(x)$ to simplify the expression to the form \\[ \\lim_{x\\rightarrow0} \\frac{-\\frac{1}{4}x^6+(\... | 6 |
[
{
"content": "Find all positive integers \\( n \\) such that both \\( 2^n - 1 \\) and \\( 2^{n+2} - 1 \\) are prime, and \\( 2^{n+1} - 1 \\) is not divisible by 7.",
"role": "user"
}
] | 3 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 5000 N |
[
{
"content": "Let $X$ and $Y$ be vector spaces, and let $f: X \\rightarrow Y$ be a surjective linear map. Suppose $M \\subset \\ker f \\subset X$ is a subspace, and $X/M$ is isomorphic to $Y$ via the map induced by $f$. Can we conclude that $M = \\ker f$?",
"role": "user"
}
] | M = \ker f |
[
{
"content": "Determine whether there exists a Fermat number (a number of the form \\( 2^{2^{m}} + 1 \\)) in the arithmetic sequence with common difference \\( k-2 \\):\n\\[ 1, k, 3k-3, 6k-8, \\cdots \\]",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether the following limit is equal to 1: \\[ \\lim_{n \\to \\infty} \\frac{1+\\sqrt{2}+\\sqrt[3]{3}+\\cdots+\\sqrt[n]{n}}{n} = 1. \\]",
"role": "user"
}
] | 1 |
[
{
"content": "Determine the smallest order of a group that contains a proper subgroup which is not commutative.",
"role": "user"
}
] | 12 |
[
{
"content": "Compute the norm of the operator $T \\in \\mathcal{L}(H)$ defined by \\[ Tf(x) = \\int_{-\\pi}^{\\pi} \\sin(x-y)f(y)\\,\\mathrm{d}y \\] where $H=L^2([-\npi,\\pi])$. \\( \\boxed{\\text{Answer}} \\)",
"role": "user"
}
] | \pi |
[
{
"content": "Find the relationship between the integrals \\( \\int_{0}^{\\frac{\\pi}{4}} x \\tan x \\, dx = -\\frac{\\pi}{8} \\ln 2 + \\frac{G}{2} \\) and \\( \\int_{0}^{\\frac{\\pi}{4}} x \\cot x \\, dx = \\frac{\\pi}{8} \\ln 2 + \\frac{G}{2} \\), where \\( G \\) is Catalan's constant, and the integral \\( \\... | 2G |
[
{
"content": "Evaluate the integral \\( \\int_{-\\infty}^{\\infty} \\frac{1}{(x^2+1)(2-2x+x^2)} \\, dx \\) using contour integration.",
"role": "user"
}
] | \dfrac{2\pi}{5} |
[
{
"content": "Find the coefficient of \\(x^{46}\\) in the generating function \\(\\frac{1}{1 - x^3 - x^4 - x^{20}}\\).",
"role": "user"
}
] | 3224 |
[
{
"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... | Balance Sheet
As of December 31, 2015
ASSETS
Current Assets
Cash $45,000
Marketable securities $25,000
Accounts receivable $125,000
Allowance for doubtful accounts ($5,000)
Merchandise inventory $250,000
Office supplies $1,200
Prepaid insurance $6,600
Total Current Assets $447,800
Non-Current Assets
Building and fixtures $1,791,000
Accumulated depreciation ($800,000)
Land $64,800
Total Non-Current Assets $1,055,800
TOTAL ASSETS $1,503,600
LIABILITIES AND STOCKHOLDERS' EQUITY
Liabilities
Accounts payable $70,000
Unearned customer deposits $8,000
Income taxes payable $75,000
Total Liabilities $153,000
Stockholders' Equity
Capital stock $1,000,000
Retained earnings $344,600
Unrealized holding gain on investments $6,000
Total Stockholders' Equity $1,350,600
TOTAL LIABILITIES AND STOCKHOLDERS' EQUITY $1,503,600 |
[
{
"content": "Find all solutions for the equation \\( \\tan(\\alpha) = \\cos(\\alpha + 33.44^\\circ) \\) within the interval \\( 0 < \\alpha < 90^\\circ \\).",
"role": "user"
}
] | 26.56^\circ |
[
{
"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... | 5559.3 m |
[
{
"content": "Find the value of \\( y_0 \\) such that the solution to the differential equation \\( y' - \\frac{3}{2}y = 3t + 2e^{t} \\) with initial condition \\( y(0) = y_0 \\) separates solutions that grow positively as \\( t \\to \\infty \\) from those that grow negatively.",
"role": "user"
}
] | -\dfrac{16}{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... | 16.99% |
[
{
"content": "Compute the limit \\( \\lim_{{n \\to \\infty}} \\sum_{{k=1}}^{{n}} \\arccos(k) \\), where the arccosine function is considered as a complex function.",
"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... | -0.41 m/s |
[
{
"content": "Given a group homomorphism $\\phi: G \\to G_0$ where $|G| = 1013$ and $|G_0| = 55$, determine the order of the image of $G$ under $\\phi$.",
"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... | 1.38e-5 T |
[
{
"content": "Evaluate the series \\( \\displaystyle \\sum^{n}_{k=0}(-4)^k\\binom{n+k}{2k} \\).",
"role": "user"
}
] | (-1)^n (2n + 1) |
[
{
"content": "Suppose $M$ is a finitely generated module over a commutative ring $R$ such that $M \\oplus R \\cong R$. Is it true that $M = 0$? Justify your answer.",
"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... | 12.7*10^5 Joule |
[
{
"content": "Let \\( p(x) \\) be a polynomial such that \\( p(x) - p'(x) = x^n \\), where \\( n \\) is a positive integer. Find the value of \\( p(0) \\).\n\nOptions:\n(A) \\( n! \\)\n(B) \\( (n-1)! \\)\n(C) \\( \\frac{1}{n!} \\)\n(D) \\( \\frac{1}{(n-1)!} \\)",
"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... | $2.18 |
[
{
"content": "Determine the limit points of the set \\( A = (0,1) \\cup \\{2\\} \\) in the lower limit topology \\( \\mathbb{R}_l \\).",
"role": "user"
}
] | [0, 1) |
[
{
"content": "Compute the degree of the line bundle \\( \\mathcal{O}_X(nX_1+mX_3)|_{X_2} \\) on the component \\( X_2 \\) of the fiber \\( X_t \\) over \\( t=0 \\), where \\( X_1, X_2, X_3 \\) are smooth irreducible components intersecting transversally in a chain, with \\( X_1 \\) intersecting \\( X_2 \\) at p... | n + m |
[
{
"content": "Consider all pairs of numbers \\((x, y)\\) that satisfy the equation\n\n\\[ x^{2} y^{2} + x^{2} - 10 x y - 8 x + 16 = 0. \\]\n\nDetermine the possible values of the product \\( xy \\).",
"role": "user"
}
] | [0, 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... | 1.36e-11 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... | 2.3e+04 N.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... | -121 ft/s |
[
{
"content": "Find a closed form expression for the series: \\[ F(s) = \\sum_{n>0} \\frac{s^n}{n} \\] for \\(|s|<1\\).",
"role": "user"
}
] | -\ln(1 - 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... | 614 |
[
{
"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... | 5.2e+2 kg |
[
{
"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... | 0.0221 m |
[
{
"content": "Let $A$ and $B$ be two $n \\times n$ primitive row-stochastic matrices, meaning all entries are non-negative, each row sums to 1, and there exists an integer $p \\geq 1$ such that all entries of $A^p$ and $B^p$ are strictly positive. Assume $A_{i,j} = 0$ if and only if $B_{i,j} = 0$ for all $i, j$... | A = B |
[
{
"content": "Let $K$ be a convex subset of a Banach space $E$. If $\\overline{\\mathrm{co}}\\,(K\\cup\\{x_0\\})$ is compact for some $x_0\\in E$, does it imply that $K$ is compact? 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... | 825.52 N |
[
{
"content": "Find the minimum value of the expression \\( \\frac{a_1}{a_2} + \\frac{a_2}{a_3} + \\cdots + \\frac{a_n}{a_1} \\) where \\(a_1, a_2, \\ldots, a_n\\) are positive real numbers.",
"role": "user"
}
] | n |
[
{
"content": "Determine the number of solutions for the equation \\(x^y - y^x = x + y\\) where both \\(x\\) and \\(y\\) are prime numbers.",
"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... | 13.92 |
[
{
"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... | 76.72 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... | 6.4 |
[
{
"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... | 15.33% |
[
{
"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... | 5.85e5 J |
[
{
"content": "Let \\( f_1, f_2, \\ldots \\) be continuous functions on \\([0,1]\\) such that \\( f_1 \\geq f_2 \\geq \\cdots \\) and \\( \\lim_{n\\to\\infty} f_n(x) = 0 \\) for each \\( x \\in [0,1] \\). Does the sequence \\( \\{f_n\\} \\) converge uniformly to 0 on \\([0,1]\\)?",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\( S = \\{(x, y) \\mid x^{2} - y^{2} \\text{ is odd}, x, y \\in \\mathbb{R} \\} \\) and \\( T = \\{(x, y) \\mid \\sin(2 \\pi x^{2}) - \\sin(2 \\pi y^{2}) = \\cos(2 \\pi x^{2}) - \\cos(2 \\pi y^{2}), x, y \\in \\mathbb{R} \\} \\). Determine the correct relationship between the sets:\n\n(A) \\(... | A |
[
{
"content": "Simplify $(3-2i)^2$. (Your answer should be of the form $a+bi$.)",
"role": "user"
}
] | 5-12i |
[
{
"content": "Find the minimum number of elements in a finite set \\( A \\) such that there exists a function \\( f: \\mathbb{N} \\rightarrow A \\) with the property that if \\( |i - j| \\) is a prime number, then \\( f(i) \\neq f(j) \\).",
"role": "user"
}
] | 4 |
[
{
"content": "Suppose $f(t)$ is a real-valued function on $\\mathbb{R}^n$ that attains its minimum at a unique point $x^*$ in the closure of a set $E$. If $x^* \\notin E$, does $f$ not attain its minimum value in $E$? Answer true or false.",
"role": "user"
}
] | A |
[
{
"content": "Let \\(n \\in \\mathbb{N}^*\\) and \\(k \\in \\{0, 1, 2, \\ldots, n - 1\\}\\). Given the expression \\(z = \\left( \\cot \\frac{(2k + 1)\\pi}{2n} + i \\right)^n\\), find the real part of \\(z\\).",
"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... | 201,042 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... | 22196 |
[
{
"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... | 382 N |
[
{
"content": "Evaluate the double integral \\( \\int_{0}^1\\int_0^{\\sqrt{2y-y^2}}\\ dxdy \\) using polar coordinates and find its value.",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "Consider a sequence of continuous functions $f_m: [0,T] \\to \\mathbb{R}$ that are non-negative and satisfy $\\int_0^T f_m(t)dt \\to 0$ as $m \\to \\infty$. Does it follow that $f_m(t)\\to 0$ as $m \\to \\infty$ for all $t \\in [0,T]$? Answer yes or no and justify 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... | 355456 |
[
{
"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... | D1 = $2.10
D2 = $2.21
D3 = $2.32 |
[
{
"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... | 3.56e+22 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... | 30048 N |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\prod_{r=2}^{n} \\cos\\left(\\frac{\\pi}{2^{r}}\\right) \\]",
"role": "user"
}
] | \dfrac{2}{\pi} |
[
{
"content": "Find the sum of the slope and $y$-intercept of the line through the points $(7,8)$ and $(9,0)$.",
"role": "user"
}
] | 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... | Manipulating the aging of the uncollectible or outstanding accounts receivable deliberately for making the aging of accounts receivable appear better for tax benefits. By comparing the aging for accounts receivable to the actual and original invoices or bills for a sample or evidence of outstanding receivable. |
[
{
"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... | 0.006 radians, 0.3 degrees |
[
{
"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... | $34,267.50 |
[
{
"content": "The vector $\\begin{pmatrix} 1 \\\\ 2 \\\\ 2 \\end{pmatrix}$ is rotated $90^\\circ$ about the origin. During the rotation, it passes through the $x$-axis. Find the resulting vector.",
"role": "user"
}
] | \begin{pmatrix} 2 \sqrt{2} \\ -\frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \end{pmatrix} |
[
{
"content": "In how many ways can 20 distinct students be assigned to four distinct dorms, A, B, C, and D, such that each dorm has at least one student?",
"role": "user"
}
] | 1085570781624 |
[
{
"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... | 1.17kgm^2s^-1 |
[
{
"content": "Determine whether there exists an unbounded sequence \\( y_n \\) for \\( n \\geq 0 \\) such that \\( y_{n+m} \\geq \\frac{y_n + y_m}{2} \\) for all \\( n, m \\geq 0 \\).",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the set of integers \\( x \\) such that \\( \\frac{7x+2}{x+5} \\) is a perfect cube.",
"role": "user"
}
] | -38 |
[
{
"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... | E_n = -13.6 eV * (9/n^2) |
[
{
"content": "Find the maximum value of the expression \\( a + 2b + c + 2d - ab - bc - cd - da \\) given that the numbers \\( a, b, c, d \\) belong to the interval \\([-9.5, 9.5]\\).",
"role": "user"
}
] | 380 |
[
{
"content": "Determine the structure of a Sylow $2$-subgroup of the symmetric group $S_4$.",
"role": "user"
}
] | D_4 |
[
{
"content": "Let $\\mathcal{FG}$ be the category of finite groups, and let $S$ be a full subcategory of $\\mathcal{FG}$. Assume that $G \\in \\mathcal{FG}$ and $P \\in S$ is a subgroup of $G$. We say that $P$ is $S$-maximal if there is no object $P' \\in S$ with $P \\subset P' \\subset G$. Assume that $S$ sati... | Yes |
[
{
"content": "Given that $a, b, c \\in \\mathbb{R}$ and $a^2 + b^2 + c^2 = 2$, find the maximum value of $a + 2b + 3c$. Use any appropriate method to determine this maximum value.",
"role": "user"
}
] | 2\sqrt{7} |
[
{
"content": "Find the value of \\(a\\) for which the integral \\(\\int_{0}^{\\infty}\\frac{dx}{a^2+\\left(x-\\frac{1}{x}\\right)^2}\\) equals \\(\\frac{\\pi}{5050}\\) for \\(a \\geq 2\\).",
"role": "user"
}
] | 2525 |
[
{
"content": "Find the value of \\(a \\in \\mathbb{R}\\) such that the function \\(f(x)\\) is differentiable at \\(x = 0\\), where:\n\\[\nf(x) = \\begin{cases} \n x + xe^{\\frac{1}{x}}, & x < 0, \\\\\n 0, & x = 0, \\\\\n \\frac{a-2\\cos x}{\\sin x}, & x > 0.\n\\end{cases}\n\\]",
"role": "user"
}
] | 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... | 22% |
[
{
"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... | 4.9 g |
[
{
"content": "Determine if the function \\(g_n(x)=\\begin{cases}n(1-|nx|)&x \\in \\left(-\\dfrac{1}{n},\\dfrac{1}{n}\\right)\\\\[6pt] 0 & \\text{otherwise} \\end{cases}\\) is a nascent delta function at 0.",
"role": "user"
}
] | Yes |
[
{
"content": "Calculate the sum of the infinite series \\( \\sum\\limits_{x=1}^\\infty x^2\\cdot\\left(\\frac{1}{2}\\right)^{x+1} \\).",
"role": "user"
}
] | 3 |
[
{
"content": "Find the radius of convergence of the series \\( \\sum_{n=0}^{\\infty} \\frac{z^{2n}}{(2+i)^n} \\).",
"role": "user"
}
] | \sqrt[4]{5} |
[
{
"content": "Evaluate the integral \\( \\int_0^a\\!\\!\\!\\int_0^a\\frac{dx\\,dy}{(x^2+y^2+a^2)^{\\frac32}} \\) using polar coordinates.",
"role": "user"
}
] | \dfrac{\pi}{6a} |
[
{
"content": "At 1 minute to midnight, 10 apples fall into a sack. The same happens at half a minute to midnight, then at a quarter minute to midnight, and so on. At each such event, you remove an apple randomly from the ones still present in the sack. What is the probability that the sack will be empty at midn... | 1 |
[
{
"content": "Compute the limit: \\[ \\lim_{n\\to\\infty}\\frac{1}{\\log n}\\sum_{k=1}^n\\left(1-\\frac{1}{n}\\right)^k\\frac{1}{k} \\] given that \\[ \\log x = \\sum_{k=1}^{\\infty}\\left(1-\\frac{1}{x}\\right)^k\\frac{1}{k},\\quad x\\ge \\frac{1}{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... | 0.164 atm |
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