prompt list | ground_truth string |
|---|---|
[
{
"content": "Evaluate \\(12(c - \\alpha)\\) given that \\(\\lim_{n\\to{\\infty}}\\frac{e(1-\\frac{1}{n})^n-1}{n^{\\alpha}}=c\\), where \\(c\\) is a non-zero constant.",
"role": "user"
}
] | 6 |
[
{
"content": "Given complex numbers \\( z_1 \\) and \\( z_2 \\) such that \\( \\left|z_1\\right| = \\left|z_1 + z_2\\right| = 3 \\) and \\( \\left|z_1 - z_2\\right| = 3\\sqrt{3} \\), find the value of:\n\n\\[ \\log_{2} \\left| \\left(z_1 \\overline{z_2}\\right)^{2000} + \\left(\\overline{z_1} z_2\\right)^{2000}... | 4000 \log_{2} 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... | 1s^(2)2s^(2)2p^(6)3s^(2)3p^4 |
[
{
"content": "Find the maximum value of \\(72\\int\\limits_{0}^{y}\\sqrt{x^4+(y-y^2)^2}dx\\) for \\(y\\in[0,1].\\)",
"role": "user"
}
] | 24 |
[
{
"content": "In triangle $ABC$, $BK$ and $CM$ are the angle bisectors of angles $B$ and $C$, respectively. Let $L$ be the intersection point of $BK$ and $CM$. Given that $AB=39$, $BC=42$, and $CA=45$, find the area of triangle $CKL$. Express your answer in square units.",
"role": "user"
}
] | 140 |
[
{
"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... | 240 m/s |
[
{
"content": "Evaluate the limit \\( \\lim_{n \\to \\infty} \\left( \\frac{a}{1} + \\frac{a^2}{2} + \\ldots + \\frac{a^n}{n} \\right) \\) for \\( a > 1 \\) without using L'Hôpital's rule or integration.",
"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... | 3.8 V |
[
{
"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... | 6911 years |
[
{
"content": "Does a group $G$ of order $p^n$ necessarily have an element of order $p^2$, where $p$ is a prime number and $n$ is an integer?",
"role": "user"
}
] | No |
[
{
"content": "Let $X$ and $Y$ be continuous random variables with the joint probability density function:\n\\[ f_{X,Y}(x,y) = \\begin{cases} \\frac{10-x y^2}{64} & \\text{for } 2 \\leq x \\leq 10 \\text{ and } 0 \\leq y \\leq 1 \\\\ 0 & \\text{otherwise} \\end{cases} \\]\nCompute the expected value of $Y$, deno... | \dfrac{7}{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... | :B≡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... | 26.67 |
[
{
"content": "Given the differential equation \\( xy'' + y' = 0 \\) and a known solution \\( y_1 = \\ln(x) \\), find the second linearly independent solution.",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to0}\\frac{\\displaystyle\\int_0^x(x-t)\\sin t^2\\ dt}{x\\sin^3x}. \\]",
"role": "user"
}
] | \dfrac{1}{12} |
[
{
"content": "Find the partial derivative \\( \\frac{\\partial E}{\\partial y_i} \\) of the expression \\( E = \\sum_{k} \\frac{e^{y_k}}{\\sum_{j} e^{y_j}} \\), where \\( j \\) is the index of summation in the denominator.",
"role": "user"
}
] | 0 |
[
{
"content": "Is there an uncountable set of algebraically independent transcendental numbers?",
"role": "user"
}
] | Yes |
[
{
"content": "A bag contains 100 unique marbles. With replacement, you pick 10 marbles at a time, at random. How many times must you pick the marbles to have a 95% probability of having seen every unique marble at least once?",
"role": "user"
}
] | 76 |
[
{
"content": "Given that \\( z - \\frac{1}{z} = i \\), find the value of \\( z^{2014} + \\frac{1}{z^{2014}} \\) in terms of \\( 1, -1, i, \\) or \\( -i \\).",
"role": "user"
}
] | 1 |
[
{
"content": "The sequence $\\{a_{n}\\}$ satisfies $a_{1}=\\frac{1}{4}$, $a_{2}=\\frac{1}{5}$, and $a_{1} a_{2}+a_{2} a_{3}+\\cdots+a_{n} a_{n+1}=n a_{1} a_{n+1}$ for any positive integer $n$. Determine the value of $\\frac{1}{a_{1}}+\\frac{1}{a_{2}}+\\cdots+\\frac{1}{a_{97}}$. \\(\\text{Express your answer as ... | 5044 |
[
{
"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... | 1.273e6 turns per meter |
[
{
"content": "Let $G$ be a finite group, and let $p$ be a prime number that divides the order of $G$. Determine whether the number of elements in $G$ with order coprime to $p$ is necessarily coprime to $p$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit for a continuous function $f \text{ on } [-1,1]$:\n$$\\lim_{h \\to 0} \\frac{ \\int^{h}_{-h} f(t) \\, dt }{h} $$",
"role": "user"
}
] | 2f(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... | 0.0536 |
[
{
"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... | 13.4 g |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty}{\\left\\{\\left(1+\\frac{1}{n}\\right)^n-\\left(1+\\frac{1}{n}\\right)\\right\\}}^{-n}$$",
"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... | 22/3 m/s^2 |
[
{
"content": "In a Parliamentary chamber with 201 seats, how many different ways can these seats be allocated among three political parties such that no party has more than half the seats?",
"role": "user"
}
] | 5050 |
[
{
"content": "Determine the number of rotations in \\( \\mathbb{R}^3 \\) that map the cube \\( [-1,1]^3 \\) onto itself.",
"role": "user"
}
] | 24 |
[
{
"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... | Increased demand for health insurance, increased demand for other products due to increased disposable income, and potential decrease in the supply of premium insurance over time. |
[
{
"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... | 86 degrees |
[
{
"content": "Find the interval of uniform convergence for the series \\( \\sum_{n=1}^{\\infty}(\\sum_{k=0}^{m}kn^k)x^n \\), where \\( m \\) is a positive integer.",
"role": "user"
}
] | (-1, 1) |
[
{
"content": "Given the sets \\(A=\\left\\{1, \\frac{x+y}{2}-1\\right\\}\\) and \\(B=\\{-\\ln(xy), x\\}\\), if \\(A = B\\) and \\(0 < y < 2\\), find the value of \n\n\\[ \n\\left(x^{2} - \\frac{1}{y^{2}}\\right) + \\left(x^{4} - \\frac{1}{y^{4}}\\right) + \\cdots + \\left(x^{2022} - \\frac{1}{y^{2022}}\\right).... | 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... | 0.65 |
[
{
"content": "Let $T: \\mathbb{R}^5 \\rightarrow \\mathbb{R}^5$ be a linear transformation such that $T^4(x) = 0$ for all $x \\in \\mathbb{R}^5$. Given that $T^k \\neq 0$ for $k \\leq 3$, determine the rank of $T$. Choose the correct option: (a) $\\text{Rank}(T) = 2$, (b) $\\text{Rank}(T) = 3$. \\(\\text{Rank}(... | B |
[
{
"content": "Tom has a red marble, a green marble, a blue marble, and three identical yellow marbles. How many different groups of two marbles can Tom choose?",
"role": "user"
}
] | 7 |
[
{
"content": "The general term of a sequence is given by \\( a_{n} = b[\\sqrt{n+c}] + d \\). The sequence is calculated successively as follows: \\( 1, 3, 3, 3, 5, 5, 5, 5, 5, \\cdots \\). In this sequence, each positive odd number \\( m \\) appears exactly \\( m \\) times consecutively. Determine the value of ... | 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... | 545.6 N |
[
{
"content": "Find the positive value of \\( x \\) for which the integral \\( \\int_0^{\\sqrt{x}} \\sin \\left(\\frac{2\\pi t}{t+2}\\right) dt \\) attains its maximum.",
"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... | 5.01e-8 N |
[
{
"content": "A deli sandwich consists of one type of bread, one type of meat, and one type of sauce. The deli offers wheat, rye, and white bread; ham, turkey, roast beef, and tuna; ranch and southwest chipotle sauce. Javier is allergic to rye bread, tuna, and southwest chipotle sauce. He tells the server to as... | \frac{3}{4} |
[
{
"content": "Given an integer \\( n \\geq 2 \\), for any pairwise coprime positive integers \\( a_1, a_2, \\ldots, a_n \\), let \\( A = a_1 + a_2 + \\ldots + a_n \\). Denote by \\( d_i \\) the greatest common divisor (gcd) of \\( A \\) and \\( a_i \\) for \\( i = 1, 2, \\ldots, n \\). Denote by \\( D_i \\) the... | (n-1)^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... | 1.507e-17 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... | 693.78 mL |
[
{
"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... | 277.94 |
[
{
"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... | Long-term debt: $85,000,000
Total Liabilities: $85,000,000
Shareholder's Equity:
Equity Capital: $17,000,000
Capital Surplus: $114,000,000
Retained Earnings: $25,200,000
Total Shareholder's Equity: $156,200,000
Total Liabilities and equity: $241,200,000 |
[
{
"content": "Let \\( m = 2023^2 + 2^{2023} \\). Determine the last digit of \\( m^2 + 2^m \\).",
"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... | $8,309.70 |
[
{
"content": "Let $G$ be a group and let $H$ and $K$ be two subgroups of $G$, each with 12 elements. Determine which of the following numbers cannot be the cardinality of the set $HK = \\{hk: h \\in H, k \\in K\\}$. Choose from the options: (a) 72, (b) 60, (c) 48, (d) 36.",
"role": "user"
}
] | B |
[
{
"content": "Consider a local neighborhood \\(U\\) in a manifold centered around a point \\(p\\) with a chart \\(X=(x_1,x_2,\\cdots,x_n):U\\rightarrow V\\subset\\mathbb{R}^n\\). The vector fields \\(\\frac{\\partial}{\\partial x_i}\\) for \\(i=1,\\cdots,n\\) are defined by the curves \\(c_i:(-\\epsilon,\\epsil... | Yes |
[
{
"content": "Consider theories $T$ and $T'$, where $T$ is a subset of $T'$, in the same language $L$. Also, let $L'$ be a subset of the language $L$. Is the reduct of $T$ with $L'$ a subset of the reduct of $T'$ with $L'$?",
"role": "user"
}
] | Yes |
[
{
"content": "Find all pairs of prime numbers \\(p\\) and \\(q\\) such that \\(p^{q+1} + q^{p+1}\\) is a perfect square.",
"role": "user"
}
] | (2, 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... | 15.29% |
[
{
"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... | $19.01 |
[
{
"content": "Given an arbitrary closed subset $E$ of $\\mathbb{R}$, determine whether there exists a function $f: \\mathbb{R} \\to \\mathbb{R}$ such that the set of points where $f$ is unbounded is exactly $E$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Find the limit of the expression as \\( n \\) approaches infinity: \\[ \\lim_{n \\to \\infty} \\frac{(2n-1)!}{(2n)^n} \\]",
"role": "user"
}
] | \infty |
[
{
"content": "Evaluate the integral \\( \\int_0^2\\int_1^3\\min(x,y)\\,dx\\,dy \\) given that \\( x < y \\).",
"role": "user"
}
] | \dfrac{2}{3} |
[
{
"content": "Find the smallest positive integer \\( M \\) such that if \\( \\left|a x^{2}+b x+c\\right| \\leq 1 \\) for all \\( |x| \\leq 1 \\), then \\( |2 a x+b| \\leq M \\) for all \\( |x| \\leq 1 \\).",
"role": "user"
}
] | 4 |
[
{
"content": "Let $(X,Y)$ be a random vector uniformly distributed in the unit square $[0,1]^2$. Define the random variables $U = \\min(X,Y)$ and $V = \\max(X,Y)$. Calculate the covariance $\\text{Cov}(U,V)$. Provide your answer as a single numerical value.",
"role": "user"
}
] | \dfrac{1}{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... | -3.874e-24 Jmol-1K-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... | 1131.82 KN |
[
{
"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... | 0.549 T |
[
{
"content": "Calculate the spectrum of the Dirichlet Laplacian and the Neumann Laplacian on the domain $[0, \\infty)$. The Dirichlet Laplacian operator $T$ is defined on $L^2[0, \\infty)$ as $Tf = -f''$ with domain $D(T) = H^2 \\cap H_0^1$. The Neumann Laplacian operator $S$ is defined on $L^2[0, \\infty)$ as ... | [0, \infty) |
[
{
"content": "Determine the type of singularity and calculate the residue of the function \\( g(z) = \\frac{\\tan(z)}{(z-\\frac{3\\pi}{2})} \\) at \\( z = \\frac{3\\pi}{2} \\).",
"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... | 1.23e4 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... | 12086 |
[
{
"content": "Determine all positive integers \\( n \\) such that the equation \\( nx^4 + 4x + 3 = 0 \\) has real roots.",
"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... | 90 N upwards |
[
{
"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... | 5.4% |
[
{
"content": "Find the minimum value of the expression:\n\\[ \\sum_{cyc} \\sqrt{a^2 + b^2 + ab - 2a - b + 1} \\]\nwhere \\(0 < a, b, c < 1\\).",
"role": "user"
}
] | \dfrac{3}{2} |
[
{
"content": "Find the coefficient of \\(x^6\\) in the expansion of \\((e^x - 1)^8\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Find the derivative of the function \\( f(x) = \\begin{cases} 1 - \\cos \\left(x \\sin \\frac{1}{x}\\right), & x \\neq 0 \\\\ 0, & x = 0 \\end{cases} \\) at the point \\( x = 0 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Given the vector function \\( r(t) = (3t + 4\\sin(t) + 4\\cos(t)) \\), use the curvature formula \\( \\frac{|r' \\times r''|}{|r'|^3} \\) to find the curvature. Verify your calculations to ensure the result is \\( \\frac{4}{25} \\).",
"role": "user"
}
] | \dfrac{4}{25} |
[
{
"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... | 80.4 Wb |
[
{
"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... | 385.95 K |
[
{
"content": "Determine whether the following statement is true or false: For any sets $X, Y \\subseteq \\Bbb A^n(k)$, $I(X) = I(Y)$ if and only if $X = Y$. Provide a justification for your answer.",
"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... | $47,418.87 |
[
{
"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... | 29 rad/s |
[
{
"content": "Let $I$ be a finite nonempty set, and $D$ a nonvoid compact subset of an N-dimensional Euclidean space. Consider continuous functions $V:D\\rightarrow\\mathbb{R}$ and $u_i:D\\rightarrow\\mathbb{R}$ for each $i \\in I$. These functions satisfy the condition: if $u_i(p) = u_i(q)$ for all $i\\in I$, ... | 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... | Date Accounts Debit Credit
a)Cash 17,500
_____Doerr's Capital 17,500
b)Supplies 1,000
_____Accounts Payable 1,000
c)Cash 12,250
_____Sales Commission 12,250
d) Rent Expense 3,800
_____Cash 3,800
e) Accounts Payable 600
_____Cash 600
f) Dividends 3,000
_____Cash 3,000
g)Automobile Expense 1... |
[
{
"content": "Evaluate the double integral \\( \\int_{-1}^{1} \\int_{-1}^{1} |x^2 - y| \\, dy \\, dx \\).",
"role": "user"
}
] | \dfrac{12}{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... | 235.7 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... | 4/5*atm |
[
{
"content": "Let \\( R \\) be the set of points \\((x, y)\\) such that \\(\\lfloor x^2 \\rfloor = \\lfloor y \\rfloor\\) and \\(\\lfloor y^2 \\rfloor = \\lfloor x \\rfloor\\). Compute the area of region \\( R \\). Recall that \\(\\lfloor z \\rfloor\\) is the greatest integer that is less than or equal to \\(z\... | 4 - 2\sqrt{2} |
[
{
"content": "Determine the range of the real number \\( a \\) such that the inequality \\( \\ln x - a \\left(1 - \\frac{1}{x}\\right) \\geq 0 \\) holds for all \\( x \\geq 1 \\).",
"role": "user"
}
] | (-\infty, 1] |
[
{
"content": "Find the inverse of the function \\( h(x) = 1 + x - \\sqrt{1+2x} \\) for \\( x > 0 \\).",
"role": "user"
}
] | x + \sqrt{2x} |
[
{
"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... | $26,000 |
[
{
"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... | 1.96e-6 N, upward |
[
{
"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... | 14.48 m |
[
{
"content": "Let $f: \\overline{\\mathbb{D}} \\to \\mathbb{C}$ be a function that is analytic on the open unit disc $\\mathbb{D}$ and continuous on its closure $\\overline{\\mathbb{D}}$. Suppose there are infinitely many distinct points $z_n \\in \\mathbb{D}$ that converge to the boundary of $\\mathbb{D}$, wit... | 0 |
[
{
"content": "Determine whether there exists a natural number $x$ such that $x^3 = \\sum_{i=1}^{2000} a_i^3$ for some natural numbers $a_i$. The $a_i$'s need not be distinct.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the degree of the splitting field of the polynomial $x^3 - 7$ over the rational numbers $\\mathbb{Q}$. The roots of the polynomial are $\\sqrt[3]{7}e^{\\frac{2\\pi ik}{3}}$ for $k=0,1,2$. Which extension is the splitting field, and what is its degree?",
"role": "user"
}
] | 6 |
[
{
"content": "Given \\( a > b > c > d > 0 \\), and\n\\[\n\\begin{array}{l}\nx=\\sqrt{ab}+\\sqrt{cd}, \\quad y=\\sqrt{ac}+\\sqrt{bd}, \\\\\nz=\\sqrt{ad}+\\sqrt{bc} ,\n\\end{array}\n\\]\ndetermine the order of \\( x, y, z \\).\n\n(A) \\( x < z < y \\) \n(B) \\( y < z < x \\) \n(C) \\( x < y < z \\) \n(D) \\( z... | 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... | 18.75% |
[
{
"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... | 890 2-year bonds |
[
{
"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... | $130,900 |
[
{
"content": "A hyperbola is centered at the origin and opens either horizontally or vertically. It passes through the points $(-3, 4),$ $(-2, 0),$ and $(t, 2).$ Find $t^2.$",
"role": "user"
}
] | \frac{21}{4} |
[
{
"content": "Determine the minimum value of the integral\n\\[ \\mathcal{I}[y] = \\int_{-\\infty}^{\\infty} \\left( \\frac{1}{2} y'^2 + 1 - \\cos y \\right) \\, dx \\]\nsubject to the boundary conditions \\(y(-\\infty) = 0\\) and \\(y(\\infty) = 2\\pi\\).",
"role": "user"
}
] | 8 |
[
{
"content": "Find the minimum value of \\((\\gamma-3+2|\\sin \\theta|)^2+(\\gamma+4|\\cos \\theta|)^2.\\)",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\sum_{k=1}^{n^2} \\frac{1}{\\sqrt{n+k}} \\]",
"role": "user"
}
] | \infty |
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