prompt list | ground_truth string |
|---|---|
[
{
"content": "Show that \\( \\sqrt{n}[\\log(X_{(1)}) - \\log(\\alpha)] \\overset{p}{\\to} 0 \\) for \\( X_1, \\dots, X_n \\sim \\text{Pareto}(\\alpha, \\beta) \\) with \\( n > \\dfrac{2}{\\beta} \\) being independent. Use only the definitions of convergence in probability and distribution, the Delta method, the... | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.0 mol |
[
{
"content": "Find the area of the small loop of the graph given by the polar equation \\( r = 2 + 2 \\sqrt{2} \\sin \\theta. \\)",
"role": "user"
}
] | 2\pi - 6 |
[
{
"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... | 2.67 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... | 718.75 ft-lb |
[
{
"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... | $14.03 |
[
{
"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 days |
[
{
"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... | 36.9 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... | 19 degrees Celsius |
[
{
"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... | 0.160 M |
[
{
"content": "Consider a system described by the equation \\( \\dot{x} = f(x) \\), where \\( x \\in \\mathbb{R}^1 \\) and \\( f \\) is continuous. Given that there are two asymptotically stable equilibrium points, determine if the statement \"the number of all equilibria must be at least three\" is true. Justif... | True |
[
{
"content": "Determine whether there exists an unbounded above function \\( f: \\mathbb{R} \\to \\mathbb{R} \\) such that there is some \\( M > 0 \\) for which \\( f(x) < \\log(x) \\) for all \\( x > M \\).",
"role": "user"
}
] | Yes |
[
{
"content": "Find the number of roots of the polynomial \\( q(z) = 2z^5 + 8z - 1 \\) within the annulus defined by \\( 1 < |z| < 2 \\).",
"role": "user"
}
] | 4 |
[
{
"content": "The line from the origin to the point \\((1, \\tan 75^\\circ)\\) intersects the unit circle at point \\(P\\). Determine the slope of the tangent line to the circle at point \\(P\\).",
"role": "user"
}
] | \sqrt{3} - 2 |
[
{
"content": "How many distinct ordered pairs of positive integers $(m,n)$ are there so that the sum of the reciprocals of $m$ and $n$ is $\\frac14$?",
"role": "user"
}
] | 5 |
[
{
"content": "Find the maximum value of \\( \\int_0^1 f^3(x) \\, dx \\) given that \\(-1 \\le f(x) \\le 1\\) and \\(\\int_0^1 f(x) \\, dx = 0\\).",
"role": "user"
}
] | \dfrac{1}{4} |
[
{
"content": "Let \\( f:D(0, 2)\\to\\,\\mathbb{C} \\) be an analytic function such that \\( f\\left(\\frac{2}{n}\\right)=\\frac{1}{n+1} \\) for \\( n = 2, 3,\\ldots \\). Calculate \\( f\\left(\\frac{3}{5}\\right) \\).",
"role": "user"
}
] | \dfrac{3}{13} |
[
{
"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... | Methods used by manufacturers to stop or reduce oxidation of food to prevent its spoilage include:
1. Addition of a poisonous chemical substance such as sulfur dioxide to kill microbial growths.
2. Removal of oxygen to prevent aerobic oxidation of the food substances.
3. Keeping food dry to remove the water necessary f... |
[
{
"content": "Calculate the determinant of the matrix \\([d_{i,j}]_{n \\times n}\\), where for every \\(i,j \\in \\{1,2,...,n\\}\\), \\(d_{i,j} = \\min\\{i,j\\}\\).",
"role": "user"
}
] | 1 |
[
{
"content": "Determine whether the following statement is true or false: If a function \\( f: D \\subseteq \\mathbb{R}^n \\to \\mathbb{R} \\) is continuous at a point \\( a \\in D \\), then there exists a \\( \\delta > 0 \\) such that \\( f \\) is continuous on the open ball \\( B(a, \\delta) \\).",
"role"... | B |
[
{
"content": "Consider a sequence ${(a_n)}_{n=1}^{\text{∞}}$ of complex numbers and a bijective map $\\sigma: \\mathbb{N} \\rightarrow \\mathbb{N}$. Is it possible for the infinite products $\\prod_{n=1}^{\\infty} (1+|a_n|)$ and $\\prod_{n=1}^{\\infty} (1+|a_{\\sigma(n)}|)$ to both equal a finite value $r$, whi... | No |
[
{
"content": "Find a number \\(a < 0\\) such that \\(\\int_a^0 \\frac{e^{\\frac{1}{x}}}{x^2} \\, dx = \\frac{e^{\\frac{1}{a}}}{a^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... | 0.40 |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty}\\sqrt[n]{n}\\cdot\\sqrt[n+1]{n+1}\\cdots\\sqrt[2n]{2n}$$",
"role": "user"
}
] | \infty |
[
{
"content": "Determine whether the imaginary part of the non-trivial zeros $z_n$ of the Riemann zeta function approaches infinity as $n$ approaches infinity.",
"role": "user"
}
] | Yes |
[
{
"content": "If \\( c = \\binom{99}{0} - \\binom{99}{2} + \\binom{99}{4} - \\binom{99}{6} + \\cdots + \\binom{99}{96} - \\binom{99}{98} \\), find \\( \\log_2{(-c)} \\).",
"role": "user"
}
] | 49 |
[
{
"content": "Given the equation \\(\\int_{0}^{\\frac{\\pi}{2}}(\\sin x + a\\cos x)^3dx - \\frac{4a}{\\pi - 2}\\int_{0}^{\\frac{\\pi}{2}}x\\cos x dx = 2\\), where the second integral evaluates to \\(\\frac{\\pi - 2}{2}\\), find the sum of the roots \\(a_1 + a_2 + a_3\\) that satisfy the equation.",
"role": ... | -\dfrac{3}{2} |
[
{
"content": "Determine the number of intersection points between the ellipses \\(\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1\\) and \\(\\frac{x^2}{b^2} + \\frac{y^2}{a^2} = 1\\), given that \\(a \\neq b\\).",
"role": "user"
}
] | 4 |
[
{
"content": "A teacher has 2010 exam questions and wants to divide them into three folders, each containing 670 questions. Each folder is to be given to a student who has solved all 670 questions in that folder. Determine the minimum number of students required in the class, given that for any question, at mos... | 3 |
[
{
"content": "Let $x_1,$ $x_2,$ $\\dots,$ $x_{101}$ be positive real numbers such that $x_1^2 + x_2^2 + \\dots + x_{101}^2 = 1.$ Find the maximum value of\n\\[x_1 x_2 + x_1 x_3 + \\dots + x_1 x_{101}.\\]",
"role": "user"
}
] | 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... | The heights attained by the masses m1 and m2 are 13.89m and 0.55m respectively. |
[
{
"content": "Determine which number is greater: $2^\\sqrt{2}$ or $e$. Justify your answer.",
"role": "user"
}
] | e |
[
{
"content": "Suppose $(a_n)$ and $(b_n)$ are positive sequences such that $(a_n)$ decreases monotonically to $0$. If $\\lim\\limits_{n\\to\\infty}\\frac{b_n}{a_n}=1$, does it imply that $(b_n)$ also decreases monotonically to $0$? Answer yes or no.",
"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... | $504,000 |
[
{
"content": "Given two compact surfaces, A and B, if their connected sum A#B is homeomorphic to the sphere \\( S^2 \\), determine if both A and B are individually homeomorphic to \\( S^2 \\).",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty}\\left(\\binom{n}{0}\\binom{n}{1}\\dots\\binom{n}{n}\\right)^{\\frac{1}{n(n+1)}} \\]",
"role": "user"
}
] | \sqrt{e} |
[
{
"content": "Let \\( p \\) and \\( q \\) be two distinct odd prime numbers. Find the multiplicative order of \\( 1 + pq \\) modulo \\( p^2 q^3 \\).",
"role": "user"
}
] | p q^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... | 4.94e6 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... | 1.42 V |
[
{
"content": "In the context of the Riemann integral, consider a function $f(x)$ defined over the interval $[0,1]$. What is the cardinality of the set of points where $f(x)$ is evaluated when computing the integral $\\int_{0}^{1} f(x) \\, dx$? Assume the continuum hypothesis holds.",
"role": "user"
}
] | \aleph_1 |
[
{
"content": "Determine whether the equation \\(x^{-17^{ 17^{-x}}}=17\\) has a solution.",
"role": "user"
}
] | Yes |
[
{
"content": "Show that the derivative of the function \\( f(M) = \\text{Tr}(M \\log M - M) \\) with respect to the positive definite matrix \\( M \\) is \\( \\log M \\).",
"role": "user"
}
] | \log 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... | 117.8 cm |
[
{
"content": "Calculate the sum of the number of fixed points of all permutations of the set \\( \\{1, 2, 3, \\ldots, n\\} \\) multiplied by the sign of the permutation, for \\( n \\geq 4 \\).",
"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... | 0.256 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... | molar per second |
[
{
"content": "Let \\( f(x) = \\sin x \\), and denote by \\( f^n = f \\circ f \\circ \\ldots \\circ f \\) the \\( n \\)-th iteration of the function \\( f \\). Find the limit (if it exists): \\[ \\lim_{n \\to \\infty} n \\cdot f^n(n^{-1}) \\]",
"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... | $101,000 |
[
{
"content": "Two distinct numbers are selected simultaneously and at random from the set $\\{1, 2, 3, 4, 5\\}$. What is the probability that the smaller one divides the larger one? Express your answer as a common fraction.",
"role": "user"
}
] | \frac{1}{2} |
[
{
"content": "In the expansion of \\((1+\\sqrt{11})^{214}\\) using the binomial theorem, determine the value of \\( k \\) for which the term \\(C_{214}^{k}(\\sqrt{11})^{k}\\) is maximized.",
"role": "user"
}
] | 165 |
[
{
"content": "Given the parabola \\( y^2 = 4x \\) with focus \\( F \\) and vertex \\( O \\), let \\( M \\) be a variable point on the parabola. Determine the maximum value of \\( \\frac{|MO|}{|MF|} \\). Choose from the following options:\n\nA. \\( \\frac{\\sqrt{3}}{3} \\)\n\nB. \\( \\frac{2\\sqrt{3}}{3} \\)\n\n... | B |
[
{
"content": "Only nine out of the original thirteen colonies had to ratify the U.S. Constitution in order for it to take effect. What is this ratio, nine to thirteen, rounded to the nearest tenth?",
"role": "user"
}
] | 0.7 |
[
{
"content": "Given the final amount $A$, the principal amounts $P_1, P_2, \\ldots, P_n$ invested at the beginning of each year, and the number of years $n$, determine the average annual rate of return $x\\%$ that satisfies the equation:\n\n\\[ A = P_1(1+x\\%)^1 + P_2(1+x\\%)^2 + \\cdots + P_n(1+x\\%)^n \\]\n\n... | 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... | ma^2/12 |
[
{
"content": "Evaluate the limit \\(\\lim_{h\\to0} \\frac{\\int_{0}^{3} e^{(x+h)^3}dx-\\int_{0}^{3} e^{x^3}dx}{h}\\) using the definition of the derivative.",
"role": "user"
}
] | e^{27} - 1 |
[
{
"content": "Given the recursive sequences defined by:\n\\[ B_n = \\sum_{i=0}^{n-1} (B_i B_{n-i-1}) \\quad \\text{with initial conditions} \\quad B_1 = 1, \\; B_0 = 1 \\]\nand\n\\[ M_n = \\sum_{i=1}^{n-1} (M_i M_{n-i}) \\quad \\text{with initial conditions} \\quad M_2 = 1, \\; M_1 = 1 \\]\nprove that \\( B_{n-... | B_{n-1} = M_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... | 6.64% |
[
{
"content": "Determine whether it is possible for a ring with identity to have $A^m$ isomorphic to $A^n$ as left $A$-modules, where $m \\neq n$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "In the 2008 Super Mega Ultra Galactic Thumbwrestling Championship (SMUG TWC), there are 2008 professional thumb-wrestlers. The championship is structured so that any set of three participants can discuss a bout between some pair of them. What is the minimum number of bouts, \\( m \\), required to ... | 1007012 |
[
{
"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... | 3.62 |
[
{
"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... | 2.824 |
[
{
"content": "Evaluate the limit \\( \\lim_{x \\to \\frac{\\pi}{2}}\\frac{b(1-\\sin x) }{(\\pi-2x)^2} \\) using trigonometric techniques, without applying L'Hôpital's rule.",
"role": "user"
}
] | \dfrac{b}{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... | $17,479 |
[
{
"content": "Determine whether the following statement is true or false: If $A$ is a dense subset of a metric space $X$ and $B$ is an open subset of $X$, then every point in $B$ is a limit point of $A \\cap B$. 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... | Na_2CO_3 + 4C + N_2 -> 2NaCN + 3CO |
[
{
"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... | 4.62 m/s at 64.9 degrees |
[
{
"content": "Calculate the limit: $$\\lim_{t \\rightarrow 0}\\frac{f(t^2+t)-f(t)}{t^2},$$ where the function \\(f(x)\\) is defined as: $$f(x)=\\begin{cases}\\exp\\left(-\\frac{1}{x^2}\\right) & x\\not=0 \\\\ 0& x=0\\end{cases}.$$ Does this limit equal zero?",
"role": "user"
}
] | 0 |
[
{
"content": "Let \\( F \\) be the left focus of the ellipse \\( E: \\frac{x^{2}}{3}+y^{2}=1 \\). A line \\( l \\) with a positive slope passes through point \\( F \\) and intersects \\( E \\) at points \\( A \\) and \\( B \\). Through points \\( A \\) and \\( B \\), lines \\( AM \\) and \\( BN \\) are drawn su... | \sqrt{6} |
[
{
"content": "Wanda is trying to locate the Fermat point $P$ of $\\triangle ABC$, where $A$ is at the origin, $B$ is at $(8,-1)$, and $C$ is at $(5,4)$ (the Fermat point is the point such that the sum of its distances from the vertices of a triangle is minimized). She guesses that the point is at $P = (4,2)$, a... | 8 |
[
{
"content": "In the Middle Sanctuary, there are four doors: X, Y, Z, and W. At least one of these doors leads to the Inner Sanctuary, while the others lead to a fire-breathing dragon. Eight priests, A, B, C, D, E, F, G, and H, each either a knight or a liar, provide the following statements:\n\nA: X is the doo... | X |
[
{
"content": "Determine whether the total variation of a function $F$, which is of bounded variation and continuous on $[a, b]$, is a continuous function.",
"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... | Banc One\nBalance Sheet\nAssets\nCash held as reserves (1,000,000 * 20%) 200,000\nCash 800,000\nTotal Assets 1,000,000\nLiabilities and Stockholders' Equity\nDeposit Liabilities 1,000,000 |
[
{
"content": "There exist nonzero integers $a$ and $b$ such that the quadratic\n\\[(ax - b)^2 + (bx - a)^2 = x\\]has one integer root and one non-integer root. Find the non-integer root.",
"role": "user"
}
] | \frac{1}{2} |
[
{
"content": "Evaluate the expression \\( \\sqrt{7\\sqrt{7\\sqrt{7\\sqrt{7\\cdots}}}} \\). Determine the value of this infinite nested radical.",
"role": "user"
}
] | 7 |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{\\frac{\\pi}{2}} M(a) \\sin (2a) \\ da \\), where \\( M(a) \\) is the maximum value of the function \\( f(x) = \\int_{0}^{\\pi} \\sin (x-t) \\sin (2t-a) \\ dt \\) for \\( a \\in \\mathbb{R} \\).",
"role": "user"
}
] | \dfrac{28}{27} |
[
{
"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... | 6.48e-3 J |
[
{
"content": "Determine the number of independent parameters required to specify all possible conditional probabilities for a set of $n$ events $X_1, X_2, ..., X_n$. Consider all probabilities such as $P(X_i)$, $P(X_i, X_j)$ for $i \\neq j$, $P(X_i | X_j)$ for $i \\neq j$, $P(X_i, X_j, X_k)$ for $i \\neq j \\ne... | 2^n - 1 |
[
{
"content": "Alex is at the candy store buying jellybeans. He wants to buy at least 100 jellybeans. He wants to buy the least amount such that he would have exactly $11$ leftover after dividing the jellybeans evenly among $13$ people. How many jellybeans should Alex buy?",
"role": "user"
}
] | 102 |
[
{
"content": "Evaluate $$\\lceil\\sqrt{5}\\rceil + \\lceil\\sqrt{6}\\rceil + \\lceil\\sqrt{7}\\rceil + \\cdots + \\lceil\\sqrt{29}\\rceil$$Note: For a real number $x,$ $\\lceil x \\rceil$ denotes the smallest integer that is greater than or equal to $x.$",
"role": "user"
}
] | 112 |
[
{
"content": "Evaluate the sum \\( \\sum^{\\infty}_{n=1} \\frac{n^2}{6^n} \\).",
"role": "user"
}
] | \dfrac{42}{125} |
[
{
"content": "Calculate the limit of the series as \\( n \\) approaches infinity: \\[ \\lim_{n\\to \\infty} \\frac{\\frac{2}{1}+\\frac{3^2}{2}+\\frac{4^3}{3^2}+\\dots+\\frac{(n+1)^n}{n^{n-1}}}{n^2}. \\]",
"role": "user"
}
] | \dfrac{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... | 1.97 revolutions |
[
{
"content": "A point has rectangular coordinates $(x,y,z)$ and spherical coordinates $\\left(2, \\frac{8 \\pi}{7}, \\frac{2 \\pi}{9} \\right).$ Find the spherical coordinates of the point with rectangular coordinates $(x,y,-z).$ Enter your answer in the form $(\\rho,\\theta,\\phi),$ where $\\rho > 0,$ $0 \\l... | \left( 2, \frac{8 \pi}{7}, \frac{7 \pi}{9} \right) |
[
{
"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... | CoO2 |
[
{
"content": "Suppose that $f: \\mathbb{R}^2 \\rightarrow \\mathbb{R}^2$ is a differentiable function. The best linear approximation of $f$ near $(a,b)$ is the affine function $$T(x,y) = Df(a,b)(x-a, y-b) + f(a,b),$$ where $Df(a,b)$ is the $2\\times2$ Jacobian matrix for $f$ at $(a,b)$. What should we expect th... | |\det Df(a,b)| |
[
{
"content": "Find the value of \\( x \\) such that \\( \\sin^{-1}\\left(x - \\frac{x^2}{2} + \\frac{x^4}{4} + \\cdots \\right) + \\cos^{-1}\\left(x^2 - \\frac{x^4}{2} + \\frac{x^6}{4} + \\cdots \\right) = \\frac{\\pi}{2} \\), where \\( 0 < |x| < 1.414 \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Find the function $f: \\mathbb{R}^+ \\to \\mathbb{R}^+$ that satisfies the equation $f(xf(y)) = yf(x)$ for all $x, y \\in \\mathbb{R}^+$ and the condition $\\lim_{x \\to \\infty} f(x) = 0$. Determine $f(x)$. \\( \\boxed{} \\)",
"role": "user"
}
] | \dfrac{1}{x} |
[
{
"content": "Evaluate the contour integral \\(\\oint_\\gamma\\frac{1}{z^2 + 4}\\:dz\\) where \\(\\gamma\\) is the circle \\(\\lvert z - i \\rvert = 2\\). Use a method to handle the singularity at \\(z = 2i\\) without using residues.",
"role": "user"
}
] | \dfrac{\pi}{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... | 12 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... | 2.23*10^-5 M |
[
{
"content": "For some positive integers $a$ and $b$, the product \\[\\log_a(a+1) \\cdot \\log_{a+1} (a+2) \\dotsm \\log_{b-2} (b-1) \\cdot\\log_{b-1} b\\]contains exactly $870$ terms, and its value is $2.$ Compute $a+b.$",
"role": "user"
}
] | 930 |
[
{
"content": "Evaluate the integral: \\[ \\int_{-2}^{2} \\frac{x^2}{1+5^x} \\, dx \\]",
"role": "user"
}
] | \dfrac{8}{3} |
[
{
"content": "Find the minimum value of\n\\[(12 - x)(10 - x)(12 + x)(10 + x).\\]",
"role": "user"
}
] | -484 |
[
{
"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... | $\frac {dv}{dm} = \frac gc -\frac {g}{e^{ct/m}} \left( \frac 1c + \frac tm \right)$ |
[
{
"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... | 170596.25 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... | CC = 0.81, cc = 0.01, Cc = 0.18 |
[
{
"content": "Suppose \\(f(z)\\) is an analytic function that maps the unit disc into itself and has zeros at \\(\\frac{i}{2}\\) and \\(-\\frac{i}{2}\\). Determine the largest possible value of \\(|f(1/2)|\\) and provide an example of a function \\(f\\) that attains this upper bound.",
"role": "user"
}
] | \dfrac{8}{17} |
[
{
"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... | 0.028 |
[
{
"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... | $5,750,000 |
[
{
"content": "Calculate the following limit, given that the function \\( f(x,y) \\) is continuous at \\((0,0)\\) and integrable in the vicinity of \\((0,0)\\): $$\\lim\\limits_{r\\to 0^+}\\dfrac{1}{\\pi r^2}\\iint\\limits_{|(x,y)|\\le r} f(x,y) \\, dS.$$",
"role": "user"
}
] | f(0,0) |
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