prompt list | ground_truth string |
|---|---|
[
{
"content": "Let $E$ be a compact topological space. Consider the set $C(E) = \\{f: E \\rightarrow \\mathbb{C} \\text{ continuous}\\}$, which is a subset of $B(E) = \\{f: E \\rightarrow \\mathbb{C} \\text{ continuous and bounded}\\}$. Given that $B(E)$ is a Banach space, is $C(E)$ also a Banach space?",
"r... | Yes |
[
{
"content": "Determine the values of $p$ for which the integral \\( \\int_1^\\infty \\frac{1}{x^p(1+\\ln x)} \\, dx \\) converges.",
"role": "user"
}
] | p > 1 |
[
{
"content": "What is the sum of the digits when $527_{10}$ is converted to base 4? (When you add the digits, express the sum in base 10.)",
"role": "user"
}
] | 8 |
[
{
"content": "The triangle \\( \\triangle ABC \\) is inscribed in the unit circle, and the internal angle bisectors of angles \\( A \\), \\( B \\), and \\( C \\) intersect the circle again at points \\( A_{1} \\), \\( B_{1} \\), and \\( C_{1} \\) respectively. Find the value of \n$$\n\\frac{A A_{1} \\cdot \\cos... | A |
[
{
"content": "Evaluate the expression: \\[\\left(\\frac{1}{2}+\\cos \\frac{\\pi}{20}\\right)\\left(\\frac{1}{2}+\\cos \\frac{3\\pi}{20}\\right)\\left(\\frac{1}{2}+\\cos \\frac{9\\pi}{20}\\right)\\left(\\frac{1}{2}+\\cos \\frac{27\\pi}{20}\\right)\\]",
"role": "user"
}
] | \dfrac{1}{16} |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\rightarrow 0}\\frac{(1+x)^{1/x}-e+\\frac{ex}{2}}{x^2}$$",
"role": "user"
}
] | \dfrac{11e}{24} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1380 rpm |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Magnesium (Mg) |
[
{
"content": "Determine the cardinality of the set \\( T = \\{ X \\in P(\\mathbb{Q}) \\mid X \\cup \\mathbb{N} = \\mathbb{Q} \\} \\).",
"role": "user"
}
] | 2^{\aleph_0} |
[
{
"content": "Let $R = C[x_1, \\ldots, x_n]$ and $M$ be a maximal ideal of $R$ such that $R/M$ is a finite-dimensional $C$-algebra. Is $R/M^n$ for $n > 1$ also a finite-dimensional $C$-algebra? Justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 146 degrees Celsius |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to 0} \\int_0^x \\frac{\\sin t}{t} \\mathrm{d}t$$ without using L'Hopital's rule or series expansion.",
"role": "user"
}
] | 0 |
[
{
"content": "Calculate Spearman's rho for the comonotonicity copula, defined as \\( Cm(u_1, u_2) = \\min(u_1, u_2) \\). Spearman's rho is given by the formula:\n\\[ \\rho = 12 \\int_{0}^{1}\\int_{0}^{1} \\min(u_1, u_2) \\, du_1 \\, du_2 - 3 \\]\nSolve the integral to find the value of \\( \\rho \\).",
"rol... | 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... | $49,085.25 |
[
{
"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... | -$9.46 |
[
{
"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... | Kp = (P_CH2Cl2^4 * P_H2^2) / (P_CH3Cl^4 * P_Cl2^2) |
[
{
"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... | $3.08 |
[
{
"content": "Given the functional equation \\( f\\left(x + \\frac{1}{x}\\right) = x^3 + x^{-3} \\), find the expression for \\( f(x) \\).",
"role": "user"
}
] | x^3 - 3x |
[
{
"content": "Determine the value of $b$ for which the series \\( \\sum_{r=2}^{\\infty}\\left(\\frac{8}{3(br+1)}-\\frac{3}{2(r+1)}+\\frac{1}{6(r-1)}\\right) \\) is convergent using the integral test.",
"role": "user"
}
] | 2 |
[
{
"content": "What cardinal number is associated with the ordinal number \\( \\omega + 1 \\) if \\( \\aleph_0 \\) is associated with \\( \\omega \\)?",
"role": "user"
}
] | \aleph_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... | 1.98 X 10^22 electrons |
[
{
"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... | 202 |
[
{
"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... | 4972 |
[
{
"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... | 0.0800 M HCl |
[
{
"content": "Determine whether the following statement is true or false: For any increasing function \\( f : [a, b] \\rightarrow \\mathbb{R} \\), the set of its discontinuities has measure zero. Provide a proof if true, or a counterexample if false.",
"role": "user"
}
] | True |
[
{
"content": "Calculate the coefficient of $x^{20}$ in the generating function $F(x) = \\frac{(x^2-x^{10})(1-x^{10})^3}{(1-x)^4}$.",
"role": "user"
}
] | 552 |
[
{
"content": "How many three digit numbers are there such that the digits are non-zero, the tens digit is a multiple of the hundreds digit, the units digit is a multiple of the tens digit, and none of the digits are the same?",
"role": "user"
}
] | 7 |
[
{
"content": "Determine the function \\( y(x) \\) that makes the integral \\( \\int_{0}^{1} (x^{2} + y'^2) \\, dx \\) stationary, subject to the constraints \\( \\int_{0}^{1} y^2 \\, dx = 2 \\), \\( y(0) = 0 \\), and \\( y(1) = 0 \\).",
"role": "user"
}
] | 2 \sin(\pi 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... | 3.66 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... | 592 nm |
[
{
"content": "Compute the intersection number \\( \\iota^*\\mathcal{O}(2,3) \\cdot \\iota^*\\mathcal{O}(2,3) \\) for the smooth subvariety \\( \\iota:X=V_+(f)\\subset\\mathbb{P}^2\\times\\mathbb{P}^1 \\) where \\( f\\in H^0(\\mathcal{O}(1,2)) \\).",
"role": "user"
}
] | 20 |
[
{
"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... | $19,020 |
[
{
"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... | 400 K |
[
{
"content": "Suppose that the joint distribution of $X$ and $Y$ is the uniform distribution on the circular disk defined by $x^2 + y^2 < 1$. Find the conditional expectation $E(X|Y)$. ",
"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... | Cost of debt |
[
{
"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... | 1833 |
[
{
"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... | In-113 |
[
{
"content": "Determine whether \\(f = \\Theta(g)\\), \\(f = \\omega(g)\\), or \\(f = o(g)\\) for the following functions:\n\n\\[f(n)=\\left( \\frac {\\ln(n^2+1)}{\\ln(n)}\\right)^{\\ln(n^2)}, \\quad g(n)= 2^{\\sqrt{\\ln(n)}}\\]",
"role": "user"
}
] | f = \omega(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... | 0.577 |
[
{
"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... | 0.00290 M |
[
{
"content": "Determine whether a surjective linear operator on $\\mathbb{Z}^n$ is always bijective. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.72e8 A |
[
{
"content": "Given a real polynomial $P(x)$ of degree 4, satisfying $P(1) = 0$ and having relative extrema at $x = 2$ and $x = 3$ with the same value of 3, compute $P(4)$ without explicitly finding the polynomial $P$. Provide your answer as a single number.",
"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... | 5590.5 Hz |
[
{
"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... | 0.947 m to the left of Q1 |
[
{
"content": "Determine all non-negative integers \\( n \\) for which there exists a non-periodic function \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) satisfying the functional equation \\( f(x+1) + f(x-1) = \\sqrt{n} f(x) \\) for all \\( x \\in \\mathbb{R} \\).",
"role": "user"
}
] | n \geq 4 |
[
{
"content": "Determine the cardinality of the set of all epsilon numbers less than $\\omega_1$. Provide your answer as a cardinal number.",
"role": "user"
}
] | \aleph_1 |
[
{
"content": "Evaluate the integral \\( I = \\int_{|z|=1} \\left( \\tan z + \\frac{\\csc z}{z} \\right) \\, dz \\) without using the residue theorem.",
"role": "user"
}
] | 0 |
[
{
"content": "Find the trace of a $227 \\times 227$ matrix $A$ with entries in $\\mathbb{Z}_{227}$, given that all its eigenvalues are distinct.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the sum \\( \\sum_{k=0}^{2013} 2^k a_k \\) for the sequence \\( a_0, a_1, a_2, \\ldots, a_n \\) defined by \\( a_0 = 2013 \\) and \\( a_n = -\\frac{2013}{n} \\sum_{k=0}^{n-1} a_k \\) for \\( n \\geq 1 \\).",
"role": "user"
}
] | -2013 |
[
{
"content": "If $a,b \\in \\mathbb{Z}$ are such that $\\gcd(a,b)=1$ and $a^2+3b^2$ is a cube in $\\mathbb{Z}$, determine whether $a+\\sqrt{-3}b$ and $a-\\sqrt{-3}b$ are both cubes in $\\mathbb{Z}[\\sqrt{-3}]$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Calculate the value of $i^i$.",
"role": "user"
}
] | e^{-\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... | 6.67e-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... | $471 |
[
{
"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... | overstimulation caused by Glutamate |
[
{
"content": "A hot air balloon is ascending at a rate of \\(10\\frac{ft}{sec}\\). How fast is the distance from the balloon to the horizon increasing when the balloon is \\(1,000\\) feet high? Assume the Earth is a sphere with a radius of \\(4000\\) miles.",
"role": "user"
}
] | 1028 |
[
{
"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... | $5.38 |
[
{
"content": "Determine the minimum number of throws required to ensure that no boxes are empty when distributing balls into boxes.",
"role": "user"
}
] | n |
[
{
"content": "Evaluate the limit: \\[ \\lim_{q\\to\\infty} \\int_{0}^{\\infty} \\left(e^{i(q+i\\delta)r} - e^{-i(q-i\\delta)r}\\right) dr \\]",
"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... | 6.3 |
[
{
"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... | 414 m/s^2 |
[
{
"content": "Petya thinks of a natural number \\( x \\) such that the sum of its digits is 2012. Vasya can select any natural number \\( a \\) and learn the sum of the digits of the number \\( |x-a| \\). What is the minimum number of moves Vasya needs to make to guarantee determining \\( x \\)?",
"role": "... | 2 |
[
{
"content": "Let \\(f(x) = x^4 + ax^3 + bx^2 + cx + d\\) be a polynomial with real coefficients and real roots. Given that \\(|f(\\iota)| = 1\\), find the value of \\(a + b + c + d\\).",
"role": "user"
}
] | 0 |
[
{
"content": "What is the probability that for two randomly chosen natural numbers \\(a\\) and \\(b\\), the fraction \\(\\frac{a}{b}\\) is a natural number?",
"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... | -$585.19 |
[
{
"content": "Determine whether the following statement is true or false: A closed and bounded subset of a complete metric space is compact. Provide a justification for your answer.",
"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... | 11.62% |
[
{
"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... | 34871 g |
[
{
"content": "Determine the maximum number of distinct sets that can be generated from a single set \\( B \\subseteq \\mathbb{R} \\) by taking a finite number of closures and interiors.",
"role": "user"
}
] | 7 |
[
{
"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... | 1.26e+4g |
[
{
"content": "What is the specific term for a path that connects two points on a torus by forming a straight line when the torus is represented as a square with opposite edges identified?",
"role": "user"
}
] | geodesic |
[
{
"content": "Altitudes $\\overline{AX}$ and $\\overline{BY}$ of acute triangle $ABC$ intersect at $H$. If $\\angle AHB = 132^\\circ$, then what is $\\angle ACB$?",
"role": "user"
}
] | 48^\circ |
[
{
"content": "Given a sequence of non-negative continuous functions $f_n(x)$ on $[0,1]$ for all $n \\geq 1$, where $\\lim_{n \\to \\infty} \\int^1_0 f_n (x)dx=0$, determine which of the following statements is necessarily true:\n\nA. $f_n$ converges to $0$ uniformly on $[0,1]$.\nB. $f_n$ may not converge unifor... | D |
[
{
"content": "Determine the value of $k$ such that the equation \\( \\left(\\binom{n}{2}-\\binom{k+1}{3}\\right)x=\\binom{k+1}{2}-1 \\) has no solution for \\( x \\in \\mathbb{N} \\), where \\( n \\in \\mathbb{N} \\) and \\( n > 4 \\).",
"role": "user"
}
] | 4 |
[
{
"content": "Suppose that $3$ is a factor of $a$, $a$ is a divisor of $12$, and $a$ is positive. What is the number of possible values of $a$?",
"role": "user"
}
] | 3 |
[
{
"content": "Find all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) such that \\( f\\left(x^2+f(y)\\right) = y + f(x)^2 \\) for all real numbers \\( x \\) and \\( y \\).",
"role": "user"
}
] | f(x) = 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... | 9.9% |
[
{
"content": "Determine the one-point compactification of the space $S^n \\times \\mathbb{R}$. Provide your answer in terms of a well-known topological space.",
"role": "user"
}
] | S^{n+1} |
[
{
"content": "The figure below consists of four semicircles and the 16-cm diameter of the largest semicircle. What is the total number of square cm in the area of the two shaded regions? Use 3.14 as an approximation for $\\pi$, and express your answer as a decimal to the nearest tenth.\n\n[asy]import graph;\nsi... | 62.8 |
[
{
"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... | The accounting concept violated is the matching principle. The matching principle requires that expenses be matched with the revenues they generate. In this case, the interest expense on the loan should be matched with the period in which the warehouse is used to generate revenue. The interest expense is not matched with the revenues generated by the warehouse if it reduces the cash and bank loan amount at the time of repayment of the loan.
The correct treatment is to debit the warehouse for $660,000 and credit the bank loan payable for $600,000 and interest payable for $60,000. This would ensure that the interest expense is matched with the period in which the warehouse is used to generate revenue and fulfill the matching principle. |
[
{
"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... | 0.0347 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... | 820080 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... | Deep ocean trench and continental volcanic mountains |
[
{
"content": "Does Lebesgue's differentiation theorem hold at every point for a continuous function $f: \\mathbb{R}^n \\to \\mathbb{R}$, such that \\[\\lim_{r\\to0}\\frac{1}{|B(x,r)|}\\int_{B(x,r)}f(y) \\, dy = f(x)\\] for every point $x$ in $\\mathbb{R}^n$, where $B(x,r)$ denotes the ball of radius $r$ centere... | 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... | Product F: $29.05 per unit
Product G: $8.16 per unit |
[
{
"content": "Let \\( X \\) be a Hilbert space with scalar product \\((\\cdot,\\cdot)\\). For a given angle \\( \\varphi' \\in [0, \\pi) \\) and a unit vector \\( v' \\in X \\), is there a unit vector \\( w' \\in X \\) such that \\((v', w') = \\cos(\\varphi')\\)?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the range of a uniform distribution from which N random numbers are drawn such that, on average, half of the unique numbers drawn are smaller than M/2. Assume that only unique numbers smaller than M/2 are counted.",
"role": "user"
}
] | [0, 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... | 2.24 |
[
{
"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... | v_1c_1=v_2c_2 |
[
{
"content": "Suppose \\(A \\subset \\mathbb{R}\\) is the domain of a function \\(f(x)\\) on which it is uniformly continuous. The function \\(f(x)\\) satisfies the property that for each \\(x_0 \\in \\overline{A} - A\\), \\(\\lim_{x \\to x_0^-} f(x) = \\lim_{x \\to x_0^+} f(x)\\). Extend the domain of \\(f(x)\... | Yes |
[
{
"content": "Given that a valid loop (derangement) has already been formed with 33 participants, calculate the probability of forming a perfect loop.",
"role": "user"
}
] | \dfrac{e}{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... | 0.017 m, 17 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... | 27.2 m/s |
[
{
"content": "Find the second derivative of the function given by the infinite series:\n\\[ f(x) = \\sum \\limits_{n=0}^{\\infty} \\frac{(-1)^{n+1}2x^{2n+1} }{(2n+1)!} \\]",
"role": "user"
}
] | 2 \sin x |
[
{
"content": "Given that \\( f(x) = 2x^3 + 7x - 1 \\) and \\( g(x) \\) is the inverse of \\( f(x) \\), find \\( g'\\left(\\frac{191}{4}\\right) \\) without explicitly solving for \\( x \\) in the cubic equation.",
"role": "user"
}
] | \dfrac{2}{89} |
[
{
"content": "Does there exist a Hilbert space $\\mathcal{H}$ such that a sequence $\\{ x_n \\} \\subset \\mathcal{H} $ is weakly convergent, but not convergent in the Hilbert space norm?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the values of $n$ for which all the nonzero roots of the polynomial $(z+1)^n - z^n - 1$ lie outside the unit circle.",
"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... | v(t) = sin(t) + 0.0206 |
[
{
"content": "Let $X_i,\\ i \\geq 1$ be i.i.d. discrete random variables with mean $\\mu$ and variance $\\sigma^2.$ Let $k > 1.$ Define the sequence \\[ Y_n := \\dfrac {X_1 X_2 \\cdots X_k + X_2 X_3 \\cdots X_{k+1} + \\cdots + X_{n-k+1} X_{n-k+2} \\cdots X_n} {n}. \\] Find \\(\\lim\\limits_{n \\to \\infty} n^{\... | 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... | 1.56 * 10^-7 M |
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