prompt list | ground_truth string |
|---|---|
[
{
"content": "Let $f(z)$ and $g(z)$ be entire functions such that $g(z)$ does not vanish anywhere in the complex plane and $|f(z)| \\le |g(z)|$ for all $z$. Determine which of the following statements is true:\n1) $f$ does not vanish anywhere.\n2) $f(0)=0$.\n3) $f$ is a constant function.\n4) $f(z)=C \\cdot g(z... | D |
[
{
"content": "Is the set of compactly supported functions $C^\\infty_c((0,1])$ dense in the space $\\{u \\in H^1(0,1) \\mid u(0)=0\\}$?",
"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... | 66272 |
[
{
"content": "Find the value of $a$ such that the following limit holds:\n\n$$\\lim_{n\\to\\infty}\\frac{1}{{2n+1}}\\sqrt{(25-a^2)n^2+2n+3}=2$$",
"role": "user"
}
] | \pm 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... | 4.18% |
[
{
"content": "Determine if there are any other semiprime numbers of the form obtained by concatenating the first even numbers, such as $2468101214$, up to a certain limit.",
"role": "user"
}
] | 2468101214 |
[
{
"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... | -0.3 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... | 600 |
[
{
"content": "Determine whether every bounded subset $A \\subset \\mathbb{R}$ with the property that $\\mu^*(A \\cap I) < \\mu(I)$ for all open intervals $I$ necessarily has measure $0$. Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"content": "Find the number of Hamiltonian cycles in the complete bipartite graph $K_{3,3}$. Provide a justification for your answer.",
"role": "user"
}
] | 6 |
[
{
"content": "Let \\( f = \\sum_{i \\ge m} a^i t^t \\) (for \\( m \\in \\mathbb{Z} \\)) be a formal Laurent power series. Consider its formal differential \\( df \\). What is the coefficient of degree \\(-1\\) of the differential form \\( \\frac{df}{f} \\)? Provide an explicit expression for this coefficient.",... | m |
[
{
"content": "Given a function \\( f(x) = f_1(x_0, x_1) + f_2(x_0, x_2) + \\cdots + f_T(x_0, x_T) \\), where each \\( f_i(x_0, x_i) \\) is non-negative and jointly concave over \\( (x_0, x_i) \\), consider the function \\( f_c(x) = f_1(c, x_1) + f_2(c, x_2) + \\cdots + f_T(c, x_T) \\) with \\( x_0 = c \\) fixed... | 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... | $26.30 |
[
{
"content": "Given that \\( \\lim_{x \\to a} \\frac{f(x) - f(a)}{x-a} = 2 \\), find \\( \\lim_{h \\to 0} \\frac{f(a+h) - f(a-h)}{h} \\) without using derivatives and integrals.",
"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... | 0.760 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... | 477 nm |
[
{
"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... | 3.900 ton ore |
[
{
"content": "Evaluate the integral \\( \\int_{-\\infty}^{\\infty}\\delta(y-x)\\delta(y-z)\\,dy \\).",
"role": "user"
}
] | \delta(x - z) |
[
{
"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... | 1,205,493.80 |
[
{
"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... | 2.403 N to the left |
[
{
"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... | 143701.28 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $21,495,333 |
[
{
"content": "Evaluate the following logarithmic expression and choose the correct answer from the options provided:\n\n$$\\frac{\\log _{10}\\left(1+\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{8}+...\\right)}{\\log _{10}\\left(2+\\frac{2}{3}+\\frac{2}{9}+...\\right)} \\cdot \\left(\\log _2\\left(3\\right)+\\log _3\\lef... | 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... | 20.68 |
[
{
"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... | 0.224*10^-3 V |
[
{
"content": "Let $Y_{n} > 0$ for all $n \\in \\mathbb{N}$, with $\\sum Y_{n} = +\\infty$. Prove that if $\\lim_{n \\rightarrow \\infty} \\frac{X_{n}}{Y_{n}} = a$, then $\\lim_{n \\rightarrow \\infty} \\frac{X_{1} + X_{2} + \\dots + X_{n}}{Y_{1} + Y_{2} + \\dots + Y_{n}} = a$.",
"role": "user"
}
] | a |
[
{
"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": "Let the point \\((a, b)\\) move along the curve defined by \\(\\arcsin x = \\arccos y\\). Consider the ellipse given by the equation \\(a x^{2} + b y^{2} = 1\\). This ellipse is outside, including the tangent case, of the circle \\(x^{2} + y^{2} = \\frac{2}{\\sqrt{3}}\\). Determine the range of \\... | C |
[
{
"content": "Find the limit of the complex function \\( \\arccos(z) \\) as \\( z \\) approaches \\(-1\\) from a direction where the imaginary part of \\( \\arccos(z) \\) is zero.",
"role": "user"
}
] | \pi |
[
{
"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... | {eq}T_f = \frac {7197 + 93.60m_w}{4.184m_w + 5.136}{/eq} |
[
{
"content": "Consider a metric space $(X,d)$ and a natural number $m$. Equip $X \\times X^m$ with the product Borel $\\sigma$-algebra and $\\{1,...,m\\}$ with the $\\sigma$-algebra $2^{\\{1,...,m\\}}$. Determine if the map \\( f: X \\times X^m \\to \\{1,...,m\\} \\) defined by \\( (x,x_1,...,x_m) \\mapsto \\mi... | Yes |
[
{
"content": "Suppose for any vector $y$ in $\\mathbb{R}^n$, there exists a vector $x$ such that the linear system $Ax = y$ is satisfied, where $A$ is an $n \\times n$ real matrix. Can we conclude that the rank of $A$ is $n$?",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | loss of $40,290 |
[
{
"content": "What is the 100th digit to the right of the decimal point in the decimal representation of $\\frac{13}{90}$?",
"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... | 9.511 |
[
{
"content": "Given a polynomial function $f(x)$ that satisfies the equation \\( f(x) \\cdot f\\left(\\frac{1}{x}\\right) + 3f(x) + 3f\\left(\\frac{1}{x}\\right) = 0 \\) and the condition \\( f(3) = 24 \\), find the value of \\( f(2) + f(-2) \\).",
"role": "user"
}
] | 18 |
[
{
"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... | You would need 26.2 mL of 0.50 M sodium acetate and 23.8 mL of 0.50 M acetic acid to prepare the buffer solution. |
[
{
"content": "Let $F$ be a finite set with two operations, addition \"+\" and multiplication \"\\cdot\", satisfying the field axioms except for the requirement that for every $a \\in F\\setminus\\{0\\}$, there exists an element $b\\in F$ such that $b\\cdot a =1$. Instead, it is required that for all $a,b \\in F... | F |
[
{
"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... | 1.25 m/s |
[
{
"content": "Compute the probability \\( P\\left [ X+Y > \\frac{1}{2}\\right ] \\) given the joint density function \\( f(x,y) = 1I(0<x<2)I(0<y<1-\\frac{x}{2}) \\).",
"role": "user"
}
] | \dfrac{7}{8} |
[
{
"content": "Evaluate the integral \\( \\int_{-\\infty}^{\\infty}\\frac{\\cos x}{1+x^2}dx \\) using methods from complex analysis.",
"role": "user"
}
] | \dfrac{\pi}{e} |
[
{
"content": "Find all positive integers \\( n \\) such that the sum-of-divisors function \\( \\sigma(n) \\) equals \\( n \\) itself. Are there any solutions other than \\( n = 1 \\)?",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the limit: $$\\lim_{t\\to\\infty}\\frac{t^3}{2e^{\\frac{t^2}{y^2}}}$$ where $y$ is a nonzero constant.",
"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... | -4/75 ms^-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... | 115.80 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... | $16,767.68 |
[
{
"content": "Determine the convergence and find the limit of the sequence \\( b_n = \\sqrt[n]{n^2 \\cdot 3^{4n}+4^{3n}} \\).",
"role": "user"
}
] | 81 |
[
{
"content": "Determine the space of sections of the vector bundle \\( \\pi: \\mathfrak{g} \\longrightarrow \\{e\\} \\), where \\( \\mathfrak{g} \\) is a Lie algebra and \\( \\{e\\} \\) is a single point. What is the space of sections of this bundle?",
"role": "user"
}
] | \mathfrak{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... | 39 kg LiOH |
[
{
"content": "Evaluate the integral \\( \\int_0^{\\pi} [\\cot(x)] \\, dx \\), where \\([\\cdot]\\) denotes the greatest integer function.",
"role": "user"
}
] | -\dfrac{\pi}{2} |
[
{
"content": "Determine the maximal ideal of the ring \\( \\frac{\\mathbb{R}[x]}{ (x^2)} \\).",
"role": "user"
}
] | (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... | 0.151 atm |
[
{
"content": "The total claim amount for a health insurance policy follows a distribution with the density function \\( f(x)= \\frac{1}{1000}e^{-\\frac{x}{1000}} \\) for \\( x>0 \\). The premium for the policy is set at the total expected claim amount plus 100. If 100 policies are sold, calculate the approximat... | 0.1587 |
[
{
"content": "An ant is sitting at the center of a circle with a radius of 3 meters. Every minute, the ant randomly chooses a direction and moves 1 meter in that direction. On average, how many minutes does it take for the ant to leave the circle? Assume that if the ant reaches the edge of the circle within a m... | 9 |
[
{
"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... | 54959.31 |
[
{
"content": "Given the differential equation \\((1-x^2)y''+2xy'-2y = 0\\) for \\(-1 < x < 1\\), and a known solution \\(y_1(x) = x\\), use the reduction of order method to find a second linearly independent solution \\(y_2(x)\\).",
"role": "user"
}
] | 1 + x^2 |
[
{
"content": "Calculate the flux of the vector field \\( \\bar{F} = \\langle x, y, x \\rangle \\) across the plane defined by \\( 1 - (x + y + z) = 0 \\) in the first octant, using the unit normal vector \\( \\langle \\frac{1}{\\sqrt{3}}, \\frac{1}{\\sqrt{3}}, \\frac{1}{\\sqrt{3}} \\rangle \\).",
"role": "u... | \dfrac{1}{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... | (sigma_(1s))^2(sigma_(1s)^*)^2(sigma_(2s))^2(sigma_(2s)^*)^2(pi_(2p_x))^2(pi_(2p_y))^2(sigma_(2p_z))^2(pi_(2p_x)^*)^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... | 10.40 |
[
{
"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... | 9.59e6 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... | 25.3 m/s |
[
{
"content": "Determine whether the following inequality is true for $u \\in W^{s,p}_0(\\Omega)$:\n\\[ \\|u\\|_{L^p(\\Omega)} \\leq c \\left(\\iint_{\\Omega\\times\\Omega}\\frac{|u(x)-u(y)|^p}{|x-y|^{n+ps}}\\,dx\\,dy\\right)^{1/p} \\]\nwhere $W^{s,p}_0(\\Omega)$ is the closure of $C^{\\infty}_c(\\Omega)$ in $(W... | True |
[
{
"content": "Determine whether every non-zero Lie algebra over an algebraically closed field has an ad-nilpotent element.",
"role": "user"
}
] | Yes |
[
{
"content": "Calculate the Jacobian determinant for the transformation from variables $(u, v)$ to $(p, r)$, where $u = p + ir$ and $v = p - ir$.",
"role": "user"
}
] | \dfrac{i}{2} |
[
{
"content": "Let \\(|A| = 5\\). How many closed binary operations on \\(A\\) are commutative? Provide your answer as a power of 5.",
"role": "user"
}
] | 5^{15} |
[
{
"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... | 0.474 rev/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... | 1.12 L |
[
{
"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 and +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... | 0.06 M |
[
{
"content": "Let $X \\sim \\mathcal{N}(0,1)$ and $Y \\sim \\text{Unif}(0,1)$. Assuming $X$ and $Y$ are independent, calculate $\\text{Var}(XY)$. Express your answer in terms of integrals if necessary.",
"role": "user"
}
] | \dfrac{1}{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... | 12.24% |
[
{
"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... | 0.081 L |
[
{
"content": "Is there an algorithm to determine the rank of a group generated by side-pairing isometries of a compact convex polyhedron in \\( \\mathbb{H}^3 \\)?",
"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... | 15.39 grams |
[
{
"content": "Let \\( f(x) \\) be a function that is thrice differentiable on \\((a,b)\\). Define the function \\( \\phi(x) = f(b) - f(x) - (b-x)f'(x) - \\frac{(b-x)^2}{2}f''(x) - (b-x)^3 \\lambda \\). Suppose \\( \\phi(x) \\) satisfies all the conditions of Rolle's Theorem on \\([a,b]\\). If there exists some ... | \dfrac{1}{6} |
[
{
"content": "Determine the number of non-isomorphic graphs with 30 vertices and 432 edges.",
"role": "user"
}
] | 5 |
[
{
"content": "Given an affine subspace $H \\subset \\mathbb{R^3}$ defined by the equation $x_1 + 2x_2 - x_3 = 2$, and a line $L$ in $\\mathbb{R^3}$ with the parametric equation $(0,1,1) + t(2,0,1)$, find the equation of the plane $f(H_1)$, where $H_1$ is another plane defined by the equation $2x_1 - x_2 + x_3 =... | 7x_1 - x_2 + 2x_3 = 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... | W = \int_{0}^{15} \frac{62.5 (1000)}{3}\pi dx
Units: lb * ft |
[
{
"content": "Compute the limit: $$\\lim _{x \\rightarrow \\frac{\\pi}{2}} \\frac{e^{\\sin 2 x}-e^{\\tan 2 x}}{\\ln \\left(\\frac{2 x}{\\pi}\\right)}$$",
"role": "user"
}
] | -2\pi |
[
{
"content": "Determine the values of \\(x \\in \\mathbb{R}\\) for which the series \\(\\sum_{n=1}^{\\infty} (-1)^n \\frac{x^2+n}{n^2}\\) converges.",
"role": "user"
}
] | \mathbb{R} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.0e+8 mph |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 16 m/s^2 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 8.49% |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\to 0} \\left(\\ln\\left(\\frac{\\cos x}{1+\\sin x}\\right)\\right)^{\\frac{1}{x}}$$",
"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... | 0.597 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... | /cimages/multimages/16/octa2740634590884863664.jpg |
[
{
"content": "Evaluate the expression \\( \\frac{f(3)f(5)f(7)f(9)f(11)f(13)}{f(2)f(4)f(6)f(8)f(10)f(12)} \\) where \\( f(n) = n^4 + n^2 + 1 \\).",
"role": "user"
}
] | 61 |
[
{
"content": "Evaluate the integral \\( \\int_{-\\infty}^\\infty (f(x+1) - f(x)) \\, dx \\) given that \\( \\lim_{x \\to \\infty} f(x) = 2 \\) and \\( \\lim_{x \\to -\\infty} f(x) = 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... | 2370 m^3 |
[
{
"content": "What is the value of \\( \\ln \\left(e^{2 \\pi i}\\right) \\)?",
"role": "user"
}
] | 0 |
[
{
"content": "Find all incongruent roots modulo 13 of the polynomial \\(x^4 + x^2 + x + 1\\).",
"role": "user"
}
] | 6 |
[
{
"content": "Let \\( f: \\mathbb{R} \\to \\mathbb{R} \\) be a continuous function such that the equation \\( f(x) = x \\) has no real solution. Is it true that the equation \\( f(f(x)) = x \\) also has no real solution?",
"role": "user"
}
] | 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... | 501 pm |
[
{
"content": "Determine whether any line on the Riemann sphere can be transformed into a geodesic using an appropriate transformation. Answer with 'Yes' or 'No'.",
"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... | 36.03 mm |
[
{
"content": "Determine the type and order of the singularities of the function \\( f(z) = \\frac{\\sin z}{z^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... | 275.6 nm |
[
{
"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... | 0.820% |
[
{
"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": "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... | P: $13 million
Q: $12 million
R: $0 million |
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