prompt list | ground_truth string |
|---|---|
[
{
"content": "Find the minimal distance between the origin and the intersection of the surfaces given by the equations \\(x^2 = 2yz\\) and \\(x^2 + 3y^2 + 2z^2 = 30\\).",
"role": "user"
}
] | \sqrt{10} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "If Luis pulls straight down on the end of a wrench that is tilted 30 degrees above the horizontal and is 39 cm long, what force must he apply to exert a torque of -13 Nm?",
"role": "user"
}
] | 38.24 N |
[
{
"content": "Let $G$ be a group of order $p^n$, where $p$ is a prime number, and let $H$ be a subgroup of $G$ with order $p^{n-2}$ that is not normal in $G$. Determine whether the index $[G:N_G(H)]$ is equal to $p$.",
"role": "user"
}
] | p |
[
{
"content": "Let \\( Q \\in \\mathbb{Z}[x] \\) be a polynomial that defines an injective function from \\( \\mathbb{Z} \\) to \\( \\mathbb{Z} \\). Determine whether there exists a prime \\( p \\) such that \\( Q \\) defines an injective function from \\( \\mathbb{Z}/p\\mathbb{Z} \\) to \\( \\mathbb{Z}/p\\mathbb{Z} \\).",
"role": "user"
}
] | Yes |
[
{
"content": "Given a function $f$ such that $f^2$ is Riemann integrable, determine whether $f$ is always Riemann integrable. Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "A company borrows $400,000 in order to build a new office building.\n\nThe original terms of the loan were that it would be paid off in 120 monthly payments at a nominal interest rate of 12%, compounded monthly.\n\nHowever after 40 payments the company wishes to pay off the loan immediately.\n\nHow much will that payment be?",
"role": "user"
}
] | $314,994.42 |
[
{
"content": "Given that $y_1 = \\sin(t^2)$ is a solution to the differential equation $t y'' - y' + 4t^3 y = 0$, use the method of reduction of order to find a second linearly independent solution.",
"role": "user"
}
] | \cos(t^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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "A solid disk with a radius of #5 m# and mass of #4 kg# is rotating on a frictionless surface. If #3 W# of power is used to increase the disk's rate of rotation, what torque is applied when the disk is rotating at #4 Hz#?",
"role": "user"
}
] | 0.12 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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "Determine the pH at the equivalence point when {eq}25.00\\ mL\n\n{/eq} of {eq}\\rm 0.1056\\ M\\ CH_3COOH\n\n{/eq} is titrated with {eq}\\rm 0.1056\\ M\\ NaOH\n\n{/eq}.",
"role": "user"
}
] | 8.73 |
[
{
"content": "Determine the radius of convergence for the power series solution of the differential equation \\( u''+(x+1)u'-u=0 \\) with initial conditions \\( u(-1)=2 \\) and \\( u'(-1)=0 \\).",
"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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "An object is placed {eq}\\rm\n\n12.0\\ cm\n\n{/eq} to the left of a diverging lens of focal length, {eq}\\rm\n\n-6.00\\ cm\n\n{/eq}. A converging lens of focal length, {eq}\\rm\n\n12.0\\ cm\n\n{/eq} is placed a distance {eq}\\rm\n\nd\n\n{/eq} to the right of the diverging lens. Find the distance, {eq}\\rm\n\nd\n\n{/eq}, so that the final image is at infinity.",
"role": "user"
}
] | 8.00 cm |
[
{
"content": "Evaluate the integral \\( \\int_{\\Lambda}\\left(\\sin\\frac{1}{z}+\\sin^2\\frac{1}{z}\\right)dz \\), where \\( \\Lambda \\) is a circle centered at zero in the complex plane excluding zero. Choose the correct value from the following options:\n\n(a) \\( 0 \\) \n(b) \\( \\pi i \\) \n(c) \\( 2\\pi i \\) \n(d) \\( 4\\pi i \\)",
"role": "user"
}
] | 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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "Consider the following cash flows:\n\nYear : Cash Flow\n\n0 : -$33,000\n\n1 : $14,900\n\n2 : $16,800\n\n3 : $12,300\n\nWhat is the IRR of the above set of cash flows?",
"role": "user"
}
] | 16.40% |
[
{
"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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "A point moves in a straight line so that its displacement x (in meter) at time t (in seconds) is given by {eq}x^{2} = t^{2} + 1\n\n{/eq}. What is its acceleration (in {eq}\\frac{m}{s^{2}}\n\n{/eq}) at time t?",
"role": "user"
}
] | \frac{1}{(\sqrt{t^{2}+1})^3} ~\rm{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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "Tyler Company applies manufacturing overhead to production at the rate of $5.7 per direct labor hour and ended August with $13,700 under-applied overhead. Actual manufacturing overhead incurred for August amounted to $131,690.\n\nHow many direct labor hours did Tyler Company incur during August?",
"role": "user"
}
] | 20,700 direct labor hours |
[
{
"content": "Suppose $M$ is a complex manifold and $\\Omega$ is a bounded pseudoconvex domain in $M$. Let $u:M\\setminus\\Omega\\to\\mathbb{R}$ be a pluriharmonic function. Determine whether $u$ can be extended as a pluriharmonic function to the entire manifold $M$. Assume $\\dim_{\\mathbb{C}}(M)\\geq 2$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the conditions under which the identity \\(A^2 - B^2 = (A+B)(A-B)\\) holds true for matrices \\(A\\) and \\(B\\).",
"role": "user"
}
] | AB = BA |
[
{
"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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "What type of solvent could be used to dissolve an oil stain?",
"role": "user"
}
] | A nonpolar solvent such as hexane or carbon tetrachloride |
[
{
"content": "Let $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ be a differentiable function such that $f'(x) = [f(x)]^2$ for all $x \\in \\mathbb{R}$ and $f(0) = 0$. Prove that $f(x) = 0$ for all $x \\in \\mathbb{R}$.",
"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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "Eco Wet, Inc., a manufacturer of gears for lawn sprinklers, is thinking about adding a new fully automated machine. This machine can produce gears that the company now produces on its third shift. The machine has an estimated useful life of ten years and will cost $500,000. The residual value of the new machine is $50,000. Gross cash revenue from the machine will be about $420,000 per year, and related operating expenses, including depreciation, should total $400,000. Depreciation is estimated to be $80,000 annually. Management has decided that only capital investments that yield at least an 8 percent return will be accepted.\n\nUsing the accounting rate-of-return method, decide whether the company should invest in the machine.",
"role": "user"
}
] | 7.27% |
[
{
"content": "Find the limit of the integral \\( \\lim_{n\\rightarrow \\infty} \\int_0^\\infty n f(x) e^{-n|x|} \\, dx \\), where \\( f: [0,\\infty) \\rightarrow \\mathbb{R} \\) is Lebesgue integrable and \\( \\lim_{x\\rightarrow 0} f(x) = 2016 \\), using the Dominated Convergence Theorem.",
"role": "user"
}
] | 2016 |
[
{
"content": "Let \\(a_n\\) and \\(b_n\\) be two sequences such that \\(a_n \\xrightarrow{n\\to\\infty} a > 1\\) and \\(b_n \\xrightarrow{n\\to\\infty} b > 1\\). Find \\(\\lim_{n\\to\\infty}\\sqrt[n]{a_n^n+b_n^n+2018}\\).",
"role": "user"
}
] | \max\{a, 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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "Consider a position consisting of a $200,000 investment in asset A and a $150,000 investment in asset B. Assume that the daily volatilities of both assets are 1% and that the coefficient of correlation between their returns is 0.35. What is the 5-day 99% value at risk for the portfolio?",
"role": "user"
}
] | $15,168 |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty}\\left(\\frac{1}{n}+\\frac{1}{2^n-1}\\sum_{k=1}^{n} \\frac{n \\choose k}{k}\\right)$$",
"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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "Calculate the wavelength, in nanometers, of the spectral line produced when an electron in a hydrogen atom undergoes the transition from the energy level n = 7 to the level n = 1.",
"role": "user"
}
] | 93 nm |
[
{
"content": "Find the quadratic factor \\( f(x) = x^2 + bx + c \\), where \\( b, c \\in \\mathbb{R} \\), that is common to both quartic polynomials \\( x^4 + 6x^2 + 25 \\) and \\( 3x^4 + 4x^2 + 28x + 5 \\). Determine \\( f(x) \\).",
"role": "user"
}
] | x^2 - 2x + 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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "The Dakota Corporation had a 2015 taxable income of $33,000,000 from operations after all operating costs but before (1) interest charges of $9,300,000; (2) dividends received of $860,000; (3) dividends paid of $5,800,000; and (4) income taxes.\n\nWhat are Dakota's average and marginal tax rates on taxable income?",
"role": "user"
}
] | The marginal tax rate is 35%, and the average tax rate is 35%. |
[
{
"content": "Given the following system of equations, where \\(q, r, s,\\) and \\(t\\) are different constant values, find the sum \\(q + r + s + t\\):\n\\[\n\\frac{1}{qa+1} + \\frac{1}{qb+1} + \\frac{1}{qc+1} + \\frac{1}{qd+1} = 1\n\\]\n\\[\n\\frac{1}{ra+1} + \\frac{1}{rb+1} + \\frac{1}{rc+1} + \\frac{1}{rd+1} = 1\n\\]\n\\[\n\\frac{1}{sa+1} + \\frac{1}{sb+1} + \\frac{1}{sc+1} + \\frac{1}{sd+1} = 1\n\\]\n\\[\n\\frac{1}{ta+1} + \\frac{1}{tb+1} + \\frac{1}{tc+1} + \\frac{1}{td+1} = 1\n\\]",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the statement \"If a finite abelian group $G$ has multiple elements of order 3, then the product of all elements in $G$ is the identity element\" is true or false.",
"role": "user"
}
] | False |
[
{
"content": "Given an $n \\times n$ matrix $B$, define $e^B$ by \\[ e^B = \\sum_{j=0}^{\\infty} \\frac{B^j}{j!} \\] Let $p$ be the characteristic polynomial of $B$. Determine the matrix $e^{p(B)}$. Provide your answer in terms of $B$ and its properties.",
"role": "user"
}
] | I |
[
{
"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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "A friend says that she expects to earn 13.50% on her portfolio with a beta of 1.50. You have a two-asset portfolio including stock X and a risk-free security. The expected return of stock X is 10.50% and the beta is 1.10. The expected return on the risk-free security is 3.50%. You construct a portfolio to match your friend's return. What is the beta of your portfolio?",
"role": "user"
}
] | 1.57 |
[
{
"content": "Each outcome on the spinner below has equal probability. If you spin the spinner three times and form a three-digit number from the three outcomes, such that the first outcome is the hundreds digit, the second outcome is the tens digit and the third outcome is the units digit, what is the probability that you will end up with a three-digit number that is divisible by 4? Express your answer as a common fraction.\n\n[asy]\ndraw(Circle((0,0),10));\ndraw((0,0)--(8.7,-5));\ndraw((0,0)--(-8.7,-5));\ndraw((0,0)--(0,10));\nlabel(\"1\",(7,7),SW);\nlabel(\"3\",(-7,7),SE);\nlabel(\"2\",(0,-2),S);\ndraw((0,0)--(3,5),Arrow);\n[/asy]",
"role": "user"
}
] | \frac{2}{9} |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{\\pi}\\left\\{\\int_{y^2}^{\\pi}\\frac{y\\sin x}{x} \\, dx\\right\\}dy \\).",
"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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "Suppose the price elasticity of demand for a good is -0.2. If there is a 5% increase in the price of the good, by what percentage will the demand for the good go down?",
"role": "user"
}
] | 1% |
[
{
"content": "Determine the smallest integer \\( n \\) such that the inequality \\[ \\sum_{i=n}^{100} \\binom{100}{i} \\left(\\frac{80}{100}\\right)^i \\left(\\frac{20}{100}\\right)^{100-i} \\geq 0.9 \\] holds.",
"role": "user"
}
] | 75 |
[
{
"content": "Evaluate the integral \\( \\int_\\gamma \\frac{1}{1+z^2} \\, dz \\) using Cauchy's Theorem, where \\( \\gamma \\) is the circular contour defined by \\( t \\mapsto 2+e^{it} \\) for \\( 0 < t < 2\\pi \\).",
"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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "You are standing at the edge of a swimming pool and shining a laser pointer onto the water surface. The laser beam hits the water at an angle of 36.0 degrees above the horizontal. Some of the light reflects off the water surface, and some enters the water. Find the angle between the reflected and refracted rays.",
"role": "user"
}
] | 88.6 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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "The radius of the moon is 27% of the earth's radius and its mass is 1.2 % of the earth's mass. Find the acceleration due to gravity on the surface of the moon?",
"role": "user"
}
] | 1.62 m/s^2 |
[
{
"content": "Find the value of \\( p \\) such that the limit \\( \\lim_{x \\to \\infty} \\left( x^p \\left( \\sqrt[3]{x+1} + \\sqrt[3]{x-1} - 2\\sqrt[3]{x} \\right) \\right) \\) is a finite, non-zero number.",
"role": "user"
}
] | \dfrac{5}{3} |
[
{
"content": "Find all continuous functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) that satisfy the equation \\( f(xy) = f\\left(\\frac{x^2 + y^2}{2}\\right) + (x-y)^2 \\) for all real numbers \\( x \\) and \\( y \\).",
"role": "user"
}
] | f(x) = -2x + C |
[
{
"content": "Suppose $b > 1$ is a real number such that $\\log_5 (\\log_5 b + \\log_b 125) = 2$. Find the value of $\\log_5 \\left(b^{\\log_5 b}\\right) + \\log_b \\left(125^{\\log_b 125}\\right)$. \\(\\boxed{\\text{Your answer here}}\\)",
"role": "user"
}
] | 619 |
[
{
"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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "A sample of 2.4 kg of lead is at a temperature of 20 degrees Celsius. 154 kJ of energy is added to the lead. What is the phase and temperature of the resulting sample?",
"role": "user"
}
] | 327 degree Celsius, Liquid phase |
[
{
"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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "Use additivity of moments and the symmetry principle to find the centroid of the region formed by the circle {eq}(x-6)^2+(y-5)^2 = 4\n\n{/eq} and the rectangle with vertices {eq}(2,-1), (2-5), (6,-5),\n\n{/eq} and {eq}(6,-1)\n\n{/eq}. Let {eq}\\rho\n\n{/eq} be the constant density. Also include graph of shaded region.",
"role": "user"
}
] | (4.88, 0.52) |
[
{
"content": "Evaluate whether the following limit holds true:\n\\[ \\lim\\limits_{n\\to\\infty} \\int_{-\\infty}^{+\\infty}f(x)\\sin(nx)dx = 0 \\]\nwhen \\( \\int_{-\\infty}^{+\\infty}f(x)dx \\) is convergent but not absolutely convergent.",
"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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "Based on the following electron configuration, Xe6s2, this element would most likely .........",
"role": "user"
}
] | Barium |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\sum_{r=1}^n \\frac{r^4}{4r^2-1} \\]",
"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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "A 4.5-g object moving to the right at 22 cm/s makes an elastic head-on collision with a 9-g object that is initially at rest. What is the velocity of the 4.5-g mass after the collision?",
"role": "user"
}
] | -7.33 cm/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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "How to calculate the rate of inflation, specifically the percentage change in the CPI-U?",
"role": "user"
}
] | Rate of Inflation = (Current CPI-U - Previous CPI-U) / Previous CPI-U * 100 |
[
{
"content": "Evaluate the line integral of the conservative vector field \\(G(x,y,z) = (y^2+z^2)i + (2xy+z)j + (2xz+y)k\\) along the parametric curve \\(r(t) = (t^2, t^3, t)\\) from \\(t=0\\) to \\(t=1\\).",
"role": "user"
}
] | 3 |
[
{
"content": "Determine whether the sequence of operators \\( A_{n}x(t) = t^{n}(1-t)x(t) \\) converges in the operator norm on \\( C[0, 1] \\).",
"role": "user"
}
] | Yes |
[
{
"content": "For a real symmetric matrix with repeated eigenvalues, is it always possible to find an orthogonal matrix \\( P \\) such that the eigen decomposition satisfies a given equation?",
"role": "user"
}
] | Yes |
[
{
"content": "Provide an example of a linear operator $T \\in L(V)$ on the vector space $V$ of infinite sequences of finite length, such that:\n\n1. $T$ admits an adjoint, i.e., there exists $T^*$ such that $\\langle Ta, b \\rangle = \\langle a, T^*b \\rangle$ for all $a, b \\in V$.\n2. $T^*T$ is the identity operator on $V$.\n3. $TT^*$ is a projection operator but not the identity operator, meaning $(TT^*)^2 = TT^*$ and $TT^* \\neq I$. \n\nConsider whether the left-shift operator $T: (a_1, a_2, a_3, \\ldots) \\to (a_2, a_3, \\ldots)$ satisfies these conditions.",
"role": "user"
}
] | T |
[
{
"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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "Calculate the enthalpy change, for the process in which 10.5g of water is converted from liquid at 6.7 Celsius to vapor at 25.0 Celsius.\n\nFor water, delta H_vap = 44.0kJ/mol at 25.0 Celsius and s = 4.18 J/(g * degree Celsius) for water .",
"role": "user"
}
] | 26.4 kJ |
[
{
"content": "Calculate the limit: $$\\lim_{x \\to 0^+}\\frac{\\tan x \\cdot \\sqrt {\\tan x}-\\sin x \\cdot \\sqrt{\\sin x}}{x^3 \\sqrt{x}}$$",
"role": "user"
}
] | \dfrac{3}{4} |
[
{
"content": "Calculate the sum of the infinite series: \\[ \\sum_{j=0}^{\\infty} (j+1) \\left(\\frac{1}{1.05}\\right)^{j+1} \\]",
"role": "user"
}
] | 420 |
[
{
"content": "Let $A$ be a full triangulated subcategory of $B$, and let $u: A \\rightarrow B$ be the corresponding embedding. Consider a triangulated functor $f: B \\rightarrow A$ satisfying the following conditions:\n\n1. $f \\circ u = \\text{id}$\n2. For any $b \\in B$, if $f(b) = 0$, then $b = 0$.\n\nDetermine whether $K_0(A) = K_0(B)$. Provide a proof or counterexample.",
"role": "user"
}
] | K_0(A) = K_0(B) |
[
{
"content": "Let $f$ be a continuous function such that $f(f(x)) = 1 + x$. Determine the value of $f'(1)$. ",
"role": "user"
}
] | 1 |
[
{
"content": "The area of one lateral face of a right pyramid with an equilateral triangular base is 75 square meters. If the slant height is 30 meters, what is the length of the side of its base, in meters?",
"role": "user"
}
] | 5 |
[
{
"content": "Let $f:(G,*) \\rightarrow (H,\\circ)$ be a homomorphism between groups $(G,*)$ and $(H,\\circ)$. If $a \\in G$ and $f(a)$ has an inverse in $H$, does it follow that $a$ has an inverse in $G$? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "In the figure, $ABCD$ and $BEFG$ are squares, and $BCE$ is an equilateral triangle. What is the number of degrees in angle $GCE$?\n\n[asy]\n\ndraw(rotate(32)*shift((-.48,.85))*unitsquare); draw(unitsquare);\ndraw( (-.85, .46) -- (0,0));\nlabel(\"$C$\", (-.85, .46), SW); label(\"$E$\", (0, 0), S); label(\"$F$\", (1, 0),SE );label(\"$G$\", (1, 1),NE ); label(\"$B$\", (0.05, 1),N);\nlabel(\"$D$\", (-2, 1.5), 3*E+2*SE); label(\"$A$\", (-.5, 2));\n\n[/asy]",
"role": "user"
}
] | 45^\circ |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "During the year, Carson Egg Processing sold assets with a total book value of $25,600 and also recorded $20,745 in depreciation expense. Carson had beginning net fixed assets of $85,642 with ending net fixed assets of $124,512. The market value of assets sold was $37,500 and the market value of ending net fixed assets is $154,250.\n\nHow much did the company spend to buy new fixed assets?",
"role": "user"
}
] | $85,215 |
[
{
"content": "For a linear transformation $T$ on an $n$-dimensional vector space $V$, if there exists a vector $\\mathbf{v}$ such that the set $\\{\\mathbf{v}, T(\\mathbf{v}), T^2(\\mathbf{v}), \\ldots, T^{n-1}(\\mathbf{v})\\}$ is linearly independent, does it imply that each eigenvalue of $T$ has a unique eigenvector (up to scalar multiplication)? Answer yes or no and 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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "Suppose you are buying your first home, a condominium, for $300,000. The down payment is $15,000 and you have arranged for a 30-year amortized mortgage at a 6.5% nominal rate of interest with the first payment due in one-month. If you pay an additional $500 per month in addition to the monthly payment, how soon will the mortgage be paid-off?",
"role": "user"
}
] | 206 months or 17.18 years |
[
{
"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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "How can atoms in a flask of methane be calculated? What equation should be used?",
"role": "user"
}
] | n=PV/RT |
[
{
"content": "Determine the value of \\( y_0 \\) such that for the initial value problem (IVP) given by \\( y'+\\frac{2x^2-4xy-y^2}{3x^2}=0, x>0, y(1)=y_0 \\), the condition \\( \\lim_{x\\to 0}y'(x) \\neq 1 \\) is satisfied, while \\( \\lim_{x\\to 0}y'(x) = 1 \\) for all other values of \\( y_0 \\). What is this value of \\( y_0 \\)?",
"role": "user"
}
] | -2 |
[
{
"content": "Suppose \\(x_1, x_2, x_3, x_4,\\) and \\(x_5\\) are the roots of the equation \\(x^5 - 2x^4 + x^3 + 1 = 0\\). Define a matrix \\(A\\) with elements \\(a_{ij} = 1 + x_i x_j\\) when \\(i = j\\), and \\(a_{ij} = x_i x_j\\) otherwise, where \\(1 \\leq i, j \\leq 5\\). Compute the determinant of matrix \\(A\\).",
"role": "user"
}
] | 3 |
[
{
"content": "Piravena must make a trip from $A$ to $B$, then from $B$ to $C$, then from $C$ to $A$. Each of these three parts of the trip is made entirely by bus or entirely by airplane. The cities form a right-angled triangle as shown, with $C$ a distance of 3000 km from $A$ and with $B$ a distance of 3250 km from $A$. To take a bus, it costs Piravena $\\$0.15$ per kilometer. To take an airplane, it costs her a $\\$100$ booking fee, plus $\\$0.10$ per kilometer. [asy]\n\npair A, B, C;\n\nC=(0,0);\n\nB=(0,1250);\n\nA=(3000,0);\n\ndraw(A--B--C--A);\n\nlabel(\"A\", A, SE);\n\nlabel(\"B\", B, NW);\n\nlabel(\"C\", C, SW);\n\nlabel(\"3000 km\", (A+C)/2, S);\n\nlabel(\"3250 km\", (A+B)/2, NE);\n\ndraw((0,125)--(125,125)--(125,0));\n\n[/asy] Determine the distance she travels for her complete trip.",
"role": "user"
}
] | 7500 |
[
{
"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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "Suppose a gangster sprays Superman's chest with {eq}3 \\ g\n\n{/eq} bullets at the rate of {eq}100\n\n{/eq} bullets/min, and the speed of each bullet is {eq}500 \\ m/s\n\n{/eq}. Suppose too that the bullets rebound straight back with no change in speed. What is the magnitude of the average force on Superman's chest?",
"role": "user"
}
] | 5 N |
[
{
"content": "Let $p$ be a prime number. Determine the order of the group $G$, which consists of all $2 \\times 2$ matrices over $\\mathbb{Z}_p$ with determinant $1$, under matrix multiplication.",
"role": "user"
}
] | p(p^2 - 1) |
[
{
"content": "Calculate the expected distance between two points chosen randomly and independently on the circumference of a circle with radius 1.",
"role": "user"
}
] | \dfrac{4}{\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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "An air conditioner is a type of refrigerator designed to remove heat from a room and exhaust heat to a warmer environment. The design for a new building calls for an air conditioner to keep the building interior at {eq}22^\\circ \\rm C\n\n{/eq}. The air conditioner needs to remove heat from the building interior at a rate of {eq}24\\ \\rm kW\n\n{/eq}. Find the change in entropy per second of the building interior.",
"role": "user"
}
] | 81 J/s.K |
[
{
"content": "Given that $a > 0$, if $f(g(a)) = 8$, where $f(x) = x^2 + 8$ and $g(x) = x^2 - 4$, what is the value of $a$?",
"role": "user"
}
] | 2 |
[
{
"content": "Find the greatest common divisor (gcd) of the numbers $2020^{1830} + 2$ and $2020^{1830} - 2$. Provide your answer as a single integer.",
"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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "You are given 2.078 g of a mixture of {eq}KClO_3\n\n{/eq} and KCl. When heated, the {eq}KClO_3\n\n{/eq} decomposes to KCl and {eq}O_2\n\n{/eq},\n\n{eq}2KClO_3 (s) \\rightarrow 2KCl (s) + 3O_2 (g)\n\n{/eq},\n\nand 338 mL of {eq}O_2\n\n{/eq} is collected over water at {eq}27^{\\circ}C\n\n{/eq}. The total pressure of the gases in the collection flask is 762 torr. What is the weight percentage of {eq}KClO_3\n\n{/eq} in the sample?\n\nThe formula weight of {eq}KClO_3\n\n{/eq} is 122.55 g/mol. The vapor pressure of water at {eq}27^{\\circ}C\n\n{/eq} is 26.7 torr.",
"role": "user"
}
] | 52.1% |
[
{
"content": "Calculate the area under the curve \\( y = \\sqrt{x} \\) on the interval \\([0, 1]\\) using Darboux sums.",
"role": "user"
}
] | \dfrac{2}{3} |
[
{
"content": "A $24 \\times 8$ table is filled such that each row contains a permutation of the numbers $1, 2, \\ldots, 8$. In each column, the numbers are multiplied together. Determine the minimum possible sum of all the column products.",
"role": "user"
}
] | 8 \times (8!)^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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "G Mart, Inc., is considering the acquisition of equipment to expand its sales. The initial cost of the equipment is $100,000. However, the production manager has estimated the expansion program will increase cash operating costs by $20,000. Assume straight line depreciation to a zero salvage value, a tax rate of 40%, and a cost of capital of 10%.\n\nHow much will the additional cash revenue during the 10 year life of the asset have to be to cause the IRR of the project to be equal to k?",
"role": "user"
}
] | $40,457.42 |
[
{
"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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "The volume flow Q over a certain dam is a function of dam width b, gravity g, and the upstream water depth H above the dam crest. It is known that Q is proportional to b. If b = 120 ft and H = 15 in, the flow rate is {eq}600 ft^3/s\n\n{/eq}. What will be the flow rate if H = 3ft?",
"role": "user"
}
] | 2230.84 ft^3/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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "A solid sphere of mass 8.6 kg, made of metal whose density is 3400 kg/m{eq}^3\n\n{/eq}, hangs by a cord. When the sphere is immersed in a liquid of unknown density, the tension in the cord is 38 N. What is the density of the liquid?",
"role": "user"
}
] | 1867 kg/m^3 |
[
{
"content": "Determine the number of distinct terms in the expansion of \\(\\left( x + \\frac{1}{x} + x^2 + \\frac{1}{x^2} \\right)^{20}\\).",
"role": "user"
}
] | 81 |
[
{
"content": "Determine the largest natural number \\( m \\) such that the polynomial \\( P(x) = x^5 - 3x^4 + 5x^3 - 7x^2 + 6x - 2 \\) is divisible by \\((x-1)^m\\).",
"role": "user"
}
] | 3 |
[
{
"content": "Consider a homogeneous Markov chain $(X_n)_{n \\in \\mathbb{N}}$ with the following transition probabilities: the only missing arrow is from state 0 to state 0 with $\\mathbb{P}(X_1=0 \\mid X_0=0) = 1/5$. Compute $h_0^6$, the probability of starting from state 0 and eventually hitting state 6. Given that $h_1=h_2=h_3=0$ because $\\{1,2,3\\}$ is a closed class, solve the following system of equations:\n\n$$\\begin{aligned}\nh_0 &= \\frac{1}{5} h_0 + \\frac{1}{5} h_4, \\\\\nh_4 &= h_5, \\\\\nh_5 &= h_6, \\\\\nh_6 &= h_4.\n\\end{aligned}$$\n\nFind the value of $h_0^6$. Provide your answer as a single probability value.",
"role": "user"
}
] | \dfrac{1}{4} |
[
{
"content": "Find a 4-tuple \\((a, b, n, k) \\in \\mathbb{N}^4\\) with \\(a, b, n, k > 1\\) and \\(a \\neq b\\) such that the reverse of \\(a^n\\) is equal to \\(b^k\\).",
"role": "user"
}
] | (12, 21, 2, 2) |
[
{
"content": "Let $K = \\mathbb{Q}(\\sqrt[3]{3}, \\eta)$ where $\\eta = (e^{\\frac{\\pi}{3}i})^2$. Determine the degree of the field extension $[K: \\mathbb{Q}(\\sqrt[3]{3})]$. \\(\\eta\\) is a primitive sixth root of unity.",
"role": "user"
}
] | 2 |
[
{
"content": "Let \\( g(t) = t \\) for all \\( t \\in [-1,1] \\) and \\( f(t) = t^{1/n} \\), where \\( n \\) is a large odd integer. Consider the operator \\( T: C[-1,1] \\rightarrow \\mathbb{R} \\), defined by \\( T_g(f) = \\int_{-1}^1 g(t)f(t)dt \\). Given the norm \\( \\|f\\| = \\max_{t \\in [-1,1]} |f(t)| \\), find \\( \\|T\\| \\).",
"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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "How many grams of carbon dioxide can be produced by burning 8.08 moles of propane?",
"role": "user"
}
] | 1.066e+3 g |
[
{
"content": "Evaluate the infinite series: \\( \\sum_{k=1}^{\\infty} \\frac{(2k + 1)^2}{2^k} \\)",
"role": "user"
}
] | 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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "A jet dives vertically at a speed {eq}v = 271 \\;m/s {/eq}, before pulling out of the dive along a circular arc. The pilot can survive an acceleration of {eq}a = 8.8 \\;g {/eq}. His mass is {eq}m = 99 \\;kg {/eq}.\n\nWhat is the minimum radius of the circular arc, in meters, that the pilot can make without injury?",
"role": "user"
}
] | 852 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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "Two machines have specified noise levels of 95 db(A) and 105 db(A); background noise is measured at 100 db(A) when the machines are not running. What is the expected noise level when the machines are running?",
"role": "user"
}
] | 107 dB |
[
{
"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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "Spherical aluminum ball of mass {eq}1.22 \\ kg\n\n{/eq} contains an empty spherical cavity that is concentric with the ball. The ball just barely floats in water.\n\nCalculate the radius of the cavity.",
"role": "user"
}
] | 5.69e-2 m |
[
{
"content": "Given a sequence \\( \\{f_n\\} \\) of conformal, one-to-one maps from the unit disk \\( \\mathbb{D} \\) onto the right half-plane \\( \\mathbb{A} = \\{ z \\in \\mathbb{C} : \\mathfrak{R} z > 0 \\} \\), suppose \\( \\{f_n\\} \\) converges to \\( f \\) uniformly on compact subsets of \\( \\mathbb{D} \\) and \\( f \\) is not one-to-one. Determine the value of \\( \\mathfrak{R} f(0) \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Given the function \\( f(x) = 1 + \\int_a^x{(x-t)^n f(t)\\,dt} \\) with the condition \\( f(a) = 1 \\), find a formula for \\( f^{(n+1)}(a) \\).",
"role": "user"
}
] | 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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "Buffer capacity is a measure of a buffer solution's resistance to changes in pH as strong acid or base is added. Suppose that you have 185 mL of a buffer that is 0.240 M in both formic acid and its conjugate base. Calculate the maximum volume of 0.490 M HCl that can be added to the buffer before its buffering capacity is lost. Show your complete solution.",
"role": "user"
}
] | 74.1 mL |
[
{
"content": "Let \\( f: \\mathbb{C} \\rightarrow \\mathbb{C} \\) be a differentiable function such that \\( |f(z)-z|^{4/3} \\le 1 + 3|f(z)-z|^{6/5} \\). Find \\( f'(-i) \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Determine the Lebesgue measure of the set $B = \\{(x,y) \\in [0,1] \\times [0,1] : \\forall n (\\{x_n, y_n\\} \\subset \\{1,5\\}) \\}$, where $x = 0.x_1x_2 \\ldots$ and $y = 0.y_1y_2 \\ldots$ are the decimal representations of $x$ and $y$, respectively.",
"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> </answer> tags, respectively, i.e., <think> reasoning process here </think> <answer> answer here </answer>.",
"role": "system"
},
{
"content": "How long would it take for a radio wave with a frequency of {eq}7.25 \\times 10^5\n\n{/eq} Hz to travel from Mars to Earth?",
"role": "user"
}
] | 750 seconds or 12.5 minutes |
[
{
"content": "Let $P$ be a point in coordinate space, where all the coordinates of $P$ are positive. The line between the origin and $P$ is drawn. The angle between this line and the $x$-, $y$-, and $z$-axis are $\\alpha,$ $\\beta,$ and $\\gamma,$ respectively. If $\\cos \\alpha = \\frac{1}{3}$ and $\\cos \\beta = \\frac{1}{5},$ then determine $\\cos \\gamma.$\n\n[asy]\nimport three;\n\nsize(180);\ncurrentprojection = perspective(6,3,2);\n\ntriple I = (1,0,0), J = (0,1,0), K = (0,0,1), O = (0,0,0);\ntriple V = (3,2,2), P;\n\nP = (2.5*I + 2.5*V/abs(V))/2;\ndraw(1.1*I..1.5*P/abs(P)..1.5*V/abs(V));\nlabel(\"$\\alpha$\", 1.5*P/abs(P), NW);\nP = (2.5*J + 2.5*V/abs(V))/2;\ndraw(1.5*J..1.5*P/abs(P)..1.5*V/abs(V));\nlabel(\"$\\beta$\", 1.5*P/abs(P), NE);\nP = (2.5*K + 2.5*V/abs(V))/2;\ndraw(1.5*K..1.5*P/abs(P)..1.5*V/abs(V));\nlabel(\"$\\gamma$\", 1.5*P/abs(P), E);\n\ndraw(O--5.5*V/abs(V));\ndraw(O--3*I, Arrow3(6));\ndraw(O--3*J, Arrow3(6));\ndraw(O--3*K, Arrow3(6));\n\nlabel(\"$x$\", 3.2*I);\nlabel(\"$y$\", 3.2*J);\nlabel(\"$z$\", 3.2*K);\ndot(\"$P$\", 5.5*V/abs(V), NE);\n[/asy]",
"role": "user"
}
] | \frac{\sqrt{191}}{15} |
[
{
"content": "Compute the following limit without using L'Hopital's rule: $$\\lim _{x\\to 0}\\left(\\frac{xe^x-2+2\\cos x-x}{\\left(\\sin x\\right)^2 \\tan\\left(2x\\right)+xe^{-\\frac{1}{\\left|x\\right|}}}\\right)$$",
"role": "user"
}
] | \dfrac{1}{4} |
[
{
"content": "Find the smallest positive real number \\( C \\) such that for any positive integer \\( k \\), the following inequality holds:\n\\[ S = \\sum_{i=k-\\lfloor \\frac{k-1}{2} \\rfloor+1}^k \\frac{1}{i} + \\sum_{i=k-\\lceil \\frac{k-1}{2} \\rceil+1}^k \\frac{1}{i} < C. \\]",
"role": "user"
}
] | 2\ln 2 |
[
{
"content": "Calculate the limit: \\[ \\lim_{x \\to 0}((-2^x+1)^{\\sin(x)}) \\]",
"role": "user"
}
] | 1 |
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