prompt list | ground_truth string |
|---|---|
[
{
"content": "Find the inverse of the polynomial \\(2x^2 + x + 1\\) in the quotient ring \\(\\frac{\\mathbb{Z}_3[x]}{\\langle x^3 + 2x + 2 \\rangle}\\).",
"role": "user"
}
] | x^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... | 3239 |
[
{
"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... | 0.2% |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.052e5 m/s |
[
{
"content": "Let $E$ be a vector space over a field $F$, and let $E^*$ be its algebraic dual space, defined as $E^* = \\{f: E \\rightarrow F \\mid f \\text{ is linear}\\}$. Assuming $E^* \\neq \\{0\\}$, does $E^*$ always have the property of separating points? In other words, for every non-zero vector $x \\in ... | 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... | $x_1x_2^2=32$ |
[
{
"content": "Determine whether the limit \\( \\lim_{n \\to \\infty} a_n \\) exists for the sequence \\( a_n = \\left(1-\\frac{1}{\\sqrt{2}}\\right)\\left(1-\\frac{1}{\\sqrt{3}}\\right)\\cdots\\left(1-\\frac{1}{\\sqrt{n+1}}\\right) \\), where \\( n \\ge 1 \\). If it exists, find its value.",
"role": "user"
... | 0 |
[
{
"content": "Evaluate the limit \\( \\lim_{n\\to\\infty}\\frac{\\ln\\left(\\frac{n+2}{n+1}\\right)}{\\ln\\left(\\frac{n+1}{n}\\right)} \\) using the Stolz-Cesaro theorem.",
"role": "user"
}
] | 1 |
[
{
"content": "Find the number of positive integer solutions \\((a, b, c)\\) that satisfy the equation:\n\\[ a^{b^c}b^{c^a}c^{a^b} = 5abc \\]",
"role": "user"
}
] | 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... | -0.79% |
[
{
"content": "Let \\( f(x) = \\left\\{ \\begin{array}{cc} 1 & 1 \\leqslant x \\leqslant 2 \\\\ x-1 & 2 < x \\leqslant 3 \\end{array} \\right. \\). For any \\( a \\,(a \\in \\mathbb{R}) \\), define \\( v(a) = \\max \\{ f(x) - a x \\mid x \\in [1,3] \\} - \\min \\{ f(x) - a x \\mid x \\in [1,3] \\} \\). Find the ... | \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... | 5.66e-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... | 2.72 |
[
{
"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... | Im{$H(\omega_0)e^{i\omega_ot}$} |
[
{
"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... | 2.17e+23 molecules H2O |
[
{
"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... | 1.11*10^{-7} C |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $5,741 |
[
{
"content": "Let $k$ be a non-zero natural number and $f:\\mathbb{R} \\to \\mathbb{R}$ be a monotonic function such that \\( f(a+b\\sqrt{3})=(a+b\\sqrt{3})^{3k-1} \\) for all integers $a$ and $b$. Determine the function $f$.",
"role": "user"
}
] | f(x) = x^{3k - 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... | Q = -W |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 86 mph |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\to +0}\\frac{1}{x}{\\int^{2022x}_{0}{t\\,\\sqrt{|\\cos(\\frac{1}{t})|}\\,dt}}$$",
"role": "user"
}
] | 0 |
[
{
"content": "Calculate the probability that the square of a geometric random variable is greater than or equal to a binomial random variable. Let $X$ be a $\\operatorname{Binomial}(4, \\frac{1}{3})$ random variable and $Y$ be a $\\operatorname{Geometric}(\\frac{1}{3})$ random variable. Find $P(Y^2 \\geq X)$. P... | \dfrac{70}{81} |
[
{
"content": "The function defined on the set of real numbers, \\(f(x)\\), satisfies \\(f(x-1) = \\frac{1 + f(x+1)}{1 - f(x+1)}\\). Find the value of \\(f(1) \\cdot f(2) \\cdot f(3) \\cdots f(2008) + 2008\\).",
"role": "user"
}
] | 2009 |
[
{
"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... | Year 1: $45,000
Year 2: $120,000 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.26 |
[
{
"content": "Determine the Galois group of the polynomial \\( f(x) = x^4 + 12x^3 + 44x^2 + 48x + 40 \\in \\mathbb{Q}[x] \\). Given that the polynomial is irreducible and its resolvents \\( R_2(x) = x^2 + 1960 \\) and \\( R_3(x) = x^3 + 10x^2 - 196x - 1960 \\) are also irreducible, decide whether the Galois gro... | S_4 |
[
{
"content": "Determine the number of non-zero ring homomorphisms from the ring \\( \\mathbb{Z}_{30} \\) to the ring \\( \\mathbb{Z}_{42} \\).",
"role": "user"
}
] | 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... | 926.70 |
[
{
"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... | 7.01% |
[
{
"content": "Evaluate the limit: \\[ \\lim_{(x,y) \\to (0,0)}\\frac{\\sin(x^2+y^2)}{\\ln(3x^2y^2+1)} \\]",
"role": "user"
}
] | \infty |
[
{
"content": "Given that $f'''(x)$ is continuous everywhere and \\(\\lim_{x \\to 0}(1+x+ \\frac{f(x)}{x})^{1/x}=e^5\\), find the value of $f^{(4)}(0)$. Express your answer as a single number.",
"role": "user"
}
] | 500 |
[
{
"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... | Merchandise Inventory
Debit $24,000
Allowance for Doubtful Accounts
Credit $3,000
Nick Rawls, Capital
Credit $9,000
Sarah Fitzpatrick, Capital
Credit $6,000
Antoine Faber, Capital
Credit $6,000 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $628.80 |
[
{
"content": "Consider a matrix with dimensions 8x10, containing numbers from 1 to 80. You randomly pick two distinct numbers to form an ordered pair. What is the expected number of picks required such that every number from 1 to 80 is used at least once in the pairs?",
"role": "user"
}
] | 199 |
[
{
"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... | $2400 |
[
{
"content": "Consider the setting where $X=\\mathbb{R}^m$, $Y=\\mathbb{R}^n$, $\\alpha:X\\to X$ is a $C^1$ diffeomorphism, and $\\beta:X\\times Y\\to Y$ is such that for all $x\\in X$, $\\beta_x:Y\\to Y$ is a $C^1$ diffeomorphism. Suppose $\\pi(x,y)=\\pi_2(y\\mid x)\\pi_1(x)$ is a continuous joint probability ... | 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... | 1.3 x 10^-34 |
[
{
"content": "Compute \\( \\lim_{x \\to -1} f(x) \\) for a function \\( f: \\mathbb{R} \\to \\mathbb{R} \\) such that \\[ 4 = \\lim_{x \\to -1} \\frac{f(x)+2 - \\frac{x}{x-1}}{x+1}. \\]",
"role": "user"
}
] | -\dfrac{3}{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... | <img>/cimages/multimages/16/4-hepten-2-yne_structural_formula2762516312447545675.png</img> |
[
{
"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... | 4.056e21 Photons/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... | 142.04 g/mol |
[
{
"content": "Evaluate the integral \\( \\int_0^\\infty \\frac{\\ln x}{1+x^2}\\ dx \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Compute the Euler Characteristic of the blowing up of $\\mathbb{C}\\mathbb{P}^2$ at $n$ points.",
"role": "user"
}
] | 3 + 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... | 5.7 mi |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty} \\int_0^1 \\frac{e^{x^{n}}}{1+x^2}\\,\\mathrm{d}x \\]",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "Evaluate the limit \\(\\lim_{x\\to \\infty} \\left(\\cfrac{x^2+x+2}{x^2+5x+3}\\right)^x\\) using the rules of de l'Hospital.",
"role": "user"
}
] | e^{-4} |
[
{
"content": "Let $f$ be a continuous function on the interval $(0,1)$. Determine whether the following statement is true or false: If $(x_n)$ is a Cauchy sequence such that $0 < x_n < 1$ for all $n$, then $f(x_n)$ is a Cauchy sequence.",
"role": "user"
}
] | B |
[
{
"content": "Evaluate the limit given that the function \\( f(x) \\) is continuous and \\( \\lim_{x \\to 0} \\frac{f(x)}{x} = 1 \\):\n\\[ \\lim_{x \\to 0} \\frac{x\\int_0^x{f(x-t)dt}}{\\int_0^x tf(x-t)dt} \\]",
"role": "user"
}
] | 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... | 1.42 |
[
{
"content": "Determine whether the sequence of functions $\\{\\phi_n\\}$ defined by $\\phi_n(x) = e^{-x^2/n^2}$ converges in the space of distributions $\\mathcal{D}'(\\mathbb{R})$. Provide an analytical approach to demonstrate convergence.",
"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... | 1.0e+2 g/mol |
[
{
"content": "Determine whether there exists a function $f$ such that the sequence of functions $f_n(x) = \\sum_{k=1}^n \\frac{1}{2^{n-k}} \\mathcal{X}_{[a_{k-1},a_k)}(x)$ converges to $f$ in measure, where $a_n = \\sum_{k=1}^n \\frac{1}{k}$ and $a_0 = 0$. Here, $\\mathcal{X}_{[a_{k-1},a_k)}(x)$ is the indicato... | 0 |
[
{
"content": "Evaluate the limit \\( \\lim_{x \\to \\infty} \\frac{(1+\\frac{1}{x})^{x^2}}{e^x} \\).",
"role": "user"
}
] | \dfrac{1}{\sqrt{e}} |
[
{
"content": "A sensor system consists of 15 sensors. The probability of a sensor detecting a signal above a threshold 'b' is given by \\( p=\\frac{1}{2\\pi }\\int_{x=b}^\\infty \\exp\\left(\\frac{-x^2}{2}\\right) \\, dx \\). Calculate the probability that at least 8 sensors detect the signal when the threshold... | 0 |
[
{
"content": "For a given integer \\( n \\), find the value of \\( x \\) such that the equation \\( \\pi(xn) = \\frac{xn}{\\frac{n}{\\pi(n)} + \\ln(x)} \\) holds, where \\( \\pi \\) is the prime counting function.",
"role": "user"
}
] | 1 |
[
{
"content": "Does there exist a function that has an infinite number of non-zero derivatives, starts at negative infinity, and approaches zero as \\( x \\) goes to infinity?",
"role": "user"
}
] | -e^{-x} |
[
{
"content": "Maximize the function \\( c = \\frac{(2-a_0) + \\sqrt{-a_0(3a_0-4)}}{2} \\) and find the corresponding value of \\( c \\) without using calculus.",
"role": "user"
}
] | \dfrac{4}{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... | 9.68e-7 W/m^2 |
[
{
"content": "If $A \\oplus B \\simeq A' \\oplus B$ for abelian groups $A$, $A'$, and $B$, does it necessarily follow that $A \\simeq A'$?",
"role": "user"
}
] | No |
[
{
"content": "Determine the minimum number of terms, $n$, required in the harmonic series $H_n$ such that the difference $H_n - \\log n$ is less than $2 \\times 10^{-3}$, given that Euler's constant $\\gamma$ is defined as $\\gamma = \\lim_{n \\to \\infty} (H_n - \\log n)$. Provide your answer as an integer $n$... | 250 |
[
{
"content": "Is the following statement true for all $0 < T \\leq 1$? For $X_t \\sim N(0, 1)$ independently defined for each $t \\in (0, 1]$, does the equality \\( P\\left(\\sup_{t \\leq T} X_t \\leq \\alpha\\right) = P(X_1 \\leq \\alpha)^T \\) hold?",
"role": "user"
}
] | No |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 3840 J/min |
[
{
"content": "Solve for \\(x\\) in the equation \\((2+\\sqrt{5})^x + (2-\\sqrt{5})^x = 4\\).",
"role": "user"
}
] | 1 |
[
{
"content": "Let $f,g: \\mathbb{R}\\rightarrow \\mathbb{R}$ be two strictly convex functions, where $f$ is differentiable, $g$ is smooth, and $f \\geq g$. Suppose that for some $x_0 \\in \\mathbb{R}$, the following conditions hold: $f(x_0) = g(x_0)$, $f'(x_0) = g'(x_0)$, and $g''(x_0) > 0$. Does there exist an... | Yes |
[
{
"content": "My three friends and I have dinner together every weekend. Each weekend, two of us cook and the other two clean up afterwards. How many different ways are there for us to choose who cooks and who cleans?",
"role": "user"
}
] | 6 |
[
{
"content": "Determine whether there is exactly one Fibonacci number that can be expressed as the sum of two positive cubes.",
"role": "user"
}
] | 2 |
[
{
"content": "Evaluate the improper integral \\( \\int_{-1}^0 \\frac{e^{1/x}}{x^2} \\, dx \\). Determine whether the integral converges or diverges.",
"role": "user"
}
] | \dfrac{1}{e} |
[
{
"content": "A point with coordinates $(x,\\ y)$ is randomly selected such that $0\\leq x \\leq10$ and $0\\leq y \\leq10$. What is the probability that the coordinates of the point will satisfy $2x+5y \\geq 20$? Express your answer as a common fraction.\n\n[asy]\n\nsize(5cm,5cm);\n\ndraw((-2,0)--(12,0),Arrows)... | \frac{4}{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... | 612.93 N |
[
{
"content": "Let $T: V \\to V$ be a linear transformation defined by $T(A) = BAB^*$, where $V$ is the vector space of all $n \\times n$ matrices over $\\mathbb{C}$ and $B$ is a fixed complex matrix. Show that $\\det T = |\\det B|^{2n}$. ",
"role": "user"
}
] | |\det B|^{2n} |
[
{
"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... | 24.3 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... | The four primary determinants of exchange rates are interest rates, economic growth, a nation's current account, and inflation rates. |
[
{
"content": "Evaluate the limit: \\[ \\lim_{h \\to 0} \\frac{f(2-h^2) - f(2)}{h^2} \\] where \\( f(x) = (x^3 + 2x) \\left\\lfloor \\frac{x}{2} \\right\\rfloor \\).",
"role": "user"
}
] | -\infty |
[
{
"content": "Evaluate the sum \\(\\sum_{n=2}^{\\infty}\\ln\\left(1-\\frac{1}{n^2}\\right)\\).",
"role": "user"
}
] | -\ln 2 |
[
{
"content": "Find the equation of the line tangent to the curve formed by the intersection of the cylinder $S = \\{ (x,y,z) | x^2 + y^2 = 2 \\}$ and the graph of the function $z = f(x,y)$ at the point $M(1,1,2)$. Given that $f(x,y)$ is differentiable everywhere and $\\nabla f(1,1) = (1,2)$, determine the equat... | \begin{cases} x = 1 - t \\ y = 1 + t \\ z = 2 + t \end{cases} |
[
{
"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... | 82g |
[
{
"content": "Calculate the contour integral \\(I = \\int_{|z|=5} f(z) \\, dz\\), where \\(f(z) = \\frac{2}{(z-1)^2} + \\frac{3}{(z-1)} + \\frac{\\sin z}{e^z}\\). Use the known results: \\(\\int \\frac{dz}{(z-a)^n} = 0\\) for \\(n \\geq 2\\) and \\(\\int \\frac{dz}{z-a} = 2\\pi i\\).",
"role": "user"
}
] | 6\pi 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> </answ... | 33 |
[
{
"content": "Calculate the limit: $$ \\lim_{x \\to 0}{\\frac{(e^{\\sin(4x)}-1)}{\\ln\\big(1+\\tan(2x)\\big)}} $$",
"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... | 16.18% |
[
{
"content": "Determine the Galois group of the polynomial \\( f(x) = x^5 - 9x + 3 \\) over the field of rational numbers, \\( \\mathbb{Q} \\).",
"role": "user"
}
] | S_5 |
[
{
"content": "Consider the series: $$2-\\frac{2^2}{x+2}+\\frac{2^3}{(x+2)^2}-\\frac{2^4}{(x+2)^3}+...+\\frac{2^r}{[-(x+2)]^{r-1}}+...$$ Determine the values of $x$ for which the series converges. If the series converges to $\\frac{10}{7}$, find the value of $x$.",
"role": "user"
}
] | 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... | 1. Banks have excess reserves.
2. Banks lend excess reserves.
3. The quantity of money increases.
4. New money is used to make payments.
5. Some of the new money remains on deposit and some is a currency drain.
6. Bank deposits increase.
7. Desired reserves increase because deposits have increased.
8. Excess reserves decrease but remain positive. |
[
{
"content": "Determine the holomorphic function \\( f: \\mathbb{C} \\rightarrow \\mathbb{C} \\) such that for all \\( x, y \\in \\mathbb{R} \\), the imaginary part of \\( f(x + iy) \\) is equal to \\( x^2 - y^2 \\).",
"role": "user"
}
] | i z^2 |
[
{
"content": "In the Poincaré disc model of hyperbolic geometry, if a triangle has all its interior angles equal, does it imply that the lengths of its sides, measured as geodesic distances between vertices, are also equal? Answer with 'Yes' or 'No' and justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Given a twice differentiable function $f(x)$ with $f(a) = 0, f(b) = 2, f(c) = -1, f(d) = 2, f(e) = 0$, where $a < b < c < d < e$, determine the minimum number of zeros of the function $g(x) = (f'(x))^2 + f(x)f''(x)$ in the interval $[a, e]$. What is this minimum number?",
"role": "user"
}
] | 6 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $1.20 per British Pound |
[
{
"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... | 8.0 mA |
[
{
"content": "Determine the number of times L'Hopital's rule must be applied to evaluate the limit \\( \\lim _{x\\to 0}\\left(\\frac{e^{x^2}+2\\cos \\left(x\\right)-3}{x\\sin \\left(x^3\\right)}\\right) \\).",
"role": "user"
}
] | 4 |
[
{
"content": "Find the sum of all real values of $y$ that satisfy the equation $$(y^2-6y+10)^{(y^2-6y+10)^{(y^2-6y+10)}} = 2018.$$",
"role": "user"
}
] | 6 |
[
{
"content": "Compute \\( E[(W_t)^2(W_s)^2] \\) where \\( W_t \\) and \\( W_s \\) are independent Brownian motions.",
"role": "user"
}
] | ts |
[
{
"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... | 30.96 sec |
[
{
"content": "Given the differential equation \\(2x^2y''+3xy' - y = 0\\), find the positive real number \\(r\\) such that the function \\(y_2(x)=x^r\\) is linearly independent of \\(y_1(x)= \\frac{1}{x}\\).",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "What is the value of $x^2+y^2-z^2+2xy$ when $x=-3$, $y=5$, and $z=-4$?",
"role": "user"
}
] | -12 |
[
{
"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... | 1388 m |
[
{
"content": "Evaluate the integral \\( \\int_{C}{\\frac{1}{z^2+4}}dz \\) where \\( C \\) is the ellipse given by \\( 4x^2+(y-2)^2=4 \\).",
"role": "user"
}
] | \dfrac{\pi}{2} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 6e14 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... | 9.7e-35 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... | 1.44e6 N/C towards the rod |
[
{
"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... | 0.30 N |
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