prompt list | ground_truth string |
|---|---|
[
{
"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... | Hardness, non-molecularity, high melting point |
[
{
"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... | 96.6 N |
[
{
"content": "Evaluate the limit \\(\\lim_{(x,y)\\to (0,0)}\\frac{x^3+y^3}{x^2+y^2}\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Given two bit vectors A and B of length N, the Jaccard similarity between them is defined as \\(\\|A \\land B\\|/\\|A \\lor B\\|\\). Determine the largest possible value of M such that M bit vectors can be selected from the set of \\(2^N-1\\) possible values (excluding the all-zero vector) such th... | N |
[
{
"content": "In a cubic graph G with q edges and a girth of at least 4, determine the number of distinct paths of length 3. Each vertex in G has a degree of 3, and the relationship 3p = 2q holds, where p is the number of such paths.",
"role": "user"
}
] | \dfrac{2q}{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... | 15.701 m |
[
{
"content": "Determine the cardinality of the equivalence class of the function \\( h(n) = n^2 \\) under the equivalence relation \\( R \\) on \\( \\mathbb{N}^{\\mathbb{N}} \\), where \\( R = \\{(f, g) \\in \\mathbb{N}^{\\mathbb{N}} \\mid \\exists b \\in \\mathbb{N}. \\forall n > b, f(n) = g(n)\\} \\).",
"... | \aleph_0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 360,000 |
[
{
"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... | r_0 |
[
{
"content": "Find the variance of $X^2$ where $X$ is a standard normal random variable $(X \\sim \\mathcal{N}(0,1))$. Express your answer as a single number.",
"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... | $110,000 |
[
{
"content": "Evaluate the limit \\( \\lim_{t \\to 0} \\frac{\\log(t)}{a+\\log(e^t-1)} \\) without using L'Hopital's Rule.",
"role": "user"
}
] | 1 |
[
{
"content": "Find the constant \\(c\\) such that \\(\\max_{a\\le n/2}\\frac{C_n^a}{\\sum_{k=0}^{\\lfloor{a/3}\\rfloor}C_n^k}=(c+o(1))^n.\\)",
"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... | \sqrt{\frac{8RT}{\pi M}} |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} (1 + \\sin x)^{\\frac{1}{x}}. \\]",
"role": "user"
}
] | 1 |
[
{
"content": "Compute the limit: \\[ \\lim_{x \\rightarrow 0}\\left(2-e^{\\arcsin^{2} \\sqrt{x}}\\right)^{\\frac{3}{x}} \\]",
"role": "user"
}
] | e^{-3} |
[
{
"content": "Let $f \\in L^2(\\mathbb R^n)$ and define its Fourier transform as $\\hat f(y)=\\int_{\\mathbb R^n} f(x) e^{-2\\pi ixy}dx$ for all $y \\in \\mathbb R^n$. Consider a closed linear subspace $M$ of $L^2(\\mathbb R^n)$ that is translation invariant, meaning $f \\in M$ implies $f_a \\in M$ for all $a \... | 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... | 0.11 % |
[
{
"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... | 3832.11 kW |
[
{
"content": "Evaluate the infinite product: \\[ \\lim_{n \\to \\infty} \\prod_{r=1}^n \\left(1+ \\frac{1}{a_r}\\right), \\] where \\( a_1=1 \\) and \\( a_r=r(1+a_{r-1}) \\) for \\( r \\geq 2 \\).",
"role": "user"
}
] | 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... | $6,290.92 |
[
{
"content": "Given that \\(\\tan^2x\\tan^2y + \\tan^2y\\tan^2z + \\tan^2z\\tan^2x + 2\\tan^2x\\tan^2y\\tan^2z = 1\\), find the value of \\(\\sin^2x + \\sin^2y + \\sin^2z\\).",
"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... | 1.5228 |
[
{
"content": "Find a lower bound for the expression \\(||a-b|-|a+b||\\) where \\(a, b \\in \\mathbb{C}\\) are complex numbers.",
"role": "user"
}
] | 0 |
[
{
"content": "Given a nonnegative random variable $X$ such that $Z = \\log(X)$, where $E(Z) = 0$ and $V(Z) = \\sigma^2 > 0$, determine whether $E(X)$ is greater than, equal to, or less than 1.",
"role": "user"
}
] | E(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... | 65.4 min |
[
{
"content": "Find the value of the limit \\( \\lim_{n \\to \\infty} \\sqrt[n]{\\sqrt[2^n]{a}-1} \\) for \\( a > 1 \\).",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Evaluate the limit \\( \\lim_{{x \\to 1}} \\frac{f(x)}{\\ln(x)} \\), where \\( f(x) = \\int_{1}^{u} e^{-\\sin(t)} \\, dt \\) and \\( u = x^2 \\). Determine if the limit is equal to 0.",
"role": "user"
}
] | 2 e^{-\sin(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... | 1.45e-8 J |
[
{
"content": "Determine whether the field extension $K = \\mathbb{Q}(\\sqrt{2}, \\sqrt{3})$ is a normal extension over $\\mathbb{Q}$. Specifically, is it true that for every irreducible polynomial $f \\in \\mathbb{Q}[X]$ which has at least one root in $K$, $f$ splits completely in $K$?",
"role": "user"
}
... | Yes |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\to 0}\\lim_{n \\to \\infty} \\frac{n}{\\left(1^{\\left(\\cot^2x \\right)}+2^{\\left(\\cot^2x \\right)}+\\ldots+n^{\\left(\\cot^2x \\right)}\\right)^{\\left(\\tan^2x \\right)}}$$",
"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> </answ... | 3.00 cm |
[
{
"content": "Show that the infinite product \\( \\prod_{n=1}^{\\infty} \\frac{R^{2^n}+z^{2^n}}{R^{2^n}} \\) converges for \\(|z|<R\\), and prove that \\( \\prod_{n=1}^{\\infty} \\frac{R^{2^n}+z^{2^n}}{R^{2^n}} = \\frac{R}{R-z} \\).",
"role": "user"
}
] | \dfrac{R}{R - 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... | 25.63 days |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 6.94 |
[
{
"content": "Given vectors \\( u \\in \\mathbb{R}^{n} \\) and \\( v \\in \\mathbb{R}^{m} \\) with unit \\( L^{2} \\) norm, i.e., \\( \\|u\\|_{2} = \\|v\\|_{2} = 1 \\), construct a rank-one matrix \\( B \\in \\mathbb{R}^{m \\times n} \\) such that \\( Bu = v \\) and \\( \\|B\\|_{2} = 1 \\).",
"role": "user"... | B = v u^\top |
[
{
"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... | $9,954.05 |
[
{
"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... | 15.1 degrees |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to \\infty}\\frac{1}{\\sqrt[3]{{n^3}+2n-1}}+\\frac{1}{\\sqrt[3]{{n^3}+2n}}+\\cdots+\\frac{1}{\\sqrt[3]{{n^3}+4n}}$$",
"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... | 12.00% |
[
{
"content": "Determine the number of terms needed in the series expansion to approximate π to within $10^{-3}$ using the formula:\n\n$$\\frac{\\pi}{4} = 4\\arctan\\left(\\frac{1}{5}\\right) - \\arctan\\left(\\frac{1}{239}\\right)$$\n\nand the Maclaurin series for the arctangent function:\n\n$$\\arctan(x)=\\sum... | 3 |
[
{
"content": "Determine whether the Lebesgue integral of a function $f: \\mathbb{N} \\to \\mathbb{R}$ is always $0$, given that the set of natural numbers $\\mathbb{N}$ has Lebesgue measure $0$. Provide a justification for your answer.",
"role": "user"
}
] | 0 |
[
{
"content": "Let the left and right foci of a hyperbola be $F_1$ and $F_2$, and the left and right vertices be $M$ and $N$. If the vertex $P$ of the triangle $\\triangle P F_1 F_2$ lies on the hyperbola, determine the position of the point where the incircle of $\\triangle P F_1 F_2$ is tangent to the side $F_... | C |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to 0} \\frac{a^{\\tan x} - a^{\\sin x}}{\\tan x - \\sin x} \\]",
"role": "user"
}
] | \ln a |
[
{
"content": "A cube is painted so that one side is blue, two sides are red, and three sides are green. How many different such cubes can be painted? Two cubes are considered the same if one cube can be rotated in any way to match the second cube.",
"role": "user"
}
] | 3 |
[
{
"content": "Evaluate the line integral \\( \\int_C \\mathbf{F} \\cdot d\\mathbf{r} \\) for the vector field \\( \\mathbf{F}(x,y) = -y \\mathbf{i} + x \\mathbf{j} \\) over the curve \\( C \\), which is the ellipse given by \\( \\frac{x^2}{4} + \\frac{y^2}{9} = 1 \\).",
"role": "user"
}
] | 12\pi |
[
{
"content": "Let $X$ and $Y$ be independent random variables, each following a standard normal distribution, $X, Y \\sim \\mathcal{N}(0,1)$. Calculate the conditional expectation $E(X \\mid X^2 + Y^2)$. Provide your answer as a function of $X^2 + Y^2$.",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the equality $G = A^{-1}A \\cup A \\cup A^{-1}$ implies $G = A^{-1}A$ for a group $G$ and a subset $A \\subseteq G$. 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> </answ... | 0.723 m/s^2 |
[
{
"content": "Given that the Net Income at time $n$ is $N_n = N_0(1+g)^n$ and the Book Value of Equity at time $n$ is $B_n = B_0 + \\sum_{t=1}^{n} N_0(1+g)^t$, determine the time $n$ at which the fraction \\( \\frac{N_n}{B_{n-1}} \\) converges to the constant growth rate $g$. Assume $N_0$, $B_0$, and $g$ are gi... | \infty |
[
{
"content": "Determine which of the following statements are true for all natural numbers $n$ given the sums $T_n = \\sum_{r = 2n}^{3n-1}\\left(\\frac{r}{r^2+n^2}\\right)$ and $S_n = \\sum_{r = 2n+1}^{3n}\\left(\\frac{r}{r^2+n^2}\\right)$:\n\n(A) $T_n > \\frac{1}{2}\\ln 2$\n\n(B) $S_n > \\frac{1}{2}\\ln 2$\n\n... | AD |
[
{
"content": "Evaluate the limit \\( \\lim_{x\\to 0^+} ~ x\\int_x^1 \\frac{f(t)}{t}dt \\), where \\( f \\) is continuous on \\([0,1]\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Can a polynomial of degree $2k-1$ and a polynomial of degree $2n$, where $k, n \\in \\mathbb{N}$, have identical graphs on a specific interval $[p, q]$?",
"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... | 0.921 A |
[
{
"content": "Suppose that $(x_n)$ is a sequence of complex numbers. If the sequence $(S_N(y))_{N\\geq1}$ with \\[S_N(y) =\\sum_{n=1}^N x_ny_n \\] converges for every $y = (y_n) \\in \\ell^1$, does it follow that $x\\in \\ell^1$? Answer yes or no.",
"role": "user"
}
] | No |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.096 J |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Common stock: $274,000 |
[
{
"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... | 3.098 kcal/g |
[
{
"content": "How many trees \\( T = \\langle V, E \\rangle \\) exist such that \\( V = \\{ v_1, v_2, \\dots, v_7 \\} \\), with \\( \\deg(v_1) = 1 \\) and \\( \\deg(v_j) \\leq 5 \\) for every \\( j = 2, \\ldots, 7 \\)?",
"role": "user"
}
] | 7770 |
[
{
"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... | 10.6 W at t=2.67s |
[
{
"content": "If a function $f$ has an approach independent derivative at a point $\\mathbf{c}$, is $f$ continuous at $\\mathbf{c}$? Answer yes or no.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\rightarrow\\infty} \\frac{1}{n}\\Big[\\log\\Big( \\frac{\\sqrt{2\\pi n} (x-q) - q(1-q) e^{-\\frac{n}{2}(\\frac{x-q}{q(1-q)})^2}}{\\sqrt{2\\pi n} (x-q)} \\Big)\\Big]$$ assuming $x\\ne q$ and $q(q-1)\\ne 0$.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty} \\frac{1}{\\ln n}\\sum_{k=1}^n \\frac{k}{k^2+1} \\]",
"role": "user"
}
] | 1 |
[
{
"content": "Given that $a(x)$ and $b(x)$ are elements of the field $\\mathbb{C}(x)$ of rational functions in $x$ with complex coefficients, and they satisfy the equation $b(x)^2 = a(x)^3 + 1$, determine whether $a(x)$ and $b(x)$ must be constant functions.",
"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... | 12 cm |
[
{
"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... | 5.84e-12 N |
[
{
"content": "Find the minimum value of $x^6 + y^6 - 54xy$ over all real numbers $x$ and $y.$",
"role": "user"
}
] | -108 |
[
{
"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... | 2e-8 M OH- |
[
{
"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... | Carrier A |
[
{
"content": "In a triangle \\(ABC\\) with an area of 1, two players take turns choosing points on the sides of the triangle. The first player selects a point \\(X\\) on side \\(AB\\), the second player selects a point \\(Y\\) on side \\(BC\\), and then the first player selects a point \\(Z\\) on side \\(AC\\).... | \dfrac{1}{4} |
[
{
"content": "Determine the boundary of the set \\(\\mathbb{Q}^n\\) when considered as a subset of \\(\\mathbb{R}^n\\) with the standard topology. Is it true that \\(\\partial \\mathbb{Q}^n = \\emptyset\\)?",
"role": "user"
}
] | \partial \mathbb{Q}^n = \mathbb{R}^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... | 0.01747 m^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... | 1.781 m/s |
[
{
"content": "Given a finite morphism \\(\\pi:X\\to Y\\) of smooth projective curves over an algebraically closed field, both of genus greater than 1, consider the maps on the level of Jacobians: \\(\\pi_\\ast : J_X \\to J_Y\\) and \\(\\pi^\\ast : J_Y\\to J_X\\), satisfying \\(\\pi_\\ast \\pi^\\ast = [\\deg \\p... | (\deg \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... | The Lorenz curve is shown in the image. The x-axis is Cumulative % Families, and the y-axis is Cumulative % Income after tax.
| Class | Families (A) | % Families | Cumulative % Families | Average class of income (B) | Average class of income after tax | Total income (A*B) | % Income | Cumulative % Income |
|---|---|---|---|---|---|---|---|---|
| 1 | 6,000 | 60% | 60% | 20,000 | 14,000 | 84,000,000 | 32.35% | 32.35% |
| 2 | 4,000 | 40% | 100% | 70,000 | 63,000 | 252,000,000 | 67.65% | 100% | |
[
{
"content": "Determine whether the statement \"Any closed set has a point on its boundary\" is true or false.",
"role": "user"
}
] | B |
[
{
"content": "Calculate the expression \\( f''_{xx} + f''_{yy} \\) for the function \\( f(x,y) = \\ln \\sqrt{(x-1)^2 + (y+1)^2} \\).",
"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... | 12.87 |
[
{
"content": "Find a basis for a generalized eigenspace of the linear operator $T$ on $P_2(\\mathbb{R})$, where $T(f(x))=2f(x)-f'(x)$, consisting of a union of disjoint cycles of generalized eigenvectors.",
"role": "user"
}
] | \{2, -2x, 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... | Assets:
Current Assets:
Cash: $16,940
Accounts Receivable: $43,250
Prepaid Insurance: $4,355
Merchandise Inventory: $28,075
Office Equipment: $28,450
Total Assets: $121,070
Liabilities and Stockholders' Equity:
Liabilities:
Accounts Payable: $8,140
Long-term notes payable: $22,300
Total Liabilities: $30,440
Stockholders' Equity:
Common Stock: $41,500
Retained Earnings: $49,130
Total Stockholders' Equity: $90,630
Total Liabilities and Stockholders' Equity: $121,070 |
[
{
"content": "Find the maximum area bounded by the curves \\(y^2=4ax\\), \\(y=ax\\), and \\(y=\\frac{x}{a}\\) for \\(a \\in [1,2]\\).",
"role": "user"
}
] | 84 |
[
{
"content": "Let $f_1, \\ldots, f_k \\in \\mathbb{C}[X_1, \\ldots, X_n]$. If the polynomials $f_i$ vanish at all points of the form $(k, 1, 1, \\ldots, 1)$ for all integers $k$, does it follow that all the polynomials $f_i$ are identically zero? Answer yes or no.",
"role": "user"
}
] | No |
[
{
"content": "Let the ellipse \\(\\Gamma: \\frac{x^{2}}{4} + y^{2} = 1\\) be in the rectangular coordinate system \\(xOy\\). Point \\(L\\) is in the third quadrant and \\(OL = 3\\sqrt{2}\\). The angle between the negative x-axis and \\(OL\\) is \\(\\frac{\\pi}{4}\\). Two tangents \\(l_{1}\\) and \\(l_{2}\\) to ... | 9 |
[
{
"content": "Evaluate the integral \\( \\int_0^{2\\pi}\\frac{\\cos 3x}{5-4\\cos x}dx \\) using complex integration techniques.",
"role": "user"
}
] | \dfrac{\pi}{12} |
[
{
"content": "The four complex roots of\n\\[2z^4 + 8iz^3 + (-9 + 9i)z^2 + (-18 - 2i)z + (3 - 12i) = 0,\\]when plotted in the complex plane, form a rhombus. Find the area of the rhombus.",
"role": "user"
}
] | \sqrt{10} |
[
{
"content": "Determine the minimal value of \\(c\\) such that the function \\(n^{cn}\\) grows faster than \\(n!\\) as \\(n\\) approaches infinity.",
"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... | $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... | 348.78 |
[
{
"content": "Let $f$ be a measurable function on $\\mathbb{R}$ such that $f$ is not almost everywhere zero. Determine whether there exists a set $S \\subset \\mathbb{R}$ with positive Lebesgue measure such that $|f(t)| \\geq c > 0$ for almost every $t \\in S$ and some constant $c > 0$. Provide a justification ... | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.6e4 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> </answ... | 2.61% |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to \\infty} \\left(\\frac{n^n}{\\prod_{k=1}^n (n+k)}\\right)^{1/n}$$",
"role": "user"
}
] | \dfrac{e}{4} |
[
{
"content": "Does there exist a continuous function \\( F: [a, b] \\rightarrow \\mathbb{R} \\) on a set \\( [a, b] \\subseteq \\mathbb{R} \\cup \\{-\\infty, \\infty\\} \\) such that the derivative \\( F'(x) := \\lim_{h\\rightarrow 0}\\frac{F(x+h)-F(x)}{h} \\) exists in \\( \\mathbb{R} \\cup \\{-\\infty, \\inft... | No |
[
{
"content": "Find integer solutions to the equation $x^6 + 5y^2 = z^2$ where $x \\neq 1$. Provide an example or prove whether such solutions exist.",
"role": "user"
}
] | (2, 4, 12) |
[
{
"content": "Let $X_1, X_2, \\ldots, X_n$ be independent and identically distributed random variables. Calculate $E[X_1 \\mid X_1 + X_2 + \\cdots + X_n = x]$. Provide your answer in terms of $x$ and any relevant parameters of the distribution.",
"role": "user"
}
] | \dfrac{x}{n} |
[
{
"content": "Given a random variable $X$ with an exponential distribution $\\mathcal{E}(2)$, where the probability density function (PDF) is $f_X(x) = 2e^{-2x}$ for $x > 0$ and $0$ otherwise, define another random variable $Y$ as follows:\n\n$$Y = \\max\\left(X^2, \\frac{X+1}{2}\\right)$$\n\nDetermine the cumu... | 0 |
[
{
"content": "Let $\\mathbf{w} = \\begin{pmatrix} 1 \\\\ 0 \\\\ -3 \\end{pmatrix}.$ The set of vectors $\\mathbf{v}$ such that\n\\[\\operatorname{proj}_{\\mathbf{w}} \\mathbf{v} = \\mathbf{0}\\]lie on a plane. Enter the equation of this plane in the form\n\\[Ax + By + Cz + D = 0,\\]where $A,$ $B,$ $C,$ $D$ ar... | x - 3z = 0 |
[
{
"content": "Suppose the function \\( f(x) \\) is bounded on the interval \\([0,1]\\) and satisfies the equation \\( f(ax) = bf(x) \\) for all \\( 0 \\le x \\le \\frac{1}{a} \\), where \\( a, b > 1 \\). Determine \\( \\lim_{x \\to 0^+} f(x) \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Express as a fraction in lowest terms: $0.\\overline{1} + 0.\\overline{01}$",
"role": "user"
}
] | \frac{4}{33} |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\rightarrow \\infty }\\frac{\\sqrt[n]{\\sum_{k=2}^{n}(k-1)!}}{n}$$",
"role": "user"
}
] | \dfrac{1}{e} |
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