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... | 5,507.47 |
[
{
"content": "Compute the limit of the sequence: \\( \\sqrt{2^\\sqrt{2^\\sqrt{2^\\sqrt{\\ldots}}}} \\)",
"role": "user"
}
] | 2 |
[
{
"content": "Compute the limit: $$\\displaystyle \\lim_{\\alpha \\to -1}\\left(2\\Gamma(-\\alpha-1)+\\Gamma\\left(\\frac{\\alpha + 1}{2}\\right) \\right)$$",
"role": "user"
}
] | -3\gamma |
[
{
"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... | 10.26 W/m^2 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... | 64% |
[
{
"content": "How many factors of 1000 can be divided by 20 without a remainder?",
"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... | s=\frac{1}{2\mu g} (\frac{mv_o}{M+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... | 11.5e23 atoms |
[
{
"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... | 15N |
[
{
"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... | 4.42e-20 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... | 30541.17 |
[
{
"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 (Prepaid Expenses) |
[
{
"content": "If two distinct members of the set $\\{ 3, 7, 21, 27, 35, 42, 51 \\}$ are randomly selected and multiplied, what is the probability that the product is a multiple of 63? Express your answer as a common fraction.",
"role": "user"
}
] | \frac{3}{7} |
[
{
"content": "Determine whether the following statement is true or false: If \\( R[x] \\) is a unique factorization domain (UFD), then \\( R \\) is also a UFD.",
"role": "user"
}
] | A |
[
{
"content": "Can a sequence be constructed such that all rational numbers are its limit points?",
"role": "user"
}
] | Yes |
[
{
"content": "Find the greatest common divisor (GCD) of $7^{3001} - 1$ and $7^{3000} + 1$. Express your answer as a single integer.",
"role": "user"
}
] | 2 |
[
{
"content": "The number of apples that Sophia has is a multiple of $6$. When she sells one apple, the number of apples is a multiple of $n$. If $n$ is a positive integer less than $10$, how many possible values are there for $n$?",
"role": "user"
}
] | 3 |
[
{
"content": "Determine the limit of the function \\( f(x,y) = \\frac{y}{x} \\) as \\((x,y)\\) approaches \\((0,0)\\) with the restriction that \\(|y| < |x^5|\\).",
"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... | 5.78% |
[
{
"content": "Determine whether the function \\( \\frac{e^{-\\frac{1}{x}}}{x} \\) is uniformly continuous on the interval \\((0, 1)\\).",
"role": "user"
}
] | Yes |
[
{
"content": "In triangle $ABC$, points $D$ and $E$ lie on sides $BC$ and $AC$, respectively, such that $\\frac{BD}{DC} = \\frac{CE}{EA} = 2$. Let $AD$ and $BE$ intersect at point $P$. Given that the area of $\\triangle BPD$ is 252 more than the area of $\\triangle APE$, find the sum of the areas of $\\triangle... | 432 |
[
{
"content": "Evaluate the product \\( \\prod_{i=1}^{89} \\tan i^\\circ = \\tan 1^\\circ \\cdot \\tan 2^\\circ \\cdots \\tan 89^\\circ \\).",
"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... | schizotypal personality disorder |
[
{
"content": "Determine the Galois group of the splitting field of the polynomial \\( X^5 - 4X + 6 \\) over \\( \\mathbb{Q} \\).",
"role": "user"
}
] | S_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... | 9.45% |
[
{
"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... | 2.82 |
[
{
"content": "Find the derivative of the function \\( F(x) = \\int_0^{x} t f(x^2 - t^2) \\, dt \\).",
"role": "user"
}
] | x f(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... | 1*10^-7 M |
[
{
"content": "Given a sufficiently large positive integer $n$, consider a set $S=\\{M_i\\}$ consisting of $n$ matrices with entries in either $\\mathbb{C}$ or $\\mathbb{Z}/m\\mathbb{Z}$. Determine if the following conditions are possible:\n\n1. For all $1 \\le i \\le n$, $M_i^2 = 0$.\n2. The product of any perm... | 2^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... | 0.12 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... | 9.36e-6 F |
[
{
"content": "Does the weak convergence of self-adjoint trace-class operators, along with the weak convergence of their absolute values and the convergence of their trace class norms, imply their trace class convergence? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | -3979.916828 |
[
{
"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... | 0.014 m |
[
{
"content": "Suppose $H$ and $K$ are groups, and there exists an isomorphism $H \\cong K \\times H$. Does it necessarily follow that $K$ is the trivial group? 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... | 1,448,617.6 |
[
{
"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... | (20, 20) |
[
{
"content": "Let $AP_1, AP_2, AP_3$ be three arithmetic progressions with common terms $m, n, r$, respectively. If $AP_1 \\cap AP_2 = AP(7, 15)$ and $AP_2 \\cap AP_3 = AP(a, d)$, find the value of $a + d$. \\( \\text{Calculate } a + d. \\)",
"role": "user"
}
] | 22 |
[
{
"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... | 54280 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... | 660 N |
[
{
"content": "Solve the equation: \\( \\sqrt[3]{5\\sqrt{2} + 7} - \\sqrt[3]{5\\sqrt{2} - 7} = 2 \\)",
"role": "user"
}
] | 2 |
[
{
"content": "Show that the centralisers \\(C_{S_6}(s)\\) and \\(C_{S_6}(t)\\) are isomorphic to \\(S_3 \\times C_3\\), where \\(s = (123)\\) and \\(t = (123)(456)\\) are elements of \\(S_6\\).",
"role": "user"
}
] | S_3 \times C_3 |
[
{
"content": "Does every non-empty open set have Lebesgue measure greater than zero?",
"role": "user"
}
] | Yes |
[
{
"content": "In a self-complementary graph G with exactly m vertices of degree k, determine the number of vertices that must have a degree of n - k - 1.",
"role": "user"
}
] | m |
[
{
"content": "Evaluate the integral \\( \\int_{-\\infty}^0 \\delta(t) \\, dt \\), where \\( \\delta(t) \\) is the Dirac delta function.",
"role": "user"
}
] | 1 |
[
{
"content": "Given the expansion \\((1+x+x^2)^n = a_{0}+a_{1}x + a_{2}x^{2} +\\cdots +a_{2n}x^{2n}\\), find the sum of the coefficients indexed by multiples of 3, i.e., find the value of \\(a_{0}+a_{3}+a_{6}+\\cdots\\).",
"role": "user"
}
] | 3^{n-1} |
[
{
"content": "Consider a set of $n$ independent and identically distributed (i.i.d.) random variables $X_i \\sim U(0,\\theta)$, where $U(0,\\theta)$ denotes the uniform distribution on the interval $(0, \\theta)$. Determine the conditional distribution of $X_1$ given that the maximum $M$ of the $X_i$'s is less ... | \frac{1}{m} |
[
{
"content": "Does there exist a finite, simple, undirected, connected graph $G=(V,E)$ such that there is a pair of non-adjacent vertices $\\{v, w\\}$ where collapsing $v$ and $w$ increases the chromatic number, but for all pairs of non-adjacent vertices $\\{a, b\\}$, adding an edge between $a$ and $b$ does not... | 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... | 0.04 m^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... | 0.92 |
[
{
"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 effective interest method amortization table for the first four semi-annual interest periods is as follows:
| Date | Interest Expense (a) | Interest Payable (b) | Amortization of Discount (a - b) | Bonds Payable |
| ----------- | -------------------- | ------------------- | -------------------------------- |... |
[
{
"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... | $8,264 |
[
{
"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... | 3.78e8 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... | $v\approx36.116\tan(0.08274(14.040-t))$ |
[
{
"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> </answ... | 33.52e-18 kg |
[
{
"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... | $675,000 |
[
{
"content": "Find the minimum distance between the curves \\( f(x) = e^x \\) and \\( g(x) = \\ln x \\).",
"role": "user"
}
] | \sqrt{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... | 256.85 nC |
[
{
"content": "Evaluate the limit: \\[ \\lim\\limits_{x\\to 0} \\frac{\\int_{0}^{x^2}\\ln(1+t+t^2)dt}{\\int_{0}^{x}\\ln(1+t^3)dt} \\]",
"role": "user"
}
] | 2 |
[
{
"content": "Let $G$ be an abelian group, $S = \\{ x^2 : x \\in G\\}$, and $T = \\{ a \\in G : a^2 = e\\}$. Prove that $G/T$ is isomorphic to $S$.",
"role": "user"
}
] | G/T \cong 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... | 0.17 m |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.024 m |
[
{
"content": "Determine whether the radius of convergence of the power series \\( (1+x)^a = \\sum_{n=0}^{\\infty} \\binom{a}{n} x^n = \\sum_{n=0}^{\\infty} \\frac{a(a-1)(a-2) \\cdots (a-n+1)}{n!}x^n \\) can be equal to 1. If so, under what conditions does this occur?",
"role": "user"
}
] | a \notin \mathbb{N} \cup \{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... | 2.67 rpm |
[
{
"content": "Determine the radius of convergence for the Taylor series expansion of the function \\( f(x) = \\frac{1}{1+x^2} \\) about a point \\( a \\in \\mathbb{R} \\).",
"role": "user"
}
] | \sqrt{a^2 + 1} |
[
{
"content": "Let $f:(-1,1)\\mapsto(-1,1)$ be a continuously differentiable function such that $f(x)=f(x^2)$ for all $x$ in $(-1,1)$ and $f(0)=\\frac{1}{2}$. Determine the value of $f(\\frac{1}{4})$. \\( \\boxed{\\text{value}} \\)",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "In an isosceles triangle \\( \\triangle ABC \\), \\( CD \\) is the altitude to the base \\( AB \\), \\( E \\) is the midpoint of \\( BC \\), and \\( AE \\) intersects \\( CD \\) at \\( F \\). Consider the following routes:\n\n(a) \\( A \\rightarrow F \\rightarrow C \\rightarrow E \\rightarrow B \\... | B |
[
{
"content": "Determine whether the following statement is true or false: If $f$ is convex on an open interval $(0, 1)$, then $f$ is continuous on $(0, 1)$. Provide a justification for your answer.",
"role": "user"
}
] | A |
[
{
"content": "Find the function \\( f(x) \\) that satisfies the functional equation \\[ f(x) = \\frac{1}{2}\\left[f(xy)+f\\left(\\frac{x}{y}\\right)\\right] \\] for all \\( x \\in \\mathbb{R^{+}} \\), given that \\( f(1)=0 \\) and \\( f'(1) = 2. \\)",
"role": "user"
}
] | 2 \ln x |
[
{
"content": "Let $f: B(0,R) \\rightarrow \\mathbb{R}$ be a continuous function. Determine whether the following limit holds:\n\\[ \\lim_{r \\rightarrow 0} \\frac{1}{\\text{area}(\\partial B(0,r))} \\int_{\\partial B(0,r)} (f(x)-f(0)) \\, dS(x) = 0, \\]\nfor $B(0,r) \\subset B(0,R)$. Provide a justification for... | 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... | 5.06% |
[
{
"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 |
[
{
"content": "Evaluate the integral \\( \\iint \\cos\\left(\\frac{x+y}{2}\\right) \\sin\\left(\\frac{x-y}{2}\\right)\\,dA \\) over the region \\( R \\) bounded by the lines \\( x=0 \\), \\( y=0 \\), and \\( x+y=2 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the sequence of operators \\( \\{T_n:H\\rightarrow H\\}_n \\) with \\( H=l^2(\\mathbb{N};\\mathbb{R}) \\) defined by \\( T_n(x_0,x_1,x_2,...)=(x_n,x_{n-1},...,x_0,0,0,...) \\) converges weakly to zero.",
"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... | 1.00 cm |
[
{
"content": "What is the minimum number of comparisons required to determine if a given number $x$ is present in a matrix $B$, where the matrix satisfies the condition $B_{ij} \\le B_{kl}$ for all $i \\ge k$ and $j \\le l$?",
"role": "user"
}
] | m + n - 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... | 0.395 m |
[
{
"content": "Determine whether there exist two subgroups of SYM($\\aleph_0$) with infinite support that are not isomorphic but have the same order.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the integral \\( \\int_0^{2\\pi}\\int_1^\\infty{\\frac{r}{\\sqrt{1+r^2}}}drd\\theta \\).",
"role": "user"
}
] | \infty |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | The two most important ones are (a) it does not conduct electricity in solid form, but does in liquid form and solution and (b) it is brittle and shatters when hit, not malleable or ductile. |
[
{
"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... | decreased breath sound |
[
{
"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... | Rw |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to +\\infty} \\sum_{k=0}^n \\left(\\sqrt[n]{k} - \\sqrt[n]{n}\\right). \\]",
"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... | 85470 min |
[
{
"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... | $12.96 |
[
{
"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... | H_2PO_4^{-} + NO_2^-
ightleftharpoons HPO_4^{2-} + HNO_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 firm would produce zero units and profits would equal -$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... | 7.4 s |
[
{
"content": "An equilateral triangle with side length $n$ is divided into smaller equilateral triangles with side length $1$. Determine the maximum number of unit segments that can be selected such that no three segments form the sides of a single small triangle.",
"role": "user"
}
] | n(n + 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... | 3.273 |
[
{
"content": "Evaluate the integral \\( \\int_C \\frac{\\sin(z)}{z^2} \\, dz \\), where \\( C \\) is the unit circle in \\( \\mathbb{C} \\). Use Cauchy's integral formula and consider the limit approach for handling the singularity at \\( z = 0 \\).",
"role": "user"
}
] | 2\pi i |
[
{
"content": "Calculate the value of \\( E[X|XY] \\) where \\( X \\) and \\( Y \\) are two independent standard normal variables.",
"role": "user"
}
] | 0 |
[
{
"content": "The letters of the alphabet are given numeric values based on the two conditions below.\n\n$\\bullet$ Only the numeric values of $-2,$ $-1,$ $0,$ $1$ and $2$ are used.\n\n$\\bullet$ Starting with A and going through Z, a numeric value is assigned to each letter according to the following pattern... | -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... | 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... | 20.1234% |
[
{
"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... | 221.7 m |
[
{
"content": "Determine whether there exists at least one positive integer \\( n \\ge 2 \\) such that for every integer \\( N \\ge 2 \\), it is possible to arrange all the numbers from 1 to \\( n \\) in a row so that the sum of every two adjacent numbers is a power of \\( N \\).",
"role": "user"
}
] | No |
[
{
"content": "Determine the order of growth of the solution to the recurrence relation $T(n) = T(\\log n) + O(1)$, given that $T(1) = 1$. Provide your answer in Big O notation.",
"role": "user"
}
] | O(\log^* n) |
[
{
"content": "Determine the cardinality of a Hilbert space $H$ that has an orthonormal basis with cardinality $\\mathfrak{c}$.",
"role": "user"
}
] | \mathfrak{c} |
[
{
"content": "Given that $x^2 + \\frac{1}{x^2} = 7$, what is the value of $x^4 + \\frac{1}{x^4}$?",
"role": "user"
}
] | 47 |
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