inclusionAI/Ring-lite-rl-data huggingface dataset:
Given that $2X^T = A^{-1}BA$, where $A$, $B$, and $X$ are $4 \times 4$ matrices, $\det(A) = 4$, and $\det(B^{-1}) = 3$, find $\det(X)$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{48}}
Compute \(d\alpha(V, W)\) at the point \((0,1)\) on \(\mathbb{R}^{2}\), where \(V = x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}\), \(W = y \frac{\partial}{\partial y}\), and the differential 1-form \(\alpha\) satisfies \(\alpha(V) = 2x - y^{2}\) and \(\alpha(W) = 2y\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of ways to choose $k$ numbers from the first $n$ natural numbers such that the longest consecutive sequence among the chosen numbers is exactly $m$ numbers long. For example, calculate the number of combinations when $k = 7$, $n = 14$, and $m = 3$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1456}
In the diagram, a circle \(\omega\) with center \(P\) is tangent to line segment \(AB\) at point \(T\), and \(\angle APB = 90^\circ\). Point \(K\) is on circle \(\omega\), and a semicircle with diameter \(BK\) intersects \(PB\) at point \(Q\). Given that the radius \(R\) of the semicircle satisfies \(4R^2 - AT \cdot TB = 10\), and \(PQ = \sqrt{2}\), find the length of \(BQ\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3}}
Let
 \( f\left( t\right)  = \frac{\left( {{10} + {9i}}\right) t - {10} + {9i}}{t + i}, \)
 where \( i = \sqrt{-1} \) . Let \( P = f\left( 0\right) ,Q = f\left( {2023}\right) \) ,and
 \( R = f\left( 1\right) \) . Determine \( {\sin }^{2}\left( {m\angle {PQR}}\right) \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\frac{1}{2}$

}
Emma's calculator has ten buttons: one for each digit $1, 2, \ldots, 9$, and one marked "clear". When Emma presses a digit button, that digit is appended to the right of the display. When she presses the "clear" button, the display is erased. Starting with an empty display, Emma presses five buttons at random, with each button having an equal chance of being chosen. The expected value of the number displayed is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $100m + n$. (An empty display represents the number 0.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6642902}
Given a set $\Omega$, let $A_1, A_2, \ldots, A_n$ be distinct subsets of $\Omega$. Determine the number of sets in the $\sigma$-field generated by $\{A_1, A_2, \ldots, A_n\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2^n}}
In a game involving two players, the following system of equations is given:

\[
\begin{aligned}
& a_{1} x_{1}+\ldots+a_{n} x_{n}=c \\
& b_{1} x_{1}+\ldots+b_{n} x_{n}=d
\end{aligned}
\]

Players take turns assigning arbitrary real numbers to the variables \(a_{i}, b_{i}, x_{i}\) (where \(i=1,2,\ldots,n\)) and the constants \(c, d\). In each turn, a player can choose any remaining variable to assign a value, but they can only assign 0 to a variable if it is the last unassigned variable in its equation.

Determine the value of \(n\) for which the first player can always ensure that both equations are satisfied, regardless of the second player's choices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the operator norms \( ||S||, ||T||, ||ST||, \) and \( ||TS|| \) for the operators \( S, T: C[0,1] \rightarrow C[0,1] \) defined by \( S(x)(z) = z \int_{0}^{1} x(t) \, dt \) and \( T(x)(z) = z \cdot x(z) \). It is known that \( ||T|| = ||S|| = ||TS|| = 1 \). Determine \( ||ST|| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
The variables \( a,b,c,d, \) traverse, independently from each other, the set of
 positive real values. What are the values which the expression
 \( S = \frac{a}{a + b + d} + \frac{b}{a + b + c} + \frac{c}{b + c + d} + \frac{d}{a + c + d} \)
 takes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
The range of \( S \) is \( (1, 2) \).

}
Determine the rank of the linear transformation $A$ defined by left multiplication by a matrix $P$ of rank $m$ on the space of all linear transformations on an $n$-dimensional vector space $V$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{mn}
Calculate the limit: \[ \lim_{n \to \infty} \sqrt[n]{|\sin n|} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Assuming \(g(x) = \int \frac{x^2 - 12}{(x^2 - 6x + k)^2}dx, \quad k \in \mathbb{N},\) where \(g(x)\) is a rational function, find the sum of all possible values of \(k\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Evaluate the limit \( \lim_{n \to \infty} \left( \sqrt{|B_{2n}|} - \sqrt{|B_{2n+2}|} \right) \), where \( B_{2n} \) and \( B_{2n+2} \) are consecutive even Bernoulli numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\infty}
Calculate the limit of the sequence \(b_n = n\left(\sqrt[n]{eb}-\sqrt[n]{b}\right)\), where \(b>0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the surface integral \( \int_{\tau} z \, d\tau \) over the surface \( S: x^2+y^2+z^2=a^2 \) with \( z \geq 0 \) using Gauss's Divergence Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi a^3}
Evaluate the integral \( \frac{2}{\pi i}\int_C \left(ze^{1/z}+\tan\left(\frac{z}{2}\right)+\frac{1}{(z-1)(z-3)^2}\right)dz \), where \( C \) is the positively oriented circle of radius 2 centered at the origin in the complex plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the contour integral \( \oint_C (4z - 1) \, dz \) along the quarter-circle \(|z| = 1\) from the point \(z_1 = 0\) to \(z_2 = 1\) in the first quadrant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the circle \(x^2 + y^2 + 2gx + 2fy + c = 0\) intersects the circle \(x^2 + y^2 - 5 = 0\) at the extremities of its diameter, and the centers of the circles \(x^2 + y^2 - 8x - 6y + 10 = 0\) and \(x^2 + y^2 - 4x + 2y - 2 = 0\) lie on the first circle, find the value of \(c + g + f\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-8}
Determine the smallest value of the constant $p$ such that the series \( \sum_{n=1}^{\infty}\frac{2^{np}(n!)^2}{(2n)!n^{n}} \) is convergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the solutions to $z^3 = -8.$  Enter the solutions, separated by commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["-2, 1 + i \\sqrt{3}, 1 - i \\sqrt{3}"]}
Evaluate the integral \( \iint_{D} \frac{\mathrm{d}x\,\mathrm{d}y}{\sqrt{1 - x^{2} - y^{2}}} \), where \( D = \{(x,y) \mid x^{2} + 4y^{2} \leq 1\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2\pi}{3}}
If \( 1 < {k}_{1} < {k}_{2} < \ldots  < {k}_{n} \) and \( {a}_{1},{a}_{2},\ldots ,{a}_{n} \) are integers such that for every integer \( I \)
 \( {k}_{i} \mid  N - {a}_{i} \) for some \( 1 \leq  i \leq  n \) ,find the smallest possible value of \( n \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
5

}
Determine the Krull dimension of the localization $S^{-1}A$, where $A$ is a commutative reduced ring and $S$ is the set of all non-zerodivisors in $A$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of distinct parametrizations for the surface defined by the equation \(|x|+|y|+|z|=1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Assume we have a problem $R \in \mathit{NP}$ that can be solved with $M(x,y)$ in $O(n^3)$ time, where $y$ is additional information no longer than $3\log n$ bits. For what function $f(n)$ can we say that $R \in \mathit{TIME}(f(n))$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^6}
In a game, Sheldon picks three non-zero real numbers, and Leonard arranges them as the coefficients of a quadratic equation \( ax^2 + bx + c = 0 \). Sheldon wins if the resulting equation has two distinct rational solutions; otherwise, Leonard wins. What is the maximum probability that Sheldon wins the game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the integral \( \int_{-\tfrac\pi2}^{\tfrac\pi2} \ln(\cos \tau) \cos \tau \, d\tau \) using the beta function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \ln 2 - 2}
Find the norm of the linear functional \( l(x) = x(0) + \int_0^1 (2t-1)x(t) \, dt \) on the space of continuous functions \( C[0,1] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
Determine the constant $c$ to which the sequence $\frac{1}{n} \sum_{i=1}^n X_i$ converges almost surely as $n$ approaches infinity, given a sequence of independent random variables $\{X_i, i\geq1\}$ with probabilities $\Bbb P(X_i=i^2 -1)=\frac{1}{i^2}$ and $\Bbb P(X_i=-1)=1-\frac{1}{i^2}$. \(X_i\) are independent random variables.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Calculate the present value of a 10-year deferred life annuity-due of one dollar per annum at Kimberly's issue age, given an interest rate of 3%. Use the formula for a deferred annuity and provide the final present value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25.55}
Let \( P(x) \) be a polynomial in \( \mathbb{R}[x] \) with a leading coefficient of 1. Determine the leading coefficient of the polynomial \( P(2x+1) - 2P(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n - 2}
Equilateral \( \bigtriangleup  {ABC} \) has side length600. Points \( P \) and \( Q \) lie outside of the plane of \( \bigtriangleup  {ABC} \) and are on the opposite sides o
 the plane. Furthermore, \( {PA} = {PB} = {PC} \) , and \( {QA} = {QB} = {QC} \) , and the planes of \( \bigtriangleup  {PAB} \) and \( \bigtriangleup  {QAB} \) form a \( {120}^{ \circ  } \)
 dihedral angle (The angle between the two planes). There is a point \( O \) whose distance from each of \( A,B,C,P \) and \( Q \) is \( d \) .
 Find \( d \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
450
}
Evaluate the integral \( \int_0^{+\infty}\cos 2x\prod_{n=1}^{\infty}\cos\frac{x}{n}dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
In an acute angle triangle $ABC,$ let $D$ be the foot of the altitude from $A,$ and $E$ be the midpoint of $BC.$ Let $F$ be the midpoint of $AC.$ Suppose $\angle{BAE}=40^o. $ If $\angle{DAE}=\angle{DFE},$ What is the magnitude of $\angle{ADF}$ in degrees $?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\angle ADF=40^{\circ}$}
Determine the dimension of the complex vector space generated by the set \( E_n = \{ e^{2 \pi i k x} \mid -n \leq k \leq n \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n + 1}
For positive integers \( m, n \geq 2 \), and a real number \( x \) such that \( 0 \leqslant x < n \), define the function:

\[ f(n, x) = (1 - \{x\}) \cdot \binom{n}{\lfloor x \rfloor} + \{x\} \cdot \binom{n}{\lfloor x \rfloor + 1} \]

where \( \lfloor x \rfloor \) is the greatest integer less than or equal to \( x \), and \( \{x\} = x - \lfloor x \rfloor \). Given that:

\[ f\left(m, \frac{1}{n}\right) + f\left(m, \frac{2}{n}\right) + \cdots + f\left(m, \frac{mn - 1}{n}\right) = 123, \]

find the value of:

\[ f\left(n, \frac{1}{m}\right) + f\left(n, \frac{2}{m}\right) + \cdots + f\left(n, \frac{mn - 1}{m}\right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74}
Two boats, Boat 1 and Boat 2, start simultaneously from docks A and B on a river, moving towards each other. Boat 1's speed in still water is twice that of Boat 2. Boat 1 leaves a floating cargo box on the water upon departure. After 20 minutes, the boats meet, and Boat 1 leaves another identical cargo box. Later, Boat 1 realizes it is missing cargo and turns around to search for it. When Boat 1 finds the second cargo box, Boat 2 encounters the first cargo box. How many minutes have passed since Boat 1 departed when it realizes its cargo is missing? Assume the time for turning around is negligible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
A manufacturer of car radios ships them to retailers in cartons of \( n \) radios. The profit per radio is \$59.50,\) less a shipping cost of \$25\) per carton, resulting in a profit of \( 59.5n - 25 \) per carton. To promote sales by assuring high quality, the manufacturer promises to pay the retailer \( 200X^2 \) if \( X \) radios in the carton are defective. Radios are produced independently, with a \( 5\% \) defect rate. Determine the number of radios that should be packed per carton to maximize the expected net profit per carton.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Given a triangle \(\triangle PQR\) with sides 16, 30, and 34, let the incircle touch the sides of \(\triangle PQR\) at points \(X, Y,\) and \(Z\). The ratio of the area of triangle \(\triangle XYZ\) to the area of triangle \(\triangle PQR\) can be expressed as \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Evaluate the improper integral \( \int_{0}^{1} \ln(e^{1/x}+1)\, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the value of \( \limsup_{n \to \infty}(n!|a_n|)^{1/n} \) for an entire function \( f(z) \) that satisfies \( |f(z)| \le e^{(2\pi M+\epsilon)|z|} \) for any \( \epsilon > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi M}
Determine $w^2+x^2+y^2+z^2$ if
\[ \begin{array}{l} \displaystyle \frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1 \\ \displaystyle \frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1 \\ \displaystyle \frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1 \\ \displaystyle \frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1 \\ \end{array}  \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Solve the equation: \(5x = 0.01^x\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.1168}
Determine the set of positive numbers \( b \) such that \( b^x \geq 1+x \) for all real numbers \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
Given $m,n\geq 2$.Paint each cell of a $m\times n$ board $S$ red or blue so that:for any two red cells in a row,one of the two columns they belong to is all red,and the other column has at least one blue cell in it.Find the number of ways to paint $S$ like this.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(n+1)^m+n^{m+1}-n^2$}
Calculate the expected value \( E\left(\frac{X_1}{X_2+1}\right) \) where \( X_1 \) and \( X_2 \) are independent random variables with the density function \( P[X_1=k]=P[X_2=k]=pq^k \) for \( k=0,1,2,\ldots \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\ln p}
Two players are playing a game. One player thinks of a sequence of integers \((x_{1}, x_{2}, \ldots, x_{n})\), which are single-digit numbers, both positive and negative. The other player can ask for the value of the sum \(a_{1} x_{1}+\ldots+a_{n} x_{n}\), where \((a_{1}, \ldots, a_{n})\) is any chosen set of coefficients. What is the minimum number of questions needed for the guesser to determine the thought-of sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of different quadratic functions of the form \( x \mapsto ax^2 + bx + c \pmod{12} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{864}
Frank alternates between flipping two weighted coins. The first coin has a \(\frac{2}{3}\) chance of landing heads and a \(\frac{1}{3}\) chance of landing tails. The second coin has a \(\frac{1}{4}\) chance of landing heads and a \(\frac{3}{4}\) chance of landing tails. Starting with the first coin, what is the probability that Frank sees two heads in a row before he sees two tails in a row?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{33}}
Consider n balls, each labeled with a number from 1 to 365, placed in a row. A ball is selected at random. Let P be the probability that the number on the selected ball is the same as the number on any one of the balls placed to its left, with all balls to the left having distinct numbers. Determine the position K, counting from left to right, that maximizes P.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
A consumer is searching for 1 unit of a product across stores with prices 1, 2, 3, ..., each occurring with probabilities 1/2, 1/4, 1/8, ..., 1/2^n respectively. The consumer incurs a search cost of 1 for each store visited. Determine the price at which the consumer should stop searching.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the possible range of values for \( a \) such that the area under the curve \( y = e^x + \cos(x) \ln(x + \sqrt{1+x^2}) \) from \( x = -a \) to \( x = a \) is less than \( \frac{3}{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, \ln 2)}
Evaluate the limit: $$ \lim_{n\to\infty}\frac{3^{3n} \binom{2n}{n}^2}{\binom{3n}{n} \binom{6n}{3n}} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of orders on the rational numbers \( \mathbb{Q} \) that are compatible with the standard field operations. Is this number finite, infinite, or uncountably infinite?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the equations:
\[ 11\sin x + 7\sin y = 3(\sin z + 3\sin w) \]
\[ 11\cos x + 7\cos y = 3(\cos z + 3\cos w) \]
find the value of \( \frac{\cos(x-z)}{\cos(w-y)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{21}{11}}
Evaluate the limit: $$\lim_{r\rightarrow \infty}r\cdot \bigg(\frac{2}{\pi}\bigg)^{r+1}\cdot \int^{\frac{\pi}{2}}_{0}e^{r\log(x)}\sin(x)\,dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Apply one step of Richardson's Extrapolation to compute the improved value for $M$ given $N(3h) = 2.7172$ and $N(h) = 2.7182$. Assume $M = N(h) + Ah^{4/3} + Bh^{7/3}$ and use $h = 0.0001$ if needed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.7185}
Determine whether there exists a function \( f \) such that \( f(f(r)\sin \theta) = f(r)\cos \theta \) for all \( \theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(r) = 0}
Find the sum of the squares of the real solutions to the equation \( \sqrt[3]{5x - 2} = \frac{1}{5}(x^3 + 2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find a number \( x \) such that for all integers \( a \) in the set \( \{0, 1, 2, \ldots, 1926\} \), the congruence \( a^x \equiv a \pmod{1926} \) holds. Exclude \( x = 1 \) from consideration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{319}
Determine the isomorphism class of the quotient group $\mathbb{R}^+/U$, where $\mathbb{R}^+$ is the group of positive real numbers under multiplication and $U$ is the unit circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Evaluate the contour integral \( \oint_{|z|=2} \frac{f'(z)}{f(z)} \, dz \), where \( f(z)=\frac{2 z^{4}-7 z^{3}-2 z-i}{(z-i)^{2}(3 z^{2}-z+9)^{2}} \). The function \( f(z) \) has no zeros or poles on the contour \( |z|=2 \), but may have poles inside the contour. Use the argument principle to determine the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-6\pi i}
Evaluate the integral \( \int _0^\infty \dfrac{\ln x}{(x^2+1)^2}dx. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{4}}
Determine the dimensions $n$ for which the existence of an $n \times n$ complex matrix $A$ with $\det(I-A^*A)<0$ implies that the operator norm of $A$ is greater than 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \geq 1}
Compute the Galois group \( \operatorname{Gal}(\mathbb{Q}(\sqrt{2},\sqrt{3})/\mathbb{Q}(\sqrt{2})) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}/2\mathbb{Z}}
For how many distinct triangles $\triangle ABC$, with $AB = 2011$, are both $\cos(2\angle A + 3\angle C)$ and $\sin(2\angle B + \angle C)$ integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the Rao-Cramer lower bound for the parameter \( \theta \) of the distribution with the probability density function given by \[ f(x;\theta)=\frac{1}{\theta}x^{\frac{1-\theta}{\theta}} \] where \( 0<x<1 \) and \( 0<\theta<\infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\theta^2}
A knight starts at position \((0, 0)\) on an infinite chessboard. In each move, the knight can move to one of the 8 possible positions: \((a \pm 1, b \pm 2)\) or \((a \pm 2, b \pm 1)\), each with equal probability. Calculate the probability that after 2023 moves, the knight is on a square \((a, b)\) such that \(a \equiv 4 \pmod{8}\) and \(b \equiv 5 \pmod{8}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{32}}
The sequence $\{a_i\}_{i \ge 1}$ is defined by $a_1 = 1$ and\[ a_n = \lfloor a_{n-1} + \sqrt{a_{n-1}} \rfloor \]for all $n \ge 2$. Compute the eighth perfect square in the sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
16384
}
Find a function \( g(x) \) that satisfies the following conditions: \( g''(0) > 0 \), \( g(x) \to -\infty \) as \( x \to \infty \) and \( x \to -\infty \), and \( g(x) = -g(-x) \) for all \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{g(x) = -x^3}
Find a permutation \( \sigma \) such that \( \sigma(1,2)(3,4)\sigma^{-1} = (5,6)(3,1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1\,5\,2\,6\,4)}
Evaluate the integral \( \int_C e^{\pi(x+iy)} \, dz \), where \( C \) is the square with vertices \( 0, 1, i, 1+i \) traversed once in the counterclockwise direction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A beacon light is located 4 miles offshore, perpendicular to a straight shoreline, and rotates at 3 revolutions per minute. Determine the speed at which the spot of light moves along the shoreline when the beam forms a 60° angle with the shoreline.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32\pi}