inclusionAI/Ring-lite-rl-data huggingface dataset:
One hundred balls labelled $1$ to $100$ are to be put into two identical boxes so that each box contains at least one ball and the greatest common divisor of the product of the labels of all the balls in one box and the product of the labels of all the balls in the other box is $1$. Determine the number of ways that this can be done.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
The answer is 2047.

}
Evaluate the limit: $$\lim_{n\to \infty}\frac{1+2^2\sqrt{2}+\dots +n^2\sqrt[n]{n}}{n^2(n+1)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Calculate the flux of the vector field \( \boldsymbol{\mathrm{F}} = x\boldsymbol{\mathrm{i}} + 4y \boldsymbol{\mathrm{j}} \) outwards across the triangle with vertices at \((0,0), (2,0), (0,2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Given 8 independent runners with finish times distributed as $X_1, X_2, \ldots, X_8 \sim \text{Exp}(1)$, find the expected value of the difference between the time of the first and second runners to finish, $E(T_2 - T_1)$, where $T_1$ is the time of the first runner to finish and $T_2$ is the time of the second runner to finish.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{7}}
Calculate the expected value of the points in the set \( P \) as \( N \to \infty \), where \( P = \{ x_i \mid x_i > 2 \} \) and \( x_i \) are independently generated from a normal distribution with mean 0 and standard deviation \( \sigma = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.05}
For \(n \geq 2\), let \(G = (V, E)\) be the loop-free undirected graph where \(V\) is the set of binary \(n\)-tuples (of 0's and 1's) and \(E = \{ \{v, w\} : v, w \in V, \text{ and } v, w \text{ differ in exactly two positions} \}\). Determine the number of connected components, \(\kappa(G)\), of the graph \(G\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Consider the set $X = \mathbb{N} \times \mathbb{Q}$ with the subspace topology induced by the usual topology on $\mathbb{R}^2$. Let $P = \{(n, \frac{1}{n}) : n \in \mathbb{N}\}$. Determine whether the set $P$ is open, closed, both, or neither in this topology.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{closed}
Find the minimum positive real number \( l \) such that the inequality \( 7\sqrt{a} + 17\sqrt{b} + l\sqrt{c} \geq 2019 \) holds, given that \( a + b + c = 1 \) and \( a, b, c > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2019^2 - 338}}
Given positive real numbers $a, b, c$ such that $a + b + c = 6$, determine the maximum value of the expression $A = a^2bc + a^2 + 2b^2 + 2c^2$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Evaluate the integral \( \iint_{x^2+y^2<\sqrt{x}}\sqrt{x^2+y^2}\,dxdy \) over the region where \( x^2 + y^2 < \sqrt{x} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Evaluate the limit using improper integrals: \[ \lim _{ n\to \infty } \int _{ |x|<n }{ \int _{ |y|<n }{ \sin(x^2+y^2)\,dx\,dy } } \] Consider using a coordinate transformation to simplify the calculation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Find the limit of the sequence \(\left(\frac{N_{2r}}{N_r}\right)_{r\in\mathbb{N}}\), where \(N_r\) represents the number of unit squares with integer coordinates that intersect a circle of radius \(r\) centered at the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the line integral of the vector field \( F = (2x\sin(\pi y) - e^z, \pi x^2\cos(\pi y) - 3e^z, -xe^z) \) along the curve \( C \) from \((0,0,0)\) to \((1,1,\ln(2))\), where the curve is defined by the parametric equations \( z = \ln(1+x) \) and \( y = x \). Compute \( \int_{C} F \cdot dr \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{13}{2}}
Compute the limit \( \lim_{n\to+\infty}n\int_{0}^{\frac{\pi}{2}}xf(x)\cos^n x\,dx \) for a continuous function \(f: [0, \frac{\pi}{2}] \to \mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(0)}
Given $a$, $b$, and $c$ as fixed positive integers, there are $a+b+c$ ducks sitting in a circle, each choosing either rock, paper, or scissors. Specifically, $a$ ducks choose rock, $b$ ducks choose paper, and $c$ ducks choose scissors. A move is defined as follows:

- A duck picking rock sitting behind a duck picking scissors switches places.
- A duck picking paper sitting behind a duck picking rock switches places.
- A duck picking scissors sitting behind a duck picking paper switches places.

Determine the maximum number of moves that can occur, in terms of $a$, $b$, and $c$, over all possible initial configurations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ab + bc + ca}
Let $A$ be a random variable uniformly distributed over the interval $(0, 2)$, and $B$ be a random variable uniformly distributed over the interval $(1, 3)$. Define $W = \min(A, B)$ and $Z = \max(A, B)$. Find the covariance $\text{Cov}(W, Z)$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{25}{576}}
Evaluate the integral \( \oint_C \frac{\sin z}{z^2 + \pi^2} \, dz \) along the closed contour \( |z - 2i| = 2 \). Use Cauchy's integral formula to find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{i \sinh \pi}
Find the surface area of the 3D shape defined by the equation \((x^2+y^2+z^2+A^2-B^2)^2 = 4A^2(x^2+z^2)\), given that \(0 < B < A\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi^2 A B}
Find the coefficient of $x^{20}$ in the expansion of $$(x^2 + x^3 + x^4)(x + x^2 + x^3 + \cdots + x^6)^4.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{309}
Find the minimal value of the function
\[\begin{array}{c}\ f(x) =\sqrt{15 - 12 \cos x} + \sqrt{4 -2 \sqrt 3 \sin x}+\sqrt{7-4\sqrt 3 \sin x} +\sqrt{10-4 \sqrt 3 \sin x - 6 \cos x}\end{array}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
6.}
Determine the values of $x$ for which the series \( \sum_{n=1}^{\infty} \frac{(2n)!\,x^n}{2^{2n}(n!)^2}\) is convergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, 1)}
A group $G$ of order 275 (where 275 = $5^2 \times 11$) acts on a set of size 18. Determine the minimum number of orbits of length 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the function \( z = f(x, y) \) and \( g(u, v) = uf(u^2, 2uv) \), calculate \( \frac{\partial^2 g}{\partial u \, \partial v}(1, 1) \). Use the following information: \( f(1, 2) = 4 \), \( \nabla f(1, 2) = (3, -1) \), \( \frac{\partial^2 f}{\partial x^2}(1, 2) = \frac{\partial^2 f}{\partial y^2}(1, 2) = 1 \), and \( \frac{\partial^2 f}{\partial x \, \partial y}(1, 2) = -1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4}
Compute the determinant of the following $6 \times 6$ matrix:
\[
\begin{pmatrix}
0 & 0 & 1 & 1 & 1 & 1\\
1 & 0 & 0 & 0 & 0 & 1\\
1 & 0 & 1 & 1 & 1 & 1\\
0 & 1 & 1 & 1 & 0 & 1\\
0 & 1 & 0 & 1 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0
\end{pmatrix}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of solutions to the equation \(\sqrt[a+bi]{1}\), where \(a\) and \(b\) are real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the limit: $$\lim_{n\to\infty}\frac{{\left(\frac{1}{1^1} \cdot \frac{1}{2^2} \cdot \frac{1}{3^3}\cdots \frac{1}{n^n}\right)}^\frac{1}{n^2}}{\sqrt{n}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the dimension of the quotient space $C^{\infty}(\mathbb{R})/W$, where $W = \{f \in C^{\infty}(\mathbb{R}): f(0) = 0, f(1) = 0\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
What is the theoretical maximum average compression rate of a reversible function applied to a random bit sequence, given that the function satisfies f'(f(s)) = s?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In a regular 2016-gon, all vertices are initially white. Determine the minimum number of vertices that must be painted black so that no right or acute triangles can be formed using only the remaining white vertices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1008}
Assume that $(X,Y)$ are independent and identically distributed standard normal random variables. Compute the probability $\mathbb{P}(1<X^2+Y^2<2)$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.2387}
The function \( g(x) \) is defined as \( \sqrt{\frac{x}{2}} \) for all positive \( x \). Evaluate the expression \( g\left(g\left(g\left(g\left(g\left(\frac{1}{2}\right)+1\right)+1\right)+1\right)+1\right) \) and express it as \( \cos(b) \) in degrees, where \( 0^\circ < b < 90^\circ \) and \( b = \frac{p}{q} \) for some relatively prime positive integers \( p, q \). Find \( p+q \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Calculate the volume of the subset \( \left\{ \begin{pmatrix} 1 & 2 & 3\\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{pmatrix}\begin{pmatrix}x \\y \\ z \end{pmatrix}:x^2+y^2+z^2 \leq 4\right\} \) of \( \mathbb{R}^3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256\pi}
A set S of distinct positive integers has the following property: for every integer x in S, the arithmetic mean of the set of values obtained by deleting x from S is an integer. Given that 1 belongs to S and that 2002 is the largest element of S, what is the greatest number of elements that S can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
30
}
Evaluate the limit: \[ \lim_{n\to \infty} \int_{0}^{n} \frac{1}{1+n^2\cos^2x} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the limit: \[ \lim_{n\to \infty}\int_0^{\pi/2}\sqrt[n]{\sin^n{x}+\cos^n{x}}\,dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Given the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a,b>0)\) with left and right foci as \(F_{1}\) and \(F_{2}\), a line passing through \(F_{2}\) with an inclination angle of \(\frac{\pi}{4}\) intersects the hyperbola at a point \(A\). If the triangle \(\triangle F_{1}F_{2}A\) is an isosceles right triangle, calculate the eccentricity of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\sqrt{2}+1"]}
Determine the largest lower bound, independent of \(\lambda\), for the expression:
\[
\frac{1+|\lambda|^2}{2} + \text{Re}\lambda - \frac{2\pi}{n+2}\text{Im}\lambda
\]
where \(|\lambda| < 1\), \(\text{Re}\lambda < 0\), \(\text{Im}\lambda > 0\), and \(n\) is a positive integer. You may choose \(n\) to be as large as necessary for the problem to make sense, but aim for the smallest possible lower bound on \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
On a straight stretch of one-way, two-lane highway, vehicles obey a safety rule: the distance from the back of one vehicle to the front of another is exactly one vehicle length for each 20 kilometers per hour of speed or fraction thereof. Suppose a sensor on the roadside counts the number of vehicles that pass in one hour. Each vehicle is 5 meters long and they can travel at any speed. Let \( N \) be the maximum whole number of vehicles that can pass the sensor in one hour. Find the quotient when \( N \) is divided by 10.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400}
In $\triangle ABC$, the internal angles $A$, $B$, and $C$ satisfy the equation $$2(\tan B + \tan C) = \frac{\tan B}{\cos C} + \frac{\tan C}{\cos B}$$. Find the minimum value of $\cos A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Find the sixth derivative of \( f(x) = e^x \left( 3\sin\frac{x}{\sqrt{3}} + 2\cos\sqrt{3}x \right) \) at \( x=0 \) without directly differentiating the function six times.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{128}
How many square inches of paint are needed to fully paint a regular $6$-sided die with side length $2$ inches, except for the $\frac13$-inch diameter circular dots marking $1$ through $6$ (a different number per side)? The paint has negligible thickness, and the circular dots are non-overlapping.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$24 - \frac{7}{12}\pi$}
Find the matrix representation of the operator \(|v_{j}\rangle \langle v_{k}|\) with respect to the orthonormal basis \(|v_{i}\rangle\) for an inner product space \(V\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{E_{jk}}
Calculate the curvature 2-form \(\Omega(X,Y)\) for the Martinet distribution, where \(X=\partial x+\frac{1}{2}y^{2}\partial t\) and \(Y=\partial y\), given that \(\Omega=y\, \mathrm{d}x \wedge \mathrm{d}y\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y}
Determine the standard deviation of the insurance payment for an automobile with a damage deductible of $250, where repair costs are modeled by a uniform random variable on the interval $(0, 1500)$. Let $X$ denote the damage, and $Y$ the insurance payment, defined as:

$$Y = \begin{cases} 0 & \text{if } X < 250 \\ X - 250 & \text{if } 250 \le X \le 1500 \end{cases}$$

Calculate the standard deviation of $Y$. Use the formula $\text{var } Y = E(Y^2) - (EY)^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{625\sqrt{15}}{6}}
Given a nilpotent endomorphism $g: W \to W$ on a $12$-dimensional vector space $W$ over a field $K$, where the dimensions of $\operatorname{ker}g^i$ are provided in the following table:

$$\begin{array}{c|c} i & 12 & 11 & 10 & 9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 \\ \hline \dim \operatorname{ker} g^i & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 12 & 11 &10 & 8 & 5 & 0 \end{array}$$

Determine the number of Jordan blocks of the nilpotent endomorphism $g$. Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the minimum number of triangles required to cover the point set \( G_{2008} = \{(i, j) \mid i=1,2, \cdots, 2008 ; j=1,2, \cdots, 2008\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the joint probability density function for random variables $X$ and $Y$:
\[ f(x, y)=\begin{cases}
2(x+y)  & \text{if } 0<x<y<1 \\
0 &  \text{otherwise}
\end{cases}. \]
Find the real number $a$ such that the conditional expectation $E(X | Y=aX)=\frac{2}{9}$. \(a\) is a real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the cardinality of the \(\sigma\)-algebra generated by the sets \(X_{i}:=\{(x_1,x_2,...,x_n):x_i > \frac{1}{2}\}\) for \(i=1,2,...,n\), where \(X=[0,1]^{n}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2^n}}
Determine the minimum number of black boxes required to pinpoint the location of the origin of a signal that caused spaceships to explode. Assume the signal spreads through space at a constant velocity uniformly in every direction, and each black box contains the spatial coordinates of the ship and the time it received the signal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let \( f: \mathbb{R}^2 \to \mathbb{R}^2 \) be defined by \( f(x,y) = (e^{2x+y}, e^{x+y}) \). Compute the density function \( \frac{df[\lambda_2]}{d\lambda_2} \) of the pushforward measure \( f[\lambda_2] \), where \( \lambda_2 \) is the Lebesgue measure on \( \mathcal{B}(\mathbb{R}^2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{uv}}
The sequence $(x_n)_{n\in N}$ satisfies $x_1 = 2015$ and $x_{n+1} =\sqrt[3]{13x_n - 18}$ for all $n \ge 1$.
Determine $\lim_{n\to \infty} x_n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{10}-1}
Evaluate the integral: $$\int_0^\infty \int_{-x}^0 \frac{e^{-x^2/2}}{\sqrt {2\pi}} \frac{e^{-y^2/2}}{\sqrt {2\pi}}\, dy\, dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{8}}
Find the minimum value of the function \( B = |x-1| + |2x-3| + |3x-5| + \cdots + |99x-197| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{285}{7}}
Calculate the limit of the sequence defined by \( a_n = \frac{1}{\sqrt{n}}\sum_{k=0}^n\sqrt{C_n^kp^k(1-p)^{n-k}} \), where \( C_n^k = \frac{n!}{k!(n-k)!} \) and \( p \in (0,1) \). Find \( \lim_{n \to \infty} a_n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the coefficient of \(x^6\) in the expansion of \((2+2x+2x^2+2x^3+2x^4+x^5)^5\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6240}
Find the volume of the solid in the first octant bounded by the surfaces $z = 1-y^2$, $y=2x$, and $x=3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{15}{8}}
Evaluate the contour integral \( \frac{1}{2\pi i}\oint_C \frac{z^2}{z^2+4}dz \), where \( C \) is the square with vertices \( \pm2 \) and \( \pm2+4i \), using Cauchy's integral formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{i}
Let $F$ be a field of order 32. Determine the number of solutions $(x,y) \in F \times F$ to the equation $x^2 + y^2 + xy = 0$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find all positive integer solutions to the equation \( \log_2(x+1) = \lfloor \log_2(x) \rfloor + 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^k - 1}
A single fair die is rolled repeatedly. What is the probability that two sixes in a row will appear before a five followed by a six?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{12}}
Let \( f(x) = 1 + \sum_{k=0}^{100}\frac{(-1)^{k+1}}{(k+1)!}x(x-1)\cdots (x-k) \). If \( S \) is the sum of all real roots of \( f \) and \( T \) is the sum of all real roots of \( f' \), find \( S-T \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Eliminate \(\alpha\) from the following equations: \(\cos(\theta-\alpha)=\lambda\sin2\theta\) and \(\sin(\theta-\alpha)=\lambda\cos2\theta\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lambda^2 = 1}
Find the volume of the region bounded by the planes $6x+5y+6z = 6$, $y=x$, $x=0$, and $z=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{6}{55}}
Determine the dimension of the symplectic group \(\operatorname{Sp}(2n,\mathbb{C})\) over the complex field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n(2n + 1)}
In a regular pentagon ABCDE, the sides are extended to their points of intersection, forming a pentagram. Let PQRST be another regular pentagon inscribed within the pentagram, such that the ratio of the length of a side of the pentagram to the length of a side of the inscribed pentagon is m:1. Determine the ratio of the area of the pentagram to the area of the inscribed pentagon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m^2}
How many normal subgroups of the symmetric group $S_4$ are isomorphic to the Klein four-group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that $X_1, X_2, \ldots, X_n$ are independent and identically distributed (i.i.d.) random variables, find the expected value of the expression $\sum (X_i - \bar{X})^2$, where $\bar{X}$ is the sample mean. Calculate this without expanding the square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n-1)\sigma^2}
Assume $a_n > 0$ and $\lim\limits_{n \to \infty} \frac{a_1 + a_2 + \cdots + a_n}{n} = a < +\infty$. Determine $\lim\limits_{n \to \infty} \frac{a_1^p + a_2^p + \cdots + a_n^p}{n^p}$ where $p > 1$. Provide your answer as a single limit value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $\mathbb{Z}$ denote the set of integers and $S \subset \mathbb{Z} $ be the set of integers that are at least $10^{100}$. Fix a positive integer $c$. Determine all functions $f: S \rightarrow \mathbb{Z} $ satisfying $f(xy+c)=f(x)+f(y)$, for all $x,y \in S$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\( f = 0 \)}
Given an $n \times n$ bipartite graph $G$ with a minimum degree of at least $n - d$, consider the following process: Select vertices from the left side randomly with probability $p = \frac{1}{d+1}$ to form set $S$. Then, form set $T$ by including all vertices on the right side that are connected to every vertex in $S$. Determine the lower bound on the probability that the complete bipartite subgraph $S \times T$ contains a specific edge $uw$. Express this probability in terms of $p$, $n$, and $d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p(1 - p)^d}
Let \( X_1, X_2, \ldots \) be independent random variables such that \( P(X_n = 1) = P(X_n = -1) = \frac{1}{2} \). Define \( t = \inf \{ n : X_1 + X_2 + \cdots + X_n = 1 \} \). Find the expected value \( E\left(\left(\frac{1}{3}\right)^t\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 - 2\sqrt{2}}
Let $ABCD$ be a cyclic quadrilateral with $|AB|=26$, $|BC|=10$, $m(\widehat{ABD})=45^\circ$,$m(\widehat{ACB})=90^\circ$. What is the area of $\triangle DAC$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
84
}
Calculate the divergence \( \nabla \cdot \mathbf{r} \) in cylindrical coordinates, where \( \mathbf{r} = \rho \hat{e}_\rho + z \hat{e}_z + \rho\theta \hat{e}_\theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The volume of a regular triangular pyramid, whose lateral face is inclined at an angle of $45^{\circ}$ to the base, is $9 \mathrm{~cm}^{3}$. Find the total surface area of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9 \sqrt{3} (1 + \sqrt{2})}
Given the determinant \( A = \begin{vmatrix} f(x) & g(x) & h(x) \\ i(x) & j(x) & k(x) \\ l(x) & m(x) & n(x) \end{vmatrix} \), where \( f(x), g(x), h(x), i(x), j(x), k(x), l(x), m(x), n(x) \) are functions of \( x \), determine the power of \( x-a \) in the factorization of \( A \) if substituting \( x = a \) makes exactly three rows identical.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given vectors \(a\), \(b\), and \(c\) such that \(|a| = |b| = 1\) and \(c = a \times b\), find the maximum value of the expression:
\[
|(a+2b+3c) \cdot ((2a+3b+c) \times (3a+b+2c))|
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}