inclusionAI/Ring-lite-rl-data huggingface dataset:
A Markov chain $X_0,X_1,X_2,\ldots$ has the transition probability matrix:
$$P=\begin{Vmatrix}
0.3 & 0.2 & 0.5\\
0.5 & 0.1 & 0.4 \\
0 & 0 & 1
\end{Vmatrix}$$
and starts in state $X_0=0$. Eventually, the process will end up in state 2. What is the probability that the time $T=\text{min}\{n\geq 0;X_n=2\}$ is an odd number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{90}{133}}
Suppose a farmer's cow gives birth to a heifer at the age of two. Subsequently, both the original cow and each heifer will produce one heifer per year starting at the age of two. Given this scenario, determine the total offspring of the original cow after 25 years.

The time is counted from the birth of the first cow, and no cattle are slaughtered within this period.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75025}
If the function $y=f(x)$ ($x \in \mathbb{R}$) satisfies $f(x+2)=f(x)$, and when $x \in [-1,1]$, $f(x)=1-x^2$, and the function $g(x)=\lg|x|$, then the number of intersection points between the graphs of $y=f(x)$ and $y=g(x)$ in the interval $[-5,5]$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
An ant starts out at \((0,0)\). Each second, if it is currently at the square \((x, y)\), it can move to \((x-1, y-1)\), \((x-1, y+1)\), \((x+1, y-1)\), or \((x+1, y+1)\). In how many ways can it end up at \((2010,2010)\) after 4020 seconds?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \binom{4020}{1005} \right)^2}
Equilateral triangle $\triangle ABC$ is inscribed in circle $\omega$ with radius $18.$ Circle $\omega_A$ is tangent to sides $\overline{AB}$ and $\overline{AC}$ and is internally tangent to $\omega.$ Circles $\omega_B$ and $\omega_C$ are defined analogously. Circles $\omega_A,$ $\omega_B,$ and $\omega_C$ meet in six points---two points for each pair of circles. The three intersection points closest to the vertices of $\triangle ABC$ are the vertices of a large equilateral triangle in the interior of $\triangle ABC,$ and the other three intersection points are the vertices of a smaller equilateral triangle in the interior of $\triangle ABC.$ The side length of the smaller equilateral triangle can be written as $\sqrt{a} - \sqrt{b},$ where $a$ and $b$ are positive integers. Find $a+b.$
Diagram
[asy] /* Made by MRENTHUSIASM */ size(250); pair A, B, C, W, WA, WB, WC, X, Y, Z; A = 18*dir(90); B = 18*dir(210); C = 18*dir(330); W = (0,0); WA = 6*dir(270); WB = 6*dir(30); WC = 6*dir(150); X = (sqrt(117)-3)*dir(270); Y = (sqrt(117)-3)*dir(30); Z = (sqrt(117)-3)*dir(150); filldraw(X--Y--Z--cycle,green,dashed); draw(Circle(WA,12)^^Circle(WB,12)^^Circle(WC,12),blue); draw(Circle(W,18)^^A--B--C--cycle); dot("$A$",A,1.5*dir(A),linewidth(4)); dot("$B$",B,1.5*dir(B),linewidth(4)); dot("$C$",C,1.5*dir(C),linewidth(4)); dot("$\omega$",W,1.5*dir(270),linewidth(4)); dot("$\omega_A$",WA,1.5*dir(-WA),linewidth(4)); dot("$\omega_B$",WB,1.5*dir(-WB),linewidth(4)); dot("$\omega_C$",WC,1.5*dir(-WC),linewidth(4)); [/asy] ~MRENTHUSIASM ~ihatemath123
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{378}
Let's call a number "elegant" if all the digits in its decimal representation are distinct and in ascending order. Are there more four-digit elegant numbers or five-digit elegant numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{equal}
Let  $ABC$  be a triangle with  $AB = 13$ ,  $BC = 14$ ,  $CA = 15$ . Let  $H$  be the orthocenter of  $ABC$ . Find the radius of the circle with nonzero radius tangent to the circumcircles of  $AHB$ ,  $BHC$ ,  $CHA$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{The radius of the circle is \(\frac{65}{16}\).}
You are trapped in ancient Japan, and a giant enemy crab is approaching! You must defeat it by cutting off its two claws and six legs and attacking its weak point for massive damage. You cannot cut off any of its claws until you cut off at least three of its legs, and you cannot attack its weak point until you have cut off all of its claws and legs. In how many ways can you defeat the giant enemy crab?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14400}
Given that \( S \) is a set consisting of \( n \) (\( n \geq 3 \)) positive numbers. If three different elements in \( S \) can form the sides of a triangle, then \( S \) is called a "triangular set." Consider the set of consecutive positive integers \(\{4,5, \cdots, m\}\). All of its 10-element subsets are triangular sets. What is the maximum possible value of \( m \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{253}
一珍稀物种出现在地球，对每个珍稀生物，每天有如下事件发生：有 $p(0 \leq p \leq 1)$ 的概率消失，有 $\frac{1-p}{3}$的概率保持不变，有 $\frac{1-p}{3}$ 的概率分裂成两个，有 $\frac{1-p}{3}$ 的概率分裂成三个。对所有新产生的生物每天也会发生上述事件。假设开始只有一个这样的珍稀生物，若希望最终这种生物灭绝的概率不超过 $\frac{1}{2}$ ，则 $p$ 至多为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Three mutually perpendicular lines intersect at a point \( O \). Points on one of these lines that are at a distance of 1 unit from \( O \) are denoted by \( X_{1} \) and \( X_{1}^{\prime} \), and points 2 units from \( O \) are denoted by \( X_{2} \) and \( X_{2}^{\prime} \). Similarly, points on the other two lines are labeled \( Y_{1}, Y_{1}^{\prime}, Y_{2}, Y_{2}^{\prime} \) and \( Z_{1}, Z_{1}^{\prime}, Z_{2}, Z_{2}^{\prime} \). Planes are formed with every possible combination of a point with a 1-index from one line and a 2-index point from each of the other two lines, for example, passing through \( X_{1}^{\prime} \), \( Y_{2} \), and \( Z_{2}^{\prime} \).

Determine the volume of the solid enclosed by these planes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8}{3}}
A grasshopper jumps either $364$ or $715$ units on the real number line. If it starts from the point $0$, find the smallest distance that the grasshopper can be away from the point $2010$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given a signal x(t) with a Laplace transform X(s) that has four distinct poles, and knowing that x(t) includes an impulse at t = 0, determine the minimum number of zeros that X(s) must have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Points R, S, and T are vertices of an equilateral triangle, and points X, Y, and Z are midpoints of its sides. How many noncongruent triangles can be drawn using any three of these six points as vertices? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the equation of the tangent line to the curve given by $$(x^2 + y^2 - 3)^3 = (xy)^3 - x^2 - y^2$$ at the point $(1, 1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = -x + 2}
Use the 0.618 method to find the optimal addition amount for an experiment. If the current optimal range is $[628, 774]$, and the good point is 718, then the addition point value to be tested at this time is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{684}
Evaluate the nested limit: $$\lim_{m\to \infty }\left[\lim_{y\to \,0}\left(1+\cos^2y+\cos^22y+\ldots+\cos^2my\right)^{1/(m^3y^2)}\right]$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Let $p, q, r$ be real numbers satisfying the equations \[\frac{(p+q)(q+r)(r+p)}{pqr} = 24\] and \[\frac{(p-2q)(q-2r)(r-2p)}{pqr} = 10.\] If $\frac{p}{q} + \frac{q}{r} + \frac{r}{p}$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. 

*Author: Alex Zhu*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Evaluate the integral \(\int_{1}^{10} [2x] \{x/2\} \, dx\), where \([2x]\) is the greatest integer function and \(\{x/2\}\) is the fractional part function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{807}{16}}
[ Equilateral (Regular) Triangle ]

In an equilateral triangle $D E F$, a circle of radius $r$ is inscribed. This circle is externally tangent to three other circles of the same radius at the points of tangency with the sides of the triangle. The centers of the external circles are $O_{1}, O_{2}, O_{3}$, respectively. Find the area of the hexagon formed by the intersection of triangles $D E F$ and $O_{1}, O_{2}, O_{3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2r^{2}\sqrt{3}}
A cyclist is walking at a constant rate of 15 miles per hour. A car traveling in the same direction along the same path at 60 miles per hour passes the cyclist. The car stops to wait for the cyclist for some time after passing her, while the cyclist continues to go at her constant rate. The car must wait 18 minutes until the cyclist catches up. How many minutes does the car initially wait for the cyclist after passing her?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.5}
Suppose the sequence $\{a\_n\}$ satisfies the following condition: for any $n\in\mathbb{N}^*$, there are only finitely many positive integers $m$ such that $a\_m < n$ holds. Let $b\_n$ denote the number of such $m$'s. This gives rise to a new sequence $\{b\_n\}$. For example, if the sequence $\{a\_n\}$ is $1$, $2$, $3$, $...$, $n...$, then the sequence $\{b\_n\}$ is $0$, $1$, $2$, $...$, $n-1$, $...$ Now, it is known that the sequence $\{a\_n\}$ is a geometric sequence with $a\_2=2$, $a\_5=16$. Find the number of positive integers $i$ such that $b\_i=2016$ in the sequence $\{b\_n\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2015}}
Consider a staircase of height \( n \), which consists of all the cells of an \( n \times n \) square that lie above the diagonal. Determine the number of ways to divide this staircase into several rectangles, such that the sides of the rectangles lie along the grid lines and all the rectangles have pairwise distinct areas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-2}}
Determine the number of five-digit sequences \(d_1d_2d_3d_4d_5\), where \(d_i\in\{0,1,\ldots,9\}\), that do not contain any palindromic subsequence of length 3 or greater.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52560}
Given the hyperbola $C: \frac{x^2}{a^2}-\frac{y^2}{a^2}=1(a>0)$, with left and right vertices $A_{1}$ and $A_{2}$ respectively. Point $P$ is a point on the hyperbola $C$ other than the vertices, and $\angle A_{1}PA_{2}=3\angle PA_{1}A_{2}$. Find $\cos 5\angle PA_{1}A_{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit: $$\lim_{x\to-\infty} x\log(-e^x + 1)+\operatorname{Li}_2(e^x)-\frac12x^2$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\infty}
Determine the value of \(a\) such that the range of the function \(f(x) = \frac{1}{3x^4 - 8ax^3 + 12a^2 + x^2 + a}\) is completely contained within the interval \([0, \frac{1}{3}]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Determine the number of unique deal combinations in the Klondike solitaire card game, taking into account both the order of the cards and the game's rules.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52!}
Let $X_1, X_2, X_3, X_4, X_5$ be a random sample from a standard normal distribution $N(0,1)$. Define $T = \sum_{i=1}^5 (X_i - \bar{X})^2$, where $\bar{X}$ is the sample mean. Calculate $E(T^2 \bar{X}^2)$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{24}{5}}
In the convex quadrilateral \(ABCD\), the diagonals \(AC\) and \(BD\) are equal. Moreover, \(\angle BAC = \angle ADB\) and \(\angle CAD + \angle ADC = \angle ABD\). Find the angle \(BAD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Find the flux of the vector field \( \mathbf{F} = 3\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \) across the part of the plane \( 1x + 3y + z = 3 \) that lies in the first octant, oriented upwards.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Calculate the limit of the following expression as $n$ approaches infinity: $$\lim_{n\rightarrow\infty}\frac{\Gamma(n+\frac{1}{2})}{\Gamma(n+1)\sqrt{n}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A standard deck contains 52 cards numbered from 1 to 52. You randomly select 26 cards without replacement. Determine the number of ways to select these 26 cards such that the sum of the reciprocals of their numbers is exactly 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Suppose \( x_{1}, x_{2}, \ldots, x_{2011} \) are positive integers satisfying
\[ x_{1} + x_{2} + \cdots + x_{2011} = x_{1} x_{2} \cdots x_{2011} \]

Find the maximum value of \( x_{1} + x_{2} + \cdots + x_{2011} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4022}
Two regular tetrahedrons  $A$  and  $B$  are made with the 8 vertices of a unit cube. (this way is unique)

What's the volume of  $A\cup B$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
Calculate the residue of the function \(-\frac{\zeta'(s)}{\zeta(s)} \frac{x^s}{s}\) at \(s = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x}
Find the smallest positive number $\lambda$, such that for any 12 points on the plane $P_1, P_2, \ldots, P_{12}$ (points can overlap), if the distance between any two of them does not exceed 1, then the inequality \( \sum_{1 \le i < j \le 12} |P_iP_j|^2 \le \lambda \) holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Determine the number of non-ordered quadruples \((a, b, c, d)\) of non-negative integers that satisfy the equation \(a + b + c + d = 18\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Calculate the flux of the vector field \( \mathbb{v} = \operatorname{curl}\mathbb{u} \) through the surface of the solid bounded by the paraboloid \( z = 1 - x^2 - y^2 \) and the planes \( x = 0 \) and \( z = 0 \). The vector field \( \mathbb{u} \) is given by \( \mathbb{u} = (xy, xz^2, x^2y) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let \( f(x,y,z,u)=\frac{24}{(1+x+y+z+u)^5}, x>0,y>0,z>0,u>0 \) be a density function. Calculate the probability \( P(X+Y+Z+U \ge 1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{15}{16}}
Evaluate the double integral \( \int_{0}^{1}\mathrm{d}y\!\int_{0}^{1}\frac{x-y}{(x+y)^3}\mathrm{d}x. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Determine the number of functions \( g \) from the set \( A = \{1, 2, 3, 4, 5, 6, 7\} \) to itself such that \( g(g(g(x))) \) is a constant function. Find the remainder when this number is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{249}
Let \( f(x) = ax^2 - bx + c \), where \( a, b, \) and \( c \) are positive integers. If \( f(p) = f(q) = 0 \) with \( 0 < p < q < 1 \), determine the smallest possible value of \( a \). The vertex of the parabola is at \( \left(-\frac{b}{2a}, \frac{b^2 - 4ac}{4a}\right) \), which lies in the interval \((0, 1)\). Additionally, \( f(0) > 0 \) and \( f(1) > 0 \), implying \( c > 0 \) and \( a - b + c > 0 \). Moreover, the discriminant \( b^2 - 4ac \) must be positive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate the double integral \( \iint_D \left(\sqrt{a^2-x^2-y^2}-\sqrt{x^2+y^2}~\right)dxdy \) where \( D_{xy} \) is the disk \( x^2+y^2\le a^2 \), using a change of variables.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given $\Delta ABC$ equilateral, with $X\in[A,B]$. Then we define unique points Y,Z so that $Y\in[B,C]$, $Z\in[A,C]$, $\Delta XYZ$ equilateral.

If $Area\left(\Delta ABC\right) = 2 \cdot Area\left(\Delta XYZ\right)$, find the ratio of $\frac{AX}{XB},\frac{BY}{YC},\frac{CZ}{ZA}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\( 2 \pm \sqrt{3} \).

}
Determine the values of $x$ for which the series \( \sum_{n=1}^\infty \frac{(-1)^{n+1}x^n}{n + 1/n} \) is convergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1]}
Let  $P$  be a regular  $2006$ -gon. A diagonal is called *good* if its endpoints divide the boundary of  $P$  into two parts, each composed of an odd number of sides of  $P$ . The sides of  $P$  are also called *good*.
Suppose  $P$  has been dissected into triangles by  $2003$  diagonals, no two of which have a common point in the interior of  $P$ . Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1003}
Given the moment generating function of a multivariate normal random variable \((X, Y, Z)\):
\[ M(t_1, t_2, t_3) = \exp(t_1 - 2t_3 + t_1^2 + t_1t_2 + \frac{3}{2}t_2^2 - t_1t_3 + \frac{5}{2}t_3^2) \]
Find the value of \(a\) such that \(X\) and \(Y + aX\) are independent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Evaluate the integral \( \int_{|z|=1} \frac{\sin(z)}{z^3} \, dz \) over the unit circle \(|z|=1\) in the complex plane without using the residue theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $ABCD$ be a cyclic quadrilateral with circumradius $100\sqrt{3}$ and $AC=300$. If $\angle DBC = 15^{\circ}$, then find $AD^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60000}
Find the $x^8$ term in the expansion of \((x+\frac{1}{\sqrt[3]{x}})^5(2x - \frac{1}{x^2})^7\) using Newton's binomial theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1280}
Let $p$, $q$, and $r$ be constants, and suppose that the inequality \[\frac{(x-p)(x-q)}{x-r} \le 0\] is true if and only if $x > 2$ or $3 \le x \le 5$. Given that $p < q$, find the value of $p + q + 2r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the number of distinct ways to color the vertices of a 12-vertex cycle graph using 8 different colors, where two colorings are considered identical if one can be transformed into the other by a rotation or reflection of the cycle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2863912668}
Solve the equation: \[ 2^{1+\sin(3x)} - 8^{\sin(x)+\frac{1}{3}} + \sin(x) \cos(2x) = \sin(x) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x = k\pi}
Let $V$ be the region defined by $\frac{1}{4} \leq x^2 + y^2 + z^2 \leq 1$ in $\mathbb{R}^3$. Consider the vector field $\vec{F} = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{(x^2 + y^2 + z^2)^2}$ for $(x, y, z) \in V$. The surface $S$ is the part of the boundary of $V$ given by $x^2 + y^2 + z^2 = \frac{1}{4}$, with $\hat{n}$ denoting the outward unit normal vector. Compute the surface integral $\int\int_S \vec{F} \cdot \hat{n} \, dS$. \(\text{Express your answer as a single number.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-8\pi}
Given a triangle ABC with points P, Q, R lying on the sides AB, BC, CA, respectively, and BP = PQ = QR = RC = 1, find the maximum possible area of triangle ABC. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the tenth digit (from the right) of the sum \( \sum_{k=1}^{30} (k!) \) using modular arithmetic.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the parabola \( y^{2} = 2 p x \) with focus \( F \) and directrix \( l \), a line passing through \( F \) intersects the parabola at points \( A \) and \( B \) such that \( |AB| = 3p \). Let \( A' \) and \( B' \) be the projections of \( A \) and \( B \) onto \( l \), respectively. If a point \( M \) is randomly chosen inside the quadrilateral \( AA'B'B \), what is the probability that \( M \) lies inside the triangle \( FA'B' \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/3}
Evaluate the contour integral \(\int_{|z|=1}\dfrac{\sin\pi z}{z^{2}\cos\pi z}dz\) by considering all the singularities of the function within the contour.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2i(\pi^2 - 8)}
Let $M$ be the real vector space of $2\times3$ matrices with real entries. Define the linear transformation $T:M\rightarrow M$ by \[ T\left(\begin{bmatrix} x_{1} & x_{2} & x_{3} \\\ x_{4} & x_{5} & x_{6}\end{bmatrix}\right)=\begin{bmatrix}-x_{6}& x_{4}& x_{1}\\ x_{3}& x_{5}& x_{2}\end{bmatrix}. \] Calculate the determinant of the linear transformation $T$. \(Note: Consider the transformation as a linear map on a vector space of dimension 6.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given that radioactive decay of an element occurs according to a Poisson process with a rate of 10,000 decays per second, calculate the approximate probability that the millionth decay occurs within 100.2 seconds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.9772}
What least number should be added to 1024, so that the sum is completely divisible by the product of the following prime numbers: 17, 23, and 29?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10315}
Consider two independent and identically distributed random variables $X_1$ and $X_2$ with a normal distribution $N(\theta, 1)$. Additionally, let $V$ be a Bernoulli random variable with $P(V=1) = \frac{1}{4}$, independent of $X_1$ and $X_2$. Define $X_3$ as follows:
- $X_3 = X_1$, if $V=0$
- $X_3 = X_2$, if $V=1$

We are testing the hypothesis $H_0: \theta = 0$ versus $H_1: \theta = 1$. Reject $H_0$ if $\frac{X_1 + X_2 + X_3}{3} > C$. Find the critical value $C$ such that the test size is 0.05.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.226}
Determine the cardinality of the set of functions from the natural numbers \( \mathbb{N} \) to the real numbers \( \mathbb{R} \), assuming the generalized continuum hypothesis (GCH).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_1}
Let $n$ be an integer greater than $1$ and $x$ be an integer between $1$ and $10^{12}$. Calculate the probability that $x^{2n+1}$ ends with the digit sequence $11$. Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{100}}
For how many postive integers $n$ less than $2013$, does $p^2+p+1$ divide $n$ where $p$ is the least prime divisor of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
173
}
Determine the number of elements of order 5 in a group of order 90.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Calculate the flux of the vector field \( \vec{F}(x, y, z) = \langle x, y, z \rangle \) across the surface of the cone defined by \( z = \sqrt{x^2 + y^2} \) for \( 0 \le z \le 1 \) using the divergence theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}