inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the norm in which the inequality \( ||g*f-g*s|| \leq C ||g|| \text{ } ||f-s|| \) holds, given that \( g \in L^p(\mathbb{R}) \) has compact support and \( 1\leq p<\infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{L^p}
Let the set $A=\{(x,y)|\frac{m}{2}\leqslant {(x-2)}^{2}+{y}^{2}\leqslant {m}^{2},x,y\in \mathbb{R}\}$, $B=\{(x,y)|2m\leqslant x+y\leqslant 2m+1,x,y\in \mathbb{R}\}$. If $A\cap B\neq \varnothing$, then the range of the real number $m$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[\frac{1}{2}, 2+\sqrt{2}\right]}
Given that the correlation between random variables X and Y is \( \rho \), and Y and Z are independent, with all three variables being standard normal, determine the correlation between X and Z.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
For a positive integer $n$, let $S_n$ be the total sum of the intervals of $x$ such that $\sin 4n x \geq \sin x$ in $0 \leq x \leq \frac{\pi}{2}$. Find $\lim_{n\to\infty} S_n$. \(60\) points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{8}}
Find a nontrivial unit polynomial \( p(x) \) in \( \mathbb{Z}_4[x] \) such that \( \deg p(x) > 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + 2x^2}
Calculate the sum of the series:

$$\sum_{n=2}^{\infty}\frac{3n-5}{n(n^2-1)}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the Fourier transform \( \mathscr{F}(f_a) \) of the function \( f_a(x) = \frac{a}{\pi(x^2+a^2)} \) for \( a > 0 \), given that \( \mathscr{F}(\mathscr{F}(f))(x) = 2\pi f(-x) \) and \( \mathscr{F}(e^{-a|x|}) = \frac{2a}{a+\xi^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-a|\xi|}}
Calculate the total surface area of the two cylinders given by the equations \(y^2 + z^2 = 1\) and \(x^2 + y^2 = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Using the method of Lagrange multipliers, determine the minimum value of the function \( f(x,y,z) = x^2y^2z^2 \) subject to the constraint \( x^2 + y^2 + z^2 = r^2 \). Given the critical point \((\pm \frac{r}{\sqrt{3}}, \pm \frac{r}{\sqrt{3}}, \pm \frac{r}{\sqrt{3}})\), identify the minimum value of the function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given that there are $3$ red cubes, $4$ blue cubes, $2$ green cubes, and $2$ yellow cubes, determine the total number of different towers with a height of $10$ cubes that can be built, with the condition that the tower must always have a yellow cube at the top.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1,260}
Given vectors $\overrightarrow{a}=(1,3)$ and $\overrightarrow{b}=(-2,4)$, calculate the projection of $\overrightarrow{a}$ in the direction of $\overrightarrow{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5}}
将一枚均匀的股子独立投郑三次, 所得的点数依次记为 $x, y, z$, 则事件 " $\mathrm{C}_{7}^{x}<\mathrm{C}_{7}^{y}<\mathrm{C}_{7}^{2}$ "发生的概率为 $\qquad$。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{28}
Find the number of positive integer solutions to the equation \(x+y+z+w=20\) such that \(x, y, z,\) and \(w\) are all different.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{552}
Danny is bringing his sailboat out to the ocean. When he uses his 24 square foot sail, he goes at a speed determined by the wind speed (w). His speed with the larger sail is 50w MPH. Similarly, when he uses his 12 square foot sail, his speed is 20w MPH. The wind speed is expected to increase linearly throughout the day, starting with no wind in the morning (w=0) and reaching a maximum wind speed of 1 by the end of the day.

During a 10-hour sailing period, Danny must decide which sail to use and plans to travel a total distance of 200 miles. How many hours faster can he complete the journey with the larger sail compared to the smaller sail, given the increasing wind speed throughout the day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given the triangle $\triangle ABC$ is an isosceles right triangle with $\angle ABC = 90^\circ$, and sides $\overline{AB}$ and $\overline{BC}$ are each tangent to a circle at points $B$ and $C$ respectively, with the circle having center $O$ and located inside the triangle, calculate the fraction of the area of $\triangle ABC$ that lies outside the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \frac{\pi}{2}}
On the lateral sides $AB$ and $CD$ of trapezoid $ABCD$, it is possible to select points $K$ and $L$ respectively, such that the segment $KL$ is not parallel to the bases and is divided into three equal parts by the diagonals. Find the ratio of the bases of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Compute the integral \( \iint_D (x^5y + xy^5) \, dA \) where \( D = \{ (x, y) : 0 < x, 0 < y, x^2 + y^2 < 4, x^2 - y^2 > 1 \} \), using the substitution \( u = x^2 + y^2 \) and \( v = x^2 - y^2 \). Assume \( x > 0 \) in the domain \( D \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{63}{16}}
A semicircle is cut out of paper with diameter \( AB \). A point \( M \) is marked on \( AB \), and two smaller semicircles with diameters \( AM \) and \( MB \) are cut out from the original semicircle. The area of the remaining figure is \( 16 \pi^3 \). Find the length of the chord passing through point \( M \) and perpendicular to \( AB \) that lies inside the remaining figure. Round your answer to two decimal places if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25.13}
A hexagon is inscribed in a circle. Five of the sides have length $81$ and the sixth, denoted by $\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A_{}^{}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$384$}
Suppose that the error, in grams, of a balance has the density function \(f(x)=\frac{1}{4}e^{\frac{-|x|}{2}}\) for \(-\infty<x<\infty\), and that 100 items are weighed independently. Use the Central Limit Theorem to approximate the probability that the absolute difference between the true total weight and the measured total weight is more than 50 grams.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.077}
Find all continuous functions $f : \mathbb{R} \to \mathbb{R}$ such that$$f(-x)+\int \limits_0^xtf(x-t)dt=x,\ \forall\ x\in \mathbb{R}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ f(x) = -\sinh(x) }
Calculate the line integral \( \int_{A}\frac{y\,dx-(x+1)\,dy}{x^2+y^2+2x+1} \) where \( A \) is the line \(|x|+|y|=4\), travelling clockwise and making one rotation. Use Green's theorem to find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Two trains of different lengths run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. The time which they take to cross each other is 12.59899208063355 seconds. If the shorter train is 160 m long, how long is the longer train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{190}
Find the positive integer $n$ such that:
\[
\underbrace{f(f(\cdots f}_{2013 \ f\text{'s}}(n)\cdots ))=2014^2+1
\]
where $f(n)$ denotes the $n$th positive integer which is not a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1015057}
How many ways are there to distribute 6 copies of one book, 7 copies of a second book, and 11 copies of a third book between two teachers if each teacher receives exactly 12 books and each teacher gets at least 2 copies of each book?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2$}
Given the set \( P = \{ x \mid x = 7^3 + a \times 7^2 + b \times 7 + c \} \), where \( a \), \( b \), and \( c \) are positive integers not exceeding 6, if \( x_1, x_2, \ldots, x_n \) are elements of set \( P \) forming an arithmetic sequence, find the maximum value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the number of points on the elliptic curve given by \(x^3 + x + 1 = y^2\) over the finite field \(\mathbb{F}_{125}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Determine the number of solutions to the equation \(x^4 - 1 = e^x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In a soap company, a soap is manufactured with some parts. For making one soap, you will get 1 part as scrap. At the end of the day, they have 251 such scraps. From that, they can manufacture 25 more soaps. How many parts are needed to manufacture one soap?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Compute the limit: $$ \lim_{n \to \infty}\int_{0}^{\pi} \frac{\sin\left(x\right)}{1 + 3\cos^{2}\left(nx\right)}\,\mathrm{d}x $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In a group of 21 people, each pair of people can communicate using a unique radio frequency. However, in every group of three people, exactly two of them do not communicate with each other. Determine the maximum number of unique frequencies required for this group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
Find the sum of the residues of the rational function \( h(z) = \frac{5z^{2015} + 7z^{2010} - 38z^5 + z^4 - 2z^3 + 5}{7z^{2016} + 2z^{2013} - 6z^8 + 3z^7 + z^5 - 4z^2 - z + 111} \) at its poles in the complex plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{7}}
Let \((X_{n})_{n \in \mathbb{N}}\) be a sequence of independent random variables, each with a Bernoulli distribution with parameter \(\frac{1}{2}\). Define the random variable \(Z_{n}\) for each \(n \in \mathbb{N}\) as follows:
\[ Z_{n} = \sum_{k=1}^{n} \frac{X_{k}}{2^k} \]
Find the limit in probability of the sequence \((Z_{n})_{n \in \mathbb{N}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{U(0,1)}
Gru and the Minions plan to make money through cryptocurrency mining. They chose Ethereum as one of the most stable and promising currencies. They bought a system unit for 9499 rubles and two graphics cards for 31431 rubles each. The power consumption of the system unit is 120 W, and for each graphics card, it is 125 W. The mining speed for one graphics card is 32 million hashes per second, allowing it to earn 0.00877 Ethereum per day. 1 Ethereum equals 27790.37 rubles. How many days will it take for the team's investment to pay off, considering electricity costs of 5.38 rubles per kWh? (20 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{165}
Calculate the expected value \( \mathbb{E}(e^{M_n}) \), where \( M_n = \max(X_1, X_2, \ldots, X_n) \) and \( X_1, \ldots, X_n \) are iid random variables each with distribution \( \mathcal{Exp}(1) \). The cumulative distribution function of \( M_n \) is given by \( F_{M_n}(x) = (1 - e^{-x})^n \) for \( x \geq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given vectors $a=(2,1)$, $b=(-3,2)$, and vector $c$ that satisfies $c⊥(a+b)$, and $b/!(c-a)$, find $c=$\_\_\_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\frac{7}{3},\frac{7}{9})}
Any positive integer $n$ can be written in the form $n = 2^aq$, where $a \ge 0$ and $q$ is odd. We call $q$ the odd part of $n$. Define the sequence $a_0,a_1,...$ as follows: $a_0 = 2^{2011}-1$ and for $m> 0, a_{m+i}$ is the odd part of $3a_m + 1$. Find $a_{2011}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$3^{2011}-1$

}
A computer program evaluates expressions without parentheses in the following way:

1) First, it performs multiplications and divisions from left to right one by one.
2) Then, it performs additions and subtractions from left to right.

For example, the value of the expression $1-2 / 3-4$ is $-3 \frac{2}{3}$. How many different results can we get if in the following expression each $*$ is independently replaced by one of the operators $+$, $-$, $/$, $\times$?

$1 * 1 * 1 * 1 * 1 * 1 * 1 * 1$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Determine the cardinality of the set of real-valued continuous functions \( f \) defined on \([0,1]\) such that \( f(x) \) is rational whenever \( x \) is rational.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
Given that \( y = f(x) \) is a monotonous function defined on \(\mathbb{R}\), with real numbers \( x_{1} < x_{2} \), \(\lambda \neq -1\), \(\alpha = \frac{x_{1} + \lambda x_{2}}{1 + \lambda}\), and \(\beta = \frac{x_{2} + \lambda x_{1}}{1 + \lambda}\). If \(\left| f(x_{1}) - f(x_{2}) \right| < |f(\alpha) - f(\beta)|\), find the range of values for \(\lambda\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lambda < 0}
If the width of a rectangle is increased by 3 cm and the height is decreased by 3 cm, its area does not change. What would happen to the area if, instead, the width of the original rectangle is decreased by 4 cm and the height is increased by 4 cm?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
A rectangular sheet of paper is folded so that two diagonally opposite corners come together. If the crease formed is the same length as the longer side of the sheet, what is the ratio of the longer side of the sheet to the shorter side?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{2}{\sqrt{5} - 1}}}
Suppose the function $\psi$ satisfies $\psi(1)=\sqrt{2+\sqrt{2+\sqrt{2}}}$ and $\psi(3x)+3\psi(x)=\psi(x)^3$ for all real $x$. Determine the greatest integer less than $\prod_{n=1}^{100}\psi(3^n)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5792}
Consider the partial differential equation (PDE): $$xu_x - yu_y = 0.$$ If two $\mathcal{C}^1$ functions, $u$ and $v$, both solve this equation and satisfy the condition $u(1, y) = v(1, y)$ for all $y \in \mathbb{R}$, determine the largest subset $A \subset \mathbb{R}^2$ on which $u$ and $v$ are equal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}^2}
If $|\overrightarrow{a}| = 2$, $|\overrightarrow{b}| = 4$, and $(\overrightarrow{a} + \overrightarrow{b})$ is perpendicular to $\overrightarrow{a}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\pi}{3}}
Irja and Valtteri are tossing coins. They take turns, Irja starting. Each of them has a pebble which reside on opposite vertices of a square at the start. If a player gets heads, she or he moves her or his pebble on opposite vertex. Otherwise the player in turn moves her or his pebble to an adjacent vertex so that Irja proceeds in positive and Valtteri in negative direction. The winner is the one who can move his pebble to the vertex where opponent's pebble lies. What is the probability that Irja wins the game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{7}}
Find the asymptotic upper bound for the recurrence relation: \( T(n) = (T(n-1))^2 \) with the initial condition \( T(n) = n \) for \( n \leq 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(2^{2^n})}
A vector in 110-dimensional Euclidean space $\mathbf{R}^{110}$ makes equal acute angles with the positive directions of the 110 coordinate axes. What is the approximate measure of that angle in degrees?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84.5}
Consider the bilinear form \(f(A, B) = n\operatorname{Trace}(AB) - \operatorname{Trace}(A)\operatorname{Trace}(B)\) defined on the space of \(n \times n\) complex matrices, \(M_n(\mathbb{C})\). Determine the dimension of the orthogonal complement \(U^\perp\), which consists of all matrices \(A\) such that \(f(A, B) = 0\) for every \(B \in M_n(\mathbb{C})\). Find \(\dim(U^\perp)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( \overline{\rm AB} \) be a diameter of a circle \( \omega \) and let \( C \) be a point on \( \omega \), different from \( A \) and \( B \). The perpendicular from \( C \) intersects \( \overline{\rm AB} \) at \( D \) and \( \omega \) at \( E (\ne C) \). The circle with center \( C \) and radius \( \rm CD \) intersects \( \omega \) at \( P \) and \( Q \). If the perimeter of \( \triangle{PEQ} \) is \( p \), find the length of \( \overline{\rm PQ} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{p}{3}}
Find the sum of the series: $$ \sum_{n=1}^\infty \frac{\sqrt{n-\sqrt{n}}}{n^{1/n + 3/2}} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Calculate the curl of the radial unit vector $e_r$ in spherical coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
$ABCDEF$ is a hexagon inscribed in a circle such that the measure of $\angle{ACE}$ is $90^{\circ}$. What is the average of the measures, in degrees, of $\angle{ABC}$ and $\angle{CDE}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
135°
}
Let $A=\{1,2,\ldots,n\}$. For a permutation $P=(P(1), P(2), \ldots, P(n))$ of the elements of $A$, let $P(1)$ denote the first element of $P$. Find the number of all such permutations $P$ so that for all $i,j \in A$:

(a) if $i < j<P(1)$, then $j$ appears before $i$ in $P$; and

(b) if $P(1)<i<j$, then $i$ appears before $j$ in $P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2^{(n-1)}$}
Evaluate the limit: $$\lim_{n\rightarrow \infty }\sum_{k=n}^{5n}\binom{k-1}{n-1}\left( \frac{1}{5} \right )^n\left( \frac{4}{5} \right )^{k-n}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
In \(\mathbb{R}^4\), consider the surface \(S\) consisting of all points that are 1 unit away from the origin. Given five points \(P_1, P_2, \ldots, P_5\) lying on \(S\), determine the maximum angle formed between any two points and the origin, denoted as \(\angle P_nOP_k\) for distinct natural numbers \(n\) and \(k\) from 1 to 5, where \(O\) is the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arccos\left(-\frac{1}{4}\right)}
In a regular tetrahedron, the center of the circumscribed sphere coincides with the center of the inscribed sphere. Determine the angle between two adjacent edges of the tetrahedron.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Find the value of \(c\) such that the areas of the regions bounded by the parabolas \(y = cx^2\) and \(y = 1 - x^2\) in the first quadrant are equal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Consider the graph $G$ obtained by deleting any edge from the complete bipartite graph $K_{7,8}$. Determine the number of spanning trees in the complement graph $\overline{G}$ of $G$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4405854208}
Calculate the line integral \( \int_l (e^x \sin y - 7y) \, dx + (e^x \cos y - 7) \, dy \) where \( l \) is the upper section of the circle \( x^2 + y^2 = 16x \) from point \( A(16,0) \) to point \( B(0,0) \), using Green's theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-224\pi}
$\prod\limits_{k=1}^{6}\left(x^{5}-5 x+k\right)=0$ 有 $\qquad$个实根。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Let $\{e_i\}$ be a basis of a $\mathbb{C}$-linear space $E$ and $h$ a Hermitian metric on $E$. Denote by $H=\{h_{ij}\}$ the matrix of $h$ for $\{e_i\}$, where $h_{ij}=h(e_i,e_j)$. Let $g\in GL(E)$ be a complex automorphism and consider the new basis $\{e'_i\}=\{ge_i\}$. Denote by $H'=\{h'_{ij}\}$ the corresponding matrix. Find the relation between $H$ and $H'$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{H' = G^* H G}
Calculate the entropy of a discrete random variable with the probability distribution given by \( p(x) = \frac{1}{2^x} \) for \( x = 1, 2, 3, \ldots \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Consider the set \( A = \{1, 2, 3, 4, 5\} \). A function \( f: A \to A \) is chosen at random from the set of all nonempty relations on \( A \). What is the probability that \( f \) is a bijective function such that \( f(x) \neq x \) for all \( x \in A \) and \( f(1) \neq 2 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{11}{40}}
Evaluate the integral \( \int_{0}^{1} x^{m-1} e^{-x} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\gamma(m, 1)}
In triangle ABC, where AB = 4024, AC = 4024, and BC = 2012, reflect the line AC over line AB to intersect the circumcircle of triangle ABC at point D (D ≠ A). Determine the length of segment CD.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3521}
A has 2 dollars and B has 3 dollars. They toss a coin with the probability of heads being \( \frac{1}{3} \). If the coin lands on heads, A gives 1 dollar to B. If it lands on tails, B gives 1 dollar to A. What is the probability that B will win all the money? Provide your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{31}}
Given that the domain of the function \( f(x) = \frac{1}{\lg \left(2^x + 4 \cdot 2^{-x} - a \right)} \) is \(\mathbf{R}\), find the range of values for the real number \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a < 3}
Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$. Find the number of real numbers $x$ that satisfy the equation $x^2 + 10,000\lfloor x \rfloor = 10,000x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{199}
Given that the variables $x$, $y$, and $z$ follow Gaussian distributions with means $\mu_x = \mu_y = \mu_z = 1,000,000$ and standard deviations $\sigma_x = \sigma_y = \sigma_z = 200,000$, calculate the probability that the function \[ f(x, y, z) = \frac{x}{1.1} + \frac{y}{1.1^2} + \frac{z}{1.1^3} - 2,000,000 > 0. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.9545}