inclusionAI/Ring-lite-rl-data huggingface dataset:
Find \( \lim_{n \to \infty} a_n^n \) where \( a_n = \frac{(n!)^2}{(n+k)!(n-k)!} \) for each \( k \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-k^2}}
Let $x_1,x_2,\ldots,x_n$ be $n$ non-negative numbers ($n>2$) with a fixed sum $S$. What is the maximum of $x_1 x_2 + x_2 x_3 + \dots + x_n x_1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{S^2}{4}\)}
Evaluate the limit: $$ \lim_{x\to 3} \frac {xe^x-3e^x-xe^3+3e^3}{\ln^2(x^2+x-11)} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{e^3}{49}}
What is the perimeter of the boundary of the region consisting of all points which can be expressed as $(2u-3w, v+4w)$ with $0\le u\le1$, $0\le v\le1,$ and $0\le w\le1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$16$}
Calculate the line integral \( \int_{dA} (x^2 \, dx + y^2 \, dy) \) over the region \( A = \{(x, y) \in \mathbb{R}^2 : -\frac{\pi}{2} \leq x \leq \frac{\pi}{2}, \ -1 \leq y \leq \cos(x)\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A right rectangular prism  $P$  (i.e., a rectangular parallelpiped) has sides of integral length  $a, b, c,$  with  $a\le b\le c.$   A plane parallel to one of the faces of  $P$  cuts  $P$  into two prisms, one of which is similar to  $P,$  and both of which have nonzero volume.  Given that  $b=1995,$  for how many ordered triples  $(a, b, c)$  does such a plane exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
A bowl contained  $10\%$  blue candies and  $25\%$  red candies.  A bag containing three quarters red candies and one quarter blue candies was added to the bowl.  Now the bowl is  $16\%$  blue candies.  What percentage of the candies in the bowl are now red?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45\%}
Evaluate the Cauchy Principal Value of the integral \( \int_{0}^{2\pi}\frac{d\theta}{1+2\cos(\theta)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A cube with side length $10$ is divided into two cuboids with integral side lengths by a straight cut. Afterwards, one of these two cuboids is divided into two cuboids with integral side lengths by another straight cut. What is the smallest possible volume of the largest of the three cuboids?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{350}
Let $a_0=1, a_1=2$, and $a_n=4 a_{n-1}-a_{n-2}$ for $n \geq 2$. Find an odd prime factor of $a_{2015}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{181}
A total of  $731$  objects are put into  $n$  nonempty bags where  $n$  is a positive integer. These bags can be distributed into  $17$  red boxes and also into  $43$  blue boxes so that each red and each blue box contain  $43$  and  $17$  objects, respectively. Find the minimum value of  $n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Unit circles $a, b, c$ satisfy $d(a, b) = 1$, $d(b, c) = 2$, and $d(c, a) = 3,$ where $d(x, y)$ is defined to be the minimum distance between any two points on circles $x$ and $y$. Find the radius of the smallest circle entirely containing $a$, $b$, and $c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{7}{2}$}
Joan's high school played 1,200 baseball games this year. Out of them, 30% were played at night, 20% were played during holidays, and 25% were played during her away days due to family visits. Additionally, Joan was sick and missed 50 games throughout the year. She attended 50% of the night games, 75% of the holiday games, and 60% of the away day games. How many baseball games did Joan miss?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{410}
Let $A, E, H, L, T$, and $V$ be chosen independently and at random from the set $\left\{0, \frac{1}{2}, 1\right\}$. Compute the probability that $\lfloor T \cdot H \cdot E\rfloor=L \cdot A \cdot V \cdot A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{55}{81}}
Let  $(a_n)_{n \geq 0}$  be the sequence of integers defined recursively by  $a_0 = 0, a_1 = 1, a_{n+2} = 4a_{n+1} + a_n$  for  $n \geq 0.$  Find the common divisors of  $a_{1986}$  and  $a_{6891}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Find the smallest natural number that cannot be written in the form \(\frac{2^{a} - 2^{b}}{2^{c} - 2^{d}}\), where \(a\), \(b\), \(c\), and \(d\) are natural numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
A bacterium develops on a $100 \times 100$ grid. It can contaminate a new cell if and only if two adjacent cells are already contaminated. What is the minimal number of initially contaminated cells required for the bacterium to be able to spread everywhere on the grid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
The product $(8)(888\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$.  What is $k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{991}
For $n \ge 3,$ if $$\sum_{k=1}^{n}x_k^m = y_m = (m+1)(n+1)-2, \quad \forall 1 \leqslant m \leqslant n,$$ then find the value of $$\sum_{k=1}^{n}x_k^{n+1} = y_{n+1}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n+1)(n+2) - 2}
Kelvin the Frog was bored in math class one day, so he wrote all ordered triples $(a, b, c)$ of positive integers such that $abc=2310$ on a sheet of paper. Find the sum of all the integers he wrote down. In other words, compute $$\sum_{\stackrel{abc=2310}{a,b,c\in \mathbb{N}}} (a+b+c),$$ where $\mathbb{N}$ denotes the positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49140}
Let $\omega = \cos\frac{2\pi}{7} + i \cdot \sin\frac{2\pi}{7},$ where $i = \sqrt{-1}.$ Find the value of the product\[\prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right).\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$24$}
Find the number of \(2 \times 2\) matrices \(A\) with entries in \(\mathbf{Z}/3\mathbf{Z}\) such that \(A^{50} = I\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
When 1 / 20 % of 8000 is subtracted from some fraction of 8000, the difference is 796. What is the fraction that needs to be subtracted?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.1}
Suppose that \[\operatorname{lcm}(1024,2016)=\operatorname{lcm}(1024,2016,x_1,x_2,\ldots,x_n),\] with  $x_1$ ,  $x_2$ ,  $\cdots$ ,  $x_n$  are distinct postive integers. Find the maximum value of  $n$ .

*Proposed by Le Duc Minh*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Three boys \( B_{1}, B_{2}, B_{3} \) and three girls \( G_{1}, G_{2}, G_{3} \) are to be seated in a row according to the following rules:
1) A boy will not sit next to another boy and a girl will not sit next to another girl,
2) Boy \( B_{1} \) must sit next to girl \( G_{1} \).
If \( s \) is the number of different such seating arrangements, find the value of \( s \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Let $N \in \left \{10, 11, \ldots, 99\right \} $ be a two-digit positive integer. Compute the number of values of $N$ for which the last two digits in the decimal expansion of $N^{21}$ are the digits of $N$ in the same order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
56
}
For the multiplicative group Z*35, determine the number of generators and list them.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Using each of the eight digits $1,3,4,5,6,7,8$ and $9$ exactly once, a three-digit number $A$, two two-digit numbers $B$ and $C$, $B<C$, and a one digit number $D$ are formed. The numbers are such that $A+D=B+C=143$. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Determine all polynomial functions \( p(x) \) with complex coefficients such that \( p(x) \) maps real numbers to real numbers and non-real complex numbers to non-real complex numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p(x) = ax + b}
Given the function f(x) = x^3 - 3x + m, and three numbers a, b, c chosen from the interval [0, 2], there exists a triangle with side lengths f(a), f(b), f(c). Calculate the range of values for m. Express your answer as an inequality (e.g., m > value).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m > 6}
A  $6 \times 6$  board is given such that each unit square is either red or green. It is known that there are no  $4$  adjacent unit squares of the same color in a horizontal, vertical, or diagonal line. A  $2 \times 2$  subsquare of the board is *chesslike* if it has one red and one green diagonal. Find the maximal possible number of chesslike squares on the board.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
The bisector $DE$ of triangle $ADC$ is extended to its intersection with the circumscribed circle at point $B$. Given that the length $BD = 1$ and $\sin \angle ADC = a$. Find the area of the quadrilateral $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2} a}
In a room illuminated by two light bulbs, each bulb has an exponential lifetime with a parameter of \( \lambda = 1 \). The bulbs are replaced only when both have burned out. What fraction of the time is the room half-lit, meaning only one bulb is functioning?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
A farmer wants to start growing watermelons. He wants to sell at least 10,000 watermelons each year. Watermelons are grown from seeds (one seed grows into one watermelon). Each watermelon can produce 250 good seeds, which can be planted the next year, but in that case, the watermelon cannot be sold. What is the minimum number of watermelon seeds the farmer must buy initially to start growing watermelons and never have to buy seeds again?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10041}
On March 1st the sun sets at 6 PM. Every day after the sun sets 1.2 minutes later. It is a certain time and 40 days after March 1st. The sun will set in 38 minutes. What is the current time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6:10}
Find the values of  $a\in [0,\infty)$  for which there exist continuous functions  $f:\mathbb{R}\rightarrow \mathbb{R}$ , such that  $f(f(x))=(x-a)^2,\ (\forall)x\in \mathbb{R}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a = 0}
Person A and person B are playing a game with the following rules: In odd-numbered rounds, the probability of A winning is $\frac{3}{4}$; in even-numbered rounds, the probability of B winning is $\frac{3}{4}$. There are no ties in any round. The game ends when one person has won 2 more rounds than the other. Determine the expected number of rounds played when the game ends.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{16}{3}}
The mean of the set of numbers $\{106, 102, 95, 103, 100, y, x\}$ is 104. What is the median of this set of seven numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{103}
Determine the order of the poles of the modular lambda function in terms of the variable $\lambda$. Provide your answer as a single number or expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Ten families have an average of 2 children per family. If exactly two of these families are childless, what is the average number of children in the families with children? Express your answer as a decimal to the nearest tenth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.5}
Calculate the volume of the region bounded by the surfaces $x^2 + z = 1$ and $y^2 + z = 1$ in the first octant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Let  $n,k$  be positive integers such that  $n>k$ . There is a square-shaped plot of land, which is divided into  $n\times n$  grid so that each cell has the same size. The land needs to be plowed by  $k$  tractors; each tractor will begin on the lower-left corner cell and keep moving to the cell sharing a common side until it reaches the upper-right corner cell. In addition, each tractor can only move in two directions: up and right. Determine the minimum possible number of unplowed cells.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n-k)^2}
If the expression $x^2 + 9x + 20$ can be written as $(x + a)(x + b)$, and the expression $x^2 + 7x - 60$ can be written as $(x + b)(x - c)$, where $a$, $b$, and $c$ are integers. What is the value of $a + b + c$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
3. On each of the lines $y=1$ and $y=12$, there are 200 points with abscissas $1,2,3, \ldots, 200$. In how many ways can three points be chosen from the 400 marked points so that they form the vertices of a right triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80268}
## Task Condition

Calculate the definite integral:

$$
\int_{\arccos (1 / \sqrt{10})}^{\arccos (1 / \sqrt{26})} \frac{12 d x}{(6+5 \tan x) \sin 2 x}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln\frac{105}{93}}
Let $B$ be a right rectangular prism (box) with edges lengths $1,$ $3,$ and $4$, together with its interior. For real $r\geq0$, let $S(r)$ be the set of points in $3$-dimensional space that lie within a distance $r$ of some point in $B$. The volume of $S(r)$ can be expressed as $ar^{3} + br^{2} + cr +d$, where $a,$ $b,$ $c,$ and $d$ are positive real numbers. What is $\frac{bc}{ad}?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
在平面直角坐标系 xOy 中，A 为椭圆 \df{x^2}{4}+y^2=1 上一点，M 为线段 OA 上的动点. 过 M 作直线与椭圆交于 P, Q 两点，若 \overrightarrow{PM}=2\overrightarrow{MQ}，求四边形 OPAQ 面积的最大值.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{5}
Evaluate the line integral \(\int_C \vec G \cdot \vec{dr}\) where \(\vec G(x,y,z) = (8xz + 5y)\vec i + (2yz + 5x)\vec j + (x^2 + 3z^2)\vec k\) and \(C\) is the curve consisting of the line segment from \((1,0,1)\) to \((3,2,0)\) followed by the parabola \(y=x^2, z=1\) from \((3,9,1)\) to \((0,0,1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{638}{3}}
In an election for the Peer Pressure High School student council president, there are 2019 voters and two candidates Alice and Celia (who are voters themselves). At the beginning, Alice and Celia both vote for themselves, and Alice's boyfriend Bob votes for Alice as well. Then one by one, each of the remaining 2016 voters votes for a candidate randomly, with probabilities proportional to the current number of the respective candidate's votes. For example, the first undecided voter David has a  $\tfrac{2}{3}$  probability of voting for Alice and a  $\tfrac{1}{3}$  probability of voting for Celia.

What is the probability that Alice wins the election (by having more votes than Celia)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1513}{2017}}
An arithmetic sequence has a common difference of 4 and each term is a real number. The square of the first term and the sum of the remaining terms do not exceed 100. What is the maximum number of terms that such a sequence can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A positive integer $n$ is magical if $\lfloor\sqrt{\lceil\sqrt{n}\rceil}\rfloor=\lceil\sqrt{\lfloor\sqrt{n}\rfloor}\rceil$ where $\lfloor\cdot\rfloor$ and $\lceil\cdot\rceil$ represent the floor and ceiling function respectively. Find the number of magical integers between 1 and 10,000, inclusive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1330}
The angle between two tangents drawn from one point to a circle is $70^{\circ}$.

Find the angular measures of the arcs enclosed between the points of tangency.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110^{\circ}, 250^{\circ}}
Given that the regular price for one backpack is $60, and Maria receives a 20% discount on the second backpack and a 30% discount on the third backpack, calculate the percentage of the $180 regular price she saved.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16.67\%}
Find the prime decomposition of 2 in the cyclotomic field \( \mathbb{Q} ( \xi_5 ) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2)}
Several sets of prime numbers, such as $\{7,83,421,659\}$ use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{207}
Xiaohua's route from home to school consists of a flat road and a downhill road. Assuming he always maintains a speed of 60 meters per minute on the flat road, 80 meters per minute on the downhill road, and 40 meters per minute on the uphill road, it takes him 10 minutes to get from home to school and 15 minutes to get from school to home. How far is Xiaohua's home from his school?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{700}
Let \(ABCDE\) be a convex pentagon such that \(AB = AE = CD = BC + DE = 1\) and \(\angle ABC = \angle DEA = 90^\circ\). Calculate the area of \(ABCDE\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In triangle $\triangle ABC$, point $D$ is a point on side $AC$, $BD=1$, $\sin \angle DBC=\frac{3}{5}$, $\cos \angle ABC=\frac{\sqrt{10}}{10}$. Find the minimum value of the area of $\triangle ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{18}{25}}
There are  $2022$  signs arranged in a straight line. Mark tasks Auto to color each sign with either red or blue with the following condition: for any given sequence of length  $1011$  whose each term is either red or blue, Auto can always remove  $1011$  signs from the line so that the remaining  $1011$  signs match the given color sequence without changing the order. Determine the number of ways Auto can color the signs to satisfy Mark's condition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{1011}}
Last summer $30\%$ of the birds living on Town Lake were geese, $25\%$ were swans, $10\%$ were herons, and $35\%$ were ducks. What percent of the birds that were not swans were geese?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40\%}
In the expression \( \left( {\_ \_  \times  \_ \_ }\right)  + \left( {\_ \_ \_  \times  \_ \_ \_ }\right) \) each blank is to be filled in with one
 of the digits\(1,2,3,\) or 4, with each digit being used once. How many different values can
 be obtained?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3

}
Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be three vectors such that $\mathbf{a}$ is a unit vector, $\mathbf{b}$ and $\mathbf{c}$ have magnitudes of 2, and $\mathbf{a} \cdot \mathbf{b} = 0$, $\mathbf{b} \cdot \mathbf{c} = 4$. Given that
\[
\mathbf{a} = p(\mathbf{a} \times \mathbf{b}) + q(\mathbf{b} \times \mathbf{c}) + r(\mathbf{c} \times \mathbf{a}),
\]
and $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = 2$, find the value of $p + q + r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Let $\Gamma_1, \Gamma_2, \Gamma_3$ be three pairwise externally tangent circles with radii $1,2,3,$ respectively. A circle passes through the centers of $\Gamma_2$ and $\Gamma_3$ and is externally tangent to $\Gamma_1$ at a point $P.$ Suppose $A$ and $B$ are the centers of $\Gamma_2$ and $\Gamma_3,$ respectively. What is the value of $\frac{{PA}^2}{{PB}^2}?$

[i]Proposed by Kyle Lee[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8}{15}}
Let $A$ and $B$ be points on a circle $\mathcal{C}$ with center $O$ such that $\angle AOB = \frac{\pi}{2}$. Circles $\mathcal{C}_1$ and $\mathcal{C}_2$ are internally tangent to $\mathcal{C}$ at $A$ and $B$ respectively and are also externally tangent to one another. The circle $\mathcal{C}_3$ lies in the interior of $\angle AOB$ and is tangent externally to $\mathcal{C}_1$, $\mathcal{C}_2$ at $P$ and $R$ and internally tangent to $\mathcal{C}$ at $S$. Find the measure of $\angle PSR$. \( \text{Express your answer in radians.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
求最大的正实数 k，使得对任意满足 a+b+c=1 的非负实数 a,b,c，均有\ds{}{}\df{a}{1+9bc+k(b-c)^2}\geq\df{1}{2}.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The number $a=\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying
\[\lfloor x \rfloor \cdot \{x\} = a \cdot x^2\]is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\}=x- \lfloor x \rfloor$ denotes the fractional part of $x$. What is $p+q$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{929}
Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positive integers. What is $a+b?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85}
Around a circle are written the natural numbers from 1 to \( N \) with \( N > 2 \), such that each number is written only once and each pair of neighboring numbers has at least one digit in common. Find the smallest \( N > 2 \) for which this is possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
In how many ways can the numbers $1,2, \ldots, 2002$ be placed at the vertices of a regular 2002-gon so that no two adjacent numbers differ by more than 2? (Rotations and reflections are considered distinct.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4004}
The largest prime factor of 101101101101 is a four-digit number $N$. Compute $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9901}
In Figure 1, $AEFD$ is a unit square. The ratio of the length of the rectangle $ABCD$ to its width is equal to the ratio of the length of the rectangle $BCFE$ to its width. If the length of $AB$ is $W$ units, find the value of $W$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1 + \sqrt{5}}{2}}
In a tetrahedron \( ABCD \), \( AB = AC = AD = 5 \), \( BC = 3 \), \( CD = 4 \), \( DB = 5 \). Find the volume of this tetrahedron.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5\sqrt{3}}