inclusionAI/Ring-lite-rl-data huggingface dataset:
Among all the five-digit numbers formed by digits 1, 2, 3, 4, 5 without repetition, count how many numbers satisfy the "wave number" pattern, where the digits follow the alternating pattern "less-greater-less-greater...less-greater". Express your answer as a single integer representing the total count of such "wave numbers".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
In a 100x60 grid, you can place only one chocolate in each square. Chocolates are considered neighbors if they are in the same row or column with no other chocolate between them. Determine the maximum number of chocolates that can be placed in the grid such that no chocolate has more than two neighbors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6000}
7. The center of a unit square coincides with the center of a circle, and the square is inside the circle. If a point is randomly selected on the circle, the probability that this point can see two complete sides of the square is $\frac{1}{2}$, then the radius of the circle is . $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{4+2\sqrt{2}}}{2}}
A group of 6 students decided to form "study groups" and "service activity groups" based on the following principle:

- Each group must have exactly 3 members.
- For any pair of students, there are the same number of study groups and service activity groups that both students are members of.

Given that there is at least one group and no three students belong to both the same study group and service activity group, find the minimum number of groups.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let \(a = \sum\limits_{r = 1}^{11} {{{\tan }^2}\left( {\frac{{r\pi }}{{24}}} \right)} \) and \(b = \sum\limits_{r = 1}^{11} {{{\left( { - 1} \right)}^{r - 1}}{{\tan }^2}\left( {\frac{{r\pi }}{{24}}} \right)} \). Find the value of \(\log_{(2b - a)}(2a - b)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
It is known that 20% of people own at least 80% of all the money in the world. What is the smallest percentage of all people that can be guaranteed to own 90% of all the money?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Dan had 97 baseball cards, and 8 were torn. Sam bought 15 of Dan's baseball cards. How many baseball cards does Dan have now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{82}
Let \( A \) be a \( 7 \times 7 \) matrix such that \( 2A^2 - A^4 = I \), where \( I \) is the identity matrix. If \( A \) has two distinct eigenvalues, each with geometric multiplicity 3, how many non-zero entries are there in the Jordan canonical form of \( A \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the number of distinct values that the difference between the principal branch of the logarithm and any other branch of the logarithm defined on a simply connected region \( \Omega \) that does not contain 0 can take.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find the Galois group of the polynomial \( g(X) = (X^2 - 5)(X^3 + 2) \) over \( \mathbb{Q} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{C_2 \times S_3}
Calculate the limit: \[ \lim_{x \rightarrow +\infty} \frac{\sqrt x + x^3 -\sqrt{x^6-2x^4} -x}{\log(1+e^{3\sqrt x})} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Evaluate the integral \( \int_{0}^{1} x f(x) \, dx \) given that \( f \in \mathcal{C}^2[0,1] \), \( f(1) = 2 \), \( \int_{0}^{1} f(x) \, dx = \frac{1}{3} \), and \( \int_{0}^{1} f'(x) \, dx = \frac{25}{3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{31}{36}}
Compute the limit \( \lim_{n \to \infty} P\left( \sigma^2 \in (A_n, B_n) \right) \), where \( A_n = \frac{(n-1)S_{n}^{2}}{\chi^{2}_{n-1,\alpha/2}} \) and \( B_n = \frac{(n-1)S_{n}^{2}}{\chi^{2}_{n-1,1-\alpha/2}} \). Assume \( X_1, X_2, \ldots, X_n, \ldots \) are i.i.d. random variables with \( \mathbb{E}(X_1^4) < \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \alpha}
In a radioactive decay process that follows a Poisson distribution with a rate of 10,000 decays per second, calculate the approximate probability that the 1,000,000th decay occurs within 100.2 seconds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.9772}
Find the fourth derivative of the inverse function \(f^{-1}(y)\) evaluated at \(y=1\), where \(f(x) = \sin{x} + x^2 + 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-140}
Calculate the line integral \( K=\oint_{l} e^{x} \, dx + z\left(x^{2}+y^{2}\right)^{\frac{3}{2}} \, dy + yz^{3} \, dz \) using Stokes' theorem, where \( l \) is the closed curve \( OCBAO \) formed by the intersection of the surfaces \( z=\sqrt{x^{2}+y^{2}} \), \( x=0 \), \( x=2 \), \( y=0 \), and \( y=1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Calculate the expected value \( \mathbb{E}\left(\frac{X}{X+Y}\right) \) where \( X \) and \( Y \) are independent random variables with exponential distributions such that \( \mathbb{E}X=1 \) and \( \mathbb{E}Y=\frac{1}{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 - 2\ln 2}
Determine the fourth-last digit of the number \(N=3^{3^{3^{3^{3^3}}}}\) (six threes) without directly computing the entire exponentiation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given the matrix \(A=\begin{pmatrix} 2 & 0 & -1 \\ 0 & 4 & 0 \\ -1 & 0 & 2 \end{pmatrix}\) and the vector sequence defined by \(y_1=\begin{pmatrix} 2 \\ 0 \\ 0 \end{pmatrix}\) and \(y_{n+1}=Ay_n\), find the limit \(\lim_{n \to \infty} \frac{1}{n}\log\|y_n\|.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 3}
Calculate the surface integral of the unit normal vector over the surface defined by $z = x^2 + y^2$, where $z$ ranges from $1$ to $4$ and $\theta$ ranges from $0$ to $2\pi$. What is the value of this integral?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\pi}
Consider a uniform random tournament with \(n\) vertices. For any two vertices \(x\) and \(y\), an edge is drawn from \(x\) to \(y\) with probability 0.5, and from \(y\) to \(x\) otherwise. Let \(S\) be the set of all out-degrees, with \(s_1\) as the largest element and \(s_2\) as the second largest (or equal to \(s_1\) if \(S\) is a singleton). Given a constant \(c\in (0,1)\), determine \(\lim_{n\rightarrow\infty}\text{Pr}[s_1-s_2<cn]\].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A plane intersects a triangular pyramid and divides the medians of the faces emanating from one vertex in the ratios \(2:1, 1:2, 4:1\) respectively, measured from the vertex. Determine the ratio in which this plane divides the volume of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{43}}
Determine the smallest integer $k$ such that the product space $S^2 \times S^2$ can be embedded in $\mathbb{R}^k$. What is this optimal dimension $k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
An encoder maps source codewords $x \in \{0,1\}^K$ to channel codewords $y \in \{0,1\}^{N'}$ with $K/N' < 1$. The corresponding decoder maps a codeword $r \in \{0,1\}^{N'}$ to a source codeword $\hat{x} \in \{0,1\}^K$. Assume this coding scheme can correctly decode codewords at the output of a BSC with transition probability $q$, satisfying $K/N' \leq 1-H(q)$. Now, consider the reverse operation: an arbitrary codeword $r \in \{0,1\}^{N'}$ is input to the decoder, outputting $\hat{x} \in \{0,1\}^K$. The codeword $\hat{x}$ is then input to the encoder, outputting $y \in \{0,1\}^{N'}$. What is the probability that a certain bit of $y$ differs from the corresponding bit of $r$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{q}
Determine the flow rate of water across a net submerged in a river. The velocity vector field of the river is given by \( \vec{v}=(x-y)\vec{i}+(z+y)\vec{j}+z^2\vec{k} \). The net is described by the surface \( y=\sqrt{1-x^2-z^2}, y\geq 0 \), and is oriented in the positive \( y \)-direction. Calculate the flow rate by evaluating the surface integral \( \iint_{S}\vec{v}\,d\vec{S} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4\pi}{3}}
Find a closed-form formula for \( \frac{d\text{vec}(X^T)}{d\text{vec}(X)} \), where \( X \) is an \( n \times n \) matrix with its entry indices increasing along every column. For example, when \( n = 2 \), \( X = \begin{pmatrix} x_1 & x_3 \\ x_2 & x_4 \end{pmatrix} \). The vectorization \( \text{vec}(X) \) is formed by stacking the columns of \( X \) into a single vector. Provide the formula for any \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{K_{n,n}}
A system consists of three components. Component 1 is in series with two parallel components, 2 and 3. Each component operates independently and has a lifetime that follows an exponential distribution with rate parameter \( \lambda = 0.002 \). Calculate the probability that the system will last more than 500 hours.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.2209}
Given the sequences \( a_m = \sum_{k=0}^{m} (\sqrt{k} + \sqrt{k+1})^{-1} \) and \( b_m = \sum_{k=0}^{m} (\sqrt{k} + \sqrt{k+2})^{-1} \), define \( c_n = \prod_{m=0}^{n} \frac{a_m}{a_m - 2(b_m - b_{m+1})} \). Find the smallest natural number \( n \) for which \( c_n < 0.01 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{139}
A 1992-digit number is written such that each two-digit number formed by adjacent digits is divisible by either 17 or 23. If the last digit of the number is 1, what is the first digit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of connected components in the graph \( H_n \), where \( H_n \) is defined for each positive integer \( n \) as a graph whose vertices are \( n \)-tuples with binary entries (0 or 1), and two vertices are adjacent if they differ in precisely two coordinates. Find the number of connected components for any \( n \geq 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the limit: $$\lim_{x\to\infty} \sum ^\infty_{n=3} \frac{n\cdot x^{n-2}\cdot(-1)^n\cdot(n-1)}{(2n)!}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{12}}
Calculate the number of possible arrangements of 5 tigers, 5 lions, and 5 cheetahs in 15 consecutive cages, given the following restrictions: tigers cannot be placed in the 5 leftmost cages, lions cannot be placed in the 5 middle cages, and cheetahs cannot be placed in the 5 rightmost cages.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2252}
A nonzero polynomial with rational coefficients has all of the numbers \(1+\sqrt{2}, \; 2+\sqrt{3}, \;3+\sqrt{4},\; \dots, \;1000+\sqrt{1001}\) as roots. What is the smallest possible degree of such a polynomial?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1970}
Given a sequence $a_n$ defined as $a_n = 2^{x_n}$, where $x_n = p_1^{p_2^{\cdots ^{p_n}}}$, express the sum of the differences between consecutive terms, $\sum\limits_{i=1}^n (a_i - a_{i-1})$, in terms of $a_n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a_n - 2}
In a city, there are 2019 metro stations, and some pairs of stations are connected by tunnels, allowing travel between any two stations. The mayor wants to organize several metro lines such that each line includes multiple distinct stations connected sequentially by tunnels, and each station is part of at least one line. The number of lines should not exceed $k$. However, the mayor's plan is impracticable. Determine the largest value of $k$ for which this situation could occur.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1008}
Determine the conditions on the parameter $p$ such that the curve $y = x^p\cos\left(\frac{\pi}{x}\right)$, for $0 < x \leqslant 1$, has a finite length.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p > 1}
Evaluate the sum \( \sum_{b=1}^r |S(r,b,\psi)|^2 \), where \( S(r,b,\psi) = \sum_{y=1}^r \psi(y) e(by/r) \) and \( \psi \) is an arbitrary character modulo \( r \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r \phi(r)}
Find the conditions on $m$ such that the sequence $\{b_n\}_{n=1}^\infty$ defined by \( b_n = \frac{n^m + 1}{n^{m+1} + 1} \) is monotonically decreasing.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \geq -1}
Given that $a + \frac{1}{b} = b + \frac{1}{c} = c + \frac{1}{a}$ for distinct real numbers $a$, $b$, and $c$, find the value of $abc$. \( \text{Express your answer as a single number.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given that \( \lim_{x\to2}\frac{\ xf(x)-4}{x-2}=3, \) find the value of \( \lim _{ x\rightarrow 2 }{ \frac { f\left( x \right) -2 }{ \sqrt { x-1 } -1 }  }. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
For any finite set $X$, let $|X|$ denote the number of elements in $X.$ Define $$S_n = \\sum |A \\cap B|,$$ where the sum is taken over all ordered pairs $(A, B)$ such that $A$ and $B$ are subsets of $\\{1, 2, 3, \u2026, n\\}$ with $|A| = |B|.$ For example, $S_2 = 4$ because the sum is taken over the pairs of subsets $$(A, B) \\in \\{ (\\emptyset, \\emptyset), (\\{1\\}, \\{1\\}), (\\{1\\}, \\{2\\}), (\\{2\\}, \\{1\\}), (\\{2\\}, \\{2\\}), (\\{1, 2\\}, \\{1, 2\\})\\},$$ giving $S_2 = 0 + 1 + 0 + 0 + 1 + 2 = 4.$ Let $\\frac{S_{2022}}{S_{2021}} = \\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find the remainder when $p + q$ is divided by $1000.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
245
}
Evaluate the line integral \( \oint_C F \cdot dr \) using Green's Theorem, where \( C \) is a closed \( C^1 \) path oriented counterclockwise, consisting of a piece of the curve \( y^2 = 2(x+2) \) and the vertical segment \( x = 2 \). The vector field \( F \) is given by \( F(x,y) = \left(\frac{-y}{x^2+y^2}, \frac{x}{x^2+y^2}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Determine the smallest grid size such that, when colored with 5 different colors, there must exist a square with cells of the same color directly above, below, to the left, and to the right. What is this minimal grid size?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Calculate the area of the region bounded by the parametric equations \( x = 32 \cos^{3} t \) and \( y = \sin^{3} t \), and the vertical line \( x = 4 \) for \( x \geq 4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
Each of the spots in a $8\times 8$ chessboard is occupied by either a black or white “horese”. At most how many black horses can be on the chessboard so that none of the horses attack more than one black horse?
Remark: A black horse could attack another black horse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
16
}
A circle with radius 1 is situated between two parallel lines, touching both lines. An isosceles triangle is positioned such that its base lies on one of the lines, and its vertex is on the other line. It is known that the triangle and the circle have exactly one common point, and this point lies on the inscribed circle of the triangle. Find the radius of the inscribed circle of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Consider permutations of the set \(\{1, 2, 3, 4, 5, 6\}\) where the third element is 4. Calculate the sum of the inversion numbers for all such permutations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{912}
Calculate the exact length of the curve given by \(x= \frac{1}{3}(y^2+2)^\frac{3}{2}\) for \(1 \leq y \leq 2\) using the arc length formula \(\int_{a}^{b}\sqrt{1+\left(\frac{\text{d}y}{\text{d}x}\right)^2}\space\space\text{d}x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{10}{3}}
A tradesman has three types of items which he deals with. For Item A, he uses a false balance that defrauds to the extent of 30% in buying and selling. For Item B, he defrauds to the extent of 20% in buying and 10% in selling. For Item C, he defrauds 10% in buying and 20% in selling. Considering he spends equal amounts on purchasing all three items, calculate his overall percentage gain on his total outlay.

Note: In each case, the defrauding percentage is applied during both buying and selling of the respective item.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
Let $ABCD$ be a trapezoid such that $AD\parallel BC$. The interior angle bisectors of the corners $A$ and $B$ meet on $[DC]$. If $|BC|=9$ and $|AD|=4$, find $|AB|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
There are seven students taking a graduation photo in a row. Among them, student A cannot stand next to student B, and students B and C must stand together. Calculate the total number of different arrangements possible under these conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1200}
In a class of 25 students arranged in five rows of five, each student must sit next to every other student at least once, either side by side, one behind the other, or diagonally. What is the minimum number of weeks required for this seating arrangement to occur?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Consider two polynomials, $P_1(x)$ and $P_2(x)$, with integer coefficients, and integers $n$ and $a$. Determine the number of distinct ordered pairs $(P_1(x), P_2(x))$ such that $(P_1(x))^2 + (P_2(x))^2 = (x^n - a)^2$, excluding the trivial solution $(x^n - a, 0)$. Assume that $x^n - a$ is irreducible over the Gaussian integers $\mathbb{Z}[i][x]$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the function \( g \) defined on the set of integers with the following conditions:

1) \( g(1) - 1 > 0 \)
2) \( g(x) g(y) + x + y + xy = g(x+y) + x g(y) + y g(x) \) for any \( x, y \in \mathbb{Z} \)
3) \( 3 g(x+1) = g(x) + 2x + 3 \) for any \( x \in \mathbb{Z} \).

Find the value of \( g(-6) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{723}
A soccer coach named  $C$  does a header drill with two players  $A$  and  $B$ , but they all forgot to put sunscreen on their foreheads.  They solve this issue by dunking the ball into a vat of sunscreen before starting the drill.  Coach  $C$  heads the ball to  $A$ , who heads the ball back to  $C$ , who then heads the ball to  $B$ , who heads the ball back to  $C$ ;
this pattern  $CACBCACB\ldots\,$  repeats ad infinitum.  Each time a person heads the ball,  $1/10$  of the sunscreen left on the ball ends up on the person's forehead.  In the limit, what fraction of the sunscreen originally on the ball will end up on the coach's forehead?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{10}{19}}
Compute the number of tuples \(\left(a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right)\) of (not necessarily positive) integers such that \(a_{i} \leq i\) for all \(0 \leq i \leq 5\) and
\[ a_{0} + a_{1} + \cdots + a_{5} = 6. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2002}
In a school, there are 1200 students. Each student must join exactly $k$ clubs. Given that there is a common club joined by every 23 students, but there is no common club joined by all 1200 students, find the smallest possible value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Consider the polynomial \( p(x) = x^3 + ax^2 + bx + c \) with roots \( \alpha, \beta, \gamma \), where \( \alpha \leq \beta \leq \gamma \) are all positive integers. Given that \( 2^2a + 2^1b + 2^0c = -2^{11} \), find the sum of all possible values of \( \gamma \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6458}
凸四边形具有面积 $30$ 和边长 $5, 6, 9,$  $7,$ 并按此顺序排列。由 $\theta$ 四边形对角线形成的锐角的量度表示。那么 $\tan \theta$ 可以写成  $\tfrac{m}{n}$ 形式 ， 其中 $m$ 和 $n$ 是相对素数的正整数。查找  $m + n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
Let $f(n)=n^2+100.$ Compute the remainder when $\underbrace{f(f(\cdots f(f(}_{2025 \ f\text{'s}}1))\cdots ))$ is divided by $10^4.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3101
}
a, b, c, and d are integers. a + b + c + d < 25, a > 8, b < 5, c is an odd number, and d is an even number. What is the smallest possible value of a - b + c - d?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4}
Evaluate the limit: \[ \lim_{x\to 0}\frac{x^{(\sin x)^{x}}-(\sin x)^{x^{\sin x}}}{x^3} \] without using Taylor's formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
A 3D cake is cut by $n$ planes, resulting in the maximum number of pieces. Given that the maximum number of pieces is defined by the Cake Number $N=\frac{1}{6}(n^3+5n+6)$, determine the number of pieces that have the cake crust as one of their boundaries. Assume the entire surface of the cake is crust.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2 - n + 2}
Determine the number of models of the real numbers up to isomorphism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Consider the following algorithm:

Step 0. Set \( n = m \).

Step 1. If \( n \) is even, divide \( n \) by two. If \( n \) is odd, increase \( n \) by one.

Step 2. If \( n > 1 \), go to Step 1. If \( n = 1 \), end the algorithm. 

How many natural numbers \( m \) exist such that Step 1 will be performed exactly 15 times when this algorithm is executed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{610}
Every edge of a square-based pyramid is 2 units. Another pyramid with edges of equal length is attached to one of the pyramid's faces as a base. The total length of the edges of the resulting solid is 18 units. What is the volume of this solid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{2}}
Equilateral triangle $ABC$ has circumcircle $\Omega$. Points $D$ and $E$ are chosen on minor arcs $AB$ and $AC$ of $\Omega$ respectively such that $BC=DE$. Given that triangle $ABE$ has area 3 and triangle $ACD$ has area 4, find the area of triangle $ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{37}{7}}
Five people are seated around a circular table. Let \( f \) represent the number of people sitting next to at least one female and \( m \) represent the number of people sitting next to at least one male. How many possible ordered pairs \( (f, m) \) are there? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}