inclusionAI/Ring-lite-rl-data huggingface dataset:
The rules for a race require that all runners start at $A$, touch any part of the 1500-meter wall, touch any part of the opposite 1500-meter wall, and stop at $B$. What is the minimum distance a participant must run? Assume that $A$ is 400 meters directly south of the first wall, and that $B$ is 600 meters directly north of the second wall. The two walls are parallel and are 1500 meters apart. Express your answer to the nearest meter.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2915}
Consider the system of quadratic equations in $n$ variables $x_1, x_2, \ldots, x_n$:

$$ax_1^2+bx_1+c=x_2$$
$$ax_2^2+bx_2+c=x_3$$
$$\vdots$$
$$ax_n^2+bx_n+c=x_1$$

where $a,b,c \in \mathbb{R}$ and $a \neq 0$. If $(b-1)^2 > 4ac$, determine the number of distinct solutions to this system.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n}
Minimize the expression \( \frac{a_1^TXa_1+a_4^TXa_4}{a_1^TXa_1 a^T_4Xa_4-a_2^TXa_2 a_3^T X a_3} \) subject to the constraints \( \text{trace}(X) = 1 \) and \( X \succeq 0 \), where \( X \) is a matrix and \( a_k \in \mathbb{R}^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find all the triples of integers $ (a, b,c)$ such that:
\[ \begin{array}{ccc}a+b+c &=& 24\\ a^{2}+b^{2}+c^{2}&=& 210\\ abc &=& 440\end{array}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$ (5, 8, 11)$}
In the $xyz$ space, consider a right circular cylinder with radius of base 2, altitude 4 such that
\[\left\{
\begin{array}{ll}
x^2+y^2\leq 4 &\quad  \\
0\leq z\leq 4 &\quad
\end{array}
\right.\]
Let $V$ be the solid formed by the points $(x,\ y,\ z)$ in the circular cylinder satisfying
\[\left\{
\begin{array}{ll}
z\leq (x-2)^2 &\quad  \\
z\leq y^2 &\quad
\end{array}
\right.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ 12\pi - \frac{80}{3} \]
}
Three fair twenty-sided dice are rolled and arranged in decreasing order. Find the sum of the numerator and denominator of the expected value of the sum of the two largest dice, expressed in simplest form as \( \frac{p}{q} \), where \( p \) and \( q \) are relatively prime positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2159}
In a table $4\times 4$ we put $k$ blocks such that
i) Each block covers exactly 2 cells
ii) Each cell is covered by, at least, one block
iii) If we delete a block; there is, at least, one cell that is not covered.
Find the maximum value of $k$.
Note: The blocks can overlap.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
The sum of the first fifty positive odd integers subtracted from the sum of the first fifty positive even integers, each decreased by 3, calculate the result.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-100}
What is the minimum number of queries required to determine the number in each cell of a $7\times7$ board with $49$ distinct numbers, given that you can query any $3$ cells at a time to find out their set of numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Calculate the determinant of the matrix \(A=\begin{pmatrix} \sin\alpha & \cos\alpha & a\sin\alpha & b\cos\alpha & ab \\ -\cos\alpha & \sin\alpha & -a^2\sin\alpha & b^2\cos\alpha & a^2b^2 \\ 0 & 0 & 1 & a^2 & b^2 \\ 0 & 0 & 0 & a & b \\ 0 & 0 & 0 & -b & a \end{pmatrix}\) where \(\alpha,a,b \in \mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^2 + b^2}
A train 1500 m long can cross an electric pole in 120 sec. Find the speed of the train.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12.5}
Each segment with endpoints at the vertices of a regular 100-sided polygon is colored red if there is an even number of vertices between the endpoints, and blue otherwise (in particular, all sides of the 100-sided polygon are red). Numbers are placed at the vertices such that the sum of their squares equals 1, and the product of the numbers at the endpoints is allocated to each segment. Then, the sum of the numbers on the red segments is subtracted by the sum of the numbers on the blue segments. What is the maximum possible result?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Calculate the value of the polynomial $f(x) = 8x^4 + 5x^3 + 3x^2 + 2x + 1$ at $x = 2$ using the Horner's method. What is the value of $v_2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
For a Brownian Motion $X_t$ starting at $0$, calculate the probability $\mathbb{P}(X_{2T}>0, X_{T}>0)$, given a time $T>0$. Provide your answer as a single probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{8}}
Find the value of \( k \) such that the length of the parabola \( p(x) = kx^2 + d \) is equal to the length of the catenary, both passing through the points \((-2, \cosh(2))\) and \((2, \cosh(2))\). The equation for the equality of lengths is given by:
\[ 2\sinh(2) = \frac{1}{2} \cdot (4k\cdot \sqrt{1+(4k)^2} + \arsinh(4k)) \]
Solve this equation for \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.868}
There are $N$  permutations $（a_1，a_2，\dots，a_{30}）$ of $1,2，\dots，30$ such that for $m\in\{2,3,5\}$, $m$  divides $a_{n+m}-a_n$ for all integers $n$ with $1\leq n <n+m\leq 30$. Find the remainder when $N$  is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
440
}
For an $n \times n$ square matrix $A$, express the determinant of its cofactor matrix in terms of $\det(A)$ and $n$. What is the determinant of the cofactor matrix of $A$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\det(A))^{n-1}}
Determine all integer values of $n$ such that $m = \sqrt{n - 5} + \sqrt{n + \sqrt{n - 5}}$ is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
In a group of 15 people, there will be five murders. Each murder involves one killer and one victim, except for the third murder, which involves one killer and two victims. After each murder, the killer and victim(s) are removed from the group. In the third murder, one of the victims can also be the killer, but the other victim must be different. The order of victims in the third murder matters. Calculate the total number of possible combinations of killers and victims for all five murders.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{76281004800}
Given point \( A(4,0) \) and \( B(2,2) \), while \( M \) is a moving point on the ellipse \(\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1\), the maximum value of \( |MA| + |MB| \) is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10 + 2\sqrt{10}}
Given complex numbers $z_1, z_2, z_3$ such that $z_1 + z_2 + z_3 = 2$, $z_1^2 + z_2^2 + z_3^2 = 3$, and $z_1z_2z_3 = 4$, evaluate the expression:

$$\frac{1}{z_1z_2 + z_3 - 1} + \frac{1}{z_2z_3 + z_1 - 1} + \frac{1}{z_3z_1 + z_2 - 1}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{2}{9}}
a circular road runs round a circular ground . if the difference between the circumferences of the outer circle and inner circle is 66 metres , the width of the road is :
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10.5}
Triangles $ABC$ and $ADF$ have areas $4014$ and $14007,$ respectively, with $B=(0,0), C=(447,0), D=(1360,760),$ and $F=(1378,778).$ What is the sum of all possible $x$-coordinates of $A$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2400}
Calculate the limit: $$ \lim_{y\to\infty}\int_{0}^{\infty}  \frac{y\cos^2(x/y)}{y+x^4} \, dx $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Let \( W_n \) be the region in the \(xy\)-plane that is bounded by the \(y\)-axis and the parabola \(x=y^2-n^2\). Determine the value of \(n\) such that the volume obtained by rotating \(W_n\) about the \(y\)-axis is 16 times the volume obtained by rotating \(W_n\) about the \(x\)-axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{15}{2}}
Find the value of \(\sum_{k=1}^{60} \sum_{n=1}^{k} \frac{n^{2}}{61-2 n}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-18910}
Let $x, y, z$ be real numbers such that $xy + yz + zx = 3$, $x + y + z = 5$, and $xyz = 2$. Calculate the value of:

$$\frac{1}{xy + z - 2} + \frac{1}{yz + x - 2} + \frac{1}{zx + y - 2}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{13}}
In the ring $B = \mathbb{Z}_2[x]/(x^3 + x^2 + x + 1)$, find an element that, when multiplied by $[x^2 + 1]$, results in zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[x + 1]}
Fifteen balls, consisting of 3 balls each of 5 different colors, are to be arranged in a triangular formation. Determine the number of distinct arrangements possible if rotations of the triangle are considered equivalent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56056080}
Find the directional derivative of the function implicitly defined by \( z + x\ln(z) + xe^{xy} - 1 = 0 \) at the point \( P = (0, 1) \) in the direction of the vector \( \mathbf{v} = \langle 4 \sqrt{3}, 3 \sqrt{3} \rangle \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{4}{5}}
Find the codimension of the subspace \( A = \overline{\operatorname{span}}\{S^n(1,4,-1,0,0,\ldots):n=0,1,2,\ldots\} \) in \( \ell_2 \), where \( S \) is the right shift operator defined by \( Se_i = e_{i+1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the smallest integer \(\alpha\) such that the error of the truncated Taylor series of \(\sqrt{e}\) at \(x=0\) with \(\alpha\) terms is less than 0.001.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given the moment generating function of a random variable X:
\[ M_X(t) = \frac{1}{(1-2500t)^4} \]
calculate the standard deviation of the claim size for this class of accidents.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5000}
Find the limit of the sum \( \frac{1}{n} \sum_{k=1}^n n^{1/k} \) as \( n \) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Two boys and three girls stand in a row for a photo. If boy A does not stand at either end, and exactly two of the three girls are adjacent, determine the number of different arrangements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
A retailer purchases shirts from a wholesaler and then sells the shirts in her store at a retail price that is 80 percent greater than the wholesale price. If the retailer decreases the retail price by a certain percentage, this will have the same effect as increasing the wholesale price by 44.000000000000014 percent. What is the percentage decrease of the retail price?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44.000000000000014\%}
Given the sequence $\sqrt{2}$, $\sqrt{5}$, $2\sqrt{2},\ldots$, find the term number of $2\sqrt{5}$ in this sequence. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given the function \( f(x) = x + \frac{2}{1 \cdot 3}x^3 + \frac{2 \cdot 4}{1 \cdot 3 \cdot 5}x^5 + \frac{2 \cdot 4 \cdot 6}{1 \cdot 3 \cdot 5 \cdot 7}x^7 + \cdots \) for all \( x \in (0,1) \), and knowing that \( f\left(\frac{1}{2}\right) = \frac{\pi}{a\sqrt{b}} \) where \( a, b \in \mathbb{R} \), find the value of \( |a+b| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A bridge can tolerate a mass without structural damage that follows a normal distribution with mean 200,000 kg and standard deviation 20,000 kg. Car weights follow a normal distribution with mean 1,300 kg and standard deviation 130 kg. Determine the minimum number of cars needed on the bridge to cause structural damage with a probability greater than 0.1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{135}
Mixture A is a certain percentage oil and the rest is material B by weight. If 2 more kilograms of oil are added to the 8 kilograms mixture A, 6 kilograms of mixture A must be added to make a 70% material B in the new mixture. What is the percentage of oil in the original mixture A?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
Determine the smallest number of lines needed such that any ray starting from a point $P$ in the plane intersects at least $k$ of these lines, where $k$ is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2k}
Find the locus of a point $P$ such that the sum of the slopes of the normals from $P$ to the hyperbola $xy = c^2$ is a constant $k$ where $k > 0$. What is the equation of this locus?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 = c^2 k}
A wall 8 feet high is located 1 foot away from a house. Determine the length \( L \) of the shortest ladder that can be placed over the wall to reach the house. Assume the ladder forms a right triangle with the wall and the ground, where the height is \( y \), the base is \( 1 + x \), and the hypotenuse is \( L \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5\sqrt{5}}
Given that the average score for six students is 92.5, the highest score is 99, and the lowest score is 76, with all scores being unique integers, calculate the minimum possible score for the student who ranks 3rd from the highest.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
Calculate the number of different deal combinations possible in the standard version of the Klondike card game, considering the order of the cards in the deck and the red-on-black stacking rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52!}
A man sold 20 articles for a certain amount and gained 20%. He should sell 29.99999625000047 articles to incur a loss of 20%. What was the initial selling price?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
For a given prime number $p$, consider the sequence defined by:

\[ a_1 = p, \]

\[ a_2 = 2p + 1, \]

\[ a_3 = 2(2p + 1) + 1 = 4p + 3, \]

\[ \ldots \]

\[ a_n = 2^{n-1}p + 2^{n-1} - 1. \]

Determine the longest sequence of consecutive terms that are all prime numbers. What is the length of this sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Evaluate the limit: $$\lim_{n \to \infty} \int_{0}^{1} \sin^2\left(\dfrac{1}{ny^2}\right) \, dy$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Among the cross-sections of a cube, the one with the most sides is a ____-sided polygon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let \( \{X(t), t \geq 0\} \) be a Brownian motion with drift parameter \( \mu = 3 \) and variance parameter \( \sigma^2 = 9 \). If \( X(0) = 10 \), find \( P(X(0.5) > 10) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.7603}
Let $a,$ $b,$ $c$ be real numbers such that $9a^2 + 4b^2 + 25c^2 = 1.$ Find the maximum value of
\[8a + 3b + 5c.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{373}}{6}}
Determine the radius of convergence for the power series \(\sum_{n=1}^\infty \left(\frac{z-i}{z+2i}\right)^n\) using the Cauchy criterion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
1614. 600 corn seeds are planted with a probability of 0.9 for each seed to germinate. Find the boundary of the absolute value of the deviation of the frequency of germinated seeds from the probability $p=0.9$, if this boundary is to be guaranteed with a probability $P=0.995$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["0.034"]}
For the integer numbers $i, j, k$ satisfying the condition $i^2 + j^2 + k^2 = 2011$, find the largest value of $i + j + k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{77}
Determine the smallest number of natives, \( N \), such that after crossing a river using a single boat, each native knows at least \( k \) additional jokes apart from their own. Each crossing involves two natives rowing to the right and one returning alone, and jokes are only shared between the two natives in the boat during each crossing.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k + 1}
A vessel with a capacity of 8 liters is filled with a mixture of oxygen and nitrogen, with oxygen comprising $16 \%$ of the vessel's capacity. Some amount of this mixture is released from the vessel and the same amount of nitrogen is added. Then, the same quantity of this mixture is released again and the same quantity of nitrogen is added again. In the new mixture, oxygen makes up $9 \%$. What quantity of the mixture is released from the vessel each time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \text{ l}}
In triangle \( \triangle ABC \), \( \angle BAC = 80^\circ \), \( \angle ABC = 60^\circ \). Point \( D \) is an interior point such that \( \angle DAB = 10^\circ \) and \( \angle DBA = 20^\circ \). Find the measure of \( \angle ACD \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x = 30^\circ}
Determine the number of elements in the ring \( \mathbb{F}_2[x]/(x^5 + x^3 + 1) \) that are divisible by \( x^2 + 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Find the largest value of \( y \) in the interval \([0, 0.1]\) such that \( P(P > y \mid X = 0) \leq 0.01 \), where \( n = 528174 \), \( X \sim \text{Binomial}(n, P) \), and \( P \sim U[0, 0.1] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.00000872}
An export company plans to open four direct sales stores in five candidate cities, with no more than two stores in the same city. Calculate the total number of different plans the company can have for establishing its direct sales stores. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Evaluate the integral \( I = \int_0^{2\pi}\sum_{n=1}^{N}a_n \sin (nt)\cos(n\xi) \sum_{n=1}^{N} a_n\sin(n\xi)\cos(n\tau) \, d\xi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In Yang's number theory class, Michael K, Michael M, and Michael R take a series of tests. Afterwards, Yang makes the following observations about the test scores:

(a) Michael K had an average test score of  $90$ , Michael M had an average test score of  $91$ , and Michael R had an average test score of  $92$ .

(b) Michael K took more tests than Michael M, who in turn took more tests than Michael R.

(c) Michael M got a higher total test score than Michael R, who in turn got a higher total test score than Michael K. (The total test score is the sum of the test scores over all tests)

What is the least number of tests that Michael K, Michael M, and Michael R could have taken combined?

*Proposed by James Lin*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{413}
The graphs of the quadratic polynomials \( f(x) \) and \( g(x) \) intersect at the point \( (3 ; 8) \). The polynomial \( f(x) + g(x) \) has a single root at 5. Find the leading coefficient of the polynomial \( f(x) + g(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
For a positive integer \( N \), we color the positive divisors of \( N \) (including 1 and \( N \) ) with four colors. A coloring is called multichromatic if whenever \( a, b \) and \( \operatorname{gcd}(a, b) \) are pairwise distinct divisors of \( N \), then they have pairwise distinct colors. What is the maximum possible number of multichromatic colorings a positive integer can have if it is not the power of any prime?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{192}
Find the quadratic variation of the stochastic process $Y(t)$ over the interval $[-1, 2]$, where $Y(t)$ is defined as follows:

$$Y(t) = \begin{cases} 0 & -1 \le t \le 0 \\\ Z(t) - tZ(1) & 0 < t < 1 \\\ 1 - t & 1 \le t \le 2 \end{cases}$$

and $Z(t)$ is a standard Brownian motion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the number of paths from the origin to the point $(a, b)$ (where $a$ and $b$ are positive integers and $a \geq b$) that consist only of rightward and upward movements and do not cross the line $y = x - \frac{1}{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\binom{a+b}{a}-\binom{a+b}{b-1}\)}
Given the joint density function \( f(x, y) = 2 \) for \( 0 < y < x < 1 \), find the probability \( P(X - Y > z) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1 - z)^2}
A whale goes on a feeding frenzy that lasts for 12 hours. For the first hour, it catches and eats x kilos of plankton. In every hour after the first, it consumes an incrementally increasing amount of additional plankton based on the sequence '2n' (2, 4, 6, ...). By the end of the frenzy, the whale will have consumed a whopping accumulated total of 960 kilos of plankton. How many kilos did the whale consume in the eighth hour?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83}
$A(-1,1)$, $B$, and $C$ are points on the hyperbola $x^2 - y^2 = 1$. If $\triangle ABC$ is a regular triangle, find the area of $\triangle ABC$. The original answer is in the form of k\sqrt{m}, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Evaluate the limit: \[ \lim_{x \to 0} \int_0^1 \frac{1}{x} \left(e^{-x^2(1+t)}(1+t)^{-1}-(1+t)^{-1}\right) dt. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $ S(n) $ be the sum of the squares of the positive integers less than and coprime to $ n $. For example, $ S(5) = 1^2 + 2^2 + 3^2 + 4^2 $, but $ S(4) = 1^2 + 3^2 $. Let $ p = 2^7 - 1 = 127 $ and $ q = 2^5 - 1 = 31 $ be primes. The quantity $ S(pq) $ can be written in the form $$ \frac{p^2q^2}{6}\left(a - \frac{b}{c} \right) $$ where $ a $, $ b $, and $ c $ are positive integers, with $ b $ and $ c $ coprime and $ b < c $. Find $ a $.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7561}
Solve the following equation in positive numbers:
$$ (2a+1)(2a^2+2a+1)(2a^4+4a^3+6a^2+4a+1) = 828567056280801 $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Given two complete subgraphs $G_a$ and $G_b$ with $a \geq 3$ and $b \geq 3$ vertices each, where they share exactly two vertices, what is the number of Hamiltonian cycles in the graph $G$ formed by these subgraphs?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(a-2)!*(b-2)!}