inclusionAI/Ring-lite-rl-data huggingface dataset:
In a darts game, Don throws 5 darts in total. The first dart hits a bullseye and is worth triple points due to a special bonus in play, totaling 150 points. The second dart completely misses the target so receives no points. His third dart is worth half the points of the bullseye, without any special bonuses. The fourth dart hits a double ring, which is twice the points of a bullseye without the bonus. The fifth dart, not only hits another bullseye but also triggers a rule called 'Golden Bullseye', which adds an additional point for every earned point so far. What is the final total score from these 5 throws?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{600}
39 students participated in a math competition. The exam consisted of 6 problems, and each problem was worth 1 point for a correct solution and 0 points for an incorrect solution. For any 3 students, there is at most 1 problem that was not solved by any of the three. Let $B$ be the sum of all of the scores of the 39 students. Find the smallest possible value of $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{165}
Determine the values of $a$ for which the inequality \( \int_0^\infty e^{2a W_s^2} \, ds < \infty \) holds almost surely, where \( W_s \) is a Wiener process.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a < 0}
Two individuals, A and B, start traveling towards each other from points A and B, respectively, at the same time. They meet at point C, after which A continues to point B and B rests for 14 minutes before continuing to point A. Both A and B, upon reaching points B and A, immediately return and meet again at point C. Given that A walks 60 meters per minute and B walks 80 meters per minute, how far apart are points A and B?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1680}
Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ g(g(x) + y) = g(x) + g(g(y) + g(-x)) - x \] for all real numbers \( x \) and \( y \).

Let \( m \) be the number of possible values of \( g(4) \), and let \( t \) be the sum of all possible values of \( g(4) \). Find \( m \times t \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4}
Given the function $y=\sin x$ and $y=\sin (2x+ \frac {\pi}{3})$, determine the horizontal shift required to transform the graph of the function $y=\sin x$ into the graph of the function $y=\sin (2x+ \frac {\pi}{3})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {\pi}{6}}
If $n$ is an odd number, find the remainder of $7^n+C_n^17^{n-1}+C_n^27^{n-2}+\ldots+C_n^{n-1}7$ when divided by 9. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Suppose \(X\) and \(Y\) have a joint density \(f_{X, Y}(x, y) = c\) for all \(x, y\) satisfying:
\(0<y<x<1\) or \(0<y<1-x\) and \(0<x<1\). Find the value of \(c\) such that the total probability is 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{3}}
Circles with centers $(2,4)$ and $(14,9)$ have radii $4$ and $9$, respectively. The equation of a common external tangent to the circles can be written in the form $y=mx+b$ with $m>0$. What is $b$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{912}{119}}
For an ordered  $10$ -tuple of nonnegative integers  $a_1,a_2,\ldots, a_{10}$ , we denote
    \[f(a_1,a_2,\ldots,a_{10})=\left(\prod_{i=1}^{10} {\binom{20-(a_1+a_2+\cdots+a_{i-1})}{a_i}}\right) \cdot \left(\sum_{i=1}^{10} {\binom{18+i}{19}}a_i\right).\] When  $i=1$ , we take  $a_1+a_2+\cdots+a_{i-1}$  to be  $0$ . Let  $N$  be the average of  $f(a_1,a_2,\ldots,a_{10})$  over all  $10$ -tuples of nonnegative integers  $a_1,a_2,\ldots, a_{10}$  satisfying
    \[a_1+a_2+\cdots+a_{10}=20.\]
    Compute the number of positive integer divisors of  $N$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{462}
Let $a_1$, $a_2$, $\cdots$ be a sequence such that $a_1 = a_2 = \frac{1}{5}$, and for $n \ge 3$,  
\[a_n = \frac{a_{n-1} + a_{n-2}}{1 + a_{n-1}a_{n-2}}.\]  
Find the smallest integer $n$ such that $a_n > 1 - 5^{-2022}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Determine the optimal constant \( C_{nr} \) such that for all matrices \( A, B \) in \( \mathbf{M}_n(\mathbb{C}) \), the inequality \( r([A,B]) \le C_{nr} r(A)r(B) \) holds, where \( r(A) \) is the numerical radius of \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Start with an angle of \(60^\circ\) and bisect it, then bisect the lower \(30^\circ\) angle, then the upper \(15^\circ\) angle, and so on, always alternating between the upper and lower of the previous two angles constructed. This process approaches a limiting line that divides the original \(60^\circ\) angle into two angles. Find the measure (degrees) of the smaller angle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given a hyperbola with one focus at $(0,-5)$ and the nearest directrix $d_1: x-3y-4=0$, and an eccentricity of $\sqrt{10}$, find the equation of the second directrix of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9x - 27y - 14 = 0}
Determine the number of integer solutions to the equation \(x^2 + 4xy = 10^{100}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19594}
Determine the minimum number of vertices for a polyhedron defined by the constraints given by the identity matrix and a vector with all entries equal to 0.5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Five months ago, Mike earned 10 times more money than Fred. At that time, Fred's salary was $1000, and he received a $500 bonus. Fred invested the entire amount (salary + bonus) in a fund that gave a 20% return over five months. Mike also got a 10% bonus on his salary and invested the entire amount (salary + bonus) in a fund that gave a 25% return over the same period. Now, Mike's salary has increased by 40%, and he is required to pay a 15% tax on his current salary. Calculate Mike's net salary (after tax) now.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$11,900}
Determine the sum function of the series $$\sum g_n$$ where $g_n(x)=\frac{(n+1)x}{(1+(n+1)^2x^2)}-\frac{nx}{[1+n^2x^2]}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-x/(1+x^2)}
Suppose that $a$ and $b$ are positive integers such that $(a-bi)^2 = 8-6i$. What is $a-bi$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 - i}
Calculate the limit using the Central Limit Theorem:
\[ \lim_{n \to \infty} \frac{8^n}{27^n}\sum_{k=0}^n\binom{3n}{k}\frac{1}{2^k} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
6. What is the group key $K$ that can be shared by four individuals with keys $k_{1}=11, k_{2}=12, k_{3}=17, k_{4}=19$ using the modulus $p=1009$ and base $a=3$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
Consider the functions $f(x) = [x]^2 - [x+6]$ and $g(x) = 3kx^2 + 2x + 4(1-3k)$, where $[x]$ denotes the floor function. Let $A = \{x \mid f(x) = 0\}$. Determine the interval $[a, b]$ for $k$ such that every element of set $A$ satisfies the inequality $g(x) \ge 0$. Calculate the value of $6b - 3a$. \( \text{Find } 6b - 3a. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
We say that a positive integer $m$ covers the number $1998$ if the digits $1, 9, 9, 8$ appear in this order as digits of $m$. For instance, $1998$ is covered by $2\textbf{1}59\textbf{9}36\textbf{98}$ but not by $213326798$.

Let $k(n)$ be the number of positive integers that cover $1998$ and have exactly $n$ digits ($n \ge 5$), with all digits different from $0$. What is the remainder of $k(n)$ when divided by $8$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Suppose you play the following game: You roll a fair die at most 60 times and get an amount of dollars equal to the last number rolled. You can decide to stop the game at any time. What is the (approximate) value of this game for N=60?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
What is the maximum number of checkers that can be placed on a $6 \times 6$ board such that no three checkers (specifically, the centers of the cells they occupy) lie on the same straight line (at any angle)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
$[a]$ denotes the greatest integer less than or equal to $a$. Given that $\left(\left[\frac{1}{7}\right]+1\right) \times\left(\left[\frac{2}{7}\right]+1\right) \times\left(\left[\frac{3}{7}\right]+1\right) \times \cdots \times$ $\left(\left[\frac{\mathrm{k}}{7}\right]+1\right)$ leaves a remainder of 7 when divided by 13, find the largest positive integer $k$ not exceeding 48.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
The negation of the proposition "$\forall x \geqslant 2, x^2 \geqslant 4$" is _______ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\exists x_0 \geqslant 2, x_0^2 < 4}
All the milk in container A, which was filled to its brim, was poured into three containers: B, C, and D. The quantity of milk in container B was 45% less than the capacity of container A, the quantity in container C was 12.5% more than the capacity of container A, and the quantity in container D was 20% less than the capacity of container A. If 150 liters was transferred from C to B and 100 liters from D to C, then all three containers, B, C, and D would have equal quantities of milk. What was the initial quantity of milk in container A?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
Four points \( B, A, E, L \) are on a straight line. \( G \) is a point off the line such that \(\angle B A G = 120^\circ\) and \(\angle G E L = 80^\circ\). Calculate the reflex angle at \( G \) in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{340}
Let $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2005}$, where $a$ and $b$ are different four-digit positive integers (natural numbers) and $c$ is a five-digit positive integer (natural number). What is the number $c$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12030}
已知函数 $f(x)=x \ln x-a x^{2}, a \in \mathbf{R}$，设函数 $F(x)=|f(x)|(x \in[1, \mathrm{e}])$ 有极小值,  $a$ 的取值范围是$\left(a, \frac{b}{\mathrm{e}}\right) \cup\left(\frac{b}{\mathrm{e}}, \frac{b}{c}\right)$，求$a+b+c$的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
14. $n^{2}(n \geqslant 4)$ positive numbers are arranged in $n$ rows and $n$ columns,
\begin{tabular}{llllll}
$a_{11}$ & $a_{12}$ & $a_{13}$ & $a_{14}$ & $\cdots \cdots$ & $a_{1 n}$ \\
$a_{21}$ & $a_{22}$ & $a_{23}$ & $a_{24}$ & $\cdots \cdots$ & $a_{2 n}$ \\
$a_{31}$ & $a_{32}$ & $a_{33}$ & $a_{34}$ & $\cdots \cdots$ & $a_{3 n}$ \\
$a_{41}$ & $a_{42}$ & $a_{43}$ & $a_{44}$ & $\cdots \cdots$ & $a_{4 n}$ \\
$\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots \cdots$ & $\cdots$ \\
$a_{n 1}$ & $a_{n 2}$ & $a_{n 3}$ & $a_{n 4}$ & $\cdots \cdots$ & $a_{n n}$
\end{tabular}

where the numbers in each row form an arithmetic sequence, and the numbers in each column form a geometric sequence, with all common ratios being equal. Given $a_{24}=1, a_{42}=\frac{1}{8}, a_{43}=\frac{3}{16}$, find $a_{11}+a_{22}+\cdots+a_{n n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2-\frac{n+2}{2^{n}}}
Let  $ ABCD$  be an isosceles trapezoid with  $ \overline{AD}\parallel{}\overline{BC}$  whose angle at the longer base  $ \overline{AD}$  is  $ \dfrac{\pi}{3}$ . The diagonals have length  $ 10\sqrt {21}$ , and point  $ E$  is at distances  $ 10\sqrt {7}$  and  $ 30\sqrt {7}$  from vertices  $ A$  and  $ D$ , respectively. Let  $ F$  be the foot of the altitude from  $ C$  to  $ \overline{AD}$ . The distance  $ EF$  can be expressed in the form  $ m\sqrt {n}$ , where  $ m$  and  $ n$  are positive integers and  $ n$  is not divisible by the square of any prime. Find  $ m \plus{} n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Find the number of ordered pairs of real numbers $(a,b)$ such that $(a+bi)^{2002} = a-bi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2004$}
It is found through experiments under the same conditions that the germination rate of mung bean seeds at the Taihu Agricultural College is $97.1\%$. Please estimate the number of seeds that cannot germinate in about $1000$ pounds of mung bean seeds in the Taihu area.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
The children went to the forest to pick mushrooms. If Anya gives half of her mushrooms to Vitya, all the children will have the same number of mushrooms. But if Anya instead gives all her mushrooms to Sasha, Sasha will have as many mushrooms as all the others combined. How many children went to pick mushrooms?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
New York recorded 5000 new coronavirus cases on a particular week. In the second week, some new coronaviruses cases were recorded by the state. In the third week, 2000 more cases were recorded in the state. The total number of recorded new coronaviruses in the state after the three weeks is 9500. What is the ratio of the number of new cases recorded in the second week to the number of new cases recorded in the first week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:4}
The area of rectangle PRTV is divided into four rectangles, PQXW, QRSX, XSTU, and WXUV. Given that the area of PQXW is 9, the area of QRSX is 10, and the area of XSTU is 15, find the area of rectangle WXUV.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{27}{2}}
## Task B-1.4.

In each cell of a $4 \times 4$ table, a number is written. For each cell, the sum of the numbers in its adjacent cells is equal to the same natural number $x$ (two cells are adjacent if they share a side). Determine the number $x$ such that the sum of all numbers in the table is 282.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
For a positive integer  $n$ , define  $n?=1^n\cdot2^{n-1}\cdot3^{n-2}\cdots\left(n-1\right)^2\cdot n^1$ . Find the positive integer  $k$  for which  $7?9?=5?k?$ .

*Proposed by Tristan Shin*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Divide the natural numbers from 1 to 8 into two groups such that the difference between the sum of the numbers in each group is equal to 16. How many different ways are there to do this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
For a function $g(x)$ defined on the interval $I$, denote $g(I) = \{y | y = g(x), x \in I\}$. Given that the function $y = f(x)$, which has a domain of $[0, 3]$, has an inverse function $y = f^{-1}(x)$, and $f^{-1}([0, 1)) = [1, 2)$, $f^{-1}((2, 4]) = [0, 1)$. If the equation $f(x) - x = 0$ has a solution $x$, then $x =$ __.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a Cartesian coordinate system \( xOy \), there are 2018 circles \( \odot A_{1}, \odot A_{2}, \cdots, \odot A_{2018} \), where the center of \( \odot A_{k} \) (for \( k=1,2, \cdots, 2018 \)) is \( A_{k}\left(a_{k}, \frac{1}{4} a_{k}^{2}\right) \) and the radius is \( \frac{1}{4} a_{k}^{2} \). Given that \( a_{1}>a_{2}>\cdots>a_{2018}=\frac{1}{2018} \), and \( \odot A_{k+1} \) is externally tangent to \( \odot A_{k} \) (for \( k=1,2, \cdots, 2017 \)), find the value of \( a_{1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{2019}}
Let \( x, y, z \) be three distinct natural numbers arranged in ascending order, and the sum of their reciprocals is an integer. Find \( x, y, z \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 3, 6)}
A function \( g(x) \) is defined for all real numbers \( x \). For all non-zero values \( x \), we have
\[ 3g(x) + g\left(\frac{1}{x}\right) = 7x + 6. \]
Let \( T \) denote the sum of all of the values of \( x \) for which \( g(x) = 2005 \). Compute the integer nearest to \( T \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{763}
In the irregular pyramid \(ABCD\), the sum of the plane angles at the vertex \(A\) is \(180^\circ\). Find the surface area of the pyramid if the area of the face \(BCD\) is \(s\) and \(AB = CD\), \(AD = BC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4s}
There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < \dots < a_k$ such that $\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + \dots + 2^{a_k}.$ What is $k?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{137}
Let $ABCD$ be a trapezoid with $AB \parallel CD$ and $AD = BD$. Let $M$ be the midpoint of $AB$, and let $P \neq C$ be the second intersection of the circumcircle of $\triangle BCD$ and the diagonal $AC$. Suppose that $BC = 27$, $CD = 25$, and $AP = 10$. If $MP = \frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2705}
Carly is a pet groomer, and today her task was to trim the nails on dogs' paws. She trimmed 164 nails in total but was faced with some irregularities among the dogs. Three of the dogs had only three legs, two dogs had only three nails on one of their paws, and one dog had an extra nail on one paw, making it five nails on that particular paw. Given these conditions, how many dogs did Carly work on today?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
[ Angles between lines and planes ]  $\left[\begin{array}{l}\text { Equation of a plane } \\ {[\text { d }}\end{array}\right]$

Given points $A(1 ; 0 ; 1), B(-2 ; 2 ; 1), C(2 ; 0 ; 3)$ and $D(0 ; 4 ;-2)$. Find the angle between the line $A B$ and the plane $B C D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\arcsin\\frac{\\sqrt{13}}{\\sqrt{101}}"]}
I live in a five-story building with four entrances, with four apartments on each landing. What is the minimum number of questions you can ask to determine which apartment I live in, if I answer all your questions truthfully, but only with "yes" or "no"?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find all integer triples \((x, y, z)\) such that \(x^3 + y^3 + z^3 - 3xyz = 2003\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(667, 668, 668)}
5 实数$a _ { 1 } ,$ $a _ { 2 } , \cdots , a _ { n } ( n > 3 )$满足：$a _ { 1 } + a _ { 2 } + \cdots + a _ { n } \geq n ,$且$a _ { 1 } ^ { 2 } + a _ { 2 } ^ { 2 } + \cdots +$ $a _ { n } ^ { 2 } \geq n ^ { 2 }$.求$\max \{ a _ { 1 } ,$ $a _ { 2 } ,$,⋯,$a _ { n } \}$的最小值．（第28届美国数学奥林匹克试题）
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $ABCD$ be a convex quadrilateral such that $m (\widehat{ADB}) = 15^{\circ}$ and $m (\widehat{BCD}) = 90^{\circ}$. The diagonals of the quadrilateral are perpendicular at $E$. Let $P$ be a point on $|AE|$ such that $|EC| = 4$, $|EA| = 8$, and $|EP| = 2$. If the answer is $x^\circ$, what is the value of $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
Determine the value of the expression $\cos (-\frac{25}{3}\pi ) + \sin (\frac{25}{6}\pi ) - \tan (-\frac{25}{4}\pi )$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Compute $\operatorname{tg} 46^{\circ}$, using the value of the function $y=\operatorname{tg}x$ at $x=45^{\circ}$ and replacing its increment with a differential.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.0350}
If a complex number $z$ satisfies $|z - 2 + i| = 1$, find the minimum value of $|z + 1 - 2i|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{2} - 1}
Calculate the area of the graph of the function $f(x,y) = xy$ over the unit circle $D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{2\pi}{3} (2\sqrt{2} - 1)\)}
Given that \( x \) and \( y \) are two coprime positive integers, find the number of pairs \( (x, y) \) that satisfy the equation:

\[ x + \frac{2016}{x} = 32y + \frac{63}{y} \]

Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Consider the ring $k[x,y]/(y-x^2,y^2-x^3)$. Find a polynomial representative of $\frac{1}{1-y}$ in this ring.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + x^2}
If $g'(x) \geq -5$ for $-2 \leq x \leq 3$ and $g(3) = 7$, what is the minimum possible value of $g(-2)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Johnny is out walking his two dogs at night, and his son joins him for the walk. Along the way, they are joined by their wheelchair-using friend and their service dog. How many legs' worth of organisms and wheels are traveling together for this walk?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Given that \(x, y,\) and \(z\) are real numbers satisfying \(x + y + z = 10\) and \(x^2 + y^2 + z^2 = 50\), find the maximum possible value of \((x + 2y + 3z)^2 + (y + 2z + 3x)^2 + (z + 2x + 3y)^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1250}
In \(\triangle ABC\), \(AB = BC\), and \(\overline{BD}\) is an altitude. Point \(E\) is on the extension of \(\overline{AC}\) such that \(BE = 10\). The values of \(\tan \angle CBE\), \(\tan \angle DBE\), and \(\tan \angle ABE\) form a geometric progression, and the values of \(\cot \angle DBE,\) \(\cot \angle CBE,\) \(\cot \angle DBC\) form an arithmetic progression. The area of \(\triangle ABC\) is in the form \(\frac{m}{n}\). Please calculate the value of \(m+n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53}
Lynne is tiling her long and narrow rectangular front hall. The hall is 2 tiles wide and 13 tiles long. She is going to use exactly 11 black tiles and exactly 15 white tiles. Determine the number of distinct ways of tiling the hall so that no two black tiles are adjacent (that is, share an edge).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{486}
Which natural numbers cannot be expressed as the sum of multiple consecutive natural numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^k}
We inscribe spheres with a radius of \(\frac{1}{2}\) around the vertices of a cube with edge length 1. There are two spheres that touch each of these eight spheres. Calculate the difference in volume between these two spheres.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{10}{3} \pi}
In how many distinct ways can you color each of the vertices of a tetrahedron either red, blue, or green such that no face has all three vertices the same color? (Two colorings are considered the same if one coloring can be rotated in three dimensions to obtain the other.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}