inclusionAI/Ring-lite-rl-data huggingface dataset:
A pyramid has a rectangular base, its side edges are equal to each other and to one type of base edge. By varying the other type of base edge, when will the volume of the pyramid be the largest?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \sqrt{\frac{3}{2}}}
A vertex-induced subgraph is a subset of the vertices of a graph together with any edges whose endpoints are both in this subset. An undirected graph contains 10 nodes and $m$ edges, with no loops or multiple edges. What is the minimum possible value of $m$ such that this graph must contain a nonempty vertex-induced subgraph where all vertices have degree at least 5?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Let $a,b$ be positive integers satisfying
$$\sqrt{\frac{ab}{2b^2-a}}=\frac{a+2b}{4b}$$.
Find $|10(a-5)(b-15)|+8$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2018}
Determine for how many positive integers $n \in \{1, 2, \ldots, 2022\}$ it holds that $402$ divides at least one of
\[ n^2 - 1, \quad n^3 - 1, \quad n^4 - 1 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Given the ellipse $$C: \frac {x^{2}}{4}+ \frac {y^{2}}{b^{2}}=1(0＜b＜2)$$, a straight line with a slope angle of $$\frac {3π}{4}$$ intersects the ellipse C at points A and B. The midpoint of the line segment AB is M, and O is the coordinate origin. The angle between $$\overrightarrow {OM}$$ and $$\overrightarrow {MA}$$ is θ, and |tanθ|=3. Find the value of b.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Let $A$ be the the area enclosed by the relation
\[x^2+y^2\le2023.\]Let $B$ be the area enclosed by the relation
\[x^{2n}+y^{2n}\le\left(A\cdot\frac{7}{16\pi}\right)^{n/2}.\]Compute the limit of $B$ as $n\rightarrow\infty$ for $n\in\mathbb{N}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$119$}
Evaluate the limit: $$\lim_{x\to0^+}\frac{e^x+e^{-x}}{-\cos x}\cdot\frac{x^2+2R_1x+R_1^2}{2x+\bar R_1(x)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine if the limit \( \lim_{x \rightarrow a} \frac{f(x) - f(2a -x)}{2(x-a)} \) always exists, given that a function \( f(x) \) is continuous in a neighborhood of \( a \) and \( f'(a) \) exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f'(a)}
Example 6 Let $a=\sqrt{17}, b=\mathrm{i} \sqrt{19}$. Try to find the maximum value of $\frac{|a-z|}{|b-z|}$, where $z$ is any complex number with a modulus of 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{3}}
Simplify first, then evaluate $2\left(x^{2}-2x^{2}y\right)-\left[3\left(x^{2}-xy^{2}\right)-\left(x^{2}y-2xy^{2}+x^{2}\right)\]$, where $x=-\dfrac{1}{3}$ and $y=-2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{2}{3}}
Given the equation $2(x^3+y^3+z^3) = 3(x+y+z)^2$, where $x, y, z$ are non-negative real numbers, find the minimum value of $x+y+z$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
Find the constant $c$ such that the random variable \( Y = \frac{c(X_1 + X_2)}{\sqrt{(X_3)^2 + (X_4)^2}} \) follows a t-distribution, given that $X_1$, $X_2$, $X_3$, and $X_4$ are independent standard normally distributed random variables.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Along the school corridor hangs a Christmas garland consisting of red and blue bulbs. Next to each red bulb, there must be a blue bulb. What is the maximum number of red bulbs that can be in this garland if there are a total of 50 bulbs?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Given a regular 2007-gon. Find the minimal number $k$ such that: Among every $k$ vertexes of the polygon, there always exists 4 vertexes forming a convex quadrilateral such that 3 sides of the quadrilateral are also sides of the polygon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1506}
Determine the number of complex numbers \( z \) with \( |z| < 1 \) that satisfy the equation \( az^4 + z^3 + bz^2 + cz + d = 0 \), given that \( a, b, c, \) and \( d \) are positive numbers such that \( a + b + c + d < 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
设集合 $S$ 是 $\{0,1, \cdots, 98\}$ 的 $m(m \geqslant 3)$元子集，满足对任意的 $x, y \in S$ ，均存在 $z \in$ $S$, 使得 $x+y \equiv 2 z(\bmod 99)$. 求 $m$ 的所有可能值的和.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{155}
Mr. Johnson is organizing the school Christmas play and needs 100 volunteers to help with decorating the auditorium. 4 students from each of the school's 5 math classes have volunteered to help. 10 teachers have also volunteered, including 3 who are skilled in carpentry. Additionally, 15 parents have agreed to volunteer, but only 6 of them have experience with lighting and sound setup. To ensure a smooth process, Mr. Johnson needs 8 more volunteers with carpentry skills and 10 more volunteers with lighting and sound experience. How many more volunteers will Mr. Johnson need in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Quadrilateral \(ABCD\) is such that \(\angle BAC = \angle CAD = 60^\circ\) and \(AB + AD = AC\). It is also known that \(\angle ACD = 23^\circ\). How many degrees is \(\angle ABC\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83}
Determine the values of the parameter \( t \) for which the linear transformation \( \psi: \mathbb{R}^4 \rightarrow \mathbb{R}^3 \) given by
\[ \psi([x_1,x_2,x_3,x_4])=[x_1+x_2+x_3+2x_4,x_1+tx_2+x_3+3x_4,2x_1+x_2+tx_3+3x_4] \]
is surjective.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t \neq 2}
A triangle in the $x y$-plane is such that when projected onto the $x$-axis, $y$-axis, and the line $y=x$, the results are line segments whose endpoints are $(1,0)$ and $(5,0),(0,8)$ and $(0,13)$, and $(5,5)$ and $(7.5,7.5)$, respectively. What is the triangle's area?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{17}{2}}
Given that 
$\frac 1{2!17!}+\frac 1{3!16!}+\frac 1{4!15!}+\frac 1{5!14!}+\frac 1{6!13!}+\frac 1{7!12!}+\frac 1{8!11!}+\frac 1{9!10!}=\frac N{1!18!}$
 find the greatest integer that is less than $\frac N{100}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{137}
Given a figure  $F: x^2+\frac{y^2}{3}=1$  on the coordinate plane. Denote by  $S_n$  the area of the common part of the  $n+1' s$  figures formed by rotating  $F$  of  $\frac{k}{2n}\pi\ (k=0,\ 1,\ 2,\ \cdots,\ n)$  radians counterclockwise about the origin. Find  $\lim_{n\to\infty} S_n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
A，B，C，D四点在同一圆周上，且BC=CD=4,E为AC与BD的交点，且AE=6，线段BE和DE的长都是整数，则BD的长等于多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given a "number array" as shown on the left, where each column forms an arithmetic sequence, and starting from the third row, each row forms a geometric sequence with a common ratio equal to that of every other row. The first column is $\frac{1}{4}$, $\frac{1}{2}$, $\frac{3}{4}$, ..., the second column is $\frac{1}{4}$, $\frac{1}{4}$, $\frac{3}{8}$, ..., and the third column is $\frac{1}{4}$, $\frac{1}{4}$, $\frac{3}{16}$, ... . Denote the element in the $i$-th row and $j$-th column as ${a_{ij}}\left( {i \geqslant j,i,j \in {N^ * }} \right)$, find the value of ${a_{83}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
The country of HMMTLand has 8 cities. Its government decides to construct several two-way roads between pairs of distinct cities. After they finish construction, it turns out that each city can reach exactly 3 other cities via a single road, and from any pair of distinct cities, either exactly 0 or 2 other cities can be reached from both cities by a single road. Compute the number of ways HMMTLand could have constructed the roads.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{875}
Kayla rolls four fair $6$-sided dice. What is the probability that at least one of the numbers Kayla rolls is greater than $4$ and at least two of the numbers she rolls are greater than $2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \frac{61}{81} \]}
求$\cos \frac { \pi } { 1 5 } \cdot \cos \frac { 2 \pi } { 1 5 } \cdot \cos \frac { 3 \pi } { 1 5 } \cdot \cdots \cdot \cos \frac { 1 4 \pi } { 1 5 }$的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$- \frac { 1 } { 1 6 3 8 4 } .$}
A 360 meter long train starts from rest and accelerates until it crosses a man standing on a platform in 20 seconds. The track has an inclination of 5 degrees to the horizontal and the train has an acceleration of 1 m/s^2. Neglecting air resistance, what is the final speed of the train when it completely passes by the man?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Problem 6. (30 points) Ten balls of the same radius are arranged in the form of a triangular pyramid such that each ball touches at least three others. Find the radius of the sphere in which the pyramid of balls is inscribed, if the radius of the sphere inscribed in the center of the pyramid of balls, touching six identical balls, is $\sqrt{2}-1$.

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\sqrt{6}+1"]}
Consider the sequence \( a_{1}, a_{2}, \ldots, a_{105} \) which is a permutation of the numbers \( 1, 2, \ldots, 105 \). This sequence satisfies the condition that for any \( m \in \{3, 5, 7\} \), and for all \( n \) such that \( 1 \leq n < n+m \leq 105 \), the difference \( a_{n+m} - a_{n} \) is divisible by \( m \). Determine the number of such distinct permutations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3628800}
Evaluate the limit of the sequence as \( n \to \infty \):

\[ \lim_{n\to \infty} \frac{((n^3+2n^2)^{1/3}-n)\cdot((n^3+2n^2)^{1/3}+n)}{(2n^2+n^3)^{1/3}+n} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
1. [15] Four standard six-sided dice are rolled. Find the probability that, for each pair of dice, the product of the two numbers rolled on those dice is a multiple of 4 .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{31}{432}}
While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$, there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$. Given that $|a-b| = 2\sqrt{3}$, $|a| = 3\sqrt{3}$, compute $|b|^2+|c|^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{132}
I2.4 Determine the unit digit, $\delta$, of $\left(\left(\gamma^{\gamma}\right)^{\gamma}\right)^{\gamma}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
There are $13$ cities in a certain kingdom. Between some pairs of the cities, two-way direct bus, train, or plane connections are established. What is the least possible number of connections to be established so that, by choosing any two means of transportation, one can travel from any city to any other without using the third kind of vehicle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Given a positive integer $n$ greater than 2004, fill the numbers $1, 2, \cdots, n^2$ into an $n \times n$ chessboard (consisting of $n$ rows and $n$ columns of squares) such that each square contains exactly one number. If a number in a square is greater than the numbers in at least 2004 other squares in its row and at least 2004 other squares in its column, then this square is called a "super square." Determine the maximum number of "super squares" on the chessboard.
(Feng Yuefeng, problem contributor)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n(n-2004)}
On Saturdays and Sundays, Suzanne makes french toast for the family. She and her husband each have some whole slices and her daughters split 1 slice. She uses a thick sliced bread that comes 12 slices per loaf. Over 52 weeks, she needs 26 loaves of bread to make french toast. How many whole slices do Suzanne and her husband each have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Circles $C_{1}, C_{2}, C_{3}$ have radius 1 and centers $O, P, Q$ respectively. $C_{1}$ and $C_{2}$ intersect at $A, C_{2}$ and $C_{3}$ intersect at $B, C_{3}$ and $C_{1}$ intersect at $C$, in such a way that $\angle A P B=60^{\circ}, \angle B Q C=36^{\circ}$, and $\angle C O A=72^{\circ}$. Find angle $A B C$ (degrees).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Find 10 distinct natural numbers with the property that their sum is divisible by each of them.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{1, 2, 3, 6, 12, 24, 48, 96, 192, 384\}}
Let $A=\{1,2,3,\cdots ,17\}$. A mapping $f:A\rightarrow A$ is defined as follows: $f^{[1]}(x)=f(x)$, $f^{[k+1]}(x)=f(f^{[k]}(x))$ for $k\in\mathbb{N}$. Suppose that $f$ is bijective and that there exists a natural number $M$ such that:

1. When $m<M$ and $1\le i\le 16$, we have $f^{[m]}(i+1)- f^{[m]}(i) \not=\pm 1\pmod{17}$ and  $f^{[m]}(1)- f^{[m]}(17) \not=\pm 1\pmod{17}$;
2. When $1\le i\le 16$, we have $f^{[M]}(i+1)- f^{[M]}(i)=\pm 1 \pmod{17}$ and $f^{[M]}(1)- f^{[M]}(17)=\pm 1\pmod{17}$.

Find the maximal value of $M$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given the curve $y = x^n + 1$ ($n \in \mathbb{N}^*$) at the point $(1,1)$, let $x_n$ be the x-coordinate of the intersection point between the tangent line and the x-axis. Determine the value of $\log_{2010} x_1 + \log_{2010} x_2 + \dots + \log_{2010} x_{2009}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Tom invites his parents, 3 siblings, 2 grandparents, and 4 cousins to his house. They each eat 4 times a day. For breakfast and lunch, there are 2 courses per meal, whereas for dinner and a late-night snack, there are 3 courses per meal. Additionally, everyone uses 2 plates per course. His parents and siblings stay for 6 days, while his grandparents stay for 4 days, and his cousins stay for 3 days. How many plates do Tom and his guests use during their respective stays?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
## Task 3 - 200923

Given two circles $k_{1}$ and $k_{2}$ with radii $r_{1}$ and $r_{2}$ respectively, where $r_{1}>r_{2}$.

Furthermore, it is assumed that both circles touch each other externally, meaning they have exactly one common internal tangent. This internal tangent intersects one of the common external tangents of both circles at $P$ and the other common tangent at $Q$.

Determine, under these conditions, the length $P Q$ from $r_{1}$ and $r_{2}$!
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{r_{1}r_{2}}}
7. On the sides $\mathrm{AB}$ and $\mathrm{AC}$ of the right triangle $\mathrm{ABC}\left(\angle B C A=90^{\circ}\right)$, right triangles АВТ and АСК are constructed externally such that $\angle A T B=\angle A K C=90^{\circ}$, $\angle A B T=\angle A C K=60^{\circ}$. On the side $\mathrm{BC}$, a point $\mathrm{M}$ is chosen such that $\mathrm{BM}=\mathrm{MC}$. Determine the degree measure of angle КМТ.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Determine the number of six-letter words where the first and last two letters are the same (e.g., "aabbaa").
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{456976}
2014 triangles have non-overlapping interiors contained in a circle of radius 1. What is the largest possible value of the sum of their areas?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
What is the minimum number of cells in a $3 \times 2016$ board that can be colored such that each cell has at least one adjacent colored cell?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
21. Let $A B C D E F$ be a hexagon such that the diagonals $A D, B E$ and $C F$ intersect at the point $O$, and the area of the triangle formed by any three adjacent points is 2 (for example, area of $\triangle B C D$ is 2 ). Find the area of the hexagon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given that $A /_{\ n }^{ 2 }=7A /_{\ n-4 }^{ 2 }$, find the value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
You need to hit a target that is $A=40 \mathrm{~m}$ away and $B=30 \mathrm{~m}$ high with a projectile launched with the smallest possible initial speed. At what angle should the projectile be launched?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha \approx 63.4^\circ}
Find the limit of the following expression as p approaches 0:

$$\lim_{p\to0} \left[p \int_{-\infty}^\infty {{{\ln}^2(1+x)}\over{ (p^2 +x^2)}^2}}\,\mathrm dx\right]$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{2}\)}
Given a positive integer $k$, let $f(k)$ be the sum of the $k$-th powers of the primitive roots of $73$. For how many positive integers $k < 2015$ is $f(k)$ divisible by $73?$

[i]Note: A primitive root of $r$ of a prime $p$ is an integer $1 \le r < p$ such that the smallest positive integer $k$ such that $r^k \equiv 1 \pmod{p}$ is $k = p-1$.[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1847}
Evaluate the limit:
$$\lim_{n\to\infty}\sum_{k=0}^n \binom n k\frac{(-1)^{k}}{k+2} \left(\ln(n)-\frac{1}{k+2}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
When a 10° angle is viewed under a 5x magnifying glass, what is the resulting angle measure in degrees? Express your answer in degrees, including the degree symbol (°).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10°}
Given a triangle \(A B C\) with \(A B = A C\) and \(\angle A = 110^{\circ}\). Inside the triangle, a point \(M\) is chosen such that \(\angle M B C = 30^{\circ}\) and \(\angle M C B = 25^{\circ}\). Find \(\angle A M C\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85^\circ}
A gambler bought $10,000 worth of chips at a casino in denominations of $50 and $200. That evening, the gambler lost 30 chips, and then cashed in the remainder. If the number of $50 chips lost was 3 times the number of $200 chips lost, what is the largest amount of money that the gambler could have received back?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$7550}
In an experiment involving two urns, Urn 1 contains 3 blue balls and 1 red ball, while Urn 2 contains 1 blue ball and 3 red balls. It is equally likely to choose either urn. Players draw a ball at random, replace it, and guess if the urn is majority blue or red based on the ball's color. Each player sees the previous player's guess but not the ball. A cascade occurs if the first two players draw the same color, influencing the third player's guess regardless of their draw. What is the probability that a wrong cascade occurs after the first two players guess?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{16}}
Let $ABC$ be a triangle with $BC = 49$ and circumradius $25$. Suppose that the circle centered on $BC$ that is tangent to $AB$ and $AC$ is also tangent to the circumcircle of $ABC$. Then \[\frac{AB \cdot AC}{-BC + AB + AC} = \frac{m}{n}\] where $m$ and $n$ are relatively prime positive integers. Compute $100m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250049}
How many pairs \((m, n)\) of positive integers with \(m<n\) fulfill the equation

\[ \frac{3}{2008}=\frac{1}{m}+\frac{1}{n}? \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let \( A \) be a \( {2n} \times  {2n} \) matrix, with entries chosen independently at random.
 Every entry is chosen to be 0 or 1,each with probability \( 1/2 \) . Find the
 expected value of \( \det \left( {A - {A}^{t}}\right) \) (as a function of \( n \) ),where \( {A}^{t} \) is the
 transpose of \( A \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(2n-1)!!}{2^n}}
Consider the fiber product \(X \times_Z X\) with two projection arrows \(X \times_Z X \rightrightarrows X\). If we extend this to the fiber product \(X \times_Z X \times_Z X\), how many arrows are associated with this fiber product? Provide the total count of arrows.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the order of the element $abc$ in the group defined by the presentation $$ G = \langle a,b,c \mid a^2=b^2=c^2=1,\, ac=ca,\, (ab)^3=(bc)^3=1 \rangle.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the number of conjugacy classes in the dihedral group $D_8$. Use the formula for the number of conjugacy classes given by \( \frac{1}{|G|}\sum|C_{G}(g)| \), where \( C_{G}(g) = \{h \in G ; gh=hg\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate the limit: $$ \lim_{x\to\infty} \left( (x+2017)^{1+\frac{1}{x}} \: -\: x^{1+\frac{1}{x+2017}} \right) $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017}
In an exchange economy with two goods, x and m, and n consumers, half of the consumers have a utility function given by \( U_1(x_1) = 5\ln{x_1} + m_1 \), while the other half have \( U_2(x_2) = 3\ln{x_2} + m_2 \). Initially, each consumer possesses 20 units of good x and 10 units of good m. Determine the maximum amount of good x that the first type of consumers (with utility function \( U_1 \)) can acquire at a Pareto Optimal allocation, given that \( m_1 > 0 \) and \( m_2 > 0 \) for all consumers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
What is the average length of a cycle in a permutation of the set \( \{1, 2, 3, \dots, n\} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{n}{H_n}}
Calculate the expected number of letters in a random string generated from the set \{A, B, C, D\} until the sequence "ABCD" appears consecutively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
Evaluate the double integral \( \iint_R xy \, dx \, dy \) over the parallelogram defined by the lines \(2x-y=1\), \(2x-y=3\), \(x+y=-2\), and \(x+y=0\). Use the variable substitution \(u=2x-y\) and \(v=x+y\) to transform the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{44}{81}}
Determine the number of decreasing sequences of positive integers \(b_1 \geq b_2 \geq b_3 \geq \cdots \geq b_7 \leq 1500\) such that \(b_i - i\) is divisible by 3 for \(1 \leq i \le 7\). Express the number of such sequences as \({m \choose n}\) for some integers \(m\) and \(n\), and compute the remainder when \(m\) is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{506}
Simplify and write the result as a common fraction: $$\sqrt[4]{\sqrt[3]{\sqrt{\frac{1}{65536}}}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{\sqrt[3]{4}}}