inclusionAI/Ring-lite-rl-data huggingface dataset:
有四所学校的学生参加一项数学竞赛，每所学校派出 3 名选手.组委会要抽选其中若干名选手做一项调研，要求任意两所学校被抽中的选手数之和至少为 1 、至多为 3 ，则不同的抽选方式数为 $\qquad$ （结果用数值表示）.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{837}
Given a sequence of independent and identically distributed (iid) random variables $X_k$, for $k \geq 0$, with a probability mass function (pmf) defined as follows:
$$p_j = P(X_1 = j), \quad 1 \leq j \leq m, \quad \sum_{j=1}^{m}p_j = 1$$
Let $T$ denote the first time the initial value is observed again, i.e., $T = \min{n > 0 : X_n = X_0}$. Determine the expected value of $T$, $E[T]$, given that $X_0$ can take any value from $\{1, 2, \ldots, m\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m}
在三棱雉 $P-A B C$ 中, 若 $P A \perp$ 底面 $A B C$, 且棱 $A B, B P, B C, C P$ 的长分别为 $1,2,3,4$ ，则该三棱雉的体积为 $\qquad$.  原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
设整数 n\geq 4，A=\{a_1,a_2,\cdots,a_n\} 是整数集，满足 2023\in A，且 A 中任意两个元素的差的绝对值是素数. 则这样的集合 A 共有_________个.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
At a joint conference of the Parties of Liars and Truth-lovers, 32 people were elected to the presidium and seated in four rows of 8 people each. During the break, each member of the presidium claimed that among their neighbors there are representatives of both parties. It is known that liars always lie, and truth-lovers always tell the truth. What is the minimum number of liars in the presidium for the described situation to be possible? (Two members of the presidium are neighbors if one of them is seated to the left, right, in front, or behind the other).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
How many integer triples $(x,y,z)$ are there satisfying the equation $x^3+y^3=x^2yz+xy^2z+2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Sasha chose five numbers from 1, 2, 3, 4, 5, 6, and 7 and informed Anya of their product. Based on this information, Anya realized that she could not uniquely determine the parity of the sum of the numbers chosen by Sasha. What number did Sasha inform Anya of?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{420}
Given a circle $C: (x-1)^{2} + (y-2)^{2} = 25$ and a line $l: mx-y-3m+1=0$ intersect at points $A$ and $B$. Find the minimum value of $|AB|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{5}}
Initially, a natural number $n$ is written on the blackboard. Then, at each minute, Neymar chooses a divisor $d > 1$ of $n$, erases $n$, and writes $n + d$. If the initial number on the board is $2022$, what is the largest composite number that Neymar will never be able to write on the blackboard?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2033}
For a sequence defined by $a_1 = 1$ and $a_n = n(1 + a_{n-1})$ for $n \geq 2$, find the value of the following limit:

$$\lim_{n\to \infty} \frac{\prod_{k=1}^{n}\left(1+\frac{1}{a_k}\right)}{\sum_{k=0}^{n-1}\frac{1}{k!}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $f(x) = \frac{x^6 - 2}{4}$. Find $f^{-1}\left(-\frac{1}{8}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{3}{2}\right)^{\frac{1}{6}}}
Max has drawn a pair of concentric circles. He begins to draw chords $\overline{PQ}$, $\overline{QR}$, and so on, of the larger circle, each chord being tangent to the smaller circle. Given that $\angle PQR = 60^\circ$, how many sections will Max draw before returning to his starting point at $P$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find a non-zero element in the set \( S = \mathbb{C} \times \mathbb{C} \) that is not invertible, where the multiplication on \( S \) is defined as \((a,b) \cdot (c,d) = (ac - db, ad + bc)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, i)}
a bookseller has two display windows . she plans to display 3 new fiction books in the left window , and 3 new non - fiction books in the right window . assuming she can put the 4 fiction books in any order , and separately , the 3 non - fiction books in any order , how many total configurations will there be for the two display windows ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Let \( r=\sqrt{\frac{\sqrt{53}}{2}+\frac{3}{2}} \). There is a unique triple of positive integers \((a, b, c)\) such that
\[
r^{100}=2 r^{98}+14 r^{96}+11 r^{94}-r^{50}+a r^{46}+b r^{44}+c r^{40}
\]
Calculate the value of \( a^{2}+b^{2}+c^{2} \). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15339}
In a puzzle, 111 = 09, 444 = 12, and 777 = 15. What number equals 15 in this puzzle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{777}
8.5. In the tetrahedron $K L M N$, the lengths of the edges are known: $K L = M N = 9$, $K M = L N = 15$, $K N = L M = 16$. Points $P, Q, R, S$ are the centers of the circles inscribed in triangles $K L M$, $K L N$, $K M N$, and $L M N$. Find the volume of the tetrahedron $P Q R S$. Round the answer to the nearest hundredth if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.85}
What is the expected number of coin flips needed to obtain more heads than tails?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Let \(f:\mathbb{R} \to \mathbb{R}\) and \(g:\mathbb{R} \to \mathbb{R}\) be two non-constant differentiable functions such that \(f'(x) - g'(x) = g'(x)(f(x))^2 - f'(x)(g(x))^2\) for all \(x \in \mathbb{R}\). Given that \(f(1) = 1\), \(g(1) = \frac{1}{3}\), and there is no \(x\) for which \(f(x) = -2\) or \(g(x) = 2\), find the minimum positive value that \((f(x) - g(x) + 3)\) may attain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{5} - 1}
Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. Calculate the area, in square yards, of the region outside of the doghouse that Spot can reach. Express your answer in terms of pi.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\pi}
Given the functions $y= \frac {1}{x^{2}}$, $y=2x$, $y=x^{2}+x$, and $y=\sqrt[3]{x^{5}}$, how many of these functions are power functions, i.e., can be expressed in the form $y=x^{\alpha}$ where $\alpha$ is a real number? Provide your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A triangle is inscribed in a circle. A second circle, concentric with the first, touches one side of the triangle and divides each of the other two sides into three equal parts. Find the ratio of the radii of these circles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{9}}
Calculate the integral \( \int_\gamma \frac{dz}{z} \) where \( \gamma = a \cos(t) + i b \sin(t) \) for \( t \in [0, 2\pi] \), using the complex logarithm function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
For a positive integer $k$, let $f_1(k)$ be the square of the sum of the digits of $k$. Define $f_{n+1} = f_1 \circ f_n$. Evaluate $f_{2007}(2^{2006})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{169}
A set of six edges of a regular octahedron is called Hamiltonian cycle if the edges in some order constitute a single continuous loop that visits each vertex exactly once. How many ways are there to partition the twelve edges into two Hamiltonian cycles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Each frog needs to eat 30 flies per day to live. Each fish needs to eat 8 frogs per day to live. Each gharial needs to eat 15 fish a day to live. Additionally, each heron needs to eat 5 frogs and 3 fish per day to live, and each caiman needs to eat 2 gharials and 2 herons per day to live.

How many flies get eaten every day in this swamp if it has 9 gharials, 12 herons, and 7 caimans?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42840}
Evaluate the limit: $$\lim_{z \to \infty}\int_{\mathbb{C}}|w|e^{-|z-w|^2}dA(w)$$ where $dA = rdrd\theta$ is the area measure in polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Let $\vartriangle ABC$ be an equilateral triangle. Points $D,E, F$ are drawn on sides $AB$,$BC$, and $CA$ respectively such that $[ADF] = [BED] + [CEF]$ and $\vartriangle ADF \sim  \vartriangle BED \sim \vartriangle CEF$. The ratio $\frac{[ABC]}{[DEF]}$ can be expressed as $\frac{a+b\sqrt{c}}{d}$ , where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a + b + c + d$.
(Here $[P]$ denotes the area of polygon $P$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
The championship finals of the Chinese Volleyball Super League adopt a seven-game, four-win system, meaning that if one team wins four games first, that team will be the overall champion and the competition will end. Now, Team A and Team B are competing in the finals. Since the two teams have similar strengths, the probability of each team winning a game is $\frac{1}{2}$. According to past data, the ticket revenue for the first game is 5 million yuan, and the ticket revenue for each subsequent game increases by 100,000 yuan compared to the previous game. The probability that the total ticket revenue for the finals is exactly 45 million yuan is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{16}}
Find the Taylor series expansion of the operator \( D = e^{bx \frac{d}{dy}} \) when it operates on an infinitely differentiable function \( g(y) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sum_{n=0}^{\infty} \frac{(bx)^n}{n!} g^{(n)}(y)}
Two parallel lines are intersected by a third line. Find the angle between the bisectors of the interior angles on the same side.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Evaluate the surface integral \( \int_Y xyz \, dS \), where \( Y \) is the part of the sphere \( x^2 + y^2 + z^2 = 1 \) that lies above the region \( \{ y \leq x, x \leq 0, 0 \leq x^2 + y^2 \leq 1 \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{16}}
A side of beef undergoes three stages of processing. In the first stage, it loses 20 percent of its weight. In the second stage, it loses 15 percent of its remaining weight. In the third stage, it loses 25 percent of its remaining weight. If the side of beef weighed 350 pounds after the third stage of processing, how many pounds did it weigh before processing?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{686.27}
Find the largest integer \( k \) that divides the product \( abcdef \) for all integer solutions \((a, b, c, d, e, f)\) to the equation \( a^2 + b^2 + c^2 + d^2 + e^2 = f^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
For a two-by-two matrix $A$, determine the flux of the linear vector field ${\bf F}(x,y)=A\begin{bmatrix}x\\ y\end{bmatrix}$ through a unit circle centered at the origin in $\mathbb{R}^2$, expressed in terms of the entries of $A$. Assume the unit circle is parameterized by $x = \cos t$ and $y = \sin t$ for $0 \leq t < 2\pi$. Provide your answer in terms of the matrix entries.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi(a + d)}
Evaluate the limit: $$\lim_{x\to\infty} \left(\int_0^1 t^{-tx} \, dt\right)^{\frac{1}{x}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{\frac{1}{e}}}
Point $P$ is taken inside the square $ABCD$ such that $BP + DP = 25$, $CP - AP = 15$, and $\angle ABP = \angle ADP$. What is the radius of the circumcircle of $ABCD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In the diagram, \(C\) lies on \(AE\) and \(AB=BC=CD\). If \(\angle CDE=t^{\circ}, \angle DEC=(2t)^{\circ}\), and \(\angle BCA=\angle BCD=x^{\circ}\), determine the measure of \(\angle ABC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Folklore

On a chessboard, $n$ white and $n$ black rooks are placed such that rooks of different colors do not attack each other. Find the maximum possible value of $n$.

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Given the cubic equation
\[
x^3 + Ax^2 + Bx + C = 0 \quad (A, B, C \in \mathbb{R})
\]
with roots \(\alpha, \beta, \gamma\), find the minimum value of \(\frac{1 + |A| + |B| + |C|}{|\alpha| + |\beta| + |\gamma|}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt[3]{2}}{2}}
The base of a prism is an isosceles trapezoid inscribed around a circle with radius $r$, and its acute angle is $\alpha$. A plane passed through one side of the base and the acute angle endpoint of the opposite side of the top plane also forms an angle of $\alpha$ with the base plane. What is the height of the prism?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 r \tan \alpha}
Determine the numbers of all squares in which a radar beam, directed along the line \( y = x + 2013 \), will detect a mysterious object moving along the trajectory \( y = \left(\left(\left(x^{5}-2013\right)^{5}-2013\right)^{5}-2013\right)^{5} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
11 个黑球， 9 个红球，依次取出，剩下全是一种颜色就结束，求最后只剩下红球的概率？  原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Mr. and Mrs. A are opposite to each other. The distance between Mr. A and Mrs. A is 120 km. Mr. A's speed is 30 kmph, and Mrs. A has a certain speed. One bee is running between Mr. A's nose to Mrs. A's nose and returning back to Mr. A's nose. The speed of the bee is 60 kmph. The bee traveled 180 km. What is the speed of Mrs. A?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Evaluate the integral: $\int \frac{1}{t\left(t^{2}-4\right)} d t $
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{8} \ln \left|1-4 / t^{2}\right|+C$}
Given the transition matrix of a Markov chain:
\[
\begin{bmatrix} 
\frac{1}{5} & \frac{4}{5} & 0 & 0 & 0 \\
\frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 \\
\frac{1}{5} & \frac{1}{5} & \frac{1}{5} & \frac{1}{5} & \frac{1}{5} \\
0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 \\
0 & 0 & 0 & 0 & 1 
\end{bmatrix}
\]
find the expected long-run proportion of time the chain spends at state $a$, given that it starts at state $b$. Use the stationary distribution to determine this proportion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{13}}
12.3.8 ** Given a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1. A sphere is constructed with vertex $A$ as the center and $\frac{2 \sqrt{3}}{3}$ as the radius. The length of the curve formed by the intersection of the sphere's surface with the cube's surface is equal to
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\sqrt{3}\pi}{6}}
Determine the minimum number of blue squares required on a $7 \times 7$ board such that each square has at least two neighboring blue squares. Two squares are considered neighbors if they share a side.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Consider the system of ordinary differential equations: \[\begin{array}{ll}\ddot y + y + \ddot x + x = 0 \\ y+\dot x - x = 0 \end{array}\] How many initial conditions are required to determine a unique solution for this system?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given point $P\left(a+1,2a-3\right)$, if the point is symmetric about the $x$-axis in the first quadrant, then the range of $a$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1 < a < \dfrac{3}{2}}
Given an $m \times n$ matrix $A$ and an $n \times p$ matrix $B$ such that $AB = 0$ and $\operatorname{dim}(\operatorname{ker}B) = 0$, find the minimum possible value of $\operatorname{dim}(\operatorname{ker}A)$. \( \text{Provide your answer as a non-negative integer.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p}
What is the sum of the value(s) of $n$ for which $|2n - 7| = 3$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
The inclination angle of the line $\sqrt {3}x-y+1=0$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac {\\pi}{3}"]}
We place a hemispherical cap on a sphere of radius $R$, where the radius of the cap is $\frac{R \sqrt{3}}{2}$. By what percentage is the surface area of the resulting shape larger than the surface area of the sphere?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12.5\%}
Suppose  $a_{1} < a_{2}< \cdots < a_{2024}$  is an arithmetic sequence of positive integers, and   $b_{1} <b_{2} < \cdots <b_{2024}$  is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences.

*Ray Li*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
A cylinder with radius 15 and height 16 is inscribed in a sphere. Three congruent smaller spheres of radius \( x \) are externally tangent to the base of the cylinder, externally tangent to each other, and internally tangent to the large sphere. What is the value of \( x \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{15 \sqrt{37}-75}{4}}
Define the derivative of the $(n-1)$th derivative as the $n$th derivative $(n \in N^{*}, n \geqslant 2)$, that is, $f^{(n)}(x)=[f^{(n-1)}(x)]'$. They are denoted as $f''(x)$, $f'''(x)$, $f^{(4)}(x)$, ..., $f^{(n)}(x)$. If $f(x) = xe^{x}$, then the $2023$rd derivative of the function $f(x)$ at the point $(0, f^{(2023)}(0))$ has a $y$-intercept on the $x$-axis of ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{2023}{2024}}
Determine the number of real roots of the equation \( x^{3} - x - \frac{2}{3 \sqrt{3}} = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In a school, there are 10 rowing athletes, 3 of whom can only row on the right side, 2 can only row on the left side, and the remaining 5 can row on both sides. Now, 6 people need to be selected to row, evenly distributed on both sides. There are $\boxed{\text{different methods}}$ to do this (without considering the order of teammates on the same side).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{675}
On a rectangular sheet of paper, a picture in the shape of a "cross" is drawn from two rectangles $ABCD$ and $EFGH$, with sides parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, and $FG=10$. Find the area of the quadrilateral $AFCH$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52.5}
What is the minimum number of non-adjacent swaps required to sort a list of distinct integers in ascending order?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - c}
Two boys start moving from the same point A on a circular track but in opposite directions. Their speeds are 5 ft. per second and 9 ft. per second. If they start at the same time and finish when they first meet at the point A again, then the number of times they meet, excluding the start and finish, is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$13$}
Determine the minimum distance between the curves $x^2 = y + 2$ and $y^2 = x + 2$ using the method of common normals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9\sqrt{2}}{4}}
Example 8. Calculate the flux of the vector field $\mathbf{a}=x \mathbf{i}+y \mathbf{j}+\sqrt{x^{2}+y^{2}-1} \mathbf{k}$ through the outer side of the one-sheet hyperboloid $z=\sqrt{x^{2}+y^{2}-1}$, bounded by the planes $z=0, z=\sqrt{3}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3}\pi}
Determine the number of solutions to the equation $t \sin(2t) = 2$ for $0 < t < 3\pi$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the value of the real number \(a\) for which the integral \(\int_0^{\infty} \frac{1-\cos x}{x^a} \, dx\) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 3)}
Given that the amount of wine in a bottle is normally distributed as \( N(1.05l, 0.004l^2) \) and the volume of a cask is normally distributed as \( N(22l, 0.16l^2) \), calculate the probability that the contents of 20 randomly chosen bottles will fit inside a randomly chosen cask.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.9794}
Determine the value of \( \alpha \) such that the integral \( \int_{0}^{\infty} x^{\alpha}\sin\left(\frac{1}{x}\right)dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, 0)}
Determine the VC-dimension of the class of parity functions defined on the boolean hypercube $\{0,1\}^N$. The class is given by $H_{N-parity} = \{h_I:I\subseteq \{1,2,..., N\}\}$, where for a binary vector $x = (x_1, x_2, ...,x_N) \in \{0,1\}^N$, the parity function $h_I(x) = \bigg(\sum_{i\in I}x_i\bigg) \mod 2$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{N}
In a game of Paper, Scissor, Rock with $n$ players $(n \geq 6)$, determine the number of strategic equilibria that exist when each player must choose at least two pure strategies.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $a=\sqrt{2}$, $b=\log_{0.5}e$, and $c=\ln{2}$. Arrange the values in increasing order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b < c < a}
Given the sequence $\{a_n\}$ with the sum of the first $n$ terms $S_n=3^{n}(\lambda-n)-6$, if the sequence $\{a_n\}$ is a decreasing sequence, then the range of $\lambda$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty,2)}
$ABCDEF$ is a regular hexagon. Let $R$ be the overlap between $\vartriangle ACE$ and $\vartriangle BDF$. What is the area of $R$ divided by the area of $ABCDEF$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{1}{3}"]}
The volume of a cylinder circumscribed around a sphere with radius $r$ is $V_{1}$, and the volume of a cone circumscribed around the same sphere is $V_{2}$. What is the minimum value of the ratio $V_{2} / V_{1}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{3}}