inclusionAI/Ring-lite-rl-data huggingface dataset:
A journalist visits a company with $\mathrm{N}$ people. He knows that there is a person $\mathrm{Z}$ in this company who knows everyone else, but no one knows him. The journalist can ask each member of the company the question: "Do you know such-and-such person?" Find the smallest number of questions required to definitively identify $\mathrm{Z}$. (Everyone answers questions truthfully. One person can be asked multiple questions.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathrm{N}-1}
If the degree measures of the angles of a triangle are in the ratio $3:3:4$, and 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{72}
On Sunday, Xiaopang found that his alarm clock was broken when he woke up. He estimated the time and set the alarm clock to 7:00 AM. Then he walked to the science museum and saw the electronic clock on the rooftop indicating 8:50 AM. He toured the science museum for an hour and a half and then returned home at the same speed. When he got home, he saw the alarm clock indicating 11:50 AM. What is the correct time that the alarm clock should be set to?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12:00}
one fourth of a solution that was 14 % salt by weight was replaced by a second solution resulting in a solution that was 16 percent sugar by weight . the second solution was what percent salt by weight ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
已知$m>0$，$m\ne \frac{\sqrt{17}-1}{2}$，直线${{l}_{1}}:y=m$与函数$y=\left| {{\log 4}}x \right|$的图像从左至右相交于点$A,B$，直线${{l}_{2}}:y=\frac{4}{m+1}$与函数$y=\left| {{\log 4}}x \right|$的图像从左至右相交于点$C,D$，记线段$AC$和$BD$在$x$轴上的投影长度分别为$a,b$，当$m$变化时，$\frac{b}{a}$的最小值是__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
We have 10 springs, each originally $0.5 \mathrm{~m}$ long with a spring constant of $200 \mathrm{~N}/\mathrm{m}$. A mass of $2 \mathrm{~kg}$ is hung on each spring, and the springs, along with the masses, are hung in a series. What is the length of the resulting chain? (Neglect the mass of the springs.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10.39}
There are  $2012$  backgammon checkers (stones, pieces) with one side is black and the other side is white.
These checkers are arranged into a line such that no two consequtive checkers are in same color. At each move, we are chosing two checkers. And we are turning upside down of the two checkers and all of the checkers between the two. At least how many moves are required to make all checkers same color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1006}
Calculate the exponential of the vector field ${\bf w} = y^2\partial_y$ using the formula $\exp(\varepsilon{\bf w}) = \sum_{n \ge 0} \frac{\varepsilon^n}{n!}{\bf w}^n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{y}{1-\varepsilon y}}
Given the functional equation $$f(x+\frac{1}{y}) + f(x-\frac{1}{y}) = 2f(x)\cdot f(\frac{1}{y}) \quad \text{for all } x,y\in\mathbb{R}-\{0\},$$ with the condition $f(0) = \frac{1}{2}$, determine the value of $f(4)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
A watch was sold at a loss of 9%. If it was sold for Rs. 220 more, there would have been a gain of 4%. What is the cost price of the watch?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1692.31}
Let  $S$  be a subset of the natural numbers such that  $0\in S$ , and for all  $n\in\mathbb N$ , if  $n$  is in  $S$ , then both  $2n+1$  and  $3n+2$  are in  $S$ .  What is the smallest number of elements  $S$  can have in the range  $\{0,1,\ldots, 2019\}$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
设 $M=\{1,2, \cdots, 1995\}, A \subseteq M$, 且当 $x \in A$ 时, $19 x \bar{\in} A$. 求 $|A|$ 的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1895}
Let $\alpha$ denote $\cos^{-1}(\tfrac{2}{3})$. The recursive sequence $a_0, a_1, a_2, \ldots$ satisfies $a_0 = 1$ and, for all positive integers $n$,  
$$a_n = \dfrac{\cos(n\alpha) - (a_1a_{n-1} + \cdots + a_{n-1}a_1)}{2a_0}.$$  
Suppose that the series  
$$\sum_{k=0}^\infty\dfrac{a_k}{2^k}$$  
can be expressed uniquely as $\tfrac{p\sqrt{q}}{r}$, where $p$ and $r$ are coprime positive integers and $q$ is not divisible by the square of any prime. Find the value of $p+q+r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Let $ABCD$ be a trapezoid with $AB||CD, AB=11, BC=5, CD=19,$ and $DA=7$. Bisectors of $\angle A$ and $\angle D$ meet at $P$, and bisectors of $\angle B$ and $\angle C$ meet at $Q$. If the area of hexagon $ABQCDP$ is $x\sqrt{3}$, what is the value of $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Given a function $y = f(x)$ defined on $\mathbb{R}$ that is an even function satisfying $f(2 + x) = f(2 - x)$, and given that the function $y = f(x)$ has at least one zero in the interval $(0, 4)$ and $f(0) = 0$, find the minimum number of zeros of the function $y = f(x)$ in the interval $(-8, 10]$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
A manager schedules a consultation session at a café with two assistant managers but forgets to set a specific time. All three aim to arrive randomly between 2:00 and 5:00 p.m. The manager, upon arriving, will leave if both assistant managers are not present. Each assistant manager is prepared to wait for 1.5 hours for the other to arrive, after which they will leave if the other has not yet arrived. Determine the probability that the consultation session successfully takes place.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Given that $a_{1}, a_{2}, \cdots, a_{10}$ are ten different positive integers satisfying  the equation $\left|a_{i+1}-a_{i}\right|=2 \text { or } 3$, where $i=1,2, \cdots, 10$, with the condition $a_{11}=a_{1}$, determine the maximum value of $M-m$, where $M$ is the maximum number among $a_{1}, a_{2}, \cdots, a_{10}$ and $m$ is the minimum number among $a_{1}, a_{2}, \cdots, a_{10}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Five points are located on a line. When the ten distances between pairs of points are listed from smallest to largest, the list reads: 2, 4, 5, 7, 8, k, 13, 15, 17, 19. Calculate the value of the missing distance k.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Imagine a dodecahedron (a polyhedron with 12 pentagonal faces) and an ant starts at one of the top vertices. The ant will randomly walk to one of three adjacent vertices, denoted as vertex A. From vertex A, the ant then walks to one of another three randomly selected adjacent vertices, signified as vertex B. Calculate the probability that vertex B is one of the five bottom vertices of the dodecahedron. Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Find the smallest nomial of this sequence that  $a_1=1993^{1994^{1995}}$  and 
\[ a_{n+1}=\begin{cases}\frac{a_n}{2}&\text{if  $n$  is even}a_n+7 &\text{if  $n$  is odd.} \end{cases} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
集合 $\mathbf{A}=\{0,1,2, \cdots, 9\}, \mathbf{B}_1, \mathbf{B}_2, \cdots, \mathbf{B}_k$ 为 $\mathbf{A}$ 的非空子集, 且 $\left|\mathbf{B}_i \cap \mathbf{B}_j\right| \leqslant 2(1 \leqslant i<j \leqslant k)$, 求 $k$ 的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{175}
Determine how many positive integer multiples of $2002$ can be represented in the form $10^{j} - 10^{i}$, where $i$ and $j$ are integers and $0 \leq i < j \leq 150$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1825}
Let $G$ be a $2k$-regular tripartite graph with $V(G) = X_1 \cup X_2 \cup X_3$ where $|X_i| = 2m+1$ for $i=1,2,3$. Determine the minimum number of matchings into which $G$ can be decomposed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2k + 1}
Ten distinct positive real numbers are given and the sum of each pair is written (So 45 sums). Between these sums there are 5 equal numbers. If we calculate product of each pair, find the biggest number $k$ such that there may be $k$ equal numbers between them.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
There were some roses in the vase. Jessica threw away 34 roses from the vase and cut some more new roses from her flower garden to put in the vase. There are now 15 roses in the vase, and she threw away 6 more roses than those she cut from her garden. How many roses were initially in the vase?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
All sides of the convex pentagon $ABCDE$ are of equal length, and $\angle A= \angle B = 90^\circ.$ If the answer is $x^\circ$, what is the degree measure of $\angle E?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
The function $y=(m^2-m-1)x^{m^2-3m-3}$ is a power function, and it is an increasing function on the interval $(0, +\infty)$. Find the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Let the function \( f(x) = \frac{\sin x}{2 + \cos x} \). If for any \( x^3 \geq 0 \), \( f(x) \leq ax \) holds, find the range of the real number \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[ \frac{1}{3}, +\infty \right)}
In trapezoid \(ABCD\) with \(BC \parallel AD\), it is known that \(AD = 3 \cdot BC\). A line intersects the non-parallel sides of the trapezoid at points \(M\) and \(N\) such that \(AM:MB = 3:5\) and \(CN:ND = 2:7\). Find the ratio of the areas of quadrilaterals \(MBCN\) and \(AMND\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9/23}
Let  $ABC$  be a triangle with  $\angle BAC = 90^o$  and  $D$  be the point on the side  $BC$  such that  $AD \perp BC$ . Let $ r, r_1$ , and  $r_2$  be the inradii of triangles  $ABC, ABD$ , and  $ACD$ , respectively. If  $r, r_1$ , and  $r_2$  are positive integers and one of them is  $5$ , find the largest possible value of    $r+r_1+ r_2$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Problem 8.6. In a $3 \times 3$ table, real numbers are arranged. It turns out that the product of the numbers in any row and any column is 10, and the product of the numbers in any $2 \times 2$ square is 3. Find the number in the central cell.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.00081}
In the group stage of the World Cup, each group has 4 teams that play in a round-robin tournament. For each match, the winning team gets 3 points, the losing team gets 0 points, and in the event of a draw, both teams get 1 point each. After the group stage, the two teams with the highest total points advance to the next round. If the total points are the same, the teams are then ranked by goal difference. What is the minimum number of points a team must score to ensure advancement?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
There exists an $x \in (0, +\infty)$ such that the inequality $mx^2 + 2x + m > 0$ holds is a false statement, then the range of values for $m$ is ___.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -1]}
The maximum number of students among whom 781 pens and some pencils can be distributed in such a way that each student gets the same number of pens and same number of pencils is 71. How many pencils are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71}
Let $f(x)$ and $g(x)$ be odd and even functions defined on $\mathbb{R}$, respectively. If for $x < 0$, $f'(x)g(x) + f(x)g'(x) > 0$ and $g(-3) = 0$, then the solution set for $f(x)g(x) < 0$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -3) \cup (0, 3)}
In a group of 80 children, there are 35 happy, 15 sad, 10 confused, 5 excited, and 15 neither happy nor sad. There are 45 boys and 35 girls. If there are 8 happy boys, 7 sad girls, 4 confused boys and 3 excited girls, how many boys are neither happy nor sad?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Triangle \(\triangle P N R\) has side lengths \(P N=20, N R=18\), and \(P R=19\). Consider a point \(A\) on \(P N\). \(\triangle N R A\) is rotated about \(R\) to \(\triangle N^{\prime} R A^{\prime}\) so that \(R, N^{\prime}\), and \(P\) lie on the same line and \(A A^{\prime}\) is perpendicular to \(P R\). Find \(\frac{P A}{A N}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{19}{18}}
In the quadrilateral $ABCD$, the angles $B$ and $D$ are right . The diagonal $AC$ forms with the side $AB$ the angle of $40^o$, as well with side $AD$ an angle of $30^o$. Find the acute angle between the  diagonals $AC$ and $BD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80^\circ}
A thin beam of light falls normally on a flat parallel glass plate. Behind the plate, at some distance from it, there is an ideal mirror (its reflection coefficient is equal to one). The plane of the mirror is parallel to the plate. It is known that the intensity of the beam passing through this system is 16 times less than the intensity of the incident beam. The reflection coefficient at the glass-air boundary is considered constant regardless of the direction of the beam. Absorption and scattering of light in both air and glass are negligible. Find the reflection coefficient at the glass-air boundary under these conditions. (10 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.5}
若 $\triangle A B C$ 的三个内角 $A, B, C$ 满足 $\cos A=\sin B=2 \tan \frac{C}{2}$, 则 $\sin A+\cos A$ $+2 \tan A$ 的值为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a cricket match, 5 batsmen A, B, C, D, and E scored an average of 36 runs, and all of them scored runs only in multiples of 4. D scored 12 more than E; E scored 8 fewer than A; B scored as many as D and E combined; and B and C scored 107 between them. Moreover, A is the opening batsman and goes on to score the maximum runs among them. How many runs did E score?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
In the diagram, \( PQ \) is perpendicular to \( QR \), \( QR \) is perpendicular to \( RS \), and \( RS \) is perpendicular to \( ST \). If \( PQ=4 \), \( QR=8 \), \( RS=8 \), and \( ST=3 \), then the distance from \( P \) to \( T \) is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
How many pairs of positive integers $(x, y)$ satisfy the equation $\sqrt{xy} - 71\sqrt{x} + 30 = 0$? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given the sample 7, 8, 9, x, y has an average of 8, and xy=60, then the standard deviation of this sample is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
A cuboctahedron is a polyhedron whose faces are squares and equilateral triangles such that two squares and two triangles alternate around each vertex. What is the volume of a cuboctahedron of side length 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5 \sqrt{2} / 3}
A sequence consists of the digits $122333444455555 \ldots$ such that each positive integer $n$ is repeated $n$ times, in increasing order. Find the sum of the 4501st and 4052nd digits of this sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Given the function y=cos(2x- π/3), derive the horizontal shift required to transform the graph of the function y=sin 2x.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{12}}
Given the sequence $\left\{a_{n}\right\}$ defined by $a_{0} = \frac{1}{2}$ and $a_{n+1} = a_{n} + \frac{a_{n}^{2}}{2023}$ for $n=0,1,2,\ldots$, find the integer $k$ such that $a_{k} < 1 < a_{k+1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2023}
Jindra collects dice, all of the same size. Yesterday he found a box in which he started stacking the dice. He managed to fully cover the square base with one layer of dice. He similarly stacked five more layers, but he ran out of dice halfway through the next layer. Today, Jindra received 18 more dice from his grandmother, which were exactly the amount he needed to complete this layer.

How many dice did Jindra have yesterday?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{234}
Teacher Li and three students (Xiao Ma, Xiao Lu, and Xiao Zhou) depart from school one after another and walk the same route to a cinema. The walking speed of the three students is equal, and Teacher Li's walking speed is 1.5 times that of the students. Currently, Teacher Li is 235 meters from the school, Xiao Ma is 87 meters from the school, Xiao Lu is 59 meters from the school, and Xiao Zhou is 26 meters from the school. After they walk a certain number of meters, the distance of Teacher Li from the school will be equal to the sum of the distances of the three students from the school. Find the number of meters they need to walk for this condition to be true.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Calculate the value of $3^{12} \cdot 3^3$ and express it as some integer raised to the third power.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{243}
Compute  $\left\lceil\displaystyle\sum_{k=2018}^{\infty}\frac{2019!-2018!}{k!}\right\rceil$ . (The notation  $\left\lceil x\right\rceil$  denotes the least integer  $n$  such that  $n\geq x$ .)

*Proposed by Tristan Shin*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2019}
Example 10 Let real numbers $x, y, z, w$ satisfy
$$
\left\{\begin{array}{l}
\frac{x^{2}}{2^{2}-1^{2}}+\frac{y^{2}}{2^{2}-3^{2}}+\frac{z^{2}}{2^{2}-5^{2}}+\frac{w^{2}}{2^{2}-7^{2}}=1, \\
\frac{x^{2}}{4^{2}-1^{2}}+\frac{y^{2}}{4^{2}-3^{2}}+\frac{z^{2}}{4^{2}-5^{2}}+\frac{w^{2}}{4^{2}-7^{2}}=1, \\
\frac{x^{2}}{6^{2}-1^{2}}+\frac{y^{2}}{6^{2}-3^{2}}+\frac{z^{2}}{6^{2}-5^{2}}+\frac{w^{2}}{6^{2}-7^{2}}=1, \\
\frac{x^{2}}{8^{2}-1^{2}}+\frac{y^{2}}{8^{2}-3^{2}}+\frac{z^{2}}{8^{2}-5^{2}}+\frac{w^{2}}{8^{2}-7^{2}}=1 .
\end{array}\right.
$$

Try to find the value of $x^{2}+y^{2}+z^{2}+w^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Suppose a biased coin gives head with probability $\frac{2}{3}$. The coin is tossed repeatedly. If it shows heads, then player $A$ rolls a fair die; otherwise, player $B$ rolls the same die. The process ends when one of the players gets a $6$, and that player is declared the winner.\n\nIf the probability that $A$ will win is given by $\frac{m}{n}$, where $m$ and $n$ are coprime, then what is the value of $m^2n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let \( ABCDEF \) be a regular hexagon. A frog starts at vertex \( A \) and can jump to one of the two adjacent vertices randomly on each jump. If the frog reaches point \( D \) within 5 jumps, it stops jumping. If the frog cannot reach \( D \) within 5 jumps, it stops jumping after completing 5 jumps. Determine the total number of different jumping sequences possible from the start until it stops.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
定义在集合 $\left\{x \in \mathbf{Z}_{+} \mid 1 \leqslant x \leqslant 12\right\}$ 上的函数 $f(x)$ 满足  
$$  
|f(x+1)-f(x)|=1(x=1,2, \cdots, 11),  
$$  
且 $f(1) 、 f(6) 、 f(12)$ 成等比数列。若 $f(1)$ $=1$ ，则满足条件的不同函数 $f(x)$ 的个数为  
$\qquad$。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{155}
Use $\max\{a, b\}$ to represent the maximum value between $a$ and $b$. Suppose the function $f(x)=\max\{-4x^{3}+kx-1, \ln x\}$ (where $x > 0$) has three zeros, then the range of the real number $k$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(3,5)}
Let $T_1$ be a triangle with side lengths $2011, 2012,$ and $2013$. For $n \ge 1$, if $T_n = \triangle ABC$ and $D, E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB, BC$, and $AC,$ respectively, then $T_{n+1}$ is a triangle with side lengths $AD, BE,$ and $CF,$ if it exists. What is the perimeter of the last triangle in the sequence $( T_n )$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1509}{128}$}
Compute \( \sum_{k=1}^{420} \gcd(k,420) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4680}
In a factory, some machines make 10 cell phones each minute. It would take 10 machines to make 50 cell phones a minute. How many cell phones do these machines make each minute?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Start by writing the integers $1, 2, 4, 6$ on the blackboard. At each step, write the smallest positive integer $n$ that satisfies both of the following properties on the board.
[list]
[*] $n$ is larger than any integer on the board currently.
[*] $n$ cannot be written as the sum of $2$ distinct integers on the board.
[/list]
Find the $100$-th integer that you write on the board. Recall that at the beginning, there are already $4$ integers on the board.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{388}
Given a sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$, where $S\_n=n^2+2a|n-2016|$ $(a > 0, n \in \mathbb{N}^+)$, find the maximum value of $a$ that ensures $a\_n \leq a\_{n+1}$ always holds true.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2016}}
There are \( n \) parking spots along a road. Each of the \( n \) drivers drives their car to their favorite spot to park. If their favorite spot is already occupied, they will park in the nearest available spot down the road. If their favorite spot and all the spots beyond it are occupied, they will drive away without parking. How many different arrays \(\left(a_{1}, a_{2}, \cdots, a_{n}\right)\) are there such that no parking spot is left empty? Here, \( a_{i} \) represents the favorite spot of the \( i \)-th driver, and \( a_{1}, a_{2}, \cdots, a_{n} \) are not necessarily distinct.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n+1)^{n-1}}
In a square \(ABCD\) with a side length of 2, \(E\) is the midpoint of \(AB\). The figure is folded along segments \(EC\) and \(ED\) so that segments \(EA\) and \(EB\) coincide. The resulting tetrahedron is \(CDEA\) (with point \(B\) coinciding with point \(A\)). What is the volume of this tetrahedron?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{3}}
正整数 $a, b, c, x, y, z$ 满足: $a x=b+c, b y=c+a, c z=a+b$ ，求 $x y z$ 的可能值的个数。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}