inclusionAI/Ring-lite-rl-data huggingface dataset:
\( a \) is a real number. \( {x}_{1} \) and \( {x}_{2} \) are the distinct roots of
 \( {x}^{2} + {ax} + 2 = x.{x}_{3} \) and \( {x}_{4} \) are the distinct roots of
 \( {\left( x - a\right) }^{2} + a\left( {x - a}\right)  + 2 = x \) . If \( {x}_{3} - {x}_{1} = 3\left( {{x}_{4} - {x}_{2}}\right) \) ,
 then \( {x}_{4} - {x}_{2} \) will be ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \frac{a}{2} \]}
There is a square with sides of 12. What is the circumference of the biggest circle that can be cut out of this square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12\pi}
Find the real solutions for \( \lambda \) that satisfy the equation \( 1 = e^{-\lambda \tau} \), given that \( \lambda = a \) is one solution to the characteristic equation \((1 - e^{-\lambda \tau})(a - \lambda) = 0\). Assume \( \tau > 0 \) and \( a \) is a constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Alice and Bob are playing in an eight-player single-elimination rock-paper-scissors tournament. In the first round, all players are paired up randomly to play a match. Each round after that, the winners of the previous round are paired up randomly. After three rounds, the last remaining player is considered the champion. Ties are broken with a coin flip. Given that Alice always plays rock, Bob always plays paper, and everyone else always plays scissors, what is the probability that Alice is crowned champion? Note that rock beats scissors, scissors beats paper, and paper beats rock.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6}{7}}
There are 47 students in a classroom with seats arranged in 6 rows $\times$ 8 columns, and the seat in the $i$-th row and $j$-th column is denoted by $(i,j)$. Now, an adjustment is made for students’ seats in the new school term. For a student with the original seat $(i,j)$, if his/her new seat is $(m,n)$, we say that the student is moved by $[a, b] = [i - m, j - n]$ and define the position value of the student as $a + b$. Let $S$ denote the sum of the position values of all the students. Determine the difference between the greatest and smallest possible values of $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Evaluate the sum: 
\[
\frac{3!+4!}{2(1!+2!)} + \frac{4!+5!}{3(2!+3!)} + \cdots + \frac{12!+13!}{11(10!+11!)}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
A passenger car traveling at a speed of 66 km/h arrives at its destination at 6:53, while a truck traveling at a speed of 42 km/h arrives at the same destination via the same route at 7:11. How many kilometers before the destination did the passenger car overtake the truck?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34.65 \text{ km}}
Rectangles \( R_{1} \) and \( R_{2} \), along with squares \( S_{1} \), \( S_{2} \), and \( S_{3} \), are arranged to form a rectangle with a length of 3322 and a width of 2020. Find the side length of square \( S_{2} \). Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{651}
The price of gas was increased three times during the first half of the year, by $5\%$, $6\%$, and $10\%$ respectively. Analysts predicted that the overall gas price would increase by 1/3 over the year. What percentage increase in price can be expected for the rest of the year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8.91\%}
Donovan Mitchell is currently averaging 26 points per game. His team has played 15 games this season. He has a goal of averaging 30 points per game for the entire 20 game season. However, based on his recent playing time, the coach announces that Donovan's minutes per game will be reduced by 20% in the remaining games. Furthermore, Donovan's teammate, Jordan Clarkson, has been on fire lately, and it's projected that he will score at least 5 points more per game going forward, directly reducing Donovan's scoring opportunities. Taking both constraints into consideration, how many points does Donovan Mitchell need to average per game to reach his goal of 30 points per game for the season?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
6. On the parade ground, 2018 soldiers are lined up in one row. The commander can order either all soldiers standing in even positions or all soldiers standing in odd positions to leave the formation. After this order, the remaining soldiers close up into one row. In how many ways can the commander issue a series of 8 orders so that exactly 7 soldiers remain in the formation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
已知在三棱锥 $D-A B C$ 中, $\angle A C B=\angle A B D=90^{\circ}, C A=C B, \angle B A D=30^{\circ}$.若点 $C$ 在平面 $A B D$ 上的射影恰好在 $A D$ 上, 则二面角 $C-A B-D$ 的平面角的正弦值大小为$\frac{\sqrt{m}}{n}$形式，请给出m + n的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Determine the number of angles $\theta$ between $0$ and $2\pi$, excluding integer multiples of $\pi/2$, such that $\sin \theta$, $\cos \theta$, and $\tan \theta$ form a geometric sequence in some order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given that $-1, a_1, a_2, -4$ form an arithmetic sequence, and $-1, b, -4$ form a geometric sequence, find the value of $\frac {a_{2}+a_{1}}{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm \frac {5}{2}}
Let $f: \{1, 2, 3, \ldots, 9\} \to \{1, 2, 3, \ldots, 9\}$ be a permutation chosen uniformly at random from the $9!$ possible permutations. Compute the expected value of $\underbrace{f(f(\cdots f(f(}_{2025 \ f\text{'s}}1))\cdots )).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{2}}
已知直线 $l$ 与单位圆 $\odot O$ 相切于点 $P$, 点 $A$ 与 $\odot O$ 在直线 $l$ 的同侧, 且 $A$ 到直线 $l$ 的距离为 $h\left(h>2\right)$, 从点 $A$ 作 $\odot O$ 的两条切线，分别与直线 $l$ 交于 $B 、 C$ 两点。求线段 $P B$ 与线段 $P C$ 的长度之乘积。请给出h=9时答案中分子与分母的和
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
In rectangle \(ABCD\), points \(E\) and \(F\) lie on sides \(AB\) and \(CD\) respectively such that both \(AF\) and \(CE\) are perpendicular to diagonal \(BD\). Given that \(BF\) and \(DE\) separate \(ABCD\) into three polygons with equal area, and that \(EF = 1\), find the length of \(BD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Evaluate the integral \( \int_{0}^{4} \frac{1}{(2x+8)\, \sqrt{x(x+8)}}\, dx \) using the substitution \( 4\sinh(z) = x + 4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{24}}
The monotonic decreasing interval of the function $f(x) = \left(\frac{1}{2}\right)^{x^2-2x}$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, +\infty)}
已知数列 $\left\{a_{n}\right\}$ 的各项均为非负实数, 且满足：对任意整数 $n \geq 2$, 均有 $a_{n+1}=a_{n}-a_{n-1}+n$. 若 $a_{2} a_{2022}=1$, 求 $a_{1}$ 的最大可能值.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{6076}
The side \( AB \) of a regular hexagon \( ABCDEF \) is equal to \( \sqrt{3} \) and serves as a chord of a certain circle, while the other sides of the hexagon lie outside this circle. The length of the tangent \( CM \), drawn to the same circle from vertex \( C \), is 3. Find the diameter of the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3}}
A candy store has $100$ pieces of candy to give away. When you get to the store, there are five people in front of you, numbered from $1$ to $5$. The $i$th person in line considers the set of positive integers congruent to $i$ modulo $5$ which are at most the number of pieces of candy remaining. If this set is empty, then they take no candy. Otherwise, they pick an element of this set and take that many pieces of candy. For example, the first person in line will pick an integer from the set $\{1, 6, \dots, 96\}$ and take that many pieces of candy. How many ways can the first five people take their share of candy so that after they are done there are at least $35$ pieces of candy remaining?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3003}
Find \( \lim_{n\to\infty} P\left(\frac{X_1 + \dots + X_{2n}}{\sqrt{2n}} \ge 1\right) \), where \( X_{1}, X_{2}, \dots \) is a sequence of independent random variables such that for \( k = 1, 2, \dots \), \( P(X_{2k-1} = 1) = P(X_{2k-1} = -1) = \frac{1}{2} \) and the probability density function of \( X_{2k} \) is \( f(x) = \frac{1}{\sqrt{2}}e^{-\sqrt{2}|x|} \), \( -\infty < x < \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \Phi(1)}
Let $a, b, x,$ and $y$ be real numbers with $a>4$ and $b>1$ such that\[\frac{x^2}{a^2}+\frac{y^2}{a^2-16}=\frac{(x-20)^2}{b^2-1}+\frac{(y-11)^2}{b^2}=1.\]Find the least possible value of $a+b.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Evaluate the double integral \( \iint_{D} \arcsin(x^2+y^2)\, dx\,dy \) where the region \( D \) is defined by the polar equation \( \rho=\sqrt{\sin \theta} \) for \( 0\le\theta\le\pi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 - \dfrac{\pi}{2}}
Given that the line $y=\frac{1}{2}x$ intersects the hyperbola $y=\frac{k}{x}$ ($k>0$) at points $A$ and $B$, and the abscissa of point $A$ is $4$. Another line $l$ passing through the origin $O$ intersects the hyperbola $y=kx$ ($k>0$) at points $P$ and $Q$ ($P$ is in the first quadrant). If the quadrilateral formed by points $A$, $B$, $P$, and $Q$ is a rectangle, then the coordinates of point $P$ are ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2,4)}
Determine the value of \( k \) such that the intersection of the sets \( A = \{ (x, y) \mid x^2 + y^2 = 2(x + y) \} \) and \( B = \{ (x, y) \mid kx - y + k + 3 \geq 0 \} \) is a unit set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2 - \sqrt{3}}
An equilateral $12$-gon has a side length of $10$ and interior angle measures that alternate between $90^\circ$, $90^\circ$, and $270^\circ$. Compute the area of this $12$-gon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{500}
Given that the sum of the first n terms of an arithmetic sequence is S_n, and the sum of the first n terms of a geometric sequence is T_n, and a_3 = b_3, a_4 = b_4, and (S_5 - S_3) / (T_4 - T_2) = 5, calculate the ratio (a_5 + a_3) / (b_5 + b_3). Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{3}{5}}
Let $C$ be the hyperbola defined by the equation $xy - 2y + x = 6$. A circle $S$ with equation $(x-2)^2 + (y+1)^2 = 25$ intersects $C$ at four points $P, Q, R, S$. Given that $O$ is the center of the hyperbola $C$, find the sum of the squares of the distances from $O$ to $P, Q, R, S$, i.e., $OP^2 + OQ^2 + OR^2 + OS^2$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Determine the order of the element $abc$ in the group $$ G = \langle a,b,c \mid a^2=b^2=c^2=1,\, ac=ca,\, (ab)^3=(bc)^3=1 \rangle.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the expectation \( \mathbb{E}(e^{tX}) \) where \( X \) is a lognormal random variable with parameters \( \mu \) and \( \sigma \), and \( t \) is a positive constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given that in an arithmetic sequence $\{a_n\}$, $a_3 \cdot a_7 = -16$ and $a_4 + a_6 = 0$, find the sum of the first $n$ terms, $S_n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-n(n - 9)}
Compute \( \lim_{n\to \infty} P\left(\left|\sum_{k=1}^n \frac{1}{X_k}\right|>\frac{\pi n}{2}\right) \), where \( \{X_k\}_{k\geq 1} \) is a sequence of independent random variables uniformly distributed on \([-1,1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
In the cells of an \( n \times n \) chessboard, there are altogether 2009 coins. The numbers of coins in any two neighboring cells (cells sharing a common edge) differ by 1 (some cells may contain no coin). Find the largest possible value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
The sum of the following seven numbers is exactly 19:  
\[a_1=2.56,\ a_2=2.61,\ a_3=2.65,\ a_4=2.71,\ a_5=2.79,\ a_6=2.82,\ a_7=2.86.\]  
It is desired to replace each $a_i$ by an integer approximation $A_i$, where $1 \le i \le 7$, so that the sum of the $A_i$'s is also 19 and the maximum of the "errors" $|A_i - a_i|$, denoted as $M$, is minimized. For this minimum $M$, what is $100M$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
Each of the integers $1, 2, \ldots, 729$ is written in its base-$3$ representation without leading zeroes. The numbers are then joined together in that order to form a continuous string of digits: $12101112202122\ldots$ How many times in this string does the substring $012$ appear?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{148}
In a set of data $7$, $8$, $8$, $9$, $11$, $13$, $15$, $17$, $20$, $22$, the $90$th percentile is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Let $\lambda(n)$ denote the total number of subgroups of $\mathbb{Z}_2^n$ of index $2$ (equivalently, of order $2^{n-1}$), for $n=1,2,3,\ldots$ Find a neat general formula for $\lambda(n)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n-1}
In a Cartesian coordinate system, given two fixed points $C(0,4)$ and $K(6,0)$, $A$ is a moving point on the line segment $OK$. Connect $AC$ and take the midpoint of $AC$ as $D$. Rotate the line segment $AD$ clockwise $90^{\circ}$ around point $A$ to get $AB$. Connect $BK$. When $BK$ reaches its minimum value, the coordinates of point $B$ are ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{26}{5}, \frac{8}{5}\right)}
How many nonzero coefficients can a polynomial $P(z)$ have if its coefficients are integers and $|P(z)| \leq 2$ for any complex number $z$ of unit length? Please provide the sum of all possible numbers of nonzero coefficients.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A railway train of a certain length moving at the rate of 46 kmph passes a telegraph post on its way in 5.008294988574827 seconds. What is the length of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
The base of a pyramid is a rhombus with an acute angle of $30^{\circ}$. The lateral faces are inclined at an angle of $60^{\circ}$ to the base plane. Find the volume of the pyramid if the radius of the circle inscribed in the rhombus is $r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8}{3} r^3 \sqrt{3}}
Determine the number of distinct equivalence classes in the semigroup defined by the presentation $A = \{a, b\}$ and $R = \{ (ab, ba), (bba, a), (bbb, b), (aaa, aa) \}$, where $R^{\sharp}$ denotes the reflexive and transitive closure of $R$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the number of permutations $( p_1, p_2, p_3 , p_4 , p_5 , p_6)$ of $1, 2 ,3,4,5,6$ such that for any $k, 1 \leq k \leq 5$, $(p_1, \ldots, p_k)$ does not form a permutation of $1 , 2, \ldots, k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{461}
A 120 m long train is traveling at 60 kmph towards east and at the same time, another 150 m long train is traveling at 45 kmph towards west. Both are about to cross a bridge 250 m long. How long does it take for both trains to completely cross the bridge considering they cross each other on the bridge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17.83}
In $\bigtriangleup ABC$, $AB = 86$, and $AC = 97$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover, $\overline{BX}$ and $\overline{CX}$ have integer lengths. What is the length of $BC$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
How many integers $n$ are there such that $(n+1!)(n+2!)(n+3!)\cdots(n+2013!)$ is divisible by $210$ and $1 \le n \le 210$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Determine the largest number $c$ for which there does not exist a natural number $n$ such that the interval $\left(n \sqrt{2}-\frac{c}{n}, n \sqrt{2}+\frac{c}{n}\right)$ contains an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6 - 4 \sqrt{2}}
If $(pq)(qr)(rp) = 16$, what is a possible value for $pqr$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
18. A circle is circumscribed around a right-angled triangle. Another circle of the same radius touches the legs of this triangle, with one of the points of tangency being the vertex of the triangle. Find the ratio of the area of the triangle to the area of the common part of the two given circles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{3}}{5\pi-3}}
Evaluate \( \mathop{\int\!\!\! \int}_{R_{1}}f(x,y)\,dA \) where \( R_{1} = [0, 5] \times [-4, 4] \), given that \( \mathop{\int\!\!\! \int}_{R_{2}}f(x,y)\,dA = 10 \) for \( R_{2} = [0, 5] \times [0, 4] \), \( \mathop{\int\!\!\! \int}_{R_{3}}f(x,y)\,dA = -24 \) for \( R_{3} = [-5, 0] \times [0, 4] \), and \( f(x,-y) = -f(x,y) \) for all \((x,y)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Two circles are externally tangent. Lines \(\overline{PAB}\) and \(\overline{PA'B'}\) are common tangents with \(A\) and \(A'\) on the smaller circle and \(B\) and \(B'\) on the larger circle. If \(PA = AB = 4\), then what is the value of \(x\) if the area of the smaller circle is \(x\pi\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
On the New Year's table, there are 4 glasses in a row: the first and third contain orange juice, and the second and fourth are empty. While waiting for guests, Valya absent-mindedly and randomly pours juice from one glass to another. Each time, she can take a full glass and pour all its contents into one of the two empty glasses.

Find the expected number of pourings for the first time when the situation is reversed: the first and third glasses are empty, and the second and fourth glasses are full.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
## Problem Statement

Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.

$M_{1}(-3 ; 4 ;-7)$

$M_{2}(1 ; 5 ;-4)$

$M_{3}(-5 ;-2 ; 0)$

$M_{0}(-12 ; 7 ;-1)$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{459}{\sqrt{2265}}}
Given an angle \( \angle BAC = \alpha \) with vertex at point \( A \). A perpendicular line \( CD \) is drawn from point \( C \) on \( AC \) to intersect \( AB \). From point \( D \), another perpendicular \( DE \) is drawn to intersect \( AC \), continuing in this fashion infinitely. What is the sum of the lengths of these perpendiculars if \( AC = b \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b \cot \left(\frac{\alpha}{2} \right)}
Hannah uses three different AdBlockers with varying effectiveness. AdBlocker A blocks 75% of ads, AdBlocker B blocks 85% of ads, and AdBlocker C blocks 95% of ads. When she tries a combination of these AdBlockers, none of the ads are blocked more than once. Among the ads that don't get blocked by any AdBlocker, 15% are actually interesting. What percentage of ads are not interesting and don't get blocked by any of the AdBlockers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.159375\%}
Arrange the numbers $2011, \sqrt{2011}, 2011^{2}$ in increasing order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2011}, 2011, 2011^{2}}
Let $ABC$ be a triangle with $AB = 15, BC = 17$, $CA = 21$, and incenter $I$. If the circumcircle of triangle $IBC$ intersects side $AC$ again at $P$, find $CP$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In a physics club, the teacher conducted the following experiment. He placed 16 weights of masses $1,2,3, \ldots, 16$ grams on a balance scale so that one of the pans was heavier. Fifteen students took turns leaving the classroom, each taking one weight with them. After each student left, the scales tipped in the opposite direction. Which weight could have remained on the scales?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 \text{ gram}}
On  $x-y$  plane,  let  $C: y=2006x^{3}-12070102x^{2}+\cdots.$  
Find the area of the region surrounded by the tangent line of  $C$  at  $x=2006$  and the curve  $C.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1003}{6}}
Hooligan Vasya loves to run on the escalator in the metro, and he runs down twice as fast as he runs up. If the escalator is not working, it takes Vasya 6 minutes to run up and down. If the escalator is running downwards, it takes Vasya 13.5 minutes to run up and down. How many seconds will it take Vasya to run up and down the escalator if it is running upwards? (The escalator always moves at a constant speed.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{324}
Let $a, b, c, d$ be real numbers. Suppose that all the roots of $z^4+az^3+bz^2+cz+d=0$ are complex numbers lying on a circle in the complex plane centered at $0+0i$ and having radius $1$. Find the sum of the reciprocals of the roots. Express your answer in terms of the coefficients of the polynomial.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-a}
In how many ways can the number 10000 be factored into three natural factors, none of which is divisible by 10? Factors that differ only in the order of multiplication are considered the same.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the value of the integral $\frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{1}{M+1} \frac{1 - \cos((M+1)t)}{1 - \cos(t)} dt$ where $M \geq 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given a right triangular prism \(ABC-A_{1}B_{1}C_{1}\) with the base being a right triangle, \(\angle ACB = 90^{\circ}\), \(AC = 6\), \(BC = CC_{1} = \sqrt{2}\), and \(P\) is a moving point on \(BC_{1}\), find the minimum value of \(CP + PA_{1}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5\sqrt{2}}
It is given that \( k \) is a positive integer not exceeding 99. There are no natural numbers \( x \) and \( y \) such that \( x^{2} - k y^{2} = 8 \). Find the difference between the maximum and minimum possible values of \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
Compute \(\lim_{n\to\infty}\int_{0}^{1}\frac{nx}{1+n^2x^4}dx\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}