inclusionAI/Ring-lite-rl-data huggingface dataset:
Given the sequence \( a_n \) defined by \( a_1 = 1111 \), \( a_2 = 1212 \), \( a_3 = 1313 \), and the recursive relation \( a_{n+3} = | a_{n+2} - a_{n+1} | + | a_{n+1} - a_n | \), find the value of \( a_{1414} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101}
Determine the values of \( x \) for which the series \( \sum_{n=1}^\infty \frac{1}{x^{\ln n}} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(e, \infty)}
For nonnegative integers $n$, let $f(n)$ be the number of digits of $n$ that are at least $5$. Define $g(n) = 3^{f(n)}$. Compute the sum:
\[ \sum_{i=1}^{1000} g(i). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8000}
Consider 9 distinct apples and 7 distinct bananas to be placed into 6 identical bins. Determine the number of ways to configure the bins such that exactly one bin contains 5 apples, exactly one bin contains 3 bananas, and the remaining 4 bins each contain 1 apple and 1 banana. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105840}
Evaluate the integral \( \int_{\mathbb{R}^3} x_3^2 e^{-\lVert x \rVert_2} \lambda_3(dx) \), where \( x = (x_1, x_2, x_3) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32\pi}
A safe has a lock with 3 wheels, each of which can be set in 8 different positions. Due to a defect, the safe will open if any two of the three wheels are in the correct position. What is the minimum number of combinations that must be tried to ensure the safe can be opened, assuming the correct combination is unknown?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Given the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{\sqrt{2}}{2}$, the long axis $AB$ is divided into $2016$ equal parts. From left to right, these points are sequentially named $M_{1}$, $M_{2}$, $…$, $M_{2015}$. A line with a slope of $k(k\neq 0)$ is drawn through point $M_{1}$, intersecting the ellipse $C$ at points $P_{1}$ and $P_{2}$, with $P_{1}$ located above the $x$-axis. Similarly, a line with a slope of $k(k\neq 0)$ is drawn through point $M_{2}$, intersecting the ellipse $C$ at points $P_{3}$ and $P_{4}$, with $P_{3}$ located above the $x$-axis. This process is repeated until a line with a slope of $k(k\neq 0)$ is drawn through point $M_{2015}$, intersecting the ellipse $C$ at points $P_{4029}$ and $P_{4030}$, with $P_{4029}$ located above the $x$-axis. Find the product of the slopes of the $4030$ lines $AP_{1}$, $AP_{2}$, $…$, $AP_{4030}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2^{-2015}}
How many ordered pairs \((\alpha_1, \alpha_2)\) of permutations in the symmetric group \(S_n\) commute, i.e., satisfy \(\alpha_1 \circ \alpha_2 = \alpha_2 \circ \alpha_1\), where \(\alpha_1, \alpha_2 \in S_n\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n! \cdot p(n)}
To guard a certain target, someone needs to be on duty day and night, thus there are day shifts and night shifts. A guard can be scheduled for a single day shift, or a single night shift, or a continuous 24-hour shift. After completing a day shift, a night shift, or a 24-hour shift, the guard must rest for at least 1 day, 1.5 days, or 2.5 days respectively. If each shift requires 1 person, what is the minimum number of people needed to participate in the duty rotation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given a function f such that f(f(n)) + f(n) = 2n + 3, and f(2013) = 2014, what is the value of f(0)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
8.4. The shorter leg $AC$ of the right triangle $ABC$ has a length of $b$. A point $D$ is chosen on the hypotenuse $AB$ such that $BD = BC$. A point $E$ is taken on the leg $BC$ such that $DE = BE = m$. Find the perimeter of the quadrilateral $ADEC$.

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Translation:
8.4. The shorter leg $AC$ of the right triangle $ABC$ has a length of $b$. A point $D$ is chosen on the hypotenuse $AB$ such that $BD = BC$. A point $E$ is taken on the leg $BC$ such that $DE = BE = m$. Find the perimeter of the quadrilateral $ADEC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2m+b}
There are five boxes \( A, B, C, D, \) and \( E \) with 9, 5, 3, 2, and 1 balls respectively. The first student identifies the box with the fewest balls and then puts one ball from each of the other boxes into this box. The second student then identifies the box with the fewest balls and repeats this process, and so on. After the 199th student has completed their turn, how many balls are in each of the five boxes \( A, B, C, D, \) and \( E \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5, 6, 4, 3, 2}
Consider a polynomial  $P(x,y,z)$  in three variables with integer coefficients such that for any real numbers  $a,b,c,$    $$ P(a,b,c)=0 \Leftrightarrow a=b=c. $$  Find the largest integer  $r$  such that for all such polynomials  $P(x,y,z)$  and integers  $m,n,$    $$ m^r\mid P(n,n+m,n+2m). $$  [i]Proposed by Ma Zhao Yu
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ r = 2 }
Evaluate the integral \( \int_{0}^{\pi}\ln\left(\sin\left(x\right)\right)\cos\left(4x\right)\,\mathrm{d}x. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{4}}
Determine whether the function \(|x|^{-r}\) is a tempered distribution. Specifically, find if there exist constants \(C\) and \(\alpha\) such that for all \(\phi \in \mathcal{S}(\mathbb{R})\), the inequality \(|\int_{\mathbb{R}} \frac{1}{|x|^r} \phi(x) \, dx| \leq C \|\phi\|_{(\alpha, \beta)}\) holds for some \(r > 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r < 1}
Evaluate$$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n}\left(\left[\frac{2 n}{k}\right]-2\left[\frac{n}{k}\right]\right)$$and express your answer in the form $\ln a-b$, with $a$ and $b$ positive integers. Please provide the value of $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $G=(V,E)$ be a planar, $3$-regular connected graph with $\chi'(G)=3$. $G$ also satisfy the following property:
For every two colors, Every path of length $6$ with those two alternating colors, is a cycle.  

Find $\vert V \vert$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In 4 weeks, Prudence has a varying sleep schedule influenced by her events and activities. 

Week 1:
From Sunday to Thursday, she sleeps 6 hours a night. On Friday, she sleeps for 9 hours. On Saturday, she sleeps 9 hours. She takes a 1-hour nap on Saturdays and Sundays. On Tuesday and Thursday, she takes a 30-minute power nap after work. However, on Wednesday night, she wakes up for 1 hour due to a thunderstorm.

Week 2:
From Sunday to Thursday, she sleeps 6 hours a night. On Friday, she attends a late-night event, reducing her sleep by 2 hours, resulting in only 7 hours of sleep. On Saturday, she sleeps 8 hours. Her napping schedule remains the same. Additionally, on Monday night, she wakes up for 30 minutes to check on a noise outside.

Week 3:
From Sunday to Thursday, she sleeps 6 hours a night. On Friday, she sleeps for 9 hours. On Saturday, she sleeps for 9 hours again. Her napping schedule remains the same. On Wednesday, she attends a late-evening yoga class and sleeps 30 minutes less that night.

Week 4:
From Sunday to Thursday, she sleeps 6 hours a night. On Friday, she attends a late-night event again, resulting in 7 hours of sleep. On Saturday, she sleeps 10 hours. Her napping schedule remains the same. On Thursday, she stays awake for an extra hour to finish a project, reducing her sleep to 5 hours that night.

With these specific events and activities affecting Prudence's sleep schedule, how much sleep does she get in 4 weeks?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{197}
Two circles are given by the equations $x^2+y^2-2mx=0$ and $x^2+(y-2)^2=1$. The circles are externally tangent. Find the value of the real number $m$. Express your answer as a numerical value, including the ± symbol if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{± \frac {3}{2}}
In a connected graph $G=(V,E)$ with $n$ vertices, where all edges form $3$-cycles, the graph has exactly $n-2$ $3$-cycles and a minimum degree of $2$. Every vertex is part of at least one $3$-cycle. Determine the maximum number of vertices with a degree of $2$ in $G$ as a function of $n$. Provide your answer as a function of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - 2}
Triangle \( ABC \) is isosceles, and \( \angle ABC = x^\circ \). If the sum of the possible measures of \( \angle BAC \) is \( 240^\circ \), find \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with probability  $\frac{1}{2}$ . What is the expected value of the number of games they will play?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Find the values of $n$ for which the differential of the square map $P:SO(n) \to SO(n)$ is not an isomorphism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \geq 3}
Given a square pyramid \(O-ABCD\) where \(OA = \sqrt{3}\) and \(BC = 2\), if a sphere is centered at \(O\) with a radius of 1, what is the volume of the intersection between the sphere and the square pyramid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2 \pi}{9}}
What is the average length in letters of a word in this question?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4
}
A $44$-gon $Q_1$ is constructed in the Cartesian plane, and the sum of the squares of the $x$-coordinates of the vertices equals $176$. The midpoints of the sides of $Q_1$ form another $44$-gon, $Q_2$. Finally, the midpoints of the sides of $Q_2$ form a third $44$-gon, $Q_3$. Find the sum of the squares of the $x$-coordinates of the vertices of $Q_3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Consider the following increasing sequence $1, 3, 5, 7, 9, \ldots$ of all positive integers consisting only of odd digits. Find the $2017^{th}$ term of the above sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39953}
Walter gets up at 6:30 a.m., catches the school bus at 7:30 a.m., has 6 classes that last 50 minutes each, has 30 minutes for lunch, and has 2 hours additional time at school.  He takes the bus home and arrives at 4:00 p.m.  How many minutes has he spent on the bus?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
A positive integer is called Tiranian if it can be written as $x^2+6xy+y^2$, where $x$ and $y$ are (not necessarily distinct) positive integers. The integer $36^{2023}$ is written as the sum of $k$ Tiranian integers. What is the smallest possible value of $k$?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2
}
Evaluate the asymptotic behavior of the integral \( \int_0^{at} \frac{e^{-u}}{(t-\frac{u}{a})^\beta} \, \mathrm{d}u \) as \( t \to \infty \), where \( \beta < 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t^{-\beta}}
6-27 The function $f(k)$ is defined on $N$ and takes values in $N$, and it is a strictly increasing function (if for any $x_{1}, x_{2} \in A$, when $x_{1}<x_{2}$, we have $f\left(x_{1}\right)<f\left(x_{2}\right)$, then $f(x)$ is called a strictly increasing function on $A$), and it satisfies the condition $f(f(k))=3k$. Try to find: $f(1)+f(9)+f(96)$.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{197}
Evaluate the line integral \( \oint_C y^2 \,dx + x \,dy \) using Green's Theorem, where the curve \( C \) is parameterized by \( r(t) = 2\cos^3(t) \mathbf{i} + 2\sin^3(t) \mathbf{j} \) for \( t \in [0,2\pi] \). Determine the region \( D \) enclosed by \( C \) as part of your solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{2}}
Let $(X,\mathscr{M},\mu)$ be a measure space and $f\colon X \to [0,+\infty[$ be a measurable function. Consider the graph of $f$, defined as ${\rm gr}(f) = \{ (x,f(x)) \mid x \in X \}$. Does the product measure $(\mu \times \mathfrak{m})({\rm gr}(f))$ always equal zero, even if the domain space is not $\sigma$-finite? Here, $\mathfrak{m}$ denotes the Lebesgue measure on $[0,+\infty[$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
设有序数组 A=(a_1,a_2,\cdots,a_{10}) 同时满足以下 4 个条件：(1) a_1,a_2,\cdots,a_{10} 是 1,2,\cdots,10 的一个排列；(2) a_1<a_2,a_3<a_4,a_5<a_6,a_7<a_8,a_9<a_{10}；(3) a_2>a_3,a_4>a_5,a_6>a_7,a_8>a_9；(4) 不存在 1\leq i<j<k\leq 10，使得 a_i<a_k<a_j.求这样的有序数组 A 的个数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Determine the time at which the solution of the differential equation \( u_t + uu_x = 0 \) with the initial condition \( u(x,0) = \cos{(\pi x)} \) becomes multi-valued.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{\pi}}
If $x^2 + bx + b + 3 = 0$ has roots of the form $\frac{-b \pm \sqrt{5}}{2}$, where $b > 0 $, then $b = m+\sqrt{n}$ for positive integers $m,n$. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Isaac is given an online quiz for his chemistry class in which he gets multiple tries. The quiz has $64$ multiple choice questions with $4$ choices each. For each of his previous attempts, the computer displays Isaac's answer to that question and whether it was correct or not. Given that Isaac is too lazy to actually read the questions, the maximum number of times he needs to attempt the quiz to guarantee a $100\%$ can be expressed as $2^{2^k}$. Find $k$.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In a $C^*$-algebra, if an element $a$ satisfies $b^*a^*ab = 0$ for all elements $b$ in the algebra, does it necessarily follow that $a = 0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In trapezoid $K L M N$, the bases $K N$ and $L M$ are equal to 12 and 3, respectively. From point $Q$, lying on side $M N$, a perpendicular $Q P$ is dropped to side $K L$. It is known that $P$ is the midpoint of side $K L$, $P M=4$, and that the area of quadrilateral $P L M Q$ is four times smaller than the area of quadrilateral $P K N Q$.

Find the length of segment $P N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Find the largest integer $k$ such that for all integers $x$ and $y$, if $xy + 1$ is divisible by $k$, then $x + y$ is also divisible by $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Positions A, B, and C are our artillery positions. A is 6km due east of B, and C is 4km from B at a bearing of 30° west of north. Position P is an enemy artillery position. At a certain moment, position A detects a signal from the enemy position, and 4 seconds later, positions B and C detect this signal simultaneously. The signal travels at a speed of 1km per second. If A fires at position P, what is the bearing of the fire?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^{\circ}}
Let $\Gamma_1$, $\Gamma_2$ be two circles, where $\Gamma_1$ has a smaller radius, intersecting at two points $A$ and $B$. Points $C$ and $D$ lie on $\Gamma_1$ and $\Gamma_2$ respectively such that point $A$ is the midpoint of the segment $CD$. Line $CB$ intersects the circle $\Gamma_2$ for the second time at point $F$, and line $DB$ intersects the circle $\Gamma_1$ for the second time at point $E$. The perpendicular bisectors of segments $CD$ and $EF$ intersect at point $P$. Knowing that $CA = 12$ and $PE = 5$, find $AP$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Find the sum: \( S = 19 \cdot 20 \cdot 21 + 20 \cdot 21 \cdot 22 + \cdots + 1999 \cdot 2000 \cdot 2001 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6 \left( \binom{2002}{4} - \binom{21}{4} \right)}
Every member of the school's entertainment team can either sing or dance, and at least one of these. It is known that there are 2 people who can sing, 5 people who can dance. If we select 2 people, let $\xi$ be the number of people selected who can both sing and dance. Given $P(\xi > 0) = \frac{7}{10}$, the total number of people in the entertainment team is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Arrange the numbers 1, 2, 3, 4, 5, 6 in a row, and denote the i-th number as a_i (i=1,2...6). If a_1!= 1, a_3!= 3, a_5!= 5, and a_1 < a_3 < a_5, calculate the total number of different arrangements. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Calculate the geometric mean of 1 and 9. Express your answer in the form ±n, where n is a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±3}
Given that the binomial coefficient of only the fourth term in the expansion of (1+2x)^n is the largest, the constant term in the expansion of the polynomial (x^2+1)(x+1/x)^n is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Find all generators of the subgroup <21> in the group $Z_{24}$. That is, determine all elements $m \in Z_{24}$ such that $<m> = <21>$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{3, 9, 15, 21}}
For the equation log 3 n + log x n, the 3 digit number n that makes it a whole number is 216. What is the value of x?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $A$ and $B$ be distinct positive integers such that each has the same number of positive divisors as 2013. Compute the least possible value of $|A - B|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let proposition $p: f(x)= \sqrt{x^{2}+mx+1}$ have a domain of $\mathbb{R}$; proposition $q$: the equation $\frac{x^{2}}{m}+\frac{y^{2}}{2}=1$ represents an ellipse with foci on the $y$-axis. If "p and q" is a false proposition, and "p or q" is a true proposition, find the range of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-2,0] \cup \{2\}}
Compute the probability $$P(T_{1} < T_{-3} < T_3)$$ for a standard Brownian motion $W_t$, where $T_a$ denotes the hitting time of level $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/12}
Find the total number of positive four-digit integers \( N \) satisfying both of the following properties:
(i) \( N \) is divisible by 7, and
(ii) when the first and last digits of \( N \) are interchanged, the resulting positive integer is also divisible by 7. (Note that the resulting integer need not be a four-digit number.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210}
Calculate the mass of the region in the first quadrant where the tile lies between $1 < xy < 3$ and $1 < x^2 - y^2 < 3$, given the density function $\sigma(x, y) = (x^2 + y^2)(x^4 + y^4)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
In quadrilateral \(ABCD\), it is given that \(\angle DAB = 150^\circ\), \(\angle DAC + \angle ABD = 120^\circ\), and \(\angle DBC - \angle ABD = 60^\circ\). Find \(\angle BDC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^
}
If $z\bar{z}'=-1$, do $z$ and $z'$ correspond to opposite points on the Riemann sphere? Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
设 a_1,a_2,\cdots,a_{21} 为 1,2,\cdots,21 的排列，满足|a_{20}-a_{21}|\geq|a_{19}-a_{21}|\geq\cdots\geq|a_1-a_{21}|.这样的排列的个数为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3070}
The negation of the proposition "_x_∈**R**, _x_<sup>2</sup> - _x_ + 3 > 0" is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{_x_∈\mathbb{R}, _x_<sup>2</sup> - _x_ + 3 \leq 0}
In a tournament bracket, a player must be defeated three times to be eliminated. If there are 512 contestants entering the tournament, what is the greatest number of games that could be played?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1535}
Determine the sign of the derivative of the function \( f(x) = \frac{x+1}{x} + \left(\frac{2x}{x+1}\right)^{\frac{\ln(6/5)}{\ln(31/25)}} \) for \( x > 10^{500} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{positive}
A cylindrical, closable container has a diameter of 82 units and a length of 225 units. How many spheres with a diameter of 38 units can be placed inside, and then the container can be closed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Determine whether a symmetric real-valued matrix with a positive determinant is necessarily a covariance matrix. Answer "yes" or "no."
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}
Determine the interval of monotonic increase for the function $y=3x-x^{3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1,1]}
Calculate the circulation of the vector $\mathbf{a}=\sqrt{1+x^{2}+y^{2}} \mathbf{i}+y[x y+\ln \left(x+\sqrt{1+x^{2}+y^{2}}\right)]$ around the circle $x^{2}+y^{2}=R^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi R^4}{4}}
A box has a square base of side $x$ and height $h$. Find the dimensions $x$ and $h$ when the volume is 12 and the surface area is as small as possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(12\right)^{1/3}, \left(12\right)^{1/3}}
Evaluate the integral of $g(z)=\cot z$ around a contour that encircles the poles at $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ exactly once.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi i}
In a non-commutative ring $R$ with unity $1$, but without assuming the uniqueness of left or right inverses, determine if every non-unit element $x$ must be contained in some maximal left ideal. Answer with 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}
Given the recursive sequence defined by a_0 = 134, a_1 = 150, and a_{k+1} = a_{k-1} - k/a_k for k = 1, 2,..., n-1, find the value of n for which a_n = 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{201}
Evaluate the Cauchy principal value of the integral \( I = \oint \frac{dz}{(z - z_1)(z - z_2)} \) where the poles \( z_1 = e^{-ik} \) and \( z_2 = e^{ik} \) lie on the unit circle contour. Determine if there are conditions under which this integral is not zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
8. What is the least number of odd numbers that can be chosen from the set of all odd numbers lying between 16 and 2016, so that no one of the chosen numbers is divisible by any other?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{672}