inclusionAI/Ring-lite-rl-data huggingface dataset:
今有茭草八千五百六十八束，欲令撒星更落一形垛之，问底子几何？（撒星更落一形的放法是第一层1束，第二层(1+4)束，第三层(1+4+10)束，···，第n层$( 1 + 4 + 1 0 + \cdots + C _ { n + 2 } ^ { 3 } )$束.）
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
The right focus of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (with $a>0$, $b>0$) is $F$, and $B$ is a point on the left branch of the hyperbola. The line segment $BF$ intersects with one asymptote of the hyperbola at point $A$, and it is given that $(\vec{OF} - \vec{OB}) \cdot \vec{OA} = 0$ and $2\vec{OA} = \vec{OB} + \vec{OF}$ (where $O$ is the origin). Find the eccentricity $e$ of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5}}
How many ordered triples $\left(a, b, c\right)$ of positive integers are there such that at least two of $a, b, c$ are prime and $abc = 11(a + b + c)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Let $\alpha = \cos^{-1} \left( \frac35 \right)$ and $\beta = \sin^{-1} \left( \frac35 \right) $.
$$\sum\limits_{n=0}^{\infty}\sum\limits_{m=0}^{\infty} \frac{\cos(\alpha n +\beta  m)}{2^n3^m}$$can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A +B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
15013
}
Use the Cauchy-Frobenius lemma to find the number of orbits of the action of $D_8$, the dihedral group of $8$ elements, on the set of $k$-colorings of the edges of a square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{8}(k^4+5k^2+2k)\)}
Dividing $x^{1951}-1$ by $P(x)=x^4+x^3+2x^2+x+1$ one gets a quotient and a remainder. Find the coefficient of $x^{14}$ in the quotient.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Determine the necessary and sufficient condition for the convergence of the integral \( \int_0^\infty \frac{x^2+1-\cos x}{(x^2+1)x^p} \, dx \), where \( p \) is a real number. Is the condition \( 1 < p \le 2 \) correct?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 < p < 3}
Solve the equation
$
3^x -5^y  = z^2.
$
in positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x,y,z)=(2,1,2)}
Lillian collects 205 candies. Lillian's father gives her 2 times the amount of candies she collected. If Lillian shares the candies equally with her 7 friends, how many candies will each friend receive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{87}
Triangle $ABC$ is inscribed in a unit circle $\omega$. Let $H$ be its orthocenter and $D$ be the foot of the perpendicular from $A$ to $BC$. Let $\triangle XY Z$ be the triangle formed by drawing the tangents to $\omega$ at $A, B, C$. If $\overline{AH} = \overline{HD}$ and the side lengths of $\triangle XY Z$ form an arithmetic sequence, the area of $\triangle ABC$ can be expressed in the form $\tfrac{p}{q}$ for relatively prime positive integers $p, q$. What is $p + q$?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
In a particular game, each of $4$ players rolls a standard $6$-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again and this process will continue until one player wins. Hugo is one of the players in this game. What is the probability that Hugo's first roll was a $5,$ given that he won the game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{41}{144}}
Let $(X, \mathcal{U})$ be a compact Hausdorff uniform space where $\bigcap \{U: U\in \mathcal{U}\}= \Delta_X$. Determine whether there exists a sequence $\{U_n\}_{n\in\mathbb{N}}$ such that $U_{n+1}\subseteq U_n$ and $\bigcap_{n\in\mathbb{N}}U_n=\Delta_X$. Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{no}
Let $f$ be an entire function of order $r = 1$. Determine the order of the derivative function $f'$. Is it finite or not?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $X$ be a smooth projective variety, and $D$ a Cartier divisor on $X$ inducing a surjective morphism $f: X \rightarrow C$, where $C$ is a curve. Can we conclude that the self-intersection number $D^2 = 0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the probability $P(1.75 < \bar{X} < 2.5)$, where $\bar{X}$ is the sample mean of $36$ independent random variables, each following a chi-squared distribution with $2$ degrees of freedom.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.7066}
Alice and Bob are playing a game with dice. They each roll a die six times and take the sums of the outcomes of their own rolls. The player with the higher sum wins. If both players have the same sum, then nobody wins. Alice's first three rolls are 6, 5, and 6, while Bob's first three rolls are 2, 1, and 3. The probability that Bob wins can be written as a fraction \( \frac{a}{b} \) in lowest terms. What is \( a + b \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3895}
ABCDE is a regular pentagon. The star ACEBD has an area of 1. AC and BE meet at P, BD and CE meet at Q. Find the area of APQD.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
A square is divided into $n^{2}$ equal smaller squares. It is known that a certain polyline passes through the centers of all these smaller squares (the polyline may intersect itself). What is the minimum number of segments in this polyline?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n - 2}
Find the magnitude of the product of all complex numbers  $c$  such that the recurrence defined by  $x_1 = 1$ ,  $x_2 = c^2 - 4c + 7$ , and  $x_{n+1} = (c^2 - 2c)^2 x_n x_{n-1} + 2x_n - x_{n-1}$  also satisfies  $x_{1006} = 2011$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given a regular tetrahedron with all four vertices lying on the surface of a sphere with radius 1, where the three vertices of the base are on a great circle of the sphere, find the volume of the regular tetrahedron.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{4}}
A point \( M \) is chosen on the diameter \( AB \). Points \( C \) and \( D \), lying on the circumference on one side of \( AB \), are chosen such that \(\angle AMC=\angle BMD=30^{\circ}\). Find the diameter of the circle given that \( CD=12 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\sqrt{3}}
Find the center of gravity of a hemispherical shell of radius $a$, assuming the density is $1$ and it is placed so that its base is on the $x-y$ plane. Use the method of slicing parallel to the $x-y$ axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{a}{2}$}
Given the numbers 1, 3, 5 and 2, 4, 6, calculate the total number of different three-digit numbers that can be formed when arranging these numbers on three cards.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
An expression arranged according to a certain rule: $-\frac{b}{a}$, $\frac{3b}{{a}^{2}}$, $\frac{5b}{{a}^{3}}$, $\frac{7b}{{a}^{4}}$, $\ldots \ldots $ The $n$th expression is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(-1\right)^{n}\frac{(2n-1)b}{{a}^{n}}}
A circle with area $40$ is tangent to a circle with area $10$. Let $R$ be the smallest rectangle containing both circles. The area of $R$ is $\frac{n}{\pi}$. Find $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Let $A \in \mathbb{R}^{n \times n}$ be a matrix where the column vectors $k_i \in \mathbb{R}^n$ for $i = 1, 2, \ldots, n$ satisfy the condition $k_i = (i+2)k_{i+2}$ for $i = 1, 2, \ldots, n-2$. Given that $n > 3$, choose one eigenvalue of $A$ and determine the smallest possible dimension of the eigenspace corresponding to the chosen eigenvalue.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A person initially has an unknown amount of money. They spend 1/3 of it on clothes, then 1/5 of the remaining on food, 15% of what's left on utilities, 1/8 of the remaining on entertainment, and finally 20% of what's still left on travel. After all these expenses, they have $3,500 left. How much money did they have in the beginning?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$11,029.41}
Let $(f_n)_{n\in \mathbb{N}}$ be a sequence of measurable functions from a finite measure space $(E, \mathcal{E}, m)$ to $\mathbb{R}$ that converges in measure to a measurable function $f:E \to \mathbb{R}$. For a given $\eta >0$, determine if the following limit holds:
\[
\lim_{n \to \infty} m\left(\{x \in E : |f_n(x)-f(x)||f_n(x)| > \eta\}\right) = 0?
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Suppose you have a function \( g : [0, \infty) \to [0, \infty) \) such that \( \lim_{x \to +\infty} (2g(2x) - g(x)) = 0 \). Additionally, for every \( M > 0 \), the restriction \( g|_{[0,M]} \) is bounded. Does it follow that \( \lim_{x \to +\infty} g(x) = 0 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the polynomial $f(x) = 12 + 35x - 8x^2 + 79x^3 + 6x^4 + 5x^5 + 3x^6$, use Horner's Rule to calculate the value of $v_4$ when $x = -4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{220}
sue ' s trail mix is 30 % nuts and 70 % dried fruit . jane ' s trail mix is 60 % nuts , 30 % chocolate chips and 10 % pretzels. Tom's trail mix is 40% nuts, 50% dried fruit, and 10% granola. If the combined mixture of sue, jane and tom's trails mix contains 45 % nuts , what percent of the combined mixture is dried fruit ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40\%}
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = 3$, $|\overrightarrow{b}| = 2\sqrt{3}$, and $\overrightarrow{a} \cdot \overrightarrow{b} = -9$, find the projection of $\overrightarrow{b}$ onto $\overrightarrow{a}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Given the function $f(x) = (m^2 - m - 1)x^{m^2 - 2m - 1}$, if it is a power function and is an increasing function on the interval $(0, +\infty)$, determine the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Two players play alternately on a 5 x 5 board. The first player always enters a 1 into an empty square and the second player always enters a 0 into an empty square. When the board is full, the sum of the numbers in each of the nine 3 x 3 squares is calculated and the first player's score is the largest such sum. What is the largest score the first player can make if the second player plays so as to minimize this score?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
What is the limit superior of:

$$ \limsup_{x\rightarrow 10} 1*\mathbb{I}_{x<10}(x)+2*\mathbb{I}_{x=10}(x)+\frac{1}{2}*\mathbb{I}_{x>10}(x) ? $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
ABC is a triangle. D is the midpoint of BC. E is a point on the side AC such that BE = 2AD. BE and AD meet at F and ∠FAE = 60°. Find ∠FEA.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Evaluate the following limit:

$$\lim\limits_{N \to \infty}{\oint_{C_N}{\frac{z}{\exp(z)-1}}\cdot\frac{dz}{z^{2\cdot k+2}}},$$

where $C_N=\{z\in \mathbb C : |z|=2\pi(N+\frac{1}{2}) \}$ and $k\in \mathbb N$ is fixed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Professor Artie, a professor of art history, is organizing an art exhibition to demonstrate the concept of artistic subjectivity and interpretation. She has selected 5 different pieces of art for the exhibition, each representing a unique interpretation of a historical event. To highlight the diversity of interpretations, she decides to display each piece in 3 different rooms, each room offering a different lighting setup that influences how the art is perceived. 

Professor Artie wants to ensure that each piece is viewed 7 times over the course of the exhibition by different groups of students. If she plans to have 2 tours per room each day, how many days will the exhibition last if she wants each piece to be viewed the desired number of times?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
已知 $[x]$ 为高斯函数，求满足 $\left[\frac{x}{2}\right]+\left[\frac{x}{3}\right]+\left[\frac{x}{5}\right]=x$ 的解的组数。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
There are three boxes of stones. Each hour, Sisyphus moves a stone from one box to another. For each transfer of a stone, he receives from Zeus a number of coins equal to the number of stones in the box from which the stone is drawn minus the number of stones in the recipient box, with the stone Sisyphus just carried not counted. If this number is negative, Sisyphus pays the corresponding amount (and can pay later if he is broke).

After 1000 years, all the stones lie in their initial boxes. What is the greatest possible earning of Sisyphus at that moment?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The quantities of rice in warehouses A and B are the same. Dad, Mom, and Yangyang each need 10 days, 12 days, and 15 days respectively to transport all the rice from one warehouse on their own. Dad and Mom start transporting the rice from warehouses A and B respectively at the same time. Yangyang first helps Mom and then helps Dad. As a result, they finish transporting all the rice from both warehouses at the same time. How many days did Yangyang help Mom?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
The polynomial $P(x)=a_0+a_1x+a_2x^2+...+a_8x^8+2009x^9$ has the property that $P(\tfrac{1}{k})=\tfrac{1}{k}$ for $k=1,2,3,4,5,6,7,8,9$.  There are relatively prime positive integers $m$ and $n$ such that $P(\tfrac{1}{10})=\tfrac{m}{n}$.  Find $n-10m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20090}
Let $k = 2^6 \cdot 3^5 \cdot 5^2 \cdot 7^3 \cdot 53$. Let $S$ be the sum of \(\frac{\text{gcd}(m,n)}{\text{lcm}(m,n)}\) over all ordered pairs of positive integers \((m, n)\) where \(mn = k\). If $S$ can be written in simplest form as \(\frac{r}{s}\), compute $r + s$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{895}
A line passing through point $P(-\sqrt{3},-1)$ and point $A(-2,0)$ intersects with the circle $x^2+y^2=1$ at a common point. The range of the slope angle of line $l$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[0, \frac{\pi}{3} \right]}
How many positive integers are within $810$ of exactly $\lfloor \sqrt{810} \rfloor$ perfect squares? 

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
57
}
Determine whether the equation $x(3y-5)=y^2+1$ has any integral solutions. If so, provide one such solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1)}
Let $\vartriangle ABC$ have $AB = 14$, $BC = 30$, $AC = 40$ and $\vartriangle AB'C'$ with $AB' = 7\sqrt6$, $B'C' = 15\sqrt6$, $AC' = 20\sqrt6$ such that $\angle BAB' = \frac{5\pi}{12}$ . The lines $BB'$  and $CC'$  intersect at point $D$. Let $O$ be the circumcenter of $\vartriangle BCD$, and let $O' $ be the circumcenter of $\vartriangle B'C'D$. Then the length of segment $OO'$ 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 $a+b+c+d$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Given that for \( |x| \leqslant 1 \), the inequality
\[ 
2px^2 + qx - p + 1 \geqslant 0 
\]
always holds, find the maximum value of \( p + q \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
棋盘上标有第 $0,1,2, \cdots, 100$ 站, 棋子开始时位于第 0 站, 棋手抛郑均匀硬币走跳棋游戏, 若掷出正面,棋子向前跳出一站; 若郑出反面, 棋子向前跳出两站, 直到跳到第 99 站（胜利大本营）或第 100 站（失败集中营) 时, 游戏结束, 设棋子跳到第 $n$ 站的概率为 $P_{n}$.
求 $P_{3}$ 的值;
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{5}{8}$}
Find all polynomials \( p(x) \) such that \( p(x+c) = p(x) + c \) for any constant \( c \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x + k}
The number  $125$  can be written as a sum of some pairwise coprime integers larger than  $1$ . Determine the largest number of terms that the sum may have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Determine a lower bound for the Hausdorff measure \( \mathcal{H}^s([0,1]) \) for \( s \in [0,1) \) using a covering strategy different from the standard one.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
A photo archive is arranged in identical albums, with exactly 4 photos on each page. The 81st photo ends up on the 5th page of one album, and the 171st photo ends up on the 3rd page of another. How many photos does each album hold?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
consider a $2008 \times 2008$ chess board. let $M$ be the smallest no of rectangles that can be drawn on the chess board so that sides of every cell of the board is contained in the sides of one of the rectangles. find the value of $M$. (eg for $2\times 3$ chessboard, the value of $M$ is 3.)

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2009}
Given the sample data: $110$, $120$, $120$, $120$, $123$, $123$, $140$, $146$, $150$, $162$, $165$, $174$, $190$, $210$, $235$, $249$, $280$, $318$, $428$, $432$, find the $75$th percentile.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{242}
Given the function $f\left( x \right)=\log_{\frac{1}{e}}\left( x^{2}+\frac{1}{e} \right)-\left| \frac{x}{e} \right|$, find the range of $x$ that makes $f\left( x+1 \right) < f\left( 2x-1 \right)$. Express your answer in interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0,2)}
函数 $f(x)=\frac{1-2 x}{\left(1-x+x^{2}\right)^{2}}$, 则 $\left[\frac{f^{(2022)}(0)}{2022!}\right]=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2023}
The coordinates of the point $\left(-2,3\right)$ with respect to the $x$-axis are ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(-2,-3\right)}
In the subtraction shown, $K, L, M$, and $N$ are digits. What is the value of $K+L+M+N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given an ellipse $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with left and right foci $F_{1}$, $F_{2}$. There exists a point $M$ in the first quadrant of ellipse $C$ such that $|MF_{1}|=|F_{1}F_{2}|$. Line $F_{1}M$ intersects the $y$-axis at point $A$, and $F_{2}A$ bisects $\angle MF_{2}F_{1}$. Find the eccentricity of ellipse $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5} - 1}{2}}
Mrs. Carter's algebra class consists of 48 students. Due to a schedule conflict, 40 students took the Chapter 5 test, averaging 75%, while the remaining 8 students took it the following day, achieving an average score of 82%. What is the new overall mean score of the class on the Chapter 5 test? Express the answer as a percent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{76.17\%}
Pipe A and Pipe B fill water into a tank of capacity 1000 litres, at a rate of 200 L/min and 50 L/min, respectively. Pipe C drains at a rate of 25 L/min. Pipe D has a varying flow rate, initially at 75 L/min for the first two cycles, then increasing by 25 L/min for each subsequent cycle. 

The process is as follows: Pipe A is open for 1 minute and closed, then Pipe B is open for 2 minutes and closed. Pipe C is opened and drains for another 2 minutes. After that, Pipe D is opened for 1 minute. This process is repeated until the tank is filled. How long will it take to fill the tank?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Given \((1+x-x^2)^{10} = a_0 + a_1 x + a_2 x^2 + \cdots + a_{20} x^{20}\), find \( a_0 + a_1 + 2a_2 + 3a_3 + \cdots + 20a_{20} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-9}
Compute Compute \( \left\lbrack  {\mathop{\sum }\limits_{{k = {2018}}}^{\infty }\frac{{2019}! - {2018}!}{k!}}\right\rbrack \) . (The notation \( \lceil x\rceil \) denotes the least integer \( n \) such that \( n \geq  x \) .)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2019
}
3. Let $O$ be the center of the circumcircle of triangle $ABC$, points $O$ and $B$ lie on opposite sides of line $AC$, $\angle AOC = 60^\circ$. Find the angle $AMC$, where $M$ is the center of the incircle of triangle $ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{165}
Find the smallest positive integer $k$ such that there exist positive integers $M,O>1$ satisfying
\[ (O\cdot M\cdot O)^k=(O\cdot M)\cdot \underbrace{(N\cdot O\cdot M)\cdot (N\cdot O\cdot M)\cdot \ldots \cdot (N\cdot O\cdot M)}_{2016\ (N\cdot O\cdot M)\text{s}}, \]where $N=O^M$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2823
}
设 a_1,a_2,\cdots,a_{2023} 是正整数，满足对任意 1\leq k\leq 2023，均有a_k=\ds{i=1}{2023}|a_k-a_i|.求 a_1+a_2+\cdots+a_{2023} 的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2046264}
Determine whether the following limit exists as \( z \) approaches \(-3\):
\[ \lim_{z\rightarrow -3}\frac{\log(z+3)}{e^{\cos(z)}}(z+3) \]
where \( \log(z) \) is the complex logarithm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x) = x^3 - ax^2 + bx - 65$. For each possible combination of $a$ and $b$, let $p_{a,b}$ be the sum of the zeroes of $P(x)$. Find the sum of the $p_{a,b}$'s for all possible combinations of $a$ and $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80
}
In a game, two players, Sam and Joe, each pick a nonnegative integer at random. The probability that the number is in any set $A$ is given by a finitely additive function $\mu$ defined on $2^X$, where $\mu(A) = 0$ for every finite set $A$ and $\mu(X) = 1$, with $X = \mathbb{N}$. The player with the larger number wins. A coin is tossed to determine whose number is revealed first. It lands on heads, so Sam's number is revealed, but Joe's number remains unknown. Based on this information, who is more likely to win?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Joe}
A loaf of sausage is marked with thin transverse rings. If you cut along the red rings, you get 5 pieces; if along the yellow rings, you get 7 pieces; and if along the green rings, you get 11 pieces. How many pieces of sausage will you get if you cut along the rings of all three colors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}