inclusionAI/Ring-lite-rl-data huggingface dataset:
设复数 $x, y, z$ 满足: $|x+2 y+3 z|=1$ 。  
求 $|x|^{2}+|y|^{2}+|z|^{2}+\left|x^{2}+y^{2}+z^{2}\right|$ 的最小值.  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{8}
4. How many sets of three consecutive three-digit natural numbers have a product that is divisible by 120?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{336}
In a 100 x 100 grid, Petya and Vasya play a game where Petya starts by coloring one cell black. On each subsequent turn, a player can color a vertical or horizontal white strip of size $1 \times n$ black, where $n$ is either equal to or one more than the number of cells colored by the previous player. The player who cannot make a move loses. Determine the winner if both play optimally.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Vasya}
In a commutative ring $R$ with unity, consider two distinct maximal ideals $I$ and $J$. Determine if the ideal generated by $I^2$ and $J$, denoted by $(I^2, J)$, is equal to the entire ring $R$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{R}
Let $g:[a,b]\to\mathbb{R}$ be a Riemann-integrable function. If there exists a dense subset $A\subseteq[a,b]$ such that $g(x)=0$ for all $x\in A$, is it always true that $\int_{a}^{b} g = 0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine whether the expression \( \frac{1^\alpha + 2^\alpha + \cdots + n^\alpha}{n^{\alpha+1}} \) diverges to infinity as \( n \to \infty \) for \( \alpha \in (-\infty, -1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
What is the sum of the six positive integer factors of 30, each multiplied by 2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
Given an arithmetic sequence $\{a_n\}$ with the first term being a positive number, and the sum of the first $n$ terms is $S_n$. If $a_{1006}$ and $a_{1007}$ are the two roots of the equation $x^2 - 2012x - 2011 = 0$, then the maximum value of the positive integer $n$ for which $S_n > 0$ holds is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2011}
What is the smallest value that the ratio of the areas of two isosceles right triangles can have, given that three vertices of one of the triangles lie on three different sides of the other triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/5}
Calculate the number of different ways to arrange 2014 points into either a triangular or a trapezoidal point lattice, where the difference in the number of points between two adjacent rows is always 1, and there are at least two layers. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In the first 3 seconds, Kevin can dribble a basketball 13 times while moving straight ahead. As he continues, he encounters a zigzag-shaped path with obstacles: during the next 5 seconds, he only dribbles 18 times while navigating the path. If Kevin continues to dribble for the remainder of the 27 seconds, slowing down at an arithmetic decreasing rate of 3 fewer dribbles for each additional 5 seconds as he navigates through three more increasingly complex obstacles, how many times will he dribble the basketball in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74}
The military kitchen needs 1000 jin of rice and 200 jin of millet for dinner. Upon arriving at the rice store, the quartermaster finds a promotion: "Rice is 1 yuan per jin, with 1 jin of millet given for every 10 jin purchased (fractions of 10 jins do not count); Millet is 2 yuan per jin, with 2 jins of rice given for every 5 jin purchased (fractions of 5 jins do not count)." How much money does the quartermaster need to spend to buy enough rice and millet for dinner?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1200}
Two swimmers start at opposite sides of a 90-foot swimming pool. One swims at 3 feet per second, while the other swims at 2 feet per second. They swim back and forth for 12 minutes without taking any time to turn around. Calculate the number of times they meet. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Problem 11-2. Mr. $A$ stood at the point with coordinates $(0,0)$ for an hour. During the same hour, moving uniformly and in a straight line, Mr. $B$ traveled from the point $(22,0)$ to the point $(2,20)$. During this same hour, Miss $C$, also moving uniformly and in a straight line, traveled from the point $(30,4)$ to the point $(0,24)$. How many times during the observation period did the area of triangle $A B C$ take on integer values? The initial and final moments are included.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53}
Consider the set of $30$ parabolas defined as follows: all parabolas have as focus the point $(0,0)$ and the directrix lines have the form $y=ax+b$ with $a$ and $b$ integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{810}
设 S 是一个 35 元集合，\mathcal{F} 是一个由一些 S 到自身的映射组成的集合. 对正整数 k，称 \mathcal{F} 具有性质 P(k)，如果对任意 x,y\in S，都存在 \mathcal{F} 中的 k 个映射f_1, f_2, \cdots, f_k（可以相同），使得f_k(\cdots(f_2(f_1(x)))\cdots)=f_k(\cdots(f_2(f_1(y)))\cdots).求最小的正整数 m，使得所有具有性质 P(2019) 的集合 \mathcal{F} 都具有性质 P(m).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{595}
The vertices of an equilateral triangle lie on the hyperbola $xy = 4$, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1728}
A middle school holds a sports meeting, with participation by grade. There are three events: high jump, long jump, and 100m sprint, with the top three in each event receiving points: 5 points for first place, 3 points for second place, and 1 point for third place. It is known that the total scores of the seventh and eighth grades are equal, and they are tied for first place. Moreover, the number of students from the eighth grade who placed in the top three is twice that of the seventh grade. What is the total score of the ninth grade?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
9. Cars A and B depart from locations $A$ and $B$ respectively at the same time, and travel back and forth between $A$ and $B$ at a constant speed. If after the first meeting, Car A continues to drive for 4 hours to reach $B$, while Car B only drives for 1 hour to reach $A$, then when the two cars meet for the 15th time (excluding meetings at $A$ and $B$), they have driven $\qquad$ hours.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{86}
If $q^k n^2$ is an odd perfect number with special prime $q$, is $\sigma(q^k)$ coprime to $\sigma(n^2)$? Assume $q \equiv k \equiv 1 \pmod{4}$ and $\gcd(q,n)=1$. The function $\sigma(x)$ denotes the sum of divisors of $x$. Determine if the conjecture $\gcd(\sigma(q^k),\sigma(n^2))>1$ holds true.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\gcd(\sigma(q^k), \sigma(n^2)) > 1}
Find all all positive integers x,y,and z satisfying the equation $x^3=3^y7^z+8$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(11, 3, 2).}
Xiao Ming places several chess pieces into a $3 \times 3$ grid of square cells. Each cell can have no pieces, one piece, or more than one piece. After calculating the total number of pieces in each row and each column, we obtain 6 different numbers. What is the minimum number of chess pieces required to achieve this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let  $a_1 = \sqrt 7$  and  $b_i = \lfloor a_i \rfloor$ ,  $a_{i+1} = \dfrac{1}{b_i - \lfloor b_i \rfloor}$  for each  $i\geq i$ . What is the smallest integer  $n$  greater than  $2004$  such that  $b_n$  is divisible by  $4$ ? ( $\lfloor x \rfloor$  denotes the largest integer less than or equal to  $x$ )
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2005}
Find the number of ordered integer pairs \((a, b)\) such that the equation \(x^{2} + a x + b = 167 y\) has integer solutions \((x, y)\), where \(1 \leq a, b \leq 2004\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2020032}
Evaluate the following double integral:

$$\int_0^{\pi / 2} \int_y^{2 y} \frac{\sin x}{x} d x d y+\int_{\pi / 2}^\pi \int_y^\pi \frac{\sin x}{x} d x d y$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the series: $$\sum\limits_{n = 1}^\infty {(2n+1)\frac{{{{(\alpha )}^{2n}}}}{{(2n)!}}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\alpha \sinh \alpha + \cosh \alpha - 1\)}
Squares $ABCD$ and $EFGH$ are congruent, $AB=10$, and $G$ is the center of square $ABCD$. The area of the region in the plane covered by these squares is

$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{175}
Minimize \(x^2 - y^2 - z^2\) subject to the constraint \(x^4 + y^4 + z^4 \leq 1\) using the Karush-Kuhn-Tucker (KKT) conditions. Determine the KKT points and find the minimum value of the objective function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\sqrt{2}}
A construction team has 6 projects (A, B, C, D, E, F) that need to be completed separately. Project B must start after project A is completed, project C must start after project B is completed, and project D must immediately follow project C. Determine the number of different ways to schedule these 6 projects.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Find the time period for which a certain sum of money was invested at 5% per annum if the amount being Rs. 896 and the principal is Rs. 799.9999999999999?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.4}
Let $ABCDEFG$ be a regular heptagon with center $O$.  Let $M$ be the centroid of triangle $ABD$.  Find $\cos^2 \angle GOM$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{8}}
How many pairs of positive integers(a,b)are there such that the roots of polynomial\( {x}^{2} - {ax} - b \) are not greater than 5 ?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
50
}
For every positive integer  $n$ , determine the biggest positive integer  $k$  so that  $2^k |\ 3^n+1$ 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ k = 2 }
Let \(E\) be a locally compact second-countable metric space, and let \(E^\ast = E \cup \{\infty\}\) denote the Alexandroff one-point extension. Consider a sequence \((x_n)_{n \in \mathbb{N}} \subseteq E^\ast\) and a point \(x \in E^\ast\) such that \(x_n \xrightarrow{n \to \infty} x\). Let \(B_n \subseteq E^\ast\) be open sets with \(x_n \in B_n\) for each \(n \in \mathbb{N}\) and \(\bigcap_{n \in \mathbb{N}} B_n = \{\infty\}\). Assume that the sequence \((B_n)_{n \in \mathbb{N}}\) is nonincreasing. Determine if it follows that \(x_0 = \infty\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Twelve chess players played a round-robin tournament. Each player then wrote 12 lists. In the first list, only the player himself was included, and in the $(k+1)$-th list, the players included those who were in the $k$-th list as well as those whom they defeated. It turned out that each player's 12th list differed from the 11th list. How many draws were there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
How many roots does the equation \(\sqrt{14-x^{2}}(\sin x-\cos 2x)=0\) have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine whether the limit \( \lim_{x\to a^+}\frac{\cos{x}\ln(x-a)}{\ln(e^x-e^a)} \) exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\cos a}
How many integer numbers are no greater than $2004$, with remainder $1$ when divided by $2$, with remainder $2$ when divided by $3$, with remainder $3$ when divided by $4$, and with remainder $4$ when divided by $5$?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Find the number of partitions of the set $\{1,2,3,\cdots ,11,12\}$ into three nonempty subsets such that no subset has two elements which differ by $1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1023}
If the circle $(x-a)^2+(x+4)^2=9$ is tangent to the y-axis, then $a=$ ___.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm3}
A large tanker can be filled by two pipes A and B in 60 minutes and some minutes respectively. It takes 29.999999999999993 minutes to fill the tanker from an empty state if B is used for half the time and A and B fill it together for the other half. How many minutes does it take for pipe B to fill the tanker alone?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Sixty $1 \times 1 \times 1$ cubes are joined face to face in a single row and placed on a table. Each cube has a total of 6 faces. When a single cube is placed on the table, the bottom face is not exposed, leaving 5 exposed faces. However, when the cubes are joined in a row, some faces between them are not exposed. Calculate the total number of exposed $1 \times 1$ faces for the 60 joined cubes. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{182}
For any positive integer \( k \), let \( f_{1}(k) \) be the square of the sum of the digits of \( k \) when written in decimal notation. For \( n > 1 \), let \( f_{n}(k) = f_{1}\left(f_{n-1}(k)\right) \). What is \( f_{1992}\left(2^{1991}\right) \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
Let $p(x)$ be defined on $2 \le x \le 10$ such that $$p(x) = \begin{cases} x + 1 &\quad \lfloor x \rfloor\text{ is prime} \\ p(y) + (x + 1 - \lfloor x \rfloor) &\quad \text{otherwise} \end{cases}$$ where $y$ is the greatest prime factor of $\lfloor x\rfloor.$ Express the range of $p$ in interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[3,7] \cup [8,9)}
A dealer purchases 15 articles of type A for Rs. 25 each and 20 articles of type B for Rs. 35 each. If he gets a discount of 5% on bulk purchase of type A articles and a 10% discount on type B articles, he then sells 12 articles of type A and 18 articles of type B. On the day of sale, due to the fluctuating market, he has to increase the selling price of type A articles by 10% which is Rs. 33 each but had to decrease the selling price of type B articles by 15% which becomes Rs. 45 each. Find the overall profit or loss percentage the dealer made from these transactions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22.28\%}
Given two permutations of the set \( \{1, 2, \ldots, N\} \) where \( 3 \leq N \leq 1000 \), find the number of permutations \( X_1, X_2, \ldots, X_N \) such that when arranged in a 3xN matrix with the two given permutations, each column contains unique elements.

For example, for \( N = 4 \):

First permutation: \( \begin{pmatrix}3 & 1 & 2 & 4\end{pmatrix} \).
Second permutation: \( \begin{pmatrix}2 & 4 & 1 & 3\end{pmatrix} \).

Determine the number of possible permutations for the third row \( \begin{pmatrix}X_1 & X_2 & X_3 & X_4\end{pmatrix} \) such that:
\( X_1 \neq 3, 2 \),
\( X_2 \neq 1, 4 \),
\( X_3 \neq 2, 1 \),
\( X_4 \neq 4, 3 \).

The answer for this example is 2, with possible permutations \( \begin{pmatrix}1 & 3 & 4 & 2\end{pmatrix} \) and \( \begin{pmatrix}4 & 2 & 3 & 1\end{pmatrix} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Five people stand in a line, each wearing a different hat numbered $1, 2, 3, 4, 5$. Each person can only see the hats of the people in front of them. Xiao Wang cannot see any hats; Xiao Zha can only see hat $4$; Xiao Tian does not see hat $3$, but can see hat $1$; Xiao Yan can see three hats, but not hat $3$; Xiao Wei can see hat $3$ and hat $2$. What number hat is Xiao Wei wearing?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
If the graph of the function $f(x)$ passes through the point $(0, 1)$, then the graph of the inverse function of $f(x+3)$ must pass through the point ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, -3)}
Find a condition on the real numbers \(a, b, c\) such that the following equality holds:

\[ \alpha = \beta \]

where:

\[ \alpha = \sqrt{a} + \sqrt{2b + 2\sqrt{a}} \]
\[ \beta = \sqrt{b + 2\sqrt{c}} + \sqrt{a + b - 2\sqrt{c} + 2\sqrt{ab - 2a\sqrt{c}}} \]

and \(a = b^2 - 4c\). Assume all square roots are well-defined.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a = b^2 - 4c}
Find the greatest integer \( n \) ,such that there are \( n + 4 \) points \( A,B,C,D,{X}_{1},\ldots ,{X}_{n} \) in the plane with \( {AB} \neq  {CD} \) that
 satisfy the following condition: for each \( i = 1,2,\ldots ,n \) triangles \( {AB}{X}_{i} \) and \( {CD}{X}_{i} \) are equal
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
The greatest integer \( n \) is 4.

}
复平面上，非零复数 $z_{1}, z_{2}$ 在以 $i$ 为圆心， 1 为半径的圆上， $\overline{z_{1}} z_{2}$ 的实部为零， $z_{1}$ 的辐角主值为 $\frac{\pi}{6}$ ，则 $z_{2}=$。请注意，$z_{2}$ 的原始答案为$-\frac{\sqrt{3}}{2}+\frac{3}{2}i$，请给出虚部分子和分母之和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
For all \( n \in \mathbb{N}^{*} \), let \( p_{n} \) be the number of sequences of \( A \) and \( B \) of length \( n \) containing neither \( AAAA \) nor \( BBB \). Calculate

$$
\frac{p_{2004}-p_{2002}-p_{1999}}{p_{2000}+p_{2001}}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine whether \( \frac{\partial}{\partial x} f(x,y(x)) \) is equal to \( x \) or \( 2x \) given that \( f(x,y) = xy^2 \) and \( y^2 = x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x}
Given \(\frac{ab}{a+b}=3\), \(\frac{bc}{b+c}=6\), and \(\frac{ac}{a+c}=9\), determine the value of \(\frac{c}{ab}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{35}{36}}
Three. (50 points) Given that the difference between any two adjacent interior angles of a convex $n$-sided polygon is $20^{\circ}$. Find the maximum value of $n$.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
A set $S$ consists of $k$ sequences of $0,1,2$ of length $n$. For any two sequences $(a_i),(b_i)\in S$ we can construct a new sequence $(c_i)$ such that $c_i=\left\lfloor\frac{a_i+b_i+1}2\right\rfloor$ and include it in $S$. Assume that after performing finitely many such operations we obtain all the $3n$ sequences of $0,1,2$ of length $n$. Find the least possible value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n + 1}
Solve the following equation for $x , y , z \in \mathbb{N}$ :
\[\left (1+ \frac{x}{y+z} \right )^2+\left (1+ \frac{y}{z+x} \right )^2+\left (1+ \frac{z}{x+y} \right )^2=\frac{27}{4}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,1,1)}
Prisha writes down one integer three times and another integer two times, with their sum being $105$, and one of the numbers is $15$. Calculate the other number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find the number of ways to assign 15 students to 5 classes such that each class has at least 2 students. Ensure that any overcounting errors are corrected in your solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9771762000}
Find the sum of all values of $x$ such that the set $\{107, 122,127, 137, 152,x\}$ has a mean that is equal to its median.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{381}
The altitudes of triangle \(ABC\) drawn from vertices \(A\) and \(C\) intersect at point \(M\). Find \(\angle AMC\) if \(\angle A = 70^\circ\) and \(\angle C = 80^\circ\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150^\circ}
Find the value of ( 85 + 32 / 113 ) × 113. What is the result?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9637}
Let $\{a_k\}_{k=1}^{2011}$ be the sequence of real numbers defined by $a_1=0.201,$ $a_2=(0.2011)^{a_1},$ $a_3=(0.20101)^{a_2},$ $a_4=(0.201011)^{a_3}$, and in general,

$$
a_k=\begin{cases}(0.\underbrace{20101\cdots 0101}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is odd,}\\(0.\underbrace{20101\cdots 01011}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is even.}\end{cases}
$$

Rearranging the numbers in the sequence $\{a_k\}_{k=1}^{2011}$ in decreasing order produces a new sequence $\{b_k\}_{k=1}^{2011}$. What is the sum of all integers $k$, $1\le k \le 2011$, such that $a_k=b_k?$



Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1341
}
Determine whether the series \(\sum_{n=1}^\infty n^r \exp\left(-k\sum_{m=1}^n \frac{1}{m}\right)\) converges when \(r-k < -1\) or \(r<k\), using the approximation \(\frac{u_{n+1}}{u_n} \approx 1 + \frac{r-k}{n}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r - k < -1}
A fair six-sided die has faces numbered $1, 2, 3, 4, 5, 6$. The die is rolled four times, and the results are $a, b, c, d$. What is the probability that one of the numbers in the set $\{a, a+b, a+b+c, a+b+c+d\}$ is equal to 4?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{343}{1296}$}
In a game involving a 5x9 rectangular grid, Petya starts by cutting the grid into two parts along the border lines of the cells, forming an "L" shape with two perpendicular segments. Vasya then cuts one of the resulting figures in the same manner, followed by Petya with one of the three resulting figures, and so on. The player who cannot make a move loses. Determine the winner of the game, Petya or Vasya, regardless of the moves made by the other player.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Petya}
Determine the number of ordered quintuples $(a,b,c,d,e)$ of integers with $0\leq a<$ $b<$ $c<$ $d<$ $e\leq 30$ for which there exist polynomials $Q(x)$ and $R(x)$ with integer coefficients such that\[x^a+x^b+x^c+x^d+x^e=Q(x)(x^5+x^4+x^2+x+1)+2R(x).\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5208}
The Minnesota Twins face the New York Mets in the 2006 World Series. Assuming the two teams are evenly matched (each has a \(0.5\) probability of winning any game), find the probability that the World Series (a best of 7 series of games which lasts until one team wins four games) will require the full seven games to determine a winner. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}