inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the norm of the linear operator $K(f)$, where $K(f)(x) = f(x) + \int_{0}^{x} f(t)dt$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the flux of the vector field \( \bar{F} = \langle x, y, x \rangle \) over the surface \( z = 1 - x - y \) in the first octant using the upward unit normal vector \( \bar{n} = \frac{1}{\sqrt{3}} \langle -1, -1, 1 \rangle \). The flux is given by the integral \( \int\int_S \bar{F} \cdot \bar{n} \, dA \), where \( dS = \sqrt{3} \, dA \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{6}}
If \((2 x+4)^{2 n}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{2 n} x^{2 n}\) for \(n \in \mathbf{N}^{+}\), then the remainder when \(a_{2}+a_{4}+\cdots+a_{2 n}\) is divided by 3 is .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A number, when divided by 44, gives 432 as quotient and a certain remainder. When dividing the same number by 38, the remainder is 8. What is the remainder in the first division?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Matrices  $A$ ,  $B$  are given as follows.
\[A=\begin{pmatrix} 2 & 1 & 0  1 & 2 & 0  0 & 0 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 2 & 0  2 & 4 & 0  0 & 0 & 12\end{pmatrix}\]
Find volume of  $V=\{\mathbf{x}\in\mathbb{R}^3 : \mathbf{x}\cdot A\mathbf{x} \leq 1 < \mathbf{x}\cdot B\mathbf{x} \}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
Evaluate

\[ \int_e^{e^{2009}} \frac{1}{x}\left\{1+\frac{1-\ln x}{\ln x\cdot \ln \frac{x}{\ln (\ln x)}}\right\}\ dx\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2008-\ln\left(2009-\ln{2009}\right)}
Find a positive integer \( n \) less than 2006 such that \( 2006n \) is a multiple of \( 2006 + n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1475}
Frieda the frog begins a sequence of hops on a \(3 \times 3\) grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite edge. For example, if Frieda begins in the center square and makes two hops "up", the first hop would place her in the top row middle square, and the second hop would cause Frieda to jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the center square, makes at most four hops at random, and stops hopping if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops? 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{57}
What is the molecular weight of 7 moles of a certain compound if the molecular weight of that compound is 1176?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8232}
Find the kernel of the evaluation map \( g: \mathbb{Z}[x] \longrightarrow \mathbb{Z}[3^{1/4}] \) where \( x = 3^{1/4} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x^4 - 3)}
Given the sequences \(a_n=\int_0^1 (1-x^2)^n \,dx\) and \(b_n=\int_0^1 (1-x^3)^n \,dx\) where \(n \in \mathbb{N}\), find the limit:
\[ \lim_{n\to \infty}(10\sqrt[n]{a_n}+5\sqrt[n]{b_n}). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Find the set of integer values of $n$ for which the inequality $3 \uparrow^{n} n < 2\uparrow^{n+2} 2$ is true.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{0, 1\}}
Determine the maximum value of \( \lambda \) such that for any distinct positive real numbers \( a, b, c \), the inequality \[ \frac{a^{3}}{(b-c)^{2}}+\frac{b^{3}}{(c-a)^{2}}+\frac{c^{3}}{(a-b)^{2}} \geqslant \lambda(a+b+c) \] holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Consider a convex polygon $\mathcal{P}$ in space with perimeter $20$ and area $30$. What is the volume of the locus of points that are at most $1$ unit away from some point in the interior of $\mathcal{P}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$60 + \frac{34\pi}{3}$}
Compare the values of $P(28, 19, 1, 2)$ and $P(30, 10, 2, 7)$, where $P(n, k, a, b)$ denotes the number of arrangements of $n$ indistinguishable balls into $k$ distinct buckets such that there are no less than $a$ and no more than $b$ balls in each bucket. Which value is higher?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{P(28, 19, 1, 2)}
A cube with dimensions \(20 \times 20 \times 20\) is divided into 8000 unit cubes. Each unit cube has a number written on it. In every row and column of 20 unit cubes, parallel to any edge of the cube, the sum of the numbers is 1. In one of these unit cubes, the number 10 is written. Three \(1 \times 20 \times 20\) layers pass through this unit cube, parallel to the faces of the cube. Calculate the sum of all the numbers outside of these three layers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{333}
Side $BC$ of quadrilateral $ABCD$ is the diameter of the circle circumscribed around this quadrilateral. Find side $AB$ if $BC=8$, $BD=4\sqrt{2}$, and $\angle DCA : \angle ACB = 2 : 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(\sqrt{6} - \sqrt{2})}
Find the derivative of the implicit function $y(x)$ defined by the equation $$ y^x = x^y - x $$ at the point $(1,1)$. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of equally spaced data points needed to uniquely specify a three harmonic Fourier series for a periodic function with a period of \(\frac{\pi}{2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine the smallest integer $k$ such that a $k$-vector with binary entries can be used to communicate any integer between 1 and 1000 to a friend, even after one of its components is altered by a function $f_i$, which sets the $i$-th component to 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Dragoons take up \(1 \times 1\) squares in the plane with sides parallel to the coordinate axes such that the interiors of the squares do not intersect. A dragoon can fire at another dragoon if the difference in the \(x\)-coordinates of their centers and the difference in the \(y\)-coordinates of their centers are both at most 6, regardless of any dragoons in between. For example, a dragoon centered at \((4,5)\) can fire at a dragoon centered at the origin, but a dragoon centered at \((7,0)\) cannot. A dragoon cannot fire at itself. What is the maximum number of dragoons that can fire at a single dragoon simultaneously?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{168}
Given that the complex number $z$ and $(z-3)^2 + 5i$ are both purely imaginary, find $z=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm 3i}
Let $G$ be a graph with 100 vertices numbered from 1 to 100. Two vertices $i$ and $j$ are adjacent if $|i-j|=8$ or $|i-j|=12$. What is the number of connected components in the graph $G$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $T:L^p[0,1]\to L^p[0,1]$ be a linear operator defined by $T(f(x))=x^kf(x)$. Determine the operator norm of $T$. \(1 \leq p < \infty\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Five positive integers are listed in increasing order. The difference between any two consecutive numbers in the list is three. The fifth number is a multiple of the first number. How many different such lists of five integers are there? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
For which values of $k$ is the blow-up $X_k = Bl_{p_1,...,p_k}\mathbb{P}^2$ of $\mathbb{P}^2$ at $k$ general points a Mori Dream Space?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k \leq 8}
Let \( \omega_1 \) and \( \omega_2 \) be circles with radii 3 and 12, respectively, and externally tangent at point \( P \). A common external tangent intersects \( \omega_1 \) and \( \omega_2 \) at points \( S \) and \( T \), respectively, and the common internal tangent at point \( Q \). Define \( X \) to be the point on \( \overrightarrow{QP} \) such that \( QX = 10 \). If \( XS \) and \( XT \) intersect \( \omega_1 \) and \( \omega_2 \) a second time at points \( A \) and \( B \), respectively, determine \( \tan\angle APB \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
My frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score. The game ends after one of the two teams scores three points (total, not necessarily consecutive). If every possible sequence of scores is equally likely, what is the expected score of the losing team?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/2}
在$\Delta ABC$中，角$A,B,C$所对的边分别为$a,b,c$，若$b\cos C=\left( 2a-c \right)\sin \left( B+\frac{\pi }{2} \right)$，且$b=\sqrt{3}$，记$h$为$AC$边上的高，则$h$的取值范围为__________．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\left( 0,\frac{3}{2} \right]$}
Let $A B C$ be a triangle and $D, E$, and $F$ be the midpoints of sides $B C, C A$, and $A B$ respectively. What is the maximum number of circles which pass through at least 3 of these 6 points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
The mass of the first cast iron ball is $1462.5\%$ greater than the mass of the second ball. By what percentage less paint is needed to paint the second ball compared to the first ball? The volume of a ball with radius $R$ is $\frac{4}{3} \pi R^{3}$, and the surface area of a ball is $4 \pi R^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84\%}
3. Given a cube and 12 colors. Find the number of ways to paint the faces of this cube using these colors (each face in one color) such that adjacent faces are of different colors. Colorings that differ by a rotation are considered different.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{987360}
Determine the order of the Galois group of the residue field extension $k_i/\mathbf{F}_q$ in the context of the theorem relating the inertia index $f$ to the order of $q$ modulo $p$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f}
Let \( S = \{ 1, 2, 3, \ldots, 280 \} \). Find the smallest integer \( n \) such that each \( n \)-element subset of \( S \) contains five numbers which are pairwise relatively prime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{217}
Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the circle. How many triangles with all three vertices in the interior of the circle are created?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Let $(b_1, b_2, b_3, \ldots, b_{10})$ be a permutation of $(1, 2, 3, \ldots, 10)$ such that $b_1 > b_2 > b_3 > b_4 > b_5$ and $b_5 < b_6 < b_7 < b_8 < b_9 < b_{10}$. An example of such a permutation is $(5, 4, 3, 2, 1, 6, 7, 8, 9, 10)$. Find the number of such permutations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{126}
What shape is impossible to obtain as a cross-section when a plane intersects a cylinder? Provide your answer as the name of the shape.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Triangle}
Find the smallest possible value of \(x^2 + y^2\) given that \(x\) and \(y\) are positive integers such that \(x(x + 1)\mid y(y + 1)\), but neither \(x\) nor \(x + 1\) divides either \(y\) or \(y + 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{596}
Today is Barbara's birthday, and Alberto wants to give her a gift by playing the following game. The numbers $0, 1, 2, \ldots, 1024$ are written on a blackboard. 

1. First, Barbara erases $2^{9}$ numbers.
2. Then, Alberto erases $2^{8}$ numbers.
3. Next, Barbara erases $2^{7}$ numbers.
4. This process continues until there are only two numbers $a$ and $b$ left.

Barbara earns $|a-b|$ euros. 

Find the maximum number of euros that Barbara can always win, regardless of Alberto's strategy.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Determine the smallest integer \( n \) such that the Chebyshev interpolation \( p_n(x) \) of the function \( f(x) = \sin(x) \) on the interval \([0, \pi/2]\) satisfies \( ||f - p_n|| < 10^{-16} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
David has a unit triangular array of 10 points, 4 on each side. A looping path is a sequence \( A_{1}, A_{2}, \ldots, A_{10} \) containing each of the 10 points exactly once, such that \( A_{i} \) and \( A_{i+1} \) are adjacent (exactly 1 unit apart) for \( i=1,2, \ldots, 10 \). Here \( A_{11}=A_{1} \). Find the number of looping paths in this array.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Consider positive integers $a \leq b \leq c \leq d \leq e$. There are $N$ lists $a, b, c, d, e$ with a mean of 2023 and a median of 2023, in which the integer 2023 appears more than once, and in which no other integer appears more than once. What is the sum of the digits of $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
How many ordered triples (x,y,z) of integers satisfy the system of equations below?
\begin{array}{l} x^2-3xy+2y^2-z^2=31 \ -x^2+6yz+2z^2=44 \ x^2+xy+8z^2=100\ \end{array}
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
What is the maximum number of squares that can be marked on a chessboard so that from each marked square, it is possible to reach any other marked square in exactly two moves of a chess knight?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Simplify: $\frac{5 \sqrt[4]{5 r^{4}}}{\sqrt[4]{8 r^{2}}}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{5 \sqrt[4]{10 r^{2}}}{2}$}
Let $a_n$ denote the remainder when $(n+1)^3$ is divided by $n^3$; in particular, $a_1=0$. Compute the remainder when $a_1+a_2+\dots+a_{2013}$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{693}
设方程 $x y z=900$ 的全部正整数解为 $\left(x_{i}, y_{i}, z_{i}\right)(1 \leqslant i \leqslant n)$. 则 $\sum_{k=1}^{n}\left(x_{k c}+y_{k}+z_{k}\right)=\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22572}
A tournament with 100 competitors, each with different skill levels, will take place. The most skilled competitor always wins against the less skilled one. Each participant plays exactly twice, with two randomly chosen opponents (once with each). A competitor who wins both matches receives a medal. Determine the minimum number of medals that can be distributed in the tournament.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The distance traveled by the center \( P \) of a circle with radius 1 as it rolls inside a triangle with side lengths 6, 8, and 10, returning to its initial position.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let $F$ be the focus of the parabola $C$: $y^{2}=3x$. A line passing through $F$ with an inclination angle of $30^{\circ}$ intersects $C$ at points $A$ and $B$. Let $O$ be the origin of the coordinates, then the area of $\triangle OAB$ is \_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{4}}
There are real numbers  $a, b, c, d$  such that for all  $(x, y)$  satisfying  $6y^2 = 2x^3 + 3x^2 + x$ , if  $x_1 = ax + b$  and  $y_1 = cy + d$ , then  $y_1^2 = x_1^3 - 36x_1$ . What is  $a + b + c + d$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
In parallelogram $ABCD$, diagonal $BD$ is equal to 2, angle $C$ is $45^\circ$, and line $CD$ is tangent to the circumcircle of triangle $ABD$. Find the area of the parallelogram $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the number of $4$-digit numbers (in base $10$) that have non-zero digits and are divisible by $4$ but not by $8$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{729}
A $4 \times 4$ square piece of paper is cut into two identical pieces along its diagonal. The resulting triangular pieces of paper are each cut into two identical pieces. Each of the four resulting pieces is cut into two identical pieces. Each of the eight new resulting pieces is finally cut into two identical pieces. Calculate the length of the longest edge of one of these final sixteen pieces of paper.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
设定点 $A 、 B 、 C 、 D$ 是以 O 点为中心的正四面体的顶点，用 $\sigma$ 表示空间以直线 $O A$ 为轴满足条件 $\sigma(B)=C$ 的旋转，用 $\tau$ 表示空间关于 $O C D$ 所在平面的镜面反射，设 l 为过 $A B$ 中点与 $C D$ 中点的直线，用 $\omega$ 表示空间以 l 为轴的 $180^{\circ}$ 旋转，设 $\sigma \circ \tau$ 表示变换的复合，先作 $\tau$ ，再作 $\sigma$ ，则 $\omega$ 可以表示为什么？答案的形式为 k \circ m \circ n \circ p \circ m \circ n，其中 k、m、n、p 是待定系数，请给出 k + m + n + p 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
As shown in the figure, $AD$ is the altitude of $\triangle ABC$. From point $D$, draw $DE \perp AB$ at point $E$. Let $F$ be a point on the extension line of $ED$ such that $AF \perp CF$. Given that $AB = 13$, $BC = 14$, and $AC = 15$, find the perimeter of $\triangle ACF$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{450}{13}}
We take $100$ consecutive natural numbers $a_{1}, a_{2}, \ldots, a_{100}$. Determine the last two digits of the number $a_{1}^{8} + a_{2}^{8} + \cdots + a_{100}^{8}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Let $q(x) = 1 - (1-x) + (1-x)^2 - \cdots + (1-x)^{10}$. Let the roots of $q(x)$ be $\beta_i, i \in \{1, 2, \ldots, 10\}$. Find the value of $\sum_{i=1}^{10} \frac{1}{\beta_i}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let \( h_{n} = \sum_{k=0}^{n} \binom{n}{k} \frac{2^{k+1}}{k+1} \). If \( S= \sum_{n=0}^{\infty} \frac{h_{n}}{n!} \), find \( \lfloor S \rfloor \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
James is a freelancer who works with international clients. In January, he made $4000 from a US client. In February, he earned 50% more than he made in January from a European client when the exchange rate was 1 Euro = 1.2 Dollars. In March, his earnings decreased by 20% from his February earnings from a Japanese client, where the exchange rate was 110 Yen = 1 Dollar. In what he earned each month, how much has James made so far this year in Dollars?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$16960}
Determine the relationship between the functions \(f(n) = \frac{\log n}{\log\log n}\) and \(g(n) = \log(n - \log n)\) using the notations \(\Omega, \Theta, O\). Express the relationship as \(f(n) = K \cdot g(n)\), where \(K\) represents one of these notations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(n) = O(g(n))}
Derek finally gets his own allowance. He puts $2 away in January, some money away in February, $8 away in March, and follows this savings pattern through to December. He has $4096 to spare and save by December. How much money did he put away in February?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$4}
求最大的正整数 $n$ ，使得存在 $n$ 个正整数 $x_{1}<x_{2}<\cdots<x_{n}$, 满足:  
$$  
x_{1}+x_{1} x_{2}+\cdots+x_{1} x_{2} \cdots x_{n}=2021  
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Mike earns $12 per hour, Phil earns $6 per hour, and Olivia earns $10 per hour. What is the average percentage difference in earnings per hour between all three individuals?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44.95\%}
There are three segments of each length $2^{n}$ for $n=0,1, \cdots, 1009$. How many non-congruent triangles can be formed using these 3030 segments? (Answer in numerical form)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{510555}
If $\sin 2x \sin 3x = \cos 2x \cos 3x$, then one value for $x$ is $x^\circ$. What is the value of $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Spencer is making burritos, each of which consists of one wrap and one filling. He has enough filling for up to four beef burritos and three chicken burritos. However, he only has five wraps for the burritos; in how many ways can he make exactly five burritos?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
For which initial values $x_0$ does a fixed point iteration exhibit third-order convergence as described by the error relation $|\alpha - x_{n+1}| = \frac{1}{4} (|\alpha - x_n|)^3$? Express the range of these initial values as an interval around $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\alpha - 2, \alpha + 2)}
The length and breadth of the floor of a room are 20 feet and some feet respectively. Square tiles of 2 feet length of different colors are to be laid on the floor. Black tiles are laid in the first row on all sides. White tiles are laid in one-third of the remaining and blue tiles in the rest. There are 16 blue tiles. What is the breadth of the room?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
The volume of a hemispherical soup bowl is 8 liters. How much soup fills the bowl up to half its height?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.5 	ext{ liters}}
For the Burgers' equation \( u_t + uu_x = 0 \), determine the breaking time \( t_B \) for the initial condition:
\[
 u(x,0) = \phi(x) = \begin{cases} 
 1, & x \leq \pi \\
 \cos x + 2, & \pi \leq x \leq 2\pi \\
 0, & x \geq 2\pi 
\end{cases}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the smallest value of $a$ such that the function \( f_a(x) = \frac{(2x)!}{(x!)^a} \) forms a bell curve shape.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Arrange the numbers 1 to 2001 in a clockwise direction on a circle. Starting from the number 2, eliminate every second number in a clockwise direction until only one number remains. Calculate the last remaining number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1955}
Mitya is 11 years older than Shura. When Mitya was as old as Shura is now, he was twice as old as she was. How old is Mitya?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}