inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the smallest number \( n \) such that, when 1000 balls of 40 different colors (25 balls of each color) are placed in a circle in any arrangement, there are always \( n \) adjacent balls that include at least 20 different colors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{476}
In a larger geoboard grid of $7 \times 7$, points are evenly spaced vertically and horizontally. Points $A$ and $B$ are at $(3,3)$ and $(5,3)$ respectively. How many of the remaining points in the grid will allow for triangle $ABC$ to be isosceles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Evaluate the expression \(A + B + C + AB + BC + AC + ABC\) where:
\[ A = \frac{2\sqrt{3}}{3}i \cdot \cos\left(\frac{\pi}{6} + \frac{i}{3}\text{arcsinh}\left(\frac{3\sqrt{3}}{2}\right)\right) \]
\[ B = \frac{2\sqrt{3}}{3}i \cdot \cos\left(\frac{5\pi}{6} + \frac{i}{3}\text{arcsinh}\left(\frac{3\sqrt{3}}{2}\right)\right) \]
\[ C = \frac{2\sqrt{3}}{3}i \cdot \cos\left(\frac{3\pi}{2} + \frac{i}{3}\text{arcsinh}\left(\frac{3\sqrt{3}}{2}\right)\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A regular triangular pyramid is circumscribed around a sphere of volume $V$. What is the smallest possible volume of this pyramid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6V \sqrt{3}}{\pi}}
Determine the number of ordered pairs of positive integers $(m,n)$ with $1 \leq m \leq 100$ and $1 \leq n \leq 100$ such that \[ \gcd(m+1,n+1) = 10\gcd(m,n). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
方程 $\sin x=\cos 2 x$ 的最小的 $20$ 个正实数解之和为$m\pi$, 请给出$m$的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{130}
In a jar, there are 3409 pieces of candy, which includes 1462 chocolate candies and 1947 gummy candies. There are also 145 secret eggs, where 98 eggs have 1 prize each, 38 eggs have 2 prizes each, and 9 eggs have 3 prizes each. Calculate the total number of items in the jar, including candies and prizes in the eggs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3610}
实数 $a_{1}, a_{2}, \cdots, a_{n}(n \geqslant 3)$ 满足
$a_{1}+a_{2}+\cdots+a_{n}=0$,
且 $2 a_{k} \leqslant a_{k-1}+a_{k+1}(k=2,3, \cdots, n-1)$.
求最小的 $\lambda(n)$, 使得对所有的 $k \in\{1$, $2, \cdots, n\}$ ，都有
$\left|a_{k}\right| \leqslant \lambda(n) \cdot \max \left\{\left|a_{1}\right|,\left|a_{n}\right|\right\}$。若$n=5$，答案格式为\frac{p}{q}，其中$p,q$互质，求$p+q$。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Teena is driving at 55 miles per hour and is currently 7.5 miles behind Loe, who is driving at a certain speed in the same direction. In 90 minutes, Teena will be 15 miles ahead of Loe. What is Loe's speed in miles per hour?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Determine the number of non-isomorphic semidirect products of the group \( \mathbb{Z}/p^2\mathbb{Z} \) by the group \( \mathbb{Z}/p\mathbb{Z} \), where \( p \) is an odd prime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a non-homogeneous Poisson process $(N_t)$ with rate function $\lambda(t)$ given by:

$$\lambda(t) = \begin{cases} 1 & t<1 \\ 2 &1\leq t < 2 \\ 3 & t\geq 2 \\ \end{cases} $$

it is known that $N_3 = 10$ events occur over the interval $[0, 3]$. Calculate the conditional probability $P(N_1 = 2 \mid N_3 = 10)$, which is the probability that exactly 2 of these events occurred in the interval $[0, 1]$. Provide your answer as a decimal rounded to four decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.2907}
Determine the number of subgroups \( G < \mathbb{Z}^3 \) such that \( \mathbb{Z}^3/G \cong \mathbb{Z}/3\mathbb{Z} \oplus \mathbb{Z}/3\mathbb{Z} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Given $((x-\sqrt{3})^{2017}=a_{0}x^{2017}+a_{1}x^{2016}+…+a_{2016}x+a_{2017})$, find the value of $((a_{0}+a_{2}+…+a_{2016})^{2}-(a_{1}+a_{3}+…+a_{2017})^{2})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2^{2017}}
若三位数$n=\overline{abc}$是一个平方数，并且其数字和$a+b+c$也是一个平方数，称$n$为超级平方数，设超级平方数的集合为$A$，$A$中元素的个数为$\left| A \right|$，$A$中所有元素之和为$S\left( A \right)$，则与$\frac{S\left( A \right)}{\left| A \right|}$最接近的整数是_______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{384}
There are 10 numbers written in a circle, and their sum is 100. It is known that the sum of any three consecutive numbers is not less than 29.

Determine the smallest number \( A \) such that in any such set of numbers, each number does not exceed \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Each day a man meets his wife at the train station after work, and then she drives him home. She always arrives exactly on time to pick him up. One day he catches an earlier train and arrives at the station some time early. He immediately begins walking home along the same route the wife drives. Eventually, his wife sees him on her way to the station and drives him the rest of the way home. When they arrive home, the man notices that they arrived 30 minutes earlier than usual. How early did the man arrive at the station?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Given that point A has coordinates (-3, 2), and F is the focus of the parabola $y^2 = -4x$, and point P is a moving point on the parabola, when $|PA| + |PF|$ takes the minimum value, the coordinates of P are.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 2)}
Express in the form of an irreducible fraction:

$$
6 \frac{16}{2015} \times 9 \frac{17}{2016}-2 \frac{1999}{2015} \times 17 \frac{1999}{2016}-27 \times \frac{16}{2015}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{17}{224}}
Consider an angular sector on a plane with its vertex at the origin and an angle of 30 degrees. Place a ball (a material point of zero diameter) on one of the sides of the angle (ray) and launch it inside the sector with a constant speed at an angle of 110 degrees to the considered ray. The ball will fly within the sector, elastically bouncing off its sides, preserving the magnitude of its speed: the angle of incidence is equal to the angle of reflection. How many times will it hit the sides of the sector before flying off to infinity?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the parabola $x^2 = 4y$ with focus $F$ and point $A(-1, 8)$, let $P$ be a point on the parabola. Find the minimum value of $|PA| + |PF|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Each of the three aluminum cans holds an integer number of liters of water. The second can holds 1.5 times more than the first, and the volume of the third can is equal to \(\frac{4^3}{3}\) times the volume of the first can. What is the total number of liters of water that the three cans together can hold, given that this number is less than 30?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Given the hyperbola $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ $(a>0, b>0)$ with the left vertex $A$, $P$ is a point on one of the asymptotes of $C$, and $Q$ is the intersection point of $AP$ and the other asymptote of $C$. If the slope of the line $AP$ is $1$ and $A$ is the trisection point of $PQ$, then the eccentricity of $C$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{\\sqrt{10}}{3}"]}
Find the number of non-abelian groups of order $p^n$ (where $n > 2$) that cannot be expressed as a direct product of any of their two subgroups.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Complex numbers \( a \), \( b \), and \( c \) form an equilateral triangle with side length 24 in the complex plane. If \( |a + b + c| = 48 \), find \( |ab + ac + bc| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{768}
Find the value of \( \lim\limits_{n\to\infty} \sup \exp\left( \frac{k}{n} \ln(n^2) \right) \) for \( k \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A train 1500 m long can cross a pole in 8 seconds. Find the speed of the train.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{187.5}
Determine the VC dimension of a Support Vector Machine (SVM) using the polynomial kernel \(k(x,x')=(1+\langle x,x'\rangle_{\mathbb{R}^2})^2\) for binary classification in \(\mathbb{R}^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
For what values of the real parameter \( t \) does the system of inequalities
\[ x \geq y^2 + t y \geq x^2 + t \]
have exactly one solution in the set of real number pairs?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Suppose we have a sequence defined by $a_1 > 0$ and the recursive relation $a_{n+1} = \ln\left[\frac{\exp(a_n)-1}{a_n}\right]$. Determine the value of the infinite sum $a_1 + a_1 a_2 + a_1 a_2 a_3 + a_1 a_2 a_3 a_4 + \cdots$, given that the sequence is decreasing and converges to zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(e^{a_1} - 1\)}
Evaluate the line integral \( \int_\gamma \left(\frac{1}{x+1}-\frac{y}{(x+y)^2}\right)dx+\left(\frac{1}{y+1}-\frac{x}{(x+y)^2}\right)dy \) where \( \gamma \) is the path along the circle from \((1,0)\) to \((0,1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the length of a normal chord in the parabola \(y^2 = 4x\) that forms a right angle with the vertex \(O(0,0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\sqrt{3}}
Three cones with vertex $A$ and a slant height of 6 touch each other externally. For two of the cones, the angle between the slant height and the axis of symmetry is $\frac{\pi}{8}$, and for the third cone, this angle is $\frac{\pi}{4}$. Find the volume of the pyramid $O_{1} O_{2} O_{3} A$, where $O_{1}, O_{2}, O_{3}$ are the centers of the bases of the cones.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9 \sqrt{\sqrt{2} + 1}}
An integer  $n\ge1$  is *good* if the following property is satisfied:
If a positive integer is divisible by each of the nine numbers  $n + 1, n + 2, ..., n + 9$ , this is also divisible by  $n + 10$ .
How many good integers are  $n\ge 1$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit: \[ \lim_{n\to\infty}\int_0^\infty nx^2\sin\left(\frac{1}{nx}\right)dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the limit: \[ \lim_{n\to\infty} \left(\sqrt[3]{n^{48}+n} - \sqrt[3]{n^{48}+n^2}\right) \left((n^3 +3)^{12} - (n^4+4n)^9\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-6}
In the tetrahedron \( D-ABC \), \( DA \) is perpendicular to the plane \( ABC \), and \( \triangle ABC \) is an equilateral triangle with side length 2. Given that the tangent of the dihedral angle \( A-BD-C \) is 2, find the volume \( V \) of the tetrahedron \( D-ABC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given \(x = 3^{a_1}5^{a_2}7^{a_3}\) and \(y = 3^{b_1}5^{b_2}7^{b_3}\), where \(a_i, b_i\) are nonnegative integers with \(a_k + b_k \leq 5\) for all \(k\), determine the sum of the greatest common divisors (gcd) of all possible pairs \((x, y)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1477773}
Given $sn(\alpha+\beta) = \frac{1}{2}$, $si(\alpha-\beta) = \frac{1}{3}$, find the value of $\lg 5 \frac{ta\alpha}{ta\beta}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In a triangle $ABC$, points $M$ and $N$ are on sides $AB$ and $AC$, respectively, such that $MB = BC = CN$. Let $R$ and $r$ denote the circumradius and the inradius of the triangle $ABC$, respectively. Express the ratio $MN/BC$ in terms of $R$ and $r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the surface integral \( \iint_\Sigma \mathbf{f} \cdot d \mathbf{a} \) where \( \mathbf{f}(x,y,z)=(x^2,xy,z) \) and \( \Sigma \) is the part of the plane \( 6x+3y+2z=6 \) with \( x,y,z\geq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{4}}
Calculate the derivative with respect to $x$ of the expression:

$$\frac{1}{\left( \int_{a}^{x+h} f(x') \, dx' \right)^{-1}}$$

where $a$ is a constant and $h$ approaches zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x)}
Find the vector $w$ such that the reflection $r_w$ about the orthogonal space $w^\perp$ maps a vector $u$ of length $l$ to a vector $v$ of the same length, given that $u \ne v$. What is the vector $w$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{u - v}
Let $S=2^3+3^4+5^4+7^4+\cdots+17497^4$ be the sum of the fourth powers of the first $2014$ prime numbers. Find the remainder when $S$ is divided by $240$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{93}
In an isosceles triangle $ABC$ with $AB = AC$, the angle bisector from $B$ intersects $AC$ at $D$ such that $AD + BD = BC$. Find the measure of angle $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100^\circ}
Given a linear operator \( \psi : M \to M \) with minimal polynomial \( m_\psi (x) = (x-2)^2x^5 \), find the minimal polynomial of \( \psi^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x - 4)^2 x^3}
The cubic polynomials $p(x)$ and $q(x)$ satisfy the following conditions:  
1. $p(1) = q(2)$  
2. $p(3) = q(4)$  
3. $p(5) = q(6)$  
4. $p(7) = q(8) + 13$  

Find $p(9) - q(10)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
Climbing a staircase with four steps, how many distinct arrangements of steps are possible, considering different step sequences as distinct? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A polygon has a ratio of 7:4 for the number of diagonals to the number of sides. If the sum of its diagonals and sides is an even number and greater than 50, find the smallest number of sides that the polygon can have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
A point $P$ is chosen uniformly at random inside a square of side length 2. If $P_{1}, P_{2}, P_{3}$, and $P_{4}$ are the reflections of $P$ over each of the four sides of the square, find the expected value of the area of quadrilateral $P_{1} P_{2} P_{3} P_{4}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Evaluate the limit as \( n \to \infty \): \[ \lim_{n\to\infty}\frac {3^{3n} \binom{2n}{n}^2} {\binom{3n}{n} \binom{6n}{3n}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the function \( \phi(d, n) = k \) that gives the minimum number of real numbers \( k \) required to define a flat subspace of dimension \( d \) in the \( n \)-dimensional Euclidean space \( \mathbb{R}^n \) in a canonical way. Here, "canonical" means using a continuous and surjective function \( f: \mathbb{R}^k \rightarrow \) set of flat subspaces of dimension \( d \) in \( \mathbb{R}^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(d + 1)(n - d)}
Determine the chromatic number of the complement of the Petersen graph.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the asymptotic behavior of the function $f(n)$, where $f(n)$ is the minimal length such that there exist $n$ binary sequences of length $f(n)$, and any pair of these sequences has a unique Hamming distance.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n^2)}
Let \( ABCDEF \) be a regular hexagon with side length 2. A circle with radius 3 and center at \( A \) is drawn. Find the area inside quadrilateral \( BCDE \) but outside the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{3} - \frac{3}{2}\pi}
Calculate the flow of fluid through the closed surface S, which is the boundary of the region bounded by the half-spheres $z = -\sqrt{4-x^{2}-y^{2}}$ and $z = \sqrt{9-x^{2}-y^{2}}$, and the plane $z = 0$. The velocity field of the fluid is given by $\vec{v}(x,y,z)= (y^{3}e^{-z^{2}} + x)\vec{i} + (ze^{x} + y^{2})\vec{j} + (\cos(x^{2}+ y^{2}) +2z)\vec{k}$. Use the divergence theorem to find the flow of fluid through the surface.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70\pi}
A street has 20 houses on each side, for a total of 40 houses. The addresses on the south side of the street form an arithmetic sequence, as do the addresses on the north side of the street. On the south side, the addresses are 4, 10, 16, etc., and on the north side they are 3, 9, 15, etc. A sign painter paints house numbers on a house for $\$1$ per digit. If he paints the appropriate house number once on each of these 40 houses, how many dollars does he collect?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Two positive integers \( x \) and \( y \) are such that:

\[ \frac{2010}{2011} < \frac{x}{y} < \frac{2011}{2012} \]

Find the smallest possible value for the sum \( x + y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8044}
In a round-robin tournament among $8$ chess players (each pair plays one match), the scoring rules are: the winner of a match earns $2$ points, a draw results in $1$ point for each player, and the loser scores $0$ points. The final scores of the players are all different, and the score of the player in second place equals the sum of the scores of the last four players. What is the score of the second-place player?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Compute the absolute value of \(p^3 + q^3 + r^3\) given that \(p, q, r\) are distinct integers such that \(p + q + r = 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Let \( x, y, z \) be the lengths of the midline segments connecting the midpoints of opposite edges of a tetrahedron \( P-ABC \). If the sum of the squares of the six edge lengths of the tetrahedron is 300, find \( x^{2} + y^{2} + z^{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
Last year, Isabella took 8 math tests and received 8 different scores, each an integer between 91 and 100, inclusive. After each test, she noted that the average of her test scores was an integer. Her score on the seventh test was 97. What was her score on the eighth test?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
Given 6 points in space such that no 4 points are coplanar, find the smallest positive integer $n$ such that whenever 10 line segments are drawn connecting these points, it is always possible to find $n$ distinct pairs of triangles, each pair sharing a common edge.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral \( \int_{-\pi/2}^{\pi/2} [\cot x] \,\text{d}x \) where \([\quad]\) represents the floor function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{2}}
Calculate the dimension of the centralizer of the matrix \( A \), where \( A = \begin{bmatrix}8 & -9 & -2 \\ 6 & -7 & -2 \\ -6 & 9 & 4 \end{bmatrix} \). The centralizer \( V(A) \) is defined as \( \{X \in M_{3}(\mathbb{R}) : AX = XA \} \), where \( M_{3}(\mathbb{R}) \) is the vector space of \( 3 \times 3 \) real matrices. Determine the dimension of \( V(A) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Compute the number of nonnegative integer triples \((x, y, z)\) which satisfy the inequality:
\[ 4x + 2y + z \leq 36 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1330}
Evaluate the infinite series: $$\sum_{n = 1}^{\infty} \frac{\ln(n^3+1)}{n^3+1} \frac{1}{4^{n-1}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{12}}
Find the maximum possible value of \( k \) for which there exist distinct reals \( {x}_{1},{x}_{2},\ldots ,{x}_{k} \) greater than 1 such that for all
 \( 1 \leq  i,j \leq  k \) ,
 \( {x}_{i}^{\left\lfloor  {x}_{j}\right\rfloor  } = {x}_{j}^{\left\lfloor  {x}_{i}\right\rfloor  }. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
The maximum possible value of \( k \) is 4.}
Given a Markov chain with the following transition matrix and times for each state:

$$\begin{bmatrix}3/4&1/4&0&0\\4/5&0&1/5&0\\6/7&0&0&1/7\\0&0&0&1\end{bmatrix}$$

State times: \(t_0 = 5\), \(t_1 = 10\), \(t_2 = 15\), \(t_3 = 20\).

Starting at state \(t_0\), calculate the expected time to reach the absorbing state \(t_3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1155}
Calculate the variance of the hitting time \( \tau = \inf \{t>0: |B_t|=1\} \) for a reflected Brownian motion \( Z_t = |B_t| \), where \( B_t \) is a standard Brownian motion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Determine the signature of the quadratic form \( f(x, y, z) = 2xy + 2yz + 2xz + x^2 - z^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 2)}
Determine the fixed subfield of the field of rational functions \( \mathbb{K}(x) \) over a field \( \mathbb{K} \) of characteristic zero, under the action of the subgroup \( G \) generated by the automorphism \( \sigma: x \mapsto x+1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{K}}
Given the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(f(x)) = 3 + 2x$ and the condition $f(0) = 3$, find the value of $f(3)$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the conditional expectation \( E(N_2 \mid N_1) \) where \( N_t \) is a Poisson process with rate \( \lambda = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{N_1 + 1}
Compute the expected value \( E[X_{n+1} | X_0, \dots, X_n] \) given that \( X_n = X_0 e^{\mu S_n} \), where \( X_0 > 0 \), \( S_n \) is a symmetric random walk, and \( \mu > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{X_n \cosh(\mu)}
Three different real numbers $a$, $b$, $c$ form an arithmetic sequence, and $a$, $c$, $b$ form a geometric sequence. Find the ratio of $a:b:c$. Express your answer in the form $a:b:c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4:1:-2}
How many solutions does the equation \((a-1)(\sin 2 x + \cos x) + (a+1)(\sin x - \cos 2 x) = 0\) (where the parameter \(a < 0\)) have in the interval \((-π, π)\) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Compute the limit \( \lim\limits_{n\to \infty }\frac {(2n)!}{(2^n(n)!)^2} \) using Stirling's formula or other methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}