inclusionAI/Ring-lite-rl-data huggingface dataset:
3. On each of the lines $x=2$ and $x=15$, there are 400 points with ordinates $1,2,3, \ldots, 400$. In how many ways can three points be chosen from the 800 marked points so that they form the vertices of a right triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["320868"]}
Find the inverse of the polynomial \(2x^2 + x + 1\) in the quotient ring \(\frac{\mathbb{Z}_3[x]}{\langle x^3 + 2x + 2 \rangle}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2}
For positive reals \( p \) and \( q \) ,define the remainder when \( p \) and \( q \) as the smallest nonnegative real \( r \) such that \( \frac{p - r}{a} \) is an integer. For
 an ordered pair(a,b)of positive integers,let \( {r}_{1} \) and \( {r}_{2} \) be the remainder when \( a\sqrt{2} + b\sqrt{3} \) is divided by \( \sqrt{2} \) and \( \sqrt{3} \)
 respectively. Find the number of pairs(a,b) such that \( a,b \leq  {20} \) and \( {r}_{1} + {r}_{2} = \sqrt{2} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16
}
In parallelogram $ABCD$, the angle bisector of $\angle DAB$ meets segment $BC$ at $E$, and $AE$ and $BD$ intersect at $P$. Given that $AB = 9$, $AE = 16$, and $EP = EC$, find $BC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Given a set of $n$ distinct positive integers, determine the number of ways to choose two disjoint subsets of size $k$ such that the absolute difference between the sums of the reciprocals of the elements in the two subsets is less than a given value $\epsilon$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the value of $a$ where the functions $\sin(x)$ and $\ln(x)$ intersect, i.e., solve $\sin(x) = \ln(x)$ for $x$. This value of $a$ is used in setting up the integral for the volume of the solid created by revolving the region bounded by $\sin(x)$, $\ln(x)$, and the $x$-axis about the line $y=5$. Provide the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.219}
Evaluate the integral \( \int_{0}^{0.1} \cos \left(100 x^{2}\right) d x \) with an accuracy of \( \alpha=0.001 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.090}
Find the greatest common divisor (GCD) of all numbers of the form \(p^8 - 1\), where \(p\) is a prime number greater than 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{480}
Evaluate the integral \( \int _0^{2\pi}\sin^2 \left(\frac{-\pi}{6}+3e^{it} \right)dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
In triangle $ABC$ , $\angle A = 45^o$ and $M$ is the midpoint of $\overline{BC}$. $\overline{AM}$ intersects the circumcircle of $ABC$ for the second time at $D$, and $AM = 2MD$. Find $cos\angle AOD$, where $O$ is the circumcenter of $ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\(\dfrac{-1}{8}\)
}
Given three periodic trigonometric functions $f_1(x)$, $f_2(x)$, and $f_3(x)$, where $x > 0$, the function $F(x) = f_1(x) \cdot f_2(x) \cdot f_3(x)$ is defined. It is known that $f_1(x)$ is positive for $\frac{2}{3}$ of its domain, $f_2(x)$ is positive for $\frac{1}{2}$ of its domain, and $f_3(x)$ is positive for $\frac{1}{2}$ of its domain. Determine the portion of the domain for which $F(x)$ is positive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine the Galois group of the polynomial \(X^5 - 1\) over the finite field \(\mathbb{F}_7\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}/4\mathbb{Z}}
Calculate the area enclosed by the curves $y = x$, $y = 2x$, $y = \frac{1}{x}$, and $y = \frac{3}{x}$ using the substitution $(x, y) = \left(\frac{u}{v}, uv\right)$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
Determine the minimum number of lottery tickets needed to ensure that at least one cell matches the winning permutation of numbers 1 through 50.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Find a function $f(x)$ such that \( \lim_{x\to\infty}\frac{f(x)}{x}=1 \) and \( \lim_{x\to0}\frac{f(x)}{x^2}=C \) for a constant \( C \neq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{C x^2}{1 + C x}}
Find the remainder when the sum of the units digits of all the terms in the sequence \(2^{2^2}+4^{4^4}+6^{6^6}+\cdots+2016^{2016^{2016}}+2018^{2018^{2018}}\) is divided by 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In quadrilateral $ABCD$, the angles are given as $\angle DAB=60^{\circ}$, $\angle ABC=90^{\circ}$, and $\angle BCD=120^{\circ}$. The diagonals $AC$ and $BD$ intersect at point $M$. If $MB=1$ and $MD=2$, determine the area of quadrilateral $ABCD$. Express your answer in simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{2}}
Calculate the gradient of the function \( L = \text{Tr}(XRX^T + YPY^T) \) with respect to the matrix \( X \), where \( R \) and \( P \) are known matrices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{X(R + R^\top)}
Evaluate the limit: $$ \lim_{x\to 0} \left(\frac{2+\cos x}{x^3\sin x}-\frac{3}{x^4}\right) $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{60}}
A frog is placed at the origin on a number line, and moves according to the following rule: in a given move, the frog advanced to either the closest integer point with a greater integer coordinate that is a multiple of 3, or to the closest integer point with a greater integer coordinate that is a multiple of 13.  A [i]move sequence[\/i] is a sequence of coordinates which correspond to valid moves, beginning with 0, and ending with 39.  For example, 0, 3, 6, 13, 15, 26, 39 is a move sequence.  How many move sequences are possible for the frog?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
169
}
Find the largest non-negative value of \(a\) such that the following limit evaluates to \(\frac{1}{4}\):
\[
\lim\limits_{x \to 1} \left(\frac{-ax + \sin(x-1)+a}{x+\sin(x-1)-1}\right)^\frac{1-x}{1-\sqrt{x}}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let \(G = C_4 \rtimes_\phi C_3\), where \(C_4 = \langle x : x^4 = 1 \rangle\) and \(C_3 = \langle y : y^3 = 1 \rangle\), and \(\phi : C_3 \to \Aut(C_4)\) is defined by \(\phi(y^i) = f^i\) for \(i \in \mathbb{Z}\) and \(f \in \Aut(C_4)\) is defined by \(f(x) = x^3\). Find the order of the element \((x, y)\) in the group \(G\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Calculate the size of the stabilizer of the permutation \((1\ 2\ 3)(4\ 5\ 6)\) in the symmetric group \(S_7\) under the action defined by \(x \cdot y = xyx^{-1}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Find one complex value of $x$ that satisfies the equation $\sqrt{3}x^7+x^4+2=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$x = \frac{\sqrt{3} + i}{2}$}
Let $ABCD$ be an isosceles trapezoid such that $CD >  AB = 4.$ Let $E$ be a point on line $CD$ such that $DE =2$ and $D$ lies between $E$ and $C.$ Let $M$ be the midpoint of $\overline{AE}.$ Given that points $A, B, C, D,$ and $M$ lie on a circle with radius $5,$ compute $MD.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{6}}
Calculate the expected value $E(X)$, where $Z$ is a standard normal random variable, and $X = \min(Z+1, 0)$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-0.0833}
Find the number of subsets of ${1,2,3,...,10}$ that contain exactly one pair of consecutive integers. Examples of such subsets are ${1,2,5}$ and ${1,3,6,7,10}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
235
}
Find the smallest positive integer \( k \) such that the sum of the digits of \( k, 2k, 3k, \ldots, 2014k \) are all equal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9999}
Calculate the line integral \( \int_{\kappa} \arctan\left(\frac{y}{x}\right) dy - dx \), where \( \kappa \) is the part of the parabola \( y = x^2 \) from the point \((1,1)\) to \((0,0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 - \dfrac{\pi}{2}}
Calculate the flux of the vector field \( F(x,y,z) = (zy^2-2x, \frac{1}{4}yz+z^2, xy+2x^2+2z) \) exiting the region \( \Omega = \{(x,y,z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 \leq 16, z \leq \sqrt{x^2+y^2}\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-8\pi}
Determine the range of values of \( \lambda \) for which the transformation \( T \) defined by \((Tf)(x) = x + \lambda \int_0^x (x-t)f(t)dt\) is a contraction with respect to the supremum norm on the space of continuous functions on the interval \([0,1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, 2)}
Two painters are painting a fence that surrounds 100 garden plots arranged in a single line. They paint one plot every other day in either red or green. The first painter is colorblind and cannot distinguish the color painted by the second painter, but he remembers which plots he painted. The first painter aims to maximize the number of transitions where a green plot borders a red plot. What is the maximum number of such transitions he can achieve, regardless of how the second painter acts?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
Calculate the value of the sum: \[ \sum_{k=1}^{10} k \binom{10}{k} \binom{20}{10-k} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100150050}
Find the radius of the inscribed sphere of a pyramid in \(\mathbb{R}^5\) with its vertex at \((1,0,0,0,0)\). The base of the pyramid is a regular 4-dimensional simplex lying in the hyperplane \(x_1=0\), which has an inscribed sphere of radius 1 centered at \((0,0,0,0,0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2} - 1}
Let \( \{ Y_t \}_{t \ge 0} \) be defined by \( Y_t := 5+2t+2B_t \), where \( \{ B_t: t \ge 0 \} \) is a standard Brownian Motion. Let \( \{ T_t \}_{t \ge 0} \) be defined by \( T_t := e^{Y_t} \). Calculate the probability \( P(T_2 > T_4, T_2 >5) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0783}
Given the three numbers $x, y=x^x, z=x^{(x^x)}$ with $.9<x<1.0$. Arranged in order of increasing magnitude, they are:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x, z, y)

}
Find the coefficient of \(x^{19}\) in the expansion of the generating function \(f(x) = \frac{(1-x^{5})^5}{(1-x)^{10}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3300000}
Evaluate \( \lim_{n \to \infty} \int_0^n \frac{dx}{1 + n^2 \cos^2 x} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
We have an a sequence such that \( {a}_{n} = 2 \cdot  {10}^{n + 1} + {19} \) . Determine all the primes \( p \) , with \( p \leq  {19} \) , for which there exists some
 \( n \geq  1 \) such that \( p \) divides \( {a}_{n} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(   \{ 3,7,{13},{17}\} \)}
Evaluate the integral: \[ \int_{0}^{1}\frac{e^{2y}(4y^2-4y+2)-2}{y^2}dy \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the number of ways to order the integers $1,2,3,4,5,6,7$ from left to right so that each integer has all its divisors besides itself appearing to the left of it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
Determine the accumulation points of the set \( A = \{ \sqrt{n} - \sqrt{m} : m, n \in \mathbb{N} \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Identify the algebraic structure defined by the following properties: Let $R$ be a ring and $G$ be a group of ring automorphisms of $R$. Consider the additive group $P=\bigoplus_{\sigma\in G}R$ and define multiplication for $x,y\in P$ as $(xy)_\omega=\sum_{\sigma\tau=\omega}x_\sigma \sigma^{-1}(y_\tau)$ for every $\omega\in G$, where the sum runs over $\sigma,\tau\in G$ such that $\sigma\tau=\omega$. What is the name of this algebraic structure?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{skew group ring}
Find the non-zero value of $k$ such that the average area between the curves $x^2$ and $ax$ for $x \ge 0$, as $a$ varies from $0$ to $k$, equals $x^2$. What is this value of $k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
In a World Cup, the Brazil team has the following probabilities for each match outcome: winning with probability \( \frac{1}{2} \), drawing with probability \( \frac{1}{3} \), and losing with probability \( \frac{1}{6} \). Points are awarded as follows: 3 points for a win, 1 point for a draw, and 0 points for a loss. During the group stage, each team plays three matches, and a team needs at least 4 points to advance to the knockout stage. In the knockout stage, if a match ends in a draw, it goes to a penalty shootout, where Brazil has a \( \frac{3}{5} \) chance of winning. Calculate the probability that Brazil wins the championship with exactly one of the four knockout stage matches decided by a penalty shootout.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{12}}
Calculate the symmetric-decreasing rearrangement of the function \( f(x) = x \) on the interval \([0, 10]\), with \( f(x) = 0 \) elsewhere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10 - 2|x - 5|}
Find the value of $n$ that maximizes the function $f(n) = \frac{\sigma_0(n)}{n}$, where $\sigma_0(n)$ is the number of divisors of $n$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that $X$ and $Y$ are normally distributed random variables with a covariance of $c$, find the covariance of $X^3$ and $Y^3$. Express your answer in terms of $c$. 

\[ \text{Cov}(X^3, Y^3) = ? \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6c^3 + 9c}
For which \(\alpha\) is the integral \(\int_Q \frac{1}{(x^2+2y^2)^{\alpha}}\,dx\,dy\) finite, where \(Q=[0,1]\times[0,1]\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha < 1}
Evaluate the triple integral: $$\int_{0}^{6} \int_{0}^{12-2y} \int_{0}^{4-\frac{2y}{3}-\frac{x}{3}} x \, dz \, dx \, dy$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
Evaluate the limit: \[ L = \lim_{x \to \infty} x^2 \times \log \left(x \cot^{-1}x\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{3}}
Evaluate the double integral: $$\int _0^{\frac{1}{4}}\int _{\sqrt{x}}^{\frac{1}{2}}\:\frac{\cos\left(\pi y\right)}{y^2}~\mathrm dy~\mathrm dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{\pi}}
Determine the set of values for \( \alpha \) for which the improper integral \( \int_1^{+\infty} \frac{x^{\alpha} \cdot \sin x}{1+x^3} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 3)}
Find the expected number of trials needed such that the sum of numbers picked from a uniform distribution over $[0, 1]$ is at least 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
Identify the group given by the presentation \( \langle a, b, c, d, e \mid cde = 1 \rangle \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F_4}
Calculate the sum \( \sum_{k \in S(m,n)} \varphi(k) \), where \( S(a,b) = \{ k \in \mathbb{Z} \mid a \equiv a \pmod{k} + b \equiv b \pmod{k} \ge k \} \) and \( \varphi \) is Euler's totient function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{mn}
Given real numbers \(a_{1}, a_{2}, \cdots, a_{n}\) which are all greater than 0 (where \(n\) is a natural number no less than 4) and satisfy the equation \(a_{1} + a_{2} + \cdots + a_{n} = 1\), find the maximum value of the sum \[ S = \sum_{k=1}^{n} \frac{a_{k}^{2}}{a_{k} + a_{k+1} + a_{k+2}} \] where \(a_{n+1} = a_{1}\) and \(a_{n+2} = a_{2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Consider a firm with quality \( \theta \in \{0, 1, \ldots, 100\} \), each equally likely. With probability \( \frac{1}{10} \), the firm can only send the message \( m = \varnothing \), and with probability \( \frac{9}{10} \), it can send either \( m = \varnothing \) or \( m = \theta \). A consumer observes the firm's message and forms a belief about the firm's expected quality \( b = E[\theta|m] \). The firm aims to maximize \( b \). Determine if there exists an equilibrium cutoff \( \theta^* \) such that: (i) every firm with \( \theta < \theta^* \) sends \( m = \varnothing \), and (ii) every firm with \( \theta \ge \theta^* \) sends \( m = \theta \) if possible. If such an equilibrium exists, find the value of \( \theta^* \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Given the vector space \( \mathbb{R}^2 \) over \( \mathbb{R} \) with the ordered basis \( \alpha = \{f_1, f_2\} \) where \( f_1 = (1,0) \) and \( f_2 = (1,1) \), consider the linear transformation \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) defined by \( T((x,y)) = (-x+2y, -x+y) \). The matrix representation of \( T \) with respect to the basis \( \alpha \) is given by:
\[ [T]^{\alpha}_{\alpha} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \]
Compute \( T^{65}(f_1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, -1)}
Determine the number of non-conjugate $6 \times 6$ complex matrices with the characteristic polynomial $(x-5)^6=0$. Consider matrices that are not diagonalizable to the diagonal matrix $D$ with all diagonal entries equal to 5, but still have the same characteristic polynomial.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Determine the largest number among ten consecutive numbers that can be obtained from an Antimagic arrangement of a 4x4 matrix, where each element is a unique number from 1 to 16. In an Antimagic arrangement, the sums of the rows, columns, and diagonals form an arithmetic sequence of ten consecutive numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
Determine the number of solutions to Langford's problem for $n=50$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A pedestrian and a cyclist are moving in one direction along a highway, while a cart and a car are moving in the opposite direction. All participants move at constant speeds. The cyclist first overtakes the pedestrian at 10 o'clock, then meets the cart, and then meets the car, with equal time intervals between these events. The car first meets the cyclist, then meets the pedestrian at 11 o'clock, and then overtakes the cart, with equal time intervals between these events. At what time did the pedestrian meet the cart?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10:40}
Determine the number of different monic polynomials of degree $q$ in a finite field $F$ where the size of the field $|F|$ is greater than $q$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|F|^q}
Find the coefficient of \(x^{10}\) in the power series expansion of the function \(\frac{1+x^3}{1-2x+x^3-x^4}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{631}
Evaluate the expression \( \left\lfloor \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{1000}} \right\rfloor \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
We roll a fair die repeatedly. Let $X$ be the number of rolls needed to obtain a 5, and $Y$ be the number of rolls needed to obtain a 6. Calculate the expected value $E(X|Y=2)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{33}{5}}
Estimate the smallest value of $r$ such that a circle of radius $r$ can contain $19$ non-overlapping circles of radius $1$. Express your answer to the nearest hundredth. For example, $11.00$, $5.60$, and $1.34$ are valid responses, but $11$ and $1.342$ are not. An invalid response will receive a score of zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4.86
}
Calculate the expected value \( \mathbb{E}(\exp(1/X)) \) where \( X \sim N(0, \sigma_x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
For every $k= 1,2, ...$ let $s_k$  be the number of pairs $(x,y)$ satisfying the equation $kx + (k+1)y = 1001 - k$ with $x$, $y$ non-negative integers. Find $s_1 + s_2 + ...+ s_{200}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
997
}
Evaluate the limit: $$ \lim_{x \rightarrow \infty} x\left(\frac{1}{x^2}+\frac{1}{(x+1)^2}+...+\frac{1}{(2x-1)^2}\right) $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
4. On a plane, an infinite number of parallel lines are drawn, each separated from the next by a distance of 1. A circle with a diameter of 1 is randomly thrown onto the plane. Find the probability that a line intersecting the circle divides it into parts such that the ratio of the areas (of the smaller to the larger part) does not exceed the number \((2 \pi - 3 \sqrt{3}) : (10 \pi + 3 \sqrt{3})\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{2-\\sqrt{3}}{2}"]}
Let \( m = 30030 \) and let \( M \) be the set of its positive divisors which have exactly 2 prime factors. Determine the smallest positive integer \( n \) such that for any choice of \( n \) numbers from \( M \), there exist three numbers \( a \), \( b \), and \( c \) among them satisfying \( abc = m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Among 10 chickens, one is false and weighs either 1kg with probability 1/2 or 3kg with probability 1/2. A machine independently picks the i-th chicken with probability 1/2 for weighing. After each weighing, all chickens are returned to the machine. After 3 weighings, the first set of chickens weighs 11kg, the second set weighs 12kg, and the third set weighs 13kg. What is the probability that the false chicken weighs 3kg?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{5}}
Evaluate the integral \( \int_0^{\infty} \frac{\log x }{(x-1)\sqrt{x}} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi^2}
Determine the number of homomorphisms from the symmetric group $S_3$ to the group $\mathbb{Z}_2 \times \mathbb{Z}_4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}