inclusionAI/Ring-lite-rl-data huggingface dataset:
Two spheres are inscribed in a dihedral angle such that they touch each other. The radius of one sphere is 1.5 times the radius of the other, and the line connecting the centers of the spheres forms an angle of $30^{\circ}$ with the edge of the dihedral angle. Determine the cosine of the dihedral angle, rounding to two decimal places if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.68}
Evaluate the integral sum given that \( a^5 - b^5 = 1 \):
\[ \int_1^a \sqrt[5]{x^5-1}\,dx + \int_0^b \sqrt[5]{x^5+1}\,dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ab}
In the expansion of \((x^3 + x^2 + x)^{36}\), how many coefficients of the resulting polynomial are not divisible by 2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate the integral \( \int_0^\infty \frac{\ln{x}}{(x^2+1)^2}dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{4}}
Find all values of $x$ such that
\[3^x + 4^x + 5^x = 6^x.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["3"]}
Evaluate the definite integral \( \int_{1}^{2} \int_{1}^{2} \int_{1}^{2} \int_{1}^{2} \frac{x_{1}+x_{2}+x_{3}-x_{4}}{x_{1}+x_{2}+x_{3}+x_{4}} \, d x_{1} \, d x_{2} \, d x_{3} \, d x_{4} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine the number of conjugacy classes of subgroups of the symmetric group $S_5$ that are isomorphic to the non-cyclic group of order 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the limit using the Taylor series approach: $$\lim_{x\to 0^+} \, \frac{-\tan ^{-1}\left(2 x^3\right)+\frac{(x+2) x^5}{x^2+3}+2 x \sin ^2(x)}{x \exp \left(-\frac{1}{x}\right)-2 x^2+\log \left(\frac{x^2+1}{1-x^2}\right)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Find the value of \(18! \mod 437\) given that \(437 = 19 \cdot 23\), \(18!\equiv -1 \pmod {19}\), and \(22!\equiv -1 \pmod {23}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{436}
Find the maximum value of $k$ for which the sum \( \sum_{i=0}^k \binom {10}{i} \binom{15}{k-i} +\sum_{i=0}^{k+1} \binom {12}{i} \binom {13}{k+1-i} \) exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Determine the number of solutions for the equation \( \cot^{-1}{\sqrt{4-x^2} + \cos^{-1}{(x^2-5)}} = \frac{3\pi}{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the maximum value of \( \alpha \) such that for any continuously differentiable function \( f: [0,1] \rightarrow \mathbb{R} \), the following inequality holds:
\[
\left|\frac{f(0)+f(1)}{2}-\int_{0}^{1} f(x) \, dx\right| \leq \alpha \max_{x \in [0,1]}\left|f^{\prime}(x)\right|.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Find the equation of the tangent line to the curve \((x^2 + y^2 - 3)^3 = (xy)^3 - x^2 - y^2\) at the point \((1, 1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = -x + 2}
Determine the order of magnitude of the number of involutions in the symmetric group $S_{256}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10^{259}}
(1901 + 1902 + 1903 + \cdots + 1993) - (101 + 102 + 103 + \cdots + 193) =
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["167400"]}
Provide an example of a Young function \( \Phi: \mathbb{R} \to \mathbb{R}^+ \) that is convex, satisfies \( \Phi(0) = 0 \), \( \Phi(-x) = \Phi(x) \), and \( \lim_{x \to \infty} \Phi(x) = +\infty \), but does not satisfy the \( \Delta_2 \) condition, which is \( \Phi(2x) \leq K\Phi(x) \) for some \( x \geq x_0 \geq 0 \) and a constant \( K > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{|x|} - 1}
Evaluate the limit \( A \) as \( n \) approaches infinity:

\[ A = \lim_{n \to \infty} \left(\binom{n}{0} \binom{n}{1} \binom{n}{2} \cdots \binom{n}{n}\right)^{\frac{1}{n(n+1)}} \]

Given that the product can be rewritten as \( \prod_{k=0}^{k=n} \prod_{r=1}^{r=k} \frac{n-k+r}{r} \), find the value of \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{e}}
Find the coefficient $c_{-2}$ of the Laurent series expansion of the function $g(z)$ in the region $\Omega=\{|z|>2\}$, where $g(z)$ is a holomorphic branch of the square root of $f(z)=(z-1)(z-2)$. Express your answer in terms of known constants or functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{3}{16}}
Consider the power set of \( \{0, 1, 2, 3\} \), and let \( C = \{ \{0\}, \{1, 2\}, \{1, 3\}, \{2, 3\} \} \). Calculate the intersection number of \( C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the number of distinct values the function \( f(z) = \sqrt{\frac{z^2-1}{\sqrt{z}}}(z-i)^{1/3} \) can take for a given value of \( z \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find all real pairs \((p, q)\) such that the inequality \(|x^2 + px + q| > 2\) has no solutions in the interval \([1, 5]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-6, 7)}
Let $z$ be a non-real complex number with $z^{23}=1$. Compute\[\sum_{k=0}^{22}\dfrac{1}{1+z^k+z^{2k}}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \frac{46}{3} \]}
Calculate the integral \( \int_{0}^{1} \sin(x^2) \, dx \) such that the error is less than or equal to \( 10^{-3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.310}
Given two normally distributed random variables, X and Y, where X has a mean of 50 and a standard deviation of 10, and Y has a mean of 60 and a standard deviation of 12, with a correlation coefficient of -0.6, calculate the probability that \(2X < 0.5Y\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0018}
Points $A$, $B$, and $C$ lie on a line, in that order, with $AB = 8$ and $BC = 2.B$ is rotated $20^\circ$ counter-clockwise about $A$ to a point $B'$, tracing out an arc $R_1$. $C$ is then rotated $20^\circ$ clockwise about $A$ to a point $C'$, tracing out an arc $R_2$. What is the area of the region bounded by arc $R_1$, segment $B'C$, arc $R_2$, and segment $C'B$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\( 2\pi \)}
Determine the domain of convergence for the series \( \sum_{n=1}^{\infty} \frac{x^n}{n^x} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1)}
Let $x, y, z$ be the sides of a triangle such that $x \neq z$. Determine the range of values for $p$ such that the equation \( t^2 + 2(x + y + z)t + 3p(xy + yz + zx) = 0 \) has real roots, given that $x, y, z$ satisfy the triangle inequality.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 1]}
What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\pi + 2"]}
Determine the maximum step size for the forward Euler method such that the solution of the system of ODEs \( a' = -a - 2b \) and \( b' = 2a - b \) with initial conditions \( a(0)=1 \) and \( b(0)=-1 \) goes to zero. Use the eigenvalues of the matrix \( \begin{pmatrix} -1-h & -2-2h \\ 2+2h & -1-h \end{pmatrix} \) to find the solution stability.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{5}}
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $(a^2 + b^2) : c^2 = 2019 : 1$, find the ratio $(\cot A + \cot B + \cot C) : (\cot A + \cot B)$. Provide your answer as a simplified ratio.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1010:1}
Evaluate the integral \( \int_a^b \delta(t-a)e^{-iwt}dt \) and determine its result in terms of the sifting property of the Dirac delta function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-i\omega a}}
Determine the number of intersection points between a circle of radius $R$ centered at the origin in 3D space and a Cassini oval defined by two fixed points $(x_{1},y_{1},z_{1})$ and $(x_{2},y_{2},z_{2})$ such that the product of the distances from any point on the oval to these two points equals a constant $C$. How many points can this circle and Cassini oval have in common?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $ n\in N^*$. A permutation $ (a_1,a_2,...,a_n)$ of the numbers $ (1,2,...,n)$ is called quadratic iff at least one of the numbers $ a_1,a_1+a_2,...,a_1+a_2+a+...+a_n$ is a perfect square. Find the greatest natural number $ n\leq 2003$, such that every permutation of $ (1,2,...,n)$ is quadratic.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1681}
Compute the value of the improper integral: $$\int_{0}^{\infty }\frac{(\ln x)^{40021}}{x}dx$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a queuing system with two facilities, A and B, each facility has a single server with service times that are negative exponential random variables. The mean service times are 1 minute for facility A and 4 minutes for facility B. A total of 60 customers per hour arrive, with arrivals following a Poisson process. Each customer independently chooses a facility without knowing the queue status, with probability \( p \) for facility A and \( 1-p \) for facility B. Determine the value of \( p \) that minimizes the expected time a customer spends in the system, assuming the system reaches a steady state.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{6}}
Calculate the number of spanning trees in a complete graph with 10 vertices, given that vertex 2 has a degree of 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{459270}
Determine the Big-O notation for the expression \( \prod_{i=2}^n \log(i) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O((\log n)^n)}
Determine the number of parameter-free definable subsets of the structure $(\mathbb{R}, *)$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Compute the quadratic variation \([X,X]_t\) of the zero-mean Ornstein-Uhlenbeck process \((X_t)_{t\geq 0}\) given by the representation:
\[ X_t = \sigma e^{-\alpha t}\int_0^t e^{\alpha s}\,dB_s \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sigma^2 t}
Let \( \{X_i\}_{i=0}^{\infty} \) be a sequence of IID random variables, which can be either discrete or continuous. Determine the expected value of \( \alpha \), where \( \alpha \) is the smallest index \( i \) such that \( X_i > X_0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given a vector space $V$ of $3 \times 3$ matrices and a matrix $A \in V$ with $\det(A) = 32$ and minimal polynomial $(x-4)(x-2)$, find the trace of the linear map $L_A: V \to V$ defined by $L_A(B) = AB$ for $B \in V$. What is the trace of $L_A$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
In an equilateral triangle ABC, point D is inside the triangle such that \(DB^2 + DC^2 + BC^2 = 120\). If the area of triangle DBC is \(6\sqrt{3}\), find the length of side BC.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\sqrt{2}}
There are 10 balls numbered 1 to 10. Two students each have two identical bags. In how many ways can the students distribute the balls into their bags such that no bag is empty?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{204630}
Find all the prime number $p, q$ and r with $p < q < r$, such that $25pq + r = 2004$ and $pqr + 1 $ is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$(p,q,r)=(7,11,79)$}
Find an ideal \( I \) of \( \mathbb{Z}[\sqrt{3}] \) generated by an integer prime such that \( \mathbb{Z}[\sqrt{3}]/I \) is not an integral domain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2)}
Determine the number of subgroups of the group \( \Bbb Z_{10} \oplus \Bbb Z_{15} \) using Goursat's Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Consider the inequality
\[(a_1+a_2+\dots+a_n)^2\ge 4(a_1a_2+a_2a_3+\cdots+a_na_1).\]
 Find all $n\ge 3$ such that the inequality is true for reals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$n=4$}
If the larger base of an isosceles trapezoid equals a diagonal and the smaller base equals the altitude, then the ratio of the smaller base to the larger base to the larger base is ：
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \frac{3}{5} \]}
In an equilateral triangle $ABC$, the distances from a fixed point $P$ on the plane to vertices $A$ and $B$ are $AP=2$ and $BP=3$. What is the maximum possible length of the line segment $PC$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the complexity of the function $f(n)$ given by the recurrence relation $f(0) = 3$, $f(1) = 3$, and $f(n) = f(\lfloor n/2 \rfloor) + f(\lfloor n/4 \rfloor) + cn$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n)}
Given the differential equation \( \frac{dP}{dt} = 0.8P \left(1 - \frac{P}{350}\right) \), where \( P \) is the number of people infected per week after the outbreak of a disease, and knowing that at the outbreak, 7 people were infected, determine the initial condition for \( P \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{P(0) = 7}
In a cube ABCD.EFGH with a side length of 12 cm, let R be the intersection of lines AD and CE. Let S be the intersection of segment ER with plane ACGE. Find the length of segment ES.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given that $(X,Y)\sim N(\mu_x=1,\mu_y=1,\sigma^2_x=4,\sigma^2_y=1,\rho=1/2)$, compute the probability $P(X+2Y\leq 4)$. Provide your answer to four decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.6136}
Let $O$ be the center of the concentric circles $C_1,C_2$ of radii $3$ and $5$ respectively. Let $A\in C_1, B\in C_2$ and $C$ point so that triangle $ABC$ is equilateral. Find the maximum length of $ [OC] $.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
8.

}
Determine the radius of convergence for the infinite power series \(\sum_{n=0}^{\infty}(\ln x)^n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given two independent random samples $X_1, \ldots, X_{100}$ and $Y_1, \ldots, Y_{50}$ from a normal distribution $N(\mu, 1)$, calculate the probability that the 80% confidence intervals for $\mu$ constructed from these samples are disjoint.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.074}
Evaluate the limit \( \lim\limits_{n\to\infty} \frac{\sum\limits_{k=1}^{n} \frac{n^{k-1}}{(k-1)!}}{e^n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine the number of 2-Sylow subgroups of the group $S_4 \times S_4$. Assume that any such subgroup is isomorphic to $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate the limit of the integral: $$ \lim_{n\to\infty}\int_A\frac{\ln(n+y^3)-\ln n}{\sin(x/n)}d\lambda_2 $$ where $A=\{(x,y)\in\mathbb{R}^2_+:1<xy<4,1<y/x<4\}$. Determine the value of this limit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{225}{8}}
Calculate the integral \( I = \int_{C} ze^{\frac{1}{z-1}} \, dz \) where \( C \) is the circle centered at 1 with radius \( \frac{1}{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\pi i}
Given distinct proper rigid transformations \(A, B \in \operatorname{SE}(n)\), determine the maximum dimension of the nullspace of \(A - B\). Specifically, find the maximum dimension of \(\operatorname{Eq}(A, B) = \{x \in \mathbb{R}^n\ |\ A(x) = B(x)\}\) for all integers \(n \geq 2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - 2}
Let $X$ and $Y$ be independent random variables, each uniformly distributed on $[0,1]$. Define $Z = \max(X, Y)$. Calculate the expected value of $X$ given $Z$, expressed as $\mathbb{E}[X|Z]$ in terms of $Z$. Express your answer as a function of $Z$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}Z}
For large values of $x$, approximate the expression $\ln \frac{1+e^{ax}}{1+e^{-bx}}$. Given that the leading term is $ax$, determine the value of $c$ in the approximation $ax + c + \frac{c_1}{x} + \dots$. What is the value of $c$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A plane contains points \( A \) and \( B \) with \( {AB} = 1 \) . Let \( S \) be the union of all
 disks of radius 1 in the plane that cover \( \overline{AB} \) . What is the area of \( S \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{${3\pi } - \frac{\sqrt{3}}{2} $}
Let a sequence be defined as follows: $a_{1}= 3$, $a_{2}= 3$, and for $n \ge 2$, $a_{n+1}a_{n-1}= a_{n}^{2}+2007$. Find the largest integer less than or equal to $\frac{a_{2007}^{2}+a_{2006}^{2}}{a_{2007}a_{2006}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
224
}
Find the angle between the vector \( v = (2,2,1,1) \) and the linear subspace \( S \) spanned by the vectors \( u_1 = (3,4,-4,-1) \) and \( u_2 = (0,1,-1,2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
Find the value of the constant \( d \) to which the sequence \( \frac{1}{n} \sum_{i=1}^n X_i \) converges almost surely, given that \( \{X_i, i \geq 1\} \) is a sequence of independent random variables with \( \Bbb P(X_i=i^3 -1)=i^{-3} \) and \( \Bbb P(X_i=-1)=1-i^{-3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Find the units digits in the decimal expansion of
$$\left(2008 + \sqrt{4032000}\right)^{2000} + \left(2008 + \sqrt{4032000}\right)^{2001} + \left(2008 + \sqrt{4032000}\right)^{2002} + \cdots + \left(2008 + \sqrt{4032000}\right)^{2007} + \left(2008 + \sqrt{4032000}\right)^{2008}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1
}
Determine the values of \(a > 0\) for which the sequence \(Y_n = \prod_{j=1}^n X_j\), where \(X_n\) are independent random variables with \(X_n \sim \text{Bern}(1-\frac{1}{n^a})\), converges in probability to zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1]}
Let \(f\) be a function on the nonnegative integers defined by the following properties:
\[ f(2n) = f(f(n)) \]
and
\[ f(2n + 1) = f(2n) + 1 \]
Determine all nonnegative integers \(k\) such that \(f(0) = 2^k\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the equation \(x^3 = (3+4i)\overline{x}\), where \(x\) is a complex number, find the sum of the moduli of its roots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{5}}
Find the coefficient of \(x^{62}\) in the expansion of \((x^3 - \frac{1}{x^5})^{50}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-37353738800}
Determine the number of mappings \( \phi: \mathbb{N} \cup \{0\} \to \mathbb{N} \cup \{0\} \) such that \( \phi(ab) = \phi(a) + \phi(b) \) for all non-negative integers \( a \) and \( b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the group obtained by imposing the relations $a^2 = e$ and $b^3 = e$ on the amalgamated free product $G = \mathbb{Z}_4 \ast_{\mathbb{Z}_2} \mathbb{Z}_6$, where $a$ and $b$ are the generators of $\mathbb{Z}_4$ and $\mathbb{Z}_6$ respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}_2 \ast \mathbb{Z}_3}
Let $A, B, C$ be three collinear points with $B$ between $A$ and $C$. Consider all circles passing through $B$ and $C$. Determine the type of curve on which the points of contact of the tangents from $A$ to these circles lie.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{circle}
Determine the computational complexity of deciding whether $a^b = c$, given that $a, c \leq 2^p$ and $b \leq 2^q$, without explicitly computing $a^b$, when the size of $q$ is not restricted.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{P}
In parallelogram \(ABCD\), the angle between diagonals \(AC\) and \(BD\) is \(30^\circ\). The ratio of \(AC\) to \(BD\) is \(2 : \sqrt{3}\). Let \(B_1\) be the reflection of vertex \(B\) across line \(AC\), and \(C_1\) be the reflection of vertex \(C\) across line \(BD\). Determine the ratio of the areas of triangle \(AB_1C_1\) to parallelogram \(ABCD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{4}}
Evaluate the limit: \[ \lim_{n\to\infty}\left(2n\int_{0}^{1}\frac{x^n}{1+x^2}dx\right)^n \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
Two angles of an isosceles triangle measure $80^o$ and $x^o$. What is the sum of all the possible values of $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
Evaluate the integral \( \int_0^\infty \frac{\sin(x)}{x^2+1} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2e}}
Find all polynomials P(x) satisfying the equation
$(2x-1)P(x) = (x-1)P(2x), \forall x.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$P(x)=c(x-1)$}
Let \( C \) be the disk of radius 1 centered at the origin in \( \mathbb{R}^2 \). Consider the linear transformation \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) given by:
\[ T(x, y) = \left( \frac{5x + 3y}{4}, \frac{3x + 5y}{4} \right) \]
What is the smallest natural number \( n \) such that \( T^n(C) \) contains at least 2019 points \( (a, b) \) where \( a, b \in \mathbb{Z} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Evaluate the limit: \[ \lim_{x \to 0} \frac{x\sin(\sin x) - \sin^2 x}{x^6} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{18}}
For what values of \( V \) does the function \( y(t) = -(V+2)e^{-3t} + (V+3)e^{-2t} \) remain nonnegative for all \( t \geq 0 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-3, \infty)}