inclusionAI/Ring-lite-rl-data huggingface dataset:
In a ring $R$, where $a^2 = a$, find the inverse of $1+a$, given that the inverse exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \frac{1}{2}a}
Evaluate the integral \( \frac{1}{5} \int_0^5 (4 + 2^{0.1t^2}) \, dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6.06}
n and r are positive integers. Find the smallest k for which we can construct r subsets \( A_1, A_2, \ldots, A_r \) of \( \{0, 1, 2, \ldots, n-1\} \) each with k elements such that each integer \( 0 \le m < n \) can be written as a sum of one element from each of the r subsets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lceil n^{1/r} \rceil}
Let \( a_1 = 1 \), and let \( a_n = \left\lfloor \frac{n^3}{a_{n-1}} \right\rfloor \) for \( n > 1 \). Determine the value of \( a_{999} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{999}
On the sides  $ AB $  and  $ AC $  of the triangle  $ ABC $  consider the points  $ D, $  respectively,  $ E, $  such that  $$  \overrightarrow{DA} +\overrightarrow{DB} +\overrightarrow{EA} +\overrightarrow{EC} =\overrightarrow{O} .  $$   
If  $ T $  is the intersection of  $ DC $  and  $ BE, $  determine the real number  $ \alpha $  so that:  $$  \overrightarrow{TB} +\overrightarrow{TC} =\alpha\cdot\overrightarrow{TA} .  $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Point  $P$  and equilateral triangle  $ABC$  satisfy  $|AP|=2$ ,  $|BP|=3$ . Maximize  $|CP|$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the range of values for \( p \) such that the interval \((1,2)\) lies between the roots of the equation \(2^x + 2^{2-x} + p = 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -5)}
Exactly at noon, Anna Kuzminichna looked out the window and saw Klava, the saleswoman of the village store, going on a break. At two minutes past one, Anna Kuzminichna looked out the window again and still saw no one in front of the closed store. Klava was gone for exactly 10 minutes, and when she returned, she found Ivan and Foma waiting at the door, with Foma evidently arriving after Ivan. Determine the probability that Foma had to wait for the store to open for no more than 4 minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.75}
Let  $P$  be the portion of the graph of  $$ y=\frac{6x+1}{32x+8} - \frac{2x-1}{32x-8} $$  located in the first quadrant (not including the  $x$  and  $y$  axes). Let the shortest possible distance between the origin and a point on  $P$  be  $d$ . Find  $\lfloor 1000d \rfloor$ .

*Proposed by **Th3Numb3rThr33***
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{433}
Consider a 1998-digit number where each pair of adjacent digits forms a two-digit number divisible by either 17 or 23. If the last digit of this number is 1, determine the first digit of the number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
The sequence starts with 12, followed by a number, then 15, 17, 111, 113, 117, 119, 123, 129, and 131. What is the second number in the sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Find the greatest real  $k$  such that, for every tetrahedron  $ABCD$  of volume  $V$ , the product of areas of faces  $ABC,ABD$  and  $ACD$  is at least  $kV^2$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9/2}
In an acute triangle with an area of 1, the altitudes divide it into six smaller triangles labeled counterclockwise as $t_1$, $t_2$, $t_3$, $t_4$, $t_5$, and $t_6$. Given that $t_1t_3t_5 + t_2t_4t_6 = 5$, determine the value of $t_1t_2t_3t_4t_5t_6$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{25}{4}}
A piece of string is cut in two at a point selected at random. The probability that the longer piece is at least x times as large as the shorter piece is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{x+1}}
Snorlax's weight is modeled by the function $w(t)=t2^t$ where $w(t)$ is Snorlax's weight at time $t$ minutes. Find the smallest integer time $t$ such that Snorlax's weight is greater than $10000.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
10
}
Select several numbers from the numbers 1, 2, 3, ..., 20, so that for any two numbers chosen, their least common multiple is also among the selected numbers. What is the maximum number of numbers that can be selected?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Two points, A and B, are 30 cm apart. Two strings, A and B, start simultaneously from A and B respectively and move towards each other. String A is 151 cm long and moves at 2 cm per second; String B is 187 cm long and moves at 3 cm per second. If both strings are ignited at their tail ends upon departure, string A burns at a rate of 1 cm per second and string B burns at a rate of 2 cm per second. How many seconds will it take for the two strings to completely pass each other from the moment they meet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40 \ \text{s}}
21 In a regular tetrahedron wooden block $ABCD$ with edge length 2, there is a point $P (AP<1)$ on the edge $AB$. A cross-section perpendicular to the edge $AB$ is to be sawed through point $P$. When the sawing stops at a certain position, it is measured that the saw cut $PM=1$ on the face $ABD$. The saw cut $PN=\frac{2}{3}$ on the face $ABC$, then $MN=$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
If the graph of the function $f(x)=\sin \omega x+\sin (\omega x- \frac {\pi}{2})$ ($\omega > 0$) is symmetric about the point $\left( \frac {\pi}{8},0\right)$, and there is a zero point within $\left(- \frac {\pi}{4},0\right)$, determine the minimum value of $\omega$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
a soccer store typically sells replica jerseys at a discount of 30 percent to 70 percent off list price . during the annual summer sale , everything in the store is an additional 20 percent off the original list price . if a replica jersey ' s list price is $ 80 , approximately what percent of the list price is the lowest possible sale price ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10\%}
Mrs. Fredrickson has 80 chickens, where 1/4 are roosters and the rest are hens. Only three-fourths of those hens lay eggs. However, 20% of the egg-laying hens have gone on vacation, and they won't contribute to egg production during this period. Additionally, 10% of the remaining egg-laying hens have fallen ill, further reducing the number of hens that lay eggs. How many chickens does Mr. Fredrickson have that do not lay eggs while considering the egg-laying hens on vacation and the ill hens?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
$ f(x)$  is a given polynomial whose degree at least 2. Define the following polynomial-sequence:  $ g_1(x)\equal{}f(x), g_{n\plus{}1}(x)\equal{}f(g_n(x))$ , for all  $ n \in N$ . Let  $ r_n$  be the average of  $ g_n(x)$ 's roots. If  $ r_{19}\equal{}99$ , find  $ r_{99}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99}
Archimedes drew a square with side length $36$ cm into the sand and he also drew a circle of radius $36$ cm around each vertex of the square. If the total area of the grey parts is $n \cdot \pi$ cm$^2$, what is the value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3888}
\[\log _{3} \log _{4} \frac{4 x-1}{x+1} - \log _{\frac{1}{3}} \log _{\frac{1}{4}} \frac{x+1}{4 x-1} < 0.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x \in \left( \frac{2}{3}, \infty \right)}
Given 5000 cards, each containing a unique natural number from 1 to 5000 (each number appears exactly once). You need to select two cards such that the sum of the numbers on them is divisible by 100. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{124950}
Find the closure of the interval \((0,1)\) in the lower-limit topology on \(\mathbb{R}\), where the topology is generated by the subbasis consisting of all half-open intervals \([a,b)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 1)}
A malfunctioning digital clock shows the time $9: 57 \mathrm{AM}$; however, the correct time is $10: 10 \mathrm{AM}$. There are two buttons on the clock, one of which increases the time displayed by 9 minutes, and another which decreases the time by 20 minutes. What is the minimum number of button presses necessary to correctly set the clock to the correct time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
If the plane vectors \(\vec{a}, \vec{b}, \vec{c}\) satisfy \(\vec{a} \cdot (\vec{a} + \vec{c}) = 0\) and \(|\vec{a} + \vec{b} - 2\vec{c}| = 2\), find the maximum value of \(\vec{a} \cdot \vec{b}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Determine the maximum variance of a distribution with support in the interval $[0, 1]$ and a fixed mean $\mu \in [0, 1]$. Is there an upper bound on this variance?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mu(1 - \mu)}
In quadrilateral \(ABCD\), the diagonals intersect at point \(M\) with an angle \(\alpha\) between them. Let \(O_{1}, O_{2}, O_{3}, O_{4}\) be the centers of the circumcircles of triangles \(ABM, BCM, CDM,\) and \(DAM\), respectively. Find the ratio of the areas of quadrilateral \(ABCD\) to quadrilateral \(O_{1} O_{2} O_{3} O_{4}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sin^2 \alpha}
Evaluate the integral \( \int_0^{2\pi} \frac{\sin^2x}{5+4\cos x} \ dx \) using complex integration techniques.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Let $b(n)$ be the number of digits in the base -4 representation of $n$. Evaluate $\sum_{i=1}^{2013} b(i)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12345}
Let \( h(x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} \) and let \( k(x) = h^{-1}(x) \). Find \( k^{(4)}(0) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that an equilateral triangle \( \triangle ABC \) has sides \( AB \) and \( AC \) that are tangent to circle \( \odot O \) at points \( B \) and \( C \) respectively, what is the ratio of the area of \( \triangle ABC \) that lies outside the circle to the total area of the triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{3} - \frac{4 \sqrt{3} \pi}{27}}
Find the critical points of the function defined by the following system of equations: \(2(x-1)=-2\lambda x\), \(2(y-3)=-2\lambda y\), \(2(z-1)=2\lambda z\), and \(z^2 - x^2 - y^2 =0\). Determine the minimum distance between these critical points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{11}}
Dean ordered 2 large pizzas that were each cut into 12 slices. One pizza was Hawaiian and the other was cheese. He invited his friends Frank, Sammy, Nancy, and Olivia to enjoy some pizza and watch a movie. The pizzeria had a promotion for a free medium veggie pizza with 10 slices if you ordered at least 2 large pizzas, so they received an additional veggie pizza as well. 

Dean was hungry and ate half of the Hawaiian pizza. Frank only ate 3 slices of Hawaiian pizza and Sammy ate a third of the cheese pizza. Nancy, who is a vegetarian, ate 2 slices of the cheese pizza and a slice of the veggie pizza. She gave her pepperoni from her veggie pizza slice to Olivia. Olivia, having received the extra pepperoni from Nancy, added it to her slice of Hawaiian pizza and then ate that slice, along with 2 slices of the veggie pizza. 

During the movie, Frank wanted to taste the cheese pizza, so he decided to swap 1 slice of his remaining Hawaiian pizza with 1 slice of cheese pizza from Sammy. Meanwhile, Dean craved another type of pizza, so he took 1 slice from the veggie pizza and exchanged it with 1 leftover slice of Hawaiian pizza with Olivia.

How many total slices were left over?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Given the equation in terms of $x$, $x^{2}+2(a-1)x+a^{2}-7a-4=0$, with roots $x_{1}$ and $x_{2}$, and it satisfies $x_{1}x_{2}-3x_{1}-3x_{2}-2=0$. Find the value of $(1+ \dfrac {4}{a^{2}-4})\cdot \dfrac {a+2}{a}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Elsa is venturing into the unknown. She stands on $(0,0)$ in the coordinate plane, and each second, she moves to one of the four lattice points nearest her, chosen at random and with equal probability. If she ever moves to a lattice point she has stood on before, she has ventured back into the known, and thus stops venturing into the unknown from then on. After four seconds have passed, the probability that Elsa is still venturing into the unknown can be expressed as $\frac{a}{b} $ in simplest terms. Find $a+b$.(A lattice point is a point with integer coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
89}
Find the following limit.

$ \lim\limits_{n\to\infty} \int_{-1}^1 |x|\left(1+x+\frac{x^2}{2}+\frac{x^3}{3}+\cdots +\frac{x^{2n}}{2n}\right)\ dx$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$ \frac{3}{2}$.

}
Find all values of $c \in \mathbb{R}$ such that for the map $F: \mathbb{R} \to \mathbb{R}^2$ satisfying the differential equation $D[F] = XF$, where $X = \begin{bmatrix} -1 & 1 \\ c & -1 \end{bmatrix}$, the limit $\lim_{t \to \infty} F(t) = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 1)}
Find the sum of the alternating series \( \sum_{n=1}^{\infty}(-1)^n \frac{16^n \pi^{2n+1}}{(2n)!} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the integral expression: $$2^{2010} \frac{\int_0^{1} x^{1004}(1-x)^{1004} \, dx}{\int_0^{1} x^{1004}(1-x^{2010})^{1004} \, dx}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4020}
Determine the values of \(a\) for which the improper integral \(I = \int_{0}^\infty x^a \ln{\left(1+\frac{1}{x^2}\right)} \, dx\) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1)}
Let $U=\{ (x,y,z) \: : \: x-z=0\}$ and $V=\{(x,y,z) \: :\: z=0\}$ be two subspaces of $\mathbb{R}^3$. Define $W=\{f:\mathbb{R}^3 \rightarrow \mathbb{R}^3 \: : f(U) \subseteq V \: \: f(V) \subseteq U \}$. Find the dimension of $W$ using its matrix representation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Batteries A and B have lifetimes that are independent and exponentially distributed with a common mean of m years. Given that the probability that battery B outlasts battery A by more than one year is 0.33, calculate the value of m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.41}
The line $l: x - 2y + 2 = 0$ passes through the left focus F<sub>1</sub> and a vertex B of an ellipse. Find the eccentricity of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\sqrt{5}}{5}}
Determine the number of altitudes of a tetrahedron that can lie outside the tetrahedron.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
What is the result of the sum of three Cantor sets, denoted as $C+C+C$, where $C$ is the usual Cantor set obtained from the interval $[0,1]$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 3]}
Find the exponential generating function (EGF) for the sequence $(a_n)_n$ where $a_0 = 1$ and $a_n = \alpha(\alpha-1)\cdots(\alpha - (n-1))$ for $n \geq 1$, with $\alpha$ being a real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1 + x)^\alpha}
Evaluate the limit: \[ \lim_{m \to \infty} m \cdot E \left[ \log\left( \frac{N+\frac{1}{2}}{m+1} \right) \right] \] where \( N \) is a Poisson random variable with mean \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Evaluate the integral using the Cauchy Principal Value: \[ \int_{-\infty}^\infty \frac{\cos(ax) - \cos(bx)}{x^2} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi(|b| - |a|)}
In a right triangle $ABC$, point $P$ is on $AC$ such that $AP = PQ = QC$. Given that $QB = 67$ and $PB = 76$, find the length of $AB$. Express your answer in terms of the given lengths.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89}
Find the limit of the integral \( \int_{0}^{1}u^k\cot{\frac{\pi(1-u)}{m}}\sin{\frac{2\pi n(1-u)}{m}}\,du \) as \( n \) approaches infinity, where \( k \) is a non-negative integer and \( m \) is a real number greater than or equal to 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{m}{2}}
Given the function defined by the following conditions:
\[
\begin{align}
f'(x) &= \frac{1}{2} f(x)\quad\text{for all } x,\\
f(0) &= 1,\\
|f(x)| &\le 6\quad\text{for all } x \in [-3, 3],
\end{align}
\]
find the higher error bound of the approximation of \(f(2)\) using the 6th degree MacLaurin polynomial \(T_6(2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{840}}
Given that $Y$ is uniformly distributed over $(0,2)$ and $X$ is exponentially distributed with mean $Y$, find the expected value of the output $XY^3$ of the process.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{16}{5}}
Let $\{e_i\}$ be a basis for a complex linear space $E$ and $h$ a Hermitian metric on $E$. Denote by $H=\{h_{ij}\}$ the matrix representation of $h$ with respect to $\{e_i\}$, where $h_{ij}=h(e_i,e_j)$. Given a complex automorphism $g\in GL(E)$ and the transformed basis $\{e'_i\}=\{ge_i\}$, find the relationship between the matrices $H$ and $H'=\{h'_{ij}\}$, representing $h$ with respect to $\{e'_i\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{H' = G^* H G}
Determine the number of distinct equivalence classes for logical statements composed of 5 variable letters, considering logical equivalence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{32}}
Find the value of \( \alpha \) that satisfies the equation: \( \sin^{-1}(2 \alpha) \sin^{-1}(60^{\circ} - 2 \alpha) \sin^{-1}(60^{\circ} + 2 \alpha) = 4 \sin^{-1}(6 \alpha) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the constant \(c\) such that \(\max_{a\le n/2}\frac{C_n^a}{\sum_{k=0}^{\lfloor{a/3}\rfloor}C_n^k}=(c+o(1))^n.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{3}}
Evaluate the limit: \[ \lim_{y \to +\infty} y \int_0^{+\infty} e^{-x^2} \sin(2xy) \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Two points rotate uniformly around two concentric circles. One completes a revolution 5 seconds faster than the other, resulting in two additional revolutions per minute. Initially, the rays from the center of the circles to these points coincide. Calculate the magnitude of the angle between the rays after 1 second.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{15}}
How many natural numbers greater than 200 are formed by using the natural numbers 0, 1, 3, 5, and 6 without overlapping?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Polycarp wrote a multiplication problem of two three-digit numbers on the board and mistakenly wrote 0 instead of the multiplication sign, resulting in a seven-digit number. This seven-digit number is an integer multiple of the product of the two numbers. Determine the exact factor by which the seven-digit number is a multiple of the product.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{73}
Find the number of positive integers less than 1000000 which are less than or equal to the sum of their proper divisors. If your answer is $X$ and the actual value is $Y$, your score will be $\max \left(0,20-80\left|1-\frac{X}{Y}\right|\right)$ rounded to the nearest integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{247548}
Given a discrete random variable $X$ that can take integer values from $0$ to $20$, and the mean of $X$ is $0.005$, determine the maximum possible variance of $X$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.099975}
Let $G$ be a group and $H$ be a normal subgroup of $G$ generated by an element $a$. If $a$ has order $n$, determine the possible orders of $bab^{-1}$ for $b \in G$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
How many $6$-digit positive integers whose digits are different from $0$ are there such that each number generated by rearranging the digits of the original number is always divisible by $7$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
133
}
Evaluate the line integral \( \int_{C_1} \frac{y\vec{e_1}-x\vec{e_2}}{x^2+y^2} d\vec{r} \), where \( C_1 \) is a circle of radius 3 centered at the origin in \( \mathbb{R}^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi}
Determine the order of the element $a^{p^{t-2}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^2}
Consider the region $D$ in $\mathbf{R}^3$ that lies above the paraboloid $z = x^2 + y^2 - 1$ and below the semi-sphere $z = \sqrt{1 - x^2 - y^2}$. Determine the volume of $D$ by evaluating the integral $\iiint z \, dV$. The intersection of the surfaces can be found by equating them: $$\sqrt{1 - x^2 - y^2} = x^2 + y^2 - 1$$ Squaring both sides to eliminate the square root: $$1 - x^2 - y^2 = (x^2 + y^2 - 1)^2$$ Expanding and simplifying: $$-1 = x^4 + y^4 - 2x^2 - 2y^2 + 1$$ Rearranging the terms: $$x^4 + y^4 - 2x^2 - 2y^2 = 0$$ This equation represents the intersection of the surfaces. Calculate the volume of the region $D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{12}}
Evaluate the limit: $$\lim_{n \rightarrow \infty} \sum_{i=(n+1)/2}^n {n \choose i} \times 0.51^i \times 0.49^{n-i}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find a lower bound for the function \( g(x) = \frac{x + (1-x)q_1}{1+3x}\cdot \frac{x + (1-x)q_2}{1+2x} \), where \( x \in (0,1) \), and \( q_1 \) and \( q_2 \) are constants such that \( q_1 + q_2 \leq 1 \) and \( q_1, q_2 > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit: \[ \lim_{k \to 0^{+}} \frac{(k+1)A-3kA+2A^3}{(k-A^2)^{\frac{3}{2}}} \] where \( A = -\sqrt{2} \frac{\Gamma(\frac{k+1}{2})}{\Gamma(\frac{k}{2})} \) and \( \Gamma \) is the Gamma function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\infty}