inclusionAI/Ring-lite-rl-data huggingface dataset:
Calculate the line integral of the vector field \( F(x,y) = \left(\frac{x}{x^2+y^2}, \frac{-y}{x^2+y^2}\right) \) along the closed curve formed by the union of the line segments from \((1,0)\) to \((0,1)\) and from \((0,1)\) to \((-1,0)\), and the upper half-circle centered at the origin from \((-1,0)\) to \((1,0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider a sequence of independent and identically distributed (i.i.d.) random variables $X_1, X_2, \dots$, where each $X_k$ follows a distribution with $\mathbb{P}(X_k \geq x) = \frac{1}{x^a}$ for $x \in \mathbb{N} = \{1,2,3,\dots\}$ and $a > 0$. Determine the range of $a \in \mathbb{R}_+$ for which the expected value $\mu$ of $X_1$ is finite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, \infty)}
Evaluate the double integral \( \iint_D e^{-x^2-4y^2} \, dx \, dy \), where \( D = \{(x,y) : 0 \leq x \leq 2y\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{16}}
Let \( X \) be a real-valued stochastic variable with density \( f(x) = 2xe^{-x^2} \) for \( x > 0 \). Let \( Y = X^2 \). Calculate \( E[Y^n] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Gamma(n + 1)}
Evaluate the sum \(\sum_{r=0}^{44} \cot^2(2r+1)\) using complex numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4005}
Consider the function \(f(x) = |x| + |x - 1| + |x - 3| + |x - 6| + \cdots + |x - (1 + 2 + \cdots + 101)|\) for all real numbers \(x\). Determine the value of \((m + n - 10)/18\), where \(m\) is the number of points at which \(f(x)\) is non-differentiable and \(n\) is the number of integer points where \(f(x)\) attains a minimum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Approximate the expression \( \frac{n^n}{{\lfloor\alpha n \rfloor}^{\lfloor\alpha n \rfloor}{(n-\lfloor\alpha n \rfloor)}^{n-\lfloor\alpha n \rfloor}} \) using the binary entropy function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n H(\alpha)}}
Find the minimum number of monomials required to generate the ideal $I(V(x_1^2, x_2^2) \cup V(x_3^2, x_4^2))$ in $\Bbb C^4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given the curve $y = rac{4	ext{π}}{3} 	imes 	ext{sin}(x)$, let $s \in \mathbb{Q}$ with $s > 1$. The arc length $L$ from $0$ to $2\pi s$ is expressed as an elliptic integral of the second kind using the formula: $L = \int_0^{2\pi s} \sqrt{1 + \cos^2 x}\, dx$. If the period is changed to $\frac{4\pi}{3}$, determine the new amplitude of $\sin x$ so that the arc length from $0$ to $2\pi s$ remains $L$. Assume the current amplitude is $|A| = 1$. What should the new amplitude be?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Given that each coordinate of a k-tuple can take one of 3 values, determine the maximum number of 2-different k-tuples that can be formed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3^{k-1}}
Find a sufficient statistic for \( \theta \) given that \( y_1, y_2, \ldots, y_n \) are independent and identically distributed as \( U([0,\theta]) \), where \( \theta > 0 \) is unknown.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\max\{y_1, y_2, \ldots, y_n\}}
Let \( A \) and \( B \) be two \( d \)-dimensional vectors with all elements initially equal to 1. We randomly select one element from each vector with replacement and change it to 0, repeating this process \( m \) times. If \( 2m \leq d \), what is the expected number of elements in \( A - B \) that are equal to 2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d \left(1 - \frac{1}{d}\right)^{2m}}
Find the coordinates $(m, n)$ of the point $P$ on the ellipse \( \frac{x^2}{8} + \frac{y^2}{18} = 1 \) such that the tangent at $P$ meets the coordinate axes at points $A$ and $B$, and the area of triangle $OAB$ is minimized, where $O$ is the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 3)}
Evaluate the integral \( \frac{1}{2\pi i}\int_{\gamma} \frac{1}{z^4(e^{2z}-1)}dz \) where \( \gamma=\{z: z=\frac{1}{2}e^{2\pi i t},0\leq t\leq1 \} \). Note that \( z=0 \) is a pole singularity of order 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{90}}
Add $537_{8} + 246_{8}$. Express your answer in base $8$, and then convert your answer to base $16$, using alphabetic representation for numbers $10$ to $15$ (e.g., $A$ for $10$, $B$ for $11$, etc.).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{205_{16}}
Two circles intersect at points $A$ and $B$ in the Cartesian coordinate system. Point $A$ is on the line $y = 3$, and point $B$ is on the line $y = 12$. The circles are tangent to the x-axis at points $P$ and $Q$, respectively. Express the distance $AB$ in terms of the distance $PQ$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4} PQ}
A cylindrical container is constructed from a material with a fixed thickness \(T\). The container has an open top and a solid circular bottom, where the bottom's radius equals the outer radius of the container. Given that the inner volume of the container is \(V\) mm\(^3\) and the volume of the material used is minimized when the inner radius is 10 mm, determine the value of \(\frac{V}{\pi T}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
Find the limit superior of the sequence \( \sin(x) + \sin(\pi x) \) as \( x \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \( f(x) = \left( \int_{0}^{x} e^{-t^{2}} \, dt \right)^{2} \) and \( g(x) = \int_{0}^{1} \frac{e^{-x^{2(1+t^{2})}}}{1+t^{2}} \, dt \). Find the value of \( f'(\sqrt{\pi}) + g'(\sqrt{\pi}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the sequence $(a_n)$ with $a_2 > 0$, $a_3 > 0$, and $a_{n+2}a_n = \left(1+\frac{1}{n}\right)^n a_{n+1}^2$ for $n \geq 2$, find the limit of $\sqrt[n^2]{a_n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{e}}
Determine the number of continuous functions \( f(x) \) defined on the interval \([0,1]\) that satisfy the following conditions:

\[ I_1 = \int_0^1 f(x) \, dx = 1, \]

\[ I_2 = \int_0^1 x f(x) \, dx = a, \]

\[ I_3 = \int_0^1 x^2 f(x) \, dx = a^2. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find the equation of the plane tangent to the surface $z = \sqrt{2x^2+y^4+1}$ that is parallel to the plane $x-y-z=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x - y - z = 0}
Evaluate the integral \(\int _{ -\infty }^{ +\infty }{ { e }^{ { (ix+b) }^{ 2 } } } dx\) where \(x\) and \(b\) are real numbers and \(b\) is a constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\pi}}
365. The lengths of two parallel sides of a rectangle are 1 cm. In addition, it is known that it can be divided into four rectangles by two perpendicular lines, three of which have an area of no less than $1 \mathrm{~cm}^{2}$, and the fourth no less than $2 \mathrm{~cm}^{2}$. What is the minimum length of the other two sides of the rectangle for this to be possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3+2\sqrt{2}}
Evaluate the limit: \[ \lim_{n\to \infty}\left[31^{1/(2n+1)}+(-93)^{1/(2n+3)}-(-75)^{1/(2n+5)}\right]^{n} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given a row of $n$ opaque boxes numbered from $1$ to $n$, a robot starts in box $x$ on the first day and moves to box $(x + ky) \mod n$ on the $k$-th day, where $y$ is the robot's speed. Determine a formula or method to calculate the minimum number of days needed to guarantee finding the robot, based on $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Find \( F''(\pi) \) if \( F(x) = \int_{x}^{\sin(x)} \left(\int_0^{\sin(t)}\sqrt{1+u^4}\,\text{d}u\right)\text{d}t \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Provide an example of a norm on \(\mathbb{R}^2\) that does not satisfy the parallelogram law.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\|(x, y)\| = |x| + |y|}
For all positive integers $ n$ less than $ 2002$, let
\[ a_n = \begin{cases} 11 & \text{if }n\text{ is divisible by }13\text{ and }14 \\
13 & \text{if }n\text{ is divisible by }11\text{ and }14 \\
14 & \text{if }n\text{ is divisible by }11\text{ and }13 \\
0 & \text{otherwise} \end{cases}
\]Calculate $ \sum_{n = 1}^{2001} a_n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
448
}
Evaluate the limit: \[ \lim_{n\to \infty} \int_0^n \left(1+\frac{x}{n}\right)^{-n}\log\left(2+\cos\left(\frac{x}{n}\right)\right)\,dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log 3}
Given a field extension $K \subseteq F$, where $K(\alpha)/K$ is an extension of degree 4 and $\alpha^2$ is not a root of the minimal polynomial $m_{(\alpha,K)}(x)$, determine the degree $[K(\alpha^2):K]$. Use the tower law for field extensions:

$$[K(\alpha^2):K] = [K(\alpha^2):K(\alpha)][K(\alpha):K]$$

It is known that $[K(\alpha):K] = 4$. Find $[K(\alpha^2):K(\alpha)]$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the cardinality of the set of all functions from \( \mathbb{N}^k \) to \( \mathbb{R} \), where \( k \geq 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Inside triangle \(ABC\), there are three circles \(k_1, k_2, k_3\), each tangent to two sides of the triangle and to its incircle \(k\). The radii of \(k_1, k_2, k_3\) are 1, 4, and 9, respectively. Determine the radius of the incircle \(k\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
The diagonal of a square is 4√2 cm. What will be the diameter of another square if their area is double that of the first square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the limiting behavior of the expression \( f(n) = n^2 \left(1 - \left(\frac{c \log n}{n}\right)^2\right)^{n^2} \) as \( n \) approaches infinity, where \( c > 0 \) is a constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let set $A=\{-1, 2, 3\}$, and set $B=\{a+2, a^2+2\}$. If $A \cap B = \{3\}$, then the real number $a=$ ___.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given a homomorphism \( h : \{a, b\}^* \to \{0, 1\}^* \) defined by \( h(a) = 01 \) and \( h(b) = 10 \), and a regular language \( L = L((00 + 1)^*) \), determine the preimage \( h^{-1}(L) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(ba)^*}
Given the function $f(x)=\cos (2x+ \frac {\pi}{4})$, if we shrink the x-coordinates of all points on the graph of $y=f(x)$ to half of their original values while keeping the y-coordinates unchanged; and then shift the resulting graph to the right by $|\varphi|$ units, and the resulting graph is symmetric about the origin, find the value of $\varphi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {3\pi}{16}}
In a ticket counter, females and males arrive according to independent Poisson processes with rates of 30 and 20 customers per hour, respectively. Given that exactly two customers arrived in the first five minutes, calculate the probability that both customers arrived in the first three minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{25}}
A circle with integer radius $r$ is centered at $(r, r)$. Distinct line segments of length $c_i$ connect points $(0, a_i)$ to $(b_i, 0)$ for $1 \le i \le 14$ and are tangent to the circle, where $a_i$, $b_i$, and $c_i$ are all positive integers and $c_1 \le c_2 \le \cdots \le c_{14}$. What is the ratio $\frac{c_{14}}{c_1}$ for the least possible value of $r$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$17$}
Given the positive integers \( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \). Any set of four numbers selected and summed results in the set \(\{44, 45, 46, 47\}\). Determine these five numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{10, 11, 11, 12, 13}.}
The rightmost nonzero digit in the decimal expansion of 101 ! is the same as the rightmost nonzero digit of $n$ !, where $n$ is an integer greater than 101. Find the smallest possible value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{103}
Consider a Markov chain denoted by $P$, where $p_{ii} = 1 - 3a$ and $p_{ij} = a$ for all $i \neq j$, with $1 \leq i, j \leq 4$. Determine the general formula for the $(1,1)$ entry of $P^n$, denoted as $P_{1,1}^n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{4}(1 + 3(1 - 4a)^n)\)}
Color the numbers $2, 3, \cdots, 9$ with one of three colors - red, green, or blue - such that each number is colored differently from its factors. Calculate the total number of different ways this coloring can be done. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{432}
The area of the orthogonal projection of a circle with a radius of 1 on the plane $\alpha$ is 1. Find the length of the orthogonal projection of this circle on a line perpendicular to the plane $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\sqrt{\pi^2 - 1}}{\pi}}
Find the largest \( k \) such that \( 1991^{k} \) divides \( A = 1990^{1991^{1992}} + 1991^{1990^{1992}} + 1992^{1991^{1990}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1991}
正方体的八个顶点中任取 3 个构成三角形，求该三角形是等腰三角形的概率。答案的形式为\(\frac{k}{m}\)，请计算\(k + m\)的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Let $S$ be the graph of $y=x^3$, and $T$ be the graph of $y=\sqrt[3]{y}$. Let $S^*$ be $S$ rotated around the origin $15$ degrees clockwise, and $T^*$ be T rotated around the origin 45 degrees counterclockwise. $S^*$ and $T^*$ will intersect at a point in the first quadrant a distance $M+\sqrt{N}$ from the origin where $M$ and $N$ are positive integers. Find $M+N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
My apologies, I misunderstood your request. I provided the answer along with the harder problem. Here is the harder problem without the solution:

Jeff has 2032 ounces of peanut butter in 16, 28, 40, and 52 ounce jars. He has an equal number of each sized jar. How many jars of peanut butter does Jeff have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
A number from 1 to 144 is chosen. You are allowed to select one subset from the set of numbers from 1 to 144 and ask whether the chosen number is in that subset. For a "yes" answer, you pay 2 rubles; for a "no" answer, you pay 1 ruble. What is the minimum amount of money necessary to guarantee guessing the number correctly?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11 \text{ rubles}}
On a sphere with a radius of 2, three circles with a radius of 1 are positioned, each touching the other two. Find the radius of a smaller circle, also located on the sphere, that touches each of the given circles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \sqrt{\frac{2}{3}}}
Through points \(A(0, 14)\) and \(B(0, 4)\), two parallel lines are drawn. The first line, passing through point \(A\), intersects the hyperbola \(y = \frac{1}{x}\) at points \(K\) and \(L\). The second line, passing through point \(B\), intersects the hyperbola \(y = \frac{1}{x}\) at points \(M\) and \(N\).

What is the value of \(\frac{AL - AK}{BN - BM}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.5}
In trapezoid $ABCD$, the angle bisectors of the exterior angles at vertices $A$ and $D$ intersect at point $M$, and the angle bisectors of the exterior angles at vertices $B$ and $C$ intersect at point $K$. Given that the distance $MK$ is 15 cm, determine the perimeter of trapezoid $ABCD$. Express your answer in centimeters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
In the convex pentagon \(ABCDE\), it is given that \(\angle A = \angle B = \angle D = 90^\circ\). If a circle can be inscribed in this pentagon, determine the measure of \(\angle ADB\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
Let  $a,b$ be integers greater than $1$. Find the largest $n$ which cannot be written in the form $n = 7a + 5b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
Given \( f(x)=a \sin ((x+1) \pi)+b \sqrt[3]{x-1}+2 \), where \( a \) and \( b \) are real numbers and \( f(\lg 5) = 5 \), find \( f(\lg 20) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given the real numbers \( a \), \( b \), and \( c \) such that the inequality \( f(x) = a \cos x + b \cos 2x + c \cos 3x \geq -1 \) holds for any real number \( x \), determine the maximum value of \( a + b + c \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find an $n<100$ such that $n\cdot 2^n-1$ is prime. Score will be $n-5$ for correct $n$, $5-n$ for incorrect $n$ ($0$ points for answer $<5$)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
n=81
}
How many positive integers less than $2023$ cannot be expressed in the form $\lfloor {x^2} \rfloor + \lfloor {x} \rfloor$, where $x$ is a positive real number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
44}
Erin the ant starts at a given corner of a cube and crawls along exactly $7$ edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point. How many paths are there meeting these conditions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
6
}
Find the determinant of the following 5x5 matrix $A$:
\[
A = \begin{pmatrix} 
d &0 &0 &0 &0 \\
d &d &0 &x &x \\
d &d &d &0 &0 \\
d &d &d &d &0 \\
d &d &d &d &d \\
\end{pmatrix}
\] where $d$ and $x$ are constants.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d^5}
Given that (1-2x)^2017 = a0 + a1(x-1) + a2(x-1)^2 +... + a2017(x-1)^2017 (x∈ℝ), find the value of a1 - 2a2 + 3a3 - 4a4 +... - 2016a2016 + 2017a2017. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4034}
Determine the best integer value of $k$ to estimate the function \( e^{\sin x} = 1+x+\frac{x^2}{2!}-\frac{x^4}{4!}+\frac{x^6}{6!}-\quad... \) for \(|x|\le\frac{\pi}{2}\) as \(x \to 0\), using o-notation and retaining three terms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
求最小的正整数 a，使得存在整数 b,c，满足二次三项式 ax^2+bx+c 有两个不超过 \df{1}{1000} 的不同正根.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1001000}
There are \( 4n \) circles of the same radius on a plane, where \( n \) is a positive integer. No two circles are tangent to each other, and each circle intersects with at least three other circles. Let \( f(n) \) represent the number of intersection points of these circles. Determine the minimum value of \( f(n) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12n}
The national security agency's wiretap recorded a conversation between two spies and found that on a 30-minute tape, starting from the 30-second mark, there was a 10-second segment of conversation containing information about the spies' criminal activities. Later, it was discovered that part of this conversation was erased by a staff member. The staff member claimed that he accidentally pressed the wrong button, causing all content from that point onwards to be erased. What is the probability that the conversation containing criminal information was partially or completely erased due to pressing the wrong button?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{45}}
设非负实数 a,b,c,x,y,z 满足 a+b+c=x+y+z=1. 求 (a-x^2)(b-y^2)(c-z^2) 的最大值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Given the inequality $x\ln x - kx > 3$, which holds true for any $x > 1$, determine the maximum value of the integer $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
A fly trapped inside a cubical box with side length $1$ meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. What is the maximum possible length, in meters, of its path?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$4\sqrt{2}+4\sqrt{3}$}
Calculate the indefinite integral:

\[ 
\int \frac{3 x^4 + 3 x^3 - 5 x^2 + 2}{x(x-1)(x+2)} \, dx 
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3x^2}{2} - \ln|x| + \ln|x-1| + \ln|x+2| + C}
30. [15] The function $f: \mathbb{Z}^{2} \rightarrow \mathbb{Z}$ satisfies
- $f(x, 0)=f(0, y)=0$, and
- $f(x, y)=f(x-1, y)+f(x, y-1)+x+y$
for all nonnegative integers $x$ and $y$. Find $f(6,12)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{77500}
Find the remainder when \( \prod_{i = 1}^{1903} (2^i + 5) \) is divided by \( 1000 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{931}
Let \( \lambda \) the positive root of the equation \( {t}^{2} - {1998t} - 1 = 0 \) . It is defined the sequence \( {x}_{0},{x}_{1},{x}_{2},\ldots ,{x}_{n},\ldots \) b)
 \( {x}_{0} = 1,{x}_{n + 1} = \left\lfloor  {\lambda {x}_{n}}\right\rfloor \) for \( n = 1,2\ldots \) Find the remainder of the division of \( {x}_{1998} \) by 1998.
 Note: \( \lfloor x\rfloor \) is the greatest integer less than or equal to \( x \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1000
}
If the graph of the function $f(x) = |\sin(\omega x - \frac{\pi}{8})| (\omega > 0)$ is shifted to the left by $\frac{\pi}{12}$ units, and the resulting graph corresponds to an even function, then the minimum value of $\omega$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
Determine the conditions under which $n^{\aleph_0}$ equals the cardinality of the continuum, where $n$ is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \geq 2}
Calculate the surface integral of the vector field \( \mathbf{G} = \begin{pmatrix} y \\ z \\ x \end{pmatrix} \) over a sphere \( S \) using the divergence theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}