inclusionAI/Ring-lite-rl-data huggingface dataset:
A deck of $2n$ cards numbered from $1$ to $2n$ is shuffled, and $n$ cards are dealt to each player $A$ and $B$. The players, $A$ and $B$, alternately discard a card face up, starting with $A$. The game ends when the sum of the discarded cards is first divisible by $2n + 1$, and the last person to discard wins. What is the probability that $A$ wins if neither player makes a mistake?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit: \[ \lim_{n \to \infty}\left(\frac{1^{1/x}+2^{1/x}+\ldots+n^{1/x}}{n}\right)^{nx} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
In $\triangle ABC$, if $\sin A:\sin B:\sin C=7:8:13$, then $C=$______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^\circ}
Let

$$
2^{x}=\left(1+\tan 0.01^{\circ}\right)\left(1+\tan 0.02^{\circ}\right)\left(1+\tan 0.03^{\circ}\right) \ldots\left(1+\tan 44.99^{\circ}\right)
$$

Find \( x \). If necessary, round the answer to the nearest 0.01.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["2249.5"]}
Given that point $P(1,-1)$ is on the terminal side of angle $φ(-π < φ < 0)$, determine the interval where the function $y=3\cos (x+φ)$, $x∈[0,π]$ is strictly decreasing.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\frac {\pi}{4},\pi]}
Find a subset of \( \mathbb{R} \) that is homeomorphic to its complement, where the complement is endowed with the subspace topology, and the subset is not open.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\bigcup_{n \in \mathbb{Z}} [2n, 2n + 1)}
$4\cos {50}^{0}-\tan {40}^{0}=$_______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
After eating a hearty meal they went to see the Buckingham palace. There, Rachel learned that 705 visitors came to the Buckingham palace that day. If there were some visitors the previous day and 457 visitors the day before that. How many visitors were there the previous day if there were 57 more visitors on that day than on the previous 245 days?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{514}
Determine how many roots of the equation \( z^4 + z^3 + 1 = 0 \) are located in the first quadrant of the complex plane using Rouche's Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In the plane of square ABCD, there is a point P such that △PAB, △PBC, △PCD, and △PDA are all isosceles triangles. How many such points P are there? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
For a real-valued random variable $X \geq 0$, we have the inequality from Jensen's inequality:

$$1 - \frac{1}{1+E \left[x\right]} \geq E \left[ 1 - \frac{1}{1+x} \right].$$

Find the tightest possible constant $\epsilon_1$ such that:

$$ \left| 1 - \frac{1}{1+E \left[x\right]} - E \left[ 1 - \frac{1}{1+x} \right] \right| \leq \epsilon_1 \cdot \text{Var}(x).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the kernel of the ring homomorphism $\psi: k[p, q, r] \rightarrow k[u^2, v^2, uv]$ defined by $\psi(p) = u^2$, $\psi(q) = uv$, and $\psi(r) = v^2$, where $k$ is a field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(q^2 - pr)}
Twenty percent off 60 is one-third more than what number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
36
}
Call an integer interesting if the integer is divisible by the sum of its digits. For example, $27$ is divisible by $2 + 7 = 9$, so $27$ is interesting. How many $2$-digit interesting integers are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
A box contains 5 balls marked with the number 1, 3 balls marked with the number 3, and 2 balls marked with the number 9. We randomly select a ball 100 times with replacement. What is the probability that the product of the numbers on the selected 100 balls is between \(3^{60}\) and \(3^{100}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.91}
Find the smallest positive real number $d,$ such that for all nonnegative real numbers $x, y,$ and $z,$
\[
\sqrt{xyz} + d |x^2 - y^2 + z^2| \ge \frac{x + y + z}{3}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
C2) Let $n$ be a strictly positive integer. Roger has a rectangular garden of dimensions $(2 n+1) \times(2 n+1)$, and he places barriers to divide his garden into several rectangular cells. Once completed, he will have thus formed exactly two horizontal rectangles $k \times 1$ and two vertical rectangles $1 \times k$ for each even integer $k$ between 1 and $2 n+1$, as well as a single $1 \times 1$ square. How many ways can Roger do this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n}
A coin is flipped multiple times until we get an odd number of heads followed by a tail. Given $n \in \mathbb{N}^{*}$, find the number of sequences of $n$ flips.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F_{n-1}}
A circle of radius 2 rolls along a flat surface without slipping. A fixed point on the circle, located 3 units away from its center, moves along a curve. Let $A$ be the midpoint of the arc where the point is at its highest point, and let $B$ be a point where the path intersects the surface. Calculate the ratio $\frac{v_A}{v_B}$, where $v_A$ and $v_B$ are the speeds of the point at $A$ and $B$, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5}}
The number 890 has the following property: by changing any of its digits by 1 (either increasing or decreasing), one can obtain a number that is a multiple of 11. Find the smallest three-digit number with the same property.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
# 8. Problem 8

Given trapezoid $A B C D$. A line parallel to the bases intersects the lateral sides $A B$ and $C D$ at points $M$ and $N$ respectively. A point $E$ is taken on the base $B C$. Segments $A E$ and $E D$ intersect $M N$ at points $S$ and $T$. The areas of triangles $A M S, S E T, T N D$ are 12, 8, and 15, respectively. What is the minimum possible area of the trapezoid under these conditions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{125}
Evaluate the limit: \[ \lim_{R\rightarrow \infty} \int_{C_R} \frac{6z^6 + 5z^5}{z^6 + z^5 + 10}dz, \] where \( C_R=\{z \in \mathbb{C} : |z|=R  \} \) for \( R>0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi i}
Compute the remainder when $$\sum_{k=1}^{30303} k^{k}$$ is divided by 101.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["29"]}
The income of a person is composed of three different sources (A, B, and C) with the following ratios to the total income: 3:2:1. The income from source A is taxed at 10%, source B at 15%, and source C is tax-free. If the person's total income is Rs. 18000, and their expenditures are in the ratio of 5:4 with their income, determine their total savings after paying taxes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6200}
Find the value of $x$ such that the following equation holds: $$\frac{2}{\sqrt{\pi}}\int\limits_{0}^{x}e^{-t^2}dt = 0.7$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.733}
Let $Y_1 < ... < Y_8$ be the order statistics of $n$ independent observations from a continuous distribution with a 70th percentile of $\pi_{0.7} = 27.3$. Calculate the probability $P(Y_7 < 27.3)$. Provide your answer as a probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.2553}
Compute the number of positive integers that divide at least two of the integers in the set $\{1^1,2^2,3^3,4^4,5^5,6^6,7^7,8^8,9^9,10^{10}\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Let $A$ be a $2n \times 2n$ matrix, with entries chosen independently at random. Every entry is chosen to be 0 or 1, each with probability $1/2$. Find the expected value of $\det(A-A^t)$ (as a function of $n$), where $A^t$ is the transpose of $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(2n)!}{4^n n!}}
Let $ABC$ be a triangle with $BC=49$ and circumradius $25$. Suppose that the circle centered on $BC$ that is tangent to $AB$ and $AC$ is also tangent to the circumcircle of $ABC$. Then
\[\dfrac{AB \cdot AC}{-BC+AB+AC} = \frac{m}{n}\]
where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250049}
Given that the origin is inside the ellipse \( k^{2} x^{2}+y^{2}-4 k x+2 k y+k^{2}-1=0 \), what is the range of the parameter \( k \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0<|k|<1}
(5) Let two regular tetrahedra $P-ABC$ and $Q-ABC$ be inscribed in the same sphere. If the dihedral angle between a lateral face and the base of the regular tetrahedron $P-ABC$ is $45^{\circ}$, then the tangent value of the dihedral angle between a lateral face and the base of the regular tetrahedron $Q-ABC$ is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the leading order behavior of \( \left(\frac{d^n}{dx^n}\right) \left.\Gamma(x)\right|_{x=1} \) as \( n \rightarrow \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1)^n n!}
In a company, some pairs of people are friends (if $A$ is friends with $B$, then $B$ is friends with $A$). It turns out that among every set of 100 people in the company, the number of pairs of friends is odd. Find the largest possible number of people in such a company.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101}
Evaluate the surface integral \( \iint_S f(x,y,z) \, dS \) where \( S \) is the part of the cone \( z = \sqrt{x^2+y^2} \) above the circle \( x^2+y^2 \leq 2x \) and \( f(x,y,z) = x^2+y^2 + z^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{2}\pi}
Given the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) with left and right foci as \(F_{1} 、 F_{2}\), a line passing through \(F_{2}\) with an inclination angle of \(\frac{\pi}{4}\) intersects the hyperbola at a point \(A\). If the triangle \(	riangle F_{1} F_{2} A\) is an isosceles right triangle, find the eccentricity of the hyperbola. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}+1}
Evaluate the integral \( \frac{1}{2\pi i}\int_\mathcal{C} |1+z+z^2+\cdots+z^{2m-1}|^2~dz \) where \(\mathcal{C}\) is the unit circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2m - 1}
Find the maximum possible value of $8 \cdot 27^{\log_6 x} + 27 \cdot 8^{\log_6 x} - x^3$ as $x$ varies over the positive real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{216}
Calculate the upper bound for the integral: $$\int_{0}^{+\infty}e^{-A\sqrt{x^2+B}}\cos(Cx)\frac{x'}{\sqrt{x'^2-B}}dx'$$ where A, B, and C are real numbers, and A and B are positive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{e^{-A\sqrt{B}}}{A}}
A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of 50 meters per second,while
 each of the spiders has a speed of \( r \) meters per second. The spiders choose their starting positions,and choose the fly’s
 starting position, with the requirement that the fly must begin at a vertex. Each bug knows the position of each other bug at all
 times, and the goal of the spiders is for at least one of them to catch the fly. What is the maximum \( c \) so that for any \( r < c \) , the
 fly can always avoid being caught?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$c = 25$}
Given a Markov chain $X$ with state space $E = \{1, 2, 3, 4\}$ and transition matrix
$$P = \begin{bmatrix}
0 & 0 & 1 & 0 \\
0 & 0.4 & 0.6 & 0\\
0.8 & 0 & 0.2 & 0\\
0.2 & 0.3 & 0 & 0.5\\
\end{bmatrix}$$
compute the expected value $E[f(X_5)f(X_6)|X_4 = 4]$, where the function $f$ has values $2, 4, 7,$ and $3$ at states $1, 2, 3,$ and $4$ respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14.41}
若方格 P 的左右两侧分别有 a,b 个方格，上下两侧分别有 c,d 个方格，其中 a,b,c,d 是正整数，且满足 (a-b)(c-d)=0，则称由这些方格所组成的图形为一个"十字星".现有一张由 2014 个方格所组成的 38\times53 方格表，求该方格表中十字星的个数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13483236}
Evaluate the double integral \( \iint_D |x-y| \, dx \, dy \), where the region \( D \) is defined by \( D= \{(x, y) \in \mathbb{R}^2 : x \ge 0, 0 \le y \le 3-2x \} \). Determine the appropriate limits of integration and handle the modulus in the integrand.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{15}{8}}
Calculate the value of the infinite product \( \prod_{n=0}^\infty a_n \), where the sequence \( \{a_n\} \) is defined recursively by \( a_0 = \frac{1}{2} \) and \( a_n = 1 + (a_{n-1} - 1)^2 \) for \( n \geq 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Determine the degree of the field extension $[F_p(X,Y) : F_p(X^p,Y^p)]$, where $F_p$ is a finite field of characteristic $p$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^2}
A list of integers has mode 32 and mean 22. The smallest number in the list is 10. The median  m of the list is a member of the list. If the list member  m were replaced by  m+10, the mean and median of the new list would be 24 and  m+10, respectively. If  m were instead replaced by  m-8, the median of the new list would be  m-4. What is  m?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Find the closure of the open interval $(0, 1)$ in the lower limit topology $\tau_l$ on $\mathbb{R}$, which is generated by the subbasis consisting of all half-open intervals $[a, b)$. Recall that the closure of a set is the smallest closed set containing that set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 1)}
Given a mapping f: {1, 2, 3} → {1, 2, 3} that satisfies f(1) < f(2) < f(3) and f(f(x)) = 3x, find the value of f(1). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Tom's algebra notebook consists of 50 pages, with pages 1 and 2 on the front and back of the first sheet of paper, pages 3 and 4 on the second sheet, and so on. One day, Tom left the notebook on the table while he went out, and his roommate took away several consecutive pages. The average of the remaining page numbers is 19. Calculate the number of pages that the roommate took away.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Determine the minimum number of questions needed to find the positions of all numbers in a 64-cell grid, where each cell contains a unique number from 1 to 64. Each question allows you to find out the list of numbers situated in the cells you choose. What is the minimum number of questions required?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Evaluate the expression: $$\frac{29\int_0^1 (1-x^4)^7\,dx}{4\int_0^1 (1-x^4)^6\,dx}$$ without expanding.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Evaluate the improper integral: $$\int_{-\infty}^{\infty} \frac{\cos^2 x}{(x^2-\pi ^2)} dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Suppose $A B C D$ is a convex quadrilateral with $\angle A B D=105^{\circ}, \angle A D B=15^{\circ}, A C=7$, and $B C=C D=5$. Compute the sum of all possible values of $B D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{291}}
In a plane, there are 2018 points such that all pairwise distances between them are distinct. For each point, mark the closest point among the remaining points. What is the minimal number of points that are marked?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the order of the pole of the product of two holomorphic functions \( h(z) \) and \( j(z) \), where \( h(z) \) has a zero of order \( m > 0 \) and \( j(z) \) has a zero of order \( n > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m + n}
Consider \(A \in \mathcal{M}_{2020}(\mathbb{C})\) such that
\[ A + A^{\times} = I_{2020} \]
\[ A \cdot A^{\times} = I_{2020} \]
where \(A^{\times}\) is the adjugate matrix of \(A\), i.e., the matrix whose elements are \(a_{ij} = (-1)^{i+j} d_{ji}\), where \(d_{ji}\) is the determinant obtained from \(A\), eliminating the row \(j\) and the column \(i\).

Find the maximum number of matrices verifying these conditions such that any two of them are not similar.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{673}
If the circle $x^2+y^2=4$ is externally tangent to the circle $(x-t)^2+y^2=1$, then the value of the real number $t$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm3}
It is now 3:15:20 PM, as read on a 12-hour digital clock. In 305 hours, 45 minutes, and 56 seconds, the time will be $X:Y:Z$. What is the value of $X + Y + Z$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Compute the integral \( \int_0^\infty \mathrm{si}(x)\mathrm{d}x \), where \( \mathrm{si}(x) = \int_x^\infty \frac{\sin(t)}{t}\mathrm{d}t \) for all \( x > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the number of ways to make change for a dollar using generating functions. Specifically, determine the coefficient of the \(x^{100}\) term in the expansion of the generating function \(\frac{1}{(x-1)(x^5-1)\cdots(x^{50}-1)}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{292}
Let $P(x)$ be a polynomial of degree at most 6 that satisfies $P(k) = k!$ for $k = 0, 1, 2, 3, 4, 5, 6$. Compute the value of $P(7)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3186}
Find the closure of the set \( M = \{(a_1, a_2, \ldots) : \text{all but finitely many } a_i = 0\} \) in the space \( l^{\infty} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c_0}
How many ways can one color the squares of a $6 \times 6$ grid red and blue such that the number of red squares in each row and column is exactly 2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["67950"]}
Find an integer $n$ such that there exist natural numbers $k \geq 2$ and positive rational numbers $a_1, a_2, \ldots, a_k$ satisfying the equation $a_1 + a_2 + \cdots + a_k = a_1 a_2 \cdots a_k = n$. What is the value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Along the diagonals of the bases \( AC \) and \( B_1D_1 \) of the cube \( ABCD A_1B_1C_1D_1 \) with side length \( a \), two ants, Gosha and Lesha, are crawling. They started simultaneously from points \( A \) and \( B_1 \) respectively, with constant speeds. Lesha's speed was twice the speed of Gosha, and they finished when Lesha reached point \( D_1 \). What was the minimum distance between Gosha and Lesha during their movement?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \sqrt{\frac{11}{10}}}
For any finite non empty set $X$ of integers, let $max(X)$ denote the largest element of $X$ and $|X|$ denote the number of elements in $X$. If $N$ is the number of ordered pairs $(A, B)$ of finite non-empty sets of positive integers, such that
$$\max(A) \times |B| = 12;$$
$$|A| \times \max(B) = 11$$
and $N$ can be written as $100a + b$ where $a, b$ are positive integers less than 100, find $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
43
}
Determine the Girsanov kernel for the process $X_t = W_t + \cos t$, where $W_t$ is a standard $\mathbb{P}$-Brownian motion, such that $X_t$ becomes a standard $\mathbb{Q}$-Brownian motion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\sin t}
Compute the intersection multiplicity of the elliptic curve given by \(x^3 + xz^2 + z^3 + y^2z + yz^2 = 0\) and the curve \(x = 0\) at the point \((0, 1, 1)\). Note: \(f(x, y, z) = x^3 + xz^2 + z^3 + y^2z + yz^2 \in \mathbb{F}_2[x, y, z]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit: $$\lim_{n \to \infty} n \int_0^\pi \{ -x \}^n \, dx$$ where $\{ x \}$ denotes the fractional part of the real number $x$. Determine the value of this limit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the integral \( \int_0^{2\pi} \frac{e^{|\sin x|}\cos(x)}{1+e^{\tan x}} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the function $f(x)= \sqrt{3}\sin 2\omega x-\cos 2\omega x$ where $\omega\in(0,1)$, if the graph of $f(x)$ passes through the point $\left( \dfrac{\pi}{6},0\right)$, then the interval of monotonic increase of $f(x)$ on the interval $[0,\pi]$ is \_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[0, \dfrac{2\pi}{3}\right]}
$\mathrm{Compute}\sum_{i=0}^{\frac{q-1}{2}}\left\lfloor\frac{ip}{q}\right\rfloor+\sum_{j=0}^{\frac{p-1}{2}}\left\lfloor\frac{jq}{p}\right\rfloor\mathrm{if}p=51\;\mathrm{and}\;q=81.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1001
}
For a set $S \subseteq \mathbb{N}$, define $f(S) = \{\left\lceil \sqrt{s} \right\rceil \mid s \in S\}$. Find the number of sets $T$ such that $\vert f(T) \vert = 2$ and $f(f(T)) = \{2\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
5043}
Determine the winding number of the function \( f(z) = z^3 + \frac{1}{(z-1)^2} \) around the origin as \( z \) moves counterclockwise along the circle \( |z| = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the limit of the solution to the initial value problem \(x'(t)=Ax(t)+g(t)\) with \(x(0)=x_0\), given that all eigenvalues of the matrix \(A\) have negative real parts (\(\mathrm{Re}(\alpha_j)<0\)), and \(\lim_{t\to\infty} |g(t)|=g_0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-A^{-1} g_0}
In cube \( ABCDA_1B_1C_1D_1 \), point \( M \) is located on edge \( AB \). A rectangle \( MNLK \) is inscribed in square \( ABCD \) such that vertex \( M \) is on \( AB \) and the other vertices \( N, L, K \) are on different sides of the square. The rectangle \( M_1N_1L_1K_1 \) is the orthogonal projection of \( MNLK \) onto the top face \( A_1B_1C_1D_1 \). Given that the diagonals of quadrilateral \( MK_1L_1N \) are perpendicular, determine the ratio \( AM:MB \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}