inclusionAI/Ring-lite-rl-data huggingface dataset:
8*. Among the numbers from 1 to 1000, how many are divisible by 4 but do not have the digit 4 in their representation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{162}
The diameter $[AC]$ of a circle is divided into four equal segments by points $P$, $M$, and $Q$. Consider a segment $[BD]$ that passes through $P$ and intersects the circle at points $B$ and $D$, such that $PD = \frac{3}{2} AP$. Given that the area of triangle $[ABP]$ is $1$ cm$^2$, calculate the area of quadrilateral $[ABCD]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Suppose that f is a function $f : R_{\ge 0} \to R$ so that for all $x, y \in R_{\ge 0}$ (nonnegative reals) we have that $$f(x)+f(y) = f(x+y+xy)+f(x)f(y).$$ Given that $f\left(\frac{3}{5} \right) = \frac12$ and$ f(1) = 3$, determine
$$\lfloor \log_2 (-f(10^{2021} - 1)) \rfloor.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10104}
$5$ 级方阵 $A$ 满足: 第一行元素均为 $1$, 第二行元素从左至右依次为 $1,-1,1,-1,1$,且 $\operatorname{trace}\left(A A^{\prime}\right)=28$. 记 $\operatorname{det} A$ 的最大值为 $M$, 则 $[M]=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Compute the minimal number of generators, \( \mu(mA_m) \), of the module \( mA_m \), where \( m \) is the maximal ideal \( (\bar{x} - 1, \bar{y} - 1) \) of \( A = \mathbb{C}[x,y]/(x^3-y^2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given integers \(a, b\) and odd prime numbers \(p, q\), find the total number of distinct integer solutions \((a, b)\) to the equation \(a^2 + 2b^2 = p^{11}q^{13}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{336}
已知 \(y^{2}=4x\)，过 \(A(-2,3)\) 做抛物线两条切线，交 y 轴于 \(B, C\) 两点，求 \(\triangle ABC\) 的外接圆方程。答案的形式为\((x + \frac{k}{2})^2 + (y - \frac{m}{2})^2 = \frac{n}{p}\)，请给出k + m + n + p的值，其中\(\frac{n}{p}\)为最简分数。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
During a certain play's run, the performances were spread across three weeks, and tickets were sold at varying ratios of full price to reduced price. During the first week, 5400 tickets were sold, all at a reduced price. In the second week, twice as many tickets were sold at full price as were sold at the reduced price, and a total of 7200 tickets were sold during the week. In the third week, seven times as many tickets were sold at the full price as were sold at reduced price, and a total of 13400 tickets were sold during the week.

Additionally, there were three levels of ticket pricing: reduced, regular full price, and premium full price. During the first week, tickets were sold only at the reduced price. During the second week, 1 out of every 10 full-priced tickets was sold at the premium price, and the rest were sold at the regular full price. In the third week, 2 out of every 15 full-priced tickets were sold at the premium price, and the rest were sold at the regular full price. 

If a total of 25200 tickets were sold during the entire play's run, what is the total number of tickets sold at full price (both regular and premium combined) during these three weeks?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16525}
Positive integers $a$, $b$, $c$, and $d$ are such that $a<b<c<d$, and the system of equations 
\[ 2x + y = 2007 \quad\text{and}\quad y = |x-a| + |x-b| + |x-c| + |x-d| \]
has exactly one solution. What is the minimum value of $d$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{504}
Out of 1500 people surveyed, $25\%$ do not like television, and out of those who do not like television, $15\%$ also do not like books. How many people surveyed do not like both television and books?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
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 $7$? Find the total number of such integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{133}
An integer, whose decimal representation reads the same left to right and right to left, is called symmetrical. For example, the number 513151315 is symmetrical, while 513152315 is not. How many nine-digit symmetrical numbers exist such that adding the number 11000 to them leaves them symmetrical?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8100}
Two given quadratic trinomials differ by exchanging the free term and the second coefficient. The sum of these trinomials has a unique root. What value does this sum take at one?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
4. The unit square in the first quadrant of the coordinate plane ( $0 \leq x, y \leq 1$ ) is divided into squares with side length $2 \cdot 10^{-4}$. How many nodes of this partition (inside the unit square) lie on the parabola $y=x^{2} ?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
Sheela deposits Rs. 4500 in her bank savings account. If this is a certain percentage of her monthly income, and her monthly income is Rs. 16071.42857142857, what is the percentage of her income that she deposits in her bank savings account?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28\%}
A shopkeeper applies an initial 10% discount on an article's cost price, followed by a 5% tax on the discounted price. Then, the shopkeeper sells the article with a 20% profit for $1800. Find the original cost price.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$1500}
There are 3,250 more red apples than green apples in a grocery store. A truck arrives and delivers another 3,600 green apples and 1,300 red apples. Simultaneously, 750 green apples and 810 red apples are sold to customers. If there were originally 1,200 green apples, how many more green apples than red apples are there in the store now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{890}
Given an arithmetic sequence $\{a_n\}$ with a common difference $d \neq 0$ and the first term $a_1 = d$, the sum of the first $n$ terms of the sequence $\{a_n^2\}$ is $S_n$. A geometric sequence $\{b_n\}$ has a common ratio $q$ less than 1 and consists of rational sine values, with the first term $b_1 = d^2$, and the sum of its first $n$ terms is $T_n$. If $\frac{S_3}{T_3}$ is a positive integer, then find the possible value of $q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Let $(a_1,a_2,a_3,\ldots,a_{15})$ be a permutation of $(1,2,3,\ldots,15)$ for which
$a_1>a_2>a_3>a_4>a_5>a_6>a_7 \mathrm{\ and \ } a_7<a_8<a_9<a_{10}<a_{11}<a_{12}<a_{13}<a_{14}<a_{15}.$
An example of such a permutation is $(7,6,5,4,3,2,1,8,9,10,11,12,13,14,15).$ Find the number of such permutations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3003}
Let point \( O \) be located inside a regular tetrahedron \( ABCD \), and let \( G \) be the centroid of the tetrahedron. The line \( OG \) intersects each face of the tetrahedron at points \( A_1, B_1, C_1, \) and \( D_1 \). What is the value of \( \frac{A_1 O}{A_1 G} + \frac{B_1 O}{B_1 G} + \frac{C_1 O}{C_1 G} + \frac{D_1 O}{D_1 G} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let \(\mu = e^{2\pi i/9}\). Define \(u_j = (\mu^j, \mu^{2j}, \mu^{3j}, \ldots, \mu^{9j})^T \in \mathbb{C}^9\) for \(j = 1, \ldots, 9\). Let \(V\) be the vector subspace of \(\mathbb{C}^9\) spanned by \(\{u_2, u_3, u_4, u_5, u_6, u_7\}\). It is known that there exists a matrix \(A \in M_{3 \times 6}(\mathbb{C})\) such that \(V = \{(x_1, \ldots, x_9) \in \mathbb{C}^9 : (x_3, x_6, x_9)^T = A(x_1, x_2, x_4, x_5, x_7, x_8)^T\}\). If the statement "On a special kind of 6-dimensional vector subspace of \(\mathbb{C}^9\)" is true, then \(A\) is unique and has the form \(A = \begin{pmatrix} a_1 & a_2 & a_3 & a_4 & a_5 & a_6 \\ a_5 & a_6 & a_1 & a_2 & a_3 & a_4 \\ a_3 & a_4 & a_5 & a_6 & a_1 & a_2 \end{pmatrix}\). Let \(L\) be the smallest subfield of \(\mathbb{C}\) generated by the entries of \(A\). It is known that \(L \subseteq \mathbb{Q}(\mu)\), and hence \(L/\mathbb{Q}\) is a Galois extension. Determine the extension degree \([L : \mathbb{Q}]\). Given that \([\mathbb{Q}(\mu) : \mathbb{Q}] = \phi(9) = 6\), the possible values for \([L : \mathbb{Q}]\) are 1, 2, 3, or 6. Find the exact value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Regular decagon `ABCDEFGHIJ` has its center at `K`. Each of the vertices and the center are to be associated with one of the digits `1` through `10`, with each digit used once, in such a way that the sums of the numbers on the lines `AKE`, `BKF`, `CKG`, `DLH` and `EJI` are all equal. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3840}
Two identical rectangles $PQRS$ and $P_{1}Q_{1}R_{1}S_{1}$ are inscribed in the triangle $ABC$ such that points $P$ and $P_{1}$ lie on side $AB$, points $Q$ and $Q_{1}$ lie on side $BC$, and points $R, S, R_{1}$, and $S_{1}$ lie on side $AC$. It is known that $PS = 12$ and $P_{1}S_{1} = 3$. Find the area of triangle $ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{225}{2}}
Given an arithmetic sequence $\{a_n\}$, let the sum of the first $n$ terms be $S_n$. If $S_{m-1} = -2$, $S_m = 0$, and $S_{m+1} = 3$, find the value of $m$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given the three-view diagrams of a solid, determine the volume of the solid. The side view is an isosceles right triangle with leg length 1. The top view is an equilateral triangle with a base length of 1 and a height of 1. The front view is an isosceles triangle with a base length of 1 and a height of 1. Calculate the volume of the solid. Express your answer as a fraction or decimal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{6}}
An $n$-sided regular polygon with side length $1$ is rotated by $\frac{180^\circ}{n}$ about its center. The intersection points of the original polygon and the rotated polygon are the vertices of a $2n$-sided regular polygon with side length $\frac{1-\tan^2 10^\circ}{2}$. What is the value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
In an acute-angled triangle $ABC$, let $H$ be the orthocenter. From points $A$ and $C$, tangents $AK$ and $CT$ are drawn to the circle with diameter $BH$. Given that the lengths of these tangents are $15$ and $19$ respectively, find the square of the smallest possible length of side $AC$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{586}
Consider a $4$ state Markov Chain with the row-stochastic transition matrix $$P = \left(\begin{array}{cccc} 1/4&1/4&1/4&1/4 \\ 1/2&0&0&1/2 \\ 1/3&1/3&1/3&0 \\ 0&0&0&1 \end{array} \right).$$ If the chain starts in state $1$, find the expected length of time intervals until absorption in state $4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
$n$ 是任意自然数，$1^{3}+2^{3}+3^{3}+\cdots+n^{3}$ 的个位数不可能是______。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral $\int_{-\pi}^{\pi} \left(\frac{1+r^2}{1 - 2r\cos \theta + r^2}-\frac{(1-r^2)^2}{(1 - 2r\cos \theta + r^2)^2}\right) d\theta$ for $r \in \mathbb{R}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the value of $n$ for which the following integral is equal to $\frac{1}{\sqrt{n}}$:

$$\int_{-\infty}^{\infty}(-1)^xe^{\frac{-\pi x}{2}(nx+1)}dx=\frac{1}{\sqrt{n}}$$

Assume that $n > 0$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{4\ln 2}}
Determine the values of $i$ for which the homology groups $H_i(K^{(p)})$ and $H_i(K)$ are isomorphic.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{i < p}
Let $z$ and $z+1$ be complex numbers such that both are $n^{\text{th}}$ roots of unity. Given that $n$ is a multiple of 5, find the minimum value of $n + z^3$. \( \text{(Express your answer as a single integer.)} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
集合 $A=\{1,2, \cdots, n\}$.若存在非空集合 $B C$, 使得 $B \cap C=\varnothing, B \cup C=A$, 且 $B$ 中元素的平方和为 $M, C$ 中元素的平方和为 $N$ ，满足 $M-N=2016$ ，求 $n$ 的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Let S<sub>n</sub> represent the sum of the first n terms of the geometric sequence {a<sub>n</sub>}. If a<sub>1</sub> = 1, S<sub>3</sub> = $$\frac{3}{4}$$, then S<sub>4</sub> = \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{8}}
A retailer buys 120 pens at the market price of 100 pens from a wholesaler. If the wholesaler offers an additional 5% discount on bulk purchases and the retailer sells these pens giving a discount of 4%, what is the profit%?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21.26\%}
Let $\mathbb{N}_0$ be the set of non-negative integers. Consider a triple $(f,a,b)$, where $f$ is a function from $\mathbb{N}_0$ to $\mathbb{N}_0$ and $a,b \in \mathbb{N}_0$ satisfying the following conditions:

1. $f(1) = 2$
2. $f(a) + f(b) \leq 2\sqrt{f(a)}$
3. For all $n > 0$, $f(n) = f(n-1)f(b) + 2n - f(b)$

Find the sum of all possible values of $f(b+100)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10201}
Three segments are drawn from a point inside an equilateral triangle, each parallel to one of the triangle's sides, with lengths 2 m, 3 m, and 6 m. Find the perimeter of the equilateral triangle. Express your answer in meters as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Find the number of matrices in \(GL_{3}(\mathbb{Z}_{3})\) such that \([1 \ 1 \ 1]^{T}\) is an eigenvector.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{864}
Provide an example of a function \( f:[a, b] \to \mathbb{R} \) such that \( f \in R[a, c] \) for all \( c \in [a, b) \), but \( f \notin R[a, b] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{b - x}}
For any positive integer $x$, let $f(x) = x^x$. Suppose that $n$ is a positive integer such that there exists a positive integer $m$ with $m \neq 1$ such that $f(f(f(m))) = m^{m^{n+2020}}$. Compute the smallest possible value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13611}
Given the matrix \( A = \begin{bmatrix}\lambda & 1 & 0 & \ldots & 0 \\
0 & \lambda & 1 & \ldots & 0 \\
\vdots & \vdots & \lambda & \ddots & 1 \\
0 & \ldots & \ldots & \ldots & \lambda \end{bmatrix} \), find the minimal polynomial of \( A \). The characteristic polynomial is given by \((\lambda-x)^n=0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x - \lambda)^n}
The value of an item is $C$ dollars. After a loss of $x \%$ of the selling price, it is sold for $100. Then, it is sold again with a profit of $x \%$ of the new selling price $S$. If the difference between $S$ and $C$ is $1 \frac{1}{9}$ dollars, what is $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Consider the discrete-time state-space system:
\[x(t+1) = Ax(t)+Bu(t), \quad y(t) = Cx(t)\]
with
\[
A =\begin{bmatrix}
0 & 1 & 0 \\
1 & 0 & 1 \\
1 & 1 & 0 \\
\end{bmatrix},  
\quad B =\begin{bmatrix}
0 \\
0 \\
1 \\
\end{bmatrix},  
\quad C =\begin{bmatrix}
1 & 0 & 0\\
\end{bmatrix}
\]
Determine the minimal time \(T \in \mathbb{N}\) such that there exists an input signal \(u : \mathbb{N} \rightarrow \mathbb{R}\) for which the state \(x(T,0,u) = x_{f}\) with \(x_{f} = \begin{bmatrix} 0 & 1 & 1\end{bmatrix} ^{T}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
At a certain university, 70% of the professors are women, and 70% of the professors are tenured. Among the tenured professors, 15% are above the age of 60. In addition, 50% of the professors are associate or full professors. If 90% of the professors are women, tenured, or both, and 80% of the associate or full professors are men, then what percent of the male associate or full professors are tenured?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50\%}
A basketball player scored 18, 22, 15, and 20 points respectively in her first four games of a season. Her points-per-game average was higher after eight games than it was after these four games. If her average after nine games was greater than 19, determine the least number of points she could have scored in the ninth game.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
在正三棱锥 $P-A B C$ 中, 已知侧棱与底面所成的角为 $45^{\circ}$ 。则相邻两侧面所成角的余弦值为 $\qquad$.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{6}
Let \(y=f(x)\) be an infinitely differentiable function on real numbers such that \(f(0) \neq 0\) and \(\frac{d^n(y)}{dx^n} \neq 0\) at \(x=0\) for \(n=1,2,3,4\). If \[\lim_{x\to0} \frac{f(4x) + af(3x) + bf(2x) + cf(x) + df(0)}{x^4}\] exists, find the value of \(25a + 50b + 100c + 500d\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
From a point given on the circumference of a circle, a diameter and a chord equal to the radius are drawn. Find the angle between them.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
John is flipping his favorite bottle, which currently contains $10$ ounces of water. However, his bottle is broken from excessive flipping, so after he performs a flip, one ounce of water leaks out of his bottle. When his bottle contains $k$ ounces of water, he has a $\frac{1}{k+1}$ probability of landing it on its bottom. What is the expected number of flips it takes for John’s bottle to land on its bottom?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Positive integers \( d, e, \) and \( f \) are chosen such that \( d < e < f \), and the system of equations 
\[ 2x + y = 2010 \quad \text{and} \quad y = |x-d| + |x-e| + |x-f| \]
has exactly one solution. What is the minimum value of \( f \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1006}
Peter is randomly filling boxes with candy. If he has 10 pieces of candy and 5 boxes in a row labeled $A$, $B$, $C$, $D$, and $E$, how many ways can he distribute the candy so that no two adjacent boxes are empty?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{771}
Given $p$: $-4 < x-a < 4$, and $q$: $(x-1)(2-x) > 0$. If $\neg p$ is a sufficient condition for $\neg q$, then the range of the real number $a$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-2, 5]}
Professor Oscar has to evaluate his students on the final exams, but this semester, he designed 5 types of exams with varying question types and components for different class sizes. Here's the breakdown of each class, the number of students, and their exams:

Class A, with 30 students:
- 12 multiple-choice questions
- 3 essay questions
- 1 individual presentation

Class B, with 25 students:
- 15 multiple-choice questions
- 5 short-answer questions
- 2 essay questions
- 1 group project with presentations, involving 5 student groups

Class C, with 35 students:
- 10 multiple-choice questions
- 3 essay questions
- 1 interactive group activity per group of 7 students

Class D, with 40 students:
- 11 multiple-choice questions
- 4 short-answer questions
- 3 essay questions
- 1 individual research paper

Class E, with 20 students:
- 14 multiple-choice questions
- 5 short-answer questions
- 2 essay questions
- 1 in-class debate per group of 4 students

Given the varying number of questions, students, and exam components for each of the 5 classes, how many evaluations must Professor Oscar review in total? Consider every component (multiple-choice, short-answer, essay, individual presentation, group project presentation, interactive group activity, individual research paper, and debate) as a separate evaluation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2680}
The set $\{(x, y) \in R^2| \lfloor x + y\rfloor \cdot \lceil x + y\rceil = (\lfloor x\rfloor + \lceil y \rceil ) (\lceil x \rceil  + \lfloor y\rfloor), 0 \le x, y \le 100\}$ can be thought of as a collection of line segments in the plane. If the total length of those line segments is $a + b\sqrt{c}$ for $c$ squarefree, find $a + b + c$. 
($\lfloor z\rfloor$ is the greatest integer less than or equal to $z$, and $\lceil z \rceil$ is the least integer greater than or equal to $z$, for $z \in R$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30202}
Calculate \(\sec \frac{2 \pi}{9} + \sec \frac{4 \pi}{9} + \sec \frac{6 \pi}{9} + \sec \frac{8 \pi}{9}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the integral \( \int_C \frac{x\mathrm{d}y-y\mathrm{d}x}{x^2+y^2} \), where \( C \) is the curve defined by the equation \( x^{48} + y^{76} = 987654321^{123456789} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
A car radiator contains 10 liters of a 30 % antifreeze solution. 2.85714285714 liters of the solution had to be replaced with pure antifreeze to achieve a certain percentage of antifreeze. What is the percentage of antifreeze in the resulting solution?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50\%}
The area of the largest regular hexagon that can fit inside of a rectangle with side lengths 20 and 22 can be expressed as $a \sqrt{b}-c$, for positive integers $a, b$, and $c$, where $b$ is squarefree. Compute $100 a+10 b+c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{134610}
8.1. Parallelogram $A B C D$ was folded along the diagonal $B D$ so that vertex $C$ remained in place, and vertex $A$ took the position $A^{\prime}$. Segments $B C$ and $A^{\prime} D$ intersected at point $K$, and $B K: K C=3: 2$. Find the area of triangle $A^{\prime} K C$, if the area of parallelogram $A B C D$ is 27.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.6}
For the numerical sequence \( \{x_n\} \), where all terms starting from \( n \geq 2 \) are distinct, the relation \( x_{n} = \frac{x_{n-1} + 198 x_{n} + x_{n+1}}{200} \) holds. Find \( \sqrt{\frac{x_{2023} - x_{1}}{2022} \cdot \frac{2021}{x_{2023} - x_{2}}} + 2022 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2023}
$M$ is the midpoint of side $BC$ in $\triangle ABC$, and $\angle MAC = 15^\circ$. Find the maximum value of $\angle B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105^\circ}
Let \(a_n\) denote the angle opposite to the side of length \(4n^2\) units in an integer right-angled triangle with side lengths \(4n^2\), \(4n^4+1\), and \(4n^4-1\), where \(n \in \mathbb{N}\). Find the value of:
\[
\lim_{p \to \infty} \sum_{n=1}^p a_n
\]
The original answer is in the format \(\frac{k\pi}{m}\). Please give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}