inclusionAI/Ring-lite-rl-data huggingface dataset:
Joseph has a refrigerator, a water heater, an electric oven, an air conditioner, and a washing machine in his house that consumes power at different rates. The total amount of money that Joseph pays for the energy used by the refrigerator is three times the amount he pays for the power used by the water heater. The electric oven uses power worth $500 in a month, twice what Joseph pays for the power the water heater uses. The air conditioner consumes power worth $300 in a month, which is half the cost of the power used by the refrigerator. The washing machine consumes power worth $100 in a month, one-fifth of the cost of the power used by the water heater. Calculate the total amount Joseph pays for the power used by all these gadgets in a month.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$1900}
Two circles are given by the equations $x^2+y^2-2mx=0$ and $(x^2+(y-2)^2=1)$. The circles are externally tangent. Find the value of the real number m. Express your answer as a numerical value, including the ± symbol if necessary to denote multiple solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{± \frac {3}{2}}
Let $X$ be a nonempty set with $m$ members. How many $\sigma$-algebras can we make on this set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$B_m$}
Let
\[f_1(x) = \frac{1}{x}\quad\text{and}\quad f_2(x) = 1 - x\]
Let $H$ be the set of all compositions of the form $h_1 \circ h_2 \circ \ldots \circ h_k$, where each $h_i$ is either $f_1$ or $f_2$.  For all $h$ in $H$, let $h^{(n)}$ denote $h$ composed with itself $n$ times.  Find the greatest integer $N$ such that $\pi, h(\pi), \ldots, h^{(N)}(\pi)$ are all distinct for some $h$ in $H$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Two brothers sold a flock of sheep, obtaining for each sheep as many rubles as there were sheep in the flock. Wishing to divide the proceeds equally, they took turns, starting with the elder brother, taking 10 rubles each from the total amount. After the elder brother took 10 rubles one more time, the younger brother was left with less than 10 rubles. To ensure an equal division, the elder brother gave the younger brother his knife. How many rubles was the knife valued at?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
对某些正整数 n>m，n 既可以表示为 2021 个加数的和，其中每个加数均为 m 的非负整数次幂，又可以表示为 2021 个加数的和，其中每个加数均为 m+1 的非负整数次幂. 求 m 的最大可能值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2021}
Given that \( F_{1} \) and \( F_{2} \) are the left and right foci of the hyperbola \( C: \frac{x^{2}}{4}-\frac{y^{2}}{12}=1 \), and point \( P \) lies on the hyperbola \( C \). Let \( G \) and \( I \) be the centroid and incenter of \( \triangle F_{1} P F_{2} \) respectively. If \( G I \) is parallel to the \( x \)-axis, find the circumradius \( R \) of \( \triangle F_{1} P F_{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
The inclination angle of the line $x-y+1=0$ can be calculated.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{\\pi}{4}"]}
若不等式$a{{x}^{2}}+bx+c>0$的解集为$\left( -1,2 \right)$，则不等式$c{{x2}}-\left( a+c \right)x-b<0$的解集为__________．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\left( -\frac{1}{2},1 \right)$}
Given the sets M = {x | x^2 ≤ 4} and N = {x | log2(x) ≤ 1}, determine the set M ∩ N. Express your answer in set notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0,2]}
log base ( 5x^3 ) root ( 7x - 3 ) / log ( 7x - 3 ) is equal to
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
In a Markov chain on the state space $\{0, 1, \ldots\}$, the transition probabilities are given by $p_{i, i-1}=q$ and $p_{i,i+1}=p$, where $p > q$. Determine the expected hitting time to reach state 0 from each state $k \geq 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Quadrilateral $A B C D$ satisfies $A B=8, B C=5, C D=17, D A=10$. Let $E$ be the intersection of $A C$ and $B D$. Suppose $B E: E D=1: 2$. Find the area of $A B C D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Find the greatest constant \( k > 0 \) such that the inequality \( \max\left(\frac{5}{5-3c}, \frac{5b}{5-3d}\right) \geq k \cdot \frac{2+3b}{5-c-2d} \) holds for all \( 0 \leq b \leq 1 \) and \( 0 \leq c \leq d \leq 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{15}{16}}
Identify the well-known topological space that is homeomorphic to the quotient space of \(\mathbb{R}^2\) induced by the equivalence relation \((a_0, b_0) \sim (a_1, b_1)\) if and only if \(a_0 + b_0^2 = a_1 + b_1^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Determine the values of $t > 1$ for which the set $F = \{a \in \mathbb{R} : a < 1\}$, together with the operations $a \boxplus b = a + b - ab$ and $a \boxdot b = 1 - t^{\log_t (1-a) \log_t (1-b)}$, forms a field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t > 1}
Evaluate the limit of the line integral \( \int_\mathrm{arc} \frac{\exp(iz)}{z+1} \, dz \) as the radius \( R \) of the semi-circular arc in the upper half complex plane goes to infinity. Does the integral converge to a fixed constant? If so, calculate this constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $ABCD$ be a cyclic quadrilateral with $BC = CD = 2$. The incenter of triangle $ABD$ is $I$, and $AI = 2$. Determine the minimum value of the length of diagonal $BD$. \( \text{Express your answer as a single number.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3}}
An insurance company pays $10,000 for the first loss, $7,500 for the second loss, and $5,000 for each successive loss. If the number of losses follows a Poisson distribution with a mean of 2.5, calculate the expected payment due to losses.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18871.33}
There are two types of electronic toy cars, Type I and Type II, each running on the same two circular tracks. Type I completes a lap every 5 minutes, while Type II completes a lap every 3 minutes. At a certain moment, both Type I and Type II cars start their 19th lap simultaneously. Calculate how many minutes earlier the Type I car started running compared to the Type II car. Express your answer in minutes as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Determine the value that \(\frac{B}{x}\) approaches, where \(B=\sqrt[3]{x\sqrt[5]{x^4\sqrt[9]{x^{24}\sqrt[17]{x^{240}\ldots}}}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $ ABCD$ be a parallelogram with $ AB=1$. Suppose that $ K$ is a point on the side $ AD$ such that $ KD=1, \angle ABK=90^{\circ}$ and $ \angle DBK=30^{\circ}$. Determine $ AD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
AD = \(2^{1/3} + 1\)

}
Determine the cardinality of the set $S$, which consists of all closed intervals on the real line with lengths being positive rational numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
Simplify the matrix expression: \[ B = (Z^TZ)^{-1} + (Z^TZ)^{-1}Z^T( I - P_z)^{-1}Z(Z^TZ)^{-1} \] where \( Z \) is an \( n \times 1 \) vector, \( I \) is the identity matrix, and \( P_z = Z(Z^TZ)^{-1}Z^T \) is the projection matrix of \( Z \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(Z^T Z)^{-1}}
In the triangular prism D-ABC, given that $\angle \mathrm{ACB} = \angle \mathrm{ABD} = 90^{\circ}$, $\mathrm{CA} = \mathrm{CB}$, and $\angle \mathrm{BAD} = 30^{\circ}$, if the projection of point $\mathrm{C}$ onto plane $\mathrm{ABD}$ is precisely on line $\mathrm{AD}$, find the sine of the plane angle of the dihedral angle $\mathrm{C}-\mathrm{AB}-\mathrm{D}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{6}}{3}}
Two trains travel in opposite directions at some speed and 45 kmph, and a man sitting in the slower train passes the faster train in 6 seconds. The length of the faster train is 135.0108 meters. What is the speed of the slower train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36.00648}
Evaluate the line integral \( \int _{C} (z^3 + 2z + \text{Re}(z))\,dz \) where \( C \) is the triangle with vertices \( z=0 \), \( z=1+2i \), and \( z=1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-i}
Given a random sample \(X_1, X_2, \dots, X_n\) from a distribution with the probability density function \(f(x, \theta) = e^{-(x-\theta)}\) for \(\theta < x < \infty\) and \(-\infty < \theta < \infty\), find the sufficient statistic for the parameter \(\theta\) using the factorization theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{X_{(1)}}
How many more hours are in $10$ years than seconds in $1$ day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1200
}
What is the maximum number of points \( P \) in the plane of the convex quadrilateral \( ABCD \) for which the areas of the triangles \( PAB \), \( PBC \), \( PCD \), and \( PDA \) are equal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of distinct ways to label the vertices of a cube with the integers 1 through 8, without repetition, such that the sum of the numbers on the four vertices of each face is the same. Arrangements that can be obtained through rotations of the cube are considered identical. How many different arrangements are possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Calculate the mass of the plane curve defined by \( y = \frac{x^{3}}{3} \) over the segment from \( x = 0 \) to \( x = 0.1 \), given that the linear density \( \rho(x) = 1 + x^{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.1003}
Given an isosceles triangle ABC, where the base points A and B are on the right branch of the hyperbola $\frac {x^{2}}{6}- \frac {y^{2}}{3}$=1, and the vertex C is on the x-axis. If AB is not perpendicular to the x-axis, find the range of the x-coordinate t of vertex C.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\frac{3}{2}\sqrt {6}, +\infty)}
Evaluate the limit of the following expression as \( r \to 0 \):

\[ \lim_{r\to0}\left(\frac{1}{\mu_1-1}\,\mathrm{K}_0(\mathrm{j}\,k_1\,r)-\frac{1}{\mu_2-1}\,\mathrm{K}_0(-\mathrm{j}\,k_2\,r)\right) \]

Given that \( \mu_1\,k_1^2=\mu_2\,k_2^2 \), \( k_1=\sqrt{k_1^2} \), \( k_2=\sqrt{k_2^2} \), \( \mathrm{Im}[k_1]<0 \), and \( \mathrm{Im}[k_2]>0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let x, y, and z be distinct integers such that w < x < y < z, where x is an even integer, y and z are odd integers, and w is a negative, non-integer multiple of 1/3. If the difference (y - x) > a positive integer k and (z - w) > a positive integer m, where k > m, what is the least possible value of z - w?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let \( A_{1}, A_{2}, \ldots, A_{11} \) be 11 points on a straight line in order, where \( A_{1} A_{11} = 56 \). Given that \( A_{i} A_{i+2} \leq 12 \) for \( i = 1, 2, \ldots, 9 \) and \( A_{j} A_{j+3} \geq 17 \) for \( j = 1, 2, \ldots, 8 \), find \( A_{2} A_{7} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Cutting equilateral triangle $BFC$ out of square $ABCD$ and translating it to the left of the square creates the shaded figure $ABFCDE$. The perimeter of square $ABCD$ was 48 inches. What is the perimeter, in inches, of $ABFCDE$?

[asy]
filldraw((0,0)--(sqrt(3),1)--(sqrt(3)+2,1)--(2,0)--(sqrt(3)+2,-1)--(sqrt(3),-1)--(0,0)--cycle,gray,linewidth(1));
draw((sqrt(3),1)--(sqrt(3),-1),linewidth(1));
draw((sqrt(3)+2,-1)--(sqrt(3)+2,1)--(2,0)--(sqrt(3)+2,-1)--cycle,dashed);

label("A",(sqrt(3),1),N);
label("B",(sqrt(3)+2,1),N);
label("C",(sqrt(3)+2,-1),S);
label("D",(sqrt(3),-1),S);
label("E",(0,0),W);
label("F",(2,0),N);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Given a sequence $\{a\_n\}$, where $a\_1=3$, $n(a_{n+1}-a_n)=a_n+1$, $n\in\mathbb{N}^*$. If for any $a\in[-1,1]$, $n\in\mathbb{N}^*$, the inequality $\frac{a_{n+1}}{n+1} < t^2-2at+1$ always holds, then the range of the real number $t$ is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -3] \cup [3, +\infty)}
$24, 2$ red balls and $11$ white balls are arranged in a row, satisfying the following conditions: the red balls are not adjacent, and in any consecutive $7$ balls, there is at least one red ball, then there are $\qquad$ ways to arrange them.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
设函数 $f(x)=\sqrt{2 x^{2}+2 x+41}-\sqrt{2 x^{2}+4 x+4}(x \in \mathbf{R})$, 则 $f(x)$ 的最大值是
$\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the number of non-unital ring homomorphisms from the ring \( \mathbb{Z}/20\mathbb{Z} \) to the ring \( \mathbb{Z}/12\mathbb{Z} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the limit \( \lim_{n\to\infty} f_n \) in the supremum norm over the interval \([-1, 1]\) for the sequence of real coefficient polynomials \(\{f_n\}\) defined recursively by \(f_0 = 0\) and \(f_{n+1}(x) = f_n(x) + \frac{x^2 - f_n^2(x)}{2}\) for \(n = 0, 1, 2, \ldots\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|x|}
Find the number of functions $f$ from $\{1,2,3,4,5\}$ to itself such that $f(f(f(x))) = f(f(x))$ for all $x \in \{1,2,3,4,5\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{756}
Evaluate the expression: $$ \frac{\tan(180^{\circ}/7)}{\tan(360^{\circ}/7)} - \frac{\tan(360^{\circ}/7)}{\tan(540^{\circ}/7)} - \frac{\tan(540^{\circ}/7)}{\tan(180^{\circ}/7)} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-7}
Evaluate the double integral \( \mathop{\int\!\!\! \int}_{R_{1}}f(x,y)\,dA \) for the rectangle \( R_{1}=[0, 5] \times [-4, 4] \), given that \( \mathop{\int\!\!\! \int}_{R_{2}}f(x,y)\,dA=10 \) for the rectangle \( R_{2}=[0, 5] \times [0, 4] \), \( \mathop{\int\!\!\! \int}_{R_{3}}f(x,y)\,dA=24 \) for the rectangle \( R_{3}=[-5, 0] \times [-4, 0] \), and \( f(-x,y)=-f(x,y) \) for all \( (x,y) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-14}
7. The cells of a $2017 \times 100$ board (2017 rows, 100 columns) are painted in a checkerboard pattern. A grasshopper standing on the board attacks all cells in its row that have the same color as the cell it stands on, as well as all cells in its column that have the opposite color. (To attack a cell, the grasshopper can jump over other pieces.) What is the maximum number of non-attacking grasshoppers that can be placed on this board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4034}
Given the differential equation \((1-x^2)\frac{d^2y}{dx^2} - 2x\frac{dy}{dx} + (\sec x)y = 0\), let \(y_1(x)\) and \(y_2(x)\) be two linearly independent solutions with a Wronskian \(W(x)\). If \(y_1(0) = 1\), \(\left(\frac{dy_1}{dx}\right)_{x=0} = 0\), and \(W\left(\frac{1}{2}\right) = \frac{1}{3}\), determine the value of \(\left(\frac{dy_2}{dx}\right)_{x=0}\) without explicitly finding \(y_1(x)\) and \(y_2(x)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
At a club, twenty gentlemen met. Some of them were wearing hats, and some were without hats. From time to time, one of the gentlemen took off his hat and put it on one of those who did not have a hat at that moment. In the end, ten gentlemen counted that each of them had given away a hat more times than they had received one. How many gentlemen came to the club wearing hats?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
The curvature $\varphi(M,N)$ of a curve $y=f(x)$ between two different points $M(x_{1},y_{1})$ and $N(x_{2},y_{2})$ on its graph, where the slopes of the tangents at $M$ and $N$ are $k_{M}$ and $k_{N}$ respectively, is defined as $\varphi(M,N)=\frac{|k_{M}-k_{N}|}{|MN|}$ (where $|MN|$ is the length of the segment $MN$). For the curve $f(x)=x^{3}+2$ with two different points $M(x_{1},y_{1})$ and $N(x_{2},y_{2})$ on it, and given that $x_{1}x_{2}=1$, the range of $\varphi(M,N)$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \frac {3 \sqrt {10}}{5})}
Evaluate the limit: $$ \lim_{n \to \infty} \left( \ \sum_{k=10}^{n+9} \frac{2^{11(k-9)/n}}{\log_2 e^{n/11}} \ - \sum_{k=0}^{n-1} \frac{58}{\pi\sqrt{(n-k)(n+k)}} \ \right) $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2018}
Given the planar vectors $\overrightarrow {a}$ and $\overrightarrow {b}$, with $|\overrightarrow {a}|=1$, $|\overrightarrow {b}|=2$, and $\overrightarrow {a} \cdot \overrightarrow {b} = 1$, find the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
Let \(\zeta_7 = e^{i\frac{2\pi}{7}}\) be the 7th root of unity. Determine the integer \(D\) such that the quadratic subfield of \(\mathbb{Q}(\zeta_7)\) can be expressed as \(\mathbb{Q}(\sqrt{D})\). Given that \(\omega = \zeta_7 + \zeta_7^{-1} = 2\cos(\frac{2\pi}{7})\) lies in \(\mathbb{Q}(\zeta_7)\) and generates the subfield \(K = \mathbb{Q}(\omega)\) with \([K(\zeta_7):K] = 2\), find the value of \(D\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-7}
Let an ordered pair of positive integers $(m, n)$ be called regimented if for all nonnegative integers $k$, the numbers $m^k$ and $n^k$ have the same number of positive integer divisors. Let $N$ be the smallest positive integer such that $(2016^{2016}, N)$ is regimented. Compute the largest positive integer $v$ such that $2^v$ divides the difference $2016^{2016} - N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10087}
If real number $a > b$, then fill in the inequality for $a^2 - ab \_\_\_\_\_\_\_\_ ba - b^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{>}
In triangle ABC, the magnitudes of the sides are a, b, and c, as shown, with c ≤ b ≤ a. Through interior point P and the vertices A, B, C, lines are drawn meeting the opposite sides in A', B', C', respectively. Let s = AA' + BB' + CC'. Find the upper bound for s in terms of a, b, and c.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a+b+c}
In a certain kennel with 45 dogs, each of the dogs is a single color. Each of the dogs in the kennel either has long fur or does not. There are 36 dogs with long fur, and 8 dogs that are neither long-furred nor brown. How many dogs are brown?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Find the value of \( a \) such that the line integral \( I = \int_\Gamma x(a+y)(ydx+xdy) \) over the path \( \Gamma \) equals 36. The path \( \Gamma \) consists of three segments: from \( O = (0,0) \) to \( A = (3,0) \), from \( A = (3,0) \) to \( B = (0,3) \), and from \( B = (0,3) \) back to \( O = (0,0) \). The segment \( AB \) is characterized by the quarter-circle \( \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 = 9, x \ge 0, y \ge 0\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The roots of the polynomial equation \( x^3 - 3x - 1 = 0 \) are \( a, b, \) and \( c \). Calculate the value of the expression:
\[ L = \frac{1}{a^2 - 2a + 2} + \frac{1}{b^2 - 2b + 2} + \frac{1}{c^2 - 2c + 2} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{39}{37}}
Evaluate the limit: \[ \lim_{x \to 0^+} \frac{1}{\ln x} \sum_{n=1}^{\infty} \frac{x}{(1+x)^n - (1-x)^n}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
17. If a square can completely cover a triangle with side lengths 3,4 and 5 , find the smallest possible side length of the square.
(2 marks)
若一個正方形可以完全覆蓋一個邊長 3、4、5 的三角形, 求正方形的最小邊長。
(2 分)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{16\sqrt{17}}{17}}
已知 $M=\{1,2, \cdots, 8\}, A 、 B$ 是集合 $M$的两个不同子集，满足：  
（1） $A$ 的元素个数比 $B$ 的元素个数少；  
（2） $A$ 中的最小元素比 $B$ 中的最大元素大。  
则所有满足条件的有序集合对 $(A, B)$ 的个数为$\qquad.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{321}
Given a right triangle $ABC$ with $AC = 3$, $BC = 4$, and $AB = 5$. Squares are constructed externally on each side of the triangle. Calculate the area of the hexagon $PQRSTU$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74}
Solve the inequality \[ 8 \cdot \frac{|x+3|-|x-5|}{|2x-11|-|2x+7|} - 9 \cdot \frac{|x+3|+|x-5|}{|2x-11|+|2x+7|} \geq -8 \] and find the sum of its integer solutions that satisfy the condition \( |x| < 90 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the domain of definition of the function \( f(x) = \int_{0}^{\infty} \frac{t^x}{1+t^4 \sin^2 t} \, dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1)}
Given the sequence \(a_n\) defined by the initial conditions \(a_1 = 1\), \(a_2 = 4\), \(a_3 = 5\), and the recurrence relation \(a_n + a_{n-1} + a_{n-2} + a_{n-3} = n^2\) for all \(n \geq 4\), find the sum of the digits of \(a_{2021}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
In a computer game, a player can choose to play as one of three factions: \( T \), \( Z \), or \( P \). There is an online mode where 8 players are divided into two teams of 4 players each. How many total different matches are possible, considering the sets of factions? The matches are considered different if there is a team in one match that is not in the other. The order of teams and the order of factions within a team do not matter. For example, the matches \((P Z P T ; T T Z P)\) and \((P Z T T ; T Z P P)\) are considered the same, while the matches \((P Z P Z ; T Z P Z)\) and \((P Z P T ; Z Z P Z)\) are different.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Determine the spectrum of the infinite-dimensional matrix with elements $a_{ij} = \sqrt{j}$ if $i = j+1$, and $0$ otherwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{C}}
A room is built in the shape of the region between two semicircles with the same center and parallel diameters. The farthest distance between two points with a clear line of sight is 12 meters. What is the area (in square meters) of the room?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18\pi}
Determine the value of the parameter \( a \) for which the magnitude \( |x + y| \), where \( (x, y) \) is a solution of the system of equations \( 4x^2 + 4y^2 = -a^2 + 16a - 32 \) and \( 2xy = a \), reaches its maximum value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}