inclusionAI/Ring-lite-rl-data huggingface dataset:
There are five cities $A, B, C, D, E$ on a certain island. Each city is connected to every other city by road. In how many ways can a person starting from city $A$ come back to $A$ after visiting some cities without visiting a city more than once and without taking the same road more than once? The order in which he visits the cities also matters: e.g., the routes $A \to B \to C \to A$ and $A \to C \to B \to A$ are different.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Find the sum of the decimal digits of the number
\[ 5\sum_{k=1}^{99} k(k + 1)(k^2 + k + 1). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Find \( \frac{a_1^3}{1 - 3a_1 + 3a_1^2} + \frac{a_2^3}{1 - 3a_2 + 3a_2^2} + \cdots + \frac{a_{101}^3}{1 - 3a_{101} + 3a_{101}^2} \), where \( a_n = \frac{n}{101} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
三位数 $(100,101, \ldots, 999)$ 共 900 个，在卡片上打印这些三位数，每张卡片上打印一个三位数，有的卡片所印的，倒过来看仍为三位数，如 198 倒过来看是 861 ；有的卡片则不然，如 531 倒过来看是,因此，有些卡片可以一卡二用，于是至多可以少打印 $\qquad$张卡片.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
What is the minimum number of terms, $n$, required in the harmonic series $H_n$ so that the difference between $H_n$ and $\log n$ is less than $2 \cdot 10^{-3}$, considering Euler's constant $\gamma$ defined as $\gamma := \lim_{n \to \infty} \left( H_n - \log n\right)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250}
The vertices \( A, B \) and \( D_1 \) of a cube \( ABCD A_1 B_1 C_1 D_1 \) lie on the lateral surface of a cylinder, the axis of which is parallel to the line \( DC_1 \). Find the radius of the base of the cylinder, if the edge of the cube is equal to \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3a\sqrt{2}}{4}}
Except for the first two terms, each term of the sequence  $1000, x, 1000-x,\ldots$  is obtained by subtracting the preceding term from the one before that.  The last term of the sequence is the first negative term encounted.  What positive integer  $x$  produces a sequence of maximum length?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{618}
Three builders are scheduled to build a house in 60 days. However, they procrastinate and do nothing for the first 50 days. To complete the house on time, they decide to hire more workers and work at twice their original speed. If the new workers also work at this doubled rate, how many new workers are needed? Assume each builder works at the same rate and does not interfere with the others.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given the function $f(x)=x-\sin x$, for any $x\in R$, the equation $f(-x^{2}+3x)+f(x-2k)\leqslant 0$ always holds true. Determine the range of values for $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k\geqslant 2}
How many group homomorphisms are there from $\Bbb Z/2\Bbb Z \times \Bbb Z/2\Bbb Z$ to $S_{3}$, where $S_{3}$ is the group of permutations of three elements?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A slanted parallelepiped has a base in the shape of a rectangle $ABCD$. The side edges are $AA_1, BB_1, CC_1$, and $DD_1$. A sphere with center at point $O$ touches the edges $BC, A_1B_1$, and $DD_1$ at the points $B, A_1$, and $D_1$, respectively. Find $\angle A_1OB$ if $AD = 4$ and the height of the parallelepiped is 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \arcsin \frac{1}{\sqrt{5}}}
Initially, an amoeba is placed in a puddle, and from that day on, each amoeba splits into two amoebas each subsequent day. However, a second amoeba is introduced into the puddle on the third day. Calculate the total number of amoebas at the end of fourteen days.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10240}
Find the value of $l$ such that the following equation holds true:
$$\int_{0}^{1}\frac{dt}{\sqrt{(1 - t^{2})(1 - l^{2}t^{2})}} = \int_{1}^{\frac{1}{\sqrt{1 - l^{2}}}}\frac{dt}{\sqrt{(t^{2} - 1)(1 - l^{2}t^{2})}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/\sqrt{2}}
There exist real numbers $a$, $b$, $c$, $d$, and $e$ such that for all positive integers $n$, we have:
\[
\sqrt{n} = \sum_{i=0}^{n-1} \sqrt[5]{\sqrt{ai^5 + bi^4 + ci^3 + di^2 + ei + 1} - \sqrt{ai^5 + bi^4 + ci^3 + di^2 + ei}}.
\]
Find the sum $a + b + c + d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1656}
Given a triangular pyramid $D-ABC$ with all four vertices lying on the surface of a sphere $O$, if $DC\bot $ plane $ABC$, $\angle ACB=60^{\circ}$, $AB=3\sqrt{2}$, and $DC=2\sqrt{3}$, calculate the surface area of sphere $O$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36\pi}
Alex is stacking his books in a pyramid. Each level of the pyramid has 80% as many books as the number of books in the previous level. He makes four levels and the top level has a certain number of books. There are 369 books in the pyramid in total. How many books are in the top level of the pyramid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Find all natural numbers whose own divisors can be paired such that the numbers in each pair differ by 545. An own divisor of a natural number is a natural divisor different from one and the number itself.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1094}
Tracy had a bag of candies, and none of the candies could be broken into pieces. She ate $\frac{1}{4}$ of them and then gave $\frac{1}{3}$ of what remained to her friend Rachel. Tracy and her mom then each ate 20 candies from what Tracy had left. Finally, Tracy’s brother took anywhere from two to six candies, leaving Tracy with ten candies. How many candies did Tracy have at the start?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{112}
There is a lamp on each cell of a  $2017 \times 2017$  board. Each lamp is either on or off. A lamp is called *bad* if it has an even number of neighbours that are on. What is the smallest possible number of bad lamps on such a board?
(Two lamps are neighbours if their respective cells share a side.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A track and field team consists of 56 male athletes and $m$ female athletes. A stratified sampling method is used to draw a sample of size $n$. The probability of randomly selecting a team captain from this sample is $\dfrac{1}{28}$, and the sample contains 4 more male athletes than female athletes. Find the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Using only pennies, nickels, dimes, quarters, and half-dollars, determine the smallest number of coins needed to pay any amount of money less than a dollar and a half.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Given an arithmetic sequence $\{a_{n}\}$ where $a_{n} > 0$ and $a_{2}a_{8}=4$, find the value of $\log _{2}a_{1}+\log _{2}a_{2}+...+\log _{2}a_{9}=\_\_\_\_\_\_\_\_\cdot$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Calculate the expectation of the cube of the absolute value of the sum of two independent normal random variables, $X$ and $Y$, where $X \sim N(\mu, \sigma^2)$ and $Y \sim N(-\mu, \sigma^2)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{8}{\sqrt{\pi}}\sigma^3\)}
Given the sequence \( x, y, z, w \) satisfying \( xy + 2yz + zw \leq A(x^2 + y^2 + z^2 + w^2) \), find the minimum value of the real number \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1+\sqrt{2}}{2}}
Compute the number of integers \( n \in\{1,2, \ldots, 300\} \) such that \( n \) is the product of two distinct primes, and is also the length of the longest leg of some nondegenerate right triangle with integer side lengths.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
An angle of $120^{\circ}$ is inscribed in a circle with a radius of 12 such that the center of the circle lies on the angle's bisector. Find the area of the part of the circle that is outside the angle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48 \pi - 72 \sqrt{3}}
When two fair 12-sided dice are tossed, the numbers $a$ and $b$ are obtained. What is the probability that both the two-digit number $ab$ (where $a$ and $b$ are digits) and each of $a$ and $b$ individually are divisible by 4?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{16}}
A rectangular chocolate bar is made of equal squares. Irena breaks off two complete strips of squares and eats the 12 squares she obtains. Later, Jack breaks off one complete strip of squares from the same bar and eats the 9 squares he obtains. Calculate the number of squares of chocolate left in the bar.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Solve for \( x \) :
\[ x\lfloor x\lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor\rfloor = 122. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{122}{41}}
A rational number written in base eight is $\underline{ab} . \underline{cd}$, where all digits are nonzero. The same number in base twelve is $\underline{bb} . \underline{ba}$. Find the base-ten number $\underline{abc}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{321}
If $\{a_1,a_2,a_3,\ldots,a_n\}$ is a set of real numbers, indexed so that $a_1 < a_2 < a_3 < \cdots < a_n,$ its complex power sum is defined to be $a_1i + a_2i^2+ a_3i^3 + \cdots + a_ni^n,$ where $i^2 = - 1.$ Let $S_n$ be the sum of the complex power sums of all nonempty subsets of $\{1,2,\ldots,n\}.$ Given that $S_8 = - 176 - 64i$ and $S_9 = p + qi,$ where $p$ and $q$ are integers, find $|p| + |q|.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$368$}
A region \( S \) in the complex plane is defined by  
\( S = \{x + iy: -1 \le x \le 1, -1 \le y \le 1\} \).  
A complex number \( z = x + iy \) is chosen uniformly at random from \( S \).  
Find the probability that \(\left(\frac{3}{4} + \frac{3}{4}i\right)z\) is also in \( S \). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
In a best-of-five game system, the probability of Player A winning each game is 2/3, and that of Player B winning each game is 1/3. Calculate the probability of Player B winning the championship. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{17}{81}}
Suppose \( a > 1 \). If the equation \( a^{x} = x \) has no real solution for \( x \), then the range of the value of the real number \( a \) is \(\qquad\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > e^{\frac{1}{e}}}
Robert's salary was first decreased by 60%, then increased by 60%. Following that, it was further decreased by 45% and later increased by 20%. Finally, the salary faced a 25% reduction. How much overall percentage did Robert's salary change after all these fluctuations?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{68.32\%}
Paco had 17 cookies. He ate 14 cookies and gave some of them to his friend. He ate 1 more cookie than he gave to his friend. How many cookies did Paco give to his friend?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Which is the three-digit (integer) number that, when increased or decreased by the sum of its digits, results in a number consisting of the same digit repeated?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
Let $T_{1}$ and $T_{2}$ be the points of tangency of the excircles of triangle $ABC$ with sides $BC$ and $AC$, respectively. It turned out that the point symmetric to the incenter of the triangle with respect to the midpoint of $AB$ lies on the circumcircle of triangle $CT_{1}T_{2}$. Find the angle $BCA$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Given the function $f(x) = |x+1| + |x+2| + \ldots + |x+2011| + |x-1| + |x-2| + \ldots + |x-2011|$ (where $x\in \mathbb{R}$), and $f(a^2 - 3a + 2) = f(a - 1)$, find the sum of all integer values of $a$ that satisfy the condition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Evaluate the limit \( \lim_{n\to\infty} \int_{-\infty}^{\infty} \frac{ne^{-x^2}}{1+n^2x^4}\ dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{\sqrt{2}}}
Find the values of natural numbers $m$ and $n$ such that the limit \( \lim_{(x,y) \to (0,0)} \frac{x^m y^n}{x^4 + y^4} \) exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m + n \geq 5}
Determine the number of permutations \( a_{1}, a_{2}, \cdots, a_{21} \) of the numbers \( 1, 2, \cdots, 21 \) such that the following condition holds:
\[
\left|a_{20}-a_{21}\right| \geqslant\left|a_{19}-a_{21}\right| \geqslant\left|a_{18}-a_{21}\right| \geqslant \cdots \geqslant\left|a_{1}-a_{21}\right|.
\]
Find the number of such permutations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3070}
Calculate the surface area of the portion of the sphere defined by \(x^2 + y^2 + z^2 = 16z\) that lies within the paraboloid \(z = x^2 + y^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16\pi}
A plane flying with a constant speed of $19 \,\text{km/min}$ passes over a ground radar station at an altitude of $10 \, \text{km}$ and climbs at an angle of $20^\circ$. At what rate is the distance from the plane to the radar station increasing $2$ minutes later?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18.5}
p1.Farmer James goes to Kristy's Krispy Chicken to order a crispy chicken sandwich. He can choose from 3 types of buns, 2
 types of sauces, 4 types of vegetables, and 4 types of cheese. He can only choose one type of bun and cheese, but can choose
 any nonzero number of sauces, and the same with vegetables. How many different chicken sandwiches can Farmer James
 order?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\({540}\)}
A bus driver is paid a regular rate of $12 per hour plus an additional $0.50 per passenger for any number of hours that does not exceed 40 hours per week. For any overtime hours worked in excess of 40 hours per week, the bus driver is paid a rate that is 50% higher than their regular hourly rate for the first 5 hours of overtime, and 100% higher for any additional hours beyond that. If last week the bus driver earned $1,280 in total compensation and carried a total of 350 passengers, how many total hours did they work that week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Given the parabola $C: y^{2}=2px$ $(p > 0)$ with focus $F$, and its directrix intersects the $x$-axis at point $M$. A line $l$ passing through point $M$ with slope $k$ intersects the parabola $C$ at two points $A,B$. If $|AM|=\frac{5}{4}|AF|$, find $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm\frac{3}{4}}
The three-digit number $999$ has a special property: It is divisible by $27$, and its digit sum is also divisible by $27$. The four-digit number $5778$ also has this property, as it is divisible by $27$ and its digit sum is also divisible by $27$. How many four-digit numbers have this property?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
What is the probability that all $N$ users are assigned to different servers when each user selects one of $K$ servers uniformly at random, ensuring that no server is left unassigned?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{K!}{K^K}}
There are three committees of bureaucrats. For each pair of bureaucrats from different committees, among the members of the remaining committee, there are exactly 10 bureaucrats who know both of them, and exactly 10 bureaucrats who do not know both of them. Determine the total number of bureaucrats in the committees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
If the line $l_{1}:ax+2y+6=0$ is parallel to the line $l_{2}:x+\left(a-1\right)y+\left(a^{2}-1\right)=0$, determine the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Determine the subfield of $K$ that corresponds to the subgroup $H_3 = \{\operatorname{id}, \sigma_3\}$ of the group $G = \{\operatorname{id}, \sigma_1, \sigma_2, \sigma_3\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Q}(\sqrt{6})}
Find the number of permutations $(a_1, a_2, a_3, a_4, a_5, a_6, a_7)$ of $(1,2,3,4,5,6,7)$ that satisfy
\[\frac{a_1 + 1}{2} \cdot \frac{a_2 + 2}{2} \cdot \frac{a_3 + 3}{2} \cdot \frac{a_4 + 4}{2} \cdot \frac{a_5 + 5}{2} \cdot \frac{a_6 + 6}{2} \cdot \frac{a_7 + 7}{2} > 7!.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5039}
Evaluate the integral \( \int_0^{2 \pi} e^{\cos x} \cos (nx - \sin x) \,dx \), where \( n \) is a natural number. Provide your answer in terms of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2\pi}{n!}}
Let $X_1, X_2, \ldots$ be i.i.d. Bernoulli random variables with parameter $\frac{1}{4}$, and let $Y_1, Y_2, \ldots$ be another sequence of i.i.d. Bernoulli random variables with parameter $\frac{3}{4}$. Let $N$ be a geometric random variable with parameter $\frac{1}{2}$, where $\mathrm{P}(N = k) = \frac{1}{2^k}$ for $k = 1, 2, \ldots$. Assume the $X_i$’s, $Y_j$’s, and $N$ are all independent. Compute the covariance $$\mathrm{Cov}\left(\sum_{i=1}^N X_i, \sum_{i=1}^N Y_i\right).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{8}}
Determine the values of $z \in \mathbb{C}$ for which the series \( \sum_{n=0}^{\infty} z^n A^n \) converges absolutely, where \( A \) is an operator in the Hilbert space \( L^2(0,2\pi) \) defined by \( (Af)(x) = \frac{1}{\pi} \int_0^{2\pi} [\cos(x)\cos(y) + \sin(x)\sin(y)] f(y) \, dy \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|z| < 1}
Consider the Markov chain with the transition matrix:
\[
P=\begin{pmatrix} 
0 & 1 & 0 & 0 & 0\\
\frac{1}{4} & 0 & \frac{1}{4} & \frac{1}{4} & \frac{1}{4}\\
0 & \frac{1}{2} & 0 & \frac{1}{2} & 0 \\
0 & \frac{1}{2} & \frac{1}{2} & 0 & 0\\
0 & 1 & 0 & 0 & 0
\end{pmatrix}
\]
Calculate the expected number of steps to first reach state 2 from state 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral \( \int_0^{2\pi} \cos(\cos x)\,dx \). If it can be evaluated, find its value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi J_0(1)}
Given a set of 18 distinct cards, each card is part of four different combinations of three cards, resulting in a total of 24 unique combinations. What is the minimum number of cards that must be drawn to ensure that at least one of these combinations is present in the drawn hand?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Evaluate the integral \( \int_{0}^{1000} \frac{e^{-10x}\sin x}{x} \text{d}x \) to within \( \pm 10^{-5} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.09967}
Find the slope of the curve given by the equation \(x^3 + y^3 = 3axy\) at the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine whether the improper integral \( \int_1^{+\infty} \frac{\ln(1+t^2)}{t^\beta} \, dt \) converges or diverges, where \( \beta \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\beta > 1}
Determine the minimum value of \( \frac{\int_0^1{x^2\left( f'(x) \right)^2 \, dx}}{\int_0^1{x^2\left( f(x) \right)^2 \, dx}} \) for all non-zero continuously differentiable functions \( f: [0,1] \to \mathbb{R} \) with \( f(1) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi^2}
我们称置换群 $S_{9}$ 中的两个元素 $\sigma_{1}, \sigma_{2}$ 是共轭的，如果存在 $S_{9}$ 中的元素 $\tau$使得 $\tau^{-1} \sigma_{1} \tau=\sigma_{2}$ .设 $\sigma=(12345) (4567)$, 则 $S_{9}$ 中有 $\qquad$个元素与 $\sigma$ 共轭.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15120}
There were 9 roses and some orchids in the vase. Jessica cut some more roses and orchids from her flower garden. There are now 13 orchids and 3 roses in the vase. There are 10 more orchids than roses in the vase now. How many orchids were there in the vase initially?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
If x, y, and z are positive integers and 3x = 4y = some multiple of z, then the least possible value of x - y + z is 19. What is the multiple of z?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
How many numbers are in the list $250, 243, 236, \ldots, 29, 22?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
In a triangle, the length of side $AB$ is $10 \mathrm{~cm}$, the length of side $AC$ is $5.1 \mathrm{~cm}$, and $\angle CAB = 58^{\circ}$. Determine $\angle BCA$ to the nearest hundredth of a degree.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91.35}
Several points, including points \( A \) and \( B \), are marked on a line. We consider all possible segments with endpoints at the marked points. Vasya counted that point \( A \) is inside 40 of these segments, and point \( B \) is inside 42 segments. How many points were marked? (The endpoints of a segment are not considered its internal points.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
There are 7 red peaches, some yellow peaches, and 8 green peaches in the basket. There is 1 more green peach than red peaches in the basket. How many yellow peaches are in the basket?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that \( PQ = 4 \), \( QR = 8 \), \( RS = 8 \), and \( ST = 3 \), if \( PQ \) is perpendicular to \( QR \), \( QR \) is perpendicular to \( RS \), and \( RS \) is perpendicular to \( ST \), calculate the distance from \( P \) to \( T \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Let the graph of the function $y=f(x)$ be symmetric to the graph of $y=2^{x-a}$ about the line $y=-x$, and $f(-2)+f(-4)=1$, then $a=$ \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
已知数列$\left\{ {{a}_{n}} \right\}$满足${{a1}}=4$，${{a}_{n+1}}=4-\frac{4}{{{an}}}$，且$f\left( n \right)=\left( {{a}_{1}}-2 \right)\left( {{a2}}-2 \right)+\left( {{a}_{2}}-2 \right)\left( {{a3}}-2 \right)+\left( {{a}_{3}}-2 \right)\left( {{a4}}-2 \right)+\cdots +\left( {{a}_{n}}-2 \right)\left( {{an+1}}-2 \right)$，若对$\forall n\ge 3\left( n\in {{\mathrm{N}}^{*}} \right)$，都有$f\left( n \right)\ge {{m2}}-2m$恒成立，则实数$\textit{m}$的最小值为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$-1$}