inclusionAI/Ring-lite-rl-data huggingface dataset:
A straight one-way city street has 8 consecutive traffic lights. Every light remains green for 1.5 minutes, yellow for 3 seconds, and red for 1.5 minutes. The lights are synchronized so that each light turns red 10 seconds after the preceding one turns red. Calculate the longest interval of time, in seconds, during which all 8 lights are green.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
In the diagram, points R and S lie on PT and PQ, respectively. If ∠PQR = 90°, ∠QRT = 158°, and ∠PRS = ∠QRS, what is the measure of ∠QSR? Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{79^\circ}
有限项等差数列的公差为 4 ，第二项起各项之和加上首项的平方小于 100 ，则该数列可能有几项？请提供可能项数的最小值与最大值之和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Three couples sit for a photograph in  $2$  rows of three people each such that no couple is sitting in the same row next to each other or in the same column one behind the other. How many such arrangements are possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
The numbers $\frac{1}{1}, \frac{1}{2}, \cdots , \frac{1}{2012}$ are written on the blackboard. Aïcha chooses any two numbers from the blackboard, say $x$ and $y$, erases them and she writes instead the number $x + y + xy$. She continues to do this until only one number is left on the board. What are the possible values of the final number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2012}
Let $A B C$ be a triangle, and let $D, E$, and $F$ be the midpoints of sides $B C, C A$, and $A B$, respectively. Let the angle bisectors of $\angle F D E$ and $\angle F B D$ meet at $P$. Given that $\angle B A C=37^{\circ}$ and $\angle C B A=85^{\circ}$, determine the degree measure of $\angle B P D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61^{\circ}}
Find the principle on a certain sum of money at 5% per annum for 6 years if the amount being Rs. 1120. What is the principle amount?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{861.54}
Provide an example of a function \( f: \mathbb{R} \to \mathbb{R} \) such that \( \rho(Af(x), Af(y)) < \rho(f(x), f(y)) \) for all \( x \neq y \), but \( f \) does not have a fixed point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = x + 1}
Given the equations \(x = 3\sqrt{y\sqrt{3\sqrt{y\cdots}}}\) and \(y = 3\sqrt{x\sqrt{3\sqrt{x\cdots}}}\), find the value of \(x + y\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{162}
Suppose that $u_1$ and $u_2$ are vectors such that $|u_1| = 2$, $|u_2| = 3$, and $u_1 \cdot u_2 = 5$. Define the vectors $u_3, u_4, u_5, u_6, \ldots$ by $u_3 = \text{proj}_{u_1} u_2$, $u_4 = \text{proj}_{u_2} u_3$, $u_5 = \text{proj}_{u_3} u_4$, and so on. Find the value of the infinite sum $$\sum_{n=1}^\infty |u_n|.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Determine the values of $p$ for which the $p$-norm of a matrix is invariant under orthogonal transformations. Specifically, find the values of $p$ such that for any orthogonal matrix $A$ and any matrix $X$, the equality $\|AX\|_p = \|X\|_p$ holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the asymptotic complexity of the recurrence relation \( T(n) = T(\sqrt{n}) + 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(\log \log n)}
Determine whether the series \(1 + \frac{1}{2} -\frac{1}{3} + \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \dots\) converges. If it diverges, specify whether it diverges to \(\infty\) or has no limit. The generic term of the series is given by:
\[a_n = \begin{cases} -\frac{1}{n}, & \text{if } 3 \mid n \\ \frac{1}{n}, & \text{otherwise} \end{cases}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given the matrix \( A = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix} \), find the sum of all elements in the matrix \( M = \sum_{i=1}^{20} A^i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2020}
For some positive integers \( p \) , there is a quadrilateral \( {ABCD} \) with positive integer side lengths, perimeter \( p \) , right angles at \( E \)
 and \( C,{AB} = 2 \) , and \( {CD} = {AD} \) . How many different values of \( p < {2015} \) are possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
31
}
Calculate the expected stopping time $E[S]$ for the first occurrence of the sequence HTH in a series of fair coin tosses using the optional stopping theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
For the recursive sequence defined by \(x_0 = 0\) and \(2x_{n+1} = x_n^2 + a\), find the range of values of \(a\) for which the sequence converges to a finite limit, considering only real-valued sequences.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-3, 1]}
You are playing a game with 3 envelopes, each containing a uniformly random amount of money between $0$ and $1000$. At each step, you can choose an envelope, look at its contents, and decide to either keep it or discard it and choose from the remaining envelopes. If you play to maximize your expected winnings, what is \( \lfloor E \rfloor \), the greatest integer less than or equal to your expected winnings?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{695}
Determine the number of elements of order 6 in the symmetric group $S_7$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1470}
Determine for which values of $n$ the ring $\mathbb{C}[x_1,\dots,x_n]/(\sum_{i=1}^n x_i^2)$ contains no nilpotent elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \geq 2}
Using Rouche's theorem, determine the number of roots of the polynomial \( z^9 + z^5 - 8z^3 + 2z + 1 \) that lie in the annular region between the circles \(|z|=1\) and \(|z|=2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
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 $S_9=p+qi,$ were $p$ and $q$ are integers, find $|p|+|q|.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
368

}
A rope of length 10 m is tied tautly from the top of a flagpole to the ground 6 m away from the base of the pole. An ant crawls up the rope and its shadow moves at a rate of 30 cm/min. How many meters above the ground is the ant after 5 minutes? (This takes place on the summer solstice on the Tropic of Cancer so that the sun is directly overhead.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2
}
Calculate the outward flux of the vector field \( F(x,y,z) = (xy+2xz, x^2+y^2, xy-z) \) through the surface of the cylinder \( x^2+y^2=4 \) bounded by the planes \( z=y-2 \) and \( z=0 \), without using the Divergence Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-40\pi}
What is the maximum number of isolated local maxima of the function \(|p(x)|\), where \(p: \Bbb R^n \to \Bbb R\) is a quadratic polynomial?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of generators in the group \((\mathbb{Z}_{100}^*, \cdot_{100})\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Compute the integral \( \int_{|z|=1} \frac{z^7}{10z^8-5z^4+2z^2+iz+1} \, dz \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi i}{5}}
Calculate the expected value of the maximum between a random variable $T$ and a constant $5$, given that $T$ follows an exponential distribution with a rate parameter of $\frac{1}{30}$, denoted as $T\sim\exp(\frac{1}{30})$. Provide your answer to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30.39}
Compute the following integral for $\sigma > 1$:
\[ \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T}\left|\zeta{(\sigma + it)}\right|^2dt .\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\zeta(2\sigma)}
In a simple graph with 239 vertices, where the graph is not bipartite and each vertex has a degree of at least 3, determine the smallest integer $k$ such that every odd cycle in the graph has a length of at most $k$. What is the value of $k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{239}
Determine the greatest integer $A$ such that in any permutation of the numbers $1, 2, \ldots, 100$, there exist ten consecutive numbers whose sum is at least $A$. What is $A$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{505}
Find a monic integer polynomial of degree $n$ whose Galois group over $\mathbb{Q}$ is $S_n$. Provide the polynomial.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^n - x - 1}
Let $M$ be the set of all holomorphic functions $f$ from the unit disk to itself such that $f(0)=\frac{2}{3}$. Determine the supremum of $|f'(0)|$ over all functions in $M$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{9}}
Two airlines each operate a flight from country A to country B. A total of 1000 passengers choose between the two airlines, each with a probability of \( \frac{1}{2} \). Each airline offers a plane with 510 seats. What is the probability that at least one passenger does not get a seat on their chosen flight?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.507}
Determine the minimal number of time steps required to reach the state $x_f = \begin{bmatrix} 1 & 1 & 0 \end{bmatrix}^T$ from the initial state $x_0 = \begin{bmatrix} 0 & 0 & 0 \end{bmatrix}^T$ in the discrete-time state-space system given by:

$$x(t+1) = Ax(t) + Bu(t), \quad y(t) = Cx(t)$$

where

$$A = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 1 & 1 \\ 2 & 0 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral \( \int_{-\infty}^{\infty} \frac{1}{(e^{-x}+1)(\pi^2+x^2)}dx \) using a complex-analytic approach.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine the number of conjugacy classes of order 3 matrices in the group \( PGL(3, \mathbb{Z}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
What is the remainder when \(41!\) is divided by \(83\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{82}
Determine the number of degrees of freedom of a rank-1 matrix $X = ab^T$, where $a \in \mathbf{R}^m$ and $b \in \mathbf{R}^n$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m + n - 1}
A bag contains 25 marbles numbered from 1 to 25. If you randomly select 12 marbles from the bag, what is the expected value of the second-largest marble?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Find a natural number whose Collatz sequence length exceeds the logarithmic trend of $\ln(x)^C$, where $C$ is a constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Find the smallest known value of $C$ such that there are infinitely many prime pairs differing by $C$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{246}
Let $a$, $b$, and $c$ be three distinct real numbers satisfying the equation \( \frac{a}{1+ab} = \frac{b}{1+bc} = \frac{c}{1+ca} \). Find the value of the product $abc$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Determine the number of solutions to the differential equation \( x'(t) = x(t) - e^{-t^2} \) that satisfy the condition \( \lim_{t \rightarrow \pm \infty} x(t) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $X$ be a linear space and $T$ be a linear operator on $X$. Define a new linear operator $S$ by $S(x) = T(T(x))$. If $T^*$ is the adjoint operator of $T$, determine the adjoint operator of $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(T^*)^2}
Evaluate the limit: $$\lim_{n\to\infty} \int\limits_{0}^{1/2} \cos(1-\cos(1-\cos(...(1-\cos(x))...))_{n\text{ times}} \ \mathrm{d}x$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine the radius of convergence of the Taylor series expansion of the function \( f(x)= \begin{cases} e^{-\frac{1}{x^2}}\quad &\text{if $x\neq 0$}\\ 0\quad &\text{if $x=0$} \end{cases} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the integral of the differential form \( \varphi = e^x dx \wedge dy + y dz \wedge dx + x dy \wedge dz \) over the surface \( S \) of the cylinder defined by \( x^2 + y^2 = 1 \) between \( z = 1 \) and \( z = 3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
Let $p(z) = z^7 + z(z-3)^3 + 1$. Find the number of zeros (including multiplicities) of $p(z)$ in the open disk $|z-3|<1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Suppose we roll a fair 6-sided die repeatedly. What is the expected number of rolls required to see 3 of the same number in succession?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
Earl and Bob start their new jobs on the same day. Earl's work schedule is to work for 3 days followed by 1 day off, while Bob's work schedule is to work for 7 days followed by 3 days off. In the first 1000 days, how many days off do they have in common?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Rupert and Ronald aced their math test, so their mother bought them a fantastic trampoline. Over the course of a week, they jumped on the trampoline every day with varying amounts of enthusiasm. On Monday, Ronald jumped 157 times while Rupert jumped 86 more times than Ronald. On Tuesday, Ronald increased his jumps to 193 times while Rupert decreased his jumps by 35 times compared to Monday. On Wednesday, both brothers doubled the number of jumps from their respective Tuesday jumps. For the remaining four days, from Thursday to Sunday, their jumps decreased by a constant rate of 20 times each day. Calculate the total number of jumps made by Rupert and Ronald throughout the week.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4411}
There is a magical tree with 123 fruits. On the first day, 1 fruit falls from the tree. From the second day onwards, the number of fruits falling each day increases by 1 compared to the previous day. However, if the number of fruits on the tree is less than the number of fruits that should fall on a given day, the falling process restarts from 1 fruit on that day and a new cycle begins. Following this pattern, on which day will all the fruits have fallen from the tree?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Marvin had a birthday on Tuesday, May 27 in the leap year 2008. In what year will his birthday next fall on a Saturday? Express your answer as a single year.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017}
Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. Find the probability that the lengths of these three segments are the three side lengths of a triangle with positive area. Provide your answer in the form \(\frac{k}{m}\) where the fraction is simplified, and give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{509}
Let $w$ and $z$ be complex numbers such that $|w+z|=2$ and $|w^2+z^2|=18$. Find the smallest possible value of $|w^3+z^3|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Find the limit of the sequence:
$$a_n = \frac{1}{n} \sqrt[n]{n(n+1)(n+2)...(2n)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{4}{e}\)}
9. Four black $1 \times 1 \times 1$ cubes and four white $1 \times 1 \times 1$ cubes can form $\qquad$ different $2 \times 2 \times 2$ cubes (cubes that are the same after rotation are considered the same type).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
## Task 3 - 331213

In a beauty contest for poodles, Asta, Benno, Cäsar, and Dolly face a jury of four members. Each jury member votes for one of the dogs by raising a card with the initial letter of the dog's name. As a rule for evaluating this voting result, it was stipulated: If two dogs clearly receive more votes than the other two dogs, they are considered qualified. However, if the voting result does not clearly show two dogs with more votes than the other two, a supplementary regulation will be made (e.g., a new vote will be scheduled).

Determine the number of all those different voting results that lead to this latter case! Voting results are considered the same only if they not only match in the number of votes for the dogs but also in which jury members voted for the respective dogs. For example, the voting results $A A B C$ and $C A B A$ are considered different.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{172}
Given that \( f(x)=a_{1} x + a_{2} x^{2} + a_{3} x^{3} + \cdots + a_{n} x^{n} \), where \( a_{1}, a_{2}, a_{3}, \cdots, a_{n} \) form an arithmetic sequence, and \( n \) is a positive even number. Also, given \( f(1) = n^{2} \) and \( f(-1) = n \), compare the value of \( f\left(\frac{1}{2}\right) \) to 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f\left(\frac{1}{2}\right) < 3}
In a rectangle, a broken line is drawn where adjacent segments are perpendicular and equal to the shorter side of the rectangle. Find the ratio of the sides of the rectangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:2}
The students were fishing. It is known that \( a_{1} \) fishermen caught at least one fish, \( a_{2} \) fishermen caught more than one fish, \( a_{3} \) - more than two fish, and so on, up to \( a_{10} \) - more than nine fish. No one caught more than 10 fish. How many fish did the students catch in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a_1 + a_2 + \cdots + a_{10}}
Bucket A is 1/4 full of sand, and bucket B, which has one-half the capacity of bucket A, is 3/8 full of sand. Bucket C has twice the capacity of bucket A and is currently 1/3 full of sand. Jana first pours the contents of bucket C into bucket B, then she pours the contents of bucket B into bucket A. If any sand overflows from bucket A during this process, how much sand would overflow in terms of bucket A's capacity?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{17}{48}}
The base of a hill located beside a river is some distance above the seabed. This depth is a quarter of the vertical distance between the riverbed and the peak of the hill. The hill is 900m high. How far above the seabed is the base of the hill?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{225}
In the Cartesian coordinate system, the vertices of $\triangle ABC$ are $A(-1, 0)$, $B(3, 0)$, and $C(0, 2)$. It is known that a moving line $y = m (0 < m < 2)$ intersects segments $AC$ and $BC$ at points $D$ and $E$ respectively, and there exists a point $P$ on the x-axis such that $\triangle DEP$ is an isosceles right triangle. Find the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the limit of the integral:
\[
I = \lim_{\epsilon\to 0} \frac{1}{2\pi\epsilon}\int_{-\infty}^\infty\int_{-\infty}^\infty f(x,y)\frac{y}{\epsilon}\exp{\frac{-(x^2+y^2)}{2\epsilon}}dxdy
\]
where \(f\) and \(\nabla f\) are continuous and bounded.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\partial f}{\partial y}(0,0)}
Find the minimum value, for \(a, b > 0\), of the expression

\[
\frac{|a + 3b - b(a + 9b)| + |3b - a + 3b(a - b)|}{\sqrt{a^{2} + 9b^{2}}}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{10}}{5}}
A train 120 meters long takes 26.997840172786177 seconds to cross a bridge of a certain length if the speed of the train is 36 kmph. How long is the bridge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{149.98}
11.175. A right parallelepiped is described around a sphere, with the diagonals of the base being $a$ and $b$. Determine the total surface area of the parallelepiped.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3ab}
Margo walks to her friend's house uphill, taking 15 minutes. She returns downhill via the same route, which takes only 10 minutes. If her walking rate uphill is 3 miles per hour and downhill is 6 miles per hour, how far is Margo's friend's house?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.75}
In how many ways can \(2^{n}\) be expressed as the sum of four squares of positive integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
若正整数 $m 、 n$ 满足 $\frac{(m+n)!}{n!}=5040$,则 $m!n$ 的值为 $\qquad$。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
Let $(X_1, \ldots, X_n) \sim \operatorname{Unif}(0,b), b>0$. Find $E\left[\sum \frac{X_i }{X_{(n)}}\right]$ where $X_{(n)} = \max_i X_i$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{n+1}{2}\)}
# Problem 6. (3 points)

Three runners are moving along a circular track at constant equal speeds. When two runners meet, they instantly turn around and start running in opposite directions.

At some point, the first runner meets the second. After 15 minutes, the second runner meets the third for the first time. Another 25 minutes later, the third runner meets the first for the first time.

How many minutes does it take for one runner to complete the entire track?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
Will was organizing his baseball cards in a binder with 3 on each page. He wanted to place the cards in consecutive order based on the year of issue. If he had 8 new cards from 2020 and 10 old cards from various years between 2015 and 2019, with 2 duplicates, how many pages would he use to arrange the cards so that no two cards from the same year share a page?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Calculate the square root of 81. Provide both positive and negative roots, if applicable. Express your answer in the form of ±n, where n is the absolute value of the root.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±9}