inclusionAI/Ring-lite-rl-data huggingface dataset:
Two spheres are inscribed in a dihedral angle in such a way that they touch each other. The radius of one sphere is 1.5 times the radius of the other, and the line connecting the centers of the spheres forms an angle of $30^{\circ}$ with the edge of the dihedral angle. Find the measure of the dihedral angle. In the answer, write down the cosine of this angle, rounding it to two decimal places if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.68}
2. 55 In decimal, find the smallest natural number: its square number starts with 19 and ends with 89.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1383}
10 In the Cartesian coordinate system, given the point set $I=\{(x, y) \mid x, y$ are integers, and $0 \leqslant x \leqslant 5,0 \leqslant y \leqslant 5\}$, then the number of different positioned squares with vertices from the set $I$ is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
Given \( A^{2}=0.012345678987654321 \times (1+2+3+\cdots+9+\cdots+3+2+1) \) and \( B^{2}=0.\dot{0}1234567\dot{9} \), find the value of \( 9 \times 10^{9} \times (1 - |A|) B \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
13th Iberoamerican 1998 Problem A3 Find the smallest number n such that given any n distinct numbers from {1, 2, 3, ... , 999}, one can choose four different numbers a, b, c, d such that a + 2b + 3c = d. Solution
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{835}
For the function $f(x) = \cos(x)$, find the degree of the Taylor polynomial needed to approximate $\cos(1/10)$ with an error less than $10^{-5}$ using the Taylor-Lagrange inequality.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find a formula for $\sum_{k=0}^{n}(-1)^k2^ks(n,k)$ using the exponential generating function of the Stirling numbers of the first kind.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Compute the number of ordered quadruples $(a, b, c, d)$ of distinct positive integers such that $\binom{\binom{a}{b}}{\binom{c}{d}} = 21$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Points $A_1, A_2, \ldots, A_{2022}$ are chosen on a plane such that no three of them are collinear. Consider all angles $A_iA_jA_k$ for distinct points $A_i, A_j, A_k$. What is the largest possible number of these angles that can be equal to $90^\circ$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2042220}
An integer is called <i>parity-monotonic</i> if its decimal representation $a_{1}a_{2}a_{3}\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is odd, and $a_{i}>a_{i+1}$ if $a_{i}$ is even. How many four-digit parity-monotonic integers are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$640$}
The plane is divided by lines into square cells with side length $a$. The cells are numbered using pairs of integers $(m, n)$ (where the cell immediately to the right has the number $(m+1, n)$, and the cell immediately above has the number $(m, n+1)$). Find the probability that a coin of radius $r < a/2$ randomly tossed onto the plane will completely fall into a cell for which $m+n$ is divisible by three.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(a-2r)^2}{3a^2}}
Find the greatest number which leaves the same remainder when it divides 1260, 2310, 30030, and 72930, which are all divisible by the first five prime numbers (2, 3, 5, 7, and 11) and their respective combinations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Evaluate the triple integral \( \int \int \int_{\Omega} xy^{2}z^{2} \, dx \, dy \, dz \) where \( \Omega \) is the region bounded by the surfaces \( z=xy \), \( y=x \), \( x=1 \), and \( z=0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{198}}
Calculate the sum $\frac{1}{2.7.12} + \frac{1}{7.12.17} + ... + \frac{1}{1997.2002.2007}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2050}{287001}}
Select 3 distinct numbers from the set {1, 2, ..., 10} such that they do not form an arithmetic sequence. How many such selections are possible? (Answer with a number).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Given an infinite number of uniquely labeled pairs of dice, you roll the pair labeled A to determine the number of times to roll pair B. The sum of pair B's rolls determines the number of times to roll pair C, and so on. For a probability p between 0 and 1 and a target sum s, determine the minimum number of pairs of dice you need to roll to have at least a p probability of reaching a total sum of s or greater.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lceil \log_7 s \rceil}
Given the sets A and B as follows:
\[
A = \left\{x^2 + x + 1, -x, -x-1\right\}, \quad B = \left\{-y, -\frac{y}{2}, y+1\right\}
\]
and knowing that A = B, find the value of x^2 + y^2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
已知平面上有 100 条直线，用 $T$ 表示由这些直线中的某三条直线围成的直角三角形的集合。求 $|T|$ 的最大值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62500}
In the country of Anchuria, a day can either be sunny, with sunshine all day, or rainy, with rain all day. If today's weather is different from yesterday's, the Anchurians say that the weather has changed. Scientists have established that January 1st is always sunny, and each subsequent day in January will be sunny only if the weather changed exactly one year ago on that day. In 2015, January in Anchuria featured a variety of sunny and rainy days. In which year will the weather in January first change in exactly the same pattern as it did in January 2015?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2047}
Determine the value of $t$ for which the polynomial $f(x) = x^3 - tx - 1$ is irreducible over $\mathbb{Q}[x]$. Provide your answer as a single value of $t$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given two geometric sequences \(\{a_{n}\}\) and \(\{b_{n}\}\) with positive terms and common ratios \(p\) and \(q\) respectively, where \(p > q\) and \(p \neq 1, q \neq 1\), define \(c_{n} = a_{n} + b_{n}\). Let \(S_{n}\) be the sum of the first \(n\) terms of the sequence \(\{c_{n}\}\). Find \(\lim _{n \rightarrow \infty} \frac{S_{n}}{S_{n-1}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\max(p, 1)}
The sides of $\triangle ABC$ measure 11,20, and 21 units. We fold it along $PQ,QR,RP$ where $P,Q,R$ are the midpoints of its sides until $A,B,C$ coincide. What is the volume of the resulting tetrahedron?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Determine the number of ordered triples of complex numbers \((x, y, z)\) such that \(x^2 - y^3\), \(y^2 - z^3\), and \(z^2 - x^3\) are rational numbers, and \(x^3(x^2+1) + y^3(y^2+1) + z^3(z^2+1) = 2[(x^2)y^3 + (y^2)z^3 + (z^2)x^3]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate the triple integral: \[ \int_{0}^{3} \int_{0}^{\frac{6-2x}{3}} \int_{0}^{\frac{6-3y-2x}{2}} xyz \, dz \, dy \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{20}}
Fill the numbers $1$ through $9$ into a $3 \times 3$ grid such that no number is repeated in any cell. Define two cells that share a common edge as a "number pair." Find the maximum number of number pairs where one number is a multiple of the other.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular $12$-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{223}{286}}
A hotel has 100 rooms and charges guests for their rooms in advance. The probability that any booked guest shows up by 10pm is \( p = \frac{19}{20} \), and guests show up independently. Let \( X \) denote the number of booked guests who show up before 10pm.

The hotel earns $50 in profit for every guest and no-show, but must pay $60 in compensation to each guest who is unable to stay due to overbooking. Assuming unlimited demand for pre-booked rooms, how many rooms should the hotel sell to maximize expected profit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Evaluate the integral \( \int_0^2\int_1^3\min(x,y)\,dx\,dy \) given that \( x < y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Determine the asymptotic upper bound for the Dirichlet divisor summatory function \( D(N) = \sum_{i=1}^N \left\lfloor\frac{N}{i}\right\rfloor \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(N \log N)}
Given the plane point sets A = {(x, y) | (y-x)(y-1/x) >= 0} and B = {(x, y) | (x-1)^2 + (y-1)^2 <= 1}, calculate the area of the plane figure represented by A ∩ B. Express your answer in terms of π.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{2}}
在$\vartriangle ABC$中，$AB=\sqrt{2},AC=2,\angle BAC=105{}^\circ $，点$\textit{D}$满足$\overrightarrow{AD}=x\overrightarrow{AB}+y\overrightarrow{AC}$且$x+y=1(x,y\in R)$，则当$|\overrightarrow{AD}|$最小时，$\frac{x}{y}$的值为________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\sqrt{3}$}
Evaluate the limit: \[ \lim_{x \to 0} \log_{\cos(3x)}\cos(2ix) \] where \( i = \sqrt{-1} \). Note: The use of L'Hospital's rule and expansion is prohibited.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{4}{9}}
A solution contains 8 parts of water for every some parts of lemonade syrup. To make the solution contain 20% lemonade syrup, 2.1428571428571423 parts of the solution should be removed and replaced with water. How many parts of lemonade syrup are there in the original solution?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the limit \( \lim_{n\to\infty}\int_{-\pi/4}^{\pi/4}\frac{n\cos(x)}{n^2x^2+1}\,dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Find the Lanczos derivative of the function \(f(x) = |x|\) at \(x = 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Three workers are digging a hole. They take turns working, with each one working as long as it takes the other two to dig half the hole. Working in this way, they finished digging the hole. How many times faster would they have finished the job if they had worked simultaneously?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A tech company has successfully developed a new product and decides to borrow 2 million yuan from a bank for production. The contract stipulates that the loan, along with an 8% interest, is to be repaid in a lump sum at the end of two years. After the product enters the market, due to strong sales, the company is able to repay the loan principal and interest, with an additional surplus of 720,000 yuan. If the company's capital increases by the same percentage each year during the production period, find this percentage.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
Solve the inequality $n \leq 8\lg n$ and find the largest integer $n$ that satisfies it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
A train 120 m long running at a certain speed takes 13.884603517432893 seconds to cross a bridge 150 m in length. What is the speed of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19.45}
Given two normally distributed variables, $X$ and $Y$, with means $E(X)=E(Y)=0$, variances $Var(X)=2$ and $Var(Y)=8$, and correlation $Corr(X,Y)=-\frac{1}{2}$, determine the smallest possible value of $E(X^5Y^3)$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3840}
4. What is the greatest length that a closed, non-self-intersecting broken line can have, which runs along the grid lines of a $6 \times 10$ cell field?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{76}
A student multiplied a number by 3/5 instead of another fraction. The percentage error is 64.00000000000001%. What fraction should the student have multiplied the number by?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5/3}
In a military exercise, there is a defensive line with 20 posts. Five different new weapons need to be distributed among the posts. The first and last posts should not be equipped with the new weapons. Additionally, among every set of 5 consecutive posts, at least one post should be equipped with a new weapon, and no two adjacent posts should be equipped simultaneously. How many possible arrangements are there to equip the posts with the new weapons?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{69600}
A group of people is lined up in [i]almost-order[/i] if, whenever person $A$ is to the left of person $B$ in the line, $A$ is not more than $8$ centimeters taller than $B$. For example, five people with heights $160, 165, 170, 175$, and $180$ centimeters could line up in almost-order with heights (from left-to-right) of $160, 170, 165, 180, 175$ centimeters.
(a) How many different ways are there to line up $10$ people in [i]almost-order[/i] if their heights are $140, 145, 150, 155,$ $160,$ $165,$ $170,$ $175,$ $180$, and $185$ centimeters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89}
Four small panda bears and five bigger panda bears eat 25 pounds and a certain amount of fresh bamboo sprouts every day, respectively. The pandas eat 2100 pounds of bamboo in a week. How many pounds of bamboo do the bigger pandas eat each day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{275}
The general term formula for the sequence $\frac {3}{5}$, $\frac {1}{2}$, $\frac {5}{11}$, $\frac {3}{7}$, $\frac {7}{17}$, ... is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {n+2}{3n+2}}
Find the sum of the series $\sum f_n$, where $f_n(x)=\frac{nx}{(1+n^2x^2)}-\frac{(n-1)x}{[1+(n-1)^2x^2]}$ for $x\in[0,1]$, using telescoping series.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
3.281. $\sin \left(\frac{5}{2} \pi+4 \alpha\right)-\sin ^{6}\left(\frac{5}{2} \pi+2 \alpha\right)+\cos ^{6}\left(\frac{7}{2} \pi-2 \alpha\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{8}\sin8\alpha\sin4\alpha}
Professor Smith is an art history professor specializing in the miniature-painting genre in England. She has a collection of 48 miniature paintings. Each painting measures 4 inches by 6 inches. She wants to arrange these paintings in a special exhibit where each painting is placed in a frame with an additional border of 2 inches on all sides. 

If she arranges the framed paintings in a single row along a gallery wall, how long will the row be in inches?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{480}
If the curve $|y| = 2^x + 1$ and the line $y = b$ have no common points, then the range of values for $b$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1)}
Given the same function $F: \mathbb{R} \rightarrow \mathbb{R}$ defined by $F(x-y) = F(x)F(y)$, determine all possible values of $F(2)$ if $F(1) = 3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
若正整数 $n$ 使得方程 $x^{3}+y^{3}=z^{n}$ 有正整数解 $(x, y, z)$, 则称 $n$ 为 "好数". 那么, 不超过 2019 的好数有 $\qquad$个.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1346}
Find all solutions to the equation

\[ x^{4} = y^{2} + 2y + 2 \]

where \( x \) and \( y \) are integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, -1), (-1, -1)}
In triangle \( ABC \), points \( A_1 \) and \( A_2 \) are on side \( BC \) such that \( BA_1 = 6 \), \( A_1A_2 = 8 \), and \( CA_2 = 4 \). On side \( AC \), points \( B_1 \) and \( B_2 \) are marked such that \( AB_1 = 9 \) and \( CB_2 = 6 \). Segments \( AA_1 \) and \( BB_1 \) intersect at point \( K \), and segments \( AA_2 \) and \( BB_2 \) intersect at point \( L \). Points \( K \), \( L \), and \( C \) are collinear. Determine the length of \( B_1B_2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where the order of the summation is important. For example, $f(4) = 6$ because $4$ can be written as:
- $4$
- $2 + 2$
- $2 + 1 + 1$
- $1 + 2 + 1$
- $1 + 1 + 2$
- $1 + 1 + 1 + 1$

Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2047}
Let $X_t$ be a Brownian motion. Calculate the probability that $X_2 > 1$ given that $X_1 = 0$. Express your answer as a single probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.1587}
Inside a circle of radius \( R \), there is a point \( A \). Two perpendicular lines are drawn through this point. Then, these lines are rotated by an angle \( \varphi \) relative to point \( A \). The chords cut by the circle from these lines sweep out a figure in the shape of a cross centered at point \( A \) during the rotation. Find the area of the cross.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \varphi R^2}
A non-abelian group $G$ of order $p^3$ has $p^2+p+1$ subgroups of order $p$. Determine the number of subgroups of order $p^2$ in $G$. \( \text{(Express your answer as a single integer.)} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p + 1}
Bernardo randomly picks 3 distinct numbers from the set $\{1,2,3,...,7,8,9\}$ and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set $\{1,2,3,...,6,7,8\}$ and also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo's number is larger than Silvia's number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{37}{56}$}
Given that the ABC stock pays no dividend and is currently priced at $S_0 = 10, with the stock price at expiry $T$ going up to $uS_0$ with probability $p$ and down to $dS_0$ with probability $1 - p$, where $d < 1 < u$, and the interest rate is zero, determine the fair value of a European call option with a strike price of $7$. Assume a European put option with a strike price of $9 is priced at $14/9$ and a European put option with a strike price of $8 is priced at $8/9$. What is the fair value of the call option?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{29}{9}}
Determine the ARIMA parameters \((p,d,q)\) for the process \(Y_t = Y_{t-1} + X_t - \phi^2 X_{t-2}\), where \(X_t\) is an \(ARMA(1,2)\) process given by \(X_t = \phi X_{t-1} + Z_t + Z_{t-2}\) and \(Z_t\) is white noise with mean 0 and variance \(\sigma^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1, 3)}
Find all positive integers \(n\) such that \(n\) divides \(a^{25} - a\) for all positive integers \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2730}
Determine the value of \(a \in \mathbb{R}\) such that the function \(f : \mathbb{R}^3 \to \mathbb{R}\) defined by \[ f(x,y,z)=\begin{cases} \frac{z}{x}+y & \text{if } x \neq 0 \\ a & \text{if } x=0 \end{cases} \] is continuous at the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A container is composed of an upright hollow frustum and a hollow cylinder, each with a base radius of $12 \,\text{cm}$ and a height of $20 \,\text{cm}$. When finely granulated sand is poured into this container, it fills the frustum and partially fills the cylinder, with the sand height in the cylindrical section measuring $5 \,\text{cm}$. If this container is then inverted, what will be the height of the sand in $\text{cm}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Find $d$, given that $\lfloor d\rfloor$ is a solution to \[3x^2 + 19x - 70 = 0\] and $\{d\} = d - \lfloor d\rfloor$ is a solution to \[4x^2 - 12x + 5 = 0.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-8.5}
Find the surface area of a sphere inscribed in a pyramid, whose base is a triangle with sides measuring 13, 14, and 15 cm, given that the apex of the pyramid is 5 cm away from each side of the base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{64\pi}{9} \text{ cm}^{2}}
What is the remainder when 71 multiplied by a certain number is divided by 9, if the remainder is known to be 8?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Within the range of 0° to 360°, find the angle(s) with the same terminal side as -120°.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Sixty-four white 1x1x1 cubes are used to form a 4x4x4 cube, which is then painted red on each of its six faces. This large cube is then broken up into its 64 unit cubes. Each unit cube is given a score as follows:

| Exact number of faces painted red | Score |
| --- | ---: |
| 3 | 3 |
| 2 | 2 |
| 1 | 1 |
| 0 | -7 |

Calculate the total score for the 4x4x4 cube.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Bob is building raised beds for his vegetable garden. Each bed is 2 feet high, 2 feet wide, and 8 feet long. The sides are going to be built of 1-foot wide planks. Bob needs a certain number of 8-foot long planks to construct the raised beds. If he needs 50 8-foot long planks in total, how many raised beds is he constructing?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
If there exist $n$ real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying $x_{1}+2 x_{2}+\cdots+ n x_{n}=2009$ and $x_{1}+x_{2}+\cdots+x_{n}=0$, where each $x_{i}= \pm 7$ for $i=1,2, \cdots, n$, determine the minimum value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Mark went to a store where he spent one-half of his money, and then spent $14 more. He then went to another store where he spent one-third of his starting money, and then spent some more. When he had no money left, he had $180 when he entered the first store. How much more did he spend in the second store after spending one-third of his starting money?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$16}
Find the number of odd numbers in the \(n\)-th row of Pascal's triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^k}