inclusionAI/Ring-lite-rl-data huggingface dataset:
In a country, there are $28$ cities, and between some cities, there are two-way flights. Each city has exactly one airport, which can be either small, medium, or big. For every route that:
1. Contains more than two cities,
2. Does not repeat any city,
3. Ends where it begins,

It must include all types of airports. What is the maximum number of flights in this country?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53}
A certain project requires 24 days for Team A to complete on their own, and 18 days for Team B to complete on their own. Now, if 6 people from Team B are transferred to Team A, then Team A alone would need 18 days to complete the project, and Team B alone would need 24 days. If Team B starts working alone for 6 days, and then Teams A and B work together for 4 days, after which Team B has another task and takes their 6 people back, leaving Team A to finish the work alone, how many more days does Team A need?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{26}{3}}
求一切实数 $p$ ，使得三次方程 $5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p$ 的三个根均为自然数。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{76}
Given real numbers \(a_{1}, a_{2}, \cdots, a_{5}\) such that 
\[
\sum_{i=1}^{5} a_{i}^{2} = 1,
\]
find the maximum value of 
\[
T = \sum_{i=1}^{5} \left|a_{i} - a_{i+1}\right| \quad \text{where} \quad a_{6} = a_{1}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
An integer is called ''parity-monotonic'' 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 lateral sides of a trapezoid lie on perpendicular lines. Find the area of the quadrilateral with vertices at the midpoints of the diagonals and at the midpoints of the bases of the trapezoid, given that its lateral sides are equal to \(a\) and \(b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}ab}
Problem 6. (30 points) Ten balls of the same radius are arranged in the form of a triangular pyramid such that each ball touches at least three others. Find the radius of the ball inscribed in the center of the pyramid, touching six identical balls, if the radius of the sphere in which the pyramid of balls is inscribed is $\sqrt{6}+1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\sqrt{2}-1)}
Find the number of three-digit numbers such that each of the first two digits is divisible by the third digit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{138}
When $x \in (0, +\infty)$, the power function $y = (m^2 - m - 1)x^m$ is a decreasing function. Find the value of the real number $m$ that satisfies this condition. Express your answer as a single real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
In a rectangular parallelepiped with dimensions AB = 4, BC = 2, and CG = 5, point M is the midpoint of EF, calculate the volume of the rectangular pyramid with base BDFE and apex M.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{40}{3}}
Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, $\widetilde{DE}$. How many different routes are there from $A$ to $B$ that use each road exactly once?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Let \( X \) be the set of consisting of twenty positive integers \( n,n + 2,\ldots ,n + {38} \) . The smallest value of \( n \) for which any three numbers \( a,b,c \in  X \) , not necessarily distinct, form the sides of an acute-angled triangle is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
92

}
What is the correct ordering of the three numbers $\frac{5}{19}$, $\frac{7}{21}$, and $\frac{9}{23}$, in increasing order?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{19} < \frac{7}{21} < \frac{9}{23}}
椭圆 $\frac{x^{2}}{6}+\frac{y^{2}}{2}=1$, 其右焦点为 $F(2,0)$. 直线 $l$ 过 $F$ 交椭圆于 $A, B$ 两点, $P$ 在 $x=3$ 上, 若 $\triangle A B P$ 为正三角形，则 $S_{\triangle P A B}$ 的最大值的原始答案形式为 \frac{k \sqrt{n}}{m}，请给出k + n + m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
In how many ways can you divide the set of eight numbers  $\{2,3,\cdots,9\}$  into  $4$  pairs such that no pair of numbers has  $\text{gcd}$  equal to  $2$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Consider a nilpotent $4 \times 4$ real matrix $T$. If the dimensions of the nullspaces of $T^2$ and $T^3$ are 3 and 4, respectively, determine the possible dimension of the nullspace of $T$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
How many permutations in $S_n$ have the first $k$ items $1,2,...,k$ all in the same cycle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{n!}{k}$}
Given that the greatest common divisor (gcd) of two numbers a and b is 1, determine the number of ways the number \(3^3\cdot 5^7 \cdot 11^{13}\) can be expressed as a product of two factors, \(m\) and \(n\), such that \(\text{lcm}(m,n)=3^3\cdot 5^7\cdot 11^{13}\) and \(\gcd(m,n)=1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In a city, 2019 metro stations have been constructed. Some pairs of stations are connected by tunnels, and it is possible to travel from any station to any other station via these tunnels. The mayor has ordered the organization of several metro lines: each line must include multiple distinct stations connected sequentially by tunnels (one tunnel can be part of multiple lines). Additionally, each station must be part of at least one line. To save costs, the number of lines should be no more than $k$.

It turned out that the mayor's order is impracticable. What is the largest $k$ for which this could happen?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1008}
How many subsets of the set \( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\} \) have the property that no two of its elements differ by more than 5? For example, count the sets \( \{3\} \), \( \{2, 5, 7\} \), and \( \{5, 6, 7, 8, 9\} \) but not the set \( \{1, 3, 5, 7\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
There is a prism whose sum of the number of edges, faces, and vertices is 50. How many edges of the pyramid have the same base shape as the prism?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
John and Mary select a natural number each and tell that to Bill. Bill wrote their sum and product in two papers hid one paper and showed the other to John and Mary.
John looked at the number (which was $2002$ ) and declared he couldn't determine Mary's number. Knowing this Mary also said she couldn't determine John's number as well.
What was Mary's Number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1001}
Let $A = {1, 2, 3, 4, 5}$. Find the number of functions $f$ from the nonempty subsets of $A$ to $A$, such that $f(B) \in B$ for any $B \subset A$, and $f(B \cup C)$ is either $f(B)$ or $f(C)$ for any $B$, $C \subset A$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the $150 regular price did he save?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
In the convex quadrilateral \(ABCD\), it is known that \(\angle BCD = 80^\circ\), \(\angle ACB = 50^\circ\), and \(\angle ABD = 30^\circ\). Find the angle \(\angle ADB\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50^\circ}
The number $m$ is a prime number between 30 and 50. If you divide $m$ by 12, the remainder is 7. What is the value of $m$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
A is located 60 kilometers west of B. Individuals A and B depart from location A while individuals C and D depart from location B at the same time. A, B, and D all travel east, while C travels west. It is known that the speeds of A, B, C, and D form an arithmetic sequence, with A having the highest speed. After n hours, B and C meet, and n hours later, A catches up to D at location C. Calculate the distance between locations B and C in kilometers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
## Task B-3.3.

Given vectors $\vec{u}=\vec{m}+3 \vec{n}, \vec{v}=7 \vec{m}-5 \vec{n}, \vec{w}=\vec{m}-4 \vec{n} \text{ and } \vec{z}=7 \vec{m}-2 \vec{n}$, where $\vec{m}, \vec{n} \neq \overrightarrow{0}$. If vectors $\vec{u} \text{ and } \vec{v}$, and $\vec{w} \text{ and } \vec{z}$ are perpendicular, determine the angle between vectors $\vec{m} \text{ and } \vec{n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
For how many rational numbers  $p$  is the area of the triangle formed by the intercepts and vertex of  $f(x) = -x^2+4px-p+1$  an integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
There are $2017$ turtles in a room. Every second, two turtles are chosen uniformly at random and combined to form one super-turtle. (Super-turtles are still turtles.) The probability that after $2015$ seconds (meaning when there are only two turtles remaining) there is some turtle that has never been combined with another turtle can be written in the form $\tfrac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1009}
Let point $O$ be the origin of a two-dimensional coordinate system, and let points $A$ and $B$ be located on positive $x$ and $y$ axes, respectively. If $OA = \sqrt[3]{54}$ and $\angle AOB = 45^\circ,$ compute the length of the line segment $AB.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54^{1/3} \sqrt{2}}
Find the number of real solutions to the equation [2x] + [3x] = 8x - 6, where [x] denotes the greatest integer less than or equal to the real number x.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
At a conference, the 2016 participants were registered from P1 to P2016. Each participant from P1 to P2015 shook hands with exactly the same number of participants as the number on their registration form. How many hands did the 2016th participant shake?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1008}
Vasya and Petya started running simultaneously from the starting point of a circular track and ran in opposite directions. While running, they met at some point on the track. Vasya ran a full lap and, continuing to run in the same direction, reached the point where they had previously met at the moment when Petya completed a full lap. How many times faster was Vasya running compared to Petya?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5} + 1}{2}}
已知直线l：(a+1)x+(2a+1)y=5a+3恒过一定点P，$F _ { 1 } , F _ { 2 }$为双曲线$\frac { x ^ { 2 } } { 2 } - \frac { y ^ { 2 } } { b } = 1$的焦点，若$\Delta F _ { 1 } P F _ { 2 }$总是钝角三角形，则b的取值范围为 ·
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$b > 3$}
Given the vertices of a regular 100-sided polygon \( A_{1}, A_{2}, A_{3}, \ldots, A_{100} \). In how many ways can three vertices be chosen that form an obtuse triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{117600}
Find the value of $(\frac{\gamma-\delta}{\gamma+\delta})^{(\frac{\gamma+\delta}{\gamma-\delta})}$ where $\gamma$ and $\delta$ are defined as:

$$\gamma=\sqrt{3}+\frac{1}{\sqrt{3}+\frac{1}{\sqrt{3}+\frac{1}{\sqrt{3}+\frac{1}{...}}}}$$

$$\delta=\sqrt{3}-\frac{1}{\sqrt{3}-\frac{1}{\sqrt{3}-\frac{1}{\sqrt{3}-\frac{1}{...}}}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let a positive integer $n$ be \textit{nice} if there exists a positive integer $m$ such that:
\[
n^3 < 5mn < n^3 + 100.
\]
Find the number of \textit{nice} positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53}
Four mayors decide to build a circular highway that passes within the limits of their cities. Since the four cities are not on the same circle, the mayors hire a company to design a project for constructing a circular highway that is equidistant from all four cities. What is the greatest number of geographically distinct projects that the company can create?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A shopkeeper purchases 20 pens, each with a different discount ranging from 5% to 25% applied on their initial cost price. After applying the discounts, the total cost price for 20 pens is equal to the total selling price for 12 pens. In addition, a sales tax of 8% is applied to the selling price of the pens. Find the gain % or loss % for the shopkeeper considering the varying discounts and the sales tax.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\%}
The distance from a fixed point $P$ on a plane to two vertices $A$ and $B$ of an equilateral triangle $ABC$ are $AP=2$ and $BP=3$. Determine the maximum possible length of the line segment $PC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
12. Six positive integers $a, b, c, d, e, f$ are arranged in alphabetical order to form a sequence, where $a=1$. If a certain positive integer is greater than 1, then the number that is one less than this positive integer must appear to its left. For example, if $d>1$, then one of $a, b, c$ must be $d-1$. Examples: $1,1,2,1,3,2$ satisfies the condition; $1,2,3,1,4,1$ satisfies the condition; $1,2,2,4,3,2$ does not satisfy the condition. The number of different permutations that satisfy the condition is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{203}
Find the distance from the point $M_{0}$ to the plane passing through the three points $M_{1}, M_{2}, M_{3}$.

$M_{1}(-3 ; 4 ;-7)$

$M_{2}(1 ; 5 ;-4)$

$M_{3}(-5 ;-2 ; 0)$

$M_{0}(-12 ; 7 ;-1)$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{459}{\sqrt{2265}}}
The arithmetic-geometric sequence {a_n} satisfies a_1 = 3 and a_1 + a_3 + a_5 = 21. Find the value of a_2 * a_4. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Given that point P (3, -1) and Q (-1, 2) are on the same side of the line $ax+2y=1$, calculate the range of the real number $a$ as an interval.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 3)}
How many unique ways are there to color the balls in a triangular array with 10 balls, arranged such that the first row has 1 ball, the second row has 2 balls, the third row has 3 balls, and the fourth row has 4 balls, using only 2 colors, considering rotations as equivalent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{352}
Example 26 Let $n(\geqslant 3)$ be a given positive integer, and for $n$ given real numbers $a_{1}, a_{2}$, $\cdots, a_{n}$, denote the minimum value of $\left|a_{i}-a_{j}\right|(1 \leqslant i<j \leqslant n)$ as $m$. Find the maximum value of the above $m$ under the condition that $a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}=$ 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{12}{n(n^{2}-1)}}}
All triangles have the same value, and all circles have the same value. Determine the sum of two circles and a triangle. The equations are as follows:
\begin{align*}
\Delta + 3 \bigcirc &= 18 \\
2 \Delta + \bigcirc &= 14 \\
\Delta + 2 \bigcirc &= \ ?
\end{align*}
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{68}{5}}
In the arithmetic sequence {a_n}, it is known that a_5 > 0 and a_4 + a_7 < 0. Find the term number n that corresponds to the maximum value of the sum of the first n terms, S_n. Express your answer in the form S_n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_5}
Given the sequence $\{a_{n}\}$ satisfies $a_{1}=1$, $({{a}\_{n+1}}-{{a}\_{n}}={{(-1)}^{n+1}}\dfrac{1}{n(n+2)})$, find the sum of the first 40 terms of the sequence $\{(-1)^{n}a_{n}\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{20}{41}}
Consider a $2 \times 2$ grid of squares. David writes a positive integer in each of the squares. Next to each row, he writes the product of the numbers in the row, and next to each column, he writes the product of the numbers in each column. If the sum of the eight numbers he writes down is 2015, what is the minimum possible sum of the four numbers he writes in the grid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{88}
Within the interval $1 \leqslant n \leqslant 10^{6}$, find the number of integers $n$ such that the equation $n = x^y$ has non-negative integer solutions $x$ and $y$ with $x \neq n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1111}
Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside of the doghouse that Spot can reach? If the area can be expressed as $x\pi$, find the value of $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
求满足$\overline { a b c } ( a + b + c ) ^ { 3 }$的所有三位数abc.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{512}
Starting with a sequence of $n 1's$, you can insert plus signs to get various sums. For example, when $n = 10$,  you can get the sum $1 + 1 + 1 + 11 + 11 + 111 = 136$, and the sum $1 + 1 + 11 + 111 + 111 = 235$.  Find the number of values of $n$ so that the sum of $1111$ is possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{121}
已知函数 $f(x)=|2 x+1|-|3 x-a|(a>0)$
若 $f(x)$ 的图像与 $x$ 轴围成的三角形的面积大于 $\frac{5}{3}$, 求 $a$ 的取值范围.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(1, +\infty)$}
A nine-digit number is formed by repeating a three-digit number three times; for example, $256256256$. Determine the common factor that divides any number of this form exactly.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1001001}
Find the maximum of the expression:
\[
| \ldots | |x_1 - x_2| - x_3| - \ldots | - x_{2023}|
\]
where \(x_1, x_2, \ldots, x_{2023}\) are distinct natural numbers between \(1\) and \(2023\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2022}
A rectangular table of size \( x \) cm \( \times 80 \) cm is covered with identical sheets of paper of size 5 cm \( \times 8 \) cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed 1 cm higher and 1 cm to the right of the previous one. The last sheet is adjacent to the top-right corner. What is the length \( x \) in centimeters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{77}
Gabriela found an encyclopedia with $2023$ pages, numbered from $1$ to $2023$. She noticed that the pages formed only by even digits have a blue mark, and that every three pages since page two have a red mark. How many pages of the encyclopedia have both colors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
An 8 by 8 grid of numbers obeys the following pattern: 1) The first row and first column consist of all 1s. 2) The entry in the $i$th row and $j$th column equals the sum of the numbers in the $(i-1)$ by $(j-1)$ sub-grid with row less than $i$ and column less than $j$. What is the number in the 8th row and 8th column?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2508}
在正整数 1,2,\cdots,20210418 中，有多少个数的数码和是 8？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5100}
For what values of $x$ in the following series, does the series converge?
\begin{align}
\sum^{\infty}_{n=1}\dfrac{(-1)^n x^n}{n[\log (n+1)]^2},\;\;-3<x<17
\end{align}
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1≤x≤1}
Given a regular quadrangular prism \(ABCD-A_1B_1C_1D_1\) with the base \(ABCD\) being a unit square, if the dihedral angle \(A_1-BD-C_1\) is \(\frac{\pi}{3}\), find \(AA_1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{6}}{2}}
A trapezoidal plot has its parallel sides measuring 60 meters and 80 meters, and the remaining two sides measuring 50 meters each. The plot is to be enclosed by wire fencing with poles placed at varying intervals: 5 meters apart along the parallel sides and 7 meters apart along the other two sides. How many poles will be needed for this fencing?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Portraits of famous scientists hang on a wall. The scientists lived between 1600 and 2008, and none of them lived longer than 80 years. Vasya multiplied the years of birth of these scientists, and Petya multiplied the years of their death. Petya's result is exactly  $ 5\over 4$  times greater than  Vasya's result. What minimum number of portraits can be on the wall? 

*Author: V. Frank*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
For how many integers $n$ between $1$ and $50$, inclusive, is  
$$\frac{(n^2-1)!}{(n!)^n}$$ 
an integer? (Recall that $0! = 1$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Two spheres are inscribed in a dihedral angle so that they touch each other. The radius of one sphere is twice 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 your answer, provide the cosine of this angle, rounded to two decimal places if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.11}
How many ways can you divide a set of $n^2$ elements into $n$ subsets?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(n^2)!}{(n!)^n}}
A farmer has twice as many pigs as cows, 4 more cows than goats, and 3 times as many chickens as pigs. Moreover, there are half as many ducks as the sum of cows and goats. If the farmer has 172 animals total, how many goats does he have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the radius of a circle in which a segment corresponding to a chord of length 6 cm contains a square with a side length of 2 cm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{10} \ \text{cm}}