inclusionAI/Ring-lite-rl-data huggingface dataset:
The light glows for every 30 seconds. Between a certain time and 3:20:47 AM, it glowed a maximum of 165.63333333333333 times. What was the starting time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:57:58}
Sean is a biologist, and is looking at a string of length 66 composed of the letters $A, T, C, G$. A substring of a string is a contiguous sequence of letters in the string. For example, the string $AGTC$ has 10 substrings: $A, G, T, C, AG, GT, TC, AGT, GTC, AGTC$. What is the maximum number of distinct substrings of the string Sean is looking at?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2100}
If the integer $a$ makes the one-variable quadratic equation $x^{2}+4x+a+2=0$ have real roots, and makes the solution of the fractional equation $\frac{a+y}{1-y}+3=\frac{1}{y-1}$ positive, then the sum of the integers $a$ that satisfy the conditions is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
a is older than b by 5 years. In a certain number of years, thrice a's age shall be equal to 4 times that of b. Their present ages are 13. How many years from now will this happen?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Determine the minimum value of the expression $|e^{(2m+1)\pi e^{i\theta}}-1|^2$ for any natural numbers $m \geq 1$ and any real number $0 \leq \theta < 2\pi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A target is a triangle divided by three sets of parallel lines into 100 equal equilateral triangles with unit sides. A sniper shoots at the target. He aims at a triangle and hits either it or one of the adjacent triangles sharing a side. He can see the results of his shots and can choose when to stop shooting. What is the maximum number of triangles he can hit exactly five times with certainty?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Triangle $ABC$ has side lengths $AB = 11, BC=24$, and $CA = 20$. The bisector of $\angle{BAC}$ intersects $\overline{BC}$ in point $D$, and intersects the circumcircle of $\triangle{ABC}$ in point $E \ne A$. The circumcircle of $\triangle{BED}$ intersects the line $AB$ in points $B$ and $F \ne B$. What is $CF$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$30$}
Evaluate \[ \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} \sum_{n=0}^\infty \frac{1}{k2^n + 1}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
To understand the performance of 1,252 students in a knowledge competition at a certain school, it was decided to use systematic sampling to draw a sample of 50. The number of individuals randomly eliminated from the total population is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
What is the radius of convergence for the function $f(z) = \frac{z - z^3}{\sin(\pi z)}$ at the point $z=0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the number of integers \( n \) that satisfy
\[ 15 < n^2 < 120. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
求方程 $x+y+z=1$ 的整数解的个数, 要求 $x, y, z$ 都大于 -5 .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly \(7\). Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice? Express your answer in the form \(\frac{k}{m}\), where the fraction is in simplest form. Please give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
The vertices of a regular nonagon ($9$-sided polygon) are to be labeled with the digits $1$ through $9$ in such a way that the sum of the numbers on every three consecutive vertices is a multiple of $3$. Two acceptable arrangements are considered to be indistinguishable if one can be obtained from the other by rotating the nonagon in the plane. Find the number of distinguishable acceptable arrangements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
A group of six Fnargs consisting of one of each possible form (one red and one blue for each number of heads) is made to line up such that no immediate neighbors are the same color nor have the same number of heads. How many ways are there of lining them up from left to right?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Two cars are traveling on a highway, 100 meters apart, and both cars are moving at a speed of 60 kilometers per hour. The highway has different speed checkpoints positioned far apart from each other. After passing the first checkpoint, the speed of each car immediately increases to 80 kilometers per hour; after the second checkpoint, the speed immediately increases to 100 kilometers per hour; and after the third checkpoint, the speed immediately increases to 120 kilometers per hour. What will be the distance between the two cars after they both pass the third checkpoint?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200 \text{ meters}}
Evaluate the integral $$\int_{C}\frac{(z-2i+1)^2 \cos(3\pi z)}{\overline{z}^{3} (z-2i)^5} \ dz$$ where $C$ is the circle $|z -2i|=1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
7. Find the smallest natural number $n$, such that in any two-coloring of $K_{n}$ there exist 5 edge-disjoint monochromatic triangles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
While moving down an escalator, Walker counted 50 steps. Trotman, who moved three times faster than Walker, counted 75 steps. If the escalator were to stop, how many steps could be counted on its visible part? It is assumed that both individuals moved at a constant speed and that the escalator's speed was constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
If $x$ is $20 \%$ of $y$ and $x$ is $50 \%$ of $z$, then what percentage is $z$ of $y$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40 \%}
Let $\Omega$ and $\omega$ be circles with radii $123$ and $61$, respectively, such that the center of $\Omega$ lies on $\omega$. A chord of $\Omega$ is cut by $\omega$ into three segments, whose lengths are in the ratio $1 : 2 : 3$ in that order. Given that this chord is not a diameter of $\Omega$, compute the length of this chord.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Find the mass percentage of Cr in K2Cr2O7. What is the mass percentage of Cr in this compound?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35.36\%}
Triangle $ABC$ is isosceles, and $\angle ABC = x^{\circ}$. If the sum of the possible measures of $\angle BAC$ is $240^{\circ}$, find $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
In a class, 20% of the students were absent for an exam. 30% failed by 20 marks and 10% just passed. The remaining students scored an average of 65 marks and the cut off for passing the exam is 40. What is the average score of the class?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
How does the closed form for $\sum_{n=1}^\infty \frac1n - \frac1{a+n}$ look like?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Psi(1+a) + \gamma}
8. The residents of the village Razumevo, located 3 km away from the river, love to visit the village Vkusnotevo, situated 3.25 km downstream on the opposite bank of the river, 1 km away from the shore. The width of the river is 500 m, the speed of the current is 1 km/h, and the banks are parallel straight lines. The residents of Razumevo have laid out the shortest route, taking into account that they always cross the river in a direction perpendicular to the shore with their own speed of 2 km/h. How long does this route take if the speed on land does not exceed 4 km/h? Enter the answer in hours in the provided field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.5}
$a$ and $b$ are positive integers. When written in binary, $a$ has 20,041 '1's, and $b$ has 20,051 '1's (not necessarily consecutive). What is the smallest number of '1's $a+b$ could possibly have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given positive integers \( x, y, z \) that satisfy the condition \( x y z = (14 - x)(14 - y)(14 - z) \), and \( x + y + z < 28 \), what is the maximum value of \( x^2 + y^2 + z^2 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{219}
给定正整数 $n$, 求最小的实数 $\lambda$, 使得对任意实数 $a_{1} \geq a_{2} \geq \cdots \geq a_{n} \geq 0$,有 $$ \sum_{k=1}^{n}\left(a_{1}+a_{2}+\cdots+a_{k}-k \sqrt[k]{a_{1} a_{2} \cdots a_{k}}\right)^{2} \leq \lambda \sum_{k=1}^{n} a_{k}^{2} $$ 请提供 $\lambda$ 的值当 $n = 10$ 时。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
What is the coefficient of $x^{60}$ in the expansion of $\prod_{m=1}^{11} (x^m - m)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given two moving points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) on the parabola \( y^2 = 6x \), where \( x_1 \neq x_2 \) and \( x_1 + x_2 = 4 \). The perpendicular bisector of segment \( AB \) intersects the \( x \)-axis at point \( C \). Find the maximum area of triangle \( \triangle ABC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{14}{3} \sqrt{7}}
设 $a, b, c$ 和 $\left(a-\frac{1}{b}\right)\left(b-\frac{1}{c}\right)\left(c-\frac{1}{a}\right)$ 均为正整数，求 $2a + 3b + 5c$ 的最大值和最小值之差。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Let $M = [m_{ij}]$ be a square, $n$ x $n$ matrix where $n \ge 2$ and all $n^2$ entries are distinct. Let $S = \{ m_{ij} : 1 \le i,j \le n \}$. In other words, $S$ is the set containing all entries in the matrix $M$ so that $|S| = n^2$. Let $T_{ij} = \{ m_{ij}, m_{ji}, m_{ii}, m_{jj} \}$ where $1 \le i < j \le n$. Note this results in $\frac{1}{2}n(n-1)$ sets. Let $R_{ij} = \{T_{ij} \cup p : p \in \mathcal P(S-T_{ij}) \}$. In other words, each element of $R_{ij}$ is the union between $T_{ij}$ and an element of the powerset of $S - T_{ij}$. What is the cardinality of the following set: $$ \bigcup_{i<j} R_{ij}$$ In other words, I am trying to derive a function $f : \{x \in \mathbb{Z} : x \ge 2 \} \rightarrow \mathbb{Z}^+$ such that $f(n) = |\bigcup_{i<j} R_{ij}|$. Thus far I've determined that $f(n) \le n(n-1)2^{n^2-5}$. Any suggestions or answers would be greatly appreciated. Thank you.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n(n-1)2^{n^2-5}}
The hour and minute hands of a clock move continuously and at constant speeds. A moment of time $X$ is called interesting if there exists such a moment $Y$ (the moments $X$ and $Y$ do not necessarily have to be different), so that the hour hand at moment $Y$ will be where the minute hand is at moment $X$, and the minute hand at moment $Y$ will be where the hour hand is at moment $X$. How many interesting moments will there be from 00:01 to 12:01?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{143}
Compare the following real numbers: $\frac{{\sqrt{3}-1}}{2}$____$\frac{1}{3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{>}
A train that is 360 meters long is running at a certain speed. It takes approximately 60 seconds to pass a bridge of 140 meters length. What is the speed of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8.33}
If $\sin\alpha = 3\sin(\alpha - 2\beta)$, then $\tan(\alpha - \beta) + 2\tan\beta = \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\tan\beta}
a train 150 meters long completely crosses a 320 meters long bridge in 40 seconds . what is the speed of the train is ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11.75}
On the radius \(AO\) of a circle with center \(O\), a point \(M\) is chosen. On one side of \(AO\) on the circle, points \(B\) and \(C\) are chosen such that \(\angle AMB = \angle OMC = \alpha\). Find the length of \(BC\) if the radius of the circle is 6 and \(\cos \alpha = \frac{2}{3}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given that the sequence $\left\{a_{n}\right\}$ has a period of 7 and the sequence $\left\{b_{n}\right\}$ has a period of 13, determine the maximum value of $k$ such that there exist $k$ consecutive terms satisfying
\[ a_{1} = b_{1}, \; a_{2} = b_{2}, \; \cdots , \; a_{k} = b_{k} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
One writes, initially, the numbers $1,2,3,\dots,10$ in a board. An operation is to delete the numbers $a, b$ and write the number $a+b+\frac{ab}{f(a,b)}$, where $f(a, b)$ is the sum of all numbers in the board excluding $a$ and $b$, one will make this until remain two numbers $x, y$ with $x\geq y$. Find the maximum value of $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1320}
Find the number of subsets of $\{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\}$ where the sum of the elements in the subset is $49$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
在一个不透明的袋子中有 1 个黑球和 2 个白球. 每一轮从中摸一个球后放回，并再在袋中放入 2 个与摸取同色的球. 记 S_n 为前 n 轮摸出黑球的次数，已知第 2 轮摸取的是黑球，在此前提下，记 S_{100} 的期望是 E，求 [E].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Given a sufficiently large positive integer \( n \), which can be divided by all the integers from 1 to 250 except for two consecutive integers \( k \) and \( k+1 \), find \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{127}
甲烷分子 $\left(\mathrm{CH}_4\right)$ 的空间结构为一正四面体, 碳原子位于该四面体的中心, 四个氢原子分别位于该四面体的四个顶点上. 若碳原子和氢原子间的距离都是 $a$, 则以四个氢原子为顶点的这个正四面体的体积是多少?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{8}{27} \sqrt{3} a^3$.}
If the ratio of the lateral surface area of a cone to the area of the section through its axis is $\frac{2 \sqrt{3}\pi}{3}$, then the size of the angle between its generatrix and axis is _________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{6}}
Simplify
\[\frac{\sin x}{1 + \cos x} + \frac{1 + \cos x}{\sin x}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \csc x}
Annie takes a  $6$  question test, with each question having two parts each worth  $1$  point. On each **part**, she receives one of nine letter grades  $\{\text{A,B,C,D,E,F,G,H,I}\}$  that correspond to a unique numerical score. For each **question**, she receives the sum of her numerical scores on both parts. She knows that  $\text{A}$  corresponds to  $1$ ,  $\text{E}$  corresponds to  $0.5$ , and  $\text{I}$  corresponds to  $0$ .

When she receives her test, she realizes that she got two of each of  $\text{A}$ ,  $\text{E}$ , and  $\text{I}$ , and she is able to determine the numerical score corresponding to all  $9$  markings. If  $n$  is the number of ways she can receive letter grades, what is the exponent of  $2$  in the prime factorization of  $n$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Given that the sequence {a_n} is a geometric sequence, if ma_6 * a_7 = a_8^2 - 2a_4 * a_9 and the common ratio q is in the interval (∛5, 2), determine the range of values for the real number m. Express your answer as an interval for m, including the endpoints.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(3, 6)}
Michael writes down all the integers between 1 and $N$ inclusive on a piece of paper and discovers that exactly $40 \%$ of them have leftmost digit 1 . Given that $N>2017$, find the smallest possible value of $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1481480}
Given points $P(\cos \alpha, \sin \alpha)$, $Q(\cos \beta, \sin \beta)$, and $R(\cos \alpha, -\sin \alpha)$ in a two-dimensional space, where $O$ is the origin, if the cosine distance between $P$ and $Q$ is $\frac{1}{3}$, and $\tan \alpha \cdot \tan \beta = \frac{1}{7}$, determine the cosine distance between $Q$ and $R$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
The circle $k_{1}$ with center $O_{1}$ and radius 2 units, and the circle $k_{2}$ with center $O_{2}$ and radius $r$, touch each other externally. The Thales circle of the segment $O_{1} O_{2}$ intersects the circle $k_{1}$ at points $A$ and $B$, and the circle $k_{2}$ at points $C$ and $D$. The reflection of $O_{1}$ over the segment $A B$ is $O_{1}^{\prime}$, and the reflection of $O_{2}$ over the segment $C D$ is $O_{2}^{\prime}$. The distance $O_{1}^{\prime} O_{2}^{\prime}$ is 1 unit.

What is the value of $r$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5+\sqrt{17}}{2}}
Three numbers have a certain ratio. The largest number value is 70, and the difference between the smallest and largest number is 40. What is the ratio of the three numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:5:7}
A travel company launched a promotion: "Buy a tour to Egypt, bring four friends who also buy the tour, and get the cost of your tour refunded." During the promotion, 13 customers came by themselves, and the rest were brought by friends. Some of them brought exactly four new clients, while the remaining 100 brought no one. How many tourists went to the Land of the Pyramids for free?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
设 $a 、 b$ 为正整数，$a^{2}+b^{2}$ 除以 $a+b$ 的商为 $q$, 余数为 $r$, 且 $q^{2}+r=2010$. 求 $a b$的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1643}
The price of Book A is increased from $300 to $360 (USD), while the price of Book B is increased from €450 to €540 (Euros), and the price of Book C is increased from £600 to £720 (GBP). After these increases, a 5% discount is applied to each book's new price, and a shipping fee of $10 (USD), €7 (Euros), and £5 (GBP) is added to each respective book. Calculate the overall percentage increase in the total cost of these three books in their respective currencies.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15.91\%}
Given the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{4} = 1$ has one asymptote equation $2x + 3y = 0$, and $F_1$, $F_2$ are the left and right foci of the hyperbola $C$, respectively. Point $P$ is on the hyperbola $C$, and $|PF_1| = 7$. Find the possible values of $|PF_2|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
The diameter of the red circle is 6 decimeters. What should be the minimum diameter (to the nearest $\frac{1}{2}$ decimeter) of five discs such that they can completely cover the circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Esmeralda has created a special knight to play on quadrilateral boards that are identical to chessboards. If a knight is in a square then it can move to another square by moving 1 square in one direction and 3 squares in a perpendicular direction (which is a diagonal of a $2\times4$ rectangle instead of $2\times3$ like in chess). In this movement, it doesn't land on the squares between the beginning square and the final square it lands on.

A trip of the length $n$ of the knight is a sequence of $n$ squares $C1, C2, ..., Cn$ which are all distinct such that the knight starts at the $C1$ square and for each $i$ from $1$ to $n-1$ it can use the movement described before to go from the $Ci$ square to the $C(i+1)$.

Determine the greatest $N \in \mathbb{N}$ such that there exists a path of the knight with length $N$ on a $5\times5$ board.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
x and y are two towns. Ganesh covers the distance from x to y at some speed. However, he covers the distance from y to x at an average speed of 34 km/hr. His average speed during the whole journey is approximately 38 km/hr. What is his average speed from x to y?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43.07}
Let \(\mathcal{S}\) be a set of 16 points in the plane, no three collinear. Let \(\chi(\mathcal{S})\) denote the number of ways to draw 8 line segments with endpoints in \(\mathcal{S}\), such that no two drawn segments intersect, even at endpoints. Find the smallest possible value of \(\chi(\mathcal{S})\) across all such \(\mathcal{S}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1430}
During a rainstorm, 15 liters of water fell per square meter. By how much did the water level in Michael's outdoor pool rise?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.5}
In a two-round election, eight candidates contested using a proportional voting system. To win, a candidate must secure at least 40% of the valid votes in the first round or participate in a second round if none of the candidates meet the threshold. In the first round, the vote percentages were as follows: Candidate A obtained 25%, Candidate B got 22%, Candidate C received 20%, Candidate D had 15%, Candidate E secured 7%, Candidate F obtained 6%, Candidate G got 4%, and Candidate H received 1% of the valid votes. As no candidate reached the 40% threshold, Candidates A, B, and C contested in the second round. In the second round, Candidate A secured 50% of the votes, Candidate B got 35%, and Candidate C received the remaining 15%. If the winning candidate had a majority of 960 votes more than the runner-up candidate in the second round, find the total number of valid votes in both rounds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12,800}
If the real numbers $m$, $n$, $s$, $t$ are all distinct and satisfy $mn=st$, then $m$, $n$, $s$, $t$ are said to have the property of "quasi-geometric progression." Now, randomly select $4$ different numbers from the $7$ numbers $2$, $4$, $8$, $16$, $32$, $64$, $128$. The probability that these $4$ numbers have the property of "quasi-geometric progression" is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{13}{35}}
The triangle \( \triangle ABC \) is inscribed in the unit circle, and the internal angle bisectors of angles \( A \), \( B \), and \( C \) intersect the circle again at points \( A_{1} \), \( B_{1} \), and \( C_{1} \) respectively. Calculate the value of 
$$
\frac{A A_{1} \cdot \cos \frac{A}{2} + B B_{1} \cdot \cos \frac{B}{2} + C C_{1} \cdot \cos \frac{C}{2}}{\sin A + \sin B + \sin C}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Inside rectangle \(ABCD\), points \(E\) and \(F\) are located such that segments \(EA, ED, EF, FB, FC\) are all congruent. The side \(AB\) is \(22 \text{ cm}\) long and the circumcircle of triangle \(AFD\) has a radius of \(10 \text{ cm}\).

Determine the length of side \(BC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}