inclusionAI/Ring-lite-rl-data huggingface dataset:
On average, how long will a traffic sniffer observe packets before it stops working, given that it observes packets arriving according to a Poisson process with rate $\lambda$ and stops working if it does not observe any packets for a period of time $T$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{e^{\lambda T}-1}{\lambda}}
In a certain math competition, there are 10 multiple-choice questions. Each correct answer earns 4 points, no answer earns 0 points, and each wrong answer deducts 1 point. If the total score becomes negative, the grading system automatically sets the total score to zero. How many different total scores are possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Jane can arrange 16 vases of flowers in a day, while her friend Mark can arrange 20 vases per day. However, Mark can only help for 3 days. If there are 248 vases of flowers to be arranged, how many more days will Jane need to finish arranging the remaining vases after Mark stops helping?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the first 1963 digits after the decimal point in the decimal expansion of the number \((\sqrt{26} + 5)^{1963}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A point $P$ lies inside an equilateral triangle $ABC$ such that $AP=15$ and $BP=8$. Determine the maximum possible value of the sum of the areas of triangles $ABP$ and $BCP$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60 + 16\sqrt{3}}
What is the largest integer less than $\log_2 \frac{3}{2} + \log_2 \frac{6}{3} + \cdots + \log_2 \frac{3030}{3029}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Determine the number of non-isomorphic semi-direct products of a cyclic group of order 5 with a cyclic group of order 31.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Initially, each cell in a $29 \times 29$ table contains the number $0$. A movement consists of selecting a $5 \times 5$ sub-table and adding $+1$ to every cell within this sub-table. Determine the maximum value of $n$ such that, after performing $1000$ movements, there exist four cells whose centers form the vertices of a square, and the sum of the values in these four cells is at least $n$. 

**Note:** The sides of the square do not necessarily have to be parallel to the sides of the table.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Suppose $A$ is a set with $n$ elements, and $k$ is a divisor of $n$. Find the number of consistent $k$-configurations of $A$ of order 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\[
\frac{n!}{(n / k)!(k!)^{n / k}}
\]}
Identify all points on the plane \((x, y)\) through which at least one curve of the family \(y = p^{2} + (2p - 1)x + 2x^{2}\) passes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y \geq x^2 - x}
The Miami Heat and the San Antonio Spurs are playing a best-of-five series basketball championship, in which the team that first wins three games wins the whole series. Assume that the probability that the Heat wins a given game is $x$ (there are no ties). The expected value for the total number of games played can be written as $f(x)$, with $f$ a polynomial. Find $f(-1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
In the land of knights (who always tell the truth) and liars (who always lie), 10 people sit at the vertices of a regular decagon at a round table, among whom there are liars. A traveler can stand somewhere and ask those seated: "What is the distance from me to the nearest liar among you?" Each person then answers the question. What is the minimum number of questions the traveler must ask to guarantee finding out who the liars are? (There are no outsiders present, the traveler cannot stand on the table, people are considered as points. Everyone, including the traveler, can measure any distance accurately.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given an ellipse Γ: x^2/a^2 + y^2/b^2 = 1 (a>b>0), there is a point P(1,1) inside the ellipse. Two lines l1 and l2 passing through point P intersect the ellipse Γ at points A, C and B, D respectively, satisfying vector AP = λ * vector PC, vector BP = λ * vector PD. If the slope of line CD is always -1/4 as λ varies, calculate the eccentricity of the ellipse Γ. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{2}}
A student travels from his university to his hometown, a distance of 150 miles, in a sedan that averages 25 miles per gallon. For the return trip, he drives his friend's truck, which averages only 15 miles per gallon. Additionally, they make a detour of 50 miles at an average of 10 miles per gallon in the truck. Calculate the average fuel efficiency for the entire trip.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16.67}
At a certain time, there were some students in the computer lab. At 15:03 and every three minutes after that, 4 students entered the lab. If at 15:10 and every ten minutes after that 8 students left the lab, there were 27 students in the computer lab at 15:44. How many students were in the computer lab initially?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
If the function $f(x)=x^{2}-e^{x}-ax$ has an increasing interval on $\mathbb{R}$, then the range of the real number $a$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty,2\ln 2-2)}
For $0<k<2$, consider two curves 
$C_1: y=\sin 2x\ (0\leq x\leq \pi),\ C_2: y=k\cos x\ (0\leqq x\leqq \pi).$
Denote by $S(k)$ the sum of the areas of four parts enclosed by $C_1,\ C_2$ and two lines $x=0,\ x=\pi$.
Find the minimum value of $S(k).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1

}
A curvilinear trapezoid is enclosed by the curve $y=e^x$, the line $y=0$, and the vertical lines $x=1$ and $x=5$. A tangent line is drawn to the curve $y=e^x$ at a point $P$ within the interval $x\in[1,5]$. This tangent line cuts a common trapezoid with the maximum area from the curvilinear trapezoid. Find the coordinates of point P.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, e^2)}
A $2$ by $2$ square is divided into four $1$ by $1$ squares.  Each of the small squares is to be painted either green or red.  In how many different ways can the painting be accomplished so that no green square shares its top or right side with any red square?  There may be as few as zero or as many as four small green squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
设定点 $A, B, C, D$ 是以点 $O$ 为中心的正四面体的顶点，用 $\sigma$ 表示空间以直线 $O A$ 为轴且满足条件 $\sigma(B)=C$ 的旋转，用 $\tau$ 表示空间关于点 $O, C, D$ 所在平面的镜面反射，设 $l$ 为过 $A B$ 的中点与 $C D$ 的中点的直线，用 $\omega$ 表示空间以 $l$ 为轴的 $180^{\circ}$ 旋转。设 $\sigma \circ \tau$ 表示变换的复合，先作 $\tau$ ，再作 $\sigma$ ，则 $\omega$ 可以表示为 $k \circ m \circ k \circ k \circ m \circ k$。请计算 $k$ 和 $m$ 出现的次数之和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
the ages of patrick and michael are in the ratio of 3 : 5, that of michael and monica are in the ratio of 3 : 4, that of monica and nathan are in the ratio of 5 : 7, and that of nathan and patrick are in the ratio of 4 : 9. If the sum of their ages is 252, what is the difference between the ages of patrick and nathan?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66.5}
Consider the equation $x^2 + 14x = 32$. Find the values of $a$ and $b$ such that the positive solution of the equation has the form $\sqrt{a}-b$, where $a$ and $b$ are positive natural numbers. Calculate $a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{88}
A point $P(5a+1,12a)$ lies inside the circle $(x-1)^{2}+y^{2}=1$. Calculate the range of values for $a$ that ensures the point remains inside the circle. Express your answer as an inequality, including absolute value if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|a| < \frac{1}{13}}
Aaron is a certain age. Henry's sister is three times as old as Aaron, and Henry is four times as old as his sister. The combined age of the siblings is 240 years. How old is Aaron?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Let $\triangle ABC$ be an equilateral triangle with side length $s$ and $P$ a point in the interior of this triangle. Suppose that $PA$, $PB$, and $PC$ are the roots of the polynomial $t^3-18t^2+91t-89$. Then $s^2$ can be written in the form $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $100m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7208}
The sum of two natural numbers and their greatest common divisor is equal to their least common multiple. Determine the ratio of the two numbers.





Translating the text as requested, while maintaining the original formatting and line breaks.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:2}
Evaluate the integral \( \int_{-\infty}^{+\infty}\int_{0}^{+\infty}xe^{-x^{2}e^{y}} \Gamma \left(0,x^2\right)\, dydx. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a scalene triangle \(ABC\), one of the angles is equal to the difference between the other two angles, and one of the angles is twice another angle. The angle bisectors of \(\angle A\), \(\angle B\), and \(\angle C\) intersect the circumcircle of the triangle at points \(L\), \(O\), and \(M\) respectively. Find the area of triangle \(LOM\) if the area of triangle \(ABC\) is equal to 8. If the answer is not an integer, round it to the nearest integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Let 100 real numbers $a_1$, $a_2$, $a_3$, …, $a_{100}$ satisfy $(n-2)a_n - (n-1)a_{n-1} + 1 = 0$ ($2 \leq n \leq 100$), and it is known that $a_{100} = 199$. Find the value of $a_1 + a_2 + a_3 + \ldots + a_{100}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10000}
Determine the number of roots of the equation, $$x^4 - 2x^2 + 4x -\sin x = 0$$ in $[-2\pi , 2\pi]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given real numbers $x$ and $y$ satisfying $4x^2 - 2\sqrt{3}xy + 4y^2 = 13$, find the range of values for $x^2 + 4y^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[10 - 4\sqrt{3}, 10 + 4\sqrt{3}]}
Triangle \(ABC\) has side lengths \(AB=231\), \(BC=160\), and \(AC=281\). Point \(D\) is constructed on the opposite side of line \(AC\) as point \(B\) such that \(AD=178\) and \(CD=153\). Compute the distance from \(B\) to the midpoint of segment \(AD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{208}
For the cubic function $$f(x)= \frac {1}{3}x^{3}- \frac {1}{2}x^{2}+3x- \frac {5}{12}$$, calculate $$f(0)+f( \frac {1}{2017})+f( \frac {2}{2017})+$$…$$+f( \frac {2015}{2017})+f( \frac {2016}{2017})+f(1)$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2018}
Given that $a > 0$, find the necessary and sufficient conditions for the function $f(x)=ax-x^{2}$ to satisfy $f(x)\leqslant 1$, where $x\in[0,1]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0 < a \leqslant 2}
Given a parallelogram $ABCD$ where $\angle B = 111^{\circ}$ and $BC = BD$. Point $H$ is marked on segment $BC$ such that $\angle BHD = 90^{\circ}$. Point $M$ is the midpoint of side $AB$. Find the angle $AMH$. Provide the answer in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{132^\circ}
Evaluate the infinite sum $\sum_{n=0}^{\infty}\binom{2 n}{n} \frac{1}{5^{n}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5}}
Among 100 young men, if at least one of the height or weight of person A is greater than that of person B, then A is considered not inferior to B. Determine the maximum possible number of top young men among these 100 young men.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Given that the sequence {a_n} is a geometric sequence and S_n = 3^n + m - 5, find the value of m. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the maximum value of the expression \((\sqrt{9-\sqrt{7}} \sin x - \sqrt{2(1+\cos 2x)} - 1) \cdot (3 + 2 \sqrt{13-\sqrt{7}} \cos y - \cos 2y)\). If the answer is not an integer, round it to the nearest whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
A mother and her baby are playing a game on a circular track with a radius of 60 meters. The mother moves at a speed of 0.4 meters per second, while the baby moves at a speed of 0.2 meters per second. At the center of the circular track, there is a cylindrical rock with a radius of 30 meters that blocks their view. Find the maximum amount of time they can see each other continuously. Express your answer in seconds, including π.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200\pi}
A cylinder with a volume of 21 is inscribed in a cone. The plane of the upper base of this cylinder cuts off a truncated cone with a volume of 91 from the original cone. Find the volume of the original cone.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{94.5}
已知 $O$ 是凸四边形 $A B C D$ 对角线 $A C$ 和 $B D$ 的交点, $\triangle A O B, \triangle B O C$, $\triangle C O D, \triangle D O A$ 的周长相等. 若 $\triangle A O B, \triangle B O C, \triangle C O D$ 的内切圆半径分别为 $3,4,6$, 则 $\triangle D O A$ 的内切圆半径的形式为\(\frac{k}{m}\)，请给出k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
For the set $A = \{1, 2, 3, \cdots, n\}$ and its every non-empty subset, define a unique "alternating sum" as follows: rearrange the subset in decreasing order, and then starting from the largest number, alternately subtract or add the succeeding number to obtain the result. For example, the "alternating sum" of the set $\{1, 2, 4, 7, 10\}$ is $10-7+4-2+1=6$, the "alternating sum" of the set $\{7, 10\}$ is $10-7=3$, and the "alternating sum" of the set $\{5\}$ is $5$, etc. Find the total sum of the "alternating sums" of all subsets of the set $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \cdot 2^{n-1}}
Given that $a$, $b$, and $c$ are unit vectors satisfying $|a-b|^2+|b-c|^2+|c-a|^2=9$, determine the angle between the vectors $2a+5b+5c$ and $a$. Express your answer in degrees or radians.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180^\circ}
Exercise 16. Let $k \geqslant 1$ be an integer. What is the smallest integer $n$ such that, no matter how $n$ points are placed in the plane, it is possible to choose a subset $S$ consisting of $k$ of these points that satisfies "for every pair $P, Q$ of points in $S$, the distance between $P$ and $Q$ is less than or equal to 2" or "for every pair $\mathrm{P}, \mathrm{Q}$ of points in $\mathrm{S}$, the distance between $\mathrm{P}$ and $\mathrm{Q}$ is strictly greater than 1."
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(k-1)^{2}+1}
Evaluate the integral \( \int_{2}^{3} \sqrt{(x^2+x+1)+\sqrt{(x^2+x+1)-\sqrt{(x^2+x+1)+\sqrt{(x^2+x+1)\cdots\cdots}}}} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{2}}
In the triangular pyramid $A B C D$ with a base $A B C$, the lateral edges are pairwise perpendicular, $D A=D B=5$, and $D C=1$. From a point on the base, a light ray is emitted. After reflecting exactly once from each of the lateral faces (without reflecting from the edges), the ray hits a point on the base of the pyramid. What is the minimum distance the ray could have traveled?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{10\sqrt{3}}{9}}
Count the number of lattice points on the region $ax+by\leq ab,x>0,y>0$, where $a,b \in \mathbb{P}\setminus{\{2\}} \ \land a\neq b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{(a-1)(b-1)}{2}\)}
How many incongruent solutions does the congruence $x^4-14x^2+36\equiv 0\pmod{2019}$ have, given that $2019=673\times 3$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
An octagon has 8 vertices, and each vertex is assigned one of the integers 1, 2, 3, or 4. The integers at the ends of every line segment are different. Given that three vertices already have the integers 1, 2, and 3 written on them, how many times will the integer 4 appear in the completed diagram? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given a $4\times4$ grid where each row and each column forms an arithmetic sequence with four terms, find the value of $Y$, the center top-left square, with the first term of the first row being $3$ and the fourth term being $21$, and the first term of the fourth row being $15$ and the fourth term being $45$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{43}{3}}
Calculate the definite integral:

$$
\int_{0}^{\operatorname{arctg} 3} \frac{4+\operatorname{tg} x}{2 \sin ^{2} x+18 \cos ^{2} x} d x
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{6} + \frac{\ln 2}{4}}
There are  $\displaystyle{2n}$  students in a school  $\displaystyle{\left( {n \in {\Bbb N},n \geqslant 2} \right)}$ . Each week  $\displaystyle{n}$  students go on a trip. After several trips the following condition was fulfiled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
If \( z \) is a complex number and \( |z| = 1 \), then the maximum value of \( \left|\frac{\sqrt{3} \mathrm{i} - z}{\sqrt{2} - z}\right| \) is?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{7}+\sqrt{5}}
Given that the foci of a hyperbola passing through point M(2,1) are F1 and F2, and a point N on the hyperbola satisfies NF1 ⊥ NF2, determine the area of △NF1F2. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A furniture store set the sticker price of a table $40$ percent higher than the wholesale price that the store paid for the table. During a special sale, the table sold for $35$ percent less than this sticker price. Find the percent the final sale price was of the original wholesale price of the table.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91\%}
Determine the number of pairs of integers $(m, n)$ such that

$$
\sqrt{n+\sqrt{2016}}+\sqrt{m-\sqrt{2016}} \in \mathbb{Q}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the smallest positive integer n such that the total number of lattice points on and inside a convex n-gon is at least n+1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
$2023$ players participated in a tennis tournament, and any two players played exactly one match. There was no draw in any match, and no player won all the other players. If a player $A$ satisfies the following condition, let $A$ be "skilled player".

[b](Condition)[/b] For each player $B$ who won $A$, there is a player $C$ who won $B$ and lost to $A$. 

It turned out there are exactly $N(\geq 0)$ skilled player. Find the minimum value of $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Six students are taking an exam. Suppose that during the exam, the six students are seated on a single bench, with aisles on both sides. They finish the exam in a random order and leave immediately. What is the probability that at least one student will have to disturb one of their remaining five classmates in order to reach the aisle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{43}{45}}
There is a sequence defined as follows:
- $a(2) = 0$
- $a(3) = 1$
- For $n \geq 4$, $a(n) = a\left(\left\lfloor \frac{n}{2} \right\rfloor\right) + a\left(\left\lceil \frac{n}{2} \right\rceil\right)$

Find $a(2014)$.

Note: $\left\lfloor \frac{n}{2} \right\rfloor$ and $\left\lceil \frac{n}{2} \right\rceil$ denote the floor function (largest integer $\leq \frac{n}{2}$) and the ceiling function (smallest integer $\geq \frac{n}{2}$), respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Let $ C_1$ and $ C_2$ be externally tangent circles with radius 2 and 3, respectively. Let $ C_3$ be a circle internally tangent to both $ C_1$ and $ C_2$ at points $ A$ and $ B$, respectively. The tangents to $ C_3$ at $ A$ and $ B$ meet at $ T$, and $ TA \equal{} 4$. Determine the radius of $ C_3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given that \( A \) and \( B \) are two distinct points on the parabola \( y = 3 - x^2 \) that are symmetric with respect to the line \( x + y = 0 \), what is the distance \( |AB| \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{2}}
Rajesh walked 10 kilometers less than 4 times the distance that Hiro walked. Rajesh and Hiro walked a certain distance together. Rajesh walked 18 kilometers. How many kilometers did they walk together?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In the Cartesian coordinate system, the length of the chord intercepted by the circle $x^2+y^2=4$ on the line $x-\sqrt{3}y+2\sqrt{3}=0$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
There are lily pads in a row numbered $0$ to $11$, in that order. There are predators on lily pads $3$ and $6$, and a morsel of food on lily pad $10$. Fiona the frog starts on pad $0$, and from any given lily pad, has a $\frac{1}{2}$ chance to hop to the next pad, and an equal chance to jump $2$ pads. What is the probability that Fiona reaches pad $10$ without landing on either pad $3$ or pad $6$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{15}{256}}
We inscribed a regular hexagon $ABCDEF$ in a circle and then drew semicircles outward over the chords $AB$, $BD$, $DE$, and $EA$. Calculate the ratio of the combined area of the resulting 4 crescent-shaped regions (bounded by two arcs each) to the area of the hexagon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:3}
The line \( L \) crosses the \( x \)-axis at \((-8,0)\). The area of the shaded region is 16. Find the slope of the line \( L \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Let $ABCD$ be a trapezoid with $AB \parallel CD$ and $\angle D=90^{\circ}$. Suppose that there is a point $E$ on $CD$ such that $AE=BE$ and that triangles $AED$ and $CEB$ are similar, but not congruent. Given that $\frac{CD}{AB}=2014$, find $\frac{BC}{AD}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{4027}}