inclusionAI/Ring-lite-rl-data huggingface dataset:
The real numbers $a_0, a_1, \dots, a_{2013}$ and $b_0, b_1, \dots, b_{2013}$ satisfy the recurrence relations:
\[
a_{n} = \frac{1}{63} \sqrt{2n+2} + a_{n-1} \quad \text{and} \quad b_{n} = \frac{1}{96} \sqrt{2n+2} - b_{n-1}
\]
for every integer $n = 1, 2, \dots, 2013$. Given the initial conditions $a_0 = b_{2013}$ and $b_0 = a_{2013}$, compute the following sum:
\[
\sum_{k=1}^{2013} \left( a_kb_{k-1} - a_{k-1}b_k \right).
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{671}
In triangle \( \triangle ABC \), the interior angles \( A, B, C \) satisfy \( \sin A = \cos B = \tan C \). Determine the value of \( \cos^3 A + \cos^2 A - \cos A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Two two-digit numbers have a total of 79. If we use the same digits to form four two-digit numbers, with each digit used twice, what is the total of these four numbers? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{158}
When Tim was Jim’s age, Kim was twice as old as Jim.  When Kim was Tim’s age, Jim was 30.  When Jim becomes Kim’s age, Tim will be 88.  When Jim becomes Tim’s age, what will be the sum of the ages of Tim, Jim, and Kim?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{221}
Determine the number of zeros of the polynomial \( p(z) = z^{2015} + 8z^{12} + 1 \) that lie in the first quadrant of the complex plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{504}
Compute \( \lim\limits_{n \to \infty} \int_0^{\frac{\pi}{2}} \frac{(\sin(x))^{n}}{1-\sin{(x)}}\,\mathrm{d}x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find the order of the group of rigid motions of a tetrahedron. Use the hint that involves determining the number of places to which a given face may be sent and the number of positions to which a vertex on that face may be sent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Determine the cardinality of the set of injective functions from the set of natural numbers, \(\mathbb{N}\), to its power set, \(\mathcal{P} \mathbb{N}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Evaluate the definite integral \( \int_{0}^{\pi}\lfloor\cot x\rfloor \, dx \), where \( \lfloor a \rfloor \) denotes the greatest integer less than or equal to \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{2}}
A large vase has a square base with a side length of \(6 \text{ cm}\), and its flat sides slope outwards at an angle of \(120^{\circ}\) with the base. Water flows into the vase at a rate of \(12 \text{ cm}^3/\text{s}\). Calculate, to three significant figures, the rate at which the height of the water is rising when the water has been flowing in for \(3\) seconds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.246}
4. The numbers $x$ and $y$ are such that the equalities $\sin y + \cos x = \sin 3x$ and $\sin 2y - \sin 2x = \cos 4x - \cos 2x$ hold. What is the smallest value that the sum $\cos y + \sin x$ can take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["-1-\\frac{\\sqrt{2+\\sqrt{2}}}{2}"]}
Determine the number of distinct arrangements of the letters \(A, A, A, A, A, B, B, B, C, C, C, D, E, E, F\) such that no two \(C\)'s are adjacent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95135040}
Consider an n-dimensional normal distribution with a mean vector of 0 and an identity covariance matrix. Determine the distribution of the distance from the mean for points drawn from this distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\chi(n)}
Find the value of $x$ that satisfies the equation $x^{x^3} = 3$. 

Express your answer as a decimal rounded to three decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.442}
Determine the number of functions \( f: \mathbb{R} \setminus \{0, 1\} \to \mathbb{R} \) that satisfy the functional equation: \[ f\left(\frac{x-1}{x}\right) + f\left(\frac{1}{1-x}\right) = 2 - 2x. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Susie Q has $2000 to invest. She invests part of the money in Alpha Bank, which compounds annually at 4 percent, and the remainder in Beta Bank, which compounds annually at 6 percent. After three years, Susie's total amount is $\$2436.29$. Determine how much Susie originally invested in Alpha Bank.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{820}
For positive integers $n$, let $f(n)$ denote the number of integers $1 \leq a \leq 130$ for which there exists some integer $b$ such that $a^b - n$ is divisible by $131$. Let $g(n)$ denote the sum of all such $a$. Find the remainder when \( \sum_{n = 1}^{130} [f(n) \cdot g(n)] \) is divided by $131$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
Evaluate the double integral \( \iint (x^2 + y^2) \, dx \, dy \) over the region enclosed by the curves \( y = 8x \), \( x + y = 6 \), \( y = 4 \), and \( y = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1301}{8}}
Given that the reflections of points \( A, B, \) and \( C \) over a line \( e \) are \( A_1, B_1, \) and \( C_1 \), respectively, and their reflections over a point \( P \) are \( A_2, B_2, \) and \( C_2 \), find the reflection of \( P \) over the line \( e \) if \( \overrightarrow{A_1 A_2} + \overrightarrow{B_1 B_2} + \overrightarrow{C_1 C_2} = \overrightarrow{0} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{P}
Determine the form of the function $f(x)$ that satisfies the equation $f(x^2 + yf(x)) = f(0)$ for all real numbers $x$ and $y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = c}
Compute the decimal expansion of \sqrt{\pi}$. Your score will be \min (23, k)$, where $k$ is the number of consecutive correct digits immediately following the decimal point in your answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.77245385090551602729816 \ldots}
Find an irreducible polynomial in \(\mathbb{Z}[x]\) that is reducible modulo 2, 3, and 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^4 + 1}
Let $x,$ $y,$ $z$ be nonzero real numbers such that $x + y + z = 0,$ and $xy + xz + yz \neq 0.$  Find all possible values of
\[
\frac{x^7 + y^7 + z^7}{xyz (xy + xz + yz)}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-7}
Determine the least ordinal \( \beta \) such that it is provable in \( \mathsf{ZFC} \) that \( 2^{\aleph_0} \leq \aleph_{\aleph_\beta} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the coefficient of \(x^{33}\) in the expansion of \((x+x^3)(1+5x^6)^{-13}(1-8x^9)^{-37}\). Use the identity \((1-x)^{-k} = \binom{k+n-1}{k-1}x^n\) to assist in your calculation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83019300}
Compute the dimension of the vector space \( \mathbb{Q}[s^{\pm 1},t^{\pm 1}]/(t^2-t+1,s^2t-st+s-1) \) over \( \mathbb{Q} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
How many different ways are there to select two numbers from the integers 1 through 20 such that their difference is greater than 10?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Evaluate the integral \( I_{n} = \int_{-1}^{1} \frac{1}{x} P_{n}(x) P_{n-1}(x) \, dx \) using the recurrence relation \((n+1)I_{n+1} + nI_{n} = 2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1 - (-1)^n}{n}}
Consider a square $\left\{(x,y): 0 \leq x,y \leq 1\right\}$ divided into $n^2$ smaller squares by the lines $x = \frac{i}{n}$ and $y = \frac{j}{n}$, where $1 \leq i,j \leq n$. Define $x_i = \frac{i}{n}$ and $d_i$ as:
$$d_i = \min_{0 \leq j \leq n} \left| \sqrt{1 - x_i^2} - \frac{j}{n}\right|.$$
Determine the limit:
$$\lim_{n\to\infty} \sum_{i=1}^n d_i$$
if it exists. This problem was posed in the American Mathematical Monthly in 1994 and has a connection to Weyl's equidistribution theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.25}
What is the minimal order of a homogeneous linear ordinary differential equation with constant coefficients that admits the solution \( y(t) = t^2 e^{-t/3} \cos(2t) \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $a$ and $b$ be randomly selected three-digit integers and suppose $a > b$.
We say that $a$ is clearly bigger than $b$ if each digit of $a$ is larger than the corresponding digit of $b$.
If the probability that $a$ is clearly bigger than $b$ is $\tfrac mn$, where $m$ and $n$ are relatively prime integers,
compute $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1061
}
Admit 4 students into 3 universities, with each university admitting at least one student. Find the total number of different admission methods. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
If the function $y=(\alpha - 1)x^{-4\alpha - 2}$ is a power function, then the value of the real number $\alpha$ is \_\_\_\_\_\_ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the smallest degree of a monic polynomial $p(x)$ with integer coefficients such that $p(n)$ is divisible by 2021 for every positive integer $n$. Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{87}
Let $X_1$, $X_2$, and $X_3$ be independent, identically distributed continuous random variables from an Exponential Distribution. Find the probability that the median of $X_1$, $X_2$, and $X_3$ is closer to the smallest value $X_{(1)}$ than it is to the largest value $X_{(3)}$. Express your answer as a single probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
What is the degree of the angle between the hour and minute hands at 8:30 on a clock?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75^\circ}
How many pairs (x, y) of non-negative integers with 0 ≤ x ≤ y satisfy the equation 5x^2 - 4xy + 2x + y^2 = 624?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine the dimensions $n$ for which the following statement holds: "For any norm $N$ in $\mathbb{R}^n$ and for any vectors $x, y \in \mathbb{R}^n$ such that $|x_i| \leq |y_i|$ for all $i$, we have $N(x) \leq N(y)$."
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $f \in L^1(\mathbb{R}) \cap C(\mathbb{R})$, meaning $f$ is integrable and continuous. For $z \in \mathbb{C}$ with $\text{Im}(z) \neq 0$, define
\[ g(z) = \int_{-\infty}^\infty \frac{f(t)}{t-z} \, dt. \]
Find \( \lim_{\text{Im}(z) \to 0} \left( g(z) - g(\bar{z}) \right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i f(x)}
Christy and her friend Tanya go to Target to buy some face moisturizer and body lotions. Tanya pays 50$ for two face moisturizers each and 60$ per body lotion, buying four of them. Christy buys three face moisturizers and five body lotions, spending twice as much as Tanya. If there is a 10% discount on face moisturizers and a 15% discount on body lotions, how much money did they spend together in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$684}
Calculate the probability of $X > Y$ using integration, where $X$ and $Y$ are independent and uniformly distributed between $300$ and $700$, and $500$ and $800$, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/6}
Given the matrix $A = (c_1,...,c_n)$ where $c_i$ are the column vectors of $A$ and $c_i \in \mathbb{R}^m$, with $c_1 = (1,...,1)$ and $c_2 = c_3 = ... = c_m = (0,...,0)$, compute the 2-norm of the matrix $A$, denoted as $||A||_2 = \sup_{x \in \mathbb{R}^m} \frac{||Ax||_2}{||x||_2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{m}}
How many ways can 4 managers and 3 employees be arranged around a round table such that no 3 managers sit together?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{288}
Consider a $4 \times 4$ square grid with 25 points. Determine the number of distinct lines passing through at least 3 of these points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Evaluate the limit: $$\lim_{n\to\infty} \frac{e^n}{1+n+\frac{n^{2}}{2!}+\cdots +\frac{n^{n}}{n!}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
3. On the surface of the cube $A B C D E F G H$, a broken line composed of four identical segments is constructed in the faces $A B F E, B C G F, C D H G$ and $G H E F$, which starts from vertex $A$ and ends at vertex $E$. Determine in what ratio this broken line divides the edge $C G$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}(\sqrt{5}-1)}
A regular triangle is constructed on the diameter of a semicircle such that its sides are equal to the diameter. Determine the ratio of the areas of the parts of the triangle lying outside and inside the semicircle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\sqrt{3} - \pi}{3\sqrt{3} + \pi}}
At the Intergalactic Math Olympiad held in the year 9001, there are 6 problems, and on each problem you can earn an integer score from 0 to 7. The contestant's score is the *product* of the scores on the 6 problems, and ties are broken by the sum of the 6 problems. If 2 contestants are still tied after this, their ranks are equal. In this olympiad, there are  $8^6=262144$  participants, and no two get the same score on every problem. Find the score of the participant whose rank was  $7^6 = 117649$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Consider the polynomial \( f(x) = x^4 - 2x^3 - x^2 + 17x + 6 \) with roots \( x_1, x_2, x_3, x_4 \). Let \( g(x) = x^5 - 3x^4 + 22x^2 - 18x - 2 \). Find the value of \( \sum_{k=1}^{4} g(x_k) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
Given an integer  $n\ge 2$ , a function  $f:\mathbb{Z}\rightarrow \{1,2,\ldots,n\}$  is called *good*, if for any integer  $k,1\le k\le n-1$  there exists an integer  $j(k)$  such that for every integer  $m$  we have 
\[f(m+j(k))\equiv f(m+k)-f(m) \pmod{n+1}. \]
Find the number of *good* functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ n \cdot \phi(n) }
A singer earns a profit of $0.75 per concert. After each concert, there is a 50% chance the singer falls into a bad mood. To recover the singer's mood, the producer can send flowers costing $x. The probability of the singer recovering from the bad mood due to flowers is $\sqrt{x}$. If the singer's career lasts for $n$ days and all profits go to the producer, determine the value of $x$ that maximizes the producer's expected profit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
A conical part has a base radius of 5 cm, and its height is three times the radius of the base. What is the volume of this part in cubic centimeters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{125\pi}
Both roots of the quadratic equation $x^2 - 63x + k = 0$ are prime numbers. Find the number of possible values of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let  $ABCD$  be a tetrahedron with  $AB=CD=1300$ ,  $BC=AD=1400$ , and  $CA=BD=1500$ . Let  $O$  and  $I$  be the centers of the circumscribed sphere and inscribed sphere of  $ABCD$ , respectively. Compute the smallest integer greater than the length of  $OI$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the directional derivative of the function \( f(x, y) \) at the point \((0, 0)\) in the direction of the vector \( \vec{v} = \langle \frac{4}{5}, \frac{3}{5} \rangle \), where the function is defined as:

\[ f(x, y) = \begin{cases} \frac{x^3 + y^3}{x^2 - y^2}, & \text{if } x^2 - y^2 \neq 0 \\ 0, & \text{if } x^2 - y^2 = 0 \end{cases} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{5}}
Determine the number of distinct ways to arrange the numbers 1 through 9 in a 3x3 grid such that the sum of the numbers in each row and each column is the same. Note that the grid cannot be rotated or reflected.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
John decides to get gym memberships so he can get in shape. The gym close to his house is close but doesn't have everything he wants, so he gets two different gym memberships. The cheap one costs $10 a month and has a sign-up fee of $50. The second gym is 3 times more expensive and it has a sign-up fee of 4 months membership. John also has to pay an annual maintenance fee of $30 at the cheap gym and $60 at the expensive gym. John refers one friend to each gym, receiving a 10% discount on the sign-up fees.

In addition, John also books personal training sessions at each gym. At the first gym, he pays $25 per session for the first 10 sessions and gets a 20% discount for the 11th to 20th sessions. At the second gym, he pays $45 per session for the first 5 sessions and then gets a 15% discount for the 6th to 15th sessions. John attends a total of 20 personal training sessions at the first gym and 15 sessions at the second gym in the first year.

How much total did he pay in the first year for gym memberships, personal training sessions, and annual maintenance fees?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1780.50}
Petya, Vasya, Kolya, and Tolya counted their catch after fishing. Tolya caught more than Kolya. Petya and Vasya together caught as much fish as Tolya and Kolya did together. Petya and Kolya together caught less fish than Vasya and Kolya. Which fisherman could have caught the most fish?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Vasya}
## Task A-3.7.

We have ten white, and one red, blue, green, yellow, and purple card each. The white cards are indistinguishable from each other. On exactly one side of each card is the symbol $X$. In how many ways can we stack these cards one on top of the other so that no two cards are facing each other with the side that has the $X$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16\cdot15\cdot14\cdot13\cdot12\cdot11}
Simplify the mathematical expression: 

$$\left(\left(\frac{a \sqrt[3]{b}}{b \sqrt{a^{3}}}\right)^{3 / 2}+\left(\frac{\sqrt{a}}{a \sqrt[8]{b^{3}}}\right)^{2}\right):\left(a^{1 / 4}+b^{1 / 4}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{a b}}
Determine the values of \(a \in \mathbb{R}_+\) for which the series \(\sum_{n>0} \frac{a(a+1)(a+2)\cdots(a+n-1)}{n!\cdot n^2}\) is convergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 2)}
Consider an equilateral triangle $T$ of side length 12. Matthew cuts $T$ into $N$ smaller equilateral triangles, each of which has side length 1,3, or 8. Compute the minimum possible value of $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Circle \( X \) has a radius of \( \pi \) . Circle \( Y \) has a circumference
 of \( {8\pi } \) . Circle \( Z \) has an area of \( {9\pi } \) . List the circles in order
 from smallest to largest radius.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
Z, X, Y
}
15 From the five numbers $-1, 0, 1, 2, 3$, select three (without repetition) to form the coefficients of the quadratic function $y=a x^{2}+b x+c$. The number of different parabolas that intersect the negative direction of the $x$-axis at least once is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Determine the minimum number of equations required to ensure that an $n \times n$ matrix has a rank of at most $n-3$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
What is the least positive integer $k$ such that $n+k \in S$ for every $n \in S$, where $S = \{ n : n3^n + (2n+1)5^n \equiv 0 \pmod{7} \}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Determine if there exists a real number \( N \) such that the double sum \[ C_N\sum_{r_1\in \mathbb{Z}}\sum_{x_1\in \mathbb{Z}}\left(1+\frac{1}{2}\left(|x_1+r_1|+|x_1|\right)\right)^{-N} \] is convergent. If so, find the range of \( N \) for which the sum converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, \infty)}
Find all the real numbers $k$ that have the following property: For any non-zero real numbers $a$ and $b$,  it is true that at least one of the following numbers: $$a, b,\\frac{5}{a^2}+\\frac{6}{b^3}$$is less than or equal to $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$k = 2$}
Evaluate the triple integral: \[ \int_0^1 \int_0^{z\pi} \int_{\frac{y}{\pi}}^z \sin\frac{y}{x} \, dx \, dy \, dz \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Petya, Sasha, and Misha are playing an elimination tennis game. An elimination game means that in each match, two players compete while the third waits. The loser of the match gives up their place to the waiting player and becomes the waiting player for the next match. Petya played a total of 12 matches, Sasha played 7 matches, and Misha played 11 matches. How many times did Petya win against Sasha?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Compute the sum of  $x^2+y^2$  over all four ordered pairs  $(x,y)$  of real numbers satisfying  $x=y^2-20$  and  $y=x^2+x-21$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{164}
Evaluate the infinite series \( \sum_{k=1}^{\infty}\frac{18[(k-1)!]^2}{(2k)!} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi^2}
On a river with a current speed of $5 \text{ km/h}$, there are docks $A$, $B$, and $C$ positioned in the direction of the current, with $B$ located halfway between $A$ and $C$. A raft, which moves with the current towards dock $C$, and a motorboat, which heads towards dock $A$ with a speed of $v \text{ km/h}$ in still water, simultaneously depart from dock $B$. After reaching dock $A$, the motorboat turns around and moves towards dock $C$. Find all values of $v$ for which the motorboat arrives at $C$ later than the raft.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5 < v < 15}