inclusionAI/Ring-lite-rl-data huggingface dataset:
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?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Consider two non-zero square matrices \( A \) and \( B \) such that \( A^2B = BA \). If \( (AB)^{10} = A^kB^{10} \), find the value of \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1023}
On New Year's Eve, Santa Claus gave the children the following task: by using all nine digits from 1 to 9 exactly once, insert either "+" or "-" between each pair of adjacent digits so that the result yields all possible two-digit prime numbers. How many such numbers can be obtained?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Ninety-nine children are standing in a circle, each initially holding a ball. Every minute, each child with a ball throws their ball to one of their two neighbors. If two balls end up with the same child, one of these balls is irrevocably lost. What is the minimum time required for the children to have only one ball left?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98}
A random simulation method is used to estimate the probability of an athlete hitting a target in a shooting event. A calculator generates random integers between 0 and 9. The numbers 0, 1, 2, and 3 represent a miss, while 4, 5, 6, 7, 8, and 9 represent a hit. Each group of 4 random numbers corresponds to the results of 4 shots. After random simulation, 20 groups of the following random numbers are generated:
7527, 0293, 7140, 9857, 0347, 4373, 8636, 6947, 1417, 4698,
0371, 6233, 2616, 8045, 6011, 3661, 9597, 7424, 7610, 4281
Estimate the probability that the athlete hits the target at least three times in four shots: $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.4}
Compute the limit for a standard normally distributed random variable \(X \sim \text{Norm}(0,1)\):
\[
\lim_{x\rightarrow\infty}\frac{x}{\overline{F}_{X}(x)}\int_{x}^{\infty}\overline{F}_{X}(s)ds
\]
where \(\overline{F}_{X}(x) = 1 - F_{X}(x)\), and \(F_{X}(x) = \int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}dx\). Use L'Hopital's rule or other appropriate techniques to solve this limit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that $\Lambda \sim \text{Exp}(1)$ and $X$ has a probability mass function $P_\lambda(X=x)=\frac{\lambda^x e^{-\lambda}}{x!}$, where $\lambda$ is a realization of the random variable $\Lambda$, find the conditional expectation $E(e^{-\Lambda}|X=1)$. Assume $f(\lambda)=e^{-\lambda}$ for $\lambda>0$ and $0$ otherwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{9}}
Let there be a point $M(x_0, 1)$. If there exists a point $N$ on the circle $O: x^2 + y^2 = 1$ such that $\angle OMN = 45^\circ$, then the range of values for $x_0$ is ________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, 1]}
A positive integer $n \geq 4$ is called *interesting* if there exists a complex number $z$ such that $|z|=1$ and \[1 + z + z^2 + z^{n-1} + z^n = 0.\] Find how many interesting numbers are smaller than $2022$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{404}
Given a graph $G$ with a minimum degree of $\delta(G)=4$, find the connectivity $\kappa(G)$ of the graph.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A certain number less than three times the number of Doberman puppies plus the difference between the number of Doberman puppies and the number of Schnauzers is equal to 90. The number of Doberman puppies is 20, and there are 55 Schnauzers. What is that certain number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In Anchuria, a checkers championship is held over several rounds, with each round taking place in a different city on a different day. There are eight cities and eight days available for the championship. To maximize the probability that no fan correctly guesses the cities and days for all the rounds, how many rounds should be scheduled in the championship?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
10. (10 points) In the Land of Dwarfs, there are 2011 dwarfs, each of whom wears either a red hat or a blue hat. When a dwarf wears a red hat, they tell the truth; when they wear a blue hat, they can lie; and they can change the color of their hat at any time. One day, every two of them met exactly once, and each said that the other was wearing a blue hat. How many times did they change the color of their hats at least? $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2009}
How many integers $n$ with $10 \leq n \leq 500$ have the property that the hundreds digit of $17n$ and $17n + 17$ are different?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
In a theater performance of King Lear, the locations of Acts II-V are drawn by lot before each act. The auditorium is divided into four sections, and the audience moves to another section with their chairs if their current section is chosen as the next location. Assume that all four sections are large enough to accommodate all chairs if selected, and each section is chosen with equal probability. What is the probability that the audience will have to move twice compared to the probability that they will have to move only once?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
How many roots does the following equation have:

$$
10^{\lg \cos x}=\frac{1}{7} x-1 ?
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Among all triangles formed by selecting any 3 of the 8 vertices of a rectangular prism, how many are acute triangles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
The height of a cone and its slant height are 4 cm and 5 cm, respectively. Find the volume of a hemisphere inscribed in the cone, whose base lies on the base of the cone.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{1152}{125} \\pi"]}
A collection of circles in the upper half-plane, all tangent to the \(x\)-axis, is constructed in layers as follows. Layer \(L_0\) consists of two circles of radii \(70^2\) and \(73^2\) that are externally tangent. For \(k \ge 1\), the circles in \(\bigcup\limits_{j=0}^{k-1}L_j\) are ordered according to their points of tangency with the \(x\)-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer \(L_k\) consists of the \(2^{k-1}\) circles constructed in this way. Let \(S=\bigcup\limits_{j=0}^{6}L_j\), and for every circle \(C\) denote by \(r(C)\) its radius. Find \(\sum\limits_{C\in S} \frac{1}{\sqrt{r(C)}}\). The original answer is in the form of \(\frac{k}{m}\), where the fraction is in its simplest form. Please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{157}
In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 18$ and $CP = 6.$  If $\tan \angle APD = 2,$ find $AB.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
9-8-1. In Midcity, houses stand along one side of a street, each house can have $1,2,3, \ldots, 9$ floors. According to an ancient law of Midcity, if two houses on one side of the street have the same number of floors, then no matter how far apart they are from each other, there must be a house with more floors between them. What is the maximum possible number of houses on one side of the street in Midcity?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{511}
Two lines pass through point \(N\) and are tangent to a certain circle with center \(O\). Point \(A\) is taken on one of these lines, and point \(B\) is taken on the other line such that \(OA = OB\) and \(OA > ON\). It is known that \(NA = a\), \(NB = b\), and \(OA = c\) where \(a \neq b\). Find \(ON\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{c^2 - ab}}
Given a function \( u(x) \) and an operator \( \hat{u} \) defined as \( \hat{u}(f(x))=f(u(x)) \) for any function \( f(x) \) over \( \mathbb{R} \), find \( \hat{u}\left(\dfrac{\partial f(x)}{\partial x}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\partial f}{\partial x}(u(x))}
A shop owner professes to sell his articles at a certain cost price, but he uses false weights with which he cheats by 15% while buying and by 20% while selling. What is his percentage profit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43.75\%}
在一节车厢里，任意$m(m\geq3)$个旅客都有唯一的公共朋友（当甲是乙的朋友时，乙也是甲的朋友.任何人不作为他自己的朋友）.问：这节车厢里有多少人？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m+1}
Consider a triangle with vertices at $(2, 1)$, $(4, 7)$, and $(7, 3)$. This triangle is reflected about the line $y=4$. Find the area of the union of the original and the reflected triangles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Two square napkins with dimensions \(1 \times 1\) and \(2 \times 2\) are placed on a table so that the corner of the larger napkin falls into the center of the smaller napkin. What is the maximum area of the table that the napkins can cover?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.75}
Find the solution interval for the equation $2^{x-2}+x=6$. Express your answer as an interval in the form (a, b).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(3, 4)}
A hiker walking at a constant rate of 7 miles per hour is passed by a cyclist traveling in the same direction along the same path at a certain speed. The cyclist stops to wait for the hiker 5 minutes after passing her, while the hiker continues to walk at her constant rate. The cyclist must wait 15 minutes until the hiker catches up. What is the cyclist's speed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
James buys 3 dirt bikes for $150 each, 4 off-road vehicles for $300 each, 2 all-terrain vehicles (ATVs) for $450 each, 5 mopeds for $200 each, and 3 scooters for $100 each. It costs him $25 to register each dirt bike and off-road vehicle, $30 to register each ATV, $15 to register each moped, and $20 to register each scooter. Additionally, he purchases maintenance packages for each vehicle: $50 for each dirt bike, $75 for each off-road vehicle, $100 for each ATV, and $60 for each moped. How much did he pay for everything?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$5170$}
Given a cube. On each edge of the cube, we write a number, either $1$ or $-1$. For each face of the cube, we multiply the four numbers on the edges of this face, and write the product on this face. Finally, we add all the eighteen numbers that we wrote down on the edges and face of the cube.

What is the smallest possible sum that we can get?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
-12
}
Sally is asked to multiply 2/3 and 1 1/2. Jane is asked to add them. Calculate the difference between Sally's answer and Jane's answer. Express your answer as a simplified fraction or mixed number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 \frac{1}{6}}
8.3. Solve the inequality $\sqrt{x^{2}+3 x-54}-\sqrt{x^{2}+27 x+162}<8 \sqrt{\frac{x-6}{x+9}}$, and find the sum of its integer solutions that belong to the interval $[-25 ; 25]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{290}
8. Given a sequence of non-negative integers $\left\{a_{n}\right\}$ satisfying
$$
a_{1}=2016, a_{n+1} \leqslant \sqrt{a_{n}},
$$

and if the number of terms is no less than 2, then any two terms are distinct. Then, the number of such sequences $\left\{a_{n}\right\}$ is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{948}
设 $\triangle A B C$ 的内角 $\angle A 、 \angle B 、 \angle C$ 的对边分别为 $a 、 b 、 c$, 且 $\angle A-\angle C=\frac{\pi}{2}, a 、 b 、 c$ 成等差数列. 则 $\cos B$ 的值为\_\_\_\_\_\_.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In the first half of the 20th century, the population of a particular country increased by 200 percent, with the birth rate at 25 per 1,000, the death rate at 12 per 1,000, and an annual immigration rate of 15,000 individuals. In the second half of the century, the population increased by 300 percent, the birth rate increased to 30 per 1,000, the death rate decreased to 8 per 1,000, and the annual immigration rate doubled to 30,000 individuals. Calculate the overall percent increase in the country's population for the 20th century as a whole, taking into account these demographic factors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1100\%}
Find all real $a$ such that there exists a function $f: R \to R$ satisfying the equation $f(\sin x )+ a f(\cos x) = \cos 2x$ for all real $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$a \in \mathbb{R} \setminus \{1\}$}
Calculate $\sqrt[3]{\frac{x}{2015+2016}}$, where $x$ is the harmonic mean of the numbers

$a=\frac{2016+2015}{2016^{2}+2016 \cdot 2015+2015^{2}}$ and $b=\frac{2016-2015}{2016^{2}-2016 \cdot 2015+2015^{2}}$.

The harmonic mean of two positive numbers $a$ and $b$ is the number $c$ such that $\frac{1}{c}=\frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2016}}
The incircle of a right-angled triangle $ABC$ ($\angle ABC =90^o$) touches $AB, BC, AC$ in points $C_1, A_1, B_1$, respectively. One of the excircles touches the side $BC$ in point $A_2$. Point $A_0$ is the circumcenter or triangle $A_1A_2B_1$, point $C_0$ is defined similarly. Find angle $A_0BC_0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Consider the sequence of numbers defined recursively by \(t_1=1\) and for \(n>1\) by \(t_n=1+t_{(n/2)}\) when \(n\) is even and by \(t_n=\frac{1}{t_{(n-1)}}\) when \(n\) is odd. Given that \(t_n=\frac{19}{87}\), find the sum of the digits of \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let \( a = 3^{100} \) and \( b = 5454 \). Determine how many integers \( z \) exist in the interval \([1, 3^{99})\) such that for every integer \( c \) with \( \gcd(c, 3) = 1 \), the congruences \( x^z \equiv c \pmod{a} \) and \( x^b \equiv c \pmod{a} \) have the same number of solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3^{95}}
Evaluate the integral \[ I = \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \delta(x_1-x_2+x_3-x_4)\delta(x_2-x_3+x_5-x_6) \, dx_1 \, dx_2 \, dx_3 \, dx_4 \, dx_5 \, dx_6. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
The set \( S \) consists of 9 distinct positive integers. The average of the two smallest integers in \( S \) is 5. The average of the two largest integers in \( S \) is 22. Calculate the greatest possible average of all the integers of \( S \). Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Determine the factor by which a random walk on a 2D lattice, with the rule that it cannot go back on itself, is sped up compared to a standard random walk without this rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the kernel of the matrix \( A = \begin{pmatrix} x_1^0 & x_1^1 & \ldots & x_1^n \\ x_2^0 & x_2^1 & \ldots & x_2^n \\ \vdots & \vdots & \ldots & \vdots \\ x_m^0 & x_m^1 & \ldots & x_m^n \end{pmatrix} \), where \( n < m-1 \) and the \( x_i \) are pairwise distinct.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{\mathbf{0}\}}
Find the value of \( \lambda > 0 \) such that the function \( f(x) = \begin{cases} 0 & x = 0 \\ |x|^{\lambda} \sin\left(\frac{1}{x}\right) & x \neq 0 \end{cases} \) is differentiable at zero and has a continuous derivative at zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Compute the standard part of the expression \(\frac{3-\sqrt{c+2}}{c-7}\) given that the standard part of \(c\) is 7 and \(c \neq 7\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{6}}
What is the name for subgroups $H < G$ that satisfy the condition $C_G(H) \cap H = Z(H)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{self-centralizing}
\(ABCD\) is a square-based pyramid with base \(ABCD\) and apex \(E\). Point \(E\) is directly above point \(A\), with \(AE = 1024\) units and \(AB = 640\) units. The pyramid is sliced into two parts by a horizontal plane parallel to the base \(ABCD\), at a height \(h\) above the base. The portion of the pyramid above the plane forms a new smaller pyramid. For how many integer values of \(h\) does the volume of this new pyramid become an integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85}
Point \(A\) lies on the line \(y = \frac{15}{8}x - 8\), and point \(B\) lies on the parabola \(y = x^2\). What is the minimum length of segment \(AB\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1823/544}
For each integer $n\ge 3$, let $f(n)$ be the number of 3-element subsets of the vertices of a regular $n$-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of $n$ such that $f(n+1)=f(n)+78$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
245
}
Determine the minimum number of terms required to approximate Euler's constant \( \gamma \) using the series definition, such that the error is less than \( 2 \times 10^{-3} \). \( \gamma \) is defined as the limit:
\[ \gamma := \lim_{n \to \infty} \left( H_n - \log n \right) \]
where \( H_n \) is the \( n \)-th harmonic number. Use the inequality:
\[ \sum_{n=k}^{\infty} \frac{1}{n} - \int_k^{\infty} \frac{1}{t} \, dt < 2 \times 10^{-3} \]
to find the smallest integer \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{251}
The positive integers are arranged in rows and columns as shown below.

| Row 1 | 1 |
| Row 2 | 2 | 3 |
| Row 3 | 4 | 5 | 6 |
| Row 4 | 7 | 8 | 9 | 10 |
| Row 5 | 11 | 12 | 13 | 14 | 15 |
| Row 6 | 16 | 17 | 18 | 19 | 20 | 21 |
|... |

More rows continue to list the positive integers in order, with each new row containing one more integer than the previous row. Calculate the number of integers less than 2000 that are in the column that contains the number 2000. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Determine the number of real roots of the equation:

$$ \mathcal{T}(e^x,5) = \frac{x^5}{5!} + \frac{x^4}{4!} + \frac{x^3}{3!} + \frac{x^2}{2!} + x + 1 $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given a sequence $\{a_n\}$ where $a_3 = 2$ and $a_7 = 1$, if the sequence $\left\{ \frac{1}{a_n+1} \right\}$ forms an arithmetic sequence, find the value of $a_{19}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the area of the figure bounded by the lines \( y = 0 \), \( y = \{\sin x\} \), \( x = -\frac{\pi}{2} \), and \( x = -\frac{\pi}{4} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{4} - \frac{\sqrt{2}}{2}}
Let $X_1, X_2, \ldots$ be a sequence of independent and identically distributed random variables. Define $M$ as the smallest index $n$ such that $X_n > X_{n-1}$. Find the expected value $\mathbb{E}[M]$. \( \text{Express your answer as a single number.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
Evaluate the integral \( \int_\gamma \frac{dz}{(z^2+1)^2} \), where \( \gamma \) is the semicircle \([-R,R]\cup\{z\in\mathbb{C}:|z|=R\ \text{and}\ \text{Im}{z}>0\}\) traced in the positive direction, and \( R>1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
An ant situated at point \( A \) decides to walk 1 foot east, then \( \frac{1}{2} \) foot northeast, then \( \frac{1}{4} \) foot east, then \( \frac{1}{8} \) foot northeast, then \( \frac{1}{16} \) foot east, and so on (that is, the ant travels alternately between east and northeast, and the distance travelled is decreased by half every time the ant changes its direction). The ant eventually reaches a certain point \( B \). Determine the distance between the ant's initial and final positions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{2}{3} \\sqrt{2 \\sqrt{2}+5}"]}
Calculate the expected number of letters needed to be output from a random sequence of the letters \( \{A, B, C, D\} \) until the sequence "ABCD" appears consecutively. Assume each letter is equally likely to appear at each position in the sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
Denote by $\left(ABC\right)$ the circumcircle of a triangle $ABC$.
Let $ABC$ be an isosceles right-angled triangle with $AB=AC=1$ and $\measuredangle CAB=90^{\circ}$. Let $D$ be the midpoint of the side $BC$, and let $E$ and $F$ be two points on the side $BC$.
Let $M$ be the point of intersection of the circles $\left(ADE\right)$ and $\left(ABF\right)$ (apart from $A$).
Let $N$ be the point of intersection of the line $AF$ and the circle $\left(ACE\right)$ (apart from $A$).
Let $P$ be the point of intersection of the line $AD$ and the circle $\left(AMN\right)$.
Find the length of $AP$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
 \( \sqrt{2} \)

}
Let $A B C$ be a triangle with $A B=2, C A=3, B C=4$. Let $D$ be the point diametrically opposite $A$ on the circumcircle of $A B C$, and let $E$ lie on line $A D$ such that $D$ is the midpoint of $\overline{A E}$. Line $l$ passes through $E$ perpendicular to $\overline{A E}$, and $F$ and $G$ are the intersections of the extensions of $\overline{A B}$ and $\overline{A C}$ with $l$. Compute $F G$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1024}{45}}
the edge of a cube is 5 a cm . find its surface ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150a^2}
A river has a right-angle bend. Except at the bend, its banks are parallel lines of distance $a$ apart. At the bend the river forms a square with the river flowing in across one side and out across an adjacent side. What is the longest boat of length $c$ and negligible width which can pass through the bend?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 2a\sqrt{2} }