inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $  l  $ be a tangent line at the point $  T\left(t, \frac{t^2}{2}\right) \ (t > 0)  $ on the curve $  C : y = \frac{x^2}{2}  $. A circle $  S  $ touches the curve $  C  $ and the $  x  $ axis. 
Denote by $  (x(t), y(t))  $ the center of the circle $  S  $. Find $ \lim\limits_{r \to +0} \int_r^1 x(t)y(t) \, dt. $
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\[ \frac{\sqrt{2} + 1}{30} \]}
求出最小正整数 $n$, 使其恰有 144 个正约数, 并且其中 10 个是连续整数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110880}
Find the limit of the complex logarithm function in the following case: $$\lim_{y\to 0} [\log(-a+iy)-\log(-a-iy)].$$ where $a>0$ and $y$ approaches 0 from above and below.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the asymptotic behavior of the expected number of flips required for the number of heads to exceed the number of tails in a sequence of fair coin flips.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Vasya has \( n \) candies of several types, where \( n \geq 145 \). It is known that if any group of at least 145 candies is chosen from these \( n \) candies (in particular, one can choose the group of all \( n \) candies), there exists a type of candy such that the chosen group contains exactly 10 candies of that type. Find the largest possible value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
Find the ratio of the number of natural numbers from 1 to 1,000,000 that can be expressed as the sum of a perfect square and a perfect cube to the number of numbers that cannot be expressed in this way.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{9}}
A team of five people is laying a 30-meter pipe into the ground. To prepare the trench, they decided to dig in turns with the condition that each person digs for the amount of time it takes the other four together to dig one-third of the trench. The 30-meter trench was completed when the last team member had worked their allotted time. How many meters of the trench could they have dug in the same time if they all worked together?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80 \text{m}}
Calculate the expectation of the stopped local time $L_t^y$ for a Brownian motion $B_t$ starting at $x$, where $x > y$ and $T_a=\inf\{ s \geq 0 \mid B_s=a\}$ with $a > x$. Express your answer in terms of $x$, $y$, and $a$. \( \boxed{\text{Your answer here}} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(a - x)}
Let \( F(x) \) be a real-valued function defined for all real numbers except \( x=0 \) and \( x=1 \). Given that \( F(x) + F\left(\frac{x-1}{x}\right) = 1 + x \), find \( F(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1+x^2-x^3}{2 x(1-x)}}
Determine the condition on the complex-valued matrix $V$ such that the determinant \( \det(I+[aA,bB]VV^*[aA,bB]^*) \) is increasing in both real positive parameters \(a\) and \(b\), where \(A\) and \(B\) are complex-valued matrices, and \(^*\) denotes the conjugate transpose.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{V_1 V_2^* = 0}
Determine the values of $p$ and $q$ for which the series \( \sum_{n=1}^\infty \frac{x}{n^p+n^qx^2} \) converges uniformly on \( \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p + q > 2}
If the area of a circle decreases by 42%, then the radius of the circle decreases by a certain percentage. What is the percentage decrease in the radius of the circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23.84\%}
The altitudes of an acute-angled triangle \(ABC\) from points \(B\) and \(C\) are extended to intersect the circumcircle at points \(B_1\) and \(C_1\). It turns out that the segment \(B_1C_1\) passes through the center of the circumcircle. Find the angle \(BAC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
Given $f(x)=2x+1$, $g(x)=x|x-2|$, if for any $x_{1}$, $x_{2}\in [0,t]$, when $x_{1}\neq x_{2}$, $\frac{{g(x_{1})-g(x_{2})}}{{f(x_{1})-f(x_{2})}}<2$ holds, then the maximum value of $t$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In an isosceles triangle with a base of \(a\) and an angle \(\alpha\) at the base, an inscribed circle is drawn. Find the radius of a circle that is tangent to both the inscribed circle and the legs of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a}{2} \tan^3 \frac{\alpha}{2}}
Given that \( T = \sin 50^{\circ} \times \left(S + \sqrt{3} \times \tan 10^{\circ}\right) \), find the value of \( T \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A $10 \times 10$ square is divided into unit squares. How many triangles are formed after drawing one diagonal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110}
The route from $A$ to $B$ is traveled by a passenger train 3 hours and 12 minutes faster than a freight train. In the time it takes the freight train to travel from $A$ to $B$, the passenger train travels 288 km more. If the speed of each train is increased by $10 \mathrm{km} / h$, the passenger train would travel from $A$ to $B$ 2 hours and 24 minutes faster than the freight train. Determine the distance from $A$ to $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360 \text{ km}}
Patty has $20$ coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have $70$ cents more. How much are her coins worth?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\textdollar 1.15$}
Square $PQRS$ lies in the first quadrant. Points $(3,0)$, $(5,0)$, $(7,0)$, and $(13,0)$ lie on lines $SP$, $RQ$, $PQ$, and $SR$, respectively. If the sum of the coordinates of the center of the square $PQRS$ is $x$, what is the value of $10x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
2. At what moment between 13:00 and 14:00 do the hour and minute hands of a clock form a straight angle $\left(180^{\circ}\right)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13:38\frac{2}{11}}
Find the coefficient of $x^{10}$ in the expansion of $(x+2)^{10}(x^2-1)$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{179}
Determine the order type of the set \( \omega \times n \) with the lexicographic order, considering the first position first.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\omega}
There are 3 stops between city A and city B for a train. How many types of tickets need to be prepared for this train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Find the minimum value of the function $$f(a,b)= 1 + a^3 + b^3 - (a + a^2 b + b^2)$$ where $a,b$ are real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A quadrilateral inscribed in a circle has side lengths $a, b, c,$ and $d$. Calculate the ratio of the lengths of the diagonals of this quadrilateral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{b c + a d}{a b + c d}}
Using the central limit theorem, determine the minimum number of 100W bulbs needed to ensure that an advertising board is lit for at least 20,000 hours with a probability of at least 0.9. Assume that the life of each bulb follows an exponential distribution with an average life of 600 hours.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Find \( f'(8.23) \) where \( f(x) = 23|x| - 37\lfloor x \rfloor + 58\{x\} + 88\arccos(\sin x) - 40\max(x,0) \). Note: For a real number \( x \), \( \{x\} = x - \lfloor x \rfloor \) denotes the fractional part of \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{129}
A large spherical model of Earth is to be cut with planar cuts. If each cut intersects the sphere, what is the maximum number of pieces the sphere can be divided into after 6 cuts?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Evaluate the limit as a definite integral: $$\lim_{N\to \infty}\frac{1}{N}\sum_{n=1}^{N}\frac{e^{2\pi i n/N}}{1-e^{2\pi i n/N}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Determine the cardinality of the set \( S = \{ (x, y, z) \mid x^2 + y^2 + z^2 = 2^{2018}, xyz \in \mathbb{Z} \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
On the board, two sums are written:

\[1+22+333+4444+55555+666666+7777777+88888888+999999999\]

\[9+98+987+9876+98765+987654+9876543+98765432+987654321\]

Determine which one is greater (or if they are equal).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["1097393685"]}
Let $X \sim \mathrm{Unif }[0,1]$ and $Y\sim \mathrm{Gamma}(2,r)$, with $r>0$, be two independent random variables. Find the covariance $\mathrm{Cov}(U,V)$ where $U = X$ and $V = XY$. Use the fact that $V \sim \textit{Exp}(r)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6r}}
Compute the fundamental group of the quotient space $G/G_x$, where $G$ is a compact, connected, and simply connected Lie group, $\mathfrak{g}$ is its Lie algebra, and $x \in \mathfrak{g}^*$. Here, $G_x$ denotes the isotropy group of $G$. What is $\pi_1(G/G_x)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Through point \( A \) located on a circle, a diameter \( AB \) and a chord \( AC \) are drawn, where \( AC = 8 \) and \( \angle BAC = 30^\circ \).

Find the length of the chord \( CM \), which is perpendicular to \( AB \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the accumulation points of the sequence \( \{x_n\} \) where \( x_n = n\sin(n) \) for \( n \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Evaluate the double integral \( \iint_R x \, dx \, dy \) using the change of variables \( x = \sqrt{u+v} \) and \( y = v \), where the region \( R \) is in the first quadrant of the \( xy \)-plane and is bounded by \( y = 0 \), \( y = 16 \), \( y = x^2 \), and \( y = x^2 - 9 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Let $S$ be the set of all rational numbers $r$, $0<r<1$, that have a repeating decimal expansion in the form\[0.abcabcabc\ldots=0.\overline{abc},\]where the digits $a$, $b$, and $c$ are not necessarily distinct. To write the elements of $S$ as fractions in lowest terms, how many different numerators are required?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
660
}
Determine the values of the parameter $x$ for which the infinite series \( \sum_{n=1}^\infty \frac{x^n}{\ln(n+1)} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, 1)}
Evaluate the expression \( \frac{\partial(\frac{d(x(t))}{dt})}{\partial x(t)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the Gaussian curvature of the surface defined by the set \( S = \{(x,y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1 \} \) with the first fundamental form given by \( ds^2 = \frac{4}{(1-x^2-y^2)^2}(dx^2 + dy^2) \). Use the formula for Gaussian curvature \( K = \frac{LN - M^2}{EG - F^2} \), where \( E = G = \frac{4}{(1-x^2-y^2)^2} \) and \( F = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Let $f(x)$ be an irreducible quartic polynomial over $\mathbb{Q}$. Let $L$ be the splitting field of $f(x)$. Suppose $\mathbb{Q}(\alpha) \cap \mathbb{Q}(\beta)=\mathbb{Q}$ for any two distinct roots $\alpha$ and $\beta$ of $f(x)$. Determine the Galois group of $f(x)$ over $\mathbb{Q}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_4}
Compute the limit: \[ \lim_{n \to \infty} \sqrt[n]{ \int_{0}^{1} (1+x^n)^n \, dx} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral \( \int^{4}_{0}\int_{0}^{\sqrt{4x-x^2}}\sqrt{x^2+y^2}\, \text{dy dx} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{128}{9}}
Evaluate the triple integral \( \iiint\limits_G (x^2 - z^2) \, dx \, dy \, dz \), where the region \( G \) is bounded by the planes \( y = -x \), \( z = x \), \( z = y \), and \( z = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{4}{15}}
Evaluate the integral \( \iint_R y\, dx\, dy \) where \( R \) is the region bounded by the \( x \)-axis and the parabolas \( y^2 = 4-4x \) and \( y^2=4+4x \). Use the transformation \( x = u^2 - v^2 \) and \( y = 2uv \) to perform the evaluation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the upper bound for the integral \( \left\lvert\int_\gamma \frac{\sin(z)}{z} \,dz\right\rvert \) where \(\gamma\) is a path that traces the circle of radius 2 centered at the origin once.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the value of the infinite sum \( \sum_{j=0}^\infty \frac{\binom{2j}{j}}{2^{2j}(1-4j^2)} \) without using Riemann sums or integrals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Calculate the value of the infinite series \( \sum_{k=1}^{\infty} \frac{(2k-1)(-3)^{k-2}}{2^{k-1}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{75}}
Calculate the volume of the region defined by the inequality \((4x+4y+2z)^2+(4x+2y+4z)^2+(2x+4y+4z)^2 \leq 1\) using the change of variables \(u=4x+4y+2z\), \(v=4x+2y+4z\), \(w=2x+4y+4z\). The Jacobian determinant of the transformation is 40. Express the volume as an integral in the new variables \(u, v, w\) and find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{30}}
Determine the smallest positive integer \( m \) for which \( m^3 - 90 \) is divisible by \( m + 9 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Twenty ants live on the faces of an icosahedron, one ant on each side, where the icosahedron have each side with length 1. Each ant moves in a counterclockwise direction on each face, along the side/edges. The speed of each ant must be no less than 1 always. Also, if two ants meet, they should meet at the vertex of the icosahedron. If five ants meet at the same time at a vertex, we call that a collision. Can the ants move forever, in a way that no collision occurs?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
no
}
Determine the matrix representation of the isomorphism \( \varphi: \text{Hom}(V,W) \to M_{m \times n} \), where \( B = (v_1, \ldots, v_n) \) and \( C = (w_1, \ldots, w_m) \) are ordered bases for the finite-dimensional vector spaces \( V \) and \( W \), respectively. Here, \( [T]_B^C \) denotes the coordinate matrix of a linear transformation \( T: V \to W \) with respect to bases \( B \) and \( C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{I_{mn}}
Find the limit of \( \frac{{{k-1}\choose n}}{{{n+k-1}\choose n}} \) as \( n, k \to \infty \) such that \( \frac{k}{n^2} \to a > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-1/a}}
14. The set $S$ is given by $S=\{1,2,3,4,5,6\}$. A non-empty subset $T$ of $S$ has the property that it contains no pair of integers that share a common factor other than 1 . How many distinct possibilities are there for $T$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Let $F \subseteq E$ be a field extension. Let $\alpha \in E$ be a root of an irreducible polynomial $f(x)$ over $F$ with multiplicity 3. If $\beta$ is any other root of $f(x)$ in $E$, what is the multiplicity of $\beta$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Right triangles \(T_1\) and \(T_2\), have areas of \(1\) and \(2\), respectively. A side of \(T_1\) is congruent to a side of \(T_2\), and a different side of \(T_1\) is congruent to a different side of \(T_2\). Find the square of the product of the lengths of the other (third) sides of \(T_1\) and \(T_2\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Mona plays a multiplayer video game on the weekends. She joins groups with four other players to complete objectives in the game. The past weekend, she joined some groups and grouped with 33 unique players. One of the groups included two players she had grouped with before that weekend, and another group included one person she had grouped with before. How many groups did Mona join that weekend?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate the integral \( \int_0^1 \left( \left\lfloor \frac{2}{x} \right\rfloor - 2 \left\lfloor \frac{1}{x} \right\rfloor \right) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \ln 2 - 1}
11. (16 points) A and B are playing a game on a $20 \times 15$ chessboard. At the beginning, a queen is placed on a square of the chessboard except the top-right corner; starting with A, the two players take turns to move the queen. Each move can be a straight or diagonal move of several squares, but only to the right, up, or up-right; the player who moves the queen to the top-right corner square wins. How many starting squares on this chessboard allow A to have a winning strategy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{287}
Consider the quotient ring \(S=\mathbb{R}[x_1,x_2,\dots,x_n]/(x_1^2-x_1,x_2^2-x_2,\dots,x_n^2-x_n)\). Determine the least degree of a polynomial \(p\in S\) that satisfies the idempotent property \(p^2=tp\) modulo the ideals \((x_i)\) and \((x_i-1)\) for all \(i=1,2,\dots,n\). Assume that finding the result for \(t=1\) is sufficient, as \(p^2=tp\Rightarrow \left(\frac{p}{t}\right)^2=\frac{p}{t}\Rightarrow q^2=q\text{ for some }q\in S.\) What is the least degree that such a polynomial \(p\) can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
If \( i^{2} = -1 \), then find the sum \( S=\cos 45^{\circ}+i \cos 135^{\circ}+\cdots+i^{n} \cos (45+90n)^{\circ}+\cdots+i^{40} \cos 3645^{\circ} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\sqrt{2}}{2}(21-20 i)\)}
Let $\alpha, \beta, \gamma \in \mathbf{R}$ satisfy the equation $\sin \alpha \cdot \cos \beta + |\cos \alpha \cdot \sin \beta| = \sin \alpha \cdot |\cos \alpha| + |\sin \beta| \cdot \cos \beta$. Then, find the minimum value of $(\tan \gamma - \sin \alpha)^2 + (\cot \gamma - \cos \beta)^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 - 2\sqrt{2}}
Solve the equation \( \ln(x) = \sin(x) \) for \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.219}
Given the Stooge sort algorithm, which runs in \( O(n^\alpha) \) where \( \alpha \) is minimal, determine the value of \( (243/32)^\alpha \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{243}
10.6  Let the insphere of a pyramid  $SABC$  touch the faces  $SAB, SBC, SCA$  at  $D, E, F$  respectively. Find all the possible values of the sum of the angles  $SDA, SEB, SFC$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360^\circ}
Evaluate the sum \( S_n = \sum_{k=n}^{2n-1} \binom{2n-1}{k} x^k (1-x)^{2n-1-k} \) and express it in a closed form involving special functions, where \(|x| \leq 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{I_x(n, n)}
If \( \alpha, \beta, \gamma \) are the roots of the polynomial \( x^3 + x^2 - x + 1 = 0 \), find the value of \( \prod \left( \frac{1}{\alpha^3} + \frac{1}{\beta^3} - \frac{1}{\gamma^3} \right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-7}
Determine all values of $a$ at which the function \( h(x) \) is differentiable, where \( h(x) = x^{3}-x^{2} \) if \( x \in \mathbb{Q} \) and \( h(x) = 2x-2x^{2} \) otherwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
A partially silvered mirror covers the square area with vertices at \((\pm 1, \pm 1)\). The fraction of incident light it reflects at \((x,y)\) is given by \(\frac{(x-y)^2}{4}\). Assuming a uniform intensity of incident light, calculate the total fraction of light reflected by the mirror.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
There are twice as many cows in Devonshire as there are hearts on a standard deck of 52 playing cards. If there are 4 hearts on a card, the total cost of the cows when they are sold at a certain price each is $83200. What is the price of each cow?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$800}
A burger at Ricky C's weighs 120 grams, of which 30 grams are filler.  What percent of the burger is not filler?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75\%}
Determine the smallest integer $A$ such that there exist consecutive integers, including $A$, which sum to 2014. An integer $A$ with this property is called "yummy."
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2013}
What is the largest value of $n$ less than 50,000 for which the expression $3(n-3)^2 - 4n + 28$ is a multiple of 7?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49999}
For a positive integer  $n$ , define  $n?=1^n\cdot2^{n-1}\cdot3^{n-2}\cdots\left(n-1\right)^2\cdot n^1$ . Find the positive integer  $k$  for which  $7?9?=5?k?$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
8 friends visited a book stall and brought 8 books. Everyone decided to contribute equally to the total bill of $600. If one of the friends had a coupon for 10% off the total bill and another friend had a coupon for 5% off the remaining amount, and if each friend still contributed equally after both coupons were applied to the bill, how much did each friend pay?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$64.13}
On an \( 8 \times  8 \) chessboard, 6 black rooks and k white rooks are placed on different cells so that each rook only attacks rooks of
 the opposite color. Compute the maximum possible value of \( k \) .
 (Two rooks attack each other if they are in the same row or column and no rooks are between them.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
In a complete graph $K_{12}$, how many distinct paths of length exactly 7 exist between vertices A and B, assuming no repeated vertices along the path?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{151200}