inclusionAI/Ring-lite-rl-data huggingface dataset:
For each set of five integers $S= \{a_1, a_2, a_3, a_4, a_5\}$, let $P_S$ be the product of all differences between two of the elements, namely:

$$P_S = (a_5-a_1)(a_4-a_1)(a_3-a_1)(a_2-a_1)(a_5-a_2)(a_4-a_2)(a_3-a_2)(a_5-a_3)(a_4-a_3)(a_5-a_4)$$

Determine the greatest integer $n$ such that given any set $S$ of five integers, $n$ divides $P_S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{288}
已知圆锥面 $x^{2}+y^{2}=\frac{z^{2}}{3}$, 记沿该圆锥面从 $P(-\sqrt{3}, 3,6)$ 到 $Q(\sqrt{3}, 0,3)$ 的曲线长度的最小为值为 I, 则 $[10 I]=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
To assess the shooting level of a university shooting club, an analysis group used stratified sampling to select the shooting scores of $6$ senior members and $2$ new members for analysis. After calculation, the sample mean of the shooting scores of the $6$ senior members is $8$ (unit: rings), with a variance of $\frac{5}{3}$ (unit: rings$^{2}$). The shooting scores of the $2$ new members are $3$ rings and $5$ rings, respectively. What is the variance of the shooting scores of these $8$ members?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{2}}
The store sells 20 items, each priced uniquely in whole numbers from 1 to 20 rubles. The store has a promotion: when purchasing any 5 items, one of them is given for free, and the customer can choose which one. Vlad wants to buy all 20 items in this store, paying as little as possible. How many rubles does he need? (Each of the 20 items is sold in a single unit.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{136}
Flora the frog starts at $0$ on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance $m$ with probability $\frac{1}{2^m}$. Find the probability that Flora will eventually land at $10$. 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{3}
Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. If the expected number of balls that occupy their original positions after these two successive transpositions is \( x \), find the value of \( 10x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
The Fibonacci sequence $F_{1}, F_{2}, F_{3}, \ldots$ is defined by $F_{1}=F_{2}=1$ and $F_{n+2}=F_{n+1}+F_{n}$. Find the least positive integer $t$ such that for all $n>0, F_{n}=F_{n+t}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Ana has \(22\) coins. She can take from her friends either \(6\) coins or \(18\) coins, or she can give \(12\) coins to her friends. She can perform these operations as many times as she wants. Find the least number of coins Ana can have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In how many ways can the nine digits from 1 to 9 be placed in a 3 × 3 grid so that the sums of the rows and the sums of the columns are all equal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
In triangle \(ABC\) with \(BC = 4\) and \(AB = 2\sqrt{19}\), it is known that the center of the circle passing through the midpoints of the sides of the triangle lies on the bisector of angle \(C\). Find \(AC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
若方程 $|x-1|=k(x-2 a)+a$ 对任意实数 $k$ 都有解, 则实数 $a$ 的取值范围是
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{3} \leqslant a \leqslant 1$}
Given that for all $x\_1 \in (0, 2]$, there exists an $x\_2 \in [1, 2]$ such that the inequality $4x\_1 \ln x\_1 - x\_1^2 + 3 + 4x\_1 x\_2^2 + 8a x\_1 x\_2 - 16x\_1 \geq 0$ holds true, find the range of values for the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-\frac{1}{8}, +\infty)}
Using the vertices of a cube as vertices, how many triangular pyramids can you form?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{58}
Determine the remainder when
\[ \sum_{i=0}^{2015} \left\lfloor \frac{2^i}{25} \right\rfloor \]
is divided by 100, where \( \lfloor x \rfloor \) denotes the largest integer not greater than \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
If \( x_{i}=\frac{i}{101} \), then calculate the value of \( S=\sum_{i=0}^{101} \frac{x_{i}^{3}}{3 x_{i}^{2}-3 x_{i}+1} \). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Four numbers w, x, y, z satisfy w < x < y < z. Each of the six possible pairs of distinct numbers has a different sum. The four smallest sums are 1, 2, 3, and 4. Find the sum of all possible values of z. Express your answer as a single value, which can be a fraction or integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{15}{2}}
What is the unit digit of the expression 
$$
\left(\frac{\sqrt{(a-2)(|a-1|)}+\sqrt{(a-2)(1-|a|)}}{1+\frac{1}{1-a}}+\frac{5a+1}{1-a}\right)^{1988},
$$
where a is a real number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given the plane point set \( A = \{ (x, y) \mid x = 2 \sin \alpha + 2 \sin \beta, y = 2 \cos \alpha + 2 \cos \beta \} \) and \( B = \{ (x, y) \mid \sin (x + y) \cos (x + y) \geq 0 \} \), find the area of the region represented by \( A \cap B \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\pi}
Find the value of the limit \( \lim_{a\searrow 0}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^a} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
14.5. 14 ** A storybook contains 30 stories, with lengths of $1, 2, \cdots, 30$ pages respectively. Starting from the first page of the book, stories are published, and each subsequent story begins on a new page. Question: What is the maximum number of stories that can start on an odd-numbered page?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Let \( X_1, X_2, \ldots, X_{50} \) be a random sample of size 50 from a distribution with probability density function \( f(x) = \frac{1}{x^2} \) for \( 1 < x < \infty \). Approximate the probability that at least 20 of these random variables have values greater than 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.1977}
Find the least positive integer $N$ such that the only values of $n$ for which $1 + N \cdot 2^n$ is prime are multiples of $12$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{556}
Determine the minimum number of terminals, \( f(n) \), required in a 3-connector network to ensure that it contains a windmill with \( n \) blades.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n + 1}
求满足如下条件的最小正整数 \(n\)：在 \(\odot O\) 的圆周上任取 \(n\) 个点 \(A_1, A_2, \cdots, A_n\)，则在 \(C_n^2\) 个角 \(\angle A_iOA_j\) \((1 \leq i < j \leq n)\) 中，至少有 2007 个不超过 \(120^\circ\)。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
Assign weights to the edges of a complete graph on $n$ vertices such that the sum of the weights is 1 and the weight of the minimum spanning tree is maximized. What is the weight of the minimum spanning tree after this assignment?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{n}}
Determine the sign of the expression \( S_n + T_n - \frac{1}{2} \), where \( S_n = \frac{1}{n}\sum_{k=1}^n \left(\frac{k}{n}\right)^3 \) and \( T_n = \frac{1}{n}\sum_{k=1}^n \left(\frac{k-1}{n}\right)^3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{+}
Determine the dimension of the real vector space $S_N$ consisting of $3 \times 3$ matrices $M$ such that $NM = 0$, where $N = \begin{bmatrix} 1 & 1 & -1 \\ 0 & 1 & 0 \\ 1 & 0 & -1 \end{bmatrix}$. \( \text{Provide your answer as a single integer.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In a restaurant, the profit is some percentage of the cost. If the cost increases by 12% but the selling price remains constant, the profit is approximately 53.333333333333336% of the selling price. What is the original percentage of the profit with respect to the cost?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{140\%}
Into how many regions do the x-axis and the graphs of \( y = 2 - x^2 \) and \( y = x^2 - 1 \) split the plane?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Given the ellipse: $\dfrac{x^2}{9} + y^2 = 1$, a line passing through the left focus $F$ with an inclination angle of $\dfrac{\pi}{6}$ intersects the ellipse at points $A$ and $B$. The length of chord $AB$ is equal to ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The volume of the tetrahedron \( ZABC \) is 5. A plane passes through the midpoint \( K \) of edge \( ZA \) and the midpoint \( P \) of edge \( BC \), and intersects edge \( ZC \) at point \( M \) such that \( ZM : MC = 2 : 3 \). If the distance from vertex \( A \) to the plane is 1, find the area of the cross-section formed by the plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Two students, A and B, are playing a chess match using a "best of five" format (the first player to win three games wins the match). Based on their past performance, the probability of A winning each game is $p$ $(0 < p < 1)$, and the outcomes of each game are independent. If the probability of A winning with a score of $3:1$ is not higher than the probability of A winning with a score of $3:2$, then the possible range of values for $p$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0,\frac{1}{2}]}
设正三棱柱 $S-A B C$ 的底面边长为 3，侧棱长为 2。求侧棱 $S A$ 与底面 $A B C$ 所成的角的大小。如果答案是 $x^\circ$，那么求 $x$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
In triangle $ABC$, $AB=AC$ and $D$ is a point on $\overline{AC}$ so that $\overline{BD}$ bisects angle $ABC$. If $BD=BC$, what is the measure, in degrees, of angle $A$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Determine the twenty-fifth number in a sequence obtained by counting in base 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100_5}
The numbers \(1, 2, 3, \ldots, 10\) are written in some order around a circle. Peter computed the sums of all 10 triples of neighboring numbers and wrote the smallest of these sums on the board. What is the maximum possible number that could have been written on the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A display screen has a row of 4 adjacent holes, each of which can display 0 or 1. If each time it displays two of the holes, but two adjacent holes cannot be displayed at the same time, the total number of signals that the display screen can show is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In the quadrilateral \(ABCD\), it is known that \(DO = 4\), \(BC = 5\), \(\angle ABD = 45^\circ\), where \(O\) is the point of intersection of the diagonals. Find \(BO\) if the area of the quadrilateral \(ABCD\) is \(\frac{1}{2}(AB \cdot CD + BC \cdot AD)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
给定正整数 $n 、 k$, 其中, $n$ 为偶数, $k \geqslant 2$,且 $n>4 k$. 在圆周上有 $n$ 个点, 若 $\frac{n}{2}$ 条在圆内互不相交的弦的端点恰为这 $n$ 个点，则称这些弦为一个"匹配".求最大的整数 $m$ ，满足对任意一个匹配，均存在圆周上连续 $k$ 个点，使得这个匹配中至少有 $m$ 条弦的所有端点均在这 $k$ 个点中. 请给出当 $k = 5$ 时，最大整数 $m$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
3.20. In the planes of the faces of the tetrahedron \(ABCD\), points \(A_1, B_1, C_1\), and \(D_1\) are taken such that the lines \(AA_1, BB_1, CC_1\), and \(DD_1\) are parallel. Find the ratio of the volumes of the tetrahedra \(ABCD\) and \(A_1B_1C_1D_1\).

## § 4. Calculation of Volume
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:3}
What is the measure of $\angle X Z Y$ if $M$ is the midpoint of $Y Z$, $\angle X M Z=30^{\circ}$, and $\angle X Y Z=15^{\circ}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75^{\circ}}
In the geometric sequence {a_n}, it is known that (a_2 = 1/3) and (a_8 = 27). Determine the value of a_5. Express your answer as a numerical value, including the ± symbol if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±3}
The number 5.6 may be expressed uniquely (ignoring order) as a product $\underline{a} \cdot \underline{b} \times \underline{c} . \underline{d}$ for digits $a, b, c, d$ all nonzero. Compute $\underline{a} \cdot \underline{b}+\underline{c} . \underline{d}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.1}
Arnold paid $5 for 1 dozen apples and 1 bunch of bananas. Tony paid a certain amount for twice the number of apples Arnold bought and 1 bunch of bananas. How much does a bunch of bananas cost?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$5}
A right triangular prism has exactly one inscribed sphere (the sphere is tangent to the two base faces and three lateral faces of the prism) and one circumscribed sphere (the sphere passes through the 6 vertices of the prism). The ratio of the surface area of the inscribed sphere to that of the circumscribed sphere is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:5}
Point \( K \) lies on side \( BC \) of parallelogram \( ABCD \), and point \( M \) lies on its side \( AD \). Segments \( CM \) and \( DK \) intersect at point \( L \), and segments \( AK \) and \( BM \) intersect at point \( N \). Find the maximum value of the ratio of the areas of quadrilateral \( KLMN \) to \( ABCD \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
In triangle $AHI$, which is equilateral, lines $\overline{BC}$, $\overline{DE}$, and $\overline{FG}$ are all parallel to $\overline{HI}$. The lengths satisfy $AB = BD = DF = FH$. However, the point $F$ on line segment $AH$ is such that $AF$ is half of $AH$. Determine the ratio of the area of trapezoid $FGIH$ to the area of triangle $AHI$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
If $\sin^{4}{\alpha} + \cos^{4}{\alpha} = 1$, then the value of $\sin{\alpha} + \cos{\alpha}$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±1}
A *T-tetromino* is formed by adjoining three unit squares to form a  $1 \times 3$  rectangle, and adjoining on top of the middle square a fourth unit square.
Determine the least number of unit squares that must be removed from a  $202 \times 202$  grid so that it can be tiled using T-tetrominoes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In a year that has  $365$  days, what is the maximum number of "Tuesday the  $13$ th" there can be?

Note: The months of April, June, September and November have  $30$  days each, February has  $28$  and all others have  $31$  days.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A company has 200 employees randomly assigned numbers from 1 to 200, and they are evenly divided into 40 groups. If employee number 22 is selected from the 5th group, find the number of the employee to be selected from the 10th group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
Analyzing the intersection of $y = x^3 - 6x + 2$ and $y = m$ where $-10 < m < 10$, define $L(m)$ as the smallest $x$ coordinate of their intersection points. Calculate the function $r = \frac{L(-m) - L(m)}{m}$ as $m$ approaches zero. Find the limit of this function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Determine the set of real numbers  $\alpha$  that can be expressed in the form \[\alpha=\sum_{n=0}^{\infty}\frac{x_{n+1}}{x_n^3}\]
where  $x_0,x_1,x_2,\dots$   is an increasing sequence of real numbers with  $x_0=1$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha \ge \frac{3\sqrt{3}}{2}}
At most how many rooks can be placed on an $8 \times 8$ chessboard so that each one is attacked by at most one other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Taequan is playing a dice game where he rolls two, fair, 6-sided dice twice. If he gets the same roll on his second try as his first, he wins. He rolled an 8 on his first try. What is the probability that he will win with his second roll?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5/36}
11. In the Cartesian coordinate system, points $A, B, C$ lie on the hyperbola $x y=1$, satisfying that $\triangle A B C$ is an isosceles right triangle. Find the minimum value of the area of $\triangle A B C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{3}}
In the regular hexagon $ABCDEF$, a point $X$ is marked on the diagonal $AD$ such that $\angle AEX = 65^\circ$. What is the degree measure of the angle $\angle XCD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Given non-zero vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ satisfy $\mathbf{a} + \mathbf{b} + \mathbf{c} = 0$, the angle between vector $\mathbf{a}$ and $\mathbf{b}$ is $60^\circ$, and $|\mathbf{a}| = |\mathbf{b}| = 1$, then the angle between vector $\mathbf{a}$ and $\mathbf{c}$ is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150^\circ}
在 1,2,\cdots,10 中随机选出三个不同的数，它们两两互素的概率为__________.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{27}
Inside triangle \(ABC\), a point \(O\) is chosen such that \(\angle ABO = \angle CAO\), \(\angle BAO = \angle BCO\), and \(\angle BOC = 90^{\circ}\). Find the ratio \(AC : OC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
A wooden cube has edges of length $3$ meters. Square holes, of side one meter, centered in each face are cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. The entire surface area including the inside, in square meters, is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Determine the radius of convergence for the following power series: C) $\sum_{n=1}^\infty \frac{z^{2^n}}{n!}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Tanya wrote numbers in the form $n^7 - 1$ for $n = 2, 3, \ldots$ and noticed that for $n = 8$, she obtained a number divisible by $337$. For what minimal $n$ did she get a number divisible by $2022$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{79}
There are 400 tons of cargo stored in a warehouse, each with a weight that is a multiple of a quintal and does not exceed 10 tons. It is known that any two pieces of cargo have different weights. What is the minimum number of trips a 10-ton truck must make to guarantee the transportation of all this cargo from the warehouse?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Given points A(3,2,0), B(0,4,0), and C(3,0,2), calculate a normal vector of the plane ABC in the format (x, y, z).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 3, 3)}
A square grid of size \(n \times n\) is initially colored in a black and white checkerboard pattern, with at least one of the corner cells being black. In one move, it is allowed to recolor all four cells in a \(2 \times 2\) square according to the following rule: each white cell is recolored black, each black cell is recolored green, and each green cell is recolored white. For which \(n\) is it possible, after a number of moves, to achieve a checkerboard coloring where the black and white colors have swapped places?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \text{ is a multiple of 3}}
The bisectors of the angles  $A$  and  $B$  of the triangle  $ABC$  intersect the sides  $BC$  and  $AC$  at points  $D$  and  $E$ . It is known that  $AE + BD = AB$ . Find the angle  $\angle C$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
If a square is divided into acute-angled triangles, what is the minimum number of parts that can be created?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A passenger train departs from point $A$ towards point $B$ at a speed of $60 \mathrm{~km/h}$. A high-speed train, which departs later, must catch up with the passenger train at point $B$ and travels at a speed of $120 \mathrm{~km/h}$. After completing $\frac{2}{3}$ of its journey, the passenger train continues at half its original speed. The high-speed train catches up with the passenger train 80 kilometers before point $B$. How far is point $B$ from point $A$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360 \text{ km}}
Consider triangle \(ABC\) with angles \(\angle BAC = 24^\circ\) and \(\angle ACB = 28^\circ\). Point \(D\) is constructed such that \(AB \parallel CD\), \(AD = BC\), and \(AD\) and \(BC\) are not parallel. Similarly, point \(E\) is constructed such that \(AE \parallel BC\), \(AB = CE\), and \(AB\) and \(CE\) are not parallel. Lines \(DE\) and \(AC\) intersect at point \(P\). Determine angle \(\angle CPE\) (in degrees).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4^\circ}
A row of ants comprised of some ants. A man walked upon it and all except 42 were crushed beneath his foot. There are 42 ants left alive. How many ants were there initially in the row?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
A person (referred to as "A") and four colleagues each have a private car with license plates ending in 0, 0, 2, 1, and 5, respectively. To comply with the local traffic restrictions from the 5th to the 9th of a certain month (cars with an odd-numbered last digit of the license plate are allowed on odd days, and those with an even number on even days), the five people discussed carpooling. Each day they can choose any car that meets the requirements, but car "A" can be used at most 1 day. Calculate the total number of different car usage plans.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
If \(\frac{(a-b)(b-c)(c-a)}{(a+b)(b+c)(c+a)}=\frac{2004}{2005}\), find \(\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4011}{4010}}
The probability that Class A will be assigned exactly 2 of the 8 awards, with each of the 4 classes (A, B, C, and D) receiving at least 1 award is $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{7}}