inclusionAI/Ring-lite-rl-data huggingface dataset:
In an equilateral triangle $\triangle{ABC}$, points $E$, $F$, and $G$ are located inside the triangle such that they form four smaller triangles of equal area. Determine the ratio $\frac{AF}{FG}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Ben flips a coin 10 times and then records the absolute difference between the total number of heads and tails he's flipped. He then flips the coin one more time and records the absolute difference between the total number of heads and tails he's flipped. If the probability that the second number he records is greater than the first can be expressed as  $\frac{a}{b}$  for positive integers  $a,b$  with  $\gcd (a,b) = 1$ , then find  $a + b$ .

*Proposed by Vismay Sharan*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{831}
Let $n = 2188 = 3^7+1$ and let $A_0^{(0)}, A_1^{(0)}, \ldots, A_{n-1}^{(0)}$ be the vertices of a regular $n$-gon (in that order) with center $O$. For $i = 1, 2, \dots, 7$ and $j=0,1,\dots,n-1$, let $A_j^{(i)}$ denote the centroid of the triangle \( \triangle A_j^{(i-1)} A_{j+3^{7-i}}^{(i-1)} A_{j+2 \cdot 3^{7-i}}^{(i-1)} \). Here the subscripts are taken modulo $n$. If \( \frac{|OA_{2014}^{(7)}|}{|OA_{2014}^{(0)}|} = \frac{p}{q} \) for relatively prime positive integers $p$ and $q$, find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2188}
Given a matrix \(A\), determine the value of \(\theta\) such that the sum of the real parts of the eigenvalues of \(A\) equals 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Given the following propositions about regular pyramids:
① A pyramid with a regular polygon as its base is a regular pyramid;
② A pyramid whose lateral edges are all equal is a regular pyramid;
③ A pyramid whose lateral edges form equal angles with the base is a regular pyramid;
④ A pyramid whose dihedral angles between the lateral faces and the base are all equal is a regular pyramid.
The definition of a regular pyramid is: the base is a regular polygon, and the projection of the apex onto the base is the center of the base polygon.
How many of these propositions are correct? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Lucky Larry's teacher asked him to substitute numbers for $a$, $b$, $c$, $d$, and $e$ in the expression $a-(b-(c-(d+e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The number Larry substituted for $a$, $b$, $c$, and $d$ were $1$, $2$, $3$, and $4$, respectively. What number did Larry substitute for $e$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The sequence \(\{\alpha_{n}\}\) is an arithmetic sequence with a common difference \(\beta\). The sequence \(\{\sin \alpha_{n}\}\) is a geometric sequence with a common ratio \(q\). Given that \(\alpha_{1}, \beta \in \mathbf{R}\), find the value of \(q\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm 1}
In a regular triangular prism $V-ABC$, the lateral edge $VC$ has a length of 3, and the base edge $AB$ has a length of 2. A section passing through the base edge $AB$ intersects the lateral edge $VC$ at point $D$. Determine the minimum area of the triangle $	riangle ABD$. Express your answer in simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Let n, m, p, q, and r be five integers such that the remainders when n is divided by 18, m is divided by 27, p is divided by 6, q is divided by 12, and r is divided by 3 are 10, 16, 4, 8, and 2 respectively. What is the remainder when ((3n + 2m) - (p + q) / r) is divided by 9?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The wholesale department operates a product with a wholesale price of 500 yuan per unit and a gross profit margin of 4%. The inventory capital is 80% borrowed from the bank at a monthly interest rate of 4.2‰, and the storage and operating cost is 0.30 yuan per unit per day. Determine the maximum average storage period for the product without incurring a loss.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Find the limit of the sequence \( b_n = \sqrt[n]{7^{3n} - 2^{n+4} - \cos(\sqrt{n})} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{343}
Given a triangle $\triangle ABC$ with side lengths $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively. If $a^{2}-a- \sqrt {3}b- \sqrt {3}c=0$ and $a+ \sqrt {3}b- \sqrt {3}c+2=0$, find the cosine value of the largest angle in $\triangle ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{- \dfrac { \sqrt {3}}{3}}
A man saves 20% of his monthly salary. His monthly expenses consist of rent, utilities, and groceries which are 40%, 30%, and 20% of his salary respectively. Due to an increase in the cost of living, he now has to increase his rent by 15%, utilities by 20%, and groceries by 10%. After these increases, he manages to still save, but his savings reduced to Rs. 180 per month. What is his original monthly salary?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3000}
Determine the values of $s \in \mathbb{R}$ for which the integral \[ \iint_{(a,b)^2}\frac{1}{|x-y|^s}\mathrm{d}x\mathrm{d}y \] converges, where $(a,b)$ is a finite open interval of $\mathbb{R}$ and $s > 0$. Provide the range of $s$ for which the integral is convergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1)}
Find the area of the subset \( S \) of \( \mathbb{R}^3 \) defined by the conditions: \( x^2 + y^2 + z^2 \leq 1 \) and \( \det A = 0 \), where \( A = \begin{bmatrix} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2 \end{bmatrix} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\pi}
Given the function \( f(x) \) defined as \( f(x) = 0 \) for \( x \leq 0 \) and \( f(x) = e^{-x} \) for \( x > 0 \), the Fourier transform \( F(\omega) \) is calculated to be \( F(\omega) = \frac{1}{\sqrt{2\pi}(1+i\omega)} \). Use this result to evaluate the integral \( \int_{0}^{\infty} \frac{\cos(2\omega) + \omega\sin(2\omega)}{1+\omega^2} \, d\omega \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{e^2}}
Gwen was organizing her book case, making room for her ever-growing collection. She had eight shelves of mystery books with six books on each shelf, five shelves of picture books with four books on each shelf, four shelves of science fiction books with seven books on each shelf, and three shelves of non-fiction books with five books on each shelf. However, she had lent out two mystery books and three science fiction books to her friends, and she had borrowed five picture books from the library. How many books did she have in her collection, excluding the borrowed books and considering the lent out books?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{106}
How do we calculate the residue of the function $$ f(x)=\frac{x^2}{(1+x^2)^2} $$ at the pole $x=i$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{4}i}
Let $p(x)$ be a nonzero polynomial of degree less than 1992 having no nonconstant factor in common with $x^{3}-x$. Let

$$
\frac{d^{1992}}{d x^{1992}}\left(\frac{p(x)}{x^{3}-x}\right)=\frac{f(x)}{g(x)}
$$

for polynomials $f(x)$ and $g(x)$. Find the smallest possible degree of $f(x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3984}
Point \( O \) is the center of the circumscribed circle of triangle \( ABC \) with sides \( AB=5, AC=8 \), and \( BC=4 \). Find the length of the vector \( \overrightarrow{OA} - 4 \overrightarrow{OB} + 3 \overrightarrow{OC} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{10}.}
Let $\mathcal{C}$ be the hyperbola $y^{2}-x^{2}=1$. Given a point $P_{0}$ on the $x$-axis, we construct a sequence of points $\left(P_{n}\right)$ on the $x$-axis in the following manner: let $\ell_{n}$ be the line with slope 1 passing through $P_{n}$, then $P_{n+1}$ is the orthogonal projection of the point of intersection of $\ell_{n}$ and $\mathcal{C}$ onto the $x$-axis. (If $P_{n}=0$, then the sequence simply terminates.) Let $N$ be the number of starting positions $P_{0}$ on the $x$-axis such that $P_{0}=P_{2008}$. Determine the remainder of $N$ when divided by 2008.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{254}
In a soccer match between Team A and Team B, the final score is a 9-9 tie. During the match, Team A had the lead at one point, and later, Team B also had the lead. How many different sequences of 18 goals can represent the course of the match, considering the order in which the goals were scored?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38896}
Calculate the sum of the infinite series \( \sum_{i=0}^{\infty} \sum_{j=0}^{\infty} \sum_{k=0}^{\infty} \frac{1}{3^i} \cdot \frac{1}{3^j} \cdot \frac{1}{3^k} \) with the condition that \( i \neq j \neq k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{81}{208}}
In \( \bigtriangleup {ABC},{PQ}//{BC} \) where \( P \) and \( Q \) are points on \( {AB} \) and \( {AC} \)
 respectively. The lines \( {PC} \) and \( {QB} \) intersect at \( G \) . It is also given
 \( {EF}//{BC} \) , where \( G \in  {EF},E \in  {AB} \) and \( F \in  {AC} \) with \( {PQ} = a \) and
 \( {EF} = b \) . Find value of \( {BC} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$BC = \frac{ab}{2a - b}$
}
Calculate the integral \( \int_{F(S^1)} \omega \) where \( F: S^1 \rightarrow S^1 \times S^1 \times S^1 \) is given by \( F(p) = (p,p,p) \) and \( \omega = dt_1 + dt_2 + dt_3 \) is a 1-form. Given that \( F^{*}\omega = 3\,dt \), find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\pi}
Let $ABCD$ be an isosceles trapezoid with $AD \parallel BC$. Points $P$ and $Q$ are placed on segments $\overline{CD}$ and $\overline{DA}$ respectively such that $AP \perp CD$ and $BQ \perp DA$, and point $X$ is the intersection of these two altitudes. Suppose that $BX=3$ and $XQ=1$. Compute the largest possible area of $ABCD$. \(\text{Express your answer as a single number.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
# Task 4.

Four brigades were developing an open coal deposit for three years, working with a constant productivity for each brigade. In the second year, due to weather conditions, work was not carried out for four months, and for the rest of the time, the brigades worked in rotation (one at a time). The ratio of the working times of the first, second, third, and fourth brigades and the amount of coal mined respectively are: in the first year 4:1:2:5 and 10 million tons; in the second year $2: 3: 2: 1$ and 7 million tons; in the third year 5:2:1:4 and 14 million tons. How much coal would these four brigades have mined in 4 months if they worked together?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Determine the operator norm \( \|T\| \) for the operator \( T: \ell^p \to \ell^p \) defined by \( Tx = (c_1 x_1, c_2 x_2, \ldots) \), where \( c_k \in \mathbb{C} \) and \( \lim_{k \to \infty} c_k = 0 \), for \( 1 \le p \le \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sup_{k} |c_k|}
Determine the value of n such that in a lottery where 5 balls are drawn without replacement from a set of balls numbered 1 through n, the probability of matching exactly 0 balls is as close as possible to the probability of matching exactly 1 ball. The goal is to create a scenario where matching at least one ball is slightly favored.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
In a Cartesian coordinate system, if the equation \(m\left(x^{2}+y^{2}+2 y+1\right)=(x-2 y+3)^{2}\) represents an ellipse, then what is the range of values for \(m\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(5,+\infty)}
What is the number of degrees in $x$? [asy]
import olympiad; import geometry; defaultpen(linewidth(0.8)); size(150);
pair A = origin, B = (Cos(50),Sin(50)), C = (Cos(50) + Sin(50)*sqrt(3),0);
draw(A--B--C--cycle);
label("$30^\circ$",C-(0.3,-.03),NW);
label("$2x$",B + (-.10,-0.1),SE);
label("$x$",A+(0.15,.03),NE);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Given the lines $l_{1}$: $\left(3+a\right)x+4y=5-3a$ and $l_{2}$: $2x+\left(5+a\right)y=8$, if $l_{1}$ is parallel to $l_{2}$, determine the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-7}
The complex numbers \( z_{1}, z_{2}, z_{3}, z_{4} \) satisfy \( \left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|=\left|z_{4}\right|=1 \), and \( z_{1}+z_{2}+z_{3}+z_{4}=0 \). What is the shape of the quadrilateral whose vertices correspond to these 4 complex numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Rectangle}
Find the smallest natural number that divides $2^n + 15$ for some natural number $n$ and can be expressed in the form $3x^2 - 4xy + 3y^2$ for some integers $x$ and $y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Find the smallest positive integer $n$, such that for any sequence of $n$ positive integers $a_{1}, a_{2}, \cdots, a_{n}$ satisfying $\sum_{i=1}^{n} a_{i}=2007$, there must be a sum of some consecutive terms equal to 30. (Provided by Tao Pingsheng)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1018}
Given a complex number z that satisfies the equation (1-i)/(1+i) - z = 1, calculate the magnitude of 2 + i times the conjugate of z. Express your answer as a numerical value, including the square root symbol if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Let $ABC$ be an equilateral triangle with each side of length 1. Let $X$ be a point chosen uniformly at random on side $\overline{AB}$. Let $Y$ be a point chosen uniformly at random on side $\overline{AC}$. (Points $X$ and $Y$ are chosen independently.) Let $p$ be the probability that the distance $XY$ is at most $\frac{1}{\sqrt[4]{3}}$. What is the value of $900p$, rounded to the nearest integer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{628}
Let \( P \) be a regular 2006-sided polygon. A diagonal of \( P \) is called a "good edge" if both parts of \( P \)'s boundary divided by this diagonal contain an odd number of edges. Each side of \( P \) is also defined as a "good edge."

Given that 2003 non-intersecting diagonals inside \( P \) divide it into several triangles, determine the maximum number of isosceles triangles with two "good edges" in this division.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1003}
已知 S_n 为数列 \{a_n\} 的前 n 项和，且 S_n=\df{9}{8}a_n-\df{3}{8}\cdot 3^n+\df{3}{8}. 则使 \ds{i=1}{n}\df{S_i}{a_i}<2024 成立的最大正整数 n 的值为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1799}
A circle passes through the point (1, 2) and has the equation $x^2+y^2=1$. The area of the quadrilateral formed by the two tangents to the circle from this point and the x-axis and y-axis is \_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {13}{8}}
Thomas buys a weight vest that weighed 60 pounds and worked well for him in the beginning. After a bit of training, he decides he wants to increase the weight by 60%. The weights come in 2-pound steel ingots. Each ingot cost $5. If you buy a certain number of ingots, you get a discount. It costs $72 to get the weight he needs. What is the discount percentage if you buy more than 10 ingots?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
Let  $S = {1, 2, \cdots, 100}.$   $X$  is a subset of  $S$  such that no two distinct elements in  $X$  multiply to an element in  $X.$  Find the maximum number of elements of  $X$ . 

*2022 CCA Math Bonanza Individual Round #3*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
The sequence \(a_{n} = b[\sqrt{n+c}] + d\) is given, where the successive terms are
\[
1, 3, 3, 3, 5, 5, 5, 5, 5, \cdots
\]
In this sequence, each positive odd number \(m\) appears exactly \(m\) times consecutively. The integers \(b\), \(c\), and \(d\) are to be determined. Find the value of \(b+c+d\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Simplify first, then evaluate: $\frac{{a}^{2}-{b}^{2}}{a}÷(a+\frac{2ab+{b}^{2}}{a})$, where $a=\sqrt{2}+1$ and $b=\sqrt{2}-1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}}{2}}
Schoolboy Alexey told his parents that he is already an adult and can manage his finances independently. His mother suggested using a duplicate bank card from her account. For participation in a charitable Christmas program, Alexey wants to buy 40 "Joy" chocolate bars and donate them to an orphanage. However, the bank, where Alexey's parents are clients, has implemented a new system to protect against unauthorized card payments. The protection system analyzes the root mean square (RMS) value of expenses for the last 3 purchases (S) using the formula \(S=\sqrt{\frac{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}{3}}\), where \(x_{1}, x_{2}\), and \(x_{3}\) are the costs of the previous purchases, and compares the value of \(S\) with the cost of the current purchase. If the cost of the current payment exceeds the value \(S\) by 3 times, the bank blocks the payment and requires additional verification (e.g., a call from mom to the call center). In the last month, payments made on the card were only for cellphone bills in the amount of 300 rubles each. Into how many minimum number of receipts should Alexey split the purchase so that he can buy all 40 "Joy" chocolate bars at a cost of 50 rubles each?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
How many different $T_0$ topologies can be defined on an infinite set, such as the set of real numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2^{2^{\mathfrak{c}}}$}
Suppose a parabola has vertex $\left(\frac{3}{2},-\frac{25}{4}\right)$ and follows the equation $y = ax^2 + bx + c$, where $a < 0$ and the product $abc$ is an integer. Find the largest possible value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Find $x$ such that $$\sum_{k=1}^{2014} k^k \equiv x \pmod {10}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two helpers painted only 24% of the house and quit at 2:12 PM. On Wednesday Paula worked by herself and finished the house by working until 7:12 P.M. How long, in minutes, was each day's lunch break?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
The sum of the coefficients in the expansion of the binomial $$(3x - \frac{1}{2 \cdot 3x})^n$$ is $\frac{1}{64}$. Find the constant term in the expansion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{5}{2}}
For an integer $30 \le k \le 70$, let $M$ be the maximum possible value of:
\[ \frac{A}{\gcd(A,B)} \]
where $A = \binom{100}{k}$ and $B = \binom{100}{k+3}$. Find the remainder when $M$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{664}
记 $I=\sum\limits_{m=1}^{+\infty} \sum\limits_{n=1}^{+\infty} \frac{1}{n!n!+m!n!}$, 则 $[10 I]=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
There is a set of points \( M \) on a plane and seven different circles \( C_{1}, C_{2}, \dots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \); circle \( C_{6} \) passes through exactly 6 points in \( M \); ..., circle \( C_{1} \) passes through exactly 1 point in \( M \). Determine the minimum number of points in set \( M \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
[Help me] Determine the smallest value of the sum M =xy-yz-zx where x; y; z are real numbers satisfying the following condition  $x^2+2y^2+5z^2 = 22$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{-55 - 11\sqrt{5}}{10}}
2. The chord $AB$ divides a circle with radius $r$ into two arcs in the ratio $1:2$. In the larger segment, a square is inscribed such that one of its sides lies on this chord. Express the length of the side of the square in terms of the radius $r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2+\sqrt{19}}{5}r}
Fill the five numbers $2015, 2016, 2017, 2018, 2019$ into the five boxes labeled " $D, O, G, C, W$ " such that $D+O+G=C+O+W$. How many different ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
The base of a pyramid is an equilateral triangle. One of the lateral faces of the pyramid is perpendicular to the plane of the base. Find the cosine of the angle between the two other lateral faces, given that they form an angle $\alpha$ with the plane of the base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1 + 3 \cos 2 \alpha}{4}}
Let $x_1, x_2, \ldots, x_n$ be distinct real numbers. Define the function $g(t)$ as the number of functions $f_k(t)$ that are equal to 1, where each $f_k(t)$ is defined based on the original question's context. Determine the maximum value of $g(t)$. \( \text{Provide your answer as a single integer.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
A fast train with a weight of $P=150$ tons travels at a maximum speed of $v=72 \frac{\text{km}}{\text{hour}}$ on a horizontal track with a friction coefficient of $\rho=0,005$. What speed can it reach on a track with the same friction conditions but with an incline having $e=0.030$? 

(Note: In this problem, $\rho$ is the friction coefficient and $e$ is the sine of the inclination angle.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10.3}
5. The set
$$
A=\left\{\left.(x, y)\left|(x-t)^{2}+y^{2} \leqslant\left(1-\frac{t}{2}\right)^{2},\right| t \right\rvert\, \leqslant 2\right\}
$$

represents a plane region with an area of $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["4\\sqrt{3}+\\frac{8\\pi}{3}"]}
For real numbers $a, b$, and $c$ the polynomial $p(x) = 3x^7 - 291x^6 + ax^5 + bx^4 + cx^2 + 134x - 2$ has $7$ real roots whose sum is $97$. Find the sum of the reciprocals of those $7$ roots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
2 cows and 10 goats are brought for Rs. 2000. If the individual prices of goats are Rs. 65, Rs. 70, Rs. 75, Rs. 80, Rs. 75, Rs. 71, Rs. 82, Rs. 64, Rs. 78, and Rs. 79, respectively, find the individual prices of each cow.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{630.50}