inclusionAI/Ring-lite-rl-data huggingface dataset:
Given points M and N on the parabola y = 4x^2, F is the focus of the parabola, and ∠MFN = 2π/3. The midpoint of chord MN is P, and the distance from P to line l (directrix): y = -1/16 is denoted as d. If |MN|^2 = λ * d^2, calculate the minimum value of λ.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In a 1000 m race, A beats B by 70 m and B beats C by some meters. In the same race, A beats C by 163 m. By how many meters does B beat C?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{93}
Find the coefficient of the $\frac{1}{z^2}$ term in the Laurent series expansion of the function $f(z) = \frac{e^{2z}}{(z-1)^3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4e^2}
In a group $G$ of order $pq^3$, where $H$ is a normal Sylow $q$-subgroup, there exists a subgroup $P$ of order $pq^2$ and a Sylow $p$-subgroup $R$ of $P$ such that $|N_P(R)|=p$. Determine the order of $N_G(R)$. \( \text{(Express your answer as a single number.)} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{pq}
On the sides $AC$ and $BC$ of triangle $ABC$, points $P$ and $Q$ are marked respectively. It turns out that $AB = AP = BQ = 1$, and the intersection point of segments $AQ$ and $BP$ lies on the incircle of triangle $ABC$. Find the perimeter of triangle $ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In triangle $\triangle ABC$, let $\overrightarrow {CA} = a$ and $\overrightarrow {CB} = b$. Point $M$ is the midpoint of $CB$, and point $N$ is the midpoint of $AB$. Lines $CN$ and $AM$ intersect at point $P$. Express $\overrightarrow {AP}$ in terms of $a$ and $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{b - 2a}{3}}
Given an integer  $n\geq 3$ . For each  $3\times3$  squares on the grid, call this  $3\times3$  square isolated if the center unit square is white and other 8 squares are black, or the center unit square is black and other 8 squares are white. 

Now suppose one can paint an infinite grid by white or black, so that one can select an  $a\times b$  rectangle which contains at least  $n^2-n$  isolated  $3\times 3$  square. Find the minimum of  $a+b$  that such thing can happen.

(Note that  $a,b$  are positive reals, and selected  $a\times b$  rectangle may have sides not parallel to grid line of the infinite grid.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4n}
Each integer in  $\{1, 2, 3, . . . , 2020\}$  is coloured in such a way that, for all positive integers  $a$  and  $b$ such that  $a + b \leq 2020$ , the numbers  $a$ ,  $b$  and  $a + b$  are not coloured with three different colours.
Determine the maximum number of colours that can be used.

*Massimiliano Foschi, Italy*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
A population consists of 20 individuals numbered 01, 02,..., 19, 20. 6 individuals are selected using the following random number table. The selection method is to start from the numbers in the 5th and 6th columns of the 1st row of the random number table and select two numbers from left to right as the individual numbers.

| 7816 | 6572 | 0802 | 6314 | 0702 | 4311 |
|------|------|------|------|------|------|
| 3204 | 9234 | 4935 | 8200 | 3623 | 4869 |

Determine the number of the 5th selected individual. Express your answer as a two-digit number (e.g., 07, 11, etc.).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
If $(a-x)^8 = a_0 + a_1x + a_2x^2 + \ldots + a_8x^8$ ($a \in \mathbb{R}$), and $a_5 = 56$, then $a_0 + a_1 + a_2 + \ldots + a_8 = \boxed{\text{\_\_\_\_\_\_}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
If 60% of (x - y) = 30% of (x + y), 40% of (x + z) = 20% of (y + z), and 50% of (x - z) = 25% of (x + y + z), then what percent of x is the sum of y and z?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0\%}
Assume that the Earth is a perfect sphere.  A plane flies between $30^\circ$ N $45^\circ$ W and $30^\circ$ N $45^\circ$ E along the shortest possible route along the surface of the Earth.  Let $\theta$ be the northernmost latitude that the plane flies over.  Compute $\sin \theta.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{10}}{5}}
A square-shaped quilt is divided into $16 = 4 \times 4$ equal squares. We say that the quilt is UMD certified if each of these 16 squares is colored red, yellow, or black, so that (i) all three colors are used at least once and (ii) the quilt looks the same when it is rotated 90, 180, or 270 degrees about its center. How many distinct UMD certified quilts are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
36
}
The sides opposite to angles $A$, $B$, and $C$ in triangle $ABC$ are $a$, $b$, and $c$ respectively. It is given that $10a\cos B=3b\cos A$, and $\cos A= \frac {5 \sqrt {26}}{26}$. Find $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {3\pi}{4}}
Find the limit as \( n \to \infty \) of the sum:
\[ S_n = \frac{1}{n}\left(\frac{1}{2} + \sum_{k=1}^{n}\cos(kx)\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $x_i$ be positive integers for $i=1,2,...,11$. If $x_i + x_{i+1} \geq 100$ and $|x_i - x_{i+1}| \geq 20$ for $i=1,2,...,10$, and $x_{11} + x_1 \geq 100$ and $|x_{11} - x_1| \geq 20$, what is the minimum possible value of $\sum_{i=1}^{11} x_i$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{580}
In a $4 \times 4$ table, numbers are written such that the sum of the neighbors of each number is equal to 1 (neighboring cells share a common side).

Find the sum of all the numbers in the table.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Problem 10.7. Olya drew $N$ different lines on the plane, any two of which intersect. It turned out that among any 15 lines, there are always two lines such that the angle between them is $60^{\circ}$. What is the largest $N$ for which this is possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
A travel agency held a promotion: "Buy a trip to Egypt, bring four friends who also buy a trip, and get the cost of your trip back." During the promotion, 13 customers came on their own, and the rest were brought by friends. Some of them brought exactly 4 new clients, while the remaining 100 did not bring anyone. How many tourists traveled to the Land of the Pyramids for free?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Let $A(x, y)$ denote the number of points ($m, n$) with integer coordinates $m$ and $n$ that satisfy $m^{2}+n^{2} \leqslant x^{2}+y^{2}$ in the plane. Let $g=\sum_{k=0}^{\infty} e^{-k^{2}}$. If
$$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} A(x, y) e^{-x^{2}-y^{2}} d x d y=\pi(ag-b)^{c}$$
Find the value of $a+b+c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
$A B C D$ is a parallelogram satisfying $A B=7, B C=2$, and $\angle D A B=120^{\circ}$. Parallelogram $E C F A$ is contained in $A B C D$ and is similar to it. Find the ratio of the area of $E C F A$ to the area of $A B C D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{39}{67}}
Find the remainder when \( \left \lfloor \frac{149^{151} + 151^{149}}{22499} \right \rfloor \) is divided by \(10^4\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7800}
The distance between location A and location B originally required a utility pole to be installed every 45m, including the two poles at both ends, making a total of 53 poles. Now, the plan has been changed to install a pole every 60m. Excluding the two poles at both ends, how many poles in between do not need to be moved?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
The hundreds digit of the 3 digit integer x is a, b is the tens digit of x, and c is the units digit of x. There is a relationship between a, b, and c such that a > 0 and the difference between the two greatest possible values of x is 241. What is the ratio of b to a?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5:7}
Find the number of ordered quadruples of positive integers $(a,b,c,d)$ such that $a,b,c,$ and $d$ are all (not necessarily distinct) factors of $30$ and $abcd>900$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1940}
Tetrahedron $A B C D$ with volume 1 is inscribed in circumsphere $\omega$ such that $A B=A C=A D=2$ and $B C \cdot C D \cdot D B=16$. Find the radius of $\omega$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{3}}
Let $F_1$ and $F_2$ be the left and right foci of the ellipse $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \ (a > b > 0)$, and let $P$ be a point on the line $x= \frac{3a}{2}$. If $\triangle F_1PF_2$ is an isosceles triangle with a base angle of $30^\circ$, find the eccentricity of the ellipse $E$. Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
Suppose there are 100 cookies arranged in a circle, and 53 of them are chocolate chip, with the remainder being oatmeal. Pearl wants to choose a contiguous subsegment of exactly 67 cookies and wants this subsegment to have exactly \( k \) chocolate chip cookies. Find the sum of the \( k \) for which Pearl is guaranteed to succeed regardless of how the cookies are arranged.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71}
Point \(A\) lies on the line \(y=\frac{12}{5} x-9\), 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{["189/65"]}
A straight line intersects parallelogram \(ABCD\) at a single point \(B\). The vertices \(A\) and \(C\) are at distances \(a\) and \(b\) from this line, respectively.

What is the distance of vertex \(D\) from this line?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a + b}
Nini's report card shows the scores of 6 exams. The average score of the 6 exams is 74 points; the most frequent score among the 6 exams is 76 points; the median score of the 6 exams is 76 points; the lowest score is 50 points; the highest score is 94 points. There is only one score that appears twice, and no score appears more than twice. Assuming her scores are all integers, how many possible values are there for the second lowest score among the 6 exams?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Determine all values of $m$ so that the domain of
\[ c(x) = \frac{3mx^2 - 4x + 1}{7x^2 - 6x + m} \]
is the set of all real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{9}{7}, \infty\right)}
Janet has 'n' number of blankets, and each blanket has a distinct color. The thickness of each blanket is initially 3 inches. For each colored blanket, she folds it 'c' times (where 'c' represents the color's numeric code, ranging from 1 to 5). Calculate the combined thickness of all the folded blankets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{186}
Evaluate the limit: $$\lim_{x\to\infty}\left((x(x+1)(x+2))^{\frac{1}{3}}-\left(2(x+1)-\left(x^x(x+1)^{x+1}(x+2)^{x+2}\right)^{\frac{1}{3x+3}}\right)^{\frac{3}{2}}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the range of values for $n$ such that a graph $G$ with $n$ vertices can have more components and edges than another graph $G'$ with the same number of vertices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \geq 5}
What is the maximum number of moves required to solve any solvable configuration of the 8-puzzle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Consider a \(7 \times 7\) field that is divided into 49 unit squares. Into this field, we want to place tiles in the shape of a Swiss cross, consisting of 5 unit squares. The edges of the crosses should lie on the lines of the field. Determine the smallest possible number of squares that must be marked on the field so that each cross, no matter where it is placed on the field, covers at least one marked square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A storage depot is a pyramid with height $30$ and a square base with side length $40$. Determine how many cubical $3\times 3\times 3$ boxes can be stored in this depot if the boxes are always packed so that each of their edges is parallel to either an edge of the base or the altitude of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
471
}
Given that the sequence $\{a\_n\}$ has positive terms, with the first term $a\_1=1$, $S\_n$ is the sum of the first $n$ terms of the sequence, and it satisfies: $a\_nS_{n+1}-a_{n+1}S\_n+a\_n-a_{n+1}= \frac {1}{2}a\_na_{n+1}$, then $\frac {3}{34}S_{12}=$ _____ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine all values of \( \alpha \) such that the inequality \( \ln(1+x) \leq x - \frac{x^2}{2} + \alpha x^3 \) holds for all \( x > -1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Distribute 5 student volunteers to participate in volunteer activities in 4 projects: Idiom Competition, Poetry Contest, Youth Song Concert, and Charity Sale. Each volunteer is assigned to only 1 project, and each project must have at least 1 volunteer. Calculate the total number of different distribution schemes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
A flower bouquet contains pink roses, red roses, pink carnations, and red carnations.  One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink.  What percent of the flowers are carnations?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
Determine the minimum number of detectors required to uniquely identify the location of a radiation point in an n-dimensional space using a given metric $d$. Provide your answer as a function of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n + 1}
Determine the relationship between $a$ and $b$ such that in any partition of the set $X = \{1, 2, 3, \ldots, a\}$ into two subsets, at least one subset contains an arithmetic progression of length $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \geq W(2, b)}
What is the weight of 6 moles of Aluminum bromide when it is utilized in the synthesis of a complex metal-organic framework compound during a specific chemical reaction?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1600.08}
Let $\{a_{n}\}$ and $\{b_{n}\}$ be arithmetic sequences with the sums of the first $n$ terms denoted as $S_{n}$ and $T_{n}$, respectively. If for any $n\in N^{*}$, $\frac{{S}_{n}}{{T}_{n}}=\frac{2n-3}{4n-3}$, then calculate $\frac{{a}_{2}}{{b}_{3}+{b}_{13}}+\frac{{a}_{14}}{{b}_{5}+{b}_{11}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{19}}
Team A has a probability of $$\frac{2}{3}$$ of winning each set in a best-of-five set match, and Team B leads 2:0 after the first two sets. Calculate the probability of Team B winning the match.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{19}{27}}
Given a regular tetrahedron \(ABCD\) with edge length 2, there is a point \(P\) on edge \(AB\) such that \(AP < 1\). A cut is made through point \(P\) perpendicular to the edge \(AB\) and continues through the faces but stops at a certain point. When the cut stops, the length of the cut on face \(ABD\) is \(PM = 1\) and the length of the cut on face \(ABC\) is \(PN = \frac{2}{3}\). Find the length \(MN\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{MN = 1}
Evaluate the surface integral \( \int\int_S z \, dx \, dy \) where \( S \) is the complete surface of the cylinder with radius \( a \) and axis \( OZ \) bounded by the planes \( z = 0 \) and \( z = 2a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi a^3}
Find the particular solution to the partial differential equation \( u_{xx}+2u_{xy}-3u_{yy}=0 \) given the general solution \( u(x,y)=F(x+y)+G(x-\frac{1}{3}y) \) and the conditions \( u(x=0,y)=y \) and \( u_x(x=0,y)=y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-x^2 + xy + y}
Find the smallest positive integer $n$ so that for any integers $a_1, a_2,..., a_{527}$,the number
$$
\left( \prod^{527}_{j=1} a_j\right) \cdot\left( \sum^{527}_{j=1} a^n_j\right)
$$
is divisible by $527$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Find the centroid of the part of a spherical shell with radius 30, centered at the origin, that lies in the first octant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(15, 15, 15)}
a group of people participate in some curriculum , 35 of them practice yoga , 20 study cooking , 15 study weaving , 7 of them study cooking only , 5 of them study both the cooking and yoga , 3 of them participate all curriculums . how many people study both cooking and weaving ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
For every integer $n \ge 1$, the function $f_n : \{ 0, 1, \cdots, n \} \to \mathbb{R}$ is defined recursively by $f_n(0) = 0$, $f_n(1) = 1$, and 
\[
(n-k) f_n(k-1) + kf_n(k+1) = nf_n(k)
\] 
for each $1 \le k < n$.  Let 
\[
S_N = f_{N+1}(1) + f_{N+2}(2) + \cdots + f_{2N}(N)
\]
Find the remainder when $\left\lfloor S_{2013} \right\rfloor$ is divided by $2011$.  (Here $\left\lfloor x \right\rfloor$ is the greatest integer not exceeding $x$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Evaluate the sum: 1 - 2 + 3 - 4 + $\cdots$ + 100 - 101
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-151}
Quadrilateral $ABCD$ satisfies $AB = 8$, $BC = 5$, $CD = 17$, and $DA = 10$. Let $E$ be the intersection of $AC$ and $BD$. Suppose the ratio $BE : ED = 1 : 2$. Find the area of quadrilateral $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
11个集合$M _ { 1 } , M _ { 2 } , \cdots , M _ { 1 1 }$，每个集合有5个元素，并且任意两个集合的交非空.求具有同一公共元素的集合数目的最大值的最小可能值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
有 10 个乒兵球选手进行单循环赛, 比赛结果显示, 没有和局, 且任意 5 人中既有 1人胜其余 4 人, 又有 1 人负其余 4 人, 则恰好胜了两场的人数为 $\qquad$个.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
It is known that the distance between any two of the given $n(n=2,3,4,5)$ points in the plane is at least 1. What is the minimum value that the diameter of this system of points can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
The water level in a rectangular swimming pool measuring 40 feet by 25 feet is to be lowered by some inches. To do this, 3750 gallons of water must be removed. By how many inches should the water level be lowered?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6.02}
In \(\triangle ABC\), \(BC = a\), \(CA = b\), \(AB = c\). If \(2a^{2} + b^{2} + c^{2} = 4\), then the maximum area of \(\triangle ABC\) is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5}}{5}}
Let an integer  $ n > 3$ . Denote the set  $ T\equal{}\{1,2, \ldots,n\}.$  A subset S of T is called *wanting set* if S has the property: There exists a positive integer  $ c$  which is not greater than  $ \frac {n}{2}$  such that  $ |s_1 \minus{} s_2|\ne c$  for every pairs of arbitrary elements  $ s_1,s_2\in S$ . How many does a *wanting set* have at most are there ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\lfloor \frac{2n}{3} \right\rfloor}
Given two sets A and B where A = {x^2, 2x - 1, -4} and B = {x - 5, 1 - x, 9}, and the intersection of B and A is {9}, find the union of sets A and B.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{-8, -7, -4, 4, 9}}
Calculate the integral:
\[ I = \int_0^{\infty} \frac{\log(x)}{(x-1)\sqrt{x}} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\pi^2\)}
在满足$y<x\leq 100$的有序正整数对（x，y）中，有多少对使得$\frac { x } { y }$与$\frac { x + 1 } { y + 1 }$都是整数？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85}
A square piece of paper is folded along its vertical centerline, and then the folded paper is cut along a dotted line, resulting in three rectangular pieces of paper: one large and two small. Calculate the ratio of the perimeter of each small rectangle to the perimeter of the large rectangle. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{6}}
Last year, the electricity price in a certain city was 0.80 yuan per kilowatt-hour, with an annual electricity consumption of $a$ kilowatt-hours. This year, it is planned to reduce the electricity price to between 0.55 yuan and 0.75 yuan per kilowatt-hour, while residential users hope for an electricity price of 0.40 yuan per kilowatt-hour (the city's cost price for electricity is 0.30 yuan per kilowatt-hour). It has been calculated that after reducing the electricity price, the increase in electricity consumption in the city is inversely proportional to the difference between the actual electricity price and the user's expected price, with a proportionality constant of $0.2a$. The question is, what is the lowest electricity price that can ensure the revenue of the power department increases by at least 20% compared to last year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.6 \text{ yuan per kilowatt-hour}}
Let \( N \) be the number of ways to distribute 52 identical balls into 4 distinguishable boxes such that no box is empty and the difference between the number of balls in any two boxes is not a multiple of 6. If \( N = 100a + b \), where \( a \) and \( b \) are positive integers less than 100, find \( a + b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{81}
Three cones with a common vertex \( A \) touch each other externally, with the angle at the vertex of the first two cones being \( \frac{\pi}{3} \). All the cones also touch a plane passing through the point \( A \) and lie on one side of it. Determine the angle at the vertex of the third cone.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{10}}
Evaluate the complex line integral \( \int_\gamma \frac{z^5}{z^7+3z-10}\,dz \), where \( \gamma \) is the boundary of \( D(0,2) \) oriented counterclockwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The number \( a \) is a root of the equation \( x^{11} + x^{7} + x^{3} = 1 \). Specify all natural values of \( n \) for which the equality \( a^{4} + a^{3} = a^{n} + 1 \) holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}