inclusionAI/Ring-lite-rl-data huggingface dataset:
Calculate the residue of the function \( f(z) = \frac{1}{z(z^2+4)(z+2i)} \) at \( z = -2i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3i}{32}}
Determine the number of square roots of matrices in a neighborhood of the identity matrix $I$ that are not reflections, for the function $\mathbf{f}: A \mapsto A^2$ where $A \in \mathbb{R}^{n \times n}$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Compute the derivative of the functional \( J[f] = \int_{0}^{1} f'(x)^2 \, \mathrm{d}x \) with respect to the function \( f \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2 f''(x)}
Evaluate the sum \(1^{2017} + 2^{2017} + \cdots + 1000^{2017} \pmod{2016}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{784}
A quadratic polynomial $f(x) = Ax^2 + Bx + C$ is small if $A, B, C$ are single-digit positive integers. It is full if there are only finitely many positive integers that cannot be expressed as $f(x) + 3y$ for some positive integers $x$ and $y.$ Find the number of quadratic polynomials that are both small and full.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
162
}
Construct a polynomial with integer coefficients \( f(x) \) such that \( f: \mathbf{Q} \rightarrow \mathbf{Q} \) is injective, but \( f: \mathbf{R} \rightarrow \mathbf{R} \) is not injective.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^3 - 2x}
Suppose you have $10,000 and invest in the stock market by buying 500 shares of DGCo at $10 each. You trade every day, buying 10 shares when the price increases by $1 and selling 10 shares when it decreases by $1. The stock price is equally likely to go up or down by $1 at any time. Calculate the expected amount of money you will have after trading for one year.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10000}
Evaluate the integral \( \iint \dfrac {\mathrm dx\mathrm dy}{\sqrt {x^{2}-3y^{2}}} \) over the region defined by \( x^{2}+y^{2}\leq 4x-3 \). Use polar coordinates to simplify the integration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2\pi}{3}}
Three identical spheres are placed inside a cone such that the diameter of the base of the cone is equal to the slant height. The spheres touch each other externally, with two spheres touching the lateral surface and the base of the cone, and the third sphere touching the lateral surface at a point in the same plane as the centers of the spheres. Determine the ratio of the radius of the base of the cone to the radius of the spheres.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + \sqrt{3}}
If $x^3-3\sqrt3 x^2 +9x - 3\sqrt3 -64=0$ find the value of $x^6-8x^5+13x^4-5x^3+49x^2-137x+2015$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1898
}
Trapezoid $EFGH$ has sides $EF=105$, $FG=45$, $GH=21$, and $HE=80$, with $EF$ parallel to $GH$. A circle with center $Q$ on $EF$ is drawn tangent to $FG$ and $HE$. Find the exact length of $EQ$ using fractions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{336}{5}}
Find the number of distinct normal lines to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) that pass through the point \((p,q)\), where \(a, b, p, q\) are real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the convergence radius of the power series \(\sum_{n=0}^\infty a_nb_nx^n\), given that the convergence radii of \(\sum_{n=0}^\infty a_nx^n\) and \(\sum_{n=0}^\infty b_nx^n\) are \(R_1>0\) and \(R_2=\infty\), respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the integral \( \int s_{4} \, d\mu \) for the Lebesgue measure \( \mu \), where \( s_{n} = \sum^{n2^{n}}_{i=1} \frac{i-1}{2^{n}} \mathbb{1}_{E_{n,i}} + n\mathbb{1}_{A_{n}} \) and \( n=4 \). The sets \( E_{n,i} \) are defined as \( E_{n,i} = \left\{ x : \frac{i-1}{2^{n}} \leq f(x) < \frac{i}{2^{n}} \right\} \) and \( A_{n} = \{ x : f(x) \geq n \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{63}{8}}
You are planning to invest in a selection of stocks:

- Stock A yields 21%, is quoted at 10%, has an annual dividend growth rate of 5%, is considered low risk, and has taxes on dividends at 10%.
- Stock B yields 15%, is quoted at 20%, has an annual dividend growth rate of 8%, is considered high risk, and has taxes on dividends at 20%.

If you invest $10,000 in Stock A and $15,000 in Stock B, taking into account their risk levels, dividend growth rates, and taxes on dividends, what will be the effective weighted average yield of your portfolio for the next year?

Note: The problem assumes the investment amounts will not change over time.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14.76\%}
Find the maximum number of planes in the space, such there are $ 6$ points, that satisfy to the following conditions:
1.Each plane contains at least $ 4$ of them
2.No four points are collinear.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
6
}
A dog has three trainers:



- The first trainer gives him a treat right away.
- The second trainer makes him jump five times, then gives him a treat.
- The third trainer makes him jump three times, then gives him no treat.

The dog will keep picking trainers with equal probability until he gets a treat. (The dog's memory isn't so good, so he might pick the third trainer repeatedly!) What is the expected number of times the dog will jump before getting a treat?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Evaluate the limit: \[ \lim_{n\to \infty} \sum_{r=0}^{n} \frac{\binom{n}{r}}{(r+4)n^r} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6 - 2e}
Given that $a > b > c$ are the roots of the polynomial $P(x) = x^3 - 2x^2 - x + 1$, find the value of $K = a^2b + b^2c + c^2a$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Calculate the volume under the surface \( z = x^2 + 2xy + y^2 \) over the triangular region with vertices \((0,0)\), \((1,0)\), and \((1,1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{12}}
A point is randomly dropped on the interval $[5, 7]$, and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-3k-4\right)x^{2}+(3k-7)x+2=0$ satisfy the condition $x_{1} \leq 2x_{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Given a three-dimensional Lie algebra \( L \) with its derived algebra \( L' \) such that \( \dim(L') = 2 \), determine the smallest natural number \( m \) such that the derived series \( L^{(m)} = 0 \), where \( L^{(1)} := L' \) and \( L^{(n)} := [ L^{(n-1)}, L^{(n-1)}] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
If the function $f(x) = (m-1)x^2 + mx + 3$ ($x \in \mathbb{R}$) is an even function, then the decreasing interval of $f(x)$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, +\infty)}
Positive integers $a, b,$ and $c$ have the property that $a^b, b^c,$ and $c^a$ end in $4, 2,$ and $9,$ respectively. Compute the minimum possible value of $a+b+c.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
17}
What is the logical negation of the proposition '$\exists x\geqslant 3$, $x^{2}-2x+3 \lt 0$'?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\forall x\geqslant 3$, $x^{2}-2x+3\geqslant 0$}
A circle with radius $r$ is contained within the region bounded by a circle with radius $R$. The area bounded by the larger circle is $a/b$ times the area of the region outside the smaller circle and inside the larger circle. Then $R:r$ equals:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt a:\sqrt{a-b}}
Let $\lfloor z \rfloor$ denote the greatest integer less than or equal to $z.$ Compute $$\sum\limits_{j=-1000}^{1000} \left\lfloor \frac{2025}{j+0.5}\right\rfloor.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-984}
Determine the eccentricity of the ellipse formed when a cylinder of radius \( r \) is cut at an angle \( \alpha \) to its axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin \alpha}
Determine the smallest cardinal \( \kappa \) such that \( \kappa^{<\kappa} = \kappa \) in models of ZFC without large cardinal assumptions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_0}
Let $ABCD$ be a convex quadrilateral such that $AB=4$, $BC=5$, $CA=6$, and $\triangle{ABC}$ is similar to $\triangle{ACD}$. Let $P$ be a point on the extension of $DA$ past $A$ such that $\angle{BDC}=\angle{ACP}$. Compute $DP^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{169}
Determine the number of index 3 normal subgroups of the group given by the presentation \( \langle a, b, c \mid aba^{-1}b^{-1}c^2 = 1 \rangle \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A teacher intends to give the children a problem of the following type. He will tell them that he has thought of a polynomial \( P(x) \) of degree 2017 with integer coefficients and a leading coefficient of 1. Then he will provide them with \( k \) integers \( n_{1}, n_{2}, \ldots, n_{k} \), and separately provide the value of the expression \( P(n_{1}) \cdot P(n_{2}) \cdot \ldots \cdot P(n_{k}) \). From this data, the children must determine the polynomial that the teacher had in mind. What is the minimum value of \( k \) such that the polynomial found by the children will necessarily be the same as the one conceived by the teacher?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017}
Determine the number of real values of $x$ that satisfy the equation: \[ \left| 2 - x \right|^{ \log_5(x^2) - 5\log_x (25)} = (2-x)^2 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the number of idempotent elements in the ring $\mathbb{Z}[i] \otimes_{\mathbb{Z}} \mathbb{Z}[i]$. An element $e$ is idempotent if $e^2 = e$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In the vertices of a convex 2020-gon, numbers are placed such that among any three consecutive vertices, there is both a vertex with the number 7 and a vertex with the number 6. On each segment connecting two vertices, the product of the numbers at these two vertices is written. Andrey calculated the sum of the numbers written on the sides of the polygon and obtained the sum \( A \), while Sasha calculated the sum of the numbers written on the diagonals connecting vertices one apart and obtained the sum \( C \). Find the largest possible value of the difference \( C - A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1010}
Let $p, q, r$ be distinct real numbers such that $ap^2 + bp + c = (\\sin(\\theta))p^2 + (\\cos(\\theta))p$, and similarly for $q$ and $r$. Here, $a, b, c \in \mathbb{R}$. Find the maximum value of the expression \( \frac{a^2 + b^2}{a^2 + 3ab + 5b^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A train 100 meters long takes 27.997760179185665 seconds to cross a bridge of a certain length if the speed of the train is 36 kmph. How long is the bridge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{179.98}
Let \( a_{0}=0 \), \( a_{1}=1 \), and then move one square at a time in a counterclockwise direction on the complex plane. If \( a_{n+1}-a_{n} = \mathrm{i}^{f(n)} \), where \( \mathrm{i} \) is the imaginary unit, find the simplest unified expression for \( f(n) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Calculate the area of the projection of the quadrilateral $PQRS$ onto the $xz$ plane, where $P(1, 1, 2)$, $Q(3, 2, 1)$, $R(4, 0, 3)$, and $S(2, -1, 4)$ are the vertices, and the acute angle between the plane containing $PQRS$ and the $yz$ plane is $\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate the surface integral \( \iint_K {x \, dS} \) over the part of the parabolic cylinder \( z = \frac{x^2}{2} \) that lies inside the first octant part of the cylinder \( x^2 + y^2 = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{8}}
Determine the equation of the curve that intersects all level curves of the function \( g(x,y) = \ln(x) + \cos(y) \) orthogonally and passes through the point (2,0).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = 0}
Evaluate the limit $$\lim_{r \to \infty} \frac {r^C \int_0^{\frac{\pi}{2}} x^r \sin(x)\, dx}{\int_0^{\frac{\pi}{2}} x^r \cos(x)\, dx}$$ where $C$ is a real constant, and determine the value of $\pi L - C$, given that $L > 0$ is the limit's value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Calculate the line integral $ \int_\gamma{\frac{d\xi}{\xi^4+1}}$ where $\gamma(t)=1+e^{it}, \,\,\,0\leq t\leq 2\pi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The region below $$y = \sqrt{\ln (x+1)}$$ and above the $x$-axis, $ 0 \leq x \leq 1$, is rotated about the $x$-axis. Find the mass centrum for this solid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{4(2\ln 2-1)}\)}
A collection of coins includes only pennies (1 cent), nickels (5 cents), dimes (10 cents), and quarters (25 cents). Using the coins in this collection, it is possible to create any amount of money less than one dollar (100 cents). What is the smallest possible number of coins in the collection? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Given a sequence of positive terms $\{a\_n\}$ that satisfies $2a\_1+3a\_2+a\_3=1$, find the minimum value of the arithmetic mean between $\frac{1}{a\_1+a\_2}$ and $\frac{1}{a\_2+a\_3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3 + 2\sqrt{2}}{2}}
In the triangle \( \triangle ABC \), if \( \log \tan A + \log \tan C = 2 \log \tan B \), what is the range of angle \( B \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3} \leq B < \frac{\pi}{2}}
The angle between vector $\overrightarrow{a}=(\sqrt{3},\;1)$ and vector $\overrightarrow{b}=(\sqrt{3},\;-1)$ is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{\\pi}{3}"]}
Consider the hyperbola $M$: ${x^{2}} - \frac{{y^{2}}}{{{b^{2}}}} = 1$. Its left and right foci are $F_{1}$ and $F_{2}$, respectively, with $|F_{1}F_{2}|=2c$. A circle with center at the origin $O$ and radius $c$ intersects with the hyperbola $M$ at point $P$ in the first quadrant. If $|PF_{1}|=c+2$, find the x-coordinate of point $P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}+1}{2}}
一年以 365 天计算, 则甲、乙、丙三人恰有两人在国庆节过生日的概率等于
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1092}{365^3}$}
Find the maximum value of the function

$$
f(x)=\sin (x+\sin x)+\sin (x-\sin x)+\left(\frac{\pi}{2}-2\right) \sin (\sin x)
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi - 2}{\sqrt{2}}}
Suppose $p$: the function $y = \log_a(x+1) (a > 0$ and $a \neq 1)$ is monotonically decreasing on $(-1, +\infty)$; $q$: the curve $y = x^2 + (2a - 3)x + 1$ intersects the $x$-axis at two distinct points. If $p \land q$ is false and $p \lor q$ is 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}{2}, 1) \cup (\frac{5}{2}, +\infty)}
In a certain table tennis singles tournament, it was originally planned for every two players to compete exactly once. However, three players each played two matches before withdrawing from the tournament, resulting in a total of 50 matches. Calculate the number of matches played among these three players. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the value of $g'(2)$ given that $g$ is a real-valued function defined on the interval $(-1,1)$ such that $$e^{-x}(g(x)-2e^x)=\int_0^x \sqrt{y^4+1}\,dy$$ for all $x\in (-1,1)$, and $f$ is another function such that $$f(g(x))=g(f(x))=x.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
The sixth graders were discussing how old their principal is. Anya said, "He is older than 38 years." Borya said, "He is younger than 35 years." Vova: "He is younger than 40 years." Galya: "He is older than 40 years." Dima: "Borya and Vova are right." Sasha: "You are all wrong." It turned out that the boys and girls were wrong the same number of times. Can we determine how old the principal is?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
Determine the sixth digit after the decimal point in the number $(\sqrt{1978} + \lfloor \sqrt{1978} \rfloor)^{20}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given that $x=\ln π$, $y=\log _{5}2$, $z=e^{- \frac {1}{2}}$, compare the values of $x$, $y$, and $z$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y < z < x}
Find the covariance between two random variables X and Y, given their joint probability density function (pdf) is \( f(x,y) = 6x \) for \( 0 < x < y < 1 \) and \( 0 \) otherwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{40}}
Every square of $1000 \times 1000$ board is colored black or white. It is known that exists one square $10 \times 10$ such that all squares inside it are black and one square $10 \times 10$ such that all squares inside are white. For every square $K$ $10 \times 10$ we define its power $m(K)$ as an absolute value of difference between number of white and black squares $1 \times 1$ in square $K$. Let $T$ be a square $10 \times 10$ which has minimum power among all squares $10 \times 10$ in this board. Determine maximal possible value of $m(T)$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
10
}
The axial section of a cone is an equilateral triangle with a side length of 1. Find the radius of the sphere that is tangent to the axis of the cone, its base, and its lateral surface.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3} - 1}{4}}
Evaluate the surface integral \( \int_{S}\nabla\times\vec{F}\cdot{\rm d}\vec{S} \) where \( \vec{F}(x,y,z) = y^2\vec{i} + 2xy\vec{j} + xz^2\vec{k} \) and \( S \) is the surface of the paraboloid \( z = x^2 + y^2 \) bounded by the planes \( x = 0, y = 0 \), and \( z = 1 \) in the first quadrant, with the surface normal pointing upward.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{7}}
Determine the cardinality of the set of subgroups of the free group on two generators, $F_2 = \mathbb{Z} * \mathbb{Z}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Calculate the limit: $$\lim_{n\to \infty}\left((1+\frac{1}{n^2})(1+\frac{2^2}{n^2})^2\cdots(1+\frac{n^2}{n^2})^n\right)^{\frac{1}{n}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{\sqrt{e}}}
Evaluate the following limit: $$\lim_{n\to \infty}\frac{1}{n^2}\sum_{i=1}^{n}\log\left(\frac{n!^n}{\left(\prod_{i=1}^{n}i!\right)\cdot\left(\prod_{i=1}^{n-1}i!\right)}\right).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
Determine the dimension of the vector space of 3x3 matrices where the sum of the elements in each column, each row, and both diagonals is zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the area of the region bounded by a function $y=-x^4+16x^3-78x^2+50x-2$ and the tangent line which is tangent to the curve at exactly two distinct points.

Proposed by Kunihiko Chikaya
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1296}{5}}
Calculate the condition number of the matrix \( A = \begin{bmatrix} 0 & 0 & -10^4 & 0 \\ 0 & 0 & 0 & -10 \\ 0 & 10^{-3} & 0 & 0 \\ 10^{-2}& 0& 0& 0 \\ \end{bmatrix} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10^7}
Evaluate the definite integral: \[ \int_{0}^{3} (x^2+1) \, d[x] \] where \([x]\) is the integer part of \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Determine the number of Sylow 2-subgroups in the symmetric group $S_5$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A four-digit positive integer is called *doubly* if its first two digits form some permutation of its last two digits. For example, 1331 and 2121 are both *doubly*. How many four-digit *doubly* positive integers are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{171}
Given that $20^{22} + 1$ has exactly four prime divisors $p_1 < p_2 < p_3 < p_4$, determine $p_1 + p_2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{490}
A broken clock moves the minute hand each minute randomly one minute ahead or one minute behind with equal probabilities, independently of previous movements. The hour hand is synchronized with the minute hand. The clock starts running at noon. Let T denote the first time that the clock shows either 1 o'clock or 11 o'clock, whichever occurs first. Compute the expected value of T, E(T).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3600}
Let $a$, $b$, $c$ be positive integers and $p$ be a prime number. Assume that 
\[ a^n(b+c)+b^n(a+c)+c^n(a+b) \equiv 8 \pmod{p} \] 
for each nonnegative integer $n$. Let $m$ be the remainder when $a^p+b^p+c^p$ is divided by $p$, and $k$ the remainder when $m^p$ is divided by $p^4$. Find the maximum possible value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{399}
Evaluate the integral \( \int_R x^4 y^7 \, dx \, dy \) over the region \( R \) in the positive quadrant bounded by the hyperbolas \( xy = 1 \), \( xy = 4 \), \( xy^2 = 1 \), and \( xy^2 = 5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{310}
实系数方程 $x^{4}+a x^{3}+b x^{2}+c x+d=0$ 的根都不是实数，其中两个根的和为 $2+\mathrm{i}$, 另两个根的积为 $5+6 \mathrm{i}$, 求 $b$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A triangle \( \triangle ABC \) has its vertices lying on a circle \( C \) of radius 1, with \( \angle BAC = 60^\circ \). A circle with center \( I \) is inscribed in \( \triangle ABC \). The line \( AI \) meets circle \( C \) again at \( D \). Find the length of the segment \( ID \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}